Dea 2012 Proceedings

DATA ENVELOPMENT ANALYSIS: THEORY AND APPLICATIONS Proceedings of the 10th International Conference on DEA

EDITED BY: Rajiv Banker, Ali Emrouznejad Ana Lúcia Miranda Lopes Mariana Rodrigues de Almeida

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Title: Data Envelopment Analysis: Theory and Applications

Subtitle (series): Proceedings of the 10th International Conference on DEA

Venue: DEA2012, August 2012, Natal, Brazil

Edited by: Rajiv Banker, Ali Emrouznejad, Ana Lúcia Miranda Lopes, Mariana Rodrigues de Almeida

Date: December 2012

Number of Page: 305pp

ISBN: 978 185449 437 5

Citation:, Banker R., A. Emrouznejad, A. L. M. Lopes, M. R. de Almeida (2012), Data Envelopment Analysis: theory and Applications: Proceedings of the 10th International Conference on DEA, August 2012, Natal, Brazil, 340pp, ISBN: 978 185449 437 5.

Data Envelopment Analysis:

Theory and Applications Proceedings of the 10th International Conference on DEA, August 2012, Natal, Brazil

EDITED BY:

Rajiv Banker Fox School of Business and Management

Temple University Philadelphia, PA 19121

USA

Ali Emrouznejad Aston Business School

Aston University Birmingham B4 7ET

UK

Ana Lúcia Miranda Lopes Federal University of Minas Gerais - UFMG

School of Economics - FACE Management Department Belo Horizonte, 31270-901

Brazil

Mariana Rodrigues de Almeida Department of Engineering and Production

University Federal do Rio Grande do Norte - UFRN Natal, 59078-970

Brazil

December 2012

ISBN: 978 185449 437 5

PREFACE: A MESSAGE FROM THE LOCAL ORGANIZERS

Dear conference participants,

It is a great honor and pleasure to us from UFMG and UFRN welcome you to the 10th

International Conference on Data Envelopment Analysis - DEA2012 in Natal, Brazil.

The main themes of the DEA2012 conference are Energy and Regulation and Health

Performance Management. In total, one workshop on Energy and Regulation of Energy

Companies, five panels, 28 sessions for paper presentations and and two poster sessions will

be available for the participants. You will find papers about applications in agriculture,

banking, logistics, education, energy regulation, health, information technology, supply chain

management, transportation, tourism and sports. Sessions about economic issues in modeling,

algorithms, price and allocative efficiency and sustainability are also presented. A hundred

seventy (170) people from 25 different countries have registered to this Conference. The

participation of Brazilian academics is substantial (54,6%). We indeed have the pleasure to

receive people from distribution and transmission energy companies that are interested in

discussing the use of the DEA methodology to the Brazilian energy regulation model.

Foreigner and Brazilian consulting companies are also registered.

We would like to take this opportunity to express our sincere thanks to the board of iDEAs

that have accepted our proposal in Thessaloniki, Greece on august 2011, granting Brazil the

opportunity of bringing this prestigious Conference for the first time to the South America. We

are grateful to the stream organizers and members of the DEA in practice and scientific

conference committee. We warmly thank to Professor Rajiv Banker and Professor Ali

Emrouznejad for their priceless dedication and hard work in setting up this Conference.

We wish you all an interesting and enjoyable time in Natal, Brazil.

Ana Lucia Miranda Lopes Universidade Federal de Minas Gerais, UFMG

Mariana Almeida Universidade Federal do Rio Grande do Norte, UFRN

Proceedings of the 10th International Conference on DEA – Brazil2012 5

TABLE OF CONTENTS

1. A genetic algorithm approach to efficiency assessments with common weights ...... 9 Valiakos Athanasios

2. A new approach to assess performance of the Brazilian national immunization program (NIP) ................................................................................................................ 16 Castro Lobo MS Menegolla IA Estellita Lins MP

3. A new approach to cross efficiency evaluation based on MILP model and measurement of energy efficiency ............................................................................... 22 Mehmet Guray UNSAL Hasan BAL H. Hasan ORKCU

4. An effectiveness analysis of different techniques for development of IT software projects .......................................................................................................................... 30 Marco Mendes Ana Lúcia Miranda Lopes Rajiv Banker

5. An IDEA model to evaluate the overall performance of Buyer-Supplier .................. 37 Zahra Yousefi Mohsen Rostamy-Malkhalifeh Somayeh Mamizadeh

6. Assessing Performance of Organized Pharmacy Retail Stores using Data Envelopment Analysis .................................................................................................. 41 G N Patel Smiti Pande

7. Behavioral effects of DEA on performance assessment ............................................ 48 Heinz Ahn Nadia Vazquez Novoa

8. Data Envelopment Analysis of the effieincy frontier for the results achived by Formula 1 drivers and teams ........................................................................................ 55 Prof. Dr. Aparecido Jorge Jubran Profa. Msc. Laura Martinson Provasi Jubran José Rubens Moura Martins Jane Leite Silva

9. Data Envelopment Analysis Type Linear and Goal Programming Models For Measuring Energy Efficiency Performance of OECD Countries ................................ 60 Hasan BAL Mehmet Guray UNSAL

10. Decentralization and productivity of the public health service in Brazil ................... 68 Aléssio Tony Cavalcanti de Almeida Carlos Eduardo Gasparini


11. Deregulation and Performance of Mexican Banking ............................................... 75 Francisco Vargas Serrano Arnulfo Castellanos Moreno Gang Cheng Panagiotis Pzervopoulos Luis Rentería Guerrero

12. Economies of scale and scope in mental health care................................................. 82 J.A. Wilschut B.L. van Hulst J.L.T Blank

13. Efficiency analysis and long run performance: a sequential model for organizational assessment ........................................................................................... 92 Frederico A. de Carvalho Marcelino José Jorge Marina Filgueiras Jorge

14. Efficiency in the industrial sectors of Brazil in terms of contributing to sustainable development .................................................................................................................. 99 Flávia de Castro Camioto Enzo Barberio Mariano Daisy Aparecida do Nascimento Rebelatto

15. Efficiency in the management of sanitation and its impacts on health promotion: an aplication of data envelopment analysis – DEA ........................................................ 106 Karlos Eduardo Arcanjo da Cruz Francisco de Sousa Ramos

16. Efficiency of Three Outliers Detection Tests on Non-Parametric Frontiers Methods ... ................................................................................................................................... 114 Victor Maia Senna Delgado Igor Viveiros Melo Souza

17. Evaluation of the Benchmarking Model Proposed by the Brazilian Electricity Regulator for Energy Distribution Companies: The Case of Tariff Revision................. ................................................................................................................................... 121 Giordano Bruno Braz de Pinho Matos Marcelo Azevedo Costa Ana Lúcia Miranda Lopes Roberta de Cássia Macedo

18. Integration of BSC, DEA and Game Theory in the performance of public health service .......................................................................................................................... 137 Marco Aurélio Reis dos Santos Fernando Augusto Silva Marins Valerio A. P. Salomon

19. Iteratively Weighted Least Squares in Stochastic Frontier Estimation Applied to the Dutch Hospital Industry .............................................................................................. 147 Jos L. T. Blank Aljar J. Meesters


20. Least distance efficiency measures satisfying strong monotonicity on the efficient frontier .......................................................................................................................... 163 Hirofumi Fukuyama Kazuyuki Sekitani Jianming Shi

21. Maximal Allocated Benefit and Minimal Allocated Cost and its Application ........... 171 Mozhgan Mansouri Kaleibar Sahand Daneshvar

22. Measurement of returns to scale using a non-radial DEA model............................. 184 Vladimir E. Krivonozhko Finn R. Førsund Andrey V. Lychev

23. Multimethodology applied to the assessment of municipalities’ health performance in Brazil ........................................................................................................................ 193 Marcos Pereira Estellita Lins Sergio Orlando Antoun Netto

24. On the Measurement of Social Efficiency in Microfinance Institutions ................... 201 Breno Sampaio Lúcio Silva

25. Performance Evaluation in Hospitals: a study on hospitals financed by the Brazilian Unified Health System ................................................................................................. 209 Antônio Artur de Souza Emerson Alves da Silva Douglas Rafael Moreira Alisson Maciel de Faria Marques Ewerton Alex Avelar Bernardo Franco Tormin

26. Performance Evaluation of expenditure in Primary Care: the Case of Brazil’s Southeastern cities. .................................................................................................... 216 Lucas Maia dos Santos Márcio Augusto Gonçalves Márcia Mascarenhas Alemão Marco Aurélio Marques Ferreira Lucas Campos Vaz Heloiza Azevedo Drummond

27. Reconceptualizing the DEA Bootstrap for improved estimations in the presence of small samples .............................................................................................................. 226 Panagiotis D. Zervopoulos Francisco Vargas Gang Cheng

28. Relative balance as a complementary measure to relative efficiency ..................... 233 Heinz Ahn Ludmila Neumann Nadia Vazquez Novoa


29. Statistical Inference and Efficient Portfolio Investment Performance ..................... 239 Shibo Liu Tom Weyman-Jones

30. Study of technical efficiency in product development in steel company, with application of Data Envelopment Analysis (DEA) ..................................................... 245 Regina Rocha de Morais Gonçalves José Edson Lara Ana Lúcia Miranda Lopes Ronaldo Lamounier Locatelli

31. Technical efficiency of Burkina Faso primary public health care centers ............... 252 Oumarou Hebie Simon Tiendrebeogo Séni Kouanda Abdel Latef Anouze

32. The efficiency of Brazilian electricity distributors during 2004 – 2009. An application using DEA corrected by environmental and stochastic factors. ................................................................................................................................... 259 Fernando Damonte Mariana De Santis

33. The production efficiency in sugarcane farms .......................................................... 268 Terezinha Bezerra Albino Oliveira Antonio Cezar Bornia Suely de Fátima Ramos Silveira Mauro Wagner de Oliveira Alexandre Matos Drumond

34. Theory of robust optimization in overall profit efficiency with data uncertainty .... 277 N. Aghayi M.A. Raayatpanah

35. The efficiency in Tourism Investment capture in relation to Highways Investment in Tourist Routes of Espírito Santo State, Brazil .......................................................... 285 Marta Monteiro da Costa Cruz Josiane Baldo

36. The investment in ports enterprises in Espirito Santo, Brazil .................................. 291 Karen Vassoler Martins Marta Monteiro da Costa Cruz

37. Benchmarking the Efficiency of Third Party Logistics in Brazil Using Data Envelopment Analysis ................................................................................................ 297 Luís Filipe Azevedo de Oliveira Mariana Rodrigues de Almeida


1. A genetic algorithm approach to efficiency assessments with common weights

Valiakos Athanasios Department of Informatics, University of Piraeus, 80 Karaoli & Dimitriou Str., 18534 Piraeus, Greece, [email protected]

Abstract

The common weights approach is one of the most prominent methods to further prioritize the subset of DEA efficient units. This approach can be modeled as a multi-objective problem, where one seeks for a common set of weights that locates the efficiency ratio of each unit as close as possible to the target score of 1. In such a setting, different metrics can be applied to measure the distance of the efficiency ratios from target, such as the L1, L2 and L∞. When L1 and L2 metrics are used the models derived are non-linear. In case of the L∞ metric, the problem can be heuristically solved by the bisection method and a series of linear programs. We investigate in this paper the ability of genetic algorithms to solve the problem for estimating efficiency scores, by using an evolutionary optimization method based on a variant of the Non-dominated Sorted Genetic Algorithm.

Keywords: Data envelopment analysis, Common weights analysis (CWA), Genetic algorithms, evolutionary optimization

Introduction

In DEA efficiency assessments, the weights for inputs and outputs are estimated to the best advantage for each unit, so as to maximize its relative efficiency. Basically, DEA provides a categorical classification of the units into efficient and inefficient ones. However, although DEA is strong in identifying the inefficient units it is weak in discriminating among the efficient units. The basic DEA model often rates too many units as efficient. This is a commonly recognized problem of DEA, which becomes more intense when the number of units is relatively small with respect to the total number of inputs and outputs. Further discrimination among the efficient units is an issue that has attracted considerable attention in the DEA literature (Angulo-Meza et al. 2002 [0]. The common weights approach is one of the most prominent methods to further prioritize the subset of DEA efficient units. This approach can be modeled as a multi-objective problem, where one seeks for a common set of weights that locates the efficiency ratio of each unit as close as possible to the target score of 1. In such a setting, different metrics can be applied to measure the distance of the efficiency ratios from target, such as the L1, L2 and L∞. When L1 and L2 metrics are used the models derived are non-linear. In case of the L∞ metric, the problem can be heuristically solved by the bisection method and a series of linear programs. We investigate in this


paper the ability of genetic algorithms to solve the problem for estimating efficiency scores, by using an evolutionary optimization method based on a variant of the Non-dominated Sorted Genetic Algorithm, Deb et al. (2002) [0].

In this paper we introduce the Evolutionary Genetic Algorithms towards the multi-criteria approach using common set of weights. The rest of the paper is organized as follows. The next section describes the global efficiency approach and a bisection method to solve it linearly. The evolutionary algorithm is introduced in the third section. The forth section presents and discusses the results obtained by applying these methods under the Chebyshev metric. Conclusions finalize this research.

Efficiency assessments with common weights

Efficiency assessment with common weights is used in many publications such as Deb et al. (2002) [0] Kao et al. (2005) [0] Liu et al. (2008) [0] Wang et al. (2010) [0] and Kao et al. (2010) [0]. Common weights approach in DEA forms a multi-objective mathematical program. With conventional DEA each DMU maximizes its ratio of weight to the other values. From a general perspective these common inputs and common outputs must have also common weights to objectively measure the

efficiency. The relative efficiencies of conventional DEA 1...nh are here converted into

calculating the ideal points optimizing each and every objective function following the multi-objective linear model:

11 1

1( , ) ( , )

11 1

max ,...,

s s

r r r rnr r

u v n u vm m

i i i ini i

u y u yh h

v x v x

= =

= =

= =∑ ∑

∑ ∑(1)

s.t.

1 10, j=1,...,n

s m

r rj i ijr i

u y v x= =

− ≤∑ ∑

, ,r ru v r iε ε≥ ≥ ∀

Within the family of LP metrics, L1 and L∞ metrics are of particular interest in the field of Multi-Objective Linear Programming. This is because they are the only LP metrics that result in linear scalar problems, when minimizing a distance of the frontier to a

reference point. Therefore, let d be the minimum distance which optimizes all efficiency scores for each DMU. Since hk is the ideal point of DMUk, the distance from ideal score 1 is minimized with different metrics.

The metrics

Within the family of LP metrics, L1 and L∞ metrics are of particular interest in the field of Multi-Objective Linear Programming. This is because they are the only LP metrics that result in linear scalar problems, when minimizing a distance of the frontier to a reference point. The L∞ metric can be calculated from the following equation,


|| || : min max1 , 1,...kkd h k n∞ = − = (2)

The Chebychev norm is a distance function where the absolute value of the largest coordinates’ difference between two points absolutely dominates. The L1 metric can be calculated by the following equation,

11

|| || : min (1 )n

kk

d h=

= −∑ (3)

The Manhattan norm hat represents the shortest distance in unit steps along each axis between two points. The case of L2 is calculated from,

22

1|| || : min (1 )

n

kk

d h=

= −∑ (4)

The Euclidean norm between two points is the length of the straight line between the two points and it is by far the most commonly used norm.

Therefore, if this metric is used in Global DEA, only the largest factors’ difference is taken into account (thus leading to the most balanced solution between achievements of different factors).

For the case of L∞, the above model Eq. (1) although not linear it can be handled through bisection search - Despotis (2002) [0] using the following equivalent form,

|| || mind z∞= (5)

s.t.

1 10, 1,...,

s m

r rj i ijr i

u y v x j n= =

− ≤ =∑ ∑

1

1

1 ( ) 0, 1,...,

s

r rjr

m

i iji

u yz j n

v x

=

=

− + ≤ =∑

∑

, ,r iu v r iε ε≥ ≥ ∀

0z ≥

For the case of L1, the model becomes

11

|| || min , j=1,...,ns

jr

d d=

= ∑ (6)

http://www.gabormelli.com/w/index.php?title=shortest&action=edit&redlink=1

http://www.gabormelli.com/RKB/distance

http://www.gabormelli.com/w/index.php?title=unit_step&action=edit&redlink=1

http://www.gabormelli.com/w/index.php?title=axis&action=edit&redlink=1


s.t.

1

1

1, j=1,...,n

s

r rjr

jm

i iji

u yd

v x

=

=

+ =∑

∑

, ,r ru v r iε ε≥ ≥ ∀

0, j=1,...,njd ≥

and for L2,

22

1|| || min , j=1,...,n

s

jr

d d=

= ∑ (7)

s.t.

1

1

1, j=1,...,n

s

r rjr

jm

i iji

u yd

v x

=

=

+ =∑

∑

, ,r ru v r iε ε≥ ≥ ∀

0, j=1,...,njd ≥

The above two models are non-linear.

GA DEA

Since Eq. (1) is not linear, it can be approached with Genetic Algorithms. Multi-objective Evolutionary Algorithms (EAs) are GAs customized to solve multi-objective problems by using specialized fitness functions. EAs, such as Non-dominated Sorted Genetic Algorithm NSGA-II can be modified to find a set of multiple non-dominated solutions in a single run.

Multiobjective Evolutionary Algorithms (EAs) are GAs customized to solve multi-objective problems by using specialized fitness functions. In order to achieve evolutionary multi-objective optimization, three different aspects must be considered: Fitness Assignment, Diversity Mechanism and Elitism. Fitness Assignment is actually the fitness function chosen. Diversity Mechanism is the way next population is generated. Elitism is whether the best dominating solutions found so far survive to the next generation. Therefore a controlled genetic algorithm, which is a variant of Non-dominated Sorted Genetic Algorithm (NSGA-II), is proposed in this research. For fitness function Pareto Ranking approach is utilized. The population is ranked according to a dominating rule. In order to maintain the Diversity, crowding distance is employed, while Elitism exists partially. An elitist GA always favours individuals with better fitness value (rank). A controlled elitist GA also favours individuals that can help increase the diversity of the population even if they have a lower fitness


value. It is important to maintain the diversity of population for convergence to an optimal Pareto front. Diversity is maintained by controlling the elite members of the population as the algorithm progresses.

The main advantage of evolutionary algorithms, when applied to solve multi-objective optimization problems, is the fact that they typically optimize sets of solutions, allowing computation of an approximation of the entire Pareto front in a single algorithm run. The main disadvantage of evolutionary algorithms is the much lower speed. Through multi-objective solution global weights are acquired, rendering some DMUs on the Pareto front efficiency frontier. This is because a DMU is (Pareto) efficient if there is no other DMU, or a non negative convex linear combination of m inputs and s outputs of some DMUs, that improves one DMU score without worsening at least one of the other DMU score for all objectives simultaneously. Finally in order to choose the best solution we apply the LP metric. In order to select one from the Pareto front Eq. (3~5) are used. Below are the parameters used to execute GA DEA.

Table 1. Parameters used in Genetic Algorithm setup

Population size 75 Maximum number of generations 150

Generation gap (GGAP) 0.9 Mutation rate 0.5 under mutationadaptfeasible

Pareto front 0.5 Crossover strategy Double point crossing

Stopping criteria Either the best solution does not improve for 20 generations or the maximum number of generations has been reached.

Genetic Algorithm approach examines each solution in a specific area. After that, a pareto front is formed with the dominating solutions. The selection of the best solution is a decision of the decision maker. Therefore, by applying the metrics, the decision maker can select the best solution minimizing the distance from ideal value 1.

Results and discussions

To examine this approach a simulation was conducted. In each case, we initially generate data from a data generation process and then conduct 200 trials. The number of DMUs for experimental purposes was 20, a relatively small number.

The inputs and the outputs of each DMU were produced from a productive function

( )xϕ . Let n be the number of DMUs and m the number of inputs and s the number of

outputs for each DMU. Supposing a Data Generation Process that generates the inputs

ijx x= . Therefore in order to produce the outputs ijy y= , the following equation was

used,

( )1

m

x ii

y x eιβϕ=

= =∑ (8)


where βi is generated independently from independent uniform random distribution on the interval [0.6, 0.8]. Variable e is a small error multiplied with the production function to emulate the inefficient DMUs.

For the simulation, mean differences were calculated for each metric. Genetic Algorithm approach is compared with bisection linear programming (bLP) algorithm (Eq. 5) using L∞ metric (Eq. 2). Similarly it was compared with nonlinear L1 (Eq. 6) and nonlinear L2 (Eq. 7) using the L1 (Eq. 3), and L2 (Eq. 4) metrics respectively. Taking different cases into account, combinations of 2-4 inputs and 2-4 outputs are tested. In Table 2, the mean differences are displayed.

Table 2. Mean differences of minimum distances using L∞, L1 and L2 metric

bLP d∞ - GAd∞ nonLP d1- GA d1 nonLP d2 - GA d2

IxO Mean Range Mean Range Mean Range

2x2 0.0571 [0.0066 , 0.2176] 0.0496 [0.0259 , 0.2491] 0.0137 [0.0096 , 0.0338]

2x3 0.0556 [0.0096 , 0.3149] 0.0567 [0.0024 , 0.2226] 0.0272 [0.0018 , 0.0478]

2x4 0.0437 [0.0169 , 0.1275] 0.0190 [0.0031 , 0.1190] 0.0318 [0.0064 , 0.0732]

3x2 0.0478 [0.0034 , 0.2140] 0.0439 [0.0167 , 0.1331] 0.0314 [0.0018 , 0.1882]

3x3 0.0515 [0.0089 , 0.1705] 0.0110 [0.0064 , 0.1516] 0.0380 [0.0038 , 0.1622]

3x4 0.0522 [0.0140 , 0.2165] 0.0950 [0.0528 , 0.2076] 0.0430 [0.0294 , 0.1700]

4x2 0.0434 [0.0062 , 0.2104] 0.0110 [0.0064 , 0.2516] 0.0625 [0.0093 , 0.2538]

4x3 0.0447 [0.0085 , 0.1693] 0.0560 [0.0019 , 0.0772] 0.0427 [0.0252 , 0.2005]

4x4 0.0431 [0.0118 , 0.1775] 0.0496 [0.0222 , 0.0537] 0.0137 [0.0083 , 0.1862]

Table 2 shows that the mean values are close to zero. The minimum distances using different metrics are obtain with a single run of the genetic algorithm. It is also important to note that the range of the values is also significant low. A further inspection shows that the overall efficiency scores from GA and bLP do not vary much.

Conclusions

In this paper, a genetic algorithm is presented to estimate the efficiency scores with common weights. No prior research is been made towards using genetic algorithms in order to solve multi objective DEA. Different metrics are used as part of the algorithm in order to minimize distance of all DMUs from efficient value 1. The general fitness framework utilizes the NSGA-II approach taking into consideration the LP metric.

A simulation is conducted in order to review the capability of the algorithm. The results indicated that the genetic algorithm is capable of minimizing the distance from optimum value 1. In other words, through optimization of the genetic algorithm there is a set of common weights that minimizes the total distance. In scenarios with various number of inputs and outputs, the data has been analyzed and metrics are computed. It is proved that using Genetic Algorithms is a viable solution to estimate efficiency scores with common weights, and is in fact less complex than nonlinear equivalent. In addition, the efficient units are slightly increased since the genetic algorithm uses pareto front optimization.


References

Dyson R.G., Thanassoulis E. (1988). Reducing weight flexibility in data envelopment analysis, Journal f the Operational Research Society (39): 563–576.

Angulo-Meza, L., Estellita Lins, M.P. Review of methods for increasing discrimination in data envelopment analysis Annals of Operations Research 116(1-4): 225-242.

Deb K., Pratap A., Agarwal S., Meyarivan T. (2002). A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation (6): 182–197.

Kao, C. , Hung, H.-T. (2005) Data envelopment analysis with common weights: The compromise solution approach Journal of the Operational Research Society 56(10): 1196-1203.

Liu, F. - H. F. , Hsuan Peng, H. (2008) Ranking of units on the DEA frontier with common weights Computers and Operations Research 35(5): 1624-1637.

Wang, Y. M. , Chin, K. S. (2010) A neutral DEA model for cross-efficiency evaluation and its extension Expert Systems with Applications 37(5): 3666-3675.

Kao, C. (2010) Malmquist productivity index based on common-weights DEA: The case of Taiwan forests after reorganization Omega 38(6): 484-491

Wu, J. , Liang, L. , Yang, F. (2009) Determination of the weights for the ultimate cross efficiency using Shapley value in cooperative game Expert Systems with Applications 36(1): 872-876

Despotis D.K. (2002). Improving the Discriminating Power of DEA- Focus on Globally Efficient Units. Journal of the Operational Research Society (53): 314-323.

Acknowledgements

This study is funded and supported by the Institute of National Funds of Greece, since one of the authors is under financial scholarship.

mailto:[email protected]






2. A new approach to assess performance of the Brazilian national immunization program (NIP)

Castro Lobo MS Federal University of Rio de Janeiro, University Hospital — HUCFF/UFRJ, University City/Fundão Island, SEAV – 5º andar, sala 5 A 26, R. Professor Rodolfo Rocco, 255, 21941-913 Rio de Janeiro, RJ, Brasil, [email protected] (corresponding author)

Menegolla IA Pan-American Health Organization – PAHO, Setor de Embaixadas Norte, Lote 19, CEP 70800-400 – Federal District, Brazil Caixa Postal 08-729, 70312-970 – Brasilia, DF, Brasil, [email protected]

Estellita Lins MP Alberto Luiz Coimbra Post-Graduation and Engineering Research Institute, Operational Research, COPPE/UFRJ, Technology Center, Block F, Room 103, University City/ Fundão Island, Rio de Janeiro, Brazil, [email protected]

Abstract

The study develops an alternative measure of efficiency to assess the Brazilian National Immunization Program, using Data Envelopment Analysis (DEA), output oriented, Variable Returns to Scale (VRS) model, in order to congregate differing indicators in a unique index and to consider the differences among the federal units when comparing the twenty six Brazilian states that have diverse socioeconomic and urbanization scenarios. The NIP program can be considered highly efficient in Brazil. The mean efficiency score for the 26 states was 93.7 % (5.4 % SD). 16 states were considered efficient. To reach the frontier of best practices, each state and region could have an individual goal for vaccine homogeneity. DEA technique evaluates homogeneity indicators for various vaccines in the same model making it possible to construct an efficiency index for “the first year of life” immunization cycle.

Keywords: Data Envelopment Analysis, Health Services Assessment, Public Health Policy, National Program of Immunization

Introduction

Immunization is the process whereby a person is made immune or resistant to an infectious disease, typically by the administration of a vaccine. Vaccines stimulate the body’s own immune system to protect the person against subsequent infection or disease. Immunization is n a proven tool for controlling and eliminating life-threatening infectious diseases and is estimated to avert between 2 and 3 million deaths each year. It is one of the most cost-effective health investments, with proven


strategies that make it accessible to even the most hard-to-reach and vulnerable populations.

The Brazilian National Immunization Program (NIP) as a health policy nowadays practiced dates from 1973. The NIP is public, universal and managed by state or provinces' health authorities. The major challenges for the program are logistics for a country that have continental dimensions and the necessity of qualified human and material resources to cope with that. NIP is organized by life cycles (children, teenagers, adults and old people) and vaccines are applied as routine or by campaign means. For children under one year-old, the vaccines are to prevent: severe tuberculosis (BCG), poliomyelitis (polio), B hepatitis (HepB), diphteria, tetanus, whooping cough and haemophilus influenzae B (tetravalent), rotaviruses, pneumococcus, meningoencephalitis, and measles, mumps and rubeola (trivalent MMR), yellow fever (in endemic States).

According to WHO, the vaccination programs´ efficiency is assessed by three main indicators, which are used for each dose, and each kind of vaccine, giving ways to long spreadsheets and datasets, turning the analysis sometimes exhaustive and difficult to summarize. They are:

a) Coverage, a key measure of immunization system´s performance, is calculated by the actual number of applications divided by expected number of applications, according to the demographic structure of the state. The index is expected to be above 90%, ideally 100%;

b) Homogeneity, that is, the proportion of municipalities inside the state with coverage above 95%. The importance of this indicator is based in its capacity for herd immunity, which means that, even if you are not vaccinated, if you live in a place where all around you are immune, you may be considered protected as well;

c) Abandon Rate, that is, for vaccines that must be repeated, as tetravalent and poliomyelitis, the difference between the number of applications for last and the first given doses, divided by the number of first doses. These are proxies to the fact that a person had access to all doses that guaranteed full protection. The expected value should be below 5%.

As there is no sense in trading off the usual goals for coverage and abandon rate, this paper intends to develop an alternative measure of efficiency to assess the Brazilian Immunization Program in 2010, using Data Envelopment Analysis (DEA) in order to congregate the differing homogeneity indicators as a unique index and to consider the different resource facilities observed when comparing the twenty six Brazilian states that together give this country a continental dimension, with diverse socioeconomic and urbanization scenarios.


Methods

Many performance and benchmark reviews approach the issue of health care assessment from the efficiency perspective. Chillingerian and Sherman, Hollingsworth and O’Neill et al. health care efficiency studies collectively provided comprehensive overview of the theme, pointing out the general advantages, concerns and limitations of applying these methods in multi-product organizations, practices in public policies or programs.

The most common used technique, Data Envelopment Analysis based on linear programming, shows which health organizations are efficient, gives the magnitude of inefficiency and indicates the means of improving efficiency by giving targets for each of the inputs and/or outputs individually. Details of the technique are presented elsewhere.

The DMUs, in our case are the Brazilian States, or the local units responsible for the NIP organization and logistics. The ones over the frontier, therefore, efficient, have an efficiency measure equals to 1.00 or 100%, while the DMUs located under the frontier are inefficient (values between 0 and 1.00, or 100%). The production model used in this work considers variable returns to scale (VRS) and is oriented to the increase of outputs to the projection in the frontier (maximization). The VRS model allows an inefficient unit to be compared only with others efficient units of similar size or operate in similar scale and is the choice to cope with the diversity of the States' sizes and social scenarios. The orientation choice (output) admits the maximum success of the results (higher homogeneity), given a fixed amount of resources.

Considering the fact that the traditional indicators used to assess NIP are already fractions, which could jeopardize the linear properties of the frontier, the authors ran a model assuming only the absolute values that generated the homogeneity indicators. The inputs were the number of births in 2009 and the number of municipalities of each State; the outputs were the number of municipalities with coverage over 95% for the studied vaccines: BCG, HepB (3rd dose), polio (3rd dose), tetravalent (3rd dose) and MMR (first). In this way, the DEA score index would comprise homogeneity for many vaccines in the same model. Once calculated the number of municipalities that needed to reach 95% coverage, the projection was used as a numerator to reconstruct the homogeneity index to be pursued as a new goal.

All demographic and vaccination data were provided by the Brazilian Ministry of Health.

Results and Discussion:

As observed in Figure I, there are clearly two distinct subgroups of Brazilian Regions, that congregate, at one side: Regions North and Northeast, less developed, with lower mean Human Development Index - HDI (0.73) and higher proportions of rural and disperse populations (mean 25%); at the other side: Regions South, Southeast and Center-West, with higher mean HDI (0.82), less rural population (mean 13%), and where the bigger cities are located (with higher populations, higher number of newborns). In a clockwise overview of Figure I, the first subgroup goes from Rondônia to Bahia and the second subgroup goes from Minas Gerais to Goiás. São


Paulo is an outlier as the State comprises almost 22% of all the Brazilian population. This demographic and socioeconomic difference has a sanitary impact, with lower values for the first subgroup, especially concerning the indicators for homogeneity as the access to remote and disperse populations can only be guaranteed by mobile teams.

Table I shows the DEA scores, the new goals established for each homogeneity output after the model was ran, and the benchmarks for the inefficient States. The mean efficiency score for the 26 states was 93.7% (5.4 % SD), the minimum score was 53.8% (Rondônia).

Given the different proportions of rural populations, each state could have an individual goal for homogeneity to each vaccine to catch the frontier. In North and Northeast Regions, the mean BCG homogeneity should raise from 40.1 % to 50.9 %; HepB homogeneity, from 52.7% to 62.4 %; polio homogeneity, from 56,7% to 65.4 %; tetravalent homogeneity, from 56.1% to 65.3 %; MMR homogeneity, from 55.4% to 64.6 %. In South, Southeast and Center-West Regions, the mean BCG homogeneity should raise from 60.4% to 63.0%; HepB homogeneity, from 66.4% to 69.2 %; polio homogeneity, from 72.4% to 75.8 %; tetravalent homogeneity, from 72.1% to 75.9 %; MMR homogeneity, from 64.6% to 66.8 %. These mobile goals give a more realistic scenario, that is, the DEA model brings an evidence-based figure of what goal should be made plausible and attainable to the health manager.

For each inefficient State, the benchmarks were defined based on the projection in the frontier of best practices. Sixteen States were efficient (number of references in parenthesis): Acre (0), Roraima (6), Pará (0), Amapá (0), Tocantins (4), Maranhão (1), Pernambuco (0), Sergipe (1), Minas Gerais (3), Espírito Santo (4), São Paulo (1), Paraná (2), Santa Catarina (3), Rio Grande do Sul (0), Mato Grosso do Sul (3), Goiás (5). As seen in administrative practice, the respective benchmarks should be explored – even beyond the studied variables - as examples to orientate changes for better results.

In the benchmarking process, care must be taken when comparing States with different demographic and socioeconomic structures as the inefficient units are projected into different parts (facets) of the best practice frontier. The risk of ranking efficiency scores and misinterpretation must be highlighted because, when you show these results to a health care authority, this is the usual first observation of the manager, which can damage the reliability and face validity of the method.

In summary, from the multi-input multi-output model perspective, the DEA score – in a unique index - presents a picture of homogeneity performance of each State that congregates many vaccine schemes offered in the first year of life; in some way, substituting lots of datasheets that can even bring contradictory results. Indicators for vaccine coverage and abandon rate should maintain the goals preconized by the World Health Organization (WHO). On the other hand, the concise DEA measure can be scrutinized to offer much more information, as: a) the new goals for each indicator, so that the DMU can project and reach the best practice frontier (an useful tool for health manager); b) the reference groups according to the facet of the frontier were the inefficient unit should be projected (the benchmarks are the vertices of theses


facets); c) the weights given to each variable, so that the model tend to prioritize the ones that behave in a more efficient way comparing to the other units.

Figure I: Comparing Brazilian States according to Demographic Characteristics

Table II – DEA Model Results: Scores, Immunization Goals and Benchmarks for the NIP in Brazilian States


Conclusions

DEA model is a potential tool to assess the Brazilian NIP that should be explored. As a multi-input multi-output index, it allows the comparison of states that have diverse resources and demographic structure and also establishes differing homogeneity goals to each one, based on their real capabilities. Also, the DEA technique evaluates homogeneity for various vaccines in the same model, giving an index that considers the linear combination of all of them. In this way, it is possible to construct an efficiency index for “the first year of life immunization cycle”, as the vaccines are given at different months according to the immunization calendar.

The NIP program can be considered highly efficient as a whole in Brazil, with a mean DEA efficiency homogeneity index of 93.7 %; mean coverage above 90% for all the studied vaccines, mean tetravalent abandon around 5, 0 %. The goals for homogeneity proposed by the model will displace all units towards the best practice frontier and, as the efficient units could also ameliorate results; new frontiers would be constructed, giving ways to technological change and a systemic development of the public policy.

References

Atkinson W., Wolfe C., Hamborski J. (2011) Epidemiology and Prevention of Vaccine-Preventable Diseases. 12th edition. The Public Health Foundation; 2011.

Barreto M.L., Teixeira M.G., Bastos F.I., Ximenes R.A.A., Barata R.A. (2011) Successes and failures in the control of infectious diseases in Brazil: social and environmental context, policies, interventions, and research needs. The Lancet; 377: 1877-1889.

Chilingerian J.A., Sherman D. (2004) Health Care Applications - From Hospitals to Physicians; From Productive Efficiency to Quality Frontiers. In: Cooper WW; Seiford LM; Zhu J. Handbook on data envelopment analysis. Boston: Kluwer Academic Publishers.

Hollingsworth B. (2003) Non-Parametric and Parametric Applications Measuring Efficiency in Health Care Health Care Management Science;6:203-218.

Hollingsworth B. (2008) The Measurement of Efficiency and Productivity of Health Care Delivery, Health Economics; 17 (10):1107-1128.

O´Neill L, Rauner M, Heidenberger K, Kraus M. (2008) A cross-national comparison and taxonomy of DEA-based hospital efficiency studies. Socio-Economic Planning Sciences 2008; 42(3):158-189.

Cooper WW, Seiford LM, Tone K. (2007) Data Envelopment Analysis-A Comprehensive Text with Models, Applications, References and DEA Solver Software. 2nd. Ed. Massachusetts: Springer

Ozcan YA. Health Care Benchmarking and Performance Evaluation: An Assessment using Data Envelopment Analysis (DEA). International Series in Operations Research and Management Science. Springer; 2008.

Lins MPE, Lobo MSC, Fiszman R, Silva ACM, Ribeiro VJP. O Uso da Análise Envoltória de Dados – DEA - para Avaliação de Hospitais Universitários Brasileiros. Revista Ciência e Saúde Coletiva 2007; 12(4): 985-998.


3. A new approach to cross efficiency evaluation based on MILP model and measurement of energy efficiency

Mehmet Guray UNSAL Gazi University, Science Faculty, Statistics Department, Ankara, TURKEY [email protected] (corresponding author)

Hasan BAL Gazi University, Science Faculty, Statistics Department, Ankara, TURKEY [email protected]

H. Hasan ORKCU Gazi University, Science Faculty, Statistics Department, Ankara, TURKEY [email protected]

Abstract

DEA has become a very popular method of performance measure, but it still suffers from some shortcomings. One of these shortcomings is the issue of having multiple optimal solutions to weights for efficient DMUs. The cross efficiency evaluation as an extension of DEA is proposed to avoid this problem. Lam (2010) is also proposed a mixed-integer linear programming formulation based on linear discriminant analysis and super efficiency method (MILP model) to avoid having multiple optimal solutions to weights. In this study, we modified MILP model to determine more suitable weight sets and also evaluate the energy efficiency of OECD countries as an application of the proposed model.

Keywords: Data envelopment analysis, discriminant analysis, cross efficiency, MILP model.

Introduction

DEA was first developed by Charnes et al. [3] that seems to be the most popular method for measuring the efficiency of homogenous decision making units. It become very popular method which is used in operations research and management science. The improvements made to the DEA technique have resulted in several new problems [2]. For example, the issue of unrealistic weights distribution, the weak discrimination power, and having multiple optimal solutions to weights for efficient DMUs.

Having multiple optimal solutions to weights affects to a great extent the consistency of operations related to weights cross efficiency method is the most frequently studied topic in DEA literature. Sexton et al. [13] developed the cross efficiency method to rate the DMUs. Their technique made use of the cross evaluation scores computed as related to all DMUs and hence identified the best DMUs [1].


Despite the extensive use of the cross efficiency method, it has some limitations arising from the classical DEA. Doyle and Green [5] stated that the non-uniqueness, i.e. having multiple solutions to optimal weights in the DEA, decreases the usefulness of the cross efficiency method. Sexton et al. [13] and Doyle and Green [5] recommended the use of a secondary objective (model) for the cross efficiency evaluation related to the non-uniqueness of optimal weights in DEA. They proposed the aggressive and benevolent models for achieving the secondary objective.

See also the following papers with applications of cross efficiency evaluation: Sexton et al. [16] to nursing homes, Oral et al. [10] to R&D projects, Doyle and Green [4] to higher education, Green et al. [5] to preference voting, Lu and Lo [9] to economic environmental performance, Liang et al. [8], Ramon et al. [12], Lam [7], Örkcü and Bal [11], Jahanshahloo et al. [6].

For having multiple optimal solutions problem in DEA, Lam [7] proposed a mixed-integer linear programming formulation based on linear discriminant analysis and the super efficiency method. In this paper, we modify this model and introduce a new model to choose suitable weight sets to be used in cross efficiency evaluation.

Lam [7] used a h constant to separate efficient and inefficient DMUs. In the Lam [7] model, h constant is determined from the super efficiency model. As h is considered as a variable in our proposed model, it is not necessary to use the super efficiency model to determine h. Lam [7] used three data scenario to illustrate his model. We also have used these data scenario for the comparison methods. With this new modification model, one can compare the efficiency scores and obtain a better picture of cross efficiency stability with respect to multiple DEA weights. The results obtained from three different data scenario in the related literature show that proposed model is compatible with the most used cross efficiency models and make valid contributions to cross efficiency evaluation.

A New Approach to Cross Efficiency Evaluation Based on the Lam Model

The cross efficiency method was developed as a DEA extension tool to be utilized for identifying the best performing DMUs, and for ranking DMUs using cross efficiency scores that are linked to all DMUs [13,15]. The basic idea of the cross efficiency method that alleviates the weak discrimination of the classical DEA model can be explained in two stages: In the first stage, the classical DEA analysis is performed, and the optimal weights of inputs and outputs are calculated for each DMU. However, the optimal weights computed by classical DEA have multiple solutions, especially for the efficient DMUs, and these solutions provide unrealistic weights, i.e., weights with extreme or zero values. In the second stage, these drawbacks are reduced, and a suitable set of weights preserving the efficiency values obtained by DEA is selected for each DMU.

In the first stage, the optimal weights of inputs and outputs are calculated for each DMU using the classical DEA formulation. Given the results of the first stage, the weights used by the DMU can be utilized for calculating the peer rated efficiency for


each of the other DMUs. The peer evaluation score, ,p jθ , indicates the efficiency score

for the DMU j using the weights obtained by the DMU p [15].

,1

,

,1

s

r p rjr

p j m

i p iji

u y

v xθ =

=

=∑

∑. (4)

Lam [7] proposed a mixed-integer linear programming (MILP) formulation based on linear discriminant analysis and the super efficiency method. The MILP formulation is summarized as follows:

In the first step, the CCR model given in equation (3) is used to determine the efficiency ratio for each DMU. Then, based on the efficiency ratios, the DMUs are

classified as either efficient ( E ) or inefficient ( E ).In the second step, an MILP model is run for each efficient DMU in E. The objective of the MILP model is to separate the efficient and inefficient DMUs while keeping the efficiency ratio of the DMU under evaluation to be the highest. The intuition of keeping the efficiency ratio of the DMU under evaluation as the highest is that the obtained weight set will then reflect the relative strengths of the efficient DMU under consideration over the other DMUs. All the obtained weight sets are used to compute cross efficiency ratios for all DMUs. The

efficient DMU under evaluation is expressed thus, eDMU . The MILP model proposed

by Lam [7] can be stated as follows:

1

n

jj

Min z=∑

. .s t

1 10, ,

s m

r rj i ij jr i

u y v x M z j E= =

− + ≥ ∈∑ ∑

1 1, ,

s m

r rj i ij jr i

u y v x M z j Eε= =

− − ≤ − ∈∑ ∑

11,

m

i iei

v x=

=∑ (5)

1,

s

r rer

u y h=

≥∑

1 10 ; 1, , , ,

s m

r rj i ijr i

u y h v x j n j e= =

− ≤ = ≠

∑ ∑

0,1 , 1, , ,jz j n∈ =

, 0, 1, , , 1, ,r iu v r s i m≥ = =

where, ε is a very small positive number, M is an extremely large positive number,

E is the efficient set which contains all the efficient DMUs, while E contains all the inefficient DMUs. The value of h is predetermined. In the MILP model proposed by


Lam [7], constant h is determined by the super efficiency model and in fact, the value of h is based on the idea of the cross efficiency calculation. In the Lam [7] model, the constant h is determined from the constraint is given in (6).

1 10 ; 1, , , , (6)

s m

r rj i ijr i

u y h v x j n j e= =

− ≤ = ≠

∑ ∑

If this constraint is organized,

1

1

; 1, , , ,

s

r rjr

m

i iji

u yh j n j e

v x

=

=

≤ = ≠∑

∑

(7)

is obtained. Lam [7] proposed to determine h with the super efficiency method due to the difficulty of fractional programming problems. Due to h>1, it seems appropriate to determine h by the super efficiency method.

The constraint given with (8) was selected instead of this constraint in this study.

1 1 ; 1, , , ,

s m

r rj i ijr i

u y v x h j n j e= =

− ≤ = ≠∑ ∑ (8)

If the constraint given with (8) is organized, (1

0m

i iji

v x=

≥∑ ),

11

1 1 1

; 1, , ,

ms

i ijr rjir

m m m

i ij i ij i iji i i

v xu yh j n j e

v x v x v x

==

= = =

− ≤ = ≠∑∑

∑ ∑ ∑

(9)

is obtained. If the constraint given with (9) is organized,

1

1 1

1 ; 1, , , ,

s

r rjr

m m

i ij i iji i

u yh j n j e

v x v x

=

= =

≤ + = ≠∑

∑ ∑

(10)

is obtained. Here, the value of h is selected as a variable, being h>1. When the constraint given

with (10) is examined, it can be seen that the efficiency ratio (1 1

s m

r rj i ijr i

u y v x= =∑ ∑ )

remains lower than a value higher than 1. In the MILP model, the efficiency ratio is of a value lower than the value of h (h>1). That is to say, the efficiency ratio is similar to the MILP model in the model that we propose also. While the model that we propose determines the threshold value of h itself for the efficiency ratio for the model as well, the value of h in the MILP model is predetermined with the super efficiency model. As h is considered as a variable in our proposed model, it is not necessary to use the super efficiency model to determine h. Besides, our proposed model is not an integer linear programming. The model suggested is given in the equation (11).


1

n

jj

Min d=∑

. .s t

1 10, ,

s m

r rj i ij jr i

u y v x d j E= =

− + ≥ ∈∑ ∑

1 1, ,

s m

r rj i ij jr i

u y v x d j Eε= =

− − ≤ − ∈∑ ∑1

1, (11)m

i iei

v x=

=∑

1,

s

r rer

u y h=

≥∑

1 1, 1, , , ,

s m

r rj i ijr i

u y v x h j n j e= =

− ≤ = ≠∑ ∑

1h ≥

0, 1, ,jd j n≥ =

, 0, 1, , , 1, ,r iu v r s i m≥ = =

where, ε is a very small positive number, E is the efficient set which contains all

efficient DMUs, while E contains all inefficient DMUs.

Application Study

According to an application study on energy data of OECD countries, we consider gross domestic product and nonfossil fuel consumption as outputs variables. CO2 Emission and fossil fuel consumption are the inputs similarly

Ramanathan’s study [14] in Table 1. We obtained the data set from International Energy Agency web page “http://www.eia.doe.gov”. The data belongs to 2008.

Table 1. Outputs and input variables for measuring of OECD Countries’ energy consumption and CO2 emission efficiency

Efficiency Measurement Approach Inputs Outputs

Energy Consumption and CO2 Emission

(Ramanathan,2005)

CO2 emissions, fossil fuel energy consumption (FOSS)

Gross domestic product (GDP), non-fossil fuel energy consumption

(NFOSS)


Table 2. Sperman’s Rank Correlations According to Rankings of DMUs

CCR SUPER CROSS MILP PROPOSED

CCR 1 0.998 0.924 0.973 0.938

SUPER 0.998 1 0.925 0.974 0.946

CROSS 0.924 0.925 1 0.910 0.857

MILP 0.973 0.974 0.910 1 0.942

PROPOSED 0.938 0.946 0.857 0.942 1

Table 3. Efficiency ratios of models for OECD Countries

Country

CCR

Eff.

Ratio Ran

k

Super

Eff.

Ratio

Ran

k

Cross

Eff.

Ratio

Ran

k

Lam

model

Eff.

Ran

k

Prop

osed

Mode

Ran

k

AUSTRALIA 0,1073 24 0,1073 24 0,3639 8 0.1556 25 0.0836 25 AUSTRIA 0,3023 12 0,3023 12 0,1929 14 0.3813 13 0.3041 13 BELGIUM 0,3505 11 0,3505 11 0,2523 12 0.5326 10 0.3622 9 CANADA 0,5986 7 0,5986 7 0,4041 6 0.8569 4 0.4582 6 CZECH REPUBLIC 0,1870 20 0,1870 20 0,1272 20 0.2626 19 0.1884 18 DENMARK 0,1613 22 0,1613 22 0,1004 24 0.1945 23 0.1626 21 FINLAND 0,6207 6 0,6207 6 0,3751 7 0.7085 7 0.6475 3 FRANCE 1 3 2,3988 2 0,8757 1 1.9039 1 1.1272 1 GERMANY 0,5910 8 0,5910 8 0,3567 9 0.7322 5 0.3724 7 GREECE 0,0658 28 0,0658 28 0,0466 27 0.0979 27 0.0676 28 HUNGARY 0,2091 18 0,2091 18 0,1237 22 0.2365 21 0.1991 17 IRELAND 0,0719 27 0,0719 27 0,0444 28 0.0854 28 0.0731 26 ITALY 0,1788 21 0,1788 21 0,1248 21 0.2660 18 0.1441 22 JAPAN 1 3 1,0875 4 0,5257 3 0.7089 6 0.3442 10 KOREA 1 3 2,1397 3 0,5106 5 0.5992 9 0.3182 11 LUXEMBOURG 0,0595 29 0,0595 29 0,0253 29 0.0521 29 0.0511 29 MEXICO 0,2202 17 0,2202 17 0,1495 18 0.3133 16 0.1682 20 NETHERLANDS 0,0898 25 0,0898 25 0,0745 25 0.1618 24 0.0937 24 NEW ZEALAND 0,3022 13 0,3022 13 0,1827 15 0.3451 14 0.3150 12 NORWAY 0,4685 10 0,4685 10 0,2814 11 0.5304 11 0.4991 5 POLAND 0,0734 26 0,0734 26 0,0571 26 0.1234 26 0.0698 27 PORTUGAL 0,2038 19 0,2038 19 0,1283 19 0.2507 20 0.2054 16 SLOVAK REPUBLIC 0,2531 15 0,2531 15 0,1528 17 0.2883 17 0.2649 14 SPAIN 0,2726 14 0,2726 14 0,2112 13 0.4570 12 0.2580 15 SWEDEN 1 3 2,0432 1 0,6001 2 1.1419 2 1.1082 2 SWITZERLAND 0,5568 9 0,5568 9 0,3353 10 0.6320 8 0.5872 4 TURKEY 0,1221 23 0,1221 23 0,1017 23 0.2208 22 0.1281 23 UNITED KINGDOM 0,2356 16 0,2356 16 0,1530 16 0.3212 15 0.1690 19 USA 1 3 4,0592 5 0,5141 4 0.9146 3 0.3651 8


Table 4. Outputs and inputs weights obtained from Lam’s MILP and proposed model (for efficient DMUs)

The proposed model produces far fewer zero weights than Lam’s MILP. As pointed out by Cooper et al [16], it is better to use weight sets which have more balanced virtual weights than extreme virtual weights in measuring the performance of the DMUs in the DEA. The results of application study are listed in Table 2-4. All of the models have significant and high correlation coefficients between each other. This solution is lead us these models have same usefulness in ranking of DMUs. Furthermore, the proposed model has an ability to find of value of h by itself. Another significant advantage of our modified MILP model is that it can effectively reduce the number of zero weights for inputs and outputs as seen in Table 3.

Conclusions

Lam [7] proposed a mixed-integer linear programming (MILP) formulation based on linear discriminant analysis and super efficiency method. In this paper, we modify this model to choose suitable weight sets to be used in cross efficiency evaluation. The weight sets obtained from the proposed model are suitable for cross-evaluation because they reflect the different strengths of the efficient DMUs. The proposed model produces far fewer zero weights than Lam’s MILP. The results are obtained from energy consumption and CO2 emmission efficiency of OECD Countries.

References

[1] T.R. Andersen, K.B. Hollingsworth, L.B. Inman, (2002) The Fixed Weighting Nature of a Cross Evaluation Model, J. Prod. Anal., 18 (1) 249–255.

[2] H. Bal, H.H. Örkcü, S. Çelebioğlu, (2010) Improving the Discrimination Power and Weight Dispersion in the Data Envelopment Analysis, Comput. Oper. Res., 37 (1) 99–107.

[3] A. Charnes, W.W. Cooper, E. Rhodes, (1978) Measuring the Efficiency of Decision Making Units, Eur. J. Oper. Res., 2 429–444.

[4] J.R. Doyle, R. Green, (1994) Efficiency and Cross Efficiency in Data Envelopment Analysis: Derivatives, Meanings and Uses, J. Oper. Res. Soc., 45 (5) 567–578.

[5] R. Green, J.R. Doyle, W. Cook, (1996) Preference voting and project ranking using DEA and cross-evaluation, Eur. J. Oper. Res., 90 (3) 461–472.

MODELSCOUNTRIES u1 u2 v1 v2 h u1 u2 v1 v2 h

FRANCE 0 0,0481 85,3197 0,0153 2,35 0 0,0262 19,8738 0,0185 1,2820JAPAN 0 0,0445 797,6734 0,0033 2,35 0 0,0126 1,5746 0,0120 1KOREA 0 0,0387 427,6736 0,0023 2,35 0,0001 0,0259 38,4121 0,0113 46,0388

SWEDEN 0 0,0348 0 0,0304 2,35 0 0,0289 6,9926 0,0262 1,9457USA 0 0,3683 5434,7830 0 2,35 0 0,0675 704,6617 0,0102 1

LAM MODEL PROPOSED MODEL


[6] G.R. Jahanshahloo, L.F. Hosseinzadeh, Y. Jafari, (2011) R. Maddahi, Selecting symmetric weights as a secondary goal in DEA cross-efficiency evaluation, Appl. Math. Model., 35, 544–549.

[7] K.F. Lam, (2010) In the determination weight sets to compute cross-efficiency ratios in DEA, J. Oper. Res. Soc., 61 134–143.

[8] L. Liang, J. Wu, W.D. Cook, J. Zhu, (2008) Alternative secondary goals in DEA cross-efficiency evaluation, Int. J. Prod. Eco., 113 (2) 1025–1030.

[9] W.M. Lu, S.F. Lo, (2007) A closer look at the economic-environmental disparities for regional development in China, Eur. J. Oper. Res., 183 (2) 882–894.

[10] M. Oral, O. Kettani, P. Lang, (1991) A methodology for collective evaluation and selection of industrial R&D projects, Manage. Sci., 37 (7) 871–885.

[11] H.H. Örkcu, H. Bal, (2011) Goal programming approaches for data envelopment analysis cross efficiency evaluation, Appl. Math. Comput., 218 346-356.

[12] N. Ramón, J.L. Ruiz, I. Sirvent, (2010) On the choice of weights profiles in cross-efficiency evaluations, Eur. J. Oper. Res., 207 (3) 1564–1572.

[13] T.R. Sexton, R.H. Silkman, A.J. Hogan, (1986) Data Envelopment Analysis: Critique and Extension. In: Silkman R.H. (Ed.), Measuring Efficiency: An Assessment of Data Envelopment Analysis, 32. Jossey-Bass, San Francisco 73-105.

[14] R. Ramanathan, (2005) An analysis of energy consumption and carbon dioxide emissions in countries of the Middle East and North Africa, Energy, 30 2831–2842.

[15] T.R. Andersen, K.B. Hollingsworth, L.B. Inman, (2002) The Fixed Weighting Nature of a Cross Evaluation Model, J. Prod. Anal., 18 (1) 249–255.

[16] W.W. Cooper, J.L. Ruiz, I. Sirvent, (2007) Choosing Weights From Alternative Optimal Solutions of Dual Multiplier Dual Models in DEA, Eur. J. Oper. Res., 180 443–458.


4. An effectiveness analysis of different techniques for development of IT software projects

Marco Mendes School of Economics - FACE - Management Department – UFMG, [email protected]

Ana Lúcia Miranda Lopes School of Economics - FACE - Management Department – UFMG, [email protected]

Rajiv Banker Fox School of Business and Management – Temple University, [email protected]

Abstract

Software development best practices claim to reduce IT projects risks and deliver better and efficient software products. Yet widely accepted on industry, these techniques are the subject of intense debate in academia. Objective: This paper presents a quantitative analysis of the impact of a set of software development techniques on the technical efficiency of software projects. A study is conducted on 105 software development projects for efficiency analysis of effort, time elapsed, productivity and defect density. The following software practices are evaluated: capability maturity models like CMMI, requirements elicitation techniques, design and architecture techniques, test techniques, project management techniques, business process management use and case tools adoption. Method: Benchmarking is performed using efficient frontier analysis with DEA BCC input oriented. Efficiency scores among software project groups are compared using DEA based hypothesis tests. Results: No single software engineering technique could explain an increase in overall performance in IT projects. Yet, we find some evidences that IT firms are more efficient than other firm’s types when delivering software projects. Conclusion: Results emphasize the importance of recognizing that optimal management techniques depend on the characteristics of the software development project, organization type and its sociotechnical environment.

Keywords: Software Economics, DEA BCC, Information Systems, CMMI, Software Architecture

Introduction

Information technology (IT) has been used for over 50 years as a plausible instrument for increasing efficiency in firms’ business processes. Several software engineering best practices such as capability maturity models, project management techniques, requirement, design and test techniques are proposed instruments to reduce software





development risks and to deliver better products. Although the value of these practices is widely accepted in industry, formal analysis of its benefits has been the subject of great debate in academia.

In this context, this paper promotes a discussion of the influence of software development practices on technical efficiency of software projects. The term technical efficiency is used according to Farrell [1] and it means the product results maximization from a predetermined set of inputs. For example, if a set of software project of the same size and effort reports different defect densities the projects with a lower defect density are classified as more efficient. The key question posed in this study is: the use of recognized software engineering practices could yield superior project performance? Economic efficiency has great significance and encourages managers in directorial search for techniques that can yield software products in less time and effort or with fewer defects.

The paper is organized as follows. Section 2 presents related work in IT projects technical efficiency. Section 3 presents the DEA based methodology and DEA based hypothesis test for comparison of groups of IT projects. Section 4 presents the results and discussion of the experiment, followed by conclusion in section 5.

Related Work

A longitudinal study conducted since 1985, compiled the report Chaos Manifesto [2], indicates that the majority of IT projects present efficiency problems regarding the quality as perceived by its users, project deadlines and costs. In a meta-analysis of investments in IT services Brynjolfsson [3] studied the IT productivity paradox, arguing that the use of IT often does not generate the expected returns on investment. The same author presents in the later work efficiency factors when information technology is linked to the innovation of business processes [4]. Since the 70s, software costs exceed hardware costs in IT in a range of up to 5:1 [5] [6]. Currently this ratio exceeds 10:1 and continues to grow. With respect to the unit of analysis for software projects, several investigations have been conducted to determine factors that guide economic efficiency. Factors related to product size, teams, technologies and construction process are more accepted in the literature. These aspects are discussed in the classic book The Mythical Man-Month [7] and its elements already indicate diseconomies of scale. Early econometric models such as SLIM and COCOMO II [6] are seminal references in determining these factors and also presents evidence of diseconomies for large projects. Yet classical analyses suggest decreasing returns to scale for software projects, other authors present divergent arguments [8]. Banker and Kemerer [9] analysed several IT project databases and noticed that some of them exhibit economies of scale. The same authors in subsequent work [10] present similar evidence in the context of software maintenance and other projects database. Over the past years, investigations by other authors show evidence of varying scales in projects [11].

Academic software economics literature encompasses many quantitative methods of analysis. Pioneering work in the 70s and 80s were based simply on multiple linear regression methods. A systematic review of measurement and prediction of productivity of software projects analyses the main quantitative methods in use in 38


selected papers [12]. The author indicates that methods like DEA (Data Envelopment Analysis), Bayesian networks and discrete event simulation are promising.

This paper investigates the effects of some software engineering practices on the performance of software projects. We analyse the following practices:

• Capability maturity models, like SEI CMMI. These maturity models claim to deliver better results.

• Software development life cycle techniques, like requirements, design, architecture and test techniques.

• Business Process Management techniques. Business Process Management is supposed to deliver better IT and business alignment.

• Case tools adoption. Case tools are used to automate modelling and reduce design errors. Again, their use is expected to improve project performance.

• Project planning and monitoring techniques. Use of managing techniques and standards, like PMBOK, is supposed to help project performance.

Methods

This paper investigates the hypothesis that software development best practices leads to better project outcomes.

To investigate this hypothesis, it is necessary to understand the relationships of efficiency in software projects and nature of returns to scale. Equation (1) is usually used to establish the relative efficiency of software development projects from the correlation between the functional size (lines of code implemented, function points, tables, databases) and their effort in hours.

𝐸𝑓𝑓𝑜𝑟𝑡 = 𝑎 ∗ 𝑆𝑖𝑧𝑒𝑏 (1)

𝐼𝑓 𝑏 < 1 𝑦𝑖𝑒𝑙𝑑𝑠 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 o𝑓 𝑠𝑐𝑎𝑙𝑒

𝑏 = 1 𝑦𝑖𝑒𝑙𝑑𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 𝑜𝑓 𝑠𝑐𝑎𝑙𝑒 𝑏 > 1 𝑦𝑖𝑒𝑙𝑑𝑠 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 𝑟𝑒𝑡𝑢𝑟𝑛𝑠 𝑜𝑓 𝑠𝑐𝑎𝑙𝑒

Methods such as COCOMO II use this equation as basis. In COCOMO II, the a term is a 14 factor composite. Although COCOMO II does not use modern software development terminology, some of its attributes are linked to aspects of software design attributes such as Application of software engineering methods, Software Complexity or Required Software Reliability of the Product.

If we assume that terms a and b are known to a certain project database in a firm, it is expected that similar size projects have similar efforts. In the presence of different efforts we can compare the efficiency of these projects. Projects that present lower efforts when compared to another project of similar size are the one that have better economic efficiencies. However, examination of project results through a single outcome (univariate analysis) such as effort in hours can be deceptive. Quality measures, effort and time elapsed should be analysed together.

We understand that the efficiency of a project should be analysed from a set of technical and managerial results (multivariate analysis). In the design of the experiment we used four variables to analyse software projects efficiency:


• Effort, defined as the total number of hours spent on the project.

• Time Elapsed, defined as the number of months spent on the project.

• Software functional size defined as function points as defined by IFPUG.

• Defect density, defined by the ratio of the number of defects by the functional size of the software. Defect density is a popular quality measure on IT projects. See equation (2)

(2)

When we compare two projects, ceteris paribus, reduced effort, reduced time elapsed, reduced defects and greater functional size suggests better efficiency.

The efficiency function in our experiment is not known, i.e., we make no assumption about the nature of this function (non-parametric function). To compare diverse software projects in a problem of multiple inputs and outputs we used DEA BCC (Data Envelopment Analysis) method.

The experiment consisted in the comparative analysis of a set of software projects. The database used was ISBSG Release Date v11, composed of 5052 software projects. The data of the projects were selected from the following criteria:

• Evaluation of data quality of a project is sound and has been formally evaluated by experts from ISBSG. (Date Quality Rating = A and B).

• Software project defects were reported.

• Functional size counting technique respects Function Point technique as defined by IFPUG (IFPUG Count = Approach) and function points project size in the range (200, 2000).

• Development of new products (Development Type = New development). Maintenance and reconstruction projects were not evaluated in this work.

One hundred and five (105) projects met these criteria and were used for efficiency comparison.

Our DEA model has two inputs (Effort and Elapsed Time) and two outputs (Functional size and Defects). Since software defects is an undesirable output, we modelled it using the reciprocal multiplicative approach [13]. In this approach, the undesirable output is modeled as being desirable: f(uik)=1/uik, where uik is one of the elements of the matrix U of the undesirable outputs i of the decision making unit k.

In our work, we are interested in the comparison of a group of IT projects which used a specific technique and a group of IT projects which didn’t use this specific technique. We use DEA-based hypothesis tests for comparing two groups of decision making units, according with [14]. We use the modified T-Test procedure for comparing the equality of means of two populations random variables.


DEA results are considered robust if the number of DMUs compared is threefold higher than the sum of inputs and outputs. Since we have 2 inputs and 2 outputs, our database sample suffices.


Table 01 shows, for each software project technique, the technical average efficiency scores (mean and standard deviation values).

Software Project Technique

Technical Efficiency Scores

Mean Value

Standard Deviation

CMMI 43.47 13.44

Business Process Management/BPM 89.19 18.83

Business Project Modeling 63.91 27.46

Project Management Techniques 46.76 18.99

Software Requirement Techniques 45.79 18.33

Software Design Techniques 47.04 16.96

Software Test Techniques 42.26 13.18

Table 02 shows the technical efficiency scores breakdown for firm type.

Firm type

Technical Efficiency Scores

Mean Value

Standard Deviation

IT company 79.26 11.58

Banking company 71.42 20.20

Telecom company 47.18 3.29


We first compared groups of software projects to check if a specific technique could indicate superior efficiency technical performance.

A t test failed to reveal a statistically reliable difference between the mean number of CMMI projects (μ = 43.47, s = 13.44) and non-CMMI projects, t(3) = -3.379, p < .05, α = .05.

A t test succeed to reveal a statistically reliable difference between the mean number of BPM projects (μ = 89.19, s = 18.83) and non-BPM projects, t (3) = 3.15, p < .05, α = .05.

A t test failed to reveal a statistically reliable difference between the mean number of projects with process modeling techniques (μ =63.91 s = 27.46) and projects without,

t (16) = 0.66, p < .05, α = .05.

A t test failed to reveal a statistically reliable difference between the mean number of projects with formal project management techniques (μ =46.76 s = 18.99) and projects

without, t (35) = -4.03, p < .05, α = .05.

A t test failed to reveal a statistically reliable difference between the mean number of projects with formal planning techniques (μ =45.79 s = 18.33) and projects without,

t (33) = -4.37 p < .05, α = .05.

A t test failed to reveal a statistically reliable difference between the mean number of projects with formal software design techniques (μ =47.04 s = 16.96) and projects

without, t (32) = -4.23 p < .05, α = .05.

A t test failed to reveal a statistically reliable difference between the mean number of projects with formal test techniques (μ =42.26 s = 13.18) and projects without,

t (17) = -5.55 p < .05, α = .05.

No single software engineering technique could deliver better results. BPM projects, however, presented some evidences of superior performance.

In the analysis of firm types, Telecom and Banking companies failed to pass on the t-test. IT companies, however, passed on the t-test. Superior technical performance of IT firms’ projects can be partially explained by the economic drivers of this type of organization. IT projects are in the core value chain of these organizations. IT projects outcomes determine the success or failures of this type of organization and therefore are continuously monitored and improved by managers and engineering teams.

Conclusions

Quantitative analysis of projects software can be better studied with multivariate analysis methods such as DEA. Results emphasize the importance of recognizing that optimal management techniques depend on the characteristics of the software development project, organization type and its sociotechnical environment. IT companies presented better results and non-IT companies, which can show how the socio-technical can influence the technical performance of projects.


References

M. J. Farrell, “The Measurement of Productivity Efficiency.” Journal of Royal Statistical Society, 1957.

Standish Group, “CHAOS Manifesto 2011,” 2011.

E. Brynjolfsson, “The productivity paradox of information technology,” CommunIcations of ACM, vol. 36, no. 12, pp. 66–77, 1993.

W. T. Lin and B. B. M. Shao, “The business value of information technology and inputs substitution: the productivity paradox revisited,” Decis. Support Syst., vol. 42, no. 2, pp. 493–507, Nov. 2006.

L. H. Putnam, “A General Empirical Solution to the Macro Software Sizing and Estimation Problem,” IEEE Transactions on Software Engineering, pp. pp. 345–361, Jul. 1978.

B. W. Boehm, Software Engineering Economics, 1st ed. Upper Saddle River, NJ, USA: Prentice Hall PTR, 1981.

F. P. Brooks,Jr., The Mythical Man-Month: Essays on Softw, 1st ed. Boston, MA, USA: Addison-Wesley Longman Publishing Co., Inc., 1978.

Kitchenham BA, “The question of scale economies in software-why cannot researchers agree?,” Information and Software Technology, 2002. .

R. Banker and C. F. Kemerer, “Scale economies in new software development,” IEEE Trans Software Eng, vol. 15, no. 10, pp. 1199–1205, 1989.

R. D. Banker, H. Chang, and C. F. Kemerer, “Evidence on economies of scale in software development,” Inf Software Technol, vol. 36, no. 5, pp. 275–282, 1994.

C. Comstock, Z. Jiang, and J. Davies, “Economies and diseconomies of scale in software development,” Journal of Software Maintenance and Evolution: Research and Practice, vol. 23, no. 8, pp. 533–548, 2011.

K. Petersen, “Measuring and predicting software productivity: A systematic map and review,” Inf. Softw. Technol., vol. 53, no. 4, pp. 317–343, Apr. 2011.

[13] E. Gomes and M. Lins, “Modelling undesirable outputs with zero sum gains data envelopment analysis models,” Journal of the Operational Research Society, pp. 616–623, 2008.

[14] R. D. Banker, N. Natarajan, N, and Z. Zeng, “DEA-based hypothesis tests for comparing two groups of decision making units.,” European Journal of Operational Research, 2010.

Acknowledgements

We authors would like to thank ISBSG for making available the ISBSG R11 data.


5. An IDEA model to evaluate the overall performance of Buyer-Supplier

Zahra Yousefi Department Mathematics, Islamic Azad University, Science and Research Branch, Tehran, Iran, [email protected]

Mohsen Rostamy-Malkhalifeh Department Mathematics, Islamic Azad University, Science and Research Branch, Tehran, Iran mohsen_rostamy @yahoo.com

Somayeh Mamizadeh Department Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran [email protected]

Abstract

After proceeding with international management, enterprises have to face the challenge of Buyer-Supplier mainly because of the paid change in the business environment and severe competition I market and customers diverse demand. In some enterprise we may come across imprecise data such as interval, ordinal and fuzzy data. And also for some Buyer-Suppliers, there are indexes that its decrease and or increase are not possible. These indexes calls non-controllable or somewhat controllable. We consider evaluating of Buyer-Supplier so that parts of inputs are non-controllable variable. We focus on the case when imprecise input-output data are represented by intervals.

Keywords: Imprecise Data Envelopment Analysis, Performance evaluation, Buyer-Supplier

Introduction

The organization is customer-oriented organization, which aims to anticipate customer demand in the issues that are most valuable to them. Customer service organization are always the cornerstone of thinking and planning organizations. In the world today that quality is driven customer-oriented, customer focus is the foundation of all commercial activities. The definitive impact of the supply chain on enterprise’s performance has been reported from many industries. A supply chain consists of organizations working in the entire supply chain. The supply chain performance evaluation problem is one of the most comprehensive strategic decision problems that need to be considered for long-term efficient operation of the whole supply chain [2]. Our approach is evaluating of the overall performance in Buyer-Supplier relationships through the measurement of intensity and effectiveness in a supply chain: we use evaluation tools such as Imprecise Data Envelopment Analysis (IDEA). In the


conventional DEA model, input and output are assumed to be exact. In recent years, various applications of DEA, there are many input and output values is unlimited. Imprecise data can be probabilistic, interval, ordinal, qualitative, or fuzzy. Therefore, some papers were presented on the theoretical development of this technique with interval data, of which we can name Inuiguchi (2011) [1]. Thus, we presented an IDEA model with intervals are shown and we consider evaluating of Buyer-Supplier so that parts of inputs are non-controllable variable. In addition, by compared with other Buyer-Suppliers, the evaluated Buyer-Supplier can be identified as efficient or inefficient. Application of classical DEA regards the supply chain as a black box and considers only the inputs from the beginning of the upstream members and final outputs at the very end of downstream members in the performance evaluation. Thus, those intermediate product are ignored. Therefore, the performance scores obtainable from the original approaches will overrate real performance of supply chain. In this paper we examine the supply chain to the black box.

Methods

Assume that there are n supply chains (SCs) each of them producing S outputs by consuming M inputs so that D represented controllable variable. The correspondences between Q1 and (Q11,Q12,Q13,Q14) and between Q2 and (Q21,Q22,Q23,Q24) are shown in Table 1.

Figure1: A supply chain

Therefore, Buyer-Supplier supply chain performance assessment model using the following model:

Buyer-Supplier X Y


Q1 Q11 Q12 Q13 Q14

L R R L

N R L L R

L L L R R

R R R L L

Q1 Q21 Q22 Q23 Q24

R L L R

N L R R L

L L L R R

R R R L L

Table 1: The correspondence between Qi and (Qi1, Qi2, Qi3, Qi4), i=1,2

1

11 12

11 12

13 14

1

s.t. if Q

X i D

X i D

Y r 1,..., if Q X , X i D

X , X

Q Qi iqQ Qi iqQ Qr rq

L L R Ri iq i iqL L Ri iq i

Min

LR

x

x

y sLRx x

x x

θ

λ θ

λ

λ

λ θ λ θ

λ λ

≠

≤ ∈

≤ ∉

≥ =

=≤ ≤ ∈

≤ ≤

21 121 22

1 121 22

1 123 24

i D

Y , Y r 1,..., if Q /

X z z i D

X z z i D

Y z z

Riq

L L R Rr rq r rq

Q Qi i iq iQ Qi i iq iQ Qr r rq r

y y sL R

x

x

y

λ λ

λ θ

λ

λ

− −

− −

+ +

∉

≥ ≥ =

≠

≤ ∈

≤ ∉

≥

21 1 2 2

1 1 2 2

2 2 1 1

r 1,..., if Q / X z z , X z z i D

X z z , X z z i D

Y z z , Y z z

L L R Ri i iq i i i iq iL L R Ri i iq i i i iq iL L R Rr r rq r r r rq r

sL R

x x

x x

y y

λ θ λ θ

λ λ

λ λ

− − − −

− − − −

+ + + +

=

=≤ ≤ ∈

≤ ≤ ∉

≥ ≥

1 2 1 2

r 1,...,

0 , e 1 , 0

z , z , z , z 0,1 .

Tq

i i r r

s

λ λ λ− − + +

=

≥ = =

∈

π

π



The proposed approach, the Buyer-Supplier performance is evaluated both qualitatively and quantitatively.

Conclusions

Today, enterprises have found that they can buy them increasingly effective in increasing the efficiency and effectiveness, and therefore buying practices have changed and try to choose an suitable manner so that they can meet its strategic objectives and purchasing. To accomplish of this subject we have to seek suitable suppliers and strategic and related with them to attain competitive advantages. To achieve this goal, the implementation of the supply chain is necessary.

References

Inuiguchi M., Mozioshita F., (2011) Qualitative and quantitative data envelopment analysis with interval data, Annals of operation research, DOI 10.1007/s10479-011-0988-y.

Yang F., Wu D., Liang L., Bi G.,& Wu D.D., (2009) Supply chain DEA: Possibility set and performance evaluation model, Annals of operation research, DOI 10.1007/s10479-008-0511-2.


6. Assessing Performance of Organized Pharmacy Retail Stores using Data Envelopment Analysis

G N Patel [email protected]

Smiti Pande [email protected]

Abstract

The purpose of the study is to evaluate the performance of a chain of organized pharmacy retail stores using a new enhanced technique, DEA. The study focuses on evaluation of technical and cost efficiency. Technical efficiency does not take into account the substitution possibilities between inputs and therefore we applied cost and allocative efficiency models. To illustrate the DEA models discussed above we are focusing upon a real-world problem of revival of loss making retail stores of an organization which deals in pharmacy retailing. This study uses secondary data collected over a time period of three years from a pharmacy retail chain located in National Capital Region. This organization has expanded itself into a chain of 46 pharmacy retail stores over a time span of three years. The study is deemed to be helpful to the retail managers in providing a framework for performance evaluation and enabling the pharmacy retail stores in gaining a competitive edge over the increasing competition faced by the emerging organized pharmacy retail market in India.

Keywords: Pharmacy, Retailing, Performance Evaluation, Data Envelopment Analysis

Introduction

In the current Indian pharmacy retailing scenario, the sector is mainly dominated by unorganised players, which comprises of, the neighbourhood chemist stores owned by small families. The rising affordability, increased consciousness and willingness to spend has not only given rise to the health and pharmacy retailing business of the society but has also demanded for a big transition. Therefore, the challenge faced by the organised sector is to position itself in such a way that they can be easily differentiated from the small and unorganised retailers of pharmacy sector. Currently there is not much differentiation in the offerings of organised players but at the same time, organised players are a big threat for dominating unorganised sector (Pande and Patel, 2011). Technology and cost are the two wheels that drive a business and therefore play a crucial role in running an efficient business. A considerable amount of attention received by the retail efficiency research is understandable but the scarcity of


research in the area of cost efficiency evaluation is a matter of apprehension. This is because companies are always eager to enquire, how effectively and efficiently their resources that incur huge costs are being utilized (Pande and Patel, 2011). As per the authors’ knowledge, the contribution of this paper to retailing research is based on the application of “allocation models” in order to show how DEA can be used to identify not only the technical efficiency but also the allocative efficiency where, just reallocation of inputs can improve the overall performance.

Theoretical Framework

According to Farrell (1957), cost efficiency gets decomposed into technical and allocative efficiency. The cost minimization problem and its decomposition can be diagrammatically represented as-

Figure 1: Cost efficiency and its decomposition

Now, to evaluate the performance of A we can use Farrell measure of technical efficiency *θ (Refer Cooper et. al., 2007 for details) which is represented in the ratio form, in the following manner-

1),(),(0 ≤≤

AOdBOd

(1.1)

The optimal point C is obtained from the following LP (1.2) (Refer Ray, 2004 for details)

jj

n

jj

n

jrjrj

n

jijij

m

iii ULyyxxtsxw λλλλλ ∀≥≤≤≥≤ ∑∑∑∑

====

0,,,..,min11

011


where, miwi ....2,1, = is the unit price for mxxx ...., 21 respectively. Based on the

optimal solution ( ** ,λx ) derived from the above linear programming the cost

efficiency of a DMU can be calculated. Allocative efficiency (α) is represented by determining the relative distance of B and D to obtain the following ratio-

1),(),(0 ≤≤

BOdDOd

(1.3)

Furthermore, another measure that is referred as cost efficiency (γ) is given as-

1),(),(0

0

*

≤=≤wxwx

AOdDOd

(1.4)

This is a measure of extent to which the originally observed values at A have fallen short of achieving the minimum cost. To put this in a way that relates all three efficiency concepts to each other, we have - α * θ = γ (1.5)

Data and Variables

Data Envelopment Analysis requires identification of input and output variables. For the purpose of this study, input and output variables are identified based on the objectives of the company and also the literature reviewed.

Table 1: Inputs and Outputs

Inputs Authors

Rent paid Joo et al. 2009

Store Size Moreno, 2008; Barros and Alves, 2003; Donthu and Yoo, 1998

Wages Moreno 2010; Joo et al. 2009

Inventory Cost Donthu and Yoo, 1998; Barros and Alves, 2003

Marketing expenses Donthu and Yoo, 1998

Maintenance expenses Moreno 2006

Other day-to-day expenses Joo et al. 2009; Barros and Alves 2003

Outputs Authors

Sales Joo et al. 2009; Barros 2006; Moreno 2006; Seller-Rubino & Mas-Ruiz 2006; Barros and Alves 2003; Donthu and Yoo 1998

Footfalls

Here, the relative unit costs of store for each DMU were also recorded in the following manner: Per unit cost of store size = Total rent / Store size. The other input variables such as maintenance, marketing and other day-to-day expenses are clubbed


under a single heading named ‘operating expenses’. The relative unit cost for operating expenses, inventory cost and wages to the employees remains equals to 1 for each DMU.

Result and Discussion

Firstly, we applied the BCC model (Cooper et. al. 2007). This model includes two outputs: sales and footfalls and four inputs: Wages of the employees, store size, inventory cost and operating expenses. Secondly, we applied the linear programming equation (1.2) to evaluate the cost efficiency of each DMU. This model records the relative unit costs of store size for each DMU, apart from the inputs and outputs considered in the previous model. Third, the study also evaluates the allocative efficiency of each DMU by using the equation (1.5). The scores of three different efficiencies obtained for each DMU for three different years are as shown in Table 2 (refer Appendix A). From the results obtained it can be observed that, the DMUs that turn out to be the most excellent or best performers are DMU 1, 3, 6, 19, 30 and 43 with all their efficiency scores equal to 1 since their inception. Apart from these best performers there are some DMUs with their technical efficiency equal to 1, but, these DMUs do not have the best cost based measures. For example: DMU 2, 5 in the year 2009. There are some that are neither technically efficient nor posses a good cost based measure, like: DMU 4, 7 in 2009. These set of DMUs also includes the ones that have worst allocative efficiency scores for example DMU 13, 14 and 15 in all the three years. All such cases identifies the need of reallocation of resources. Lastly, there are few that are neither technically efficient nor have efficient cost based measures but have attained high allocative efficiency, for example: DMU 4 and 29 in 2010.

\This case is a part of a thesis wherein in order to clearly examine the determinants of efficiency, we applied Tobit regression model regressing the BCC efficiency scores as dependent variable. As per DEA literature Coelli (1998), Tobit regression model is suitable when the dependent variable is censored. The Tobit regression model is represented as-

ii

iiiiiii

LocationAgesizeStoreensesOperatingFootfallsSales

εββββββαθ

+++++++=

)()()()exp()()(

6

54321

Where, iθ is the efficiency score for the retail store i computed from the BCC model

with categorization of age and location as non-discretionary variables.

iiii sizeStoreensesOperatingFootfallsSales ,exp,, are sales, footfalls, operating

expenses and store size of the thi retail store. The non-discretionary variables

ii LocationandAge are part of the thesis work and are not considered in this cost

efficiency section.

The 2χ test statistics (=161.7) with six degrees of freedom associated with p value

(=2.5844x10-32) shows that the model is a good fit for the data. Also we find that the value of constant 2 (e-2.49479751= 0.082513157) from the Tobit model is much less that the

standard deviation of iθ (=0.1676) which again shows that the models appears to fit


the data well. From the results obtained using Tobit model we find that, footfalls, sales, operating expenses, store size are significant contributors to the efficiency of retail store. For an increase in footfalls and sales the efficiency of the retail store will increase by 0.00164001 and 0.00105268 respectively. However, the sign of operating expenses and store size is negative as expected. This indicates that the efficiency of a retail store will fall by (-0.00465915) and (-0.00094193) for an increase in operating expenses and store size respectively. The age and location of a store were not found to be significant contributors to the efficiency.

Managerial Implications and Conclusions

There is a likelihood of huge reductions in the inputs and also, reallocation of inputs as discussed above in various cases. Second, we observe that the older the store, the higher is the cost efficiency. For majority of the stores, the analysis shows huge reduction and reallocation possibilities to attain the optimal input mix. Third, we can state that allocative models better describes the performance of chain of pharmacy retail stores because it brings number of inputs, costs and right mix of inputs altogether. If all the three considerations are well taken care of by retail managers, it may lead to enhancement of the overall performance.

References:

Banker, R. D., Charnes, A. and Cooper, W.W. (1984) ‘Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis’, Management Science, No. 30, pp.1078-1092.

Barros, C.P. and Alves, C.A. (2003) ‘Hypermarket retail store efficiency in Portugal’, International Journal of Retail & Distribution Management, Vol. 31, No. 11, pp.549-560.

Barros, C.P. (2006) ‘Efficiency measurement among hypermarkets and supermarkets and the identification of the efficiency drivers’, International Journal of Retail & Distribution Management, Vol. 34, No. 2, pp. 135-154.

Charnes, A., Cooper, W.W. and Rhodes, E. (1978), "Measuring the Efficiency of Decision Making Units”, European Journal of Operational Research, Vol. 2, 1978, pp.429-444.

Coelli, T.J., Rao, P. And Battese, G.E. (1998), An Introduction to Efficiency and Productivity Analysis, Kluwer Academic Press, Dordrecht

Cooper, W.W., Seiford, L.M. and Tone, Kaoru (2007) Data Envelopment Analysis: A comprehensive text with models, applications, references and DEA-Solver Software, Springer Science + Business Media, LLC.

Donthu, Naveen and Yoo, Boonghee (1998) ‘Retail Productivity Assessment Using Data Envelopment Analysis’, Journal of Retailing, Vol. 74(1), pp. 89-105

Farrell, M.J. (1957) ‘The Measurement of Production Efficiency’, Journal of the Royal


Statistical Society A, 120, pp.253-281

Joo, Seong-Jong, Stoeberl, P.A. and Fitzer, Kristin (2009) ‘Measuring and benchmarking the performance of coffee stores for retail operations’, Benchmarking: An International Journal, Vol. 16, No. 6, pp. 741-753

Moreno, Justo de Jorge (2006) ‘Regional regulation analysis of performance in Spanish retailing’, International Journal of Retail & Distribution Management, Vol. 34, No. 10, pp. 773-793

Moreno, Justo de Jorge (2008) ‘Efficiency and regulation in Spanish hypermarket retail trade: A cross-section approach’, International Journal of Retail & Distribution Management”, Vol. 36, No. 1, pp.71-88

Moreno, Justo de Jorge (2010) ‘Productivity growth of European retailers: a benchmarking approach’, Journal of Economic Studies, Vol. 37, No. 3, pp.288-313

Pande, Smriti and Patel, G.N. (2011) ‘Assessment of Performance using DEA: A Case on Chain of Pharmacy Retail Stores located in NCR’, in Sardana, G.D. and Thatchenkery, Tojo (Eds.), Building Competencies for Sustainability in Organizational Excellence, Macmillan publishers India Ltd., pp. 319-336

Ray, Subhash C. (2004) Data Envelopment Analysis Theory and Techniques for Economics and Operations Research, Cambridge University Press, New York, USA

Sellers-Rubio, R. and Mas-Ruiz, F. (2006) ‘Economic efficiency in supermarkets: evidences in Spain’, International Journal of Retail & Distribution Management, Vol. 34, No. 2, pp. 155-171

Sellers-Rubio, R. and Mas-Ruiz, F. (2007) ‘An empirical analysis of productivity growth in retail services: evidence from Spain’, International Journal of Service Industry Management, Vol. 18, No. 1, pp. 52-69


Appendix A Table 2: Efficiencies obtained 2009 2010 2011

S.No.

BCC Eff. Cost Eff.

Allocative Eff.

BCC Eff. Cost Eff. Allocative Eff.

BCC Eff. Cost Eff. Allocative Eff.

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 1.0000 0.8323 0.8323 1.0000 0.7856 0.7856 1.0000 0.7847 0.7847

3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

4 0.9146 0.7349 0.8036 0.7284 0.6197 0.8508 0.7022 0.6251 0.8902

5 1.0000 0.6619 0.6619 0.9290 0.5567 0.5993 0.8824 0.5562 0.6304

6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

7 0.9706 0.9027 0.9300 0.9412 0.7713 0.8195 0.9107 0.7522 0.8260

8 0.9587 0.7332 0.7648 0.9963 0.6796 0.6822 1.0000 0.6691 0.6691

9 0.9428 0.6686 0.7092 0.9247 0.6043 0.6536 0.9170 0.5883 0.6415

10 0.9657 0.8743 0.9054 0.9175 0.8566 0.9336 0.9210 0.7378 0.8010

11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9042 0.9042

12 0.8695 0.6373 0.7329 0.8815 0.6475 0.7346 0.8533 0.5364 0.6286

13 0.8349 0.4594 0.5502 0.7951 0.4026 0.5064 0.7579 0.4041 0.5331

14 0.8854 0.5706 0.6445 0.8498 0.5659 0.6660 0.8667 0.3905 0.4506

15 0.9348 0.4764 0.5097 0.8644 0.4268 0.4938 0.8088 0.3557 0.4399

16 1.0000 0.8645 0.8645 1.0000 0.7863 0.7863 1.0000 0.7603 0.7603

17 0.9012 0.7526 0.8352 0.8909 0.6790 0.7621 0.8962 0.6798 0.7585

18 1.0000 0.5625 0.5625 1.0000 0.5749 0.5749 1.0000 0.4587 0.4587

19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

20 0.7347 0.3187 0.4337 0.7345 0.3134 0.4267

21 1.0000 0.8099 0.8099 1.0000 0.8340 0.8340

22 1.0000 0.6796 0.6796 1.0000 0.5752 0.5752

23 0.8429 0.5150 0.6110 0.8582 0.5297 0.6172

24 0.8199 0.6512 0.7943 0.7461 0.4975 0.6668

25 0.8046 0.6451 0.8018 0.7906 0.6415 0.8115

26 1.0000 0.9421 0.9420 0.9249 0.7646 0.8267

27 0.7690 0.4849 0.6306 0.7592 0.4847 0.6384

28 1.0000 0.6197 0.6197 0.9740 0.4822 0.4951

29 0.7775 0.7552 0.9713 0.7607 0.7438 0.9777

30 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

31 1.0000 0.5752 0.5752 1.0000 0.5655 0.5655

32 1.0000 0.8634 0.8634 1.0000 0.8817 0.8817

33 1.0000 0.4477 0.4477 1.0000 0.4301 0.4301

34 1.0000 0.6035 0.6035 1.0000 0.6068 0.6068

35 0.8492 0.7987 0.9406 0.8156 0.8115 0.9949

36 0.7357 0.5160 0.7013 0.5891 0.5102 0.8662

37 0.9418 0.2850 0.3026

38 0.8774 0.7284 0.8301

39 0.9169 0.8111 0.8846

40 1.0000 0.5684 0.5684

41 1.0000 0.6334 0.6334

42 0.9908 0.8003 0.8077

43 1.0000 1.0000 1.0000

44 0.9906 0.6208 0.6267

45 0.8861 0.7307 0.8247

46 0.8948 0.8287 0.9262


7. Behavioral effects of DEA on performance assessment

Heinz Ahn Technische Universität Braunschweig, Germany, [email protected]

Nadia Vazquez Novoa Technische Universität Braunschweig, Germany, [email protected]

Abstract

This paper examines the Data Envelopment Analysis (DEA) from a cognitive perspective. Experimental evidence regarding the role of DEA efficiency scores as an overall non-financial performance measure in a context with multiple alternatives and attributes is outlined. The study not only confirms that the efficiency score acts as a strong performance marker when deciding on which decision making units (DMUs) should be awarded for their non-financial performance, but also shows that the score may induce a halo effect, significantly influencing a posterior financial assessment. These results have practical consequences for planning, reporting, and controlling processes that incorporate DEA efficiency scores.

Keywords: DEA, performance assessment, performance markers, halo effect, experimental study.

Introduction

DEA is a promising approach to performance evaluation that allows the simultaneous analysis of financial and non-financial performance indicators. Nevertheless, this instrument presents some shortcomings. Imminent pitfalls have been extensively discussed from a prescriptive point of view. Dyson et al. (2001), e.g., systematically outline conceptual problems of DEA and describe possibilities to cope with them. In contrast, although the “specification of cognitive processes is important to theory development” (Peters, 1993, p. 391), we have found no contributions discussing DEA applications from the descriptive point of view. This is astonishing, since a DEA-based evaluation also includes subjective components, implying that it is susceptible to behavioral influences and limitations. In fact, a (DEA-based) performance assessment aiming at identifying the best performing DMUs can be seen as a multiple attribute choice with multiple alternatives, and as such, it can be analyzed from the perspective of psychological theories.

In this study, the non-financial and the financial performance assessment tasks are considered as two choice problems. In the first one, DEA scores serve to measure the overall non-financial performance of a set of DMUs. Based on heuristics and bias literature, we hypothesize that the relatively high accessibility of DEA scores will make them act as performance markers, thus (over-)simplifying the choice problem by


focusing the analysis on a single cue (Shah and Oppenheimer, 2008; Cardinaels and van Veen-Dirks, 2010).

Literature on contingent decision making indicates that information used for a choice task has an effect on posterior choices by influencing the attributes used and their weights (Dholakia et al., 2005). Additionally, research on decision making has also shown that (irrelevant) previous information is commonly used when determining a searched value, thus producing an anchoring bias. A special case of this bias is the halo effect (Huber, et al., 1987). It refers to the influence that the information about one (often irrelevant) trait, i.e., a characteristic of a person or an object, has on the perception of another trait. In this context, the non-financial performance assessment is expected to act as a halo for the posterior financial performance assessment, improving (worsening) the financial performance assessment of those DMUs with a perceived relative high (low) non-financial performance (Tversky and Kahneman, 1974; Huber et al., 1987). We therefore hypothesize that the presence of high DEA scores will indirectly affect the posterior financial performance assessment through the non-financial performance assessment.

In the following, we depict the methodological aspects of the experiment conducted as well as its results. A discussion of the findings concludes the paper.

Method

Bachelor students (N = 72) taking introductory Management Control and Business Accounting courses at a German university were asked to answer the questionnaire during a lecture. Students were randomly assigned to one of two conditions (DEA score, N = 37; no DEA scores, N = 35). Seven cases were eliminated from the analysis since no valid answers were provided for the dependent variables defined in the general research model.

The experiment comprised two phases. The first had the aim of evaluating the role of DEA scores as performance markers on the non-financial performance assessment. The second phase was designed to investigate possible effects of DEA scores on a posterior financial performance assessment.

Participants received a table with non-financial data corresponding to 10 different DMUs and were asked to decide to which three DMUs they would assign a bonus for non-financial performance (Bnf). The non-financial data included in the report were based on a real case of a European pharmacy chain and included the following performance criteria: average number of employees, sales area in m², sales transactions, and number of customer advices. For those participants in the treatment with DEA scores (θ), this measure was also included. The treatment containing DEA scores permitted to identify three DMUs as efficient (θ = 100%), one as almost efficient (θ = 98%) and all others as inefficient, with θ = 54% being the lowest efficiency score. A brief description regarding the DEA methodology was included in the corresponding vignette.

In the second phase, participants were provided with a table containing three different financial indicators for the 10 DMUs: profit, cash flow in thousand euro, and return on capital. The financial performance indicators were designed to avoid not only a clear


domination of one DMU over its peers but also a significant correlation among DEA scores and the financial measures. Before executing any kind of analytical examination, participants were required to decide whether to assign a bonus for financial performance (Bf) to A. This question had the purpose of concentrating the attention of respondents on the performance of DMU A. In case an anchoring effect would occur, augmenting the accessibility of A should reinforce this effect.

The financial performance assessment was measured by means of two main variables. The first question required participants to decide to which three subsidiaries they would assign a Bf. The answer was considered to have nominal scale since respondents were not asked to elaborate a ranking. The second relevant question required participants to estimate the financial performance of A in an interval between 0% and 100%. Since this estimation was made only for DMU A, the general research model was evaluated exclusively for this case (i.e., a DMU with a high non-financial performance that does not clearly dominate other DMUs and with a relatively good financial performance).

Previous research has shown that a sequential presentation of alternatives affects the way they are evaluated. Consequently, the data corresponding to each DMU was randomized to avoid undesired biases related to the presentation order.

Results

We used a path analysis to prove the general research model for DMU A. The correlation matrix for the four variables included in the causal model together with the main statistics for each variable are presented in Table 1.

As expected, the presence of DEA scores is significantly correlated with the assignment of Bnf to A, and the assignment of a Bf to A is significantly correlated with the perceived score for financial performance for this DMU. It is interesting that the assignment of a Bnf to A is also significantly correlated both with the posterior assignment of a Bf to this DMU and its perceived financial performance score. This provides initial support for our hypothesis.

When developing the hypothesis it was predicted that DEA scores would serve as performance markers and therefore would have an influence on the decision of Bnf assignment. The path analysis for the case of DMU A (Fig.1) indicates that the presence of a DEA score of 100% has a positive influence on the Bnf assignment. The direct effect is positive and significant (a = 0.329, z = 2.822, p = 0.005). The contrary occurs for the case of DMU D, whose DEA score was only 98% (a = –0.419, z = –3.742, p = 0.000), therefore providing confirmatory evidence for our research hypothesis.

Following the criterion of considering an indirect effect to be significant if all the paths involved in its calculation are significant (Kline, 2011), it can be concluded that DEA scores have an effect on the posterior financial performance assessment. The presence of DEA scores has an indirect effect on the assignment of Bf to A (a×b) as well as on the perceived financial performance score (a×b×d). Similar results are obtained for DMU D (a×b significant, with b = 0.265, z = 2.292, p = 0.022) Table 2 presents the coefficients for each of the paths included in the causal model for DMU A.


Table 1: Correlation matrix for DMU A and main statistics.

DEA Bnf to A Bf to A Score A

DEA 1

Bnf to A 0.324** 1

Bf to A 0.109 0.244* 1

Score A 0.169 0.303* 0.401** 1

Mean 0.492 0.430 0.540 70.26

SD 0.504 0.499 0.502 19.38

* Significant at 0.05 level (two-tailed). ** Significant at 0.01 level (two-tailed).

The proposed model for DMU A fits with the data. All but one path coefficients were significant and several measures of goodness of fit gave support to the model (CFI = 1.000, χ²(2,65) = 0.411, p = 0.814, and SRMS = 0.034). However, other measures of fit indicate that the model should still be improved (RMSEA = 0.000 with confidential interval (0.000 – 0.150), TLI = 1.270). To augment the model fit, different modifications could be conducted adding parameters that were not considered in this experiment. Incorporating other variables to the model could help to improve the proportion of explained variance for each endogenous variable (R²Bnf = 0.108, R²Bf = 0.057, R²Score = 0.211).

b = 0.239*

c = 0.206

Assignment of Bf to A

Perceived financial

efficiency score of A

Assignment of Bnf to A

Presence of DEA scores

a = 0.329**d = 0.365***

Performance marker

Halo effect

Halo effect

* Significant at 0.05 level (one-tailed). ** Significant at 0.01 level (one-tailed). *** Significant at 0.001 level (one-tailed).

FIG. 1: PATH COEFFICIENTS FOR DMU A (N=65, SOLID LINES REPRESENT S G CA A S ² 0 21)


Conclusions

The present study offers an analysis of the DEA approach focusing on its decisional consequences. To the best of our knowledge, it constitutes the first attempt to deal with the DEA approach from a behavioral operations perspective. The results of the experiment indicate that the presence of DEA scores influences not only the non-financial performance assessment of the DMUs, but also the posterior financial one.

As expected, DEA scores calculated to measure the overall non-financial performance of a set of DMUs were found to act as performance markers. Consistent with previous research on lexicographic heuristics, decision-makers seem to concentrate their attention on the DEA scores as a manner to reduce the cognitive load and to save time.

A managerial limitation associated to DEA scores operating as performance markers arises from a methodological drawback of the basic DEA models: the weights of the performance criteria are endogenously determined, allowing zero-value weights. This may induce managers to concentrate their efforts only on some criteria when attempting to raise the efficiency score of their DMU. A bonus assignment solely based on the DEA scores may encourage this behavior, leading to a biased resource allocation. At the same time, it may decrease the motivation of managers running DMUs that are achieving all relevant performance criteria on a decent level. These undesirable effects emphasize the importance of the ongoing research on how to cope with the zero-value weights problem.

As the second main result of our study, it could be shown that the inclusion of DEA scores to measure the overall non-financial performance may affect the posterior financial assessment. For the case of DMUs with a general good performance level, this effect is mediated by the decision of awarding the respective DMU for its non-financial performance. A DEA score of 100% contributed to increment the proportion of Bnf awarded to the DMU which in turn affected its financial assessment: more Bf were assigned to this DMU and its estimated financial performance score was higher. For cases of good performers with a DEA score lower than 100% such as D (θD = 98%), the presence of the DEA score led to a lower Bnf assignment that negatively influenced the posterior financial assessment.

The unintended consequences of the presence of DEA scores can be attributed to a halo effect. A DMU being awarded for its non-financial performance (positive trait) will be perceived as attaining better financial results. Thereby, the Bnf assigned to the DMU serves as a self-generated anchor that acts as a “starting point” when assessing the financial performance.

These findings strongly suggest that including DEA scores in the performance report for facilitating decision-making may result in biased decisions. Since the DEA approach is gaining relevance for everyday management control, it results essential to investigate how to deal with this difficulty. To this respect, theories of mental protection provide some insights (Wilson et al., 2002). On the one side, instruments that prevent contaminating stimulus to enter managers’ minds could be developed. On the other side, providing adequate training regarding cognitive biases could at least help to diminish the undesirable effects.


The halo effect is also relevant from a decision influencing perspective. Incrementing the recognition for financial performance to DMUs of superior non-financial achievement level and reducing such recognition to other DMUs may be perceived as unjust. As a consequence, problems related to organizational commitment, job satisfaction, performance, trust, etc. may arise.

The results were obtained from a sample of 65 bachelor students of a German university. Even if the practical choice of resorting to students as surrogates for managers has been supported by several authors (Moore, 2005), the adequacy of undergraduate students as substitute for managers has been questioned. One of the main critics suggests that undergraduate students may differ from experts on a number of psychological and behavioral dimensions. However, this argument could be relativized, since it can be assumed that not many managers have a more comprehensive understanding of the DEA approach. Nevertheless, the current work should be replicated using different samples in order to achieve a higher validity of the results.

Further research analyzing DEA from a behavioral perspective is necessary if this instrument is to be satisfactorily applied in the business context. Besides the cognitive effects associated to the interpretation of the results, the influence of the DEA approach on fairness perception should also be considered. A thorough investigation of the behavioral aspects will enrich the analysis and provide insights into necessary modeling improvements.

References

Cardinaels E., P.M.G. van Veen-Dirks (2010) Financial versus non-financial information: the impact of information organization and presentation in a balanced scorecard, Accounting, Organizations and Society 35 (6): 565–578.

Dholakia U.M., M. Gopinath, R.P. Bagozzi (2005) The role of desires in sequential impulsive choices, Organizational Behavior and Human Decision Processes 98 (2): 179–194.

Dyson R.G., R.S. Allen, A.S. Camanho, V.V. Podinovski, C.S. Sarrico, E.A. Shale (2001) Pitfalls and protocols in DEA, European Journal of Operational Research 132 (2): 245–259.

Huber V.L., M.A. Neale, G.B. Northcraft (1987) Judgment by heuristics: effects of ratee and rater characteristics and performance standards on performance-related judgments, Organizational Behavior and Human Decision Processes 40 (2): 149–169.

Kline R.B. (2011) Principles and practice of structural equation modeling, 3rd edition, The Guilford Press: New York.

Moore D.A. (2005) Commentary: conflicts of interest in Accounting. In: D.A. Moore, D.M. Cain, G. Loewenstein, M.H. Bazerman (Eds.), Conflicts of interest. Challenges and solutions in Business, Law, Medicine, and Public Policy. Cambridge University Press: Cambridge et al., 70–73.


Peters J.M. (1993) Decision making, cognitive science and Accounting: an overview of the intersection, Accounting, Organizations and Society 18 (5): 383–405.

Shah A.K., D.M. Oppenheimer (2008) Heuristics made easy: an effort-reduction framework, Psychological Bulletin 134 (2): 207–222.

Tversky A., D. Kahneman (1974) Judgment under uncertainty: heuristics and biases, Science, New Series 185 (4157): 1124–1131.

Wilson T.D., D.B. Centerbar, N. Brekke (2002) Mental contamination and the debiasing problem. In T. Gilovich, D. Griffin, D. Kahneman (Eds.), Heuristics and biases. The psychology of intuitive judgment, Cambridge University Press: Cambridge et al., 185–200.

Table 2: Effects of the presence of DEA scores on the performance assessment of DMU A.

Path

Standardized coefficient

Coefficient (original units)

Standard error

z-stat. Sign.

Direct effects

DEA – Bnf a 0.329** 0.325 0.115 2.822 0.005

Bnf – Bf b 0.239* 0.241 0.119 2.024 0.043

Bnf – Score c 0.206 8.259 4.679 1.765 0.077

Bf – Score d 0.365*** 14.527 4.272 3.400 0.001

Indirect effects

DEA – Bf a×b 0.079* 0.078

DEA – Score a×c 0.068 2.684

a×b×d 0.029* 1.138

* Significant at 0.05 level (one-tailed).

** Significant at 0.01 level (one-tailed).

*** Significant at 0.001 level (one-tailed).


8. Data Envelopment Analysis of the effieincy frontier for the results achived by Formula 1 drivers and teams

Prof. Dr. Aparecido Jorge Jubran UNINOVE, [email protected]

Profa. Msc. Laura Martinson Provasi Jubran UNINOVE, [email protected]

José Rubens Moura Martins UNINOVE, [email protected]

Jane Leite Silva UNINOVE, [email protected]

Abstract

The current Formula 1 driver and team ranking system does not allow for an impartial and unbiased comparison between results, since the criteria used are oftentimes inconsistent. In order to make such comparison, the mathematical optimization tool DEA - Data Envelopment Analysis was used. As a result of this research, an plan was elaborated to review the efficiency frontier for the results achieved by Formula 1 drivers and teams, and a new driver and team ranking list was created.

Key words: Data Envelopment Analysis, Formula 1, Ranking.

Introduction

The objective of this paper was to design a plan to analyze the efficiency frontier for the results achieved by Formula One car drivers and teams. This topic is particularly relevant to the sports community, since the existing comparative performance measurement methods applied along decades of competitions have always lacked indicators that apply to all sports categories.

The existing car racer and racing team ranking system does not allow for an impartial and unbiased comparison, since the criteria used are oftentimes inconsistent.

This paper also aims to help shed a light on the reasons why Formula One car racers and agents constantly criticize the changes in the ranking criteria set out by FIA - Fédération Internationale de l'Automobile. By constantly introducing technological innovations, Formula 1 is today the most audacious and technologically sophisticated motorsport category in the world.

One of the problems that affect this auto racing category is its points scoring system, which generates distortions when analyzing and ranking the best drivers.

http://pt.wikipedia.org/wiki/F%C3%A9d%C3%A9ration_Internationale_de_l%27Automobile


The year 2010 marks the beginning of a new scoring scheme up to 10th place in each Grand Prix, as illustrated in Fig. 1.

Scoring system

Temp./Pos. 1º 2º 3º 4º 5º 6º 7º 8º 9º 10º

1950-1955 8 6 4 3 2

1956-1990 9 6 4 3 2 1

1991-2002 10 6 4 3 2 1

2003-2009 10 8 6 5 4 3 2 1

2010 25 18 15 12 10 8 6 4 2 1

Fig.1:Scoring system from 1950- 2010.

These issues bring a negative impact upon sponsors and manufacturers, as well as on drivers and teams, which are greatly affected. The points scoring system adopted by FIA has changed frequently along the years, making it impossible to objectively compare and analyze the results of each race and each championship along the years.

Due to the criteria currently in use, the rankings prepared and released to the public are inaccurate. The rankings measure performance by driver, nationality, constructor, engine, and tire manufacturer.

An example of these distortions can be seen in Table 1 (in APPENDICES), showing the top 10 in the 2010 championship, where the 3 and 4 positions would be reversed only if the current scoring criteria valid in 2009 continued in 2010.

Methods

The mathematical optimization tool DEA - Data Envelopment Analysis was used to prepare the reports. Data Envelopment is an analytical tool designed to identify the best practices for use of resources, which, in our study, comprise those resources available for Formula 1 teams.

According Emrouznejad (2005), Figure 2 shows a number of units P1, P2 ... P6, where each unit consumes a resource, but produce different amounts of outputs y1 and y2. Thus, for a given amount of input feature units that provide larger quantities of outputs will be considered efficient.


Fig. 2: Efficiency frontier. Source: Emrouznejad (2005).

Through this research, an analysis plan was elaborated to review the efficiency frontier for the results achieved by Formula 1 drivers and teams. It also allowed putting together a new ranking covering all Formula 1 drivers and teams, as well as other stakeholders, including engine, tires, and service suppliers.

The mathematical optimization tool DEA - Data Envelopment Analysis was used to prepare the reports.

A bibliographical research was carried out to collect data on the results of all F1 races ever – beginning with the first race in 1950.

After the results of all F1 races were known, a study was conducted to analyze the inputs and outputs for a review using the DEA method, with the following components being assessed:

• Decision Maker Units - DMUs: Teams and Drivers.

• INPUT: Grid position at qualifying session.

• OUTPUT: Position at the end of the race.


After extensive testing, the following factors were chosen for assessment of F1 results.

Inputs: The driver's position at the start of the qualifying session, i.e., weight 1 for all drivers, since it was assumed all drivers start the session under the same conditions.

Outputs: The driver's position at the end of the race, considering the arrival order according to FIA’s rules. The output weight for each driver's position was calculated based on the number of cars participating in each race.


Therefore, in a race where 20 cars cross the start line, the car crossing the finish line first is given weight 20; with weight 19 given to the second place; 18 to the third place; and so on, up to the last car completing the race. This way, all competing cars are ranked.

After efficiency was calculated using the DEA tool, each competitor was attributed an efficiency frontier value. In this case, the driver crossing the finish line first was attributed an efficiency value 1 or 100%.

The other efficiency values were attributed according to the number of competing cars, which means the efficiency frontier value increases when the number of competing cars increases.

The Figure 3 (in APPENDICES), shows the results calculated by the DEA model for the first Grand Prix (England - 1950), where it is reported the name, the position in the hallway and Rank obtained. Thus the sum of the Scores of all Grand Prix shows the champion, according DEA efficiency.

Conclusions

FIA’s criteria determine that the ranking be based on points awarded to drivers according to their position at the end of the race, with up to a number of positions scoring points, and the remaining drivers scoring no points, regardless of the number of drivers. Sometimes the scoring criteria, which is set out annually, are changed at the end of a season. By using this new scoring method, i.e., the DEA efficiency frontier, these comparisons become a reality, since the assessment is based only on the order of arrival, which is translated into an efficiency ranking.

References

FARREL, M. J. (1957) The measurement of productive efficiency. Journal of the Royal Statistic Society, London, v. 120, n. 3, p.253-290.

FIA. http://www.fia.com/en-GB/Pages/HomePage.aspx. Acessado em 30/jan/2011. Access 30/jan/2011.

EMROUZNEJAD, Ali. (1995-2000) Coventry: Warwick Business School, Data Envelopment Analysis Home Page. Disponível em: <http://www.deazone.com/index.htm>. Access 25 mai 2005.

FORMULA 1. http://www.formula1.com/default.html. Access 30/jan/2011.

JUBRAN, L.M.P. (2005) Aplicação da Análise por Envoltória de Dados: um estudo da eficiência das companhias seguradoras. 2005. 143 p. Dissertação (Mestrado) – Departamento de Engenharia Elétrica, Escola Politécnica da Universidade de São Paulo. São Paulo.

PRADO, D. L. Pontos na F1: Decisão polêmica da FIA não é a primeira na história. Disponível em < http://www.dzai.com.br/>, 2009. Access 30/jan/2011.

PUCCINI, A. L.; PIZZOLATO, N. D. (1989) Programação linear. 2.ed. Rio de Janeiro: LTC.

http://www.fia.com/en-GB/Pages/HomePage.aspx.%20Acessado%20em%2030/jan/2011



http://www.deazone.com/index.htm


Appendices

Table 1: Distortions between scoring criteria 2009 and 2010.

Fig. 3: Results calculated by the DEA model


9. Data Envelopment Analysis Type Linear and Goal Programming Models For Measuring Energy Efficiency Performance of OECD Countries

Hasan BAL Gazi University, Science Faculty Department of Statistics 06500 Ankara TURKEY, [email protected] (corresponding author)

Mehmet Guray UNSAL Gazi University, Science Faculty Department of Statistics 06500 Ankara TURKEY, [email protected]

Abstract

Models planned in this work are as follows: the goal programming DEA ranking model recognized as an alternative to multiple criteria DEA and the model which closes each weighted output (input) component to weighted output (input) sum, and hence providing a contribution to efficiency account of each output (input) component proportional to the output(input) values. The proposed models are applied to measure the energy efficiency performances of OECD countries and the results obtained are presented.

Keywords: Linear and Goal Programming, Energy Efficiency, Data Envelopment Analysis

Introduction

DEA was first developed by Charnes et al. [1] that seems to be the most popular method for measuring the efficiency of homogenous decision meaking units. Bal et al. [2] suggest goal programming approaches to improve the discrimination power of DEA. Sexton et al. [3] and Doyle and Green [4] suggest cross efficiency evalution as an extension of DEA aimed at avoiding some of the mentioned difficulties. Their technique made use of the cross evaluation scores computed as related to all DMUs and hence identified the best DMUs [5].

The problem of having multiple optimal solutions to weights for efficient DMUs affect to a great extent the consistency of operations related to weight cross efficiency method is most frequently studied topic in DEA literature. Sexton et al. [3] and Doyle and Green [4] recommended the use of a secondary objective (model) for the cross efficiency evaluation related to the non-uniqueness of optimal weights in DEA. They proposed the aggressive and benevolent models for achieving the secondary objective. Andersen et. al. [5] proved the fixed weighting nature of the cross efficiency


evalution in the case of single input and multiple outputs. Orkcu and Bal [6] proposed goal programming models that could be used in the second stage of the cross evalution.

DEA, Super Efficiency, Cross Efficiency Evaluation and Multiple Criteria DEA Models

Data Envelopment Analysis (DEA) is a mathematical programming approach that utilizes multiple inputs and outputs to measure the relative efficiencies within a group of decision making units (DMUs). In DEA, it is assumed that there are n DMUs to be

evaluated in terms of m inputs and s outputs. Let ijx ( 1, . . . , i m= ) and

rjy ( 1, . . . , r s= ) represent the input and output values of DMU j ( 1, . . . , j n= ),

respectively. Subsequently, the efficiency of DMU p can be calculated as

1

1

s

r rpr

p m

i ipi

u y

v xθ =

=

=∑

∑ (1)

where, iv ( 1, . . . , i m= ) and ru ( 1, . . . , r s= ) are the input and output weights

assigned to thi input and thr output,respectively.

1ma x

s

p r rpr

u yθ=

=∑

s.t.

11

m

i ipi

v x=

=∑ (2)

1 10

s m

r rj i ijr i

u y v x= =

− ≤∑ ∑ , 1, 2, . . . , j n=

0ru ≥ , 1, . . . , r s=

0iv ≥ , i=1, ..., m

In the above-mentioned models, DMUs is considered to be efficient if and only if θ *p=1; otherwise, it is referred to as non-efficient. DEA can be used only for ranking inefficient DMUs and in order to abolish this disadvantage various methods were developed [8]. The most commonly used method developed for ranking efficient decision units is the super efficiency model proposed by Andersen and Petersen [9].

In addition, the cross efficiency method was developed as a DEA extension tool to be utilized for identifying the best performing DMUs, and for ranking DMUs using cross efficiency scores that are linked to all DMUs [3,6]. In the first stage, the optimal weights of inputs and outputs are calculated for each DMU using the classical DEA formulation. Given the results of the first stage, the weights used by the DMU can be utilized for calculating the peer rated efficiency for each of the other DMUs. The peer


evaluation score, θp,j, indicates the efficiency score for the DMU j using the weights

obtained by the DMU p [5].

,1

,

,1

s

r p rjr

p j m

i p iji

u y

v xθ =

=

=∑

∑. (3)

If we consider the multiple criteria DEA model, CCR model could be expressed equivalently in the form given by Li and Reeves [10].

1min or max

st

s

p p r rpr

d u yθ=

=

∑

1

1 1

1 (4)

0

1, 2, . . . , , , 0 , all , and values

p

m

i iis m

r rj i ij jr i

r i j

v x

u y v x d

j nu v d r i j

=

= =

=

− + =

=≥

∑

∑ ∑

where pd is the deviation variable for DMUo. This DMU is efficient if and only if

0pd = or 1pθ = . A multiple criteria data envelopment analysis model formulation with

the minmax and minsum criteria, which minimizes a deviation variable, rather than maximizing the efficiency score, is shown as below (MCDEA):

1

1

1

1 1

min or max

min

min (5)

st

1

0

1, 2, . . . ,

o

o

s

o o r rjr

n

jj

m

i ijis m

r rj i ij jr i

j

d w u y

M

d

v x

u y v x d

j nM d

=

=

=

= =

=

=

− + =

=− ≥

∑

∑

∑

∑ ∑

0 , 1, 2, . . . ,

, , 0 , All , and r i j

j nu v d r i j

=

≥

Goal Programming DEA and Component Models

MCDEA which is given in model (6) can also be easily adapted to the weighted goal programming as follows:

(GPDEA-CCR)


1 1 2 3

1 11

2 21

1 1

min , ,

1

1 (6)

0 , 1, 2, . . . ,

j jj j

m

i ipis

r rpr

s m

r rj i ij jr i

j

a n p p n d

v x n p

u y n p

u y v x d j n

M d

=

=

= =

= + +

+ − =

+ − =

− + = =

−

∑ ∑

∑

∑

∑ ∑3 3 0 , 1, 2, . . . ,

j jn p j n+ − = =

0, 1, 2, . . . , 0, 1, 2, . . . ,

r

i

u r sv i m

≥ =≥ =

1 1 2 2

3 3

0, 1, 2, . . . ,

, , , 0 , 0 , 1, 2, . . . ,

j

j j

d j nn p n pn p j n

≥ =

≥≥ =

Where for the DMU under evaluation, 1n and 1p are the unwanted deviation variables

for the goal which constraints the weighted sum of outputs less than or equal to unity,

2n is the wanted deviation for the goal which makes the weighted sum of outputs less

than or equal to unity. 3 jn ( 1, 2, . . . , j n= ) are the unwanted deviation variables for

the goal which realizes M as the maximum deviation, and 3 jp ’ are the wanted

deviation variables for the same goal. Whereof our aim, given equal weight to the unwanted deviations, is to minimize the sum of unwanted deviations, is to minimize

thesum of unwanted deviations 1n , p (1

1m

i ioi

v x=

=∑ ) and 2p , 3 jj

n∑ and jj

d∑ .

The proposed approach is aimed at approximating each weighed output (input) component to weighted output (input) sum in order to contribute to the efficiency account of each output componenet in proportion to the output(input) values, i.e., to the extent of their greatness or smallness. It is also aimed at obtaining weights that are more appropriate when compared to those obtained by classical DEA. In the second stage of cross evalution, a model is presented (7) in which the classical DEA efficiency scores for each unit are preserved and more appropriate optimal weight values are selected for the units for which the optimal weights obtained by classical DEA in the first stage possibly have multiple and inappropriate solutions.


*

1

1

1 1

1

1

min

st

1 (7)

0 j 1, 2, . . . ,

r 1, 2, . . . , s

p

p

p p

s

r rp prm

i iis m

r rj i ijr i

s

r rj r rp pr

m

i ij i i pi

w z

u y

v x

u y v x n

u y u y z

v x v x z

θ=

=

= =

=

=

=

=

=

− ≤ =

− ≤ =

− ≤

∑

∑

∑ ∑

∑

∑ 1, 2, . . . , m

, , 0 , all , valuesr i p

i

u v z r i

=

≥

Here, *pθ is the efficiency value for p. DMU obtained from the classical DEA.

Application

In this section, the methods, which are mentioned above, are used to compare the performance of OECD countries in respect of CO2 emissions, energy consumption [10]. According to Ramanathan’s study [10], the input variables are CO2 emissions per capita (denoted as CO2 per cap hereafter), fossil fuel energy consumption (FOSS), and the output variables are gross domestic product per capita (GDP per cap) and non-fossil fuel energy consumption (NFOSS) to measure the energy consumption and CO2emmissionefficiency.

Table 1. DEA-CCR Results of Application Study


Table 2. GPDEA-CCR Results of Application Study

Table 3. Component Model Results of Application Study

Table 4. Coefficients of Variation For Weights

According to the obtained results in Table 1-4, it could be seen GPDEA and Component models have fewer efficient DMUs than classical DEA model. According to coefficients of variation for weights of inputs and outputs, the proposed models generally decrease the variation of weights of variables.


Table 5. Rankings of DMUs For Methods

Table 6. Sperman’s Rank Correlations According to Ranking of DMUs

Conclusions

It could be seen that GPDEA and Component models have fewer efficient DMUs and the proposed models generally decrease the variation coefficient of weights of variables. The models models also have significant and high correlation between each other according to ranking scores of DMUs.

References

A. Charnes, W.W. Cooper, E. Rhodes, Measuring the Efficiency of Decision Making Units, Eur. J. Oper. Res., 2 (1978) 429–444.

H. Bal, H.H. Örkcü, S. Çelebioğlu, Improving the Discrimination Power and Weight Dispersion in the Data Envelopment Analysis, Comput. Oper. Res., 37 (1) (2010) 99–107.

T.R. Sexton, R.H. Silkman, A.J. Hogan, Data Envelopment Analysis: Critique and Extension. In: Silkman R.H. (Ed.), Measuring Efficiency: An Assessment of Data Envelopment Analysis, 32. Jossey-Bass, San Francisco (1986) 73-105.

J.R. Doyle, R. Green, Efficiency and Cross Efficiency in Data Envelopment Analysis: Derivatives, Meanings and Uses, J. Oper. Res. Soc., 45 (5) (1994) 567–578.

T.R. Andersen, K.B. Hollingsworth, L.B. Inman, The Fixed Weighting Nature of a Cross Evaluation Model, J. Prod. Anal., 18 (1) (2002) 249–255.


H.H. Örkcu, H. Bal, Goal programming approaches for data envelopment analysis cross efficiency evaluation, Appl. Math. Comput., 218 (2011) 346-356.

R. Ramanathan, An analysis of energy consumption and carbon dioxide emissions in countries of the Middle East and North Africa, Energy, 30 (2005) 2831–2842.

D.L., Retzlaff-Roberts, Relating discriminant analysis and data envelopment analysis to one another, European Journal of Operational Research , (1996) 23: 311-322.

P., Andersen, N., Petersen, A procedure for ranking efficient units in Data Envelopment Analysis, Management Science, 39(10), (1993) 1261-1264.

X. B., Li, G.R., Reeves, A Multiple Criteria Approach to Data Envelopment Analysis, European Journal of Operational Research, 115: (1999) 507–517.


10. Decentralization and productivity of the public health service in Brazil

Aléssio Tony Cavalcanti de Almeida UFPB – Federal University of Paraiba, Brazil, [email protected]

Carlos Eduardo Gasparini* UFPB – Federal University of Paraiba, Brazil, [email protected]

Abstract

The scientific literature has pointed to the process of fiscal decentralization as a potential inducer of efficiency and productivity in the public sector. However, some authors have questioned whether the process in Brazil would actually generate a waste of resources and raise problems in the quality of services provided. This paper uses the Malmquist index and econometrics with panel data to empirically assess the question of the relationship between fiscal decentralization and performance of the public health service in Brazil, as well as to provide an overview of the dynamics of regional productivity in the sector. The results allow us to observe that the decentralization of health spending has a negative relationship on the productivity of these services, but fiscal responsibility has a greater influence on the performance of the local governments.

Keywords: Health, Decentralization of Expenditures, Productivity, Fiscal Responsibility.

Introduction

Worldwide, health care is of increasing concern, both politically and socio-economically. In Brazil, the health system experiences increasing pressure to improve its performance, both in regard to controlling the cost of services and ensuring greater access and better quality health care is available to the population. To analyze the current stage of this system it is necessary to understand the process of decentralization, particularly with the creation of the Sistema Único de Saúde (Unified Health System, SUS), provisions for which are contained within the 1988 Constitution, where states and municipalities gained the biggest transfers of funds and responsibilities over the provision of health services.

Oates (1977 and 2005), among others, notes that the fiscal decentralization process generates a number of benefits to society, given that local governments can provide goods and services more efficiently, which are more relevant to local preferences and demands. On the other hand, critics like Prud'homme (1995) point to the number of

* Recipient of financial aid from IPEA – Brazil.


risks involved. In an analysis of the Brazilian case, Campos (1998) points out that the SUS is a system designed to decentralize, "within the government," the management of the Brazilian public health service, i.e., transferring some powers to states and municipalities. However, he goes on to highlight the level of unpreparedness of sub-national governments, in both administrative incapacity, nepotism, lack of technical resources, inexperience, high level of corruption, etc - to assume the important responsibilities of the provision of health care.

Against this backdrop, this study aims to examine whether the increased decentralization of public health services favored or not the productive performance of the sector in Brazil. In Brazil, several studies attempted to measure the efficiency of the provision of public goods and services, e.g.: Gasparini and Ramos (2004) and Faria et al. (2008). However, this study, besides considering the growth of productivity, advances on two aspects of this issue. First, it focuses explicitly on the relationship between changes in productivity in the public health service and decentralization. In addition, it offers a regionalized analysis of the dynamics of productivity in the sector in Brazil, considering the indicators of technological change and efficiency for the years 1996 to 2007.

This paper is organized into four parts including this introduction. The next section presents a description of the underlying data and methodological procedures based on the nonparametric estimation of the technological frontier and the Malmquist index, together with the econometric approach. The third section contains the analysis of results and, finally, the fourth section summarizes the principal conclusions of this study.

Methods

To achieve the set goals, the empirical analysis of the study was divided into two stages. The first builds a dynamic index of productivity growth for public health services, using data from the municipalities aggregated at the state level, where the indicator can be divided into changes in efficiency and technical innovation. This index is intended to ascertain the best relations of efficiency and technical changes obtained during the period between 1996 and 2007. To calculate this indicator, we used the Malmquist index of productivity, with the help of non-parametric method Data Envelopment Analysis (DEA) to estimate the necessary efficiency scores. This approach was chosen as it handles simultaneously multiple inputs and outputs that are typical of the health sector, and also not to impose functional form on the production frontier1. Thus, for the development of Stage I we used the following production function of public health services:

(𝑦1, 𝑦2) = 𝑓(𝑥1, 𝑥2, 𝑥3) (1)

1 The concept of Malmquist productivity index was first introduced by Malmquist (1953) and later refined by several works, including Caves et al. (1982), Färe et al. (1994) and Thrall (2000). This index represents the growth of total factor productivity (TFP) of decision making units (DMU), which reflect two components: efficiency change and technological change over time.


here: 𝑥1 = Number of public hospital beds; 𝑥2 = Number of doctors; 𝑥3 = Number of

nurses; 𝑦1 = Number of hospitalizations; 𝑦2 = Number of consultations. These variables were taken from the DATASUS site (Ministry of Health - Brazil).

The second stage of the study evaluates, by means of an econometric approach with Panel Data, the relationship between the indicator of productivity growth in health (as calculated in the previous stage) and variables related to the issue of Brazilian fiscal federalism and other relevant socioeconomic factors. In this step, we seek to precisely evaluate the effect of decentralization on the performance of the provision of public health in Brazil. The model final to be estimated in Stage II is structured as follows:

𝑚𝑖𝑡 = 𝛽0 + 𝛽1𝐷𝑖𝑡 + 𝛽2𝐶𝑖𝑡 + 𝛽3𝐹𝑖𝑡 + 𝛽6𝑃𝑖𝑡 + 𝛽7𝐸𝑖𝑡 + 𝛼𝑗𝑅𝑒𝑔𝑗

4

𝑗=1

+ 𝛽10𝑑𝑢𝑚𝑆𝑐 + 𝑒𝑖𝑡

(2)

Where: 𝑚𝑖𝑡 = Productivity growth of public health (calculate in Stage I); 𝐷𝑠 =

Decentralization of spending on public health; 𝐶𝑓 = Cash flows; 𝐹𝑟 =Fiscal

responsibility; 𝑃𝑜 = Poverty rate; 𝐸𝑑 = Educational attainment. In notation, the subscript i denotes the different DMUs and t denotes time. The dot above the variables expresses growth rates. The Regj are dummies for each region of Brazil, dumESC represents another binary variable that includes the units that have changed the scale. The information for the second stage was collected from the Secretatia do Tesouro Nacional (National Treasury Secretariat, STN), the Instituto de Pesquisa Econômica Aplicada (Institute of Applied Economic Research, IPEA) and the EDUDATA platform from the Ministry of Education. The data covers the years 1995 to 2007. Nevertheless, in Stage II the evaluation is done by growth rates of all variables and the year 1996 becomes the starting point of analysis.

Decentralization of the health service becomes more evident from the regulation of SUS under Laws 8080 and 8142 of 1990 as well as through the largest transfer of resources from fund to fund to sub-national governments, especially from the approval of the SUS Basic Operational law of 1996. This has developed a dynamic analysis from this date. It is noted that the choice of the initial and final analysis was also dictated by the availability of official statistical databases.


(a) Productivity of the public health services

To calculate the Malmquist productivity index it was assumed that the technology employed by Brazilian cities has variable returns of scale, considering both the technical and socioeconomic heterogeneity of them. In general this indicator was negative, with an effective growth in only three years. In general, health services showed a negative growth rate in productivity. On average, the rate of productivity growth was -1.46% for the entire national territory. The region with the lowest absolute percentage of decrease was the Southeast with about -1%. On the other hand, the Midwest, with -3%, was the region that showed reduced productivity of


factors on public health. It is noteworthy that in 2004 there was a sharp decline in the growth of the Malmquist productivity index for the South and the Midwest, -16.39% and -13.43% respectively.

When looking at individual results, it is noted that the Espirito Santo had the highest total productivity growth in the sample, 1.1% on average, and that the change of efficiency to this location was mainly responsible for this result. Besides Espirito Santo, just two other places had a positive indicator, namely: Acre and Bahia, which together obtained a small 0.1% growth in productivity. The places with the worst average performance were Roraima, Piauí and Distrito Federal, with a decrease of 6.4%, 5.8% and 5.8% respectively. At the regional level, all the locations had a Malmquist score lower than one (1) indicating that there was a decline in productivity in the provision of health services provided by the public sector. The Midwest was the region which showed the greatest decrease in productivity (-3.1%) much higher than the average for Brazil (-1.5%).

The major problem showed by this analysis relates to the fact that all the locals, except Bahia, had a negative result in its technical change (TC), a sign that the technological frontier has not moved favorably within the range analyzed. On the other hand the change in efficiency (EC) has had a much better performance than MT, since only about 26% of the local governments from Brazil had a negative score of change in efficiency. We observed that average between the years 1996 and 2007, the TC component dictated the behavior established for the yield index of public health in the country. Both TC and m reveal a downward trend at the lower end of the range. On the other hand the EC component showed a more positive performance in the analysis.

(b) Impacts of fiscal decentralization

Table 1 shows the estimation results of the econometric model, which present the marginal effects of explanatory factors from the dynamics of productivity on public health in Brazil. After holding various estimations, with the inclusion or exclusion of socioeconomic and control variables, a final model that showed greater robustness was reached. It should be noted that estimation of this model showed a satisfactory fit, indicating that the variables incorporated explain adequately the phenomenon under study.

Table 1: Factors associated to the dynamics of productivity of public health

Type Variables Coefficients Standard Deviation

t-Statistic (prob.)

Constant -3.0654* 0.6790 -4.51 (0.00)

Federal Questions

Ds -0.0017*** 0.0009 -1.91 (0.06)

Fr 0.1263* 0.0117 10.78 (0.00)

Cf 0.1350* 0.0198 6.80 (0.00)

Socioeconomic Factor Po -0.3249* 0.0117 -27.87 (0.00)


Variables of control Ed 0.1854* 0.0252 7.37 (0.00)

Dummies

Dum_North 2.6715* 0.4503 5.93 (0.00)

Dum_Northeast 3.0262* 0.7635 3.96 (0.00)

Dum_Southeast 10.2140* 1.5619 6.54 (0.00)

Dum_South 9.4414* 0.9323 10.13 (0.00)

Dum_Scale -7.0935* 1.0988 -6.46 (0.00)

R² adjusted 0.9921

Observations 312

Source: Author’s calculations.

The estimation results described in the table above have been achieved considering the model of regression on panel data of fixed effects. The choice was based on the Hausman test, which revealed that the fixed effects estimator is consistent and efficient when compared with the random effects estimator. The estimation was performed with a total of 26 units in cross-section, over a period of 12 years, totalling 312 observations in the panel.

The dummy referring to the technological aspect of the municipalities (Dum_Scale) as a factor in controlling the volume produced showed that decentralization caused by SUS can negatively affect the provision of public health, revealing that the size of the hospital influences the productivity indicator. The purpose of the incorporation of this dummy was to control the issue of change in the technological pattern from the decentralization of health. The model without this binary variable also captured the productivity of the DMUs that had no change in their returns to scale in the period. The result expressed in table 1 corroborates the intuition that large hospitals with decreasing returns to scale tend to have a higher level of productivity than the units of lower scale. From a regional perspective all the dummies were significant and the indicator of the productivity of public health care possessed a better relationship with the localities in South and Southwest, compared to the ones in the Northeast, North and Midwest. This result is interesting as it highlights that the performance of health care provisions is influenced by their geographical position, a clear sign of the great technical and socioeconomic disparities faced by Brazilian regions.

Another interesting feature which helps to better understand regional differences in productivity concerns the design of Brazilian fiscal federalism. As it is shown in table 1, governments belonging to the North and Northeast have high dependence on transfers from the Union. Thus, as indicated by positive and significant coefficients of fiscal responsibility variables (Fr) and cash flows (Cf), where the DMUs that depend less on transfers tend to have greater accountability and efficiency in the provision of public goods.

The literature on health economics indicates that environmental factors, particularly socioeconomic factors, directly influence the productivity and efficiency of goods and


services provided by the public sector. In this context it is interesting to note the marginal effect of the growth in the proportion of the poor (Po) on the performance of health services. The results show that the greater level of education tends to generate less pressure on public health care through another channel: people with more education have higher profitability, which creates the possibility of them being covered by private health plans.

Concerning federal issues, the results showed that the performance of the DMUs in health provision has an inverse relationship with the decentralization of expenditures. It confirms the pessimistic expectations about the decentralization process as experienced by Brazil. The marginal effect found refers to the problems of sub-national governments in the country (either of a technical nature or a waste of resources), which probably caused the increased decentralization of expenditures, has not had the expected effect of increasing the productivity of health services.

It should be remembered, however, that Brazilian municipalities are undergoing a process of both technical and operational change. The decrease in productivity resulting from the decentralization of public health may be justified in terms of changes in returns to scale, as in recent years there has been a strong regionalization of service delivery. Another important fact captured in the estimates is that localities that have more fiscal responsibility and increased cash flows produced better results. Thus, federal units that do not have a typical behaviour of fiscal free-rider tend to have better performances, given that they tend to worry more (people in positions of power and taxpayers) about the allocation of resources, which creates greater returns in efficiency and quality. We also emphasize that the most independent of intergovernmental transfers, therefore with a more adequate balance between benefits and burdens, can waste less resources in the provision of public health than those locations that keep the balance in disequilibrium.

Conclusions

This study has tried to respond to whether increased decentralization of public health spending favored or not the productivity of this service in the country. Given the facts presented, it was observed that the performance of the public health services revealed a negative relationship with the indicator of decentralization of spending in the area. This result was contrary to the vision in the so called "decentralization theorem," which emphasizes gains in social welfare and efficiency when products are provided by local governments. In this context, the critical view of Campos (1998) on the high level of wastage, technical and administrative insufficiency, corruption, nepotism and other problems faced by the management of local governments in Brazil can be a possible explanation for the results found.

Nevertheless, one must keep in mind that the process of decentralizing the provision of public health in Brazil has brought strong technological changes, as evidenced by the change of the returns to scale. Therefore, this technical move may have acted to impose at first a lower level of productive performance, since the hospitals have become more spatially decentralized and started to operate with increasing returns to scale, which might further generate greater resource and productivity savings.


References

Campos R (1998). Na Virada do Milênio. Rio de Janeiro: Topbooks.

Caves DW et al. (1982). The economic theory of index numbers and the measurement of input, output and productivity. Econometrica 50: 1393-1414.

Färe R et al. (1994). Productivity growth, technical progress, and efficiency change in industrialized countries. The American Economic Review 84: 66-83.

Faria FP et al. (2008). Efficiency of municipal expenditure in health and education: an investigation using data envelopment analysis in the state of Rio de Janeiro, Brazil. Revista da Administração Pública 42: 155-177.

Gasparini CE, Ramos FS (2004). Relative deficit of health services in Brazilian states and regions. Brazilian Review of Econometrics 24: 75-107.

Malmquist S (1953) Index Numbers and Indifference Surfaces. Trabajos de Estadística 4: 209-242.

Oates WE (1977). Federalismo fiscal. Madri: Instituto de Estudios de Administración Local.

Oates WE (2005). Toward a Second-Generation Theory of Fiscal Federalism. International Tax and Public Finance 12(4): 349-373.

Prud’homme R (1995). The Dangers of Decentralization. The World Bank Research Observer 10(2): 201-220.

Thrall RM (2000). Measures in DEA with an application to the Malmquist Index. Journal of Productivity Analysis 13: 125-137.


11. Deregulation and Performance of Mexican Banking

Francisco Vargas Serrano Universidad de Sonora, Mexico, [email protected]

Arnulfo Castellanos Moreno Universidad de Sonora, Mexico, [email protected]

Gang Cheng [email protected], Peking University, China.

Panagiotis Pzervopoulos University of Loannina, Greece. [email protected]

Luis Rentería Guerrero Universidad de Sonora, Mexico, [email protected]

Abstract

The aim of this paper is to study the relationship between Mexican banking sector deregulation and bank performance, in addition, the effects of competition and bank risk-taking is accounted for. To this end, we develop a DEA-Malmquist productivity model of bank performance. A recent econometric advance in General Methods of Moments, such as the Arellano, Bover/Blundell-Bond estimation, to estimate bank performance, is applied. The estimated model involves bank panel data from the Mexican deregulation period, 1988-2000.

Keywords: Deregulation, banking, productivity, Mexico.

Introduction

In the last decades the Mexican banking system has undergone several transformations. Private banks were nationalized in 1982 and in the period 1991-92 the banking system was privatized. In 1994 the North America Free Trade Agreement (NAFTA) became in force. From 1994 onwards, banking investment was open to foreign capital. Several restrictions were imposed under NAFTA covenants for overseas participation in the banking sector. While such limitations were supposed to last until 2000, they in fact were lifted during the Mexican financial crises that took place in 1995. After the crisis, foreign investment became predominant in the Mexican banking sector.

The financial liberalization process encompassed several reforms in order to enhance efficiency as well as to increase productivity in the financial system. The purpose of this study is to analyze the effects of banking liberalization on bank performance. The banking performance is measured by productivity and efficiency. In addition, the effects of competition and risk-taking are accounted for. With such a purpose, an


empirical model is applied to bank panel data from 32 banks in Mexico throughout the period 1982 to 2000.

The next section deals with relevant studies on this subject. The data set is briefly described in the third section. The fourth section presents the algorithm and the econometric model. The fifth section deals with some outstanding results. Finally, some concluding remarks are displayed in the last part.

Previous studies

Selected papers have applied Malmquist techniques to asses bank performance. Some of those who have done it, include Berg, Forsund and E. Jansen (1992), Berg, et al (1993), Bukh, Forsund and Berg (1995) and Mlima (1999) for Nordic countries; Grifell-Tatje and Lovell (1995,1997), for Spanish banks; Leightner and Lovell (1998), for Thai banks; Rebelo and Mendes, for Portuguese banks (2000); Isik and Hassan, for Turkish banks; Paradi et al.(2002) for Canadian Banks, Kirikal (2004) and Kirikal, Sorg and Vensel (2004), for Estonian banks; and Galagedera, U.A. D. and Edirisuriya, P. (2004), for Indian banks; and Murillo-Melchor. C. et al. (2005) for European banks. By and large, the available empirical evidence shows a mixture on the effects of financial liberalization on productivity and efficiency. Gilbert and Wilson (1998) found that financial liberalization in Korea produced outstanding results on productivity of Korean Banks. Likewise, Isik and Hassan (2003) dealing with Malmquist DEA model showed a relevant improvement on Turkish banking productivity after liberalization. On the other hand, Yildirim (2002) studying the technical and scale efficiencies of Turkish banks between 1988 and 1999 measured with the use of nonparametric Data Envelopment Analysis found that over the sample period both pure technical and scale efficiency measures show a great variation and the sector did not achieve sustained efficiency gains.

Method

The calculated original DEA Malmquist results were adjusted by bootstrapping following the Kneip et al (2008) process. Subsequently, in order to avoid distortion on results, the adjusted Malmquist Index was purified by excluding figures that were detected as outliers. Once the previous corrections took place, the figures were incorporated to run the regressions specified below on the liberalization model.

To this end, the liberalization model with four regressions was estimated. The bank performance (p) dependent variable is regressed on four independent and four instrumental variables. Bank performance is assessed by four variables, which are total factor productivity change (TFPCH), efficiency change (EFFCH), technological change (TECHCH) and net interest margin (nim).

The explicative variables include financial liberalization index (finlib) which is calculated by Kaminsky, G, and S. Schmukler (2008). The market power (mp) stands for another explanatory variable, which is calculated following the Panzar and Rose (1987) model. In addition, a dummy variable accounting for the origin of capital: foreign or national (for) is included. Finally, the ratio of investment to GDP (invgdp),


as a macroeconomic explanatory variable, has been added in order to capture the state of business.

In order to explain bank performance, a general method of moments (GMM) model is applied to a set of 261 observations on the above mentioned variables. Table 1 shows the main results of the effects of explanatory variables on total factor productivity change.

The numerical data available encompasses the period 1982-2000. Input variables include: deposits, operating cost, and nonperforming loans. Output variables were represented by: net profit, credit and profitable assets. Such information was gathered for all 32 banks.

Hypotheses

H0 1) Financial liberalization positively affected bank performance

2) Credit, capital and liquidity risk enhanced bank performance

3) The foreign capital positively influenced bank performance

4) Investment improved bank performance.

Model

The Malmquist index is used to evaluate productivity change of banks between two time periods. Such change is called product “catch-up” and “frontier-shift” terms. The catch-up (or recovery) is related to the firm efficiency. The catch-up is a term to assess the degree at which a firm attains efficiency improvements. On the other hand, the frontier-shift (or innovation) reflects changes in the efficient frontiers surrounding an enterprise between two time periods.

Firstly, following Tone, K. (2004) a set of n banking firms (xj, yj) (j=1, …n) is established, each one has m inputs and q outputs where the vector xj ∈ Rm denotes

inputs and a set of q outputs denoted by a vector yj ∈ R q, over the periods 1 and 2. The notations (xo,yo)

1 and (xo,yo)2 are employed in order to designate decision making

units (DMUo )(o = 1, …, n) in the periods 1 and 2 respectively.

The production possibility set (X,Y)t (t = 1 and 2 ) spanned by (xj,yj)t (j=1,…,n) is

defined by:

(𝑋, 𝑌)𝑡 = (𝑥, 𝑦)| 𝑥 ≥ ∑ 𝜆𝑗𝑥𝑗𝑡 , 0 ≤ 𝑦 ≤ ∑ 𝜆𝑗 𝑦𝑗

𝑡, 𝐿 ≤ 𝑒𝜆 ≤ 𝑈, 𝜆 ≥ 0 𝑛𝑗=1 𝑛

𝑗=1 … (1),

where 𝑒 is the row vector with all elements equal to one and 𝜆 ∈ Rn is the intensity vector, and L and U are the lower and upper bounds of the sums of the intensity respectively. The production possibility set (X,Y)t is characterized by frontiers that are composed of (x,y) ∈ (X,Y)t such that is not possible to improve any element of the input x or any element of the output y without worsening some other input or output. This frontier is called the frontier technology at the period t. In the Malmquist index analysis, the efficiencies of DMUs (xo,yo)

1 and (xo,yo)2 are evaluated by the frontier

technologies 1 and 2 in several ways.

The Malmquist index (MI) is computed as the product of Catch-up and Frontier-Shift: MI = (Catch-up) x (Frontier-Shift) …(2)


Catch-up=𝛿2((𝑥𝑜,𝑦𝑜)2)𝛿1((𝑥𝑜,𝑦𝑜)1)

… (3) Frontier-Shift=𝛿1((𝑥𝑜,𝑦𝑜)1)𝛿2((𝑥𝑜,𝑦𝑜)1)

𝑋 𝛿1((𝑥𝑜,𝑦𝑜)2)𝛿2((𝑥𝑜,𝑦𝑜)2)

1/2

… (4)

MI= 𝛿1((𝑥𝑜,𝑦𝑜)2)𝛿1((𝑥𝑜,𝑦𝑜)1)

𝑋 𝛿2((𝑥𝑜,𝑦𝑜)2)𝛿2((𝑥𝑜,𝑦𝑜)1)

1/2

….. (5)

The obtained output with the DEA Malmquist algorithm was adjusted through bootstrapping techniques following Kneip et al (2008).

Panzar and Rose´s market power model

Panzar and Rose (1987) have a reduced form approach using banking information or bank level to discriminate between perfect competition, monopolistic competition and monopoly. The Panzar and Rose methodology shows how the changes in input prices are reflected in the balance of industry or revenues of a specific bank.

The methodology is applied to the banking sector through the following equation:

𝑙𝑛(𝐼𝑁𝑇𝑅𝑖𝑡) = 𝛼 + 𝛽𝑓𝑙𝑛(𝑃𝑓,𝑖𝑡)𝑓

+ 𝜆𝑘𝑋𝑘,𝑖𝑡𝑘 + 𝜖𝑖𝑡 … (6)

INTRit is ratio of total interest revenue to total assets of bank i at time t. Pf, it and Xk, it denote the input price of factor f and control variable k, respectively, of bank i at time t.

The Panzar and Rose H-statistic can be calculated as: 𝐻 = 𝛽𝑓𝑓

…(7)

Where, H is the sum of the elasticities of the (scaled) total revenue of banks with respect to their factor input prices.

Liberalization model

Following Brissimis et al (2008), the liberalization model is specified as: 𝑝𝑖𝑡 = 𝑎0 +𝑎1𝑓𝑖𝑛𝑙𝑖𝑏 + 𝑎2𝑚𝑝𝑡 + 𝑎3𝑥𝑖𝑡 + 𝑎4𝑖𝑛𝑣𝑔𝑑𝑝𝑡 + 𝑓𝑜𝑟 + 𝑢𝑖𝑡 (8) where p stands for the performance of bank i at time t; p is written as a function of a time-dependent financial liberalization variable, finlib. 𝑚𝑝 stands for an index of banking industry market power; x is a vector of bank-level variables representing credit, liquidity and capital risk; invgdp is a variable that captures the macroeconomic conditions common to all banks; and finally the error term uit.


Results

Table 1: Regression results

TFPCH EFFCH TECHCH NIM

L1. -0.033442 - .27857647*** 0.0496661 -0.162373

L2. -0.031977 -0.081132 -.11918838* .25791811**

Cap -.35856318*** -0.105165 -.33699958* 3.7375533***

L1. 0.9558624 -0.281596 1.0923063* 1.3688977

L2. 0.2210093 -0.686561 0.8996325 -0.958623

Lq -0.607796 -.42143966* 0.0209608 1.7907308***

L1. -0.158325 0.2817561 -0.224369 3.6363456***

L2. 0.1853787 .57322185* 0.0087009 -2.5883752***

Cr -.11590636** .51538691*** -.19412563** .35141742**

L1. 1.0120731 0.3834893 0.4518117 14.836783***

L2. 2.7817392* 1.020891 5.2228821*** -6.9483792***

For -0.011101 -0.166214 0.0444095 0.623175

Finlib -0.136164 -0.037475 0.0282024 0.0450181

invgdp 7.5766796** 7.8517844*** 4.3904461* -8.479457

Mp 0.0484273 0.0214329 0.0652986 0.2667224

Legend: * p<.05; ** p<.01; *** p<.001

The total factor productivity change (TFPCH) is positively affected by factors such as investment (invgdp), lagged (1) credit risk, and capital risk. On the other hand, TFPCH is negatively affected by capital risk (cap), and credit risk (cr).

It is clear that investment, capital risk, lagged (1 and 2) of capital risk, lagged (1) of capital risk and credit risk exert a positive effect for technological advance. On the opposite side, credit risk, and the lagged (2) of technological change, exert a negative effect over technological change.

Efficiency change is affected positively by investment, liquidity risk lagged (1 and 2) capital risk lagged (1), credit risk, credit risk lagged (2). The reverse is true for explanatory variables such as liquidity risk, credit risk, efficiency change lagged (1).

Net interest margin is positively affected by lagged (1 and 2) liquidity risk, capital risk, credit risk, credit risk lagged (1), and net interest margin lagged (1). On the opposite side, credit risk lagged (1), liquidity risk lagged (2), capital risk lagged (1 and 2) and net interest margin lagged (2) show an adverse effect on net interest margin.


Conclusions

Risk taking does not seem to be a factor to enhance productivity. As far as credit risk variables are concerned, the null hypothesis of a positive effect is rejected when bank performance is measured by total factor productivity change. While credit risk variable showed up with a negative sign, the lagged versions of the same variable observed positive coefficients; however, only the two periods lagged was significant.

When bank performance is measures by net interest margin, we have mixed results. If bank performance is measured by net interest margin, it turns that liquidity risk lagged (2), capital risk lagged (2), show a negative effect; the null hypothesis is rejected. However, liquidity risk lagged (1), credit risk, credit risk lagged (1) exert a positive effect. Thus, the null hypothesis is not rejected.

Likewise if bank performance is measured by efficiency change, credit risk and liquidity risk lagged (2) showed a positive effect on the dependent variable. Hence, the null hypothesis is not rejected. On the opposite side, only liquidity risk turned with negative effect. Therefore, the null hypothesis could not be rejected.

The mixture of result is also present assessing the effects of risk variables on technical change. The hypothesis about a positive effect of risk variables on technical change was not rejected for capital risk lagged (1), and credit risk lagged (2). On the other hand, capital risk, and credit risk, adversely affected banking performance measured through technical change.

Finally, capital formation seems to be a relevant factor of productivity growth. The hypothesis that investment positively affects bank performance could not be rejected for total factor productivity change, efficiency change and technological change.

References

Berg, S. A., F. Fordsund, and E. Jansen. (1992). Malmquist Indexes of Productivity Growth During the Regulation of Norwegian Banking, 1980-1989, Scandinavian Journal of Economics, 94 (Supplement), 211-228.

Berg, S. A. F. Fordsund, L. Hjalmarsson, M. Suominem. (1993). Banking Efficiency in the Nordic Countries- Journal of Banking and Finance 17, 371-88.

Brissimis, S. N., Delis, M. D. and Papanikolaou, N. I. (2008).Exploring the nexus between banking sector reform and performance: Evidence from newly acceded EU countries. Journal of Banking & Finance, 32, 2674–2683.

Bukh P.N.D., F.R. Forsund and S.A. Berg. (1995). Banking Efficiency in the Nordic Countries: A four-country Malmquist index Analysis, Bank of Norway: Working Paper Research Department.

Färe, Grosskopf, Norris and Zhang (1994). Productivity Growth, technical progress and efficiency change in industrial countries, American Economic Review, 84, 66-83.

Galagedera, U. A. D. and Eridisuriya, P. (2004). Application of Data Envelopment Analysis and Malmquist Productivity Index. Economics Working Paper.


Grifell-Tatjé, E. and C.A.K. Lovell.(1996). Deregulation and Productivity Decline: The Case of Spanish Savings Banks. European Economic Review 40(1996) 1281-1303. Elsevier Science.

Isik, I. and K. Hassan. (2003). Financial Deregulation and Total Factor Productivity Change: An Empirical Study of Turkish Commercial Banks. Journal of Banking & Finance 27. Elsevier Science.

Kaminsky, Graciela and S. Schmukler, “Short-Run Pain, Long-Run Gain: Financial Liberalization and Stock Market Cycles” Review of Finance, Vol. 12, 2008, 253-292.

Kirikal, L. (2004). Malmquist Indexes of Productivity Change in Estonian Banking. Tallinn Technical University. In Emrouznejad, A. y Podinovsky, V. (2004) Data Envelopment Analysis Performance Management. 4th International Symposium of DEA. Aston University UK.

Kneip, Alois & Simar, Léopold & Wilson, Paul W., 2008. "Asymptotics and Consistent Bootstraps For Dea Estimators In Nonparametric Frontier Models," Econometric Theory, Cambridge University Press, vol. 24(06), pages 1663-1697, December.

Leightner,J.E. and C.A.K. Lovell. (1998). The Impact of Financial Liberalization on the Performance of Thai Banks. Journal of Economics and Business 1998; 50.115-131. Elsevier Science Inc. New York.

Mlima, A. P. (1999). Productivity Change in Swedish Banks: A Comparison of Malmquist Productivity Indexes, In dissertation Four Essays on Efficiency and Productivity in Swedish Banking. Göteborgs Universitet: Economiska Studier.

Murillo-Melchor. C. et al. (2005). Productivity growth in European banking. Universitat Jaume I. Working Paper. Spain.

Panzar, J. C. y J. N. Rosse (1987); “Testing for Monopoly Equilibrium”, Journal of Industrial Economics, 35, pp. 443-456.

Paradi, C. J. et al. (2002) Performance Evaluation in an Oligopoly Environment: Combining DEA Window Analysis with the Malmquist Index Approach –A Study of the Canadian Banking Industry. Centre for Management of Technology and Entrepreneurship. University of Toronto. Working Paper. Canada.

Rebelo, J. And V. Mendes (2000) Malmquist Indexes of Productivity Change in Portuguese Banking: The Deregulation Period, IAER, 6, 3, 531-543.

Tone, K. (2004) Malmquist Productivity Index. In Cooper, W.W., Seiford, L. M., and Joe Zhu. (editors) Handbook on Data Envelopment Analysis. Kluwer Academic Publishers. The Netherlands. Pp. 203-228.


12. Economies of scale and scope in mental health care

J.A. Wilschut* Delft University of Technology, The Netherlands, [email protected]

B.L. van Hulst Delft University of Technology, The Netherlands, [email protected]

J.L.T Blank Delft University of Technology, The Netherlands, [email protected]

Abstract

Economies of scale and scope are usually derived under the assumption that the set of production possibilities are shared by all firms in an industry irrespective of whether they specialize in a single output or not. Mental health care institutions in the Netherlands vary substantially in the scale and the number of outputs. Estimation of one cost function therefore seems very restrictive and requires the allowance of zero-values. We used a translog cost function model with dummy-variables for different types of institutions, to allow for different technologies. We found evidence for differences in technologies between institutions specialized in counseling and integrated institutions that also performed other activities, expressed by the number of days in the hospital or permanent care, number of treatments in daycare or number of day activities. The marginal costs of counseling were lower for the integrated institutions than for the specialized institutions.

Keywords: economies of scale, economies of scope, stochastic frontier analysis, mental health care

Introduction

Governments seek tools to control the growth of healthcare spending. Producing at the optimal scale and full use of economies of scope can help reducing the healthcare spending. However it is important that the economies of scale and scope are correctly derived. In this study, we will determine economies of scale and scope of mental health care institutions in the Netherlands. The mental healthcare in the Netherlands is particularly interesting because of the increasing costs in mental health care due to an increasing number of patients. Furthermore, the size of the institutions and the combinations of activities offered varies widely between mental healthcare providers. If economies of scale and scope exist potential cost savings can be realized by restructuring the sector.

Economies of scale and scope are generally derived under the assumption that all firms in a certain industry operate under the same production possibilities, irrespective * Corresponding author


of whether they specialize in a single output or not (Baumol et al., 1988). In this paper we test that the assumption of the same technology for all for the mental health in the Netherlands. Mental health care institutions in the Netherlands vary widely in scale and in the number and kind of services they offer. Basically there are two types of firms; integrated firms that offer a wide range of care and ambulant firms that offer only ambulant care. Given the huge differences between institutions the possibility of a different cost function for the different types of institutions seems very appropriate. We therefore apply a cost model to the Dutch mental health care institutions that allows for such differences, in order to estimate economies of scale and scope as tools to increase the productivity level. Allowing for differences between the technologies of the two types of institutions seems a more realistic approach.

Determination of economies of scale and scope starts with the estimation of a cost function. The assumption that all firms operate under the same production possibilities implies the estimation of one cost function for all firms. Economies of scope therefore only depend on differences in cost levels and do not account for differences in cost functions between specialized and integrated firms. Moreover, estimation of the frequently used translog function, introduced by (Christensen et al., 1973), requires the handling of substantial amount of zeros for the specialized institutions in the outputs. It has been suggested to estimate separate translog cost functions (Weninger, 2003). Here, we estimate a cost function where we allow for different parameters for the specialized and integrated institutions.

Methods

Model

Institutions vary widely with respect to scale and type of treatment they offer. A substantial part of the institutions only perform ambulant care (counseling), and are likely to vary substantially from integrated institutions that not only offer counseling but also offer residence to their patients for example. We therefore estimated a cost function that allows for different technologies between different types of institutions by including dummy variables. We divided the institutions into two groups, the first group consisted of institutions that had counseling as the only output, and the second group was all other institutions. Under the assumption of a translog form (Christensen et al., 1973), we estimated the following cost function:

( ) [ ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ]

[ ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ] ε+++

++++

+++

++++=

∑∑ ∑∑

∑∑∑∑

∑∑ ∑∑

∑∑∑∑

= = = =

= ===

= = = =

= ===

TaaWYaeWWac

YYabWacYabaaD

TaWYeWWc

YYbWcYbaDC

n

i

n

j

m

i

n

jjiijjiij

m

i

m

jjiij

n

iii

m

iiiamb

n

i

n

j

m

i

n

jjiijjiij

m

i

m

jjiij

n

iii

m

iiig

11 1 1 1

1 1110

11 1 1 1

1 1110int

lnlnlnln21

lnln21lnln

lnlnlnln21

lnln21lnlnln

(1)


With:

C = Total costs;

Damb = Dummy variable for institutions that performed only counseling

Dintg = Dummy variable for integrated institutions

Yi = output i (i = 1,.., m);

Wi = Price of input i (i = 1,.., n);

T = year;

,,,,,,,,,,,, ijijijiioijijijiio aeacabacabaaecbcba parameters to be estimated.

With Shephard’s lemma we obtain share equations :

[ ( ) ( ) ]

[ ( ) ( ) ] ),..,1(lnln

lnln

1 1

1 1

njYaeWacacD

YeWccDS

n

i

m

iiijiijjamb

n

i

m

iiijiijjingj

=++

+++=

∑ ∑

∑ ∑

= =

= =

(2)

With :

Sj = Cost share equation of input j (j = 1,.., n)

The following restrictions were imposed on the parameters to impose linear homogeneity in input prices and symmetry:

jiijjiijjiijjiij acacababccbb ==== ;;;

);(0);'(0;1

);(0);'(0;1

11'

1

11'

1

maenacac

mencc

m

imi

n

iin

n

ii

m

imi

n

iin

n

ii

∀=∀==

∀=∀==

∑∑∑

∑∑∑

===

===

(3)

We first estimated the cost function under the assumption of the same technology for different types of institutions by assuming the same parameters for the ambulant institutions as for the integrated institutions. For the specialized institutions we multiplied the parameters of the zero-outputs with a dummy variable to make sure they were not estimated for that observation. The obvious restrictions that we impose in case of the assumption of same technology:

ijijijijijijiiiioo eaecacbabcacbabaaa ====== ,,,,,

Next, we estimated the model under the assumption of different technologies. We tested the hypothesis of a same technology using a loglikelihood ratio test.

The models were estimated with maximum likelihood. Moreover, we used a thick frontier approach with the first estimation over all observations and the second


estimation over the 50% most efficient observations (according to the first estimation). We also tested the required monotonicity and concavity of the model.

The estimated cost functions were used to derive economies of scale and economies of scope. The economies of scale were represented by the cost flexibility which described the increase in cost relative to the increase in output (Baumol et al., 1988), depending on the type of institution. The cost flexibility is described by the following formula:

( ) [ ( ) ( ) ]

[ ( ) ( ) ]∑∑∑∑∑

∑∑∑∑∑∑

= == ==

= == ==

++

+++=∂∂

=

n

i

n

jjij

m

i jjij

m

iiamb

n

i

n

jjij

m

i jjij

m

iig

i i

WaeYababD

WeYbbDYCv

1 11 11

1 11 11int

lnln21

lnln21

)ln(ln'

Economies of scope were deducted from the marginal cost of the overlapping outputs between specialized and integrated institutions. We determined the marginal costs at varying levels of output to account for differences in scale between institutions. The marginal costs were derived as follows:

( ) [ ( ) ( ) ]j

n

iiji

m

iiijj

jjjj Y

CWeYbbYC

YC

YCmc *lnln

)ln(ln

11∑∑==

++=∂∂

=∂∂

=

Data

The mental health care institutions in the Netherlands report to the Ministry of Health in the Netherlands. The yearly data collected for this purpose were used in this analysis over the years 2008-2010. We selected the institutions that dealt with mental health care only, so departments of psychiatry as part of general hospitals for example were not included. We selected those institutions that had valid and plausible values for all variables. In total 201 observations (institutions per year) remained (59 in 2008, 73 in 2009 and 69 in 2010).

We included four measures of treatment as output variables: the number of counsels, the number of days in residence, the number of part-time treatments and the number of day activities (Table 1). Of these institutions, 32 were specialized in the sense that they only performed counseling. The other institutions did counseling and at least one of the other treatment activities. None of the institutions was specialized in one of the other activities. We therefore used two groups to which we refer to as specialized and integrated institutions. We used two inputs, personnel and material and capital. The latter two were added and used as one variable. Costs of the inputs were available and we used the number of full time jobs to calculate the price of personnel. The prices of materiel and capital were based on an index constructed from the Consumer Price Index and a Price Index for investments of fixed activa of the government (Statistics Netherlands, www.cbs.nl).



We estimated the cost function under the assumption of a common technology for both specialized and integrated institutions and under the assumption of different technologies (Table2). The hypothesis of the same technology for both groups was rejected by the log likelihood ratio test (p< 0.001).

The parameters of the models are very different. Moreover, under the assumption of a common technology the signs of two of the outputs become negative. The parameter of output 1 (counseling) had a value that was close to the value of the specialized institutions under the assumption of different technologies, but then overestimated the total costs of the integrated institutions. Therefore, the parameters of the other outputs were given values that decrease the total costs of the integrated institutions. Under the assumption of different technologies, the costs of the specialized institutions were slightly more influenced by costs of materiel and capital than by personnel, compared to the integrated institutions. No significant change was found over time. The model fulfilled the criteria of monotonicity and concavity.

Since the model with the assumption of the same technology for both types of institutions does not result in plausible estimates, we report economies of scale and scope only for the model with different technologies. The integrated institutions operate under diseconomies of scale on average (cost flexibility 1.05). This particularly applies to the relatively small integrated institutions (Table 3). The larger institutions operate under increasing economies of scale. The specialized institutions operate under economies of scale on average (0.72). However, the cost flexibility rapidly increases for these institutions and the larger ones operate under diseconomies of scale. All the specialized institutions are smaller than 0.5 of the average size institution.

The economies of scope follow from the marginal costs of counseling in a specialized or integrated institution with the same scale (Table 4). The marginal costs of counseling are higher for the specialized institutions than for the integrated institutions, except for the very small institutions. Moreover, costs increase with scale for the specialized institutions, and decrease with scale for the integrated institutions. The marginal costs of all other products also decrease with scale, reflecting the economies of scale.

Conclusions

Specialized and integrated mental health care institutions in the Netherlands vary in the way they operate, as shown by the different technology assumption. We also found indications that increasing the scale of the institutions and that integrating counseling in institutions with other types of treatment could increase the productivity of the sector.

The assumption of the same technology did not give plausible estimates because of the negative parameter values of two of the outputs. Instead of using dummy variables, we also tried to replace the zero-values with the minima of the non-zero values and estimate the model under the assumption of the same technology for all institutions. The estimates were very similar then to the estimates of the model that


allows for different technologies, as derived for the integrated institutions alone. This approach also led to the rejection of the hypothesis of equal technologies.

The technology can differ between institutions for several reasons. These differences can be caused by the kind and severity of the disorders that are treated, and for example by the average age of the patients. We had data on the types of treatments as expressed by the outputs we used, but did not have any information about the patients or the disorders that were treated. A Norwegian study showed a substantial impact of case mix on productivity growth estimates but did not differentiate between types of institutions (Halsteinli et al., 2010). Given the data, we could only differentiate between institutions that had counseling as a single output, and all other institutions. No further differentiation was possible for example of institutions that only performed one of the other outputs.

We used the marginal costs as a proxy for the economies of scope. Due to the lack of observations we could not estimate a separate cost function without counseling. Moreover, the costs cannot be estimated by a translog function in case of a zero output. Direct estimation of the economies of scope was therefore not possible. We compared the marginal costs of specialized and integrated institutions of the same size. The differences in costs could be overestimated because the size of the integrated institution that relates to counseling is smaller and the marginal costs of counseling increase with a decreasing scale for these institutions.

We used 201 observations of mental health care institutions over a 3 year period to compare the costs of the institutions relative to their production. Unfortunately, we had to exclude another 346 observations because of missing or implausible values. A substantial part of the excluded institutions had high costs relative to the number of outputs. Most of the excluded institutions perform other activities which are not counted in the outputs we used, like for example reintegration activities or parental support. We did not have usable data to systematically exclude institutions for this reason. It is therefore likely that some of the institutions with other activities are included in the analysis. Because of the high costs relative to the output, this would underestimate the cost efficiency. Due to a lack of data of the costs of mental health care activities alone, we were not able to include 137 health care institutions with more than just mental health care, like for example the general hospitals. Summarizing, this analysis covers a substantial but selective part of the mental health care sector.

Measuring the output of health care institutions is subject of debate (Hollingsworth & Street, 2006). In this study we only used the number of treatments and not the number of patients or the effectiveness of the treatment. We used quality of care measurements from interviews to evaluate the relation between the efficiency score of the institutions and quality. We found a negative relation with patient perception of the effectiveness of the treatment and a non-significant correlation with all other quality measures (results not shown).

The plausibility of the same technology assumption between firms or institutions has been largely unexplored. The couple of studies we found came to the same conclusion that different cost functions are required for specialized and diversified firms. Differences in cost functions were for example found between general and


specialty hospitals in Vietnam (Weaver & Deolalikar, 2004), between firms that provided both freight and passenger railway services in the US and firms that offered primarily freight services (Weninger, 2003) and between the water and sewerage firms in England and the water only companies (Bottasso et al., 2011).

In conclusion, specialized and integrated mental health care institutions in the Netherlands operate under different cost functions. The issue of different technologies deserves more attention. Drawing the wrong conclusions from the possibly incorrect same technology assumption can have major policy implications on scale and scope.

References

Baumol, J., Panzar, J.C., & Willig, R.D. (1988). Contestable markets and the theory of industry structure Sydney: Marcourt Brace Jovanovich.

Bottasso, A., Conti, M., Piacenz, M., & Vannoni, D. (2011). The appropriateness of the poolability assumption for multiproduct technologies: Evidence from the English water and sewerage utilities. International Journal of Production Economics, 130(1), 112-117.

Christensen, L.R., Jorgenson, D.W., & Lau, L.J. (1973). Transcendental Logarithmic Production Frontiers. The Review of Economics and Statistics, 55(1), 28-45.

Halsteinli, V., Kittelsen, S.A., & Magnussen, J. (2010). Productivity growth in outpatient child and adolescent mental health services: the impact of case-mix adjustment. Soc Sci Med, 70(3), 439-446.

Hollingsworth, B., & Street, A. (2006). The market for efficiency analysis of health care organisations. Health Econ, 15(10), 1055-1059.

Weaver, M., & Deolalikar, A. (2004). Economies of scale and scope in Vietnamese hospitals. Soc Sci Med, 59(1), 199-208.

Weninger, Q. (2003). Estimating multiproduct costs when some outputs are not produced. Empirical Economics, 28(4), 753-765.


Table 2 Summary data of included institutions

Average stddev min max

Output (x 1.000)

Counseling (contacts) 137.7 170.8 1.1 888.8

Residence (day) 112.8 125.9 0.0 503.5

Part-time treatment 11.8 16.3 0.0 71.2

Day-activity 22.5 46.3 0.0 258.9

Costs (x 1 million euro)

Personnel 34.1 37.6 0.2 153.9

Kapital 2.1 2.8 0.0 14.9

Material 12.3 12.8 0.1 54.2

Total 48.5 52.7 0.6 215.7

Total # FTE 609.6 646.7 10.8 2602.9

Price personnel (euro) 53556 9376 15514 87747


Table 3 Parameter estimates

Shared technology Different technologies

Integrated institutions Specialized institutions

Variable Parameter

Estimate

Error

t-statistic

Estimate

Error

t-statistic

Estimate

Error

t-statistic

Constant A0 -0.59 0.09 -6.88 0.30 0.05 6.03 -1.47 0.16 -9.11

Counseling (y1)

B1 1.35 0.04 31.17 0.25 0.06 4.31 1.40 0.06 22.30

Day in residence (y2)

B2 0.22 0.10 2.18 0.34 0.05 7.20

Part-time treatment (y3)

B3 -0.28 0.09 -3.27 0.30 0.06 5.38

Dayactivity (y4)

B4 -0.17 0.05 -3.84 0.01 0.02 0.63

Personnel (w1)

C1 0.71 0.01 65.71 0.70 0.01 61.28 0.55 0.04 15.38

Materiel & capital (w2)

C2 0.29 0.01 27.39 0.30 0.01 26.10 0.45 0.04 12.46

Time A1 -0.03 0.03 -0.95 -0.02 0.02 -1.20 0.05 0.05 1.02

y1 x y1 B11 0.06 0.05 1.38 0.26 0.05 5.41 0.35 0.03 12.03

y1 x y2 B12 -0.34 0.07 -5.17 -0.15 0.03 -4.88

y1 x y3 B13 0.21 0.04 5.85 -0.17 0.04 -4.46

y1 x y4 B14 0.19 0.03 6.82 0.00 0.01 0.23

y2 x y2 B22 0.16 0.07 2.35 0.05 0.04 1.35

y2 x y3 B23 0.05 0.04 1.10 0.03 0.02 1.45

y2 x y4 B24 -0.03 0.01 -2.00 -0.01 0.01 -1.19

y3 x y3 B33 0.04 0.07 0.48 0.17 0.04 4.07

y3 x y4 B34 -0.15 0.03 -5.89 -0.02 0.01 -1.82

y4 x y4 B44 -0.01 0.01 -0.66 0.00 0.01 0.63

w1 x w1 C11 0.13 0.05 2.55 0.12 0.07 1.63 0.08 0.08 1.01

w2 x w2 C22 0.13 0.05 2.55 0.12 0.07 1.63 0.08 0.08 1.01

w1 x w2 C12 -0.13 0.05 -2.55 -0.12 0.07 -1.63 -0.08 0.08 -1.01

y1 x w1 E11 -0.01 0.01 -1.41 0.01 0.01 0.82 -0.08 0.02 -5.00

y2 x w1 E21 -0.02 0.01 -1.85 -0.01 0.01 -1.46

y3 x w1 E31 0.03 0.01 3.46 0.02 0.01 2.82

y4 x w1 E41 0.00 0.00 -0.73 -0.01 0.00 -1.84


Shared technology Different technologies

y1 x w2 E12 0.01 0.01 1.41 -0.01 0.01 -0.82 0.08 0.02 5.00

y2 x w2 E22 0.02 0.01 1.85 0.01 0.01 1.46

y3 x w2 E32 -0.03 0.01 -3.46 -0.02 0.01 -2.82

y4 x w2 E42 0.00 0.00 0.73 0.01 0.00 1.84

R2 0.98 0.99

Loglikelihood

123 206

Table 4 Economies of scale

Total costs* Specialized Integrated

0.2 1.32 1.14

0.5 1.54 1.04

1.0 0.94

2.0 0.83

*Standardized costs: the costs of an average size institution are equal to 1

Table 5 Marginal costs by type and scale of the institution

Total Costs * Specialized Integrated

Counsel Counsel Days in residence Part-time treatment Dayactivities

0.2 113 142 183 779 81

0.5 172 134 172 854 60

1.0 117 149 873 32

2.0 92 114 827 3

*Standardized costs: the costs of an average size institution are equal to 1


13. Efficiency analysis and long run performance: a sequential model for organizational assessment

Frederico A. de Carvalho* Rua Dona Maria, 303 – Javary -26900-000 Miguel Pereira RJ – Brazil; UFRJ (the Federal University at Rio de Janeiro); [email protected]

Marcelino José Jorge Avenida Brasil, 4365 – Manguinhos – 21040-360 Rio de Janeiro RJ – Brazil; IPEC / FIOCRUZ (the National Institute for Infectology / Oswaldo Cruz Foundation); [email protected]

Marina Filgueiras Jorge Avenida Padre Natuzzi, 22 – São Francisco - 24360-180 Niterói RJ – Brazil; INPI (the National Institute of Industrial Property); [email protected]

Abstract

This paper presents a sequential approach to organizational assessment from an efficiency standpoint. The empirical illustration originates from data referring to the period 2000-2007 and collected from a sample of 37 libraries affiliated to a federal university in Rio de Janeiro; this sample covers some 90% of the population. In the first and second steps efficiency scores computed from estimated DEA (Data Envelopment Analysis) models are employed to rank DMUs (libraries) as well as to provide pro-efficiency allocative corrections. The third step presents a long run evaluation that is accomplished by Markovian analysis through computing the corresponding equilibrium distribution between (efficiency) states. The Markovian approach also provides some particular durations – such as mean recurrence times and first passage times – that allow managerial interpretation. Since DMUs (libraries) are here classified as “efficient” or “inefficient” according to computed annual scores, the proposed model is “systemic” to the extent that only aggregate data are considered.

Keywords:Organizational assessment. Efficiency analysis. Markovian analysis. Academic libraries - Brazi.

Introduction

This paper presents an optimization approach to organizational evaluation from an Efficiency Analysis standpoint. The approach combines in a simple way efficiency scores computed from the estimation of selected Data Envelopment Analysis (DEA) models and a long run evaluation provided by Markovian analysis. The proposed

* Corresponding author


model relies essentially on the application of the so-called Efficiency Principle to assess organizational performance. Following the literature, “organization” may be interpreted quite broadly as meaning both public and private entities, and even nonprofit ones among the latter (VAKKURI, 2003). Hence, even though the empirical illustration employs data from a sample of Decision Making Units (DMUs) pertaining to a public organization, there is no loss of generality if and when other kind of organizations are considered.

Methods

The Efficiency Principle simply states that, when studying the production process in any organization, whenever a production unit uses the same resources but yields greater quantities of output than another unit, it should be considered “relatively more efficient” (i. e., relative to one another), no matter how formally the productivity problem is analyzed. Analogously it should be considered “relatively more efficient” if it uses fewer resources and yields the same output. From an analytical standpoint these properties correspond to evaluating an organizational unit in terms of its position vis à vis an adequately defined and computed “efficiency frontier”. There is an established body of knowledge - Data Envelopment Analysis (DEA), a class of mathematical programming models – with a now long tradition (Emrouznejad, Parker, Tavares, 2008) of being applied to a broad range of situations involving the analysis of production frontiers in a multi-unit, multi-input and multi-output framework in such a way that usual parametric restrictions are absent.

In a seminal methodological paper Tulkens and Vanden Eeckaut (1995) describe and explain the main issues relating to the role of time in nonparametric efficiency analysis, especially in what concerns alternative ways to accommodate empirical information into reference production sets that will be submitted to efficiency computations. Of particular interest here (see Table I) is their classification of the variety of forms whereby the time dimension present in panels may be treated when investigating observed productive activity.

The paper by Wang and Huang (2007) introduces two models to examine long run efficiency analysis. However they do not compute any long run solution in any of those cases. In addition, although they have modelled and specified the probability of one-step temporal transition from efficient (resp. inefficient) to inefficient (resp. efficient) state, there seems to be no indication as to how those probabilities might be used to compute long run “structural” distributions of the DMUs among the two states (“efficient” or “inefficient”).

Using direct results from finite ergodic Markov chains (Kemeny, Snell, 1972), and assuming one (estimated) aggregate transition matrix is available, it is possible to compute the long run distribution of the “system” (the set of DMUs) between the two states. This is the purpose of the third step in our model.

The model

Our proposed sequential model consists of three steps. The first two steps – involving the computation of efficiency scores and of operational plans in turn – are typically


performed through the application of Data Envelopment Analysis to empirical data on DMU performance. The third step, the novel one, incorporates the “structural” long run assessment of efficiency. The case is summarized in Table I according to the Tulkens-Vanden Eeckhaut (1995) framework. Our example is supported by a convenience sample of 37 library units, corresponding to more than 80% of total population. Data were collected from the library system’s Management Information System and relate to three inputs (number of employees, physical area and number of volumes) and four outputs (number of visits, of loans, of registers and of consultations). The solution of the appropriate linear programming problem provides numerical scores for each DMU that characterize them with respect to efficiency status. For each inefficient DMU an operation plan is also provided that indicates (re)allocative targets for the DMU to reach efficiency. Finally scores will also be needed to compute the transitions between the two states along the time period for the whole set of DMUs.

TABLE I. SUMMARY ON CASE STUDY

Case (DMUs) Number of DMUs

Number of variables

Time Period

University libraries 37 7 2000 -2007

DEA condition satisfied * DEA model

Yes BCC-O

TVE classification**

Contemporaneous

Notes: *: number of DMUs not less than two (three) times the number of variables.

**: classification of (sample) observed subsets by Tulkens-Vanden Eeckaut (1995)

As soon as a transition matrix is available, long run analysis is possible and will result from the simple computation of a fixed point for the transition matrix (KEMENY, SNELL, 1972). This fixed point is a vector containing the long run distribution of the “system”. Since we do not follow the statistical approach applied by Wang and Huang (2007), some form of combination must be chosen for the “initial” transition matrix. In our model we use the following basic result (Kemeny, Snell, 1972): when the number of time steps grows indefinitely one has

lim (1/n)(P + P2 + . . . + Pn) = [1 1 1 1]’π (2)

where n is the number of steps; Pn = (( pij (n) )) is the nth power matrix, whose (i ; j) element represents the probability of transition from state i to state j after n steps; [1 1 ;…1]’ is a column-vector with all elements equal to 1; the apostrophe means transpose and π is a constant row-vector containing the long run equilibrium distribution between states whose components are nonnegative and sum to 1 (a so-called probability vector). The “finite mean” in the left hand side of (2) suggests a way to estimate a single matrix from the seven available and then to compute the long run corresponding to this “averaging matrix”.



In this section findings are presented according to the order of proposed steps – computed efficiency scores, operation plans and long run distribution.

First step: efficiency scores are computed and DMUs may be ranked accordingly

A sample profile for the 37 DMUs is given in Table II for the last year of the period of study. The basic results for any DEA analysis – namely, computed efficiency scores – appear in Table III.

Second step: operation plans indicate optimal changes for each library along time

Operation plans are always produced as a typical result from a DEA solution and in this example they appear in a consolidated form in Table IV. In every individual matrix (not exhibited here) showing the allocative change for each (inefficient) library and each year, there are indications of resource decrease and output increase; this information is summarized in that table and deserves managerial attention.

Third step: first passages, mean recurrence and long run change

In order to compute, for the whole system of libraries, the transition matrix between the two states - “efficient” and “inefficient” - all transitions are now considered in addition to those implied in Table III, where the trivial case of DMUs that were efficient along the whole period is only quoted in a footnote.

Table 2. Sample profile for university libraries in 2007

Number Employees

1 33 8,41 8,06 95,83%

Total area ( m2 )

37 6000 865,16 1400,03 161,82%

Volumes 872 277134 35228,9 53343,4 151,42%Visits 108 137385 20974,7 33971 161,96%

Registrations 0 5603 1043,38 1115,4 106,90%

Loans 0 30191 5116,03 6578,68 128,59%Consultations 0 66638 8091,62 12228,7 151,13%

Service mix (number)

5 13 9,54 1,87 20%

Variables Coefficient of Variation

Min Max Mean Standard deviation

Given that we are working with contemporaneous reference sets (see Tulkens and Vanden Eeckhaut, 1995, and Table I), data for 2000-2007 allow to obtain 7 transition matrices, say P1, P2, … , P6, P7 . In order to apply the finite sum approach, we employ successive products of yearly transition matrices, instead of powers of the same (initial or otherwise chosen) transition matrix. Therefore we can take the seven factor average matrix A defined as

A = [P1 + P1P2 + P1P2P3 + ... + P1P2P3…P6P7) / 7 (3)

as a good candidate to be used when solving the fixed point problem, since it incorporates more information than each individual matrix, in addition to being a good picture of the successive one-step, two-step until seven-step transitions, in the


spirit of equation (2). Solving the fixed point equation πA = π we obtain the (estimated) long run distribution of the system between states as follows

πE ( percent efficient ) = 49,52%; πNE ( percent inefficient ) = 50,48%.

Note that these figures slightly differ from the mean (48,65%, equal to the median) of percents in the last line of Table III. In this sense it can be argued that long run analysis seems to be of a different nature vis à vis the arithmetic of numerical individual scores. Remember that products of transition matrices bring into play all the transitory visits to the two states along time.

The fixed point π for the equation πA = π also provides directly the mean recurrence time (Kemeny et al. 1959, p. 413) for the states of the system, that is, the mean time required before the system returns to a given state having started in that

same state. The mean recurrence time is approximately equal to 2 years in both cases, so that the period of two years seems to be critical in the sense of monitoring the return of a state to itself. In the case of inefficiency it represents a sort of “safe mean time span” for managers to try to change the operating conditions facing inefficient units, Since the operation plans already point to “optimal changes” by unit, managers may evaluate for which units those changes would be feasible within (the next) two years. Note that on average an inefficient will return to inefficiency four months before it may reach efficiency for the first time, if no managerial action is taken.


Table III. Efficiency Scores* And Yearly Averages: 2000 – 2007

SCORES SCORES SCORES SCORES SCORES SCORES SCORES2001 2002 2003 2004 2005 2006 2007

1 1 0,841 1 1 0,605 0,811 0,68 12 0,571 1 1 1 0,965 0,943 1 13 0,305 0,936 0,845 0,661 0,542 0,846 0,775 0,5744 0,989 0,96 0,769 0,783 0,829 1 1 15 1 1 1 1 1 0,947 1 16 1 0,696 0,742 0,494 0,584 0,757 0,548 0,657 1 0,731 0,87 0,452 0,353 0,127 0,466 0,6248 0,941 1 0,471 0,559 0,782 0,65 0,626 1

10 0,62 0,895 0,712 0,974 0,619 0,74 1 0,67911 0,528 0,66 1 0,779 0,727 1 0,847 0,64612 0,404 0,59 0,287 1 1 1 1 117 1 1 0,627 1 1 1 0,336 0,3718 0,604 0,815 0,696 1 1 1 0,807 119 1 1 1 1 1 0,959 1 0,92120 0,6 1 0,867 0,779 0,743 0,498 0,543 0,5621 0,401 0,302 0,396 0,109 0,138 0,371 0,145 0,11522 1 1 0,507 0,654 0,337 1 0,842 0,12124 0,391 0,501 0,492 0,387 0,395 0,931 0,319 0,3225 0,733 0,69 0,84 0,329 0,307 0,482 0,64 0,50626 0,838 1 0,467 0,683 0,236 0,562 0,384 0,86327 0,334 0,412 0,41 0,407 0,358 0,223 0,496 0,24128 0,892 0,574 1 1 1 1 1 0,94530 1 0,442 1 0,555 0,972 1 1 0,8231 0,071 0,064 0,055 0,143 0,185 0,02 0,01 0,01732 0,45 0,781 0,928 0,873 0,87 1 1 134 0,562 1 1 1 1 1 1 135 1 0,793 1 0,665 0,757 0,354 1 136 0,107 0,202 0,196 0,172 0,113 0,353 0,401 0,38137 0,359 1 1 1 0,892 1 1 1

Mean (n=37) 0,7486 0,8077 0,7886 0,7691 0,7381 0,7993 0,7801 0,7663Percent efficient 45,96% 48,65% 48,65% 48,65% 40,54% 51,35% 54,05% 51,35%

DMU SCORES2000

Note. * - All DMUs with scores equal to 1 for the whole period have been excluded

Conclusions

Upon assuming the Efficiency Principle as a guideline to organizational evaluation, this paper presented a model for organizational assessment in the short and long runs by combining two approaches – the DEA approach to efficiency analysis and the Markovian assumption that introduces a long run perspective. Results have shown that the three step model uncovers quantitative aspects that may be of assistance to managers committed to efficiency in the short and long runs. For long run assessment we rely on Markov Chains to compute an aggregate measure of the distribution of the productive system (the “organization”) between two states – efficient or inefficient.


Since DMUs (libraries) are here classified as “efficient” or “inefficient” according to computed annual scores, the proposed model is “systemic” to the extent that only aggregated data are considered. The other useful application of the Markovian approach provided better knowledge concerning the time delay required for efficiency to be attained for the first time when a prescribed operation plan happens to be adopted, as well as about the time during which an undesired (inefficient) situation will persist if that adoption is postponed.

Table IV. Average Operation Plans: 2000 - 2007

Inputs 2000 2001 2002 2003

Inputs 2004 2005 2006 2007

-1,44 -1,15 -0,76 -1,29

-0,93 -1,18 -0,61 -0,81

-60,75 - 71,04 * -29,85 -70,35

-48,94 -143,47 -88,05 -136,27

-3064,48 -3373,49 -1880,71 -4601

-6447,08 -651,77 -4720,75 -3153,65

Area (m2)

Volumes (number)

Employees (number)

Area (m2)

Volumes (number)

Employees (number)

Note * - this figure relates to a single library.

References

Emrouznejad A., B. Parker; G. Tavares (2008) Evaluation of research in efficiency and productivity: a survey and analysis of the first 30 years of scholarly literature in DEA, Socio-Economic Planning Sciences 42 (3): 151-157.

Kemeny J. G., J. L. Snell (1972) Mathematical Models in the Social Sciences. Cambridge, Mass.: The MIT Press.

Tulkens H., P. Vanden Eeckaut (1995) Non-parametric efficiency, progress and regress measures for panel data: methodological aspects, European Journal of Operational Research 80 (3): 474-499.

Vakkuri J. (2003) Research Techniques and Their Use in Managing Non-profit Organizations – an illustration of DEA analysis in NPO environments, Financial Accountability and Management 19 (3): 243-263.

Wang M. H., T. H. Huang (2007) A study on the persistence of Farrell’s efficiency measure under a dynamic framework, European Journal of Operational Research 180 (3): 1302-1316.


14. Efficiency in the industrial sectors of Brazil in terms of contributing to sustainable development

Flávia de Castro Camioto Department of Production Engineering, University of São Paulo (Brazil), [email protected]*

Enzo Barberio Mariano Department of Production Engineering, University of São Paulo (Brazil), [email protected]

Daisy Aparecida do Nascimento Rebelatto Department of Production Engineering, University of São Paulo (Brazil), [email protected]

Abstract

The purpose of this article is to analyze the efficiency of the industrial sectors in Brazil from 1996 to 2009, considering their contributions to the sustainable development of Brazil. To this end, we used the Data Envelopment Analysis (DEA), which enabled, from the additive model and the window analysis, to evaluate the ability of industries to reduce environmental impacts and increase social and economic benefits. The results of this study indicated that the Textile sector is the most efficient industrial sector in Brazil in terms of contributing to sustainable development, followed by these sectors: Foods and Beverages, Chemical, Mining, Nonmetallic, Paper and Pulp and Metallurgical.

Keywords: Industrial Sectors, Data Envelopment Analysis, Sustainable Development.

Introduction

The data presented in the last Intergovernmental Panel on Climate Change (IPCC, 2007) indicate that global warming is largely due to human activity, especially human-caused CO2 emissions. Along these lines, fossil fuel burning has been shown to be responsible for approximately 85% of all anthropogenic CO2 emission produced yearly.

Silva and Guerra (2009) explain that the use of fossil fuels has driven the world economy since the Industrial Revolution, with energy as an essential component for the social and the economic development of a nation and its supply is an essential pre-requisite to human activities.

* Address correspondence to Flávia de Castro Camioto, Department of Production Engineering, University of São Paulo, Trabalhador São-Carlense, 400, São Carlos, SP, 13566-590 Brazil. Phone: +51 16 3373 9428, Fax: +55 16 3373 9425. E-mail: [email protected]


Thus, the environmental implications of the production and the use of energy resources have been presented as a major challenge for developed and developing countries, since the production, distribution, processing and consumption of energy should be directed to ensure development, without increasing the negative effects to society and the environment.

Despite the study by La Rovere and Simões (2008), which analyzed the availability of renewable energy sources in Brazil, to conclude that Brazil’s energy matrix is relatively clean, Brazil’s internal use of renewable energy is of 43.7% (BEN, 2010), the industrial sector still has many of its activities dependent on fossil fuels. As a result, the industry ends up impacting the environment by emitting extremely high concentrations of greenhouse gases (GHG), increasing global warming, in addition to adding to the extensive mining in the form of fuel oil and coal.

Considering that the industrial sector can significantly contribute to the challenge against climate change, several studies on environmental and economic aspects in the industry have been developed (Oggiono et al., 2011, Schneider et al. 2011; Tomasula and Nutter, 2011; Wernet et al. 2011; Hamzah et al. 2010, etc.). It should be mentioned, however, that most of these studies tend to focus on particular industrial sectors, processes or products.

Despite the few works, such as that by Mao et al. (2011), which using statistical data analyzed China’s energy consumption and GHG emissions by industrial subsystem and sector, and that by Luken and Castellanos-Silveria (2011), which compared the changes in economic, environmental and social variables that occurred in the manufacturing industry, in groups of developing countries, between 1990 and 2004, there are still ample opportunities for studies covering various industrial sectors and their contribution to promoting economic development with environmental respect and social improvement.

Given this context, this article’s main objective is to analyze the efficiency of the main industrial sectors in Brazil, during 1996 to 2009, in view of their energy consumption and their contributions to the sustainable development of the country. Therefore, this work was developed to address the three basic pillars of sustainability, which are economic, social and environmental development.

Methods

To reach the goal, a mathematical programming method called Data Envelopment Analysis (DEA) was used. This method, based on the additive model and on the window analysis, enabled analyzing the performance of the industrial sectors of Brazil to reduce energy consumption and CO2 emissions from fossil fuels (inputs), while increasing the GDP by sectors, the persons employed and personnel expenses (outputs).

For this research, the main Brazilian industrial sectors were selected, for which the National Energy Balance (BEN) and the Brazilian Institute of Geography and Statistics (IBGE) provided data, and due to lack of available information from IBGE some sectors were grouped. Thus, for this work, the spatial delimitation of the Brazilian industry includes: (a) Nonmetallic, which corresponds to the cement and ceramics


sectors, (b) Mining, which corresponds to the mining and pelletizing, excluding oil, natural gas and coal exploration, (c) Metallurgical, which corresponds to the sectors of pig-iron and steel, iron alloys and non-ferrous (d) Chemical, (e) Foods and Beverages, (f) Textiles, and (g) Paper and Pulp.

In addition to the energy consumption, the variables used in this analysis were: (1) sectoral GDP, as a variable of economic growth; (b) personnel expenses in the form of salaries, withdrawals and other remunerations, as a variable related to social development; (c) persons employed in each sector, as a variable related to social development; and (d) CO2 emissions from fossil fuels, as a variable related to the environmental development.

The data related to the variables "personnel expenses" and "persons employed" were collected on the website of the Brazilian Institute of Geography and Statistics (IBGE). The variables "GDP sectoral" and "energy consumption" were collected in the report of the National Energy Balance (BEN), available on the website of the Ministry of Mines and Energy (MME). The variable "CO2 emissions from fossil fuels" was calculated using the top-down method, internationally recognized and recommended by the UN (United Nations) (IPCC, 1996). In order to calculate the carbon emissions of the Brazilian energy system, the MCT (2006) adapted the top-down method, recommended by IPCC (1996), for the particular characteristics of the Brazilian energy system. Thus, in this work, for the calculation of CO2 emissions, many of the data used were drawn from this document.

The time interval to be analyzed in this study includes a period of fourteen years (1996-2009), and the criterion used to define it was the data availability with the same calculation base.


First, in order to verify if the "CO2 emissions from fossil fuels" and "energy consumption" contributed to the formation of the output variables, we performed a statistical analysis. Thus, we constructed a correlation matrix in order to verify the linear correlations among all variables with their statistical significance. The software used to perform such analysis was Stata MP 11.

Table 1 – Correlation Matrix

Energy consumption

CO2 emissions

GDP by sector

Persons employed

Personnel expenses

Energy consumption

1,0000

p-value CO2 emissions 0.7235 1,0000 p-value 0,0000 GDP by sector 0.8833 0.4977 1,0000 p-value 0,0000 0,0000 Persons employed 0.4352 -0.2278 0.6602 1,0000 p-value 0,0000 0.0241 0,0000 Personnel expenses

0.5230 -0.0099 0.7532 0.8399 1,0000

p-value 0,0000 0.9231 0,0000 0,0000


Table 1 presents the results of the correlation matrix which showed that the all variables of inputs were significant with p-value below 10%, for at least one output and vice versa.

From the application of the variant additive model of DEA, window analysis was performed to verify the efficiency of industrial sectors from 1996 to 2009, considering both the reduction of inputs "energy consumption" and "CO2 emissions from fossil fuels", as the increase of outputs "GDP by sector ","persons employed" and "personnel expenses". The results of this study indicated that the Textile sector was the one with the highest average efficiency with respect to the variables considered, followed by the sectors: Foods and Beverages, Chemical, Mining, Paper and Pulp, Nonmetallic, and Metallurgical. It is important to remember that in the DEA-additive, the higher the value of the objective function, the greater the inefficiency of the DMU. As for the standard deviation of efficiency, the highest value was that of the Metallurgical sector, followed by the sectors: Chemicals, Foods and Beverages, Mining, Nonmetallic, Paper and Pulp, and Textile.

Thus, the Metallurgical sector, considering the variables analyzed in this work, are the least efficient, being far below the others and with greater variability. From Table2, we can see that this sector was becoming more inefficient as the oldest years were being excluded and the most recent ones being contemplated in the windows, and the average of the sum of the slacks increased from 1.99 in the first window (1996 -2002) to 2.61 in the last one (2002 - 2009).

Table 2 - Window Analysis– Sectors

Efficiency Means Windows (means)

Total Means Standard deviation 1 2 3 4 5 6 7

Metallurgical 1.99 2.23 2.30 2.20 2.44 2.63 2,61 2.34 0.10

Paper and Pulp 0.60 0.70 0.74 0.76 0.78 0.88 0.90 0.76 0.01

Nonmetallic 0.66 0.73 0.75 0.75 0.75 0.82 0.80 0.75 0.01

Mining 0.41 0.45 0.46 0.41 0.36 0.33 0.32 0.39 0.03

Chemical 0.14 0.29 0.27 0.27 0.27 0.31 0.32 0.26 0.04

Foods and Beverages 0.06 0.19 0.22 0.18 0.13 0.22 0.17 0.17 0.03

Textile 0.01 0.02 0.01 0.01 0.01 0.02 0.02 0,02 0,0004

Then, as second to last in the efficiency ranking of Brazil’s industrial sectors, there is the Paper and Pulp sector, which like the Metallurgical sector, its efficiency decreased as the oldest years were excluded from the analysis and the most recent ones were contemplated in the windows, as shown in Table 2. It is noteworthy, however, that the variability of this sector was significantly lower than the metallurgical sector, with the standard deviation equal to 0.01.

The third in the inefficiency ranking is the Nonmetallic sector. This sector also was becoming more inefficient as the oldest years were being excluded and the most


recent ones being contemplated in the windows. As the fourth sector, both in efficiency and inefficiency, the Mineral Extraction sector, presented only in the last four windows analyzed, increased efficiency as the oldest years were excluded and the youngest years were included in the analysis. Thus, it can be said that the Mining industry sector has shown improvement in recent years, in relation to their contribution to sustainable development. The results showed that this sector’s leap in quality began in 2006.

Although the Chemical sector showed a good level of efficiency, being the third in the ranking, it also showed a great variability over the years. An important feature to be considered when analyzing the results of this sector is the fact that despite it has, in general, worsened from window to window, the last year of each window was always efficient, which demonstrates the sector has shown significant and rapid improvement, with the most recent year of the window being always more efficient than the previous ones. It should be mentioned that the only exception to this fact was the years of 2008 and 2009, when the sector had a significant worsening. It is also noteworthy that the years this sector stood out most, efficient in all the windows, were the years 2004 and 2007.

Similar to the Chemical sector, the Foods and Beverages sector also showed high variability in relation to efficiencies, and also worsened from window to window. This sector, as well as the Chemical sector, showed significant and rapid improvement in recent years, which can be corroborated by the fact that the last year of each window, without exception, was efficient in relation to the others. It is important to mention that the years the Foods and Beverages sector most stood out, being efficient in multiple windows, were the years 2004 and 2006.

Finally, the sector that was most efficient by reducing inputs and increasing outputs, in this work was the Textile sector. This sector showed high average efficiency in all windows, besides having the lowest standard deviation relative to the other sectors. It is noteworthy that the years this sector most stood out, effective in multiple windows, were the years 2001, 2003 and 2004.

Conclusions

Adopting the concept of sustainability requires not only the viability of the economic approach, but also the social and environmental variables in order to achieve a better spread of the gains acquired by the use of the natural resources with minimum damages to the planet and to humanity. A fair development process demands the interaction of sustainability dimensions to harmonize different interests involving economical growth and a social and ecological perspective.

Therefore, this article analyzed through the concept of efficiency, the contribution of seven Brazilian industrial sectors for the sustainable development, which addressed the three basic pillars of the triple bottom line, which are economic, social and environmental development. Thus, we analyzed the efficiency of the Brazilian industrial sectors from 1996 to 2009, to reduce the inputs "energy consumption" and "CO2 emissions from fossil fuels" and increase the outputs "sector GDP", "persons employed" and "personnel expenses", simultaneously. The outcome of this study showed that the Textile sector is the most efficient in terms of contribution to the


sustainable development in Brazil, followed by the sectors of: Foods and Beverages, Chemical, Mining, Nonmetallic, Paper and Pulp, and Metallurgical.

These results, besides the caution needed in interpreting the trends due to limited data, must be seen as a first attempt to illustrate the contribution of the Brazilian industrial sectors to sustainable development. In this sense, besides the variables analyzed in this study that addressed the three pillars of sustainability, for future works, it is possible to consider that adopting the sustainability concept includes one of the main goals of achieving a greater fairness in income distribution, which would imply the inclusion of new variables in the study.

According to the results of this work, it is important notice that all industrial sectors, especially the Metallurgical, the Nonmetallic, and the Paper and Pulp, can still contribute significantly to the complex challenge of promoting economic development with social improvement and environmental respect. For this, the current and future developments must be closely associated to the sustainable, efficient and secure use of energy based on environmentally and economically viable approaches for the future of society in the short and long term.

References

BEN - Balanço Energético Nacional, 2010. Divulga informações relativas ao binômio oferta consumo de fontes de energia. Available at: <https://ben.epe.gov.br>. (Accessed 04.11.10).

Hamzah, M.O., Jamshidi, A., Shahadan, Z., 2010. Evaluation of the potential of Sasobit® to reduce required heat energy and CO2 emission in the asphalt industry. Journal of Cleaner Production 18: 1859-1865.

IBGE – Instituto Brasileiro de Geografia e Estatística., 2011. Pesquisa Industrial Anual – Empresa. Available at: <http://www.sidra.ibge.gov.br>. (Accessed 04.05.11).

IPCC – Intergovernmental Panel on Climate Change., 1996. Greenhouse gas inventory reporting instructions – Revised IPCC Guidelines for national greenhouse gas inventories. In: United Nations Environment Program, the Organization for Economic Co-operation and Development and the International Energy Agency, London.

IPCC - Intergovernmental Panel on Climate Change., 2007. Climate Change 2007: Mitigation. Contribution of Working Group III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. B. Metz, O.R. Davidson, P.R. Bosch, R. Dave, L.A. Meyer (eds). Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.

Luken, R., Castellanos-Silveria, F., 2011. Industrial transformation and sustainable development in developing countries. Sustainable Development 19: 167-175.

Mao, J., Du, Y., Xu, L., Zeng, Y., 2011. Quantification of energy related industrial eco-efficiency of China. Frontiers of Environmental Science and Engineering in China. doi: 10.1007/s11783-010-0289-8.


MCT – Ministério da Ciência e Tecnologia, 2006. Emissões de dióxido de carbono por queima de combustíveis: abordagem top-down. Available at: <http://www.mct.gov.br>. (Accessed 05.11.10).

Scheneider, M., Romer, M., Tschudin, M., Bolio, H., 2011. Sustainable cement production - present and future. Cement and Concrete Research 41(7): 642-650.

Silva, F. I. A., Guerra, S. M. G., 2009. Analysis of the energy intensity evolution in the Brazilian industrial sector - 1995 to 2005. Renewable and Sustainable Energy Reviews 13(9): 2589-2596.

Simões, A., La Rovere, E. L., 2008. Energy Sources and Global Climate Change: The Brazilian Case. Energy Sources Part A: Recovery, Utilization & Environmental Effects 30: 1327-1344.

Tomasula, P. M., Nutter, D. W., 2011. Mitigation of Greenhouse Gas Emissions in the Production of Fluid Milk. Advances in Food and Nutrition Research 62: 41-88.

Wernet, G., Mutel, C., Hellweg, S., Hungerbühler, K., 2011. The Environmental Importance of Energy Use in Chemical Production. Journal of Industrial Ecology 15(1): 96-107.


15. Efficiency in the management of sanitation and its impacts on health promotion: an aplication of data envelopment analysis – DEA

Karlos Eduardo Arcanjo da Cruz Universidade Federal de Pernambuco, Brazil, [email protected]

Francisco de Sousa Ramos Universidade Federal de Pernambuco, Brazil, [email protected]

ABSTRACT

This paper analyzes the public administration efficiency regarding the attenuation of infant mortality. In many Brazilian regions there is a low coverage of sanitation systems. This deficit in supply per se is directly associated with under five mortality. However, the poor quality supply of service also causes a similar effect. Thus, this paper analyzes the efficiency of the Brazilian states in the management of sanitation. To do this, it was considered as the primary objective of sanitation to improve the population welfare, which is translated by the elevation in the number of over-five-year children who were born survivors for each thousand. As a result, it was found that all the states in the South region are efficient, while the Southeast are not as good as these, but have a score close to 100%. Northeast states show a low efficiency. In the north, most of the units was ineffective, but better than the northeast units. When comparing these results with the indicator of infant mortality, it is perceived that in the Northeast the scarcity of infrastructure is complemented by the inefficiency and produces a perverse effect that makes the region have the highest Under Five Mortality Rate nationally.

Keywords: Sanitation, Infant Mortality, Efficiency, DEA.

Introduction

Services of Water Supply and Sewage Collection (SWSSC) in many developing countries is quite impaired and very far from achieving universal service (Turolla, 2002; RIVERA, 1996). Over 2.6 billion people worldwide lack access to adequate sanitation system and approximately 900 million people do not use drinking water (WHO, 2010).

According to the World Health Organization, the inappropriate use of sanitation and water is a major risk of mortality. It also has an additional adverse effect to be more connected to regions of low income - 99% of its occurrence is in developing countries. It is associated to a group of five risk factors that together account for 25%


of global mortality and 84% of these deaths associated to children (WHO, 2009). Moreover, it is directly associated to diarrhea, which is one of the major causes of mortality of children under 5 years (BLACK et al., 2003).

The United Nations Children's Fund (UNICEF) prepares a ranking of countries according to the number of children under five years who die per 1,000 born in the year, called Under five Mortality Rate - U5MR. In other words, it measures the probability that a child has to die before its fifth birthday. This indicator is a "smaller is better." In the list of 2007, Brazil has a U5MR of 22 children, which puts it in an intermediate position, the 107th. This situation is worse than Peru (with 20 children of U5MR), Colombia (20), Argentina (16) and Uruguay (14) (UNICEF, 2009), all these countries belong to Latin America.

One possible cause for this result is the low percentage of the population that is served by the essential services of sanitation: according to information from the National Sanitation held in 2008, the country has only 44% of households with access to the general network sewage and has 78.6% of households with access to the main water supply, which represents more than 12 million households without access (BRAZIL, 2008).

Furthermore, according to this research, in 2008, as regards the quality of water, 18% of the Brazilian municipalities distributed water without any treatment (flocculation, settling, filtration and disinfection) in which region that had the largest number of municipalities in this situation was the North (20.8%) followed by the Northeast (7.9%). In 23.4% of the municipalities water rationing occurred, and the Northeast region with the highest percentage of water rationed, with 40.5% of its municipalities in rationing.

Regarding to efficiency, on average, 40% of water injected into the network is lost - are not billed (SNIS, 2008). This indicator reflects the number of leaks occurring in the country, the low rate of micro-measurement and underpricing. It is a sign of high inefficiency in the sector (WHO, UNICEF, 2000 p 25), which may be due to the prevailing institutional structure, the federal conflicts and cross-subsidy.

Thus, the fact is that the coverage index of the national SWSSC is far from desired. However, this is not the only problem in the sector: the quality of provision of this service is another problem because it has not proven satisfactory, since the percentage of municipalities that do not treat the water supplied is considerable – or even those who work under the regime of rationing - and that the loss rate of billing is high. This makes clear that the sector problem is not only the deficit of the service, but also in efficient management of sanitation. Therefore, this level of efficiency needs to be measured, so that public agents can interfere with the sector to ensure that resources are used in the best possible way and more lives can be saved.

The aim of this study is to measure how effective are the Brazilian state governments in the management of sanitation to improve the population welfare with the reduction of child mortality for children under five years old. The methodology chosen for this was the DEA.


Methodology

In Brazil, the value of the U5MR is high and uneven across geographic regions. In 2007, 24 of every 1,000 children born, died (DATASUS, 2007). This indicator is more pronounced in the poorest regions of the country, North and Northeast. It is a fact that this rate has decreased over the years - mainly in the 2000s – as is shown in Graph 1. However, this inter-regional inequality is not necessarily a management issue, it may exist due to scarce resources, given that the same regions are the ones with the worst indicators of coverage for water supply networks (WS) and Coverage by and sewage collection networks (SC).

Brazilian states, as making-decision units (DMU), use the existing infrastructure to combat the mortality of children under 5 years. The mechanism by which these resources are managed can be described by a production function, even if unknown. However, the product of this function is, by definition, non-decreasing in relation to inputs. Thus, the U5MR will not be used as the output, but as the number of children who survive (U5SR), defined by equation 1.

U5SR = 1.000 – U5MR. (01)

Once the indicator U5SR defined as output, define the possible inputs to remaining possible inputs. Sanitation is one of these because there is a significant association between the quality of it and the product. It is composed of four elements: (1) water supply, (2) wastewater collection and treatment, (3) solid waste management and (4) storm drainage. WS and SC indicators will be inputs of the production function, since they account for items 1 and 2. For items 3 and 4, Degree of Urbanization (DU) will be the used indicator as a proxy for them, since in urban areas is expected that these services are performed.

The aim of the work is to analyze the efficiency of sanitation, however, other indicators may be important for this analysis, since the product is an indicator that involves economic and social factors. Thus, two possible inputs are the number of doctors by a group of 1,000 inhabitants (DBI), as a proxy for hospital infrastructure, since the reduction of infant mortality depends on the medical monitoring in the early days of life and even vaccinations and medicines, and Gross Domestic Product per capita (GDPPC), because they believe that wealthier states may use more healthcare resources and the richest people can have access to better conditions for their children.

To determine the importance of each variable, an analysis of correlation between them was performed, which is shown in Table 1. From the table, there is a positive association between the indicator U5SR and the other variables.


Graph 1 – Mortality rate evolution in less than 5 years old.

Source: elaborated by the author by means of data extracted from DATASUS (2007)

There is a strong correlation between the indicators DU and DBI. This perhaps be due to the more urbanized regions hold a greater attractive power, and therefore it is natural that there is a greater concentration of physicians in these areas. Also there is a strong correlation between DBI and GDPPC, which may occur for the same reason: the richest regions can attract more doctors, both in the private or public

On the other hand, according to UNICEF (2010, p. 14), there is a significant relationship between GU and underweight, so that, in developing countries, there is a concentration twice higher for underweight children in rural areas compared to urban areas. The underweight is directly linked to four factors: food deficit in terms of quality and quantity; sanitation and inadequate health services, lack of care and feeding practices.

It is also observed, from the table, that the WS and U5SR correlation is low, despite being significant. This may indicate that the coverage of water supply systems has no significant effect on infant mortality, or that these systems are operating inefficiently in many states, which possibly reduces the positive effect. The second option is more likely, and hence the indicator will be remained in the model.

The main object of study is the efficiency of sanitation on health promotion, so that the model used in the research will be called Model 1 and it will have as inputs WS, SC and DU. The method used was the BCC. As a guideline for the optimization, it was considered a maximization of the product, given the levels of available inputs.

Região Norte; 2000; 33,35








Região Nordeste; 2000; 48,81 Região Nordeste;

2001; 45,56 Região Nordeste; 2002; 43,28 Região Nordeste;

2003; 40,97 Região Nordeste; 2004; 38,87

Região Nordeste; 2005; 37,3



Região Sudeste; 2000; 22,12








Região Sul; 2000; 19,93

Região Sul; 2001; 19,46

Região Sul; 2002; 18,65

Região Sul; 2003; 18,82

Região Sul; 2004; 17,53

Região Sul; 2005; 16,1

Região Sul; 2006; 15,84

Região Sul; 2007; 15,11

Região Centro-Oeste; 2000; 25,12








Brasil; 2000; 31,99 Brasil; 2001; 30,56 Brasil; 2002; 29,13 Brasil; 2003; 28,09 Brasil; 2004; 26,62 Brasil; 2005; 25,36 Brasil; 2006; 24,77 Brasil; 2007; 24,07

Região Norte

Região Nordeste

Região Sudeste

Região Sul

Região Centro-Oeste

Brasil

U5MR


Table 1– Analysis of correlation between variable

U5SR WS SC DU DBI GDPPC

SOB5 1.0000

CA 0.3911 1.0000

CE 0.5667 0.5656 1.0000

GU 0.7156 0.6527 0.4622 1.0000

MPH 0.6121 0.6443 0.6652 0.7563 1.0000

PIBPC 0.7228 0.4882 0.6009 0.6961 0.8607 1.0000

Source: elaborated by the author by means of the softwareEviews 7.0.

Results and discussion

The result of linear programming for the chosen model is shown in Table 3, in which the Local Technical Efficiency is presented by the BCC.

Table 3 – Local Technical Efficiency of the states in management in sanitation.

Rank State BCC (%) U5SR Rank State BCC (%) U5SR

1 DF 100 986.69 11 AP 99.43 975.38

1 RO 100 975.07 12 MG 99.41 979.28

1 PR 100 984.49 13 AM 99.31 974.08

1 SC 100 985.43 14 AC 99.02 966.51

1 MT 100 977.62 15 CE 98.97 969.17

2 RS 99.98 985 16 PI 98.97 966.49

3 ES 99.89 983.22 17 BA 98.7 965.62

4 GO 99.87 979.12 18 MA 98.68 965.25

5 MS 99.85 976.72 19 SE 98.58 965.16

6 SP 99.84 984.84 20 RN 98.52 965.62

7 RJ 99.67 982.85 21 PE 98.41 965.2

8 RR 99.59 979.64 22 PB 98.25 963.03

9 PR 99.57 973.32 23 AL 97.34 950.79

10 TO 99.46 973.06

Source: elaborated by the author


Thus, one can assume that there is a clear need for improvement in the quality of sanitation services in the northeastern states, for they have not made good use of existing infrastructure. Possibly, there must be elements that affect efficiency, but were not considered in the model. The DBI, which is a proxy for hospital infrastructure and was neglected in Model 1, may be affecting the efficiency of the state. Similarly, the GDPPC also exogenously may be affecting the efficiency of states.

One factor to be considered is that water availability in the Northeast is far from a considerable extent in other regions of Brazil. This element can also influence the efficiency of the state to providing the essential elements for reducing infant mortality. Years of studies for over 25 years - which, as a proxy, can be interpreted as the level of knowledge of health education and knowledge of rights and duties - have a great inequality between geographic regions, so that the North and Northeast are below on average.

For an analysis of the elements that are possibly impacting the efficiency of the States, it was considered the linear regression model, which has TE (technical efficiency) as the dependent variable, and as independent variables, the average of years of studies for people over 25 years of age (ST), obtained from the IPEA; the relation between the municipalities population under the rule of state and the total population of the state (CONT), with the information on the population obtained in the SNIS (2007) and IBGE, the Population Density (PD), obtained from the IBGE; Per Capita Water availability (PCWA) (m3/inhabitant.ano) obtained in Lima (2001), and GDPpc. The model is shown in equation 4.

𝑇𝐸 = 𝑐 + 𝛽1 𝑆𝑇+

+ +𝛽2 𝑅𝑈𝐿𝐸+

+ 𝛽3 𝑃𝐷+

+ 𝛽4 𝑃𝐶𝑊𝐴+

+ 𝛽5 𝐺𝐷𝑃𝑃𝐶+

+𝛽6 𝐷𝐵𝐼+

(04)

The regression result is shown in Table 4, in which it is possible to verify that the indicator ST had no significant effect on the model or on the indicator PCWA. Thus, it cannot join the issues of water availability the inefficiency model nor the education of the population.

Table 4 – Linear Regression of TE in relation to the socio-economic indicators

Indicator Coefficient Test-t p-value

GCPPC 0.0001 2.9580 0.0075

DBI 0.7105 2.1013 0.0479

PD -0.0079 -4.2991 0.0003

PCWA 0.0000 0.4214 0.6777

ST 0.0051 0.1298 0.8980

C 97.9028 347.2115 0.0000

Source: elaborated by the author with the use of Eviews 7.0 / R2 adjusted= 0,6

It is noteworthy, however, that the PCWA used here may not measure the situation well, because the water shortage problem is a micro-regional one, and aggregating by states, this information may not be well captured by the indicator. The attention


given to health can be important for the efficiency of sanitation systems, since the DBI indicator shows a positive effect on the dependent variable, which indicates that not enough investment in sanitation, but it is also necessary to improve the hospital infrastructure so that the sanitarian policies have a higher efficiency.

The indicator population density showed a negative relation with efficiency. This information can be an indication that a concentration of population in excess can hinder the carrying out of the work of health teams, impeding the company of working optimally. The GDPpc has a positive effect on efficiency, which may point to the richest states get greater efficiency in the management of sanitation. One possibility for this is that a richer state may involve mitigation measures for sanitation that were not captured in the model, or a better buying power allows people to protect themselves against the inefficiency of sanitation.

Conclusion

In this study, we analyzed the technical efficiency of the state governments in administering the sewerage, which is directly associated with reduced infant mortality. For this analysis, we used the DEA methodology oriented to the maximization of the product. As a result, states with 100% of technical efficiency are divided into three geographic regions of the country, in a total of five: two in the South, one North and two in the Midwest. In the North, states have not been efficient, but efficiency achieved scores above the northeastern states, even having an infrastructure below to these, either through coverage of basic sanitation, by number of physicians or by per capita income. This result clearly shows the inefficiency of the Northwest states, which make them to have the highest U5MR.

Thus, we perceive the need for quality improvement in the provision of sanitation services, for its whole service whole is not limited to providing water and sewage services to citizens. A better control is essential over the quality of this offer, since when comparing the effectiveness of services between the states, it is noted that there is an inequality, especially among geographic regions, which further contributes to the social inequality in the country.

References

BLACK, R. E.; MORRIS, S. S.; BRYCE, J (2003). Where and Why are 10 million children dying every year? The Lancet. V. 361,p. 2.226-2.234.

BRASIL (2008). Ministério das Cidades. Pesquisa Nacional de Saneamento Básico, Brasília, DF.

DEPARTAMENTO DE INFORMÁTICA DO SUS – DATASUS (2008). Informações de saúde epidemiológica e de morbidade – causa por local de Residência. Disponível em: www.datasus.gov.br.

LIMA, J. E. F. W. (2001) Recursos hídricos no Brasil e no Mundo. Ministério da Agricultura Pecuária e abastecimento. Planaltina: Embrapa Cerrados.


RIVERA, D. (1996). Private sector participation in the water supply and wastewater sector, lessons for six developing countries. Word Bank, Washington, DC.

SISTEMA NACIONAL DE INFORMAÇÕES EM SANEAMENTO – SNIS (2008). Diagnóstico dos serviços de água e esgoto. Brasília.

TUROLLA, F. A. (2002). Política de saneamento: avanços recentes e opções futuras de políticas públicas. Brasília. Ipea.

UNITED NATIONS CHIDREN’S FUND – UNICEF (2010). Progress for children: Achieving the MDGs with equity. UNICEF, New York.

UNICEF (2009).The States of The World’s Children 2009. UNICEF, New York.

WORLD HEALTH ORGANIZATION – WHO (2010).water global annual assessment of sanitation and drinking-water (GLAAS) 2010: targeting resources for

better results. WHO Library, Switzerland.

WORLD HEALTH ORGANIZATION – WHO (2009). Global health risks: mortality and burden of disease attributable to selected major risks. WHO Library, Switzerland.

WHO; UNICEF (2000). Global Water supply and sanitation assessment 2000 report. Switzerland.


16. Efficiency of Three Outliers Detection Tests on Non-Parametric Frontiers Methods

Victor Maia Senna Delgado Al. Acácias, 31275-150, Belo Horizonte, MG, Brazil - FJP, [email protected] (corresponding author)

Igor Viveiros Melo Souza R. Catete, 35420-970, Ouro Preto, MG, Brazil - UFOP, [email protected]

Abstract

One of the most important problems concerning non-parametric frontiers is the detection of outliers, influential information that distorts statistical analysis. This topic has been present for a long time in the literature of non-parametric statistics. Wilson (1995) used the test proposed by Andrews and Pregibon (1978) to apply one of the first empirical statistical tests to detect outliers on convex sets of non-parametric frontiers. Since then, a series of proposals for the detection and treatment of outliers has emerged in the field of non-parametric frontiers. The purpose of this research is to highlight three methods of outlier detection: Wilson (1995), Simar (2003), Sousa and Stosic (2005), and also answer which one is more efficient in statistical terms, lower bias and variance.

Keywords: Outliers Detection Methods; Non-Parametric Frontiers; Data Envelopment Analisys; mask effect; Monte-Carlo simulation.

Introduction

One of the most important problems concerning non-parametric frontiers is to control its borders to outliers, influential information that distorts statistical analysis. Any Outlier Detection Method (ODM) must help the researcher to isolate wheat from the weeds, and because most of them grow together, many times, it’s very hard to separate which one observation is not part of a particular bunch of data, is necessary a queer eye-detector to do it right with no help of detection procedures.

The ODMs has been present for a long time in the non-parametric statistics literature. Wilson (1995) used the test proposed by Andrews and Pregibon (1978) to apply one of the first empirical statistical tests to detect outliers in convex sets of non-parametric frontiers.

Since then, a series of proposals for the detection and treatment of outliers has emerged in the field of non-parametric frontiers. The purpose of this research is to highlight three methods of outlier detection: the already mentioned Wilson (1995), and also Simar (2003) and Sousa and Stosic (2005). The purpose is to answer which one is more efficient in statistical terms, lower bias and variance.


In order to make comparisons between different methods, we used the same Data Generator Process (DGP) to construct the frontier, and the outliers were kept fixed, they are the same between the three methods. The methods were compared with optimal threshold outlier detection value for each. The first results show advantages of Wilson (1995) and Sousa-Stosic (2005) to obtain large single outliers, but with small efficacy to detect outliers altogether, Sousa-Stosic has very large variance at some points. Simar (2003) is the more efficient test to detect outliers altogether and to minimal variance, but with small efficacy to detect particular individual outliers.

Methods

Three outliers detection methods (ODM) applied to non-parametric frontiers will be studied: the AP-Wilson statistic, named APW for short; the statistic from Simar Test (ST) developed by the approach of Cazals, Florens and Simar (2002) and Simar (2003). And the statistics of Sousa-Stosic (SS) developed by Sousa and Stosic (2005). These three methods have considerably creative and different approaches to the problem of detecting outliers.

These three statistics are:

APW: [ ] [ ] )1(**'det**'det )( XXXX iLS

iLSR −− =

ST:

)2(),(~1),(ˆ1

0000 ∑=

=B

b

bm

bm yx

Byx λλ

SS: ( )

)3(1

;1

2,

−

−=

∑≠=

n

n

isssis

i

λλς

Where X* is the [XY] matrix of information, with X representing the set of inputs and Y representing the set of outputs. The dataset element, or decision maker units (DMU’s), is S = 1, 2, …, n, which i represents any particular observation from S. L is a subset of S (L⊂ S) and det[X*´X*]i(S-L) represents the determinant of X* for one particular i in S without the subset L.

The Simar's statistic test is just efficiency index (λi), usually attained with the distance from DMU i to its frontier projection DMU´ (i´). Where x0 and y0 are values for x and y for one arbitrary point, m is the sample size (trimming parameter), and b is the bootstrap index of one particular replication with maximum of B. In this paper, for sake of simplicity, m = B = 100. Also, j is the number of outputs in the Y variable set (n × p). Where n is the maximum number of observations pertaining to S, and p is the number of outputs.

Usually the Shephard-efficiency (λ=1.00) indicates maximum efficiency and λ ≥ 1.00 is reserved to inefficiency. But in this case, with trimming parameter, it’s possible to obtain super-efficiency (λ<1.00), values of λ ± δ (efficiency index plus or minus a small delta) are used to indicate the outliers.

The ςi is the statistic SS mentioned above, s is any element of S-set described above. The λs,i is the efficiency index without the subject information i, also belonging to S, (s ≠ i). If the DMU ‘i’ is influent the sum of (λs,i – λs)2 will be positive and greater


than zero. The statistical ςi removes the observations one by one and verifies it influence, greater values of ςi denotes potential outliers.

Equipped with the three statistics listed in the previous section, one natural question that emerges is “Which one to pick up?” The reasonable answer must state that the better statistic is the one with greater aim and accuracy, in statistical terms, less bias and small variance.

To control the precision it’s necessary to know the Data Generating Process (DGP) of the real data and the generator of the fake ones (Outliers Generator Process – OGP). The more intersection exists between these two process, more difficult is to separate the wheat from the weeds. Following the process suggested by Simar (2003, p. 405), we have the following production function technology:

Y = Xβ *exp(-U) (4)

Where Y and X are outputs and inputs, here and in all paper, these are vectors are (n×1) dimension. The β = 0.5 and U is uniform on (0,1), so, exp(-U) have mean μ = ⅓, and the output-efficiency mean is 3∕2. The OGP is also here arbitrary and somewhat similar to Simar’s, but apart from paper of Simar, we chose 90 observations of process above and 10 outliers, summing up to 100 DMU’s and a proportion of 0.1 outliers in database (Table 1 brings the outliers values).

Table 1 - Values of Outlier Generating Process

Outlier id. Y X

91 0.004 0.100

92 0.025 0.190

93 0.035 0.240

94 0.100 0.700

95 0.500 0.900

96 0.700 1.000

97 0.750 1.050

98 1.000 1.000

99 1.200 1.000

100 1.100 1.100

For now on, we made means to compare different ODMs, called statistic ρ*. This statistic is based on simple proportion of all outliers detected by the method (ωi) over all ‘true’ outliers (Ω):

)5(1* 11

Ω⋅

−= ∑∑ ==

n

i iin

i i Tn

ωωρ

The term in first parenthesis is useful for one penalty to over identification. On the limit, a weak threshold could identify all ‘true’ outliers identifying all observations as outliers. To avoid this, the penalty was suggested, if one ODM detects all n


observations as outliers, the ρ* will become zero. Another change from the previous equation is the Identification vector (Ti) that enables the term in second parenthesis capture only true outliers.2

With the previous statistic is possible to compare the three different ODMs, but to be fair with all three methods, is necessary to find out the best possible threshold value to each one, the value that maximizes the identification with no penalty loss.

To identify the maximum threshold, all the other parameters must be fixed. It was used all data, with one particular realization of equation 6 for all 90 “non-outliers” points, and 10 fixed outliers. If we had continuity and second order condition satisfied, to discover the maximum we need the derivative of ρ* with respect the threshold value (τ): ∂ρ*(τ)∕∂τ = 0.

Table 2 presents the values of threshold and the maximum value of ρ* for each method, the first is computational statistic of maximum, the second is obtained by optimize function of R language.

Table 2 - Optimum values for the threshold of each Method.

Computational Optimize

APW 0.965 0.966

ST 1.085 1.095

SS 0.020 0.121

To compare three different methods with only one database is insufficient to build confidence intervals and pronounce about reliability of indicators. One idea to compare empirically three different methods is Monte Carlo simulations. The simulation was composed by 2000 replications of equation (4) with ninety observations each (DGP) and the ten outliers were kept fixed.

In the first procedure, the outliers were added one by one until they sum up 10. This procedure identifies the method capacity of identify all outliers and to control mask effects. The routine is described:

• Run the equation (6) and obtain 90 observations for each.

• Append to each replication the first outlier kept fixed (from table 1).

• Replicate step 1 and 2 2000 times.

• Save the mean and sampled standard deviation (σ) of the ρ* statistic.

• Obtain the confidence interval (CI) with 5% for α (error type I and supposing normal distribution).

2 If the ODM is really unbiased the term in second parenthesis will be 1.00, and with n = 100, the first penalty term should not trespass 0.9, therefore, the maximum ρ* is 0.9 with ten outliers. For only one outlier this maximum is 0.99.


• Do steps above more 10 times, adding to step 2: first up to second, first up to third, and so on until the tenth outlier.

In the second procedure, outliers were added one (and only one) by time until all ten were analyzed. This procedure identifies the method capacity to point one outlier per time: “if that was the only outlier, how the ODM should be doing?” The routine is very similar to the previous described, with exception of sixth step, which adds one outlier per time.


As results for first procedure we have that the SS method is the best result to this first Monte Carlo procedure, mean of 0.883 for first outlier (91th DMU) and 0.869 to second (92th). But for all three first outliers the confidence interval of SS method is very large, maybe because its leverage properties of being in frontier, this introduces greater variance because more sparsely data in frontier region.

The Simar’s ODM greatly improves from the fourth outlier beyond, it’s the better method to identify outliers from 94 to 100, with relativity small variance and mean over 0.8 for statistic (ρ*). The APW method does not captures the first three outliers. Despite that, the APW method improves at each outlier added, with very small, but increasing, variance from 94th to 97th. Although the maximum value at 0.549 is far below the ST at final outliers, with relatively same variance.

As discussed by Wilson (1995), the mask effect should be important here for APW and SS methods. Table 3 shows the results for three ODMs pointing the central statistic and standard deviation with confidence interval of α = 5% (1.96).

In Table 4, the capacity of identify each outlier per time is investigated. Remembering that the maximum ρ* value is .99 to next simulation. The capacity of SS method for identify the seven first outliers (from 91 to 97) is very high, dropping drastically for 98th and 99th. These two observations should be not declared as outliers by the SS method. Again, the confidence interval for SS method is very abroad to 91, 92, 93, 98 and 99.

The APW shows better results identifying the outliers one per time, with the exception of the first three (91, 92, 93) which are note declared outliers by this method. Our hypothesis is that the identification one per time excludes the mask effect that prejudices APW and SS ODMs. Meanwhile, ST do not so well to identify one per time, probably because there is over identification of false outliers, and it increases the penalty for ST method (maybe one different threshold should be applied to this part). Next section investigates this possibility in detail.


Table 3 - Results for Outliers added one by one until sum up ten

Ap.wilson Simar test Sousa-Stosic

Outliers ρ* σ CI_min CI_max ρ* σ CI_min CI_max ρ* σ CI_min CI_max

91 0.000 0.000 0.000 0.000 0.728 0.037 0.656 0.800 0.883 0.268 0.358 1.407

92 0.000 0.000 0.000 0.000 0.738 0.036 0.667 0.810 0.869 0.197 0.483 1.254

93 0.000 0.000 0.000 0.000 0.738 0.035 0.669 0.808 0.583 0.123 0.342 0.824

94 0.206 0.006 0.194 0.218 0.845 0.027 0.791 0.898 0.485 0.003 0.480 0.490

95 0.335 0.010 0.315 0.355 0.888 0.023 0.842 0.934 0.579 0.003 0.573 0.585

96 0.423 0.012 0.399 0.447 0.903 0.018 0.868 0.939 0.484 0.001 0.482 0.486

97 0.485 0.012 0.461 0.509 0.903 0.015 0.873 0.933 0.415 0.001 0.413 0.417

98 0.530 0.013 0.505 0.555 0.893 0.016 0.862 0.924 0.363 0.001 0.362 0.365

99 0.564 0.015 0.535 0.593 0.883 0.015 0.853 0.913 0.323 0.001 0.322 0.324

100 0.589 0.019 0.552 0.626 0.875 0.015 0.846 0.904 0.325 0.045 0.236 0.414

Table 4 - Results for Outliers added one per time

Ap.wilson Simar test Sousa-Stosic

Outliers ρ* σ CI_min CI_max ρ* σ CI_min CI_max ρ* σ CI_min CI_max

91 0.000 0.000 0.000 0.000 0.726 0.037 0.653 0.799 0.899 0.243 0.422 1.375

92 0.000 0.000 0.000 0.000 0.731 0.037 0.659 0.803 0.954 0.097 0.765 1.144

93 0.000 0.000 0.000 0.000 0.737 0.038 0.661 0.812 0.966 0.011 0.945 0.987

94 0.814 0.026 0.764 0.864 0.831 0.033 0.766 0.895 0.972 0.007 0.959 0.986

95 0.805 0.027 0.753 0.858 0.781 0.034 0.714 0.849 0.969 0.010 0.949 0.990

96 0.804 0.027 0.752 0.857 0.760 0.034 0.692 0.827 0.968 0.010 0.950 0.987

97 0.805 0.026 0.755 0.856 0.758 0.036 0.687 0.829 0.969 0.010 0.949 0.988

98 0.792 0.026 0.741 0.843 0.713 0.038 0.638 0.788 0.738 0.408 -0.062 1.537

99 0.790 0.026 0.740 0.840 0.714 0.040 0.636 0.791 0.277 0.435 -0.576 1.131

100 0.795 0.025 0.746 0.844 0.713 0.039 0.637 0.788 0.954 0.101 0.755 1.152

Conclusions

It’s hard to be conclusive to answer the question “Which ODM is better?” There is some indication that Simar’s method (2003) is the better one for detect bunch of outliers, and do it with great confidence, because small interval and small misidentification. With correct threshold, the SS is the better one to detect outliers 91, 92 and 93 in two Monte-Carlo procedures and the more powerful method to detect


one outlier per time. With exception of observations 98 and 99, that SS method virtually does not detect.

As SS, the APW method does better for identify outliers one per time. With the advantage of have smaller variance than the others for 91, 92 and 93 and for 98, 99 and 100. The APW appears to be the better method to see super inefficient outliers, points very internal in production set. This also appears in ROC analysis and suggests powerful performance for large outliers in large data. Due to the ‘mask’ effect, both methods (APW and SS) did not dispense visual inspection of researcher, which becomes a limitation for large databases.

One additional future advance for this research is to investigate how these three methods behave for more dimensions (more than 1-output and 1-input framework) and for other technology process, increasing and mixed returns. It’s also an extra topic to apply this analysis to other well-know databases that could also lead to interesting results.

And, finally, it seems, for us, that a promising method can blend the best qualities of each ODM discussed here. One possibility of doing this is to take the maximum expectancy of each method in its best extend of outlier detection.

References

Andrews, D., D. Pregibon (1978) Findings the outliers that Matter, Journal of the Royal Statistical Society, series B, 40 (1): 85-93.

Cazals, C., J. Florens, L. Simar (2002) Nonparametric frontier estimation: a robust approach, Journal of Econometrics106 (1): 1-25.

Simar, L. (2003) Detecting outliers in frontier models: a simple approach, Journal of Productivity Analisys, 20 (3): 391-424.

Sousa, M.C.S., B. Stosic (2005) Technical efficiency of the Brazilian municipalities: correcting nonparametric frontier measurements for outliers, Journal of Productivity Analysis 24 (2): 157-181.

Wilson, P.W. (1995) Detecting Influential Observations in Data Envelopment Analysis, Journal of Productivity Analysis 6 (1): 27-45.

Acknowledgements

The authors thank the support of Centro de Estudos de Políticas de Públicas of Fundação João Pinheiro (CEPP/FJP) for technical and finance support to the presentation in the 10th DEA international congress. Like many others Operational Research Professionals, we also regret the passing of Professor W.W. Cooper, one of those who made DEA meeting possible. All possible remaining flaws are responsible exclusively by the authors and not their institutions and collaborators.


17. Evaluation of the Benchmarking Model Proposed by the Brazilian Electricity Regulator for Energy Distribution Companies: The Case of Tariff Revision

Giordano Bruno Braz de Pinho Matos CEPEAD, UFMG, Brazil, [email protected]

Marcelo Azevedo Costa UFMG, Brazil, [email protected]

Ana Lúcia Miranda Lopes CEPEAD, UFMG, Brazil, [email protected]

Roberta de Cássia Macedo CEPEAD, UFMG, Brazil, [email protected]

Abstract

In 2011 the Brazilian Electricity Regulator - ANEEL implemented a benchmarking model to evaluate the operational efficiency of the electrical power utilities in Brazil. This model is based on two benchmarking methods widely applied by other regulators: Data Envelopment Analysis - DEA and Corrected Ordinary Least Square - COLS. The aim of this paper is to identify the cause of discrepancies between the results obtained by applying DEA and COLS models and also to discuss the use of a non- decreasing returns to scale(NDRS) DEA model by the regulator. It is shown that the differences between the parametric (COLS) and non-parametric (DEA) models are mainly due to the unsuitability of the Cobb-Douglas model as a cost function, by the effect of the sample size which shifts the COLS's efficiency scores towards smaller values, and because of the DEA model type NDRS.

Introduction

In September 10, 2010, the Brazilian National Electric Energy Agency - ANEEL started to debate with the society about the rules and methodologies for defining the revenues of electricity distribution utilities in the 3rd Periodic Tariff Review Cycle (3PTRC), through the public hearing 040/2010 (AP040). By means of the technical note 265/2010 the regulator proposed a full review on the model which calculates regulatory operational costs. The definition of efficient operating costs is a central point in the incentive regulation, because it has been chosen for regulating natural


monopolies in the Brazilian electricity sector after their privatization in the nineties. The incentive regulation requires the definition of a level of revenue or rate for a fixed period of time, which is defined in a formal contract. Given the level of rates or revenues defined by the regulator, the companies are encourage to reduce their costs to lower levels in order to achieve higher financial returns. After the concession period defined in the agreement contract, the costs of the companies are revised, i.e., the regulator defines new levels which are considered more efficient. In this case, these new efficiency levels are proposed by the regulator for the benefit of consumers. Therefore, it is crucial for both the regulated company as for consumers a consistent methodology for the definition of operational costs.

The Data Envelopment Analysis (DEA) benchmarking method was first considered by ANEEL because it has been successfully applied by other regulatory agencies from Austria, Britain, Belgium, Finland, Netherlands, among others. Therefore, it is of interest of ANEEL to replace previous benchmarking models by a concise and robust approach, like DEA. By applying DEA, the operating cost of each company is compared to the costs/results of the remaining companies and, by means of a linear system of equations, an efficiency frontier is calculated. The final results of this model are efficiency scores that indicate the efficiency of each company in transforming inputs (operational cost) in outputs (electricity consumption, number of customers and network extension), when compared with similar companies.

Previous studies apply DEA to assess efficiency and productivity change of electrical power distribution companies (e.g. Arocena, 2008; Bagdadioglu, Price, and Weyman-Jones, 1995; Hjalmarsson and Veiderpass, 2002; Kumbhakar and Hjalmarsson, 1998; Pacudan and Guzman, 2002). Estellita et al. (2007) presents a case study of electrical power distribution companies in which two different DEA models are proposed. The first modelassess efficiency using previous information from regulators and the second model assess efficiency using previous information from companies. Thegrell, Bogetoft and Tind (2005) proposed a multi-period and multi-output model based on costs related to productivity analysis and a theoretical model of the company (power distribution companies). Zhou, Ang, and Poh (2008) present results of a survey which applies DEA in the energy sector and environment.

Based on previous literature review, the aim of this paper is to evaluate the benchmarking model proposed by ANEEL and to discuss major inconsistencies of the methodology. Specifically, we aim at identifying the causes of the discrepancy between DEA and COLS efficiency scores. We also evaluate the use of non-decreasing returns to scale in the DEA model presented by ANEEL

Methodology

The most commonly used benchmarking frontier techniques include Data Envelopment Analysis (DEA), Corrected Ordinary Least Squares (COLS) and Stochastic Frontier Analysis (SFA) (Haney and Pollitt 2009). DEA is a non-parametric method which requires assumptions of increasing and concave production function (Banker, Charnes and Cooper 1984). It provides a very flexible function, robust to errors of misspecification. Both SFA and COLS are parametric methods which require the specification of a functional mathematical equation, such as Cobb-Douglas or translog


functions. The main difference between SFA and COLS is that COLS implies that deviations from the frontier are due to inefficiency only, while SFA considers that deviations from the efficiency frontier are due to technical inefficiencies and to a random noise. Although SFA may have a theoretical advantage over COLS model, it is hard to fit SFA in small samples (CEPA, 2003). Recently, advances in DEA models allowed deviations from the efficiency frontier to be related to both, inefficiency and random noise, as in stochastic frontier (BANKER and NATAJARAN, 2008).

The Data Envelopment Analysis - DEA methodology was first introduced by Charnes, Cooper and Rhodes in 1978 (CHARNES, COOPER and RHODES, 1978) and then extended by Banker, Charnes and Cooper in 1984 (BANKER, CHARNES and COOPER, 1984). This methodology has been widely used for estimating technical efficiencies of Decisions Making Units (DMU). DEA is a mathematical programming method that provides a single measure of efficiency. It is calculated with the use of multiple inputs and multiple outputs information and results in a frontier which represents the best practice. From this best efficiency frontier, the relative efficiency of DMUs is calculated. For each DMU, DEA presents an efficiency score, typically ranging between zero and 1, which indicates inefficiencies. Furthermore, the DEA efficiency frontier can be used as a guideline so that inefficient companies can improve their inputs and outputs and reach the efficiency frontier.

The following DEA models calculate the efficiency score of each DMU j by imposing

different returns to scale. Let be the efficiency score of the reference DMU j. Let yrk

be the variable that represents the output r (r = 1, ..., R) and xik be the input variable

(i = 1, ...I) for each DMU k. Let be the weights of the observations used as benchmarking. Model (1) is the CCR model (CHARNES, COOPER and RHODES, 1978), which assumes constant returns to scale (CRS). Model (2) assumes variable returns to scale (VRS or BCC) (BANKER, CHARNES and COOPER, 1984). Model (3) assumes non-increasing returns to scale (NIRS), while model (4) assumes non-decreasing returns to scale (NDRS).

(1)

(2)

1

1

max

. . θ , 1,..

, 1,... ;

0, 1,..

CRSj

N

k rk rjk

N

k ik ijk

k

S t y y r R

x x i I

k N

θ θ

λ

λ

λ

=

=

=

≥ ∀ = ;∑

≤ ∀ =∑

≥ ∀ =

1

1

1

max

. . , 1,..

, 1,... ;

1, 0, 1,..

VRSj

N

k rk rjk

N

k ik ijk

N

k kk

S t y y r R

x x i I

k N

θ θ

λ θ

λ

λ λ

=

=

=

=

≥ ∀ = ;∑

≤ ∀ =∑

= ≥ ∀ =∑


(3)

(4)

ANEEL Proposal

ANEEL proposed, in a first stage of the public hearing (040/2010), two DEA models organized in two stages. Both models for the first stage used the operational cost as their input variable and the network extension (km) as the output variable. Model 1 includes the number of customers as the second output variable, while model 2 includes the energy consumption (MWh) as the second output variable. Figure 1 shows the inputs and outputs of each model. Both models assume a non-decreasing returns to scale (NDRS).

Inputs Outputs

Model 1 Operational Costs (R$)

Number of Customers

Network Extension (km)

Model 2 Operational Costs

(R$)

Energy Consumption (MWh)

Network Extension (km)

Figure 1. Proposed model by ANEEL in the first stage of the Public Hearing 040

The data used was from 2003-2010. The electrical distribution companies were split into two groups. Group A is composed by companies with an annual energy consumption greater than 1 Terawatt-hour (TWh), whereas group B is composed by companies with an annual energy consumption below 1 TWh. The DEA method was applied separately on each group in order to estimate the final efficiency scores.

After to reach the efficient operational cost for each distribution energy company ANEEL proposed to adjust these scores by environmental variables in a two stage model.

1

1

1

max

. . , 1,..

, 1,... ;

1, 0, 1,..

NIRSj

N

k rk rjk

N

k ik ijk

N

k kk

S t y y r R

x x i I

k N

θ θ

λ θ

λ

λ λ

=

=

=

=

≥ ∀ = ;∑

≤ ∀ =∑

≤ ≥ ∀ =∑

1

1

1

max

. . θ , 1,..

, 1,... ;

1, 0, 1,..

NDRSj

N

k rk rjk

N

k ik ijk

N

k kk

S t y y r R

x x i I

k N

θ θ

λ

λ

λ λ

=

=

=

=

≥ ∀ = ;∑

≤ ∀ =∑

≥ ≥ ∀ =∑


After public contributions, ANEEL proposed one DEA model in the first stage, aggregating all previous output variables. In addition, the regulator proposed a second benchmarking model known as Corrected Ordinary Least Squares (COLS), and presented in Technical Note 101/2011. COLS is a parametric model which fits a regression model by means of ordinary least squares. In sequence, the regression model is shifted towards the smallest observed value of operational cost, creating the lower bound or the efficiency frontier of the operational cost. In this case, the regression model is the Cobb-Douglas production function.

This method is currently used by a few regulators (Denmark and Great Britain), and it is known to be more restrictive, i.e., it strongly penalizes companies which are not on the frontier (Bogetoft & Otto, 2011). To overcome this limitation, ANEEL proposed to average the DEA and COLS efficiency scores and to use the mean value as the first stage outcome.

In the final decision(november 2011), the energy consumption (MWh) output variable was replaced by a weighted energy consumption variable which aggregates high, medium and lower voltage energy consumptions. The weights were chosen to be proportional to the amount of consumption in the markets of high, medium, and lower voltage, of each company.

Figure 2 shows the different models presented by the ANEEL for the estimate of operational costs in each stage of the public hearing.

Figure 2. Phases and models suggested by the Brazilian regulator

Although the methodology proposed by ANEELs was based on the experience of leading European regulatory agencies, it was subject to criticisms and suggestions from the Brazilian community and power distribution companies. One of the major concerns was to use the DEA model with non-decreasing returns to scale (NDRS) as a replacement to the most commonly used model, the DEA with variable returns to scale -VRS (BANKER, 1984). Later, a technical report (BANKER, 2011) was submitted

Method Input Varibles Output VariablesDEA Operational Costs (R$) Network lenght (km), Number CustomersDEA Operational Costs (R$) Network lenght (km), Electricity Consumption

Method Input Varibles Output VariablesDEA Operational Costs (R$) Network lenght (km), Number Customers, Elec. Consumption (Mwh)

COLS Operational Costs (R$) Network lenght (km), Number Customers, Elec. Consumption (Mwh)

Method Input Varibles Output VariablesDEA Operational Costs (R$) Network lenght (km), Number Customers, Weighted Consumption (Mwh)

COLS Operational Costs (R$) Network lenght (km), Number Customers, Weighted Consumption (Mwh)

Technical Note 265/2010




to ANEEL, providing proper evidence that the VRS model is the appropriate model. Briefly, even if economic theory argues that non-decreasing returns to scale prevails in situations of natural monopoly, empirical evidence strongly suggests that VRS achieves better fit. This is because a mathematical model is an abstract representation, and without the proper choice of production function, outputs and environmental variables, such model should not hold on strong assumptions, which is the case of NDRS

A separate analysis of the estimated efficiency scores of DEA and COLS, presented by ANEEL, shows that the efficiency scores present major inconsistencies. The efficiency scores using the COLS model are outstandingly smaller than DEA estimates for all companies, except for one. Figures 3 and 4 show that the differences between the results of COLS and DEA reach up to 21% in Group A, and 41% in Group B.

Figure 3. Differences en percentage between COLS and DEA scores for Group A.

-1%

-4%-5%

-8%-9%

-1%

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-12%-11%

-15%

-3%-2%

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0%

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-6% -6%

1%

-18%

-9%

-7%

-12%

-2%

-9%

-23%

-18%

-13%

-8%

-3%

2%


Figure 4. Differences en percentage between COLS and DEA scores for Group B.

Analysis of inconsistencies of the methodology presented by ANEEL

The COLS model, presented in Technical Note 101/2011, is known to be more restrictive than other benchmark methodologies, as illustrated in Figure 5. Succinctly, COLS generates an efficiency frontier which is generally more distant to the data and, therefore provides smaller efficiency scores for the DMUs.

Figure 5. Comparison of Benchmarking Methods

(SYRJANEN, M., BOGETOFT, P., AGRELL, P. Efficiency benchmarking project B: Analogous efficiency measurement model based on Stochastic Frontier Analysis, Final Report, 2006.)

-10%

-19%

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-8%

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0%


Analysis of the Cobb-Douglas functional

The unsuitability of the Cobb-Douglas function as a cost function is already known in the literature. Bogetoft and Otto (2011) present the results of a benchmarking case study of the energy sector in Germany. The authors were evaluating different functional forms for both Stochastic Boundary (SFA) and DEA frontiers and concluded that a Normed Linear function provided better results. The work also concludes that the Cobb-Douglas function is not a cost function but rather a production function.

“We could, of course, have handled this heteroscedasticity problem using a log-linear specification, but we did not do so to avoid the specifications curvature problem; the output-isoquants in a log-linear specification curve the opposite way than do usual output-isoquants. This is not surprising since the log-linear model corresponds to a Cobb-Douglas model, which is really a production function and not a cost function” (BOGETOFT e OTTO, 2011, p. 31).

Syrjanen, Bogetoft and Agrell (2006), pointed this problem in the report to the Finnish Regulator[1]: "the Cobb–Douglas function is not a cost function". After concluding that the linear function has heteroscedasticity problems that can be solved with the use of a log-linear function (Cobb-Douglas), the authors argue that this has significant conceptual problems suggesting not using it in the regulatory cost modeling for Finnish distributors. “Due to conceptual problems related to the log-linear model we suggest discarding this model"(p. 50).

Figure 6 shows that although in the log scale, the Cobb-Douglas function seems to fit properly the data (figure in the left side), in the original space it produces a curve taking an opposite direction (blue curve) than what it should be expected as a cost function (pink curve). Therefore, the Cobb-Douglas is not appropriate as a cost function.

Figure 6 . Analysis of Log-linear function as a cost function SYRJANEN, M., BOGETOFT, P., AGRELL, P. Efficiency benchmarking project B: Analogous efficiency measurement model based on Stochastic Frontier Analysis,

Final Report, 2006.


Furthermore, Cobb-Douglas function has an increasing marginal rate for the outputs which contradicts economic theory. In fact, based on economin theory, production functions are required to be convex.

The problem of the sample size

According to ANEEL, the COLS efficieny scores are calculated in three steps. In the first step, the companies are grouped into two groups: Group A has 29 distribution companies which presented a total electricity consumption larger (or equal) than 1 TWh in year 2003. Group B has 30 distribution companies which presented a total electricity consumption smaller than 1 TWh in year 2003. In the second step, a multiple regression model is fit separately to each group (A and B). The response, or dependent variable, is the total operational cost of each company, for years 2003 to 2010. The predictors are: (1) total number of consumer units, (2) the total length of the network, (3) total weighted consumption (for consumers of high, medium and low voltage), for years 2003 to 2010. The model applies the logarithm (or log) scale which adjusts the heteroscedasticity within each group. In the third step, for each group, the estimated regression model is shifted towards the company with the smallest residual. The efficiency scores are calculated with respect to this new shifted regression model.

In can be showed that the efficiency scores estimated using the model COLS Cobb-Douglas are extremely sensitive to the sample size. The larger the sample size, the smaller the efficiency scores of all companies, except the company which achieved the smallest residual in each group. The company with the smallest residual has an efficiency score of 1 (100%). By accounting the operational cost over a period of time, say year 2003 to 2010, the sample size increases. Therefore, the larger the time period, the larger is the sample size. In this case, there might exist temporal dependencies which were not accounted in the Cobb-Douglas model. As a consequence of the sample size, most of the efficiency scores were between 20% and 50% for most companies. For instance: Boa Vista company (Group B) achieved an efficieny score of 19%, João Cesa company (Group B) achieved an efficiency score of 21% and Cemig company (Group A) achieved and efficiency score of 39%. If those efficiency scores were estimated for a single year, say 2010, most of the scores would be, on average, close to 60%, which is incredible higher than current estimates. In brief, with the increase of the sample sample, the scores are mostly between 20% and 50%, as can be shown in Figure 7. Figure 7 shows that by shifting the regression model towards the smallest residual, the larger the sample size the farther is the shift of the regression model from the data and, as a consequence, the smaller are the estimated efficiency scores. Therefore, the efficiency scores using the score COLS Cobb-Douglas are inapropriate for benchmarking. This asymmetric distribution of the efficiency scores, in the presence of a larger sample size, is known in the literature as the extreme value distribution, and it is caused primarily by the minimal operator in the COLS procedure.


Figure 7. Effect of the minimum operator on the distribution of independent and identically distributed normal random variables.

Furthermore, by assuming some basic statistical assumptions on the distribution of the residuals, the estimates of the regression model can be used to generate confidence intervals for the efficiency scores. Table 1 shows the efficiency score of Cemig company and its confidence intervals (lower and upper limits) using COLS Cobb-Douglas model and the sample size adjustment. It worh mentioning that without the sample size adjusment the efficiency score estimate is 39%.

Table 1: Confidence intervals for the efficiency score of Cemig using the adjusted multiple regression model

Description Efficiency Score

Estimated Value 0.4519956

Lower Limit 0.3412855

Upper Limit 0.5986191

amostra de tamanho n

Sample size n


Hypothesis testing of the correlation between DEA and COLS Cobb–Douglas efficiency scores for groups A and B

ANEEL reports that DEA-NDRS and COLS Cobb-Douglas efficiency scores have a correlation coefficient of 0.94 (94%) (paragraph 68, PG. 13) and therefore, they are similar.

Figure 8 shows the relationship between the estimated using DEA and COLS for groups A and B. For Group A the linear correlation coefficient is 0.9539 (95.38%). For Group B, the linear correlation coefficient is 0.8369 (83.69%). To test the hypothesis that the DEA and COLS scores are statistically equivalent, it is possible to fit a simple linear regression model with no intercept for each group of the form: 𝐶𝑂𝐿𝑆𝑖 = 𝛼 ∙𝐷𝐸𝐴𝑖, and generate confidence intervals for the parameter α . If the confidence interval contains the unit value (α=1 ) it is possible to conclude that there is statistical evidence that DEA and scores are similar.

(a)

(b)

Figure 8. Scatter plots of DEA and COLS efficiency score for groups A and B.

Table 2 presents the estimated values for the parameter α for groups A and B, and the respective confidence intervals of 95% and 99%. The value α=1 is not included in any of the confidence intervals. From these results it can be concluded that the DEA efficiency scores and COLS are not statistically similar. On average, for Group A, the efciciency scores calculated by COLS is 10.33% smaller than DEA efficiency scores. For Group B the efficiency scores calculated by COLS is, on average, 21.66% smaller than those generated by DEA.

0.5 0.6 0.7 0.8 0.9 1.0

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Grupo A

DEA

CO

LS

0.2 0.4 0.6 0.8 1.0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Grupo B

DEA

CO

LS


Table 2: Hypothesis testing for the similarity between DEA and COLS efficiency scores, for groups A and B.

Group A

Estimated value for intercept: α

0,89662

CI95%(α) [0,8648902 a 0,9283498]

CI99%(α) [0,8538171 a 0,9394229]

Group B

Estimated value for intercept: α

0,78342

CI95%(α) [0,7280965 0,8387435]

CI99%(α) [0,7088598 0,8579802]

Analisys of Outliers in Efficiency Scores estimated by COLS

Particularly, the UHENPAL company achieved in year 2004 an efficiency score of 100% and, based on the current model (Technical Note 294/2011), this company is the reference company for group B. Because of current efficiency score methodology, after 2004, the efficiency scores of UHENPAL have gradually decreased reaching an efficiency score of 45.79% in 2009. It is worth noting that an observation from 2004 still has a great impact on the scores estimates in five years later, as shown in Table 3. In short, the UHENPAL 2004 results negatively impacts UHENPAL 2009 score because of the current methodology.

Table 3: Analysis of the UHENPAL efficiency scores. The company achieved 100% efficiency score in 2004 but because of current methodology, most

recent efficiency scores have been compromised

Ano Efficiency score Operational Cost

2003 0.7953737 R$ 2.508.824

2004 1 R$ 2.050.212

2005 0.7477445 R$ 2.916.724

2006 0.7001319 R$ 3.163.022

2007 0.3701561 R$ 5.962.437

2008 0.3654717 R$ 6.182.374

2009 0.4579384 R$ 5.038.483


Therefore, it can be concluded that the claim, presented by ANEEL (Technical Note 294/ 2011, p. 8), that the COLS methodology is robust to outliers is flawed. In fact, COLS is highly sensitive to outliers .

It should be noted, also, that it was not found in the literature any regulator that applies an average result of two different methodologies such as DEA and COLS. The Austrian regulator, for example, uses a weighted average of DEA and MOLS, though MOLS is a more flexible form of translog function. CEPA (2003) apud NERA (2003) suggest a combined methodology of DEA and COLS in which DEA would be used to calculate efficiency scores while COLS would evaluate the products chosen and the significance of scores from DEA. The British regulator deploys COLS by applying the model to the data of the last year of British companies (2009). Many models that use more than a single methodology use the best (best-off) of the scores (Germany, for example).

Problems related to the DEA model applied by ANEEL

In technical note No. 101/2011, ANEEL proposed a DEA model having one variable as input (operational costs) and three outputs (consumer number, network length and total consumption). The model adopted was DEA-NDRS.

To justify the use of NDRS, ANEEL relies on the theory of natural monopolies which exists only for the sectors where economies of scale are not decreasing. Banker, in its technical report submitted to the second stage of public auction 040, points the inappropriateness of DEA model using NDRS for the calculation of operational costs:

“The arguments presented in note 101/2011 from page 18 to page 20 are not correct when translating the theoretical assumption of absence of scale diseconomies in natural monopolies to the empirical estimation of a production frontier to benchmark electricity distribution companies…

Technical note 101/2011 makes an error in interpreting this assertion in my earlier report about why we need to use a VRS model rather than a NDRS model. This note seems to suggest that the use of the VRS model implies that the true underlying production technology and the cost function exhibit decreasing returns to scale. This interpretation is not correct, as noted in my last report, and elaborated further in this report. We do not propose that the true cost function has decreasing returns to scale. We assert that ANEEL’s proposed simplified model distorts the true production relationship such that the NDRS assumption valid for the true function cannot be sustained for estimating the simplified production frontier. Even when the true cost function exhibits increasing returns to scale, if the estimation model does not capture all the complexities of interrelations between multiple inputs and multiple outputs, then the estimation model based on VRS significantly outperforms in accuracy a corresponding estimation model based on NDRS.” (BANKER, 2011, p.2).

So, as stated by Banker(2011), for empirical analyses, the VRS model is the correct model, even when economic theory argues that NDRS prevails in situations of natural monopoly. This happens because an empirical model is only an abstraction of reality


and, unless you have a complete specification of the production function between all inputs, outputs and environmental variables, it is not appropriate to impose NDRS.

The regulator conducted statistical tests showing that the VRS model had the best fit to the frontier calculated for the large companies group with data in cross section. This result can be seen in Table 6 of the technical note No. 101/2011. The regulator contradicts itself in the justification for the maintenance of the NDRS model because states that the results are quite sensitive to the withdrawal of the two largest companies in the sample. ANEEL still points to a serious problem of modeling: "the problem may lie in the lack of a significant sample of larger companies, and not in a characteristic of the activity distribution of electricity (ANEEL, 2011, p. 20).

The statistical tests for returns to scale pointed VRS in the case of large companies probably due to the existence of the two companies with very distinctive size. In this way, the model applied by ANEEL with few variables, is not sufficiently robust to represent with the complexity of the selected sample, so that the test points to a better fit with the VRS frontier. By the sensitivity of the test to the withdrawal of the two largest companies in the sample, these companies are the most harmed by the regulator model.

Still, the regulator argues that there were not appropriate justifications to explain the presence of diseconomies of scale:

Thus, it lacked to contributions received in the public auctions 040/2010 the theoretical reasons which would lead to diseconomies of scale. Furthermore, in two past reviews cycles, in the reference company model for the distribution activity, it was assumed that larger companies have lower costs with administrative structure which would imply higher returns to scale. (ANEEL, 2011, p. 20)

The rationale for the VRS model is not the presence of diseconomies of scale in the distribution sector in Brazil, but the simplicity of the proposed model, which imposes diseconomies of scale for some companies that, in the real theoretical frontier where all variables are properly measured and considered, it would not present this feature. So that the model penalizes these companies because it does not allow diseconomies of scale in a simplified model that does not reflect correctly the reality.

Conclusions

The evaluation and implementation of benchmarking models are certainly a difficult task, primarily due to the difficulty of getting the best data and then due to the wide range of possibilities for methodologies and modeling. However, the available techniques have been already applied for several years and the literature on the subject is quite wide, having been already discussed the main problems and benefits associated with each tool.

In order to determine the best method, it is important to consider the problem to be solved, but it is also essential to avoid methodological problems that might disqualify the results.

It is known that the goal of mathematical and statistical models is, through simplified simulations, capture the most important aspects of reality. To consider all the variables


involved in any phenomenon is very difficult and often impossible. However, any simplification cannot leave aside key variables.

The methodology presented by ANEEL demonstrates flaws that compromise the results. The problems presented may be affecting negatively quite a few companies and benefiting other precisely because they do not sufficiently reflect the reality. The main methodological problems presented in this paper can be summarized in the following items:

1. The use of the COLS associated with a Cobb-Douglas function, which is inadequate for estimating cost functions;

2. The sample size issue related to the use of the COLS, which reduces efficiency scores as it increases the sample size. This problem is further aggravated by the impact of outliers in this method;

3. The use of NDRS in the DEA model in a highly simplified model.

The purpose of this paper was to demonstrate some problems of the model presented by ANEEL. As demonstrated, the model should be improved, but, in addition, there must be a more complete study to identify the main variables that explain the costs in the electricity distribution service in Brazil.

Acknowledgments

This work was supported by Fundação de Amparo à Pesquisa de Minas Gerais – FAPEMIG, Project PPM-00543-11 and Fundação de Amparo à Pesquisa de Minas Gerais – FAPEMIG, Agência Nacional de Energia Elétrica- ANEEL and Centrais Elétricas de Minas Gerais – CEMIG, Project SHA-APQ-03165-11.

References

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http://www.aneel.gov.br/aplicacoes/audiencia/arquivo/2010/040/contribuicao/rajiv_banker_and_ana_lopes_report.pdf

http://www.aneel.gov.br/aplicacoes/audiencia/arquivo/2010/040/contribuicao/rajiv_banker_and_ana_lopes_report.pdf

http://www.ofgem.gov.uk/

http://lattes.cnpq.br/2759596279174236

http://lattes.cnpq.br/5742885316028139


18. Integration of BSC, DEA and Game Theory in the performance of public health service

Marco Aurélio Reis dos Santos Department of Production Engineering at Sao Paulo State University (UNESP), [email protected]*

Fernando Augusto Silva Marins Department of Production Engineering at Sao Paulo State University (UNESP), [email protected]

Valerio A. P. Salomon Department of Production Engineering at Sao Paulo State University (UNESP), [email protected]

Abstract

We evaluate the efficiency of the primary public health service in a typical Brazilian medium size town. This paper describes an integration of Network DEA model, Nash bargaining model, and BSC. The Network DEA allows establishing a hierarchical structure that corresponds to the BSC perspectives by adopting a sequence of stages where a set of indicators at one stage impacts in a subsequent stage. We used the Nash bargaining model in order to negotiate the desired levels of input-output from one stage to another. Public health services were compared as well as the benchmarks were identified for the inefficient services.

Keywords: Network DEA, Game Theory, BSC, Bargaining Problem.

Introduction

Many researchers have focused attention on the measurement of efficiency in the healthcare. According to Emrouznejad et al. (2008), the DEA (Data Envelopment Analysis) has been applied in the evaluation of the health services performance with good results. Various DEA approaches have been widely developed and applied in both public and private sectors (HADAD et al., 2011).

Here, we evaluate the efficiency of the primary public health service in a typical Brazilian medium size town, which has as the main challenge to find way to efficiently achieve its goals.

Moreover, this paper describes a method that integrates Network DEA model (FÄRE and GROSSKOPF, 2000), Nash bargaining model, and BSC - Balanced Scorecard





(Kaplan and Norton, 1992) with the intent of extending this analysis beyond only one organizational perspective. The Network DEA allows establishing a hierarchical structure that corresponds to the BSC perspectives by adopting a sequence of stages where a set of indicators at one stage impacts in a subsequent stage. The DEA-BSC model was proposed by Eilat et al. (2008), and extends the original CCR model and quantifies some concepts that are embedded in BSC. Amado et al. (2012) proposed a conceptual framework based on BSC and multistage DEA approach. However, it is evident that there are conflicts among the objectives of each perspective. We used the Nash bargaining model in order to negotiate the desired levels of input-output from one stage to another.

Among specific objectives of this paper are the elaboration of a model of performance evaluation by using DEA and Game theory under the guidance of a Strategic Management System, as for instance, the application of the BSC framework, and finally to test the applicability of proposed model in primary healthcare service by comparing the similar aspects of healthcare services offered in different locations.

Method

With the help of the Health Department personnel of the studied city, were identified which resources (inputs) and service levels (outputs) would be considered in the modeling: the medical specialties (internal medicine, gynecology and pediatrics) offered by a typical HBU (Health Basic Units) were defined as being the DMU. Also, the DEA variables were designated for each BSC perspective customized for the Health System: Patient Perspective; Internal Processes Perspective, Learning and Growth Perspective; and Financial Perspective.

As the next step, a survey was planned and performed in each HBU to obtain data information about the "Patient Perspective", "Learning and Growth Perspective" and other organizational characteristics the DMU. The Key Performance Indicators considered were: Financial Perspective - Number of physician, Number of functionaries, and Expenses; Learning & Growth Perspective - Working conditions; Internal Process Perspective - Medical service time (minutes), Wait time (minutes), Percentage of available medications, and Capacity used; and Patient Perspective - Percentage of patient satisfied.

Finally we developed three models in order to evaluate the performance in each medical speciality. Each model was applied in three phases. Consider a three-stage process shown in Figure 1. Suppose we have k DMU, and that each DMU has the vector of inputs (that are indicators for the Financial Perspective) to the first stage and the vector of outputs from this stage that are indicators for the Learning & Growth Perspective. These outputs (Learning & Growth Perspective) then become the inputs to the second stage. Indicators assigned to Internal Process Perspective denote the vector of outputs from second stage; this vector (Internal Process Perspective) becomes the vector of inputs to third stage. Finally, indicators assigned to Patient Perspective denote the vector of outputs from the third stage.


Figure 1. BSC in three-stage process

Note that the vector of outputs and inputs in each stage can be regard as two players in Nash bargaining theory. We briefly describe the game bargaining approach (Nash, 1950).

Denote the set of all individuals players by N=1, 2, 3,…, n, and a payoff vector is an element of the payoff space RN , which is the n-dimensional Euclidean space indexes by the set of individual players. A feasible set S is a subset of the payoff space, and a disagreement point d is an element of payoff space. A bargaining problem can be defined as three vectors (N, S, d), which respectively consist in individual players, a feasible set and a disagreement point. Nash (1950) required that the feasible set is compact, convex, and contain some payoff vector that each individual’s payoff is at least greater than the individual’s disagreement point. The solution proposed by Nash is the maximization of (1):

∏=≥∈

−n

iii

duSudu

1,)(max

(1)

The optimum value of function F(N, S, d) satisfy four properties: Pareto efficiency, Invariance with respect to affine transformation, Independence of irrelevant alternatives, and Symmetry.

In the first model (2), we used the Nash Bargaining problem in order to negotiate the desired levels of input-output in each stage of Network DEA, as expressed below. This model (2) is the first phase of our analysis.

321max zzz ++ ..ts (2.1)

111

1 λβ PP Yyo

≤ (2.2)

)()1( 1222

2 λλβ −≤− PP Yyo

(2.3)

)()1( 2333


(2.4)

344 λα PP Xxo

≥ (2.5)

12211 ))(( z=−− δβδβ (2.6)

23322 ))(( z=−− δβδβ (2.7)

3433 ))(( z=−− αδδβ (2.8)

1,1,1 321 === λλλ (2.9)

4332211 ,,, δαδβδβδβ ≤≥≥≥ (2.10)

.0,0,0,0 321 ≥≥≥≥ αλλλ (2.11)


In this model, P1 is the set of indicators assigned to Patient Perspective, P2 is the set of indicators assigned to Internal Processes Perspective, P3 is the set of indicators assigned to Learning & Growth Perspective, P4 is the set of indicators assigned to Financial Perspective, yo is the vector of output generated by the DMUo, xo is the vector of input used by the DMUo, Y is the vector of outputs generated by the set of DMU, X is the vector of inputs used by the set of DMU, λ is the vector of intensities variables for the set of DMU, α is the scalar coefficient that promotes the maximum

equiproportional reduction of all inputs and 3,2,1: =iiβ , are scalar coefficients that

promotes the maximum equiproportional increase all outputs. We add the convexity constraints (2.9) for each stage in order to adopt the variable returns to scale assumption. From the optimum solution obtained by the first model, we proposed the second model in order to negotiate the desired levels of input-output from one stage to another. This is the second phase of our analysis.

∏≠

−iii

ii fδβ

αδβ*:

)()(max ..ts

(3.1)

=≠−

=.,1

),()( *4

*44

αδαδαδα

ififfwhere (3.2)

111

1 λβ PP Yyo

≤ (3.3)

)()1( 1222


(3.4)

)()1( 2333


(3.5)

344 λα PP Xxo

≥ (3.6)

1,1,1 321 === λλλ (3.7)

4332211 ,,, δαδβδβδβ ≤≥≥≥ (3.8)

.0,0,0,0 321 ≥≥≥≥ αλλλ (3.9)

In this model, *α and *iβ is the value of optimum solution for the 1st model. Note

that if some *iβ and some *α do not satisfy ii δβ >* and iδα <* , then the respective

output associate to *iβ and, the input associate to *α do not participate as a player in

model (3).

In the third phase, we used the model (4) in order to optimise iβ and α that do not

participate as a player in second model (3), in order words, this model (4) optimise

the values of *iβ and *α if ii δβ =* and iδα =* in the first model.

αβββ −++ 321max ..ts (4.1)

111

1 λβ PP Yyo

≤ (4.2)

)()1( 1222


(4.3)

)()1( 2333


(4.4)


344 λα PP Xxo

≥ (4.5)

1,1,1 321 === λλλ (4.6)

**33

*22

*11 ,,, ααββββββ ≤≥≥≥

(4.7)

.0,0,0,0 321 ≥≥≥≥ αλλλ (4.8)

In this model (4), *α and *iβ are the optimum solution for the 2nd model, but it does

not provide information on the efficiency of each individual stage. We calculated the efficiency of each stage by the multiplier model (5).

..

min*

4*33

*22

*11

**3

*2

33

22

tsdddd

vuuyuyu Po

Po

αβββ +−−−

++++

(5.1)

111

1 =− dyu Po , 12

22 =− dyu P

o (5.2)

133

3 =− dyu Po , 14

4 ≤− dvxPo (5.3)

0*2

22

11 ≤−− uYuYu PP (5.4)

0*3

33

22 ≤−− uYuYu PP (5.6)

0,0,0,0 321 ≥≥≥≥ vuuu (5.7)

0,0,0,0 4321 ≥≥≥≥ dddd (5.9)

This last model is the dual to the model (4). According to Cook et al. (2010), due to the usual procedure of adjusting the virtual inputs or virtual outputs by the efficiency scores obtained from model (5), the model (4) does not necessarily supply information about frontier projection for intermediate outputs. Thus, we considered maxyo

P2u2; yoP2u* and maxyo

P3u3; yoP3u* as targets for intermediate outputs. We

calculated the efficiency for each stage according to:

*2

22

11

3 uyuyue P

o

Po

+= stage 3 (5.1)

*3

33

22

2 uyuyue P

o

Po

+= stage 2 (5.2)

*4

33

1 vvxyue P

o

Po

+= stage 1 (5.3)

According to Cook et al. (2010) and Chen et al. (2009), because of extra free-in-sign variable in the VRS-DEA model becomes difficult the use of geometric mean of the efficiency scores of the three individual stages (KAO and HWANG, 2008), once the geometric efficiency decomposition of the overall efficiency is restricted to CRS situations. Thus, we adopted Additive efficiency decomposition (arithmetic mean) approach (CHEN et al. 2009).



The models were run in the software GAMS (General Algebraic Modeling System),

and Excel 2010, using a notebook with processor Intel Core i5-2410M, CPU with 2.30 GHz, and 6.00 GB RAM. The studied city had five HBU, and they will just be referred as HBU A, B, C, D and E. The

Table 6 presented the obtained data.

Table 6.The data of medical specialities for five Health Basic Units

In our preliminary study, we considered as disagreement point 1=iδ , for i=1, 2, 3, 4,

thus it is possible guarantee that S was a feasible set. We obtained the efficiency scores (Table 2) for each stage by equations from (5.1) to (5.3).

DMU Satisfaction Medical service time Wait time Medicine Available Capacity usedA-Internal medicine 86.22% 6.1 77.8 61.83% 81.25%A-Pediatrics 95.14% 7.8 116.5 84.38% 81.25%A-gynecology 77.38% 9.4 106.3 53.27% 64.58%B-Internal medicine 74.13% 5.3 108.6 44.94% 65.25%C-Pediatrics 58.87% 7.7 268.0 82.59% 68.75%D-Internal medicine 80.38% 7.7 75.6 100.00% 68.75%E-Internal Medicine 67.46% 14.2 171.2 62.92% 62.50%E-Pediatrics 46.41% 15.2 96.8 54.64% 56.25%E-Ginecology 80.02% 21.5 66.3 100.00% 25.00%DMU Working conditions Physicians Functionaries ExpensesA-Internal medicine 68.57% 6 40 R$ 1240.18A-Pediatrics 68.57% 2 40 R$ 1132.34A-gynecology 68.57% 4 40 R$ 1186.26B-Internal medicine 23.18% 4 13 R$ 929.76C-Pediatrics 25.70% 2 12 R$ 513.17D-Internal medicine 14.39% 4 15 R$ 1688.13E-Internal Medicine 27.29% 2 16 R$ 588.04E-Pediatrics 27.29% 2 16 R$ 588.04E-Ginecology 27.29% 3 16 R$ 620.71


Table 7. Efficiency scores for each stage.

Table 8 presents the value of scalar coefficients corresponded to the perspectives of BSC. Note that, although the DMU “B-Internal medicine” was considered efficient at third stage, according to

Table 7, the proposed model can identify an improvement opportunity.

Table 4 shows the targets for medical specialties.

Table 8. The value of scalar coefficients

DMU Stage 3 Stage2 Stage1 Overall (arithmetic mean)A-Internal medicine 0.9487 0.6863 0.9018 0.8456A-Pediatrics 1 1 1 1A-gynecology 0.9183 0.6359 0.9484 0.8342B-Internal medicine 1 0.6633 0.8432 0.8355C-Pediatrics 0.7033 0.7628 1 0.8220D-Internal medicine 0.9621 1 0.4749 0.8123E-Internal Medicine 0.7094 1 0.8837 0.8644E-Pediatrics 0.5043 0.9675 0.8837 0.7851E-Ginecology 1 0.6590 0.8234 0.8275

DMU Beta 1 Beta 2 Beta 3 AlphaA-Internal medicine 1 1 1.4277 0.6584A-Pediatrics 1 1 1 1A-gynecology 1 1.1115 1.3968 0.6687B-Internal medicine 1.0085 1.1199 1.4882 0.9231C-Pediatrics 1.4192 1 1.4402 1D-Internal medicine 1 1.0383 1.7864 0.8000E-Internal Medicine 1.1129 1.1284 1.4557 1E-Pediatrics 1.6140 1.1278 1.4812 1E-Ginecology 1 1.0446 1.4146 0.8268


Table 9. Targets for medical specialities.

The dual values of λk that represents the weights for a linear combination (virtual DMU that is a benchmark) of medical specialties that are considered efficient. Making an analogy with the BSC theory, it would be interesting to establish objectives, measures and targets for the medical specialties, as well to develop the Strategic Map. The BSC allows the identification of good initiatives, which should be adopted by the inefficient DMU, based on implemented actions by a DMU that is a benchmark.

Table 10. Benchmarks for each medical specialty.

DMU Satisfaction Medical service time Wait time Medicine Available Capacity usedA-Internal medicine 86.22% 9.2 77.8 91.42% 81.25%A-Pediatrics 95.14% 7.8 116.5 84.38% 81.25%A-gynecology 77.38% 10.4 89.1 89.94% 71.78%B-Internal medicine 74.76% 7.7 75.6 100.00% 73.08%C-Pediatrics 83.55% 7.7 75.6 100.00% 68.75%D-Internal medicine 80.38% 8.0 72.8 103.83% 71.39%E-Internal Medicine 75.08% 16.1 91.7 88.33% 70.53%E-Pediatrics 74.91% 17.2 85.8 90.33% 67.12%E-Ginecology 80.02% 22.5 63.4 104.46% 68.75%

DMU Working conditions Physicians Functionaries ExpensesA-Internal medicine 97.90% 2 25 R$ 816.53A-Pediatrics 68.57% 2 40 R$ 1132.34A-gynecology 95.77% 2 24 R$ 793.29B-Internal medicine 34.49% 2 12 R$ 513.17C-Pediatrics 37.02% 2 12 R$ 513.17D-Internal medicine 25.70% 2 12 R$ 513.17E-Internal Medicine 39.73% 2 15 R$ 588.04E-Pediatrics 40.43% 2 15 R$ 588.04E-Ginecology 38.61% 2 12 R$ 513.17

Stage 1 A-Pediatrics C-PediatricsA-Internal medicine 49.0% 51.0%A-Pediatrics 100.0% 0%A-gynecology 45.2% 54.8%B-Internal medicine 0% 100.0%C-Pediatrics 0% 100.0%D-Internal medicine 0% 100.0%E-Internal Medicine 12.1% 87.9%E-Pediatrics 12.1% 87.9%E-Ginecology 0% 100.0%

Benchmarks


Conclusions

This research demonstrated how is possible the use of the DEA to measure (and to rank) the relative efficiency of the medical specialties offered by Health Basic Units (HBU) in a medium size town of Brazil. The ranking of efficiencies provides important information to the Health Secretary decides regarding actions to distribute better the available resources, making more efficient the HBU, therefore improving the overall performance of the Health System as a whole.

Regarding the set of medical specialties analyzed, it should be noted that the results obtained are based on selected inputs and outputs and their priorities. Sometimes, the targets proposed by the DEA technique cannot be applied in practice or they are very difficult of achieve them, and, in these cases, managers ought to take them as a reference. Each medical specialty must be analyzed according to its reality or organizational context; therefore, even those, which achieved 100% of relative efficiency, sometimes do not have the performance desired by the organization.

This occurs because the efficiency’s concept represents the best relation between the benefits obtained and the available resource used, but does not means these benefits are the desired results.

References

Amado C.A.F., Santos S. P., Marques P.M. (2012) Integrating the Data Envelopment Analysis and the Balanced Scorecard approaches for enhanced performance assessment, Omega (40): 390-403.

Chen Y, et al. (2009) Additive efficiency decomposition in two-stage DEA, European Journal of Operational Research (196): 1170-1176.

Stage 2 A-Pediatrics D-Internal medicine E-Internal MedicineA-Internal medicine 0 76.9% 23.1%A-Pediatrics 100.0% 0% 0%A-gynecology 0% 72.9% 27.1%B-Internal medicine 0% 100.0% 0%C-Pediatrics 0% 100.0% 0%D-Internal medicine 0% 100.0% 0%E-Internal Medicine 0% 68.5% 31.5%E-Pediatrics 0% 73.9% 26.1%E-Ginecology 0% 100.0% 0%Stage 3 A-Pediatrics B-Internal medicine E-GinecologyA-Internal medicine 53.1% 30.9% 16.0%A-Pediatrics 100.0% 0% 0%A-gynecology 10.6% 71.8% 17.6%B-Internal medicine 0% 89.3% 10.7%C-Pediatrics 42.5% 49.3% 8.2%D-Internal medicine 27.3% 64.0% 8.7%E-Internal Medicine 0% 83.9% 16.1%E-Pediatrics 0% 86.8% 13.2%E-Ginecology 26.7% 68.6% 4.7%


Cook, et al. (2010) Measuring performance of two-stage network structures by DEA: A review and future perspective, Omega: The International Journal of Management Science (38): 423-430.

Eilat H. et al. (2008) R&D project evaluation: An integrated DEA and balanced scorecard approach. Omega (36): 895-912.

Emrouznejad A. et al. (2008) Evaluation of research in efficiency and productivity: A survey and analysis of the first 30 years of scholarly literature in DEA, Socio-Economic Planning Science (42): 151-157.

Färe R., Grosskopf S. (2000) Network DEA, Socio-Economic Planning Sciences (34): 35-49.

Hadad S. et al. (2011) Determinants of healthcare system’s efficiency in OECD countries, The European Journal of Health Economics 2011; DOI 10.1007/s10198-011-0366-3.

Kao C., Hwang S-N. (2008) Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan, European Journal of Operational Research (185): 418-429.

Kaplan R.S., Norton D.P. (1992) The Balanced Scorecard: measures that drive performance, Harvard Business Review (70): 71-79.

Nash, J. F. (1950) The bargaining problem, Econometrica (18): 155-16.


19. Iteratively Weighted Least Squares in Stochastic Frontier Estimation Applied to the Dutch Hospital Industry

Jos L. T. Blank Delft University of Technology, Netherlands, [email protected]

Aljar J. Meesters University of Groningen, Netherlands, [email protected]

Abstract

This paper proposes an alternative class of stochastic frontier estimators. Instead of making distributional assumptions on the error and efficiency component in the econometric specification of a cost function model (or any other model), this class is based on the idea that some observations contain more information about the true frontier than others. If an observation is likely to contain much information, it will get a large weight in the regression analysis. In order to establish the weights, we propose an iterative procedure. In each step weights can be determined by the residuals obtained earlier and a user-specified weighting function. In each step the weights will be updated and a next stage WLS regression will be carried out.

The advantages of this approach are its high transparency, the easy application to a full-specified model and its flexibility. It allows to directly observing which observations determine the frontier for a large part. The easy expansion to a full-specified model refers to a model that includes a cost function and its corresponding share equations. Its flexibility refers to the use of several alternative weighting functions and the easiness of testing for the sensitivity of the outcomes.

The model has been applied to a set of Dutch hospital data. The outcomes of this application are promising. The model converges rather quickly and presents reliable estimates for the parameters, the cost efficiencies and the error components.

Key words: weighted least squares. SFA, hospitals

Introduction

The methodology Stochastic Frontier Analysis (SFA), suggested by Aigner, Lovell and Schmidt (1977) and Meeusen and Van den Broek (1977), has become a standard in econometric estimation of production and cost (or any other value) function. It is based on the idea that empirically, production (or cost) can be described as a function of a number of inputs (or outputs and input prices), a stochastic term, reflecting errors, and another stochastic term, reflecting efficiency. With maximum likelihood techniques, the parameters of the function and the parameters of the distribution of




the stochastic components can be estimated. For extensive discussions on this technique, see e.g. Kumbakhar & Lovell (2000) and Fried, Lovell & Schmidt(2008).

Since the 1977 publications, SFA has become very popular and has been applied in many empirical work (for extensive literature reviews, see also Fried, Lovell and Schmidt (2008) and Blank (2000)). Nevertheless, the approach has also widely been criticized. The criticisms focus on two major points. The first type of criticism debates the a priori specification of the production (or cost) function, the second type concerns the assumptions on the distribution of the stochastic terms. Although both criticisms can, to a certain extent, be overcome, by using flexible forms and using different assumptions on the distribution of the stochastic variables in the analysis, the rigidity might be seen as a problem. Although not mentioned very often, there is a third type of criticism, which can be considered to be of a conceptual nature. In a rather complex econometric framework, the methodology suggests to observe an unobservable (the efficiency), which can be distracted from another unobservable (the measurement and specification error). Those, who try to explain this approach to non-initiated persons, such as managers and policy makers, will be confronted with scepticism and unbelief. A technique like Data Envelopment Analysis, which is actually seeking for observations that form the envelope, is far more appealing and more transparent. This explains that in real-life problems, DEA has become a very popular tool in applied work. Another conceptual framing of SFA may tackle the problem and make the technique more accessible for non-experts.

The original work of Aigner, Lovell and Schmidt (1977) derives the stochastic frontier approach in case of a single equation model. In a single equation model, we can only estimate the technical or cost efficiency. If we are not only interested in technical or cost efficiencies, but also in allocative efficiencies, a multiple equations approach, allowing to derive the under- or overutilization of inputs, is needed. However, the estimation of a multiple equations model, with a far going decomposition of the underlying stochastic variables for measurement errors, technical and allocative efficiency, is very troublesome. In particular, the theoretical linkage between the cost function and the input demand equations is extremely difficult to handle (the so-called Greene problem). Although some interesting solutions have been proposed by applying shadow cost models (see Blank, 2009; Kumbhakar, 1997) or using Bayesian estimation techniques, new estimation problems occur. Obviously, these approaches suffer even more from a lack of transparency.

An alternative for the original SFA approach is the thick frontier analysis (TFA), developed by Berger & Humphrey (1991). This approach allows to estimate a single equation or a multiple equation. The technique is based on the selection of firms in the top 10% (or any other percentage) and the bottom 10%. For both subsamples, the production (or cost) function is estimated separately. Consequently, cost efficiencies are derived by taking the ratio of the average cost of the worst practice firms and the best practice firms. TFA has a number of advantages. Seemingly Unrelated Regression allows for a straightforward estimate of a system of a cost function and the corresponding share equations. TFA doesn’t require any rigid assumptions on the distributions of the error components and it does not suffer from the Greene problem. Conceptually, it is a very transparent and appealing approach. On the other hand, it


also has some serious drawbacks. It does not provide firm-specific cost efficiencies, but only rather general efficiency scores. From an econometric point of view, there is a loss of information, due to the discard of a large subset of observations. It is questionable whether the researcher has the luxury of losing so many degrees of freedom.

Estimating a production, cost, or profit frontier (hereinafter: frontier) would become trivial if all firms would operate with full efficiency. One could use ordinary least squares to estimate the parameters of the model. However, in reality some firms are inefficient, which makes the estimation of the frontier a challenging task. This problem could be solved by only taking account of efficient firms for the estimation of the frontier, and by neglecting all inefficient firms. However, this method implies the requirement of a priori knowledge about whether or not a firm is efficient and obviously, knowledge about the efficient firms is, in general, not available prior to the estimation of a production frontier and therefore, other methods for addressing this problem have been proposed.

One method that is often used for estimating a frontier is the aforementioned stochastic frontier analysis (SFA). In this approach, a stochastic term is added to an OLS equation, where it is assumed that it follows a distribution with a non-negative support. This stochastic term is supposed to pick up the inefficiency for each firm. Although SFA includes the concept of inefficiency while estimating frontiers, it does have its shortcomings. Firstly, it is often being criticized for its distributional assumption for the efficiency component (see e.g. Ondrich & Ruggiero, 2001). Secondly, although SFA allows for the estimate of cost functions, the concept of cost efficiency doesn’t seem to fit in with this type of estimation. Cost efficiency is built on technical and allocative efficiency, and yet, under the SFA specification of a cost function, all firms should be completely allocatively efficient (Greene, 1980).

Another approach for estimating a frontier was provided by Wagenvoort and Schure (2006). They show the way in which efficient firms can be identified, if panel data are available. They use a recursive procedure, by dropping the most inefficient firm at each iteration. In each step, the firm-specific efficiency is calculated by averaging the residuals of a firm over the whole time period. Their final step consists of using the fully efficient firms for estimating the frontier. Our approach is similar to this idea; however, we argue that the fact whether or not a firm is fully efficient doesn’t concern a zero-one-probability. We show the way in which a weighting scheme can be implemented, in order to determine which firms are likely to be efficient and which firms are likely to be inefficient. This concept also translates to a cross-section setting so as to avoid the necessity of panel data. This also implies that we don’t need to assume that inefficiency is time-invariant, which can also be regarded as a rather restrictive assumption in many efficiency models that are based on panel data.

In this paper, we propose an alternative that is related to the concept of stochastic frontier analysis, while being far more appealing conceptually. As in TFA, it can also be applied to multiple equation systems, while avoiding the Greene problem. Our alternative incorporates information from all the data available. It allows decomposing cost efficiency into technical and allocative efficiency without any additional distributional assumptions. The method is based on an Iterative Weighted Least


Squares (IWLS) method and can easily be programmed in standard econometric software.

The outline of the paper is as follows. In Section 2, we discuss a few conceptual issues of our method. In Section 3, we introduce a formal description of the model and the estimation procedure. In Section 4, we apply the method to a set of Dutch hospital data. Section 5 concludes the paper.

Methods

We start from the cost function, although the method may be applied to any other model (see e.g. Färe & Primont, 1995). We assume that the firm is cost-minimizing and that the total cost can be represented by a cost function c(y,w) that meets all the requirements it entails. Input demand equations xn(y,w) can be derived from the cost function, by applying Shephard’s Lemma. For reasons of convenience, we rewrite the cost equations and input demand equations in terms of logarithms and cost shares, and add an error term.

ln(𝐶) = 𝑐(ln(𝑦) , ln(𝑤)) + 𝜀0 (1)

𝑆𝑛 = 𝑠𝑛(ln(𝑦) , ln(𝑤)) + 𝜀𝑛 (2)

With:

C = total costs;

y = vector of outputs;

w = vector of input prices;

Sn = optimal cost share for input n (n = 1,.., N).

ε0, εn error terms

Equations (1) and (2) can be estimated by a certain minimum distance estimator or, if one wants to check for heterogeneity, with fixed or random effects, which will result in consistent estimates of the parameters if 𝐸[𝜀|𝑦, 𝑤] = 0. However, if some firms are inefficient, i.e. they have a higher cost than what can be explained, the cost function or random noise 𝐸[𝜀] > 0, causing biases in the parameters of Equations (1) and (2).

Our suggestion for reducing these biases consists of estimating Equations (1) and (2) with weighted least squares and assigning the ‘ill-behaving’ observations with a low weight, while the ‘well-behaving’ observations will be assigned with a high weight. Weighted least squares (WLS), which is also referred to as generalized least squares (GLS), is a widely used econometric technique; however, since the weights are generally not observable, they have to be estimated (see e.g. Verbeek, 2012). Our proposed weighting scheme is based on the residuals obtained after Equations (1) and (2) have been estimated with LSQ3, as we know that firms that are highly inefficient, and thus likely to bias the results, will have a large residual 𝜀, where 𝜀 is the estimate

of 𝜀.

3 If Equation (1) and (2) are estimated with fixed effects, the weights can also be based on the fixed effects, which would make our estimator to a generalized version of the estimator, suggested by Wagenvoort and Schure (2006).


Obviously, the way in which 𝜀 should be transformed into weights is as debatable as the distributional assumption for the efficiency component in SFA. The weighting scheme should reflect the tradeoff between noise and inefficiency. If one expects all firms to be efficient, the deviation from the frontier, captured by𝜀, ismostly determined

by noise. If a weighting scheme, reducing the weight rapidly if 𝜀 is increasing, is being used, the assessment of the level of the frontier will prove to be overly optimistic, since firms that perform very well due to luck will get a larger weight. On the other hand, if many firms are inefficient, while a weighting scheme that is virtually flat for all 𝜀 is being used, the estimation of the frontier will prove to be too low, since firms that effectively are very inefficient will still be considered to be quite efficient. One way to determine the amount of noise versus inefficiency is to examine the skewness of the LSQ residuals. It is easy to implement other weighting schemes and see if the results differ. This is another advantage of our approach over the SFA approach, which requires to calculate the convolution of two random variables and to derive the maximum likelihood, if one wants to use another distribution.

Since the weighting scheme depends on 𝜀,which can be updated after the second step, an iterative reweighted least squares procedure can be implemented. This procedure is used for some robust regression estimators, such as the Huber W estimator (Guitton, 2000). This similarity is no coincidence, since our suggested estimator can also be considered as a robust type of regression. This implies that, after each WLS estimation, new 𝜀s are calculated, which are then used to generate new weights, which, on their turn, are used in a next stage WLS estimation, until the convergence criterion upholds. The convergence criterion we use requires that the parameter estimates don’t differ more than one percent from the previous stage.

Normally, we also want to know the levels of inefficiency and not only the parameters of the cost function. In our suggested model, it is not obvious how these levels can be calculated since, for the calculation of E[𝜇|𝜀], where 𝜇 is the level of efficiency, we

need at least the probability density distributions of 𝜇 and 𝜀. However, our estimator

doesn’t require making an assumption for the distribution of 𝜇. This doesn’t mean that we are not able to say something about the efficiency of each firm.

Ondrich and Ruggiero (2001), for instance, show that, if a normal distribution is assumed for the noise, the ranking of 𝜀 is equal to the ranking of 𝜇 and therefore, our model enables us to specify the efficiency ranking for each firm.

Although the distributional assumptions about the efficiency term are not necessary for the estimation, we still might use them for deriving the efficiency scores. Therefore, we introduce the two usual unobservables u and v, representing the (in) efficiency and the error term, respectively. We simplify the original problem of Aigner et al. (1977), by only estimating the distribution of the error term, instead of both components simultaneously. Since we did identify the cost frontier, we are able to select a subsample of observations that satisfy u = 0, i.e. all observations with observed cost lower than or equal to frontier cost (v ≤ 0). Note that we are not able to identify observations that satisfy u = 0 and v ≥ 0, i.e. efficient firms with observed cost greater than frontier cost. Therefore, we assume that |v| in the subsample is distributed as 𝑁+(0, 𝜎𝑣

2). The variance 𝜎𝑣2 can now be estimated by the sum of


squared residuals, divided by the number of observations in the subsample (denoted as 𝜎𝑣

2). Furthermore, in the full sample, we assume that the subsample is representative with respect to the variance of the random errors and that random errors are distributed as 𝑁(0, 𝜎𝑣

2). Since we now have an estimate for the variance of the random errors, we are also able to conditionally derive the expected efficiency from the residuals, by applying, for instance, Materov’s formula:

𝑀(𝑢𝑖|𝜀) = 𝜀 𝜎𝑢2

𝜎𝜀2 if 𝜀 ≥ 0; = 0 otherwise (3)

with:

𝜎𝑢2 = 𝜎𝜀

2 − 𝜎𝑣2

The efficiency score then equals:

𝐸𝑓𝑓𝑖 = exp ( −𝑀(𝑢𝑖|𝜀) (4)

Obviously, there are other alternatives available (see e.g. Kumbakhar & Lovell, 2000). Note that, in comparison with the original Jondrow et al. (1982) paper, in the model, we have swapped the roles of the random error and efficiency components. It is important to stress that we don’t apply the distributional assumptions to the errors and efficiency components in the estimation procedure, but only in the derivation of the efficiency scores. However, this procedure has some consequences for the weighting scheme in the estimation procedure. Since we assume that all negative residuals indicate efficient firms and that the residuals only reflect random noise, there is no reason for assigning different weights to these observations; therefore, for reasons of consistency, we use a weighting scheme that assigns a weight equaling 1 to all the observations in the efficient subset.


Model specification

We apply the well-known translog cost function model (Christensen et al., 1973; Christensen & Greene, 1976). The cost function model consists of a translog cost function and the corresponding cost share equations. The model includes first and second order terms, as well as cross-terms between outputs and input prices, on the one hand, and a time trend, on the other hand. These cross-terms with a time trend represent the possible different natures of technical change. Cross-terms with outputs refer to output-biased technical change and cross-terms with input prices refer to input-biased technical change.


( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ε++

+++

+++

+++++=

∑∑

∑∑∑∑

∑∑ ∑∑∑∑

∑∑∑∑∑

==

= == =

= = = == =

= ====

N

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M

mmi

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o

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mmoij

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o

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n

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n

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m

N

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ooooonnnn

M

m

M

mmmmm

O

ooo

N

nnn

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mmm

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ThYZgWZf

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11

11

01 11 1

1 1' 1 110' 1''''

1 1'''

1110

lnln

lnln21lnln

lnlnlnln21lnln2

1

lnln21lnlnlnln

(1)

With:

C = total costs;

Ym = output m (m = 1,.., M);

T = year of observation;

Wn = price of input n (n = 1,.., N);

Zo = fixed input o (o = 1,.., O).

nmomonmnoonnmmonmo jihgfedcbdcba 110''' ,,,,,,,,,,,, parameters to be estimated.

By applying Shephard’s Lemma, we see that the optimal cost share functions can be presented as:

( ) ( ) ( )∑ ∑ ∑= = =

=⋅++++=N

n

M

m

O

oinoonmmnnnnnn NnTjZfYeWccS

1' 1 1' ),..,1(lnlnln

(2)

With:

Sn = optimal cost share for input n (n = 1,.., N)

Homogeneity of degree one in prices and symmetry is imposed by applying constraints to some of the parameters to be estimated. In formula:

oooonnnnmmmm ddccbb '''''' ;; ===

∑∑∑∑∑=====

=∀=∀=∀==N

nn

O

oon

N

nmn

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11

111'

10;)(0);(0);'(0;1

(3)

Equations (1) and (2) can be estimated by OLS or if one wants to control for heterogeneity, with fixed or random effects, which will result in consistent estimates of the parameters if 𝐸[𝜀|𝑌, 𝑊] = 0. Yet, if some firms are inefficient, i.e. they have higher

cost than what can be explained, the cost function or random noise 𝐸[𝜀] > 0, causing

a bias in 𝛼0. Moreover, if the input mix of a firm is partly determined by the firm’s

expectations of its efficiency level we even have 𝐸[𝜀|𝑌, 𝑊] ≠ 𝜇 where 𝜇 is a constant. This causes biases, also in the other parameters of Equations (1) and (2).

Data

The data for this study cover the period 2003-2009 and were obtained from the Dutch Hospitals Association. Annual financial, patient and personnel data were collected by means of surveys. The surveys cover all the general hospitals in the Netherlands. For


the purpose of this study, the data were checked for missing or unreliable data. Various consistency checks were performed on the data, in order to ensure that changes in average values and in the distribution of values across time were not excessive. In particular, observations with unit value (for instance the personnel costs per FTE for each type of personnel) less than 0.5 times or exceeding 2 times the median value were identified. After the elimination of observations, whose dataset contained inaccurate or missing values, we obtained an unbalanced panel dataset of 406 observations over the 5 years of study. There are approximately 80 observations for each year. The year 2005 has the highest coverage with 84 observations out of 89, 2007 has the lowest coverage with 75 observations out of 86.

The main service delivery of hospitals consists of the treatment of patients. Therefore, the output of hospitals is measured by the number of discharges, including day-care patients and outpatients (not followed by an admission). The discharges have been divided into over 30 medical specialties, in order to measure case-mix. Since it is not possible to use such a large number of categories, they have been aggregated into three categories, on the basis of average stay homogeneity and the distinction between surgery/non-surgery specialties. Therefore, we distinguish the following three groups of specialties:

• Surgery and Non-surgery with average stay less than 4 days;

• Non-surgery with average stay more than 4 days;

• Surgery with average stay more than 4 days.

This results in four types of output, three types of inpatients (including day-care patients) and outpatients. Although these four types of production explain a very large part – as we’ll see later – of variations in cost, the services are much more nuanced than just the number of outpatients and discharges. The health outcome of patients seems to be a particularly important component of hospital production. Nevertheless, it seems reasonable to assume that the quality has not decreased, as it is constantly monitored, for instance, by the health inspectorate, patient associations and media, and subjected to quality-improving interventions by physicians and hospital management. Therefore, the estimates of productivity change can be regarded as a lower bound.

Resources include staff, administrative and maintenance personnel (including security and cleaning), nursing personnel, paramedical personnel (such as lab technicians), material supplies, maintenance and capital. Physicians are not included in these personnel variables, in order to ensure that hospitals with hospital-employed physicians and hospitals with self-employed physicians are treated equally. The costs of physicians (wages) are not included in the cost or price variables either.

Material supplies include items such as medical supplies, food and general cost. Maintenance includes energy costs, costs related to grounds and buildings. The maintenance costs are rather low and have, for reasons of simplicity, been added to the material supplies. Personnel and material supplies are treated as variable resources, since the hospital can change these in the short term.

Capital refers to capital assets, such as buildings and medical equipment. The volume of the capital is measured as a weighted aggregate of beds, intensive care beds,


psychiatric beds, square meters and number of radiotherapists (a proxy for the number of linear accelerators and cobalt units).

There are data on the costs and the quantity for each resource personnel category. For each region and time period, wages are defined as the average wage per full time equivalent. This is considered to be the market price for labour; qualitative differences between hospitals are included in the volume of labour.

Since for material supplies, there is no natural unit of measurement, they are presented by means of a circumventing concept. The price of material supplies is a weighted index, based on components of the consumer index, calculated for the Netherlands by Statistics Netherlands. The weights are derived from cost shares.

The price of capital is defined as a unit value, derived from capital costs and the aforementioned volume of capital.

Estimation results

The models are estimated as multivariate regression systems with various equations with a joint density, which we assume to be a normal distribution. Because disturbances are likely to be cross-equation-correlated, Zellner’s Seemingly Unrelated Regression method is being used for estimation (Zellner, 1962). As usual, as the shares add up to one, causing the variance–covariance matrix of the error terms to be singular, one share equation in the direct cost function model is eliminated. Since we are dealing with a relatively large number of cross-sectional units and a limited number of periods, we ignore the fact that we are dealing with a panel data (with respect to intra-firm correlations). It is obvious that the between variance is far more important than the within variance.

In our analysis, we use the following weighting scheme:

𝑤 = 1

1+ 𝜀𝜎𝐿𝑆𝑄

if 𝜀 > 0, else 𝑤 = 1 (4)

With:

𝜎𝐿𝑆𝑄 = the standard deviation of the LSQ residuals.

An interesting aspect of this weighting scheme is that the weights are directly related to the efficiency scores. Efficient firms have weights equal to 1 and inefficient firms have efficiency scores equaling the weights multiplied by a constant (equal to the ratio of variances).

However, it is easy to implement other weighting schemes and see if the results differ. This is another advantage of our approach over the SFA approach, which requires to calculate the convolution of two random variables and to derive the maximum likelihood, if one wants to use another distribution. As it turns out, our results were quite robust for another weighting scheme, based on rank numbers. In case of IWLS estimation, the procedure will stop if the maximum change in the parameters is less than 1%. The number of iterations in our application equals 12. Besides the imposed


theoretical requirements, there are a few others requirements that also have to be fulfilled, such as monotonicity and concavity in resource prices (Färe & Primont, 1995). These requirements can be tested posteriorly. An estimated cost function is monotonic in resource prices, if the fitted cost shares are positive. A necessary condition for concavity of the cost function is that the own partial elasticities of substitution are less than zero for all resources; a sufficient condition is that the matrix of partial elasticities of substitution is negative semi-definite. The matrix is negative semi-definite if all eigenvalues are less than or equal to zero. These requirements are tested for the average firm.

Figure 2 Estimates frontier cost function by SUR and IWLS

Variable Parameter Esti-mate

St. Error T-value Esti-mate St. Error T-value

LSQ-estimates IWLS-estimates

Constant A0 0.150 0.022 6.849 0.070 0.017 4.208

Year=2004 A2 -0.050 0.010 -4.901 -0.043 0.008 -5.392

Year=2005 A3 -0.081 0.012 -7.003 -0.069 0.009 -7.871

Year=2006 A4 -0.112 0.013 -8.506 -0.094 0.010 -9.350

Year=2007 A5 -0.142 0.015 -9.373 -0.122 0.012 -10.520

Year=2008 A6 -0.162 0.016 -9.833 -0.137 0.012 -11.064

Year=2009 A7 -0.186 0.018 -10.260 -0.165 0.014 -11.547

Discharges 1 B1 0.228 0.044 5.216 0.187 0.032 5.845

Discharges 2 B2 0.519 0.050 10.309 0.497 0.043 11.612

Discharges 3 B3 0.190 0.058 3.273 0.239 0.046 5.221

Discharges 4 B4 0.293 0.037 7.814 0.307 0.027 11.562

Discharges 1 x discharges 1 B11 -0.157 0.101 -1.550 0.006 0.086 0.070

Discharges 1 x discharges 2 B12 0.031 0.110 0.280 0.074 0.099 0.746

Discharges 1 x discharges 3 B13 -0.045 0.117 -0.390 -0.213 0.098 -2.165





Discharges 3 x discharges 3 B33 0.000 0.237 -0.001 0.384 0.176 2.184



Price management C1 0.095 0.006 15.648 0.095 0.006 16.591

Price nursing C2 0.342 0.008 42.832 0.344 0.007 46.729

Price medical C3 0.041 0.003 14.388 0.037 0.003 14.079

Price support C4 0.096 0.007 14.450 0.094 0.006 16.306

Price materials C5 0.292 0.005 57.940 0.291 0.005 62.965

Price capital C6 0.134 0.002 64.125 0.139 0.002 74.551

Price management x price management

C11 -0.016 0.026 -0.627 -0.063 0.025 -2.518

Price management x price nursing

C12 -0.005 0.029 -0.169 0.014 0.028 0.486

Price management x price medical

C13 0.013 0.008 1.555 -0.009 0.008 -1.130

Price management x price support

C14 0.023 0.028 0.795 0.027 0.024 1.128


Price management x price materials

C15 -0.042 0.032 -1.320 -0.028 0.028 -0.985

Price management x price capital

C16 0.027 0.020 1.334 0.060 0.018 3.383

Price nursing x price nursing C22 0.124 0.066 1.891 0.072 0.060 1.207

Price nursing x price medical C23 -0.016 0.012 -1.292 -0.002 0.012 -0.147

Price nursing x price support C24 -0.067 0.051 -1.320 -0.062 0.043 -1.435

Price nursing x price materials

C25 0.020 0.053 0.376 0.077 0.047 1.654

Price nursing x price capital C26 -0.056 0.034 -1.670 -0.099 0.028 -3.459

Price medical x price medical C33 -0.018 0.006 -2.766 -0.015 0.006 -2.496

Price medical x price support C34 0.031 0.011 2.913 0.032 0.010 3.293

Price medical x price materials

C35 -0.027 0.013 -1.988 -0.009 0.012 -0.714

Price medical x price capital C36 0.016 0.007 2.403 0.003 0.006 0.522

Price medical x price support C44 -0.016 0.058 -0.277 -0.057 0.047 -1.232

Price medical x price materials

C45 0.036 0.050 0.711 0.043 0.041 1.038

Price medical x price capital C46 -0.006 0.039 -0.157 0.017 0.032 0.547

Price materials x price materials

C55 0.064 0.053 1.198 -0.026 0.049 -0.523

Price materials x price capital C56 -0.051 0.003 -14.844 -0.058 0.003 -17.838

Price capital x price capital C66 0.071 0.002 29.539 0.076 0.002 35.626

Radiology D1 0.034 0.007 5.220 0.031 0.005 6.320

Radiology x radiology D11 0.015 0.004 3.557 0.012 0.003 4.013

Discharges 1 x price management

E11 0.004 0.003 1.377 0.004 0.003 1.215

Discharges 1 x price nursing E12 0.006 0.005 1.207 0.008 0.005 1.834

Discharges 1 x price medical E13 -0.001 0.003 -0.269 -0.002 0.002 -0.767

Discharges 1 x price support E14 -0.022 0.004 -5.646 -0.022 0.003 -6.565

Discharges 1 x price materials E15 0.012 0.005 2.350 0.016 0.004 3.598

Discharges 1 x price capital E16 0.000 0.003 0.059 -0.005 0.002 -1.971


E21 -0.011 0.005 -2.353 -0.013 0.005 -2.938

Discharges 2 x price nursing E22 -0.029 0.007 -4.045 -0.017 0.007 -2.588

Discharges 2 x price medical E23 0.016 0.004 4.347 0.008 0.003 2.542

Discharges 2 x price support E24 0.027 0.006 4.749 0.024 0.005 4.835

Discharges 2 x price materials E25 0.021 0.007 2.845 0.018 0.007 2.777

Discharges 2 x price capital E26 -0.023 0.004 -5.976 -0.019 0.003 -5.637


E31 -0.001 0.005 -0.155 0.002 0.005 0.367

Discharges 3 x price nursing E32 0.024 0.007 3.259 0.008 0.007 1.227

Discharges 3 x price medical E33 -0.005 0.004 -1.223 0.006 0.003 2.073

Discharges 3 x price support E34 -0.021 0.006 -3.681 -0.013 0.005 -2.502

Discharges 3 x price materials E35 -0.002 0.008 -0.217 -0.010 0.007 -1.483

Discharges 3 x price capital E36 0.004 0.004 1.018 0.006 0.004 1.733


E41 0.014 0.004 3.249 0.013 0.004 3.298

Discharges 4 x price nursing E42 -0.004 0.007 -0.641 0.004 0.006 0.679

Discharges 4 x price medical E43 0.008 0.003 2.473 0.007 0.003 2.440

Discharges 4 x price support E44 0.005 0.005 0.934 0.000 0.004 -0.079

Discharges 4 x price materials E45 -0.038 0.007 -5.635 -0.039 0.006 -6.778

Discharges 4 x price capital E46 0.015 0.004 4.220 0.016 0.003 5.239


Radiology x price management

F11 0.000 0.000 0.158 0.000 0.000 -0.115

Radiology x price nursing F12 -0.003 0.001 -4.114 -0.004 0.001 -7.439

Radiology x price medical F13 0.000 0.000 1.189 0.001 0.000 3.064

Radiology x price support F14 -0.001 0.000 -1.388 0.000 0.000 -0.037

Radiology x price materials F15 0.002 0.001 3.085 0.002 0.001 3.968

Radiology x price capital F16 0.001 0.000 2.448 0.001 0.000 4.395

Discharges 1 x discharges 1 G11 0.020 0.012 1.696 0.002 0.008 0.231

Radiology x discharges 2 G12 -0.012 0.014 -0.855 -0.010 0.012 -0.905

Radiology x discharges 3 G13 0.026 0.014 1.813 0.033 0.012 2.843

Radiology x discharges 4 G14 -0.031 0.011 -2.682 -0.020 0.009 -2.297

Time x price management J11 0.003 0.032 0.102 0.001 0.035 0.034

Time x price nursing J12 0.128 0.046 2.755 0.153 0.048 3.156

Time x price medical J13 0.003 0.016 0.196 -0.008 0.017 -0.479

Time x price support J14 0.041 0.035 1.155 0.029 0.035 0.822

Time x price materials J15 -0.245 0.040 -6.189 -0.275 0.039 -7.007

Time x price capital J16 0.070 0.015 4.555 0.100 0.016 6.274

Table 1 shows that most parameter estimates are significant at the 5% level. For most variables, the estimated parameters also obtain the expected signs. We have calculated the theoretical conditions for monotonicity and concavity for the average firm. Since the fitted cost shares are positive for the average firm, the theoretical condition for monotonicity is satisfied for all inputs. A necessary condition for concavity of the cost function is that the own partial elasticities of substitution are less than zero for all inputs. This condition also upholds for all inputs. A sufficient condition is that the matrix of partial elasticities of substitution is negative semi-definite. This condition unfortunately does not uphold, since one of the eigenvalues is (slightly) positive. All other eigenvalues are negative. Therefore, the sufficient condition is too tight. These statements uphold for the outcomes of both estimation procedures.

Comparing the outcomes of the plain LSQ estimates and the iterative weighted LSQ, a number of the estimated parameters are quite similar. Especially the estimates of the parameters, corresponding to input prices and fixed resources, show great similarities. On the other hand, there are also a few striking differences, in particular in respect of the trend parameters. The IWLS estimated parameters a2-a7, representing the frontier shift from year to year, are lower than the parameter estimates from the plain LSQ estimation, implying that technical change is slower in comparison to the average cost function, which may also take account of some cost efficiency changes. The parameters, corresponding to the services produced, also show some substantial differences. However, the calculated cost flexibilities for the average firm are identical

up to the third decimal ( 230.1=∑ mb ). Bigger differences can be found between the

parameters of the cross-terms of services produced. However, the LSQ estimates (and partly also the IWLS-estimates) are rather unreliable. One of the most striking results is that, apart from very few exceptions, the parameter estimates according to the IWLS-estimation are far more efficient.


In order to underline the plausibility of the estimates, we derive a few other economically relevant outcomes. The first relevant outcome concerns the cost efficiency scores. Figure 2 shows the distribution of the efficiency scores in 2009, based on the IWLS estimation.

Figure 3 Distribution of cost efficiency scores, 2009

Figure 2 shows that, in 2009, approximately one quarter of the hospitals were (almost) efficient. Furthermore, the inefficient hospitals show a plausible pattern of inefficiencies. The average efficiency equals 95% with a standard deviation of 5%. The minimum efficiency score equals 81%. When comparing efficiency scores between the years, it appears that they are very robust (not presented in the figure). In 2003, the average efficiency is a little bit lower (94%) and in 2008, it is a little bit higher (96%).

One of the serious drawbacks of the thick frontier approach is that it requires sampling from a stratified sample. Since, in this procedure, we haven’t stratified the sample at all, it is questionable whether, regardless of certain characteristics, each hospital has an equal probability of being identified as an efficient hospital. Obvious characteristics that may affect the probability of being (in)efficient are the size and the year. Therefore, we inspect the distribution of the efficiency scores, related to year and size. Figure 3 reflects the number of efficient hospitals in each year of the sample.


Figure 4 Number of efficient hospitals by year

Figure 3 shows that the final selection of efficient hospitals is quite uniformly distributed over the years, varying between 18 and 29. This shows that the procedure does not tend to favour a particular year.

Another potential selection bias may occur with respect to the size of the hospitals. Figure 4 reflects the frequency distribution with respect to the size (divided in four quartiles with respect to the number of beds).

Figure 5 Number of efficient hospitals by size


Figure 4 also shows that all the size categories are well represented by a substantial number of efficient firms, although there seems to be a tendency for small hospitals to be somewhat overrepresented and for large hospitals to be underrepresented.

Conclusions

This paper proposes an alternative class of stochastic frontier estimators. Instead of making distributional assumptions on the error and efficiency component in the econometric specification of a cost function model (or any other model), this class is based on the idea that some observations contain more information about the true frontier than others. If an observation is likely to contain much information, it will be assigned a large weight in the regression analysis. In order to establish the weights, we propose an iterative procedure. Since no a priori information is available, the first step consists of running a standard Least Squares method. Subsequently, weights can be determined by the residuals obtained and a user-specified weighting function. The weights obtained allow for Weighted Least Squares (WLS) to be applied. Since the WLS residuals will differ from the LSQ residuals, new weights will be determined by means of an iterative procedure. In each step, the weights will be updated and a new WLS regression will be estimated. Since the negative residuals, by definition, represent the error component, the variance of these errors can easily be calculated and used as an estimator of the variance of the normal distribution of the noise. Similar to SFA, (expected) inefficiency and noise can be derived for all the other observations. The iterative procedure stops as soon as the change in the parameters between two iterations is less than a given threshold value.

The advantages of this approach are its high transparency, the easy application to a full-specified model and its flexibility. It allows to directly observe which observations determine the frontier for a large part. The easy expansion to a full-specified model refers to a model that includes a cost function and its corresponding share equations. Its flexibility refers to the use of several alternative weighting functions and the easiness of testing for the sensitivity of the outcomes.

The model is applied to a set of Dutch hospital data (including about 550 observations). The outcomes of this application are promising. The model converges rather quickly and presents reliable estimates for the parameters, the cost efficiencies and the error components. About 25% of the hospitals are designated as efficient. The average efficiency score is approximately 93%.

References

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20. Least distance efficiency measures satisfying strong monotonicity on the efficient frontier

Hirofumi Fukuyama Fukuoka University, Japan

Kazuyuki Sekitani Shizuoka University, Japan

Jianming Shi Muroran Institute of Technology, Japan,

Abstract

In this study, we introduce and investigate least distance p-norm efficiency measures that satisfy strong monotonicity on the efficient frontier.

Keywords: DEA, least distance measures, p-norm, strong monotonicity, shadow profit-based dominated set, free disposal input-output set, efficient frontier

Introduction

In data envelopment analysis (DEA), mathematical programming is applied to observed input-output data in order to assess the efficiency performance and provide target information for managers of entities such as banks, hospitals and business units. Each entity responsible for transforming multiple inputs to multiple outputs is called a decision-making unit (DMU). In DEA, there are basically two frameworks for the efficiency assessment and targeting: the greatest and the least distance frameworks. Greatest distance measures generally provide efficiency targets that are obtained as the farthest projections from the DMU to be assessed. Such measures include Tone’s (2001) slacks-based measure of efficiency (SBM) and Cooper et al.’s (1999) range-adjusted measure of efficiency (RAM). These greatest distance efficiency measures and projections are often used because of their computational ease. However, closest or least distance projections are often more relevant than greatest distance projections from the perspective of private or public managers. This is so because closer efficiency targets may be reached with less effort. In a p-norm framework, RAM and SBM are considered to be greatest distance 1-norm measures, i.e., they are obtained

by maximizing the 1-norm from the DMU being evaluated to the efficient frontier E .

It is well-known that the standard measures of CCR (Charnes-Cooper-Rhodes) and BCC (Banker-Charnes-Cooper) do not satisfy strong monotonicity. This is due to the fact that these measures identify improvement targets on the weakly efficient frontier


WE of the production possibility set, i.e., the one-step procedure of CCR and BCC

may identify targets on WE , but not on E, where WE E⊆ . CCR and BCC require two

steps to identify an efficiency target: measuring an efficiency score in the first step and finding the target on E in the second step. However, these measures are not strongly monotonic.

Briec (1998) proposed a family of least distance inefficiency measures under an arbitrary p-norm, [1, ]p∈ ∞ , and showed that the p-norm least distance efficiency

measure is reduced to the CCR measure when p is equal to positive infinity. Therefore, Briec’s (1999) inefficiency measure satisfies weak monotonicity. His inefficiency measure may find an efficiency target on the weakly efficient portion of the boundary.

Building upon the previous contributions, the present study develops least distance p-norm inefficiency/efficiency measures that satisfy not only strong monotonicity over the efficient frontier, but also some other desirable properties. For the development, we exploit and extend an input-output set dominated by the evaluated DMU’s (decision-making unit’s) input-output vector. This dominated set was utilized to define non-radial supper-efficiency measures by Cooper et al. (2007), and was also utilized to define a family of least distance p-norm inefficiency measures satisfying weak monotonicity over the efficient frontier (Ando et al. 2012).

2. Axiomatic approach to least distance based DEA

2.1 Weak monotonicity

Let LR+ be the L -dimensional nonnegative orthant, and let LR++ be the L -dimensional

positive orthant. We assume that there are J decision-making units (DMUs), each

DMUj ( = 1, , )j J of which transforms N different inputs Nj R+∈x into M different

outputs Mj R+∈y . We assume that the production technology available to the DMUs

can be characterized by the production possibility set:

( , ) | can produce .N MT R ++≡ ∈x y x y (1)

In this research we utilize two DEA representations of T: the constant returns to scale reference technology expressed by

=1 =1

( , ) , , ,J J

c N Mj j j j

j jT R λ λ+

+

≡ ∈ ≤ ≥ ≥

∑ ∑x y x x y y λ 0 (2)

and the variable returns to scale reference technology expressed by

=1 =1 =1

( , ) , , = 1, ,J J J

v N Mj j j j j

j j jT R λ λ λ+

+

≡ ∈ ≤ ≥ ≥

∑ ∑ ∑x y x x y y λ 0 (3)

where 0 is a zero vector with appropriate dimension. The weakly efficient frontier or

boundary of (1) is defined by


( , ) | ( , ) < ( , ) ( , ) .WE T T′ ′ ′ ′≡ ∈ − − ⇒ ∉x y x y x y x y (4)

From this definition, the weakly efficient frontier is the set of all the inputs and outputs that are not strongly dominated. The efficient frontier of (1) is defined by

( , ) ( , )

( , ) ( , ) .( , ) ( , )

E T T′ ′ − ≤ −

′ ′≡ ∈ ⇒ ∉ ′ ′− ≠ −

x y x yx y x y

x y x y (5)

A real valued function ( , )f x y defined on T is an inefficiency measure. The

desirable properties that ( , )f x y should have are expressed as follows:

Axiom A: ( , ) E∈x y if and only if ( , ) 0f =x y .

Axiom B. For any ( , ) T∈x y , 0 ( , )f≤ x y .

Axiom C. For ( , )a a T∈x y and ( , )b b T∈x y , ( , ) ( , )a a b b− ≥ −x y x y and

( , ) ( , )a a b b− ≠ −x y x y imply ( , ) > ( , )a a b bf fx y x y .

In this paper, we discuss efficiency measures with respect to p-norm p

⋅ defined by

1/

1if [1, )

max , , if

pn p

ll

p

l l

z p

z z p

=

∈ ∞ =

= ∞

∑z

(6)

Armed with (6), we define a general least distance p-norm measure by

( , ) min ( , ) ( , ) ( , )ppf E′ ′ ′ ′≡ − ∈x y x y x y x y (7)

which is an inefficiency measure that identifies the closest point on E from an activity

( , )x y . Pastor and Aparicio (2010) showed that an efficiency measure defined by

21 min ( , ) ( , )Z E′ ′ ′ ′− − − ∈x x y y x y (8)

is not a strongly monotonic efficiency measure that seeks closest points over the

efficient frontier E . Recently, Ando et al. (2012) considered the following axiom

(Axiom C′ ) by relaxing Axiom C.

Axiom 'C : For ( , )a a T∈x y and ( , )b b T∈x y , ( , ) ( , )a a b b− ≥ −x y x y implies

( , ) ( , )a a b bf f≥x y x y .

Ando et al. (2012) developed a least distance inefficiency measure satisfying weak

monotonicity over E by modifying (7) as follows:


( ) , min ( , ) ( , ) ( , ) , ( , ) ( , )pp

f E D′ ′ ′ ′= − ∈ ∈x y x y x y x y x y x y, (9)

where ( , ) = ( , ) ( , ) ( , ) .D − ≥ −x y x y x y x y (10)

The situation where an activity ( , )′ ′x y belongs to ( , )D x y means that ′x can product

′y if ( , )x y is feasible. Hence, ( , )D x y represents a free disposal input-output set of

an activity ( , )x y . ( , )D x y is utilized to define a slacks-based supper-efficiency

measure in Cooper et al. (2007, p. 313). By incorporating ( , )D x y into

min ( , ) ( , ) ( , )p E′ ′ ′ ′− ∈x y x y x y , Ando et al. (2012) shows that pf , defined in (9), is

weakly monotonic over E for all [1, ]p∈ ∞ .

2.2 An extension of ( , )D x y

This study extends (9) to a least distance efficiency measurement setting, in which strong monotonicity is satisfied, i.e., all of Axioms A, B and C are satisfied. For this

purpose, we replace ( , )D x y of (10) by by ( , )Dε x y with the following ε ( 0≥ )

dependent set of ( , )x y :

1( , ) = ( , ),1

( , ) = ( , ) , ( )0 ( )

1

x y

x

y

D IIε

ε εε

εεε

ε ε

+ − = ≤

x y x d y dx y x y d

d

. (11)

In what follows, we show the profits associated with any activities in ( , )Dε x y do not

exceed the profit of ( , )x y . That is, we utilize shadow prices of inputs and outputs of

a DMU in place of observed prices to interpret ( , )Dε x y where ( , )x y attains the

maximum shadow profit. Therefore, we call ( , )Dε x y the shadow profit-based

dominated set.

( , )Dε x y extends ( , )D x y by expanding the latter by a fixed ε . Let ( , )pfε x y be

min ( , ) ( , ) ( , ) , ( , ) ( , )p

E Dε′ ′ ′ ′− ∈ ∈x y x y x y x y x y , (12)

then it follows from 0 ( , ) = ( , )D Dx y x y that 0 ( , ) = ( , )p pf fx y x y and hence, ( , )pfε x y

is a natural extension of (9).

Now, let us explain how ( , )Dε x y and prices in dual space are related to ε . For

this purpose, we define, for a given 0>ε , the following set of allowable multiplier weights (shadow prices):


( ) ( ) ( ) ( )

1 1

, 1, , , , 1, ,1 1 1 1,

1

n m

N Mn mn m

v n N w m MN M N M

v w

ε εε εε

= =

≥ = ≥ = + + − + + −Ω ≡

+ = ∑ ∑v w

(13)

We assume each component of ( ),v w in (13) is positive. The following proposition

establishes an equivalent relationship between ( , )Dε x y and ( )Ω ε ．

Proposition 1: ( ) ( ), ,Dε∈ ⇔ − ≥ −x y x y wy vx wy vx ( ) ( ),∀ ∈Ω εv w .

Proposition 1 indicates that, in ( ),Dε x y , there does not exist an activity, whose profit

exceeds that of the activity ( , )x y for any price vector ( ),v w in ( )Ω ε . Therefore, any

activities in ( ),Dε x y are no better than ( , )x y from the viewpoint of shadow profit

values as far as ( ),v w is in ( )Ω ε . At this point, it is appropriate to state that we have

restricted the existence region of allowable shadow prices to ( )εΩ , and thus we need

to explicitly specify ( )εΩ in relation to each fully efficient DMU. We suggest

constructing ( )εΩ so that ( ), T∈x y satisfies the following condition:

( ) ( ) ( )( ) ( ), , such that 0 , ( )

, such that 0 ( )

E a

b

∈ ⇔ ∃ ∈Ω = −

∀ ∈Ω ≥ −

x y v w wy vx

v w wy vx

ε

ε (14)

It is known that, for any efficient activity ( , ) E∈x y , there is at least one positive

shadow price vector ( ),v w satisfying ( ), T− ≥ − ∀ ∈wy vx wy vx x y . Furthermore,

since the efficient frontier is the union of finitely many faces, we can limit to finitely many positive shadow price vectors for each efficient activity. Consequently, if we

select a sufficiently small 0ε > , then the existence of ( )εΩ is guaranteed. The

collection of shadow price vectors contains at least one positive price vector ( ),v w

satisfying (a) in (14) for any efficient activity ( , ) E∈x y . That is, ( )εΩ contains

information on shadow prices enough to describe the efficient frontier E . For all

shadow price vectors ( ) ( ), ∈Ω εv w , any activities ( ),x y in ( ),Dε x y cannot have

higher shadow profits than the evaluated DMU ( , )x y . Mathematically, we can

express this situation as follows:

( ) ( )

( ) ( ) ( )

( ) ( )( ) ( )

( ), ,

, , ,,

max 0 , , ,

max 0 ,D

D

T

Eε

εε

ε∈

∈∈Ω

− = − ≤ ∀ ∈Ω ∀ ∈

− = ∀ ∈

x y x y

x y x yv w

wy vx wy vx v w x y

wy vx x y


3. Least distance inefficiency measure satisfying strong monotonicity

Consider two cones translating the origin to ( , )x y , ( , )D x y and ( , )Dε x y , that satisfy

ˆ ˆ( , ) ( , )D∈x y x y ˆ ˆ( , ) ( , ) ( , )D intDε ε⇒ ⊆x y x y x y (15)

for any positive ε , where intS is the interior of S .

Lemma 1: For any > 0ε , (15) holds.

Lemma 2: Choose ( , ) T∈x y and > 0ε arbitrarily and let * * * *( , , , )′ ′x y x y be an

optimal solution of (12). If ( , ) =E Dε∩ ∅x y or ( , ) = ( , )E Dε∩ x y x y , then

* *( , ) ( , ).intDε∉x y x y (16)

Lemma 3: Choose ( , ) T∈x y and > 0ε arbitrarily and let ˆ ˆ( , ) ( , ) \ ( , )D∈x y x y x y .

If ( , ) =E Dε∩ ∅x y or ( , ) = ( , )E Dε∩ x y x y , then

ˆ ˆ( , ) < ( , ) for all [1, ].p pf f pε ε ∈ ∞x y x y (17)

The inequality (17) means that ( , )pfε x y satisfies strong monotonicity over E under

the assumption ( , ) ( , )E Dε∩ ⊆x y x y . For a given positive value ε , any [0, )∈ε ε

and the cone ( , )Dε x y satisfy the two following conditions:

( , ) ( , ) = ( , )E D Eε∈ ⇒ ∩x y x y x y (18)

( , ) \ ( , ) = .T E D Eε∈ ⇒ ∩ ∅x y x y (19)

The existence and choice of the positive value ε were discussed and provided in our previous work (Fukuyama and Sekitani 2012) in a different context. The existence of

> 0ε guarantees Axiom C as well as Axiom A. For any face F of T , we define

1 =1= 1, = 1, ,

( ) ( , ) .= ( , )

N M

n m j jn m

v w j JVW F

F=

+ ≥ ∀ ≡

∀ ∈

∑ ∑ vx wyv w

vx wy x y

(20)

A face F is called a maximal efficient face if F E⊆ and a face G with G F∩ ≠∅

and G E⊆ implies G F⊆ . Since E consists of a finite number of maximal efficient

faces, we have L maximal efficient faces, say 1, , LF F , such that =1= Ll lE F∪ . For any

( , ) E∈x y there exists an index 1, , l L∈ such that ( , ) lF∈x y . Consider ( , ) T∈x y .

There exists ( , ) ( )lVW F∈v w such that >vx wy if and only if ( , ) lF∉x y . Hence,

( , ) \T E∈x y if and only if for every = 1, ,l L there exists ( , ) ( )lVW F∈v w satisfying


>vx wy . Let

*1 1

=1, , ( , ) ( )= min , , , , ,maxmin N M

l L VW Fl

v v w w∈v w

η . (21)

It follows from 1 1min , , , , , 1/ ( )N Mv v w w N M≤ + , the definition of E and Motzkin's

theorem of the alternative that *0 < 1/ ( )N M≤ +η . Let the maximal ε be obtained by

( ) 1* 1= 1N Mε η−− − − + , then we have 0 < 1≤ε .

Figure 1 shows a graphical representation of two dominated sets of ( ),x y ,

represented by ( , )D x y and ( , )Dε x y . ( )B B,N x y and ( )C C,N x y are normal cones of

( )B B,x y and ( )C C,x y , respectively. The set of all allowable shadow prices ( )εΩ is

shown in the shaded area in two places. ε is chosen so that ( )B B,N x y and

( )C C,N x y are contained in ( )εΩ . The shadow profit-based dominated set ( , )Dε x y ,

which is a proper superset of ( , )D x y for any 0ε > and is shown as the area

described by two heavy broken lines, can be constructed from the set ( )εΩ . For

example, 3 3( , )D x yε is obtained by moving the normal cone of ( )εΩ to 3 3( , )x y .

Figure 1. Two Dominated Sets of ( , )x y

Lemma 4: Choose ( , ) T∈x y and (0, )∈ε ε arbitrarily and let ( , ) ( , )Dε∈x y x y , then

for every = 1, ,l L , there exists ( , ) ( )l llVW F∈v w such that

0.l l l l− ≥ − ≥v x w y v x w y (22)

Moreover, = 0l l−v x w y if and only if ( , ) = ( , ) lF∈x y x y .

Lemma 5: Choose ( , ) T∈x y and (0, )∈ε ε arbitrarily. If ( , ) E∈x y , then

( )E E,D x y

( )E E,Dε x y

( )C C,Dε x y

( )B B,Dε x y( )C C,N x y

( )Ω ε ( )B B,N x y

1x

2x


( , ) = ( , )D Eε ∩x y x y . If ( , ) \T E∈x y , then ( , ) = .D Eε ∩ ∅x y

Proposition 2: Choose (0, )∈ε ε and [1, ]p∈ ∞ arbitrarily. Then, ( , )pfε x y satisfies

strong monotonicity over E .

4. Summary and Conclusions

The present paper has developed a least distance p-norm efficiency (inefficiency) measure satisfying strong monotonicity over the efficient frontier as well as efficiency indication, zero-unity requirement, units invariance and translation invariance. For the development of our measures, we exploit the shadow profit-based dominated set, which has the interpretation of shadow prices and profits for the DMU being evaluated. The shadow profit-based dominated set is an extension of the free disposal input-output set that is utilized in the development of super-efficiency measures. The fundamental difference between the two dominated sets is that the use of the first set enables us to construct strongly monotonic least distance inefficiency measures, whereas the use of the latter only guarantees weak monotonicity.

References

Ando K., A. Kai, T. Maeda, K. Sekitani (2012) Least Distance Based Inefficiency Measures in the Pareto-Efficient Frontier in DEA, Journal of the Operational Research Society of Japan 55: 73-91.

Briec, W. (1998) Hölder Distance Function and Measurement of Technical Efficiency, Journal of Productivity Analysis 11: 111-130.

Cooper W.W., K. S. Park, d J. T. Pastor (1999) RAM: A Range Adjusted Measure of Inefficiency for Use with Additive Models and Relations to Other Models and Measures in DEA, Journal of Productivity Analysis 11: 5-42.

Cooper W.W., L.M. Seiford, K. Tone (2007) Data Envelopment Analysis, A Comprehensive Text with models, applications, references and DEA-Solver Software, second ed., Springer, New York

Fukuyama H., K. Sekitani (2012) An Efficiency Measure Satisfying the Dmitruk-Koshevoy Criteria on DEA Technologies, Journal of Productivity Analysis 38: 131-143.

Pastor J.T., J. Aparicio (2010) The Relevance of DEA Benchmarking Information and the Least-Distance Measure: Comment, Mathematical and Computer Modelling 52: 397-399.


21. Maximal Allocated Benefit and Minimal Allocated Cost and its Application

Mozhgan Mansouri Kaleibar* Young Researchers Club, Tabriz Branch, Islamic Azad University, Tabriz, Iran, [email protected]

Sahand Daneshvar Tabriz Branch, Islamic Azad University, Tabriz, Iran, [email protected]

Abstract

In this paper, we investigate the problems of consensus-making among institution in stock exchange with multiple criteria for evaluating performance when the players (institutions) are supposed to be egoistic and the score for each criterion for a player is supposed to be a positive score. Each player sticks to his superiority regarding the criteria. This paper introduces the models for computing minimal cost ratio and the maximal benefit for institutions.

Keywords: Cooperative Game, DEA, Game Theory, Stock Exchange, Weight selection.

Introduction

Let us suppose n players each have m criteria for evaluating their competency or ability, which is represented by a positive score for each criterion. As with usual classroom examination, the higher the score for a criterion is, the better player is judged to perform that criterion. For example, there are three students A, B and C, with three criteria of linear algebra, real analysis and numerical analysis. The scores are their records for the three subjects, measured by positive cardinal numbers. All players are supposed to be selfish in the sense that they insist on their own advantage on the scores. Similar situations exist with many societal problems. We now present some of the potential applications of DEA game. In the literature on cooperative game theory, there have been many applications to cost or benefit sharing problems. The proposed DEA game models are in sharp contrast to them, in that we can deal with these problems under multi – criteria environments that are common to real conflicts in our society. This paper through allocating and imputing the given benefit [6] propose a new scheme for computing maximal allocated benefit and minimal allocated cost for the institutions under the framework of game theory and data envelopment analysis (DEA). The different sections of this paper are sequenced in the following order:



In section 2, the basic models are described and some properties of problem are proved. In section 3, the extensions to the basic model and computing the ratios by proposed models are discussed. In section4, a numerical presentation of real data from a Stock Exchange of Tehran is elaborated on. Finally, section 5 includes conclusions and some remarks.

Basic models of the game

In this section, we introduce the basic models and structures of the game.

Selfish behavior

Let ( ) m nijX x R ×

+= ∈ be the score matrix, scores of player j in the criterion i for

1,...,i m= and 1,...,j n= and 0ijx > . It is assumed that the higher score for a criterion

is, the better player is judged to perform as regard to that criterion. Each person k has a right to choose two sets of nonnegative weights 1( ,..., )k k k

mw w w= to the criteria that

are most preferable to the player. Using the weight kw , the relative scores of player k to the total score

are defined as follows:

1

1 1

( 1 )( )

mk

i iki

m nk

i iji j

w x

w x

=

= =

∑

∑ ∑

The denominator represents the total score of all players as measured by player k s weight selection, while the numerator indicates player k self evaluation using the same weight selection. Hence, the expression (1) demonstrates player k relative importance

(share) under the weight (or value) selection kw . We assume that the weighted scores are transferable. Player k wishes to maximize this ratio by selecting the most preferable weights, thus resulting in the following fractional program.

( )

1

1 1

( 2 )( )

. 0

k

mk

i iki

m nw ki ij

i j

ki

w xMax

w x

s t w i

=

= =

≥ ∀

∑

∑ ∑

The motivation behind this program is that player k aims to maximize his relative value as measured by the ratio: the weighted sum of his records vs. the weighted sum of all players' records. This arbitrary weight selection is the fundamental concept underlying DEA initiated by Charnes et al. [2]. DEA terms this as variable weight, in contrast to a prior fixed one. Refer to cooper et al. [10] for further explanation of this issue.

Before continuing, we reformulate the problem as follows, without losing generality. We normalize the fuzzy data set X so that it is row-wise normalized, i.e.,

1

1 ( )nijj

x i=

= ∀∑ .


For this purpose, we divide the row 1( ,.., )i inx x by the row-sum 1

n

ijj

x=∑ for 1,...,i m= .

The program (2) is not affected by this operation. Thus, using the Charnes-Cooper transformation scheme, the fractional program (2) can be expressed using a linear program as follows:

1

1

( )

. 1, 0 (3)

mk

i ikim

k ki i

i

c k Max w x

s t w w i

=

=

=

= ≥ ∀

∑

∑

Apparently, the optimal solution is given by assigning 1 to ( )k

i kw for the criterion i(k)

such that ( ) 1,.....,i k ikx Max x i m= = and assigning 0 to the weight of remaining

criteria. We denote this optimal value by c(k).

( ) 1,...,ikc k x k n= = (4)

The c(k) indicates the highest relative score for player k which is obtained by the optimal weight selecting behavior. The optimal weight ( )

ki kw may differ from one

player to another.

Theorem 1. 1

( ) 1 (5)n

kc k

=

≥∑

That is, sum of maximized scores is greater or equivalent 1.

Proof. Let the optimal weight for player k is * * * *1 ( )( ..., ), 1k k mk i k kw w w w= = and

0 ( ( ))ikw i i k∗ = ∀ ≠ then, we have *( ) 1

1 1 1 1 1( ) 1

n n m n n

i k ik i k k kk k i i k

c k w x x x= = = = =

= = ≥ =∑ ∑∑ ∑ ∑

The inequality above follows from ( ) 1i k k kx x≥ and the last equality follows from the

row–wise normalization.

This theorem asserts that, if each player sticks to his egoistic sense of value and insists on getting the portion of the benefit as designated by c(k), the sum of shares usually exceeds 1 and hence c(k) cannot fulfil the role of division of the benefit. If eventually, the sum of c(k) turns out to be 1, all players will agree to accept the division c(k), since this is obtained by the players most preferable weight selection. The latter case will occur when all players have the same and common optimal weight selection. More concretely, we have the following theorem.

Theorem 2.

The equality 1

( ) 1n

kc k

=

=∑ holds if and only if our data satisfies the condition

1 2 ....k k mkx x x= = = , 1,...,k n∀ = .

That is, each player has the same score with respect to the m criteria.

Proof. The (if) part can be seen as follows:


Since 1( ) kc k x= for all k, we have: 11 1

( ) 1n n

kk k

c k x= =

= =∑ ∑

The (only if) part can be proved as follows. Suppose 11 21x x> then there must be

column 1h ≠ such that 1 2h hx x< , otherwise the second row sum cannot attain 1. Thus

we have 11 2 1(1) , ( ) h hc x c h x x≥ ≥ > and 1( ) ( 1, )jc j x j h≥ ∀ ≠ . Hence it holds that

1 2 11 1, 1

( ) 1n n n

j h jk j h j

c k x x x= = ≠ =

≥ + > =∑ ∑ ∑

This leads to a contradiction. Therefore player1 must have the same score in all criteria. The same relation must hold for the other players.

In the above case, only one criterion is needed for describing the game and the division proportional to this score is a fair division.

However, such situations might only occur in rare instances. In the majority of cases,

we have1

( ) 1n

kc k

=

≥∑ .

Coalition with additive property

Let the coalition S be a subset of player set (1,..., )N n= . The record for coalition S is

defined by ( ) ( 1,..., ) (6)i ijj S

x S x i m∈

= =∑

These coalitions aim to maximize the outcomes c(S).

1

1

( ) ( )

. 1 , 0 (7)

m

i ii

m

i ii

c S Max w x S

s t w w i

=

=

=

= ≥ ∀

∑

∑

The c(S) with ( ) 0c ϕ = , defines a characteristic function of the coalition S. Thus this

game is represented by (N,c).

Definition 1. A function f is called sub – additive if for any S⊂N and T⊂N with S

T

=φ the following statement holds: ( ) ( ) ( )f S T f S f T≤ + .

Definition 2.A function f is called super – additive if for any S⊂N and T⊂N with S

T =φ the following statement holds: ( ) ( ) ( )f S T f S f T≥ + .

Theorem 3. The characteristic function of c is sub – additive, for any S⊂N and T⊂N

with S

T =φ we have ( ) ( ) ( ) (8)c S T c S c T≤ +

Proof. By renumbering the indexes, we can assume that 1,..., , 1,..., S h T h k= = +

and 1,...,S T k= . For these sets, it holds that

1 1 1( ) ( ) ( )

k h k

ij ij iji i ij j j hc S T Max x Max x Max x c S c T

= = = +

= ≤ + = +∑ ∑ ∑

Theorem 4. ( ) 1c N = .


Proof. 1 1 1

( ) 1m n m

i ij ii j i

c N w x w= = =

= = =∑ ∑ ∑ .

A DEA minimum game

The opposite side of the game can be constructed by (N,d) as follows :

1

1

( )

. 1, 0 (9)

mk

i iki

mk k

i ii

d k Min w x

s t w w i

=

=

=

= ≥ ∀

∑

∑

The optimal value d(k) assures the minimum division that player k can expect from the game .

Theorem 5. 1

( ) 1n

kd k

=

≤∑ (10)

Analogous to the max game, for the coalition S⊂N, we define

1

1

( ) ( )

. 1 , 0 (11)

m

i ii

m

i ii

d S Min w x S

s t w w i

=

=

=

= ≥ ∀

∑

∑

Theorem 6. The min game (N,d) is super – additive we have ( ) ( ) ( )d S T d S d T≥ + for

each S, T ⊂N with S T φ=

Proof. By renumbering the indexes, we have 1,..., , 1,..., S h T h k= = + and

1,...,S T k= . For these sets it holds that

1 1 1( ) ( ) ( )

k h k

ij ij ijj j j h

d S T Min x Min x Min x d S d T= = = +

= ≥ + = +∑ ∑ ∑

Thus this game starts from d(k)>0, 1,...,k n= and enlarge the gains by the coalition until the grand coalition N with d(N)=1 is reached .

Theorem 7. ( ) ( \ ) 1d S c N S S N≠

+ = ∀ ⊂

Proof. By renumbering the indexes, we can assume that 1,..., , 1,..., S h N n= = and

\ 1,...,N S h n= + .For this sets, it holds that

1 1 1 1 1

1 1 1 1

( ) ( \ ) ( )

(1 ) 1 1

h n n n n

ij ij ij ij iji ii ij j h j j h j h

n n n n

ij ij ij iji i i ij h j h j h j h

d S c N S Min x Max x Min x x Max x

Min x Max x Max x Max x

= = + = = + = +

= + = + = + = +

+ = + = − +

= − + = − + =

∑ ∑ ∑ ∑ ∑

∑ ∑ ∑ ∑


Extensions

In this section, we extend the basic model to maximal allocated benefit and minimal allocated cost.

Maximal allocated benefit

Suppose that there are s criteria for representing benefits. Let ( 1,..., )ijy i s= be the

benefits of player ( 1,..., )j j n= where u 1( ,..., )su u is the virtual weights for benefits.

Analogous to the expression (1) we define the relative score of player j to the total scores as:

1

1 1

(12)( )

s

i iji

s n

i iji j

u y

u y

=

= =

∑

∑ ∑

Player j wishes to maximize his benefits. We can express this situation by the linear program below:

1

1 1 1. ( ) 1, 0 ( 1,..., )

0 (13)

s

i iji

s n s

i ij i iji j i

i

Max u y

s t u y u y j n

u i

=

= = =

= ≥ =

≥ ∀

∑

∑ ∑ ∑

The weights of benefits are nonnegative. A characteristic function of the coalition S is defined by the linear program below:

1

1 1 1

( )

. ( ) 1, 0 ( 1,..., )

0 (14)

s

i iji j S

s n s

i ij i iji j i

i

c S Max u y

s t u y u y j n

u i

= ∈

= = =

=

= ≥ =

≥ ∀

∑ ∑

∑ ∑ ∑

In the program (14), the benefits of all players are nonnegative. Since the constraints of program (14) are the same for all coalitions, we have the following theorem.

Theorem 8. The maximal allocated benefits game satisfies a sub- additive property.

Proof. For any S N⊂ andT N⊂ withS T φ= , we have:

1 1

1 1 1 1

( ) ( )

( ) ( ) ( ) ( )

s s

i ij i ij iji j S T i j S j T

s s s s

i ij i ij i ij i iji j S i j T i j S i j T

c S T Max u y Max u y y

Max u y u y Max u y Max u y c S c T

= ∈ = ∈ ∈

= ∈ = ∈ = ∈ = ∈

= = +

= + ≤ + ≤ +

∑ ∑ ∑ ∑ ∑

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑


Minimal allocated cost

Suppose that there are m criteria for representing costs. Let ( 1,..., )ijx i m= be the costs

of player ( 1,..., )j j n= where v 1( ,..., )mv v is the virtual weights for costs. Player j

wishes to minimize his costs then we have:

1

1 1 1. 1, 0 ( 1,..., )

0 (15)

m

i ijim n m

i ij i iji j i

i

Min v x

s t v x v x j n

v i

=

= = =

= ≥ =

≥ ∀

∑

∑ ∑ ∑

The weights of costs are nonnegative. A characteristic function of the coalition S is defined by the linear program below:

1

1 1 1

( )

. 1, 0 ( 1,..., )

0 (16)

m

i iji j S

m n m

i ij i iji j i

i

d S Min v x

s t v x v x j n

v i

= ∈

= = =

=

= ≥ =

≥ ∀

∑ ∑

∑ ∑ ∑

In the program (16), the costs of all players are nonnegative. Minimal allocated costs game satisfies a super–additive property.

Theorem 9. The maximal allocated benefit game (N,c) and min game (N,d) are dual

games, for any S⊂N, we have ( ) ( \ ) 1d S c N S+ = .

Proof.

1 \ 1 1 1

1 1

( \ ) ( ( ( )) ( )

(1 ) 1 ( ) 1 ( ).

s s s s

i ij i ij ij i ij i iju u ui j N S i j N j S i j N i j S

s s

i ij i ijuu i j S i j S

c N S Max u y Max u y y Max u y u y

Max u y Min u y d S

= ∈ = ∈ ∈ = ∈ = ∈

= ∈ = ∈

= = − = −

= − = − = −

∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

In above programs we presented a scheme for computing maximal benefit ratio and minimal cost ratio for coalitions. Also we can compute maximal benefit ratio and minimal cost ratio for members of coalition, using extended programs in this paper. These results are applied by the following section.

Avoiding occurrence of zero weights and setting preference on weights

In above programs we presented a scheme for determining the weights through the program (13), (15). Some weight may happen to be zero for all optimal solutions. This means that the corresponding criterion is not accounted for in the solution of the game at all. Let us suppose that all players agree to put preference on certain criteria. The zero weight issue can thus be solved in this way. If all players agree to incorporate preference regarding criteria, we can apply the following "assurance region method". For example , we set constraints on the ratio w1, wi (i=2,…,m) as:


1

ii i

wL Uw≤ ≤ , (i=2,…,m) where Li and Ui denote lower and upper bounds of the ratio

1

iww , respectively. These bounds must be set by agreement among all players. The

program (3) is now modified as:

1

1 1

( ) ( )

. 1, ( 2,..., )

0 (17)

m

i ii

mi

i i ii

i

c S Max w x s

ws t w L U i mw

w i

=

=

=

= ≤ ≤ =

≥ ∀

∑

∑

Similarly, we can avoid occurrence of zero weight in linear programs of maximal allocated benefits and minimal allocated costs. Then, we have:

1

1 1

1

1

. ( ) 1,

0 ( 1,..., ) (18)

( 2,..., ), 0

s

i iji

s n

i iji j

s

i iji

ii i i

Max u y

s t u y

u y j n

uL U i s u iu

=

= =

=

=

≥ =

≤ ≤ = ≥ ∀

∑

∑ ∑

∑

1

1 1

1

1

. 1,

0 ( 1,..., )

( 2,..., ), 0 (19)

m

i ijim n

i iji j

m

i iji

ii i i

Min v x

s t v x

v x j n

vL U i m v iv

=

= =

=

=

≥ =

≤ ≤ = ≥ ∀

∑

∑ ∑

∑

The (13), (15) are modified in (18), (19) respectively.

1

1 1 1

1

( )

. 1, 0 ( 1,..., )

( 2,..., ) 0 (20)

s

i iji j S

s n s

i ij i iji j i

ii i i

c S Max u y

s t u y u y i n

uL U i s u iu

= ∈

= = =

=

= ≥ =

≤ ≤ = ≥ ∀

∑ ∑

∑ ∑ ∑

1

1 1 1

1

( )

. 1, 0 ( 1,..., )

( 2,..., ), 0 (21)

m

i iji j S

m n m

i ij i iji j i

ii i i

d S Min v x

s t v x v x i n

vL U i m v iv

= ∈

= = =

=

= ≥ =

≤ ≤ = ≥ ∀

∑ ∑

∑ ∑ ∑

The (14), (16) are modified in (20), (21) respectively.


The minimal allocated cost and maximal allocated benefit in stock exchange of Tehran

We now apply this approach to some institution in stock exchange of Tehran. There are 15 capital institutions in this district. Clearly, using new programs shows the better (successful) institution. Table 1 shows specifications of companies. Our sample covers the period from 1 March 2010 to 1 September 2010. The data were obtained from the stock exchange of Tehran.

Table 1: specifications of companies

company First price

Stocks Remaining capacity

Purchase

price

Change rate

Selling capacity

Selling price

1 7000 4780617160 466334 6900 2.66 466334 7000

2 1428 2924848280 10606 1450 4 10606 1482

3 2406 62871272 210306 2406 3.97 1100 2441

4 1300 1271112850 5208061 1300 4 4000 1388

5 1399 741384210 168805 1399 3.93 19800 1419

6 2920 57660229402 5949865 2920 3.98 7500 3110

7 1661 172844210 4132704 1661 3.94 10000 1800

8 921 1343516580 389782 931 3.95 3168 926

9 1957 2278030900 4162032 1957 3.98 2207 2090

10 371 626186510 282201 371 3.92 1000 380

11 3396 14947079 35831910 3396 3.98 30000 3500

12 3500 4800709200 28682483 3500 3.98 1700 3699

13 955 590990420 1421379 930 3.91 1421379 955

14 1116 1970649670 5552335 1116 3.91 19960 1145

15 3139 861200190 475297 3139 3.97 10000 3189

Now we compute the inferiority and superiority criteria for institution.

Each institution uses 2 inferiority and 2 superiority criteria. In Tables 2 and 3 inferiority and superiority for these institutions (players) are given, respectively. Also in Table 4, the results of approach are presented.


Table 2: Inferiority criteria for the 15 institutions of stock exchange

Player j

x1j= First price Stocks

2jselling capacityx selling price=

1 3.3464320121013 67.5846

2 4.334617031011 7.314482759

3 1.51126828041011 87.40897756

4 1.6524467051012 4006.200769

5 1.037196511012 120.6611866

6 1.6836805851013 2037.625

7 2.87094423281011 2488.081878

8 1.237378771012 418.670247

9 4.4581064711012 2126.74093

10 2.3231519521011 760.6495957

11 5.0760281641012 10551.21025

12 1.680248221013 8194.995143

13 5.6439585111011 1528.364516

14 2.1992450321012 4975.210573

15 2.7033073961012 151.4166932


Table 3: superiority criteria for the 15 institutions of stock exchange

Player j y1j= Change rate 2j

remaining capacityy purchase price=

1 2.66 66.619143

2 4 7.156545209

3 3.97 0.450634985

4 4 2.88184438

5 3.93 13.95348837

6 3.98 2.411575563

7 3.94 5.5555

8 3.95 3.421166307

9 3.98 1.055980861

10 3.92 2.631578947

11 3.98 8.571428571

12 3.98 0.45958367

13 3.91 1488.354974

14 3.91 17.17030568

15 3.97 3.13577924


Table 4: Maximal allocated benefit and Minimal allocated cost for Institutions of stock exchange

Player j

Maximal allocated benefit

Minimal allocated cost

1 0.0458 0.3859

2 0.0689 0.0052

3 0.0684 0.0017

4 0.0689 0.0190

5 0.0677 0.0151

6 0.0685 0.2478

7 0.0678 0.0329

8 0.0680 0.0254

9 0.0685 0.0512

10 0.0675 0.0027

11 0.0685 0.0583

12 0.0685 0.1928

13 0.9166 0.9166

14 0.0673 0.0065

15 0.0684 0.0351

First company has the minimal allocated benefit and the 13th has the maximal allocated benefit. Clearly, this scheme is a way for attaining reasonable and fair division for institutions.

Conclusions

In this paper, we have studied the common weight issues that connect the game solution with arbitrary weight selection behaviors of the players (institutions). In this regard, we have proposed a method for computing maximal allocated benefit and minimal allocated costs for institutions. An extension of problem, in which both superiority and inferiority criteria could be considered to players (institutions), has been discussed. Furthermore a numerical example, in which some institutions of stock exchange evaluated with the proposed ways, has been considered.


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M. Sakawa, Fuzzy sets and Interactive Multiobjective Optimization, Plenum Press, New York, 1993.

R. Allen, A.D. Athanassopoulas , R.D. Dyson , E. Thanassoulis, Weights restriction and value judgment in data envelopment analysis, Annals of Operations Research. 73 (1997) 13-34.

S. Daneshvar, M. Mansouri Kaleibar, The minimal cost-benefit ratio and maximal benefit-cost ratio, presented at the Int Conf. Engineering System Management and Application Sharjah, UAE, 2010.

W.W. Cooper, L.M. Seiford, K. Tone, Data envelopment analysis, a comprehensive text with models application references and DEA- solver software, Boston: Klawer Academic Publishers, 2000.

Acknowledgements

The authors acknowledge the support from Young Researchers Club. This research is supported by Tabriz Branch, Islamic Azad University research budget and sponsorship.


22. Measurement of returns to scale using a non-radial DEA model

Vladimir E. Krivonozhko National University of Science and Technology «MISiS», Russia, [email protected]

Finn R. Førsund, Department of Economics, University of Oslo, Norway, [email protected]

Andrey V. Lychev National University of Science and Technology «MISiS», Russia, [email protected]

Abstract

There are some specific features of the non-radial DEA (data envelopment analysis) models which cause some problems under the returns to scale (RTS) measurement. In the scientific literature on DEA, some methods were suggested to deal with the RTS measurement in the non-radial DEA models. These methods are based on using strong complementary slackness conditions (SCSC) in the optimization theory. In this paper, we propose and substantiate a direct method for the RTS measurement in the non-radial DEA models. Our computational experiments documented that the proposed method works reliably and efficiently on the real-life data sets.

Keywords: Data envelopment analysis; Efficiency; Non-radial models; Returns to scale

Introduction

The measurement of scale properties of frontier functions estimated using DEA models may in some cases be problematic. In particular the non-radial DEA models (Banker et al., 2004) can possess some specific features that give rise to estimation problems. First, multiple reference sets may exist for a production unit. Second, multiple supporting hyperplanes may occur on optimal units of the frontier. Third, multiple projections (a projection set) may occur in the space of input and output variables. All these features cause certain difficulties under measurement of returns to scale of production units.

Banker et al. (2004) proposed a two-stage approach to determine returns to scale in the non-radial models. Sueyoshi and Sekitani (2007) showed that this approach may generate incorrect results in some cases. An interesting approach was proposed for measurement of returns to scale based on using strong complementary slackness conditions (SCSC) in the non-radial DEA models (Sueyoshi and Sekitani, 2007).

However, our theoretical consideration and computational experiments show that the SCSC non-radial model may not be efficient from the computational point of view. The SCSC non-radial model generates ill-conditioned basic matrices during the


solution process, which results in “strange results” that do not coincide with the optimal solution of the corresponding non-radial DEA model. This naturally contradicts the optimization theory.

In our work we propose a two-stage approach to measure returns to scale in the non-radial DEA models. At first stage, an interior point, belonging to the optimal face, is found using a special elaborated method. In Krivonozhko et al. (2012) it was proved that any interior point of a face has the same returns to scale as any other interior point of this face. At the second stage, we propose to determine the returns to scale at the interior point found in the first stage with the help of Banker et al. (2004) method or using the direct method of Førsund et al. (2007).

Our computational experiments documented that the proposed approach is reliable and efficient for solving real-life DEA problems.

The plan of the paper is to state the problem to be investigated in the next section. Then a direct method for discovering all units belonging to the minimum face is developed. After this, some of the numerical experiments with the SCSC non-radial model specification are reported.

Problem statement

The non-radial DEA model can be written in the following form (Banker et al., 2004; Sueyoshi and Sekitani, 2007)

)(max −−++ += SCSCh TT

subject to

,0,0

,,,1,0,1

,

,

1

1

1

≥≥

=≥=

=−

=+

−+

=

+

=

−

=

∑

∑

∑

SS

nj

YSY

XSX

j

n

jj

o

n

jjj

o

n

jjj

λλ

λ

λ

(1)

where ),,( 1 mjjj xxX = and ),,( 1 rjjj yyY = represent the observed inputs and

outputs of production units ),( jj YX , nj ,,1= , ),,( 1−−− = mssS

and ),,( 1+++ = rssS

are vectors of slack variables. The superscript «T» indicates a vector transpose. The

components of the objective-function vectors +C and −C are specified as follows:

.,,1,),,1|min

,,1|(max)(

,,,1,),,1|min

,,1|(max)(

1

1

1

1

rinjy

njyrmc

mknjx

njxrmc

ij

iji

kj

kjk

==−

=+=

==−

=+=

−

−+

−

−−

The model (1) is also called range-adjusted model (RAM) (Cooper et al., 2000).


In the model (1), an efficiency score for unit ),( oo YX is evaluated, where ),( oo YX is

any production unit from the set ),( jj YX , nj ,,1= . If the optimal value *h of the

model is equal to zero, then unit ),( oo YX is considered efficient, if 0* >h , then the

unit is inefficient (Banker et al., 2004).

The dual problem to the model (1) is written in the form:

0min uYuXv oT

oT +−

subject to

,,

,,,1,00

+− ≥≥

=≥+−

CuCv

njuYuXv oT

jT

(2)

where ),( 1 mvvv = and ),,( 1 ruuu = are vectors of dual variables associated with the

first and the second group of constraints of problem (1), 0u is a free variable

associated with the convex constraint.

In the papers (Sueyoshi and Sekitani, 2007, 2009), it was proposed to use strong complementary slackness conditions from the optimization theory in order to find the set of optimal solutions in the primal and dual space. The SCSC non-radial model (Sueyoshi and Sekitani, 2007) is written in the following form

=≥−+=≥−+

=

≥+−++−=+

≥≥=≥+−

≥≥=≥

==−

=+

++

−−

++−−

+−

−+=

+

=

=

−

∑∑

∑

ricusmkcvs

njuYuXv

uYuXvSCSCCuCv

njuYuXvSSnj

YSY

XSX

iii

kkk

jT

jT

j

oT

oTTT

jT

jTj

n

jjo

n

jjj

o

n

jjj

,,1,,,,1,

,,,1

,,

,,,,,1,0,0,0,,,1,0

,1,

,

max

0

0

0

11

1

ηη

ηλ

λ

λλ

λ

η(3)

In paper (Sueyoshi and Sekitani, 2007), the problem (3) is used in order to find the minimum face that contains the set of optimal solutions (a projection set) on the efficient hyper-surface of production possibility set T in the space of input and output variables. Next, two additional fractional-linear optimization problems are determined for measurement of returns to scale.

The SCSC non-radial model is very interesting as a theoretical idea. However, our computational experiments show that the model (3) may generate strange results even for medium-size problems using well-reputed optimization software. The size of the model (3) increases significantly in comparison with the model (1). Indeed, the size of

the model (3) is equal to )222()2233( +++×+++ nrmnrm , where m is the number

of inputs, r is the number of outputs, and n is the number of production units.


Remember that the size of the model (1) is equal to )()1( nrmrm ++×++ , and n is

usually much greater than )( rm + in real-life models.

In order to investigate our suspicions about what will happen when using larger real datasets, we conducted computational experiments using two middle-sized models. For the first model, call it Model 1, we took the data for electricity utilities in Sweden 1987; see Førsund et al. (2007). For Model 2 we took the data from 920 Russia bank’s financial accounts for January 2009.

In the computational experiments we used the well-reputed optimization software CPLEX, the software Mathematica that is very popular among mathematicians, and the software FrontierVision, a specially elaborated program for DEA models that enables one to visualize the multidimensional frontier with the help of constructing two- and three-dimensional sections of the frontier.

The main discrepancies in solving model (1) and model (3) with the help of the program CPLEX (though theoretically optimal solutions of these problems have to coincide) are as follows: a) efficiency scores of model (1) and model (3) may differ significantly; b) reference sets obtained in the solution of model (3) may contain inefficient units.

Direct method for discovering all units belonging to the minimum face

In the linear programming problem (1), the set of optimal points *Λ (the set of

projections of unit ),( oo YX on the frontier) are situated on the boundary (frontier) of

the production possibility set T. The boundary consists of a number of faces.

The solution set *Λ cannot belong to different faces that do not have common points, otherwise interior points of T would belong to the solution set. Thus, optimal

solutions of set *Λ can belong only to the intersection of some faces of the setT .

Lemma 1. Let two different faces 1Γ and 2Γ of the setT intersect. Then faces 1Γ and 2Γ

do not have common interior points, i.e. ∅=Γ∩Γ 21 riri .

Corollary 1. Let two different faces 1Γ and 2Γ of the set T intersect. Then only one

from the following cases occurs:

(i) one face belongs to the other face entirely, to be precise let 21 Γ⊂Γ , and set 21 Γ∩Γ is

a part of the boundary of 2Γ ;

(ii) set 21 Γ∩Γ is a part of the boundary of the face 1Γ and face 2Γ , moreover the set

21 Γ∩Γ is also a face and its dimension is less than the dimensions of the face 1Γ or the

face 2Γ .

From the assertions written above, it follows that faces can intersect only along the boundaries of these faces. Taking into account also that the number of faces of the

set T is finite, we obtain that there exists a face of minimum dimension minΓ


containing set *Λ . By virtue of the non-negativity constraints on slack variables and the

specific objective function associated with all slack variables in problem (1), set *Λ

may constitute only some part of face minΓ .

Now, we proceed to the construction of the procedure that finds all production units

belonging to the minimum face minΓ and to the set *Λ .

Let problem (1), respectively problem (2), be solved by the simplex-method (Dantzig

and Thapa, 2003) and optimal primal variables ;,,1, * njj =λ ;,,1,* mksk =−

,,1,* risi =+ and dual variables ;,,1, * mkvk = ;,,1, *0

* uriui = be obtained.

Determine the following index sets for primal variables

.,,1,0|

,,,1,0|

,,,1,0|

*

*

**

risiImkskI

njjI

iy

kx

j

=>=

=>=

=>=

++

−−

λ

(4)

Introduce the index sets associated with the dual variables

.,,1,|

,,,1,|

,,,1,0|

*

*

*0

***

ricuiJmkcvkJ

njuYuXvjJ

iiu

kkv

jT

jT

===

===

==+−=

+

− (5)

Since the optimal solution is obtained with the help of the simplex-method, every non-basic variable is equal to zero. However, some basic variables may be equal to zero also, then the optimal solution is considered degenerate.

For the dual problem (2) all indices belonging to the set )( *uv JJJ ∪∪ contain a basic

set of indices. However, the set )( *uv JJJ ∪∪ may also contain non-basic indices, in

this case the dual problem (2) is considered degenerate. Thus, the following relations hold

)()( **uvByx JJJJIII ∪∪⊆⊆∪∪ +− ,

where BJ is a set of optimal basic variables of the problem (1).

Variables *jλ , *Ij∈ determine only one point on the minimum face. To find all points

belonging to the face minΓ it is necessary to solve additional problems.

Problem olQ ( *Jl∈ ):

llf λ=max

subject to


,,0,,0

,,0,1

,

,

*

*

*

*

uivk

jJj

j

oJi

iiJj

jj

oJk

kkJj

jj

JisJks

Jj

YseY

XseX

u

v

∈≥∈≥

∈≥=

=−

=+

+−

∈

∈

+

∈

∈

−

∈

∑

∑∑

∑∑

λλ

λ

λ

(6)

where mk Ee ∈− and r

i Ee ∈+ are identity vectors associated with variables −ks and +

is ,

respectively.

Notice that problem (6) includes only those variables for which the corresponding dual constraints hold strictly equations for optimal dual variables (5). According to the dual theorems of linear programming, this means that optimal variables of problem (6) will also be optimal variables of problem (1).

The Procedure that finds all production units belonging to the minimum face minΓ and

to the set *Λ is described as follows:

Initialize sets ∅=oJ , *JJH = , ∅=1J . If the set JH is not empty, then go to the

next step. If set JH is empty then go to step 3.

Choose index JHl∈ , if the set JH is empty, then go to step 3. Solve the problem (6).

If 0* >lf , then determine lJJ oo ∪= . If 1* =lf , then lJJ ∪= 11 . Delete index l from

the set lJHJH \= . Go to the beginning of the step.

If 0* =lf , then delete index l from the set lJHJH \= . Go to the beginning of step 2.

Set oJ determines the set of units belonging to the face minΓ . Set 1J determines the set

of units belonging to the set *Λ .

The Procedure is completed.

The standard present-day optimization software generates only one point in the multidimensional space as an optimal solution. However, this may be not sufficient in order to determine returns to scale on the whole minimum face, since different

vertices of the face may display different returns to scale. Any unit from set *J may

belong to the minimum face. The standard software generates set *J as a by-product.

So, the Procedure enables one to check whether some unit from set *J belongs to the minimum face or not. The validity of this assertion is based on the theorems given below.

After running the Procedure, the minimum face minΓ , containing the optimal set *Λ ,

can be written in the form:


.,0,1

,,),(

∈≥=

===Γ

∑

∑∑

∈

∈∈

ojJj

j

Jjjj

Jjjjmin

Jj

YYXXYX

o

oo

λλ

λλ(7)

However, some points of the set minΓ may not belong to the set *Λ on the frontier.

The set *Λ is written as:

.)(

)(

,,,),(),(

*0

*

*

*

uYCuXCvXCYC

YYXXYXYX

oT

oTTT

oomin

+−

−−=−

≥≤Γ∈=Λ

+

−−+ (8)

The Procedure enables one to find all units belonging to the face minΓ and to the set *Λ . The validity of this assertion is based on the following theorems.

Theorem 1. Let unit rmt EZ +∈ be an interior point of polyhedron rmE +⊂Γ , let also

unit rmp EZ +∈ be any point of this polyhedron, which is distinct from point tZ . Then

unit tZ can be represented as a convex combination of )1( ++ rm units of set Γ and

unit pZ enters this combination with a nonzero coefficient.

Theorem 2. The optimal value of problem (6) is strictly positive 0* >lf if and only if

unit ),( ll YX belongs to the minimum face minΓ that contains the set *Λ .

In essence, Theorem 1 says that, if some unit pZ is a vertex of face minΓ or belongs to

the face, then it is necessary that there exists such solution that variable *pλ enters this

solution with a nonzero coefficient.

Corollary 3. If the optimal value of problem (6) 1* =lf , then unit ),( ll YX belongs to

the set *Λ .

If 1* =lf , then 1* =lλ , this means that *lλ is the only non-negative λ -variable in the

optimal basis, hence unit ),( ll YX belongs to the set *Λ .

It was proved in (Krivonozhko et al., 2012) that interior points of a face have the same returns to scale, so it is sufficient to determine returns to scale at any interior point of

this face. An interior point ),( YX of the face minΓ can be chosen as a strong convex

combination of units from the set oJ , that is

.,0,1

,,

ojJj

j

Jjjj

Jjjj

Jj

YYXX

o

oo

∈>=

==

∑

∑∑

∈

∈∈

λλ

λλ

Returns to scale of unit ),( YX can be measured by two methods at least. In the first

(indirect) method (Banker et al., 2004) the BCC model is solved at the first step, the

dimension of this problem is equal to )()1( nrmrm ++×++ , then at the second step


two additional problems are solved, the dimension of these two problems coincides with the dimension of the BCC problem. Returns to scale is determined with the help of dual variables.

In the second (direct) method (Førsund et al., 2009), an intersection of the set T and two-dimensional plane is constructed with the help of some algorithms at the first step. At the second step returns to scale of any point on the graph is measured by using derivatives of this graph.

Computational experiments

However, the question arises: “What is the ‘payment’ for discovering all vertices of the

face minΓ ?”

We calculated the number of iterations accomplished by CPLEX software in order to

solve problem (1) for all units ),( jj YX , nj ,,1= of the Model 1. This number is

equal to 2876=α iterations. Next, we calculated the number of iterations accomplished by CPLEX in order to solve problems (6) for all units ),( jj YX ,

nj ,,1= . This number makes up αβ 62.01782 == iterations in the problems of the

type (6). That is really not a heavy burden, even for an ordinary notebook.

Finding. The non-radial DEA models possess some specific features. However this is a not a problem in order for find returns to scale of the set of optimal points. For this purpose it is sufficient to find an interior point of the minimum face that contains the set of optimal solutions of problem (1) and to determine returns to scale of this interior point. Such solution requires much less computations than to solve problem (1) for the specific unit.

Conclusions

In Sueyoshi and Sekitani (2007; 2009), a method was proposed in order to measure returns to scale in the non-radial DEA models using strong complementary slackness conditions. However, the size of the SCSC non-radial model (3) increases significantly in comparison with the model (1). In particular, for the banks data set the size of basic matrices during the solution process becomes ( 18401840× ) instead of ( 77× ) in the model (1). In addition, some constraints in model (3) do not make sense from an economic point of view. Unreliable solutions may follow due to these reasons.

In our method, it is sufficient to solve several problems of the form of model (1), however such problems have much less variables than problem (1). Thus, the proposed approach is reliable and efficient for solutions of real-life problems. Moreover, it was stressed in Sueyoshi and Sekitani (2007) that the method of Banker et al. (2004) cannot always generate reliable results because of difficulties described above in the non-radial DEA models. However, the method of Banker et al. (2004) can also be used to measure returns to scale from our point of view. For this purpose, it is sufficient to take an interior point of the minimum face, which can be found by the method proposed in this paper, after this one can use the method proposed in Banker et al. (2004).


References

Banker, R. D., Cooper, W. W., Seiford, L. M., Thrall, R. M., and Zhu, J. (2004) Returns to scale in different DEA models. European Journal of Operational Research 154(2), 345–362.

Cooper, W. W., Park, K. S., and Pastor, J. T. (2000) RAM: A range adjusted measure of efficiency. Journal of Productivity Analysis 11: 5–42.

Dantzig, G. B. and Thapa, M. N. (2003) Linear programming 2: Theory and Extensions. New York: Springer-Verlag.

Førsund, F. R., Hjalmarsson, L., Krivonozhko, V. E., and Utkin, O. B. (2007) Calculation of scale elasticities in DEA models: direct and indirect approaches. Journal of

Productivity Analysis 28: 45–56.

Førsund, F. R., Kittelsen, S. A. C., and Krivonozhko, V. E. (2009) Farrell revisited – Visualizing properties of DEA production frontiers. Journal of the Operational

Research Society 60: 1535–1545.

Krivonozhko, V. E., Førsund, F. R., and Lychev, A. V. (2012) Returns-to-scale properties in DEA models: the fundamental role of interior points. Journal of Productivity Analysis 38(2): 121–130.

Sueyoshi, T. and Sekitani, K. (2007) Measurement of returns to scale using a non-radial DEA model: A range-adjusted measure approach. European Journal of Operational Research 176: 1918–1946.

Sueyoshi, T. and Sekitani, K. (2009) An occurrence of multiple projections in DEA-based measurement of technical efficiency: theoretical comparison among DEA models from desirable properties. European Journal of Operational Research 196: 764–794.

Acknowledgements

The research is carried out with financial support of the Programme of Creation and Development of the National University of Science and Technology «MISiS». The reported study was partially supported by RFBR, research projects No.11-07-00698 and No.12-07-31136.


23. Multimethodology applied to the assessment of municipalities’ health performance in Brazil

Marcos Pereira Estellita Lins Production Engineering Program POLI-COPPE/UFRJ, Post Office Box 68507, CEP 21945-970 - Rio de Janeiro, RJ, Brasil, [email protected]

Sergio Orlando Antoun Netto Department of Cartography/Engineering School/UERJ - COPPE/UFRJ, 524 São Francisco Xavier Street, 5st floor - Rio de Janeiro – CEP 20550-013, [email protected] (corresponding author)

Abstract

This work presents a strategic approach to the formulation and structuring of health problem in 5565 Brazilian municipalities using Concept Map and Data Mining. In addition, using the Operational Research Method called DEA (Data Envelopment Analysis) for the determination of counties goals and performance indicators and the establishment of benchmarks for regulating the public health sector in Brazil. The analytical results will corroborate to a progressive incentive for greater productivity in the Brazilian municipalities health.

Keywords: Data Envelopment Analysis (DEA), Data Mining, Concept Map, Multimethodology, Public Health

Introduction

Data Envelopment Analysis (DEA) has been used to support public policy in regulatory processes because of its ability to handle multidisciplinary problems and deliver efficient targets. However, public policy is also characterized by its inherent complexity, which requires taking a particular arbitrary perspective regarding the system purpose and relevant factors.

As a comparative method in its essence, DEA requires that homogeneity prevail as one of the key issues for the Decision Making Units under assessment. Nevertheless, not much research has been devoted to either complexity or homogeneity in DEA applications. One reason could be the small size of most databases in regulatory problems, which does not allow the application of clustering procedures to data.

This work presents a strategic approach to the formulation and structuring of the health problem, taking advantage of the large size database comprising 5565 Brazilian municipalities. First it uses Concept Maps to represent the many factors that characterize the complex issues involved in health performance management in Brazil. Then heterogeneity is investigated and a Data Mining technique is applied to find appropriate clusters of around a thousand municipalities.


Actually, the philosophy behind these two approaches are emphasized by the US and UK Operational Research schools, as shown in the respective societies websites; the first prescribes the use of analytics in order to explore different approaches to the database, while the second includes a critical step that consists of the analysis of different approaches to problem structuring. Thus the use of multiple methodologies is an inherent trend in both schools of OR.

This work employed a strategic approach to the formulation and structuring of problems through the use of conceptual maps. Actually, in solving problems of systems, "Soft" and "Hard" tools can be employed, and both may be used in a complementary manner (Oral & Reisman, 2005). It improves the modeling of efficiency in many ways, since stakeholders can acquire an accounting of the whole system, while putting the quantitative indicators in a comprehensive and still accessible context.

The application to municipality health comprises main causes of death in three different age ranges, each one assessed by a different DEA model. Different profiles according to life threatening and care provided by governmental authorities emerge from the analysis.

Methods

According to Rosenhead and Mingers (2001), Concept Map is important to make explicit when we are seeking proposals for dealing with complex problems and not just solving a simplified part of a problem under a particular perspective. In this former approach the structuring of matters, issues and situations is one of the stages of the modeling at the very beginning of the decision-making process.

For Weis and Indurkhya (1999), data mining is the search for valuable information in large databases, as a cooperative effort between men and computers. Men design databases, describe problems and define goals. Computers check data and look for patterns that match the goals established by men.

The Data Envelopment Analysis (DEA) has been used in the calculation of performance indicators and to establish benchmarks for regulation of public sectors. The method lends itself to use in multidisciplinary issues and multiagents may be used in the estimation of production frontier functions or for incorporating the opinion of specialists, like a multicriteria method.

Multimethodology is the "art" of use, combined, more than a methodology or part of methodologies, in order to consider, in the best way, the various problems, as proposed by Mingers and Brocklesby (1997). The multimethodology approach assumes that there is a specific method that is more appropriate, but that all methods have advantages and disadvantages that can be balanced.


We developed the formulation and structuring of Public Health in Brazilian municipalities using information from Public Health experts reported in legal documents as Special Checking Accounts and press interviews with government authorities as Federal Prosecutors, National Health Council, Parliament House, Minister


for Science, Technology and Innovation, Comptroller General of the Union, Board of Policies, Performance and Competitiveness Management linked to the Governing of the Republic Presidency Council and members of non-governmental sectors as SUS National Audit Department, SUS Auditors National Union and Brazilian Collective Health Association. The formulation and structuring of Public Health in Brazilian municipalities can make use of the concept maps shown in Antoun Netto (2012).

To implement the clustering technique we employed the Profile of Brazilian Municipalities Database (MUNIC), from Brazilian Institute of Geography and Statistics (IBGE), covering the year 2009. It consists of a survey of Municipal basic information published annually by IBGE with information about 16 themes of 5565 Brazilian municipalities, such as: human resources, legislation and municipal planning tools, education, culture, sports, housing, transportation, health, environment, among others.

The software used for clustering was the Waikato Environment for Knowledge Analysis (WEKA), a set of algorithms and programs produced by the University of Waikato in New Zealand. The Weka is implemented in the Java language, which has as main characteristic its portability, this way you can use it on different operating systems. The WEKA is free software, i.e. is under GPL license and is available at http: //www.cs.waikato.ac. nz /ml/weka.

In the Table 1 is a summary with the main feature of each grouping set by clustering technique, using the WEKA.

Table 1: Main feature of each cluster

Cluster Main feature

1 High Public consortia participation with the Federal and State Government

2 Participation of variables near the average of the municipalities examined

3 Municipalities have an improved infrastructure in the area of health compared to those of other groups, though not necessarily the ideal

4 Considerable participation of public Consortium between municipalities

5 Participation of the variables below average of the municipalities examined

Antoun Netto (2012) has more details about the clustering technique in Brazilian municipalities.

For the determination of municipal development targets and indicators in the area of health we will consider clusters 1 and 3. Such a choice results from the following facts:

a. Regarding Cluster 1, we want to investigate if municipalities that present high participation of State and Federal Governments’ resources are employing those resources efficiently and effectively.


b. Analysis of Cluster 3 is particularly interesting because this cluster is composed of municipalities with better infrastructure, compared to the other clusters. In this work we will considered only the municipalities with population greater than or equal to 50,000 for cluster 1 and 100,000 for cluster 3, aiming at minimizing possible anomalies in the DEA models to be implemented.

To analyse the level of human development in a particular country we need to conduct studies about various social indicators. A very important indicator for the analysis is infant mortality, which corresponds to the number of children who die before reaching one year of age. According to the IBGE, there has been a 50% reduction in child mortality between 1990 and 2008, from 47 to 23.3 per 1,000 live births. The proportion, however, is still not low according to the standards of the World Health Organization (less than 20 per thousand).

It is worth mentioning that in Nations like Sweden, Norway and Canada the ratio is, respectively, 3, 10.4 and 4.63 per 1,000 live births. The index can vary between different cities, States and regions, as well as on the basis of income, i.e. the low-income sector presents always higher child mortality.

Another important indicator refers to diseases of the circulatory system that are the leading causes of death in Brazil and in the world. The two main causes of mortality within this group are ischemic heart disease and cerebrovascular disease.

The risk factors for cardiovascular diseases are very prevalent in urban populations, currently being cardiovascular risk factors: smoking, overweight, excess meat consumption of fats, physical inactivity, alcohol abuse, etc.

Mortality ratios are not distributed homogeneously in space. The inequality in mortality from diseases of the circulatory system, whether on a national, regional or local basis, is strongly associated with social and economic inequality and the distribution of health equipment.

Finally, in view of violence growth in Brazilian municipalities, search in this study consider the indicator of mortality from external causes, which include, according to the World Health Organization, all types of accidents, suicides and homicides. High rates of mortality are associated with higher prevalence of risk factors specific to each type of external cause. Traffic accidents, homicides and suicides respond together for about two-thirds of deaths from external causes in Brazil. The rates are considerably higher in the young adult population, especially males.

It is worth emphasizing that the country's economic growth in recent years and spending on public security higher than in some developed countries, such as Germany and Spain, have not contributed to the reduction of homicides in Brazil, according to the Yearbook of Brazilian Public Security. Given the above, the choice of variables sought to contemplate the dimensions of child mortality, external causes and diseases of the circulatory system. It is important to report that restrictions on weights were not introduced in the templates.

Table 2 shows the variables to be considered within the model. The choice of variables in DEA is a key issue for the determination of efficiency.


Table 2 – Variables identified in the case study

DIMENSION

External Causes Circulatory Diseases

Infant Mortality

INPUT

Deaths Chapter XX - External causes of morbidity and mortality

Deaths Chapter IX - Circulatory Diseases

Deaths Chapter XVI - Some disorders originating in the perinatal period

Deaths Chapter XVII - Congenital malformations, deformities and chromosomal abnormalities

OUTPUT Resident Population

20 to 29 years old

Resident Population

40 to 59 years old

Resident Population

< 1 year old

In determining targets and indicators of municipal development in the area of health, each municipality is represented as a DMU (Decision Making Unit) endowed with autonomy. We adopted a design of hierarchical modeling in 2 (two) stages, as proposed by Lins et al. (2007) and Ozcan et al. (2010), namely:

• In the first stage, we considered the output-oriented VRS basic models for the three dimensions: infant mortality, circulatory diseases and external causes.

• In the second stage, we determined the final efficiencies for these municipalities, using the weighted average.

In Table 3, we will present the first stage efficiency of municipalities in various dimensions, by cluster, through the DEAFrontier software, developed by Professor Joe Zhu of WPI (Worcester Polytechnic Institute).


Table 3- Efficient Municipalities by dimension in the first stage

Dimension

Circulatory Diseases External Causes Infant Mortality

Clu

ster

1

Manaus (AM)

Abaetetuba (PA)

Barcarena (PA)

Monte Santo (BA)

Manaus (AM)

Santarém (PA)

Monte Santo (BA)

Manaus (AM)

Abaetetuba (PA)

Belém (PA)

Frutal (MG)

Porto Ferreira (SP)

Almirante Tamandaré (PR)

Concórdia (SC)

Laguna (SC)

Santa Cruz do Sul (RS)

Viamão (RS)

Aparecida de Goiânia (GO)

Clu

ster

3

Santana (PA)

Belo Horizonte (MG)

Uberlândia (MG)

São Paulo (SP)

Valinhos (SP)

Jaraguá do Sul (SC)

Juiz de Fora (MG)

Pouso Alegre (MG)

São Paulo (SP)

Juiz de Fora (MG)

Macaé (RJ)

Ourinhos (SP)

Salto (SP)

São Paulo (SP)

Curitiba (PR)

Porto Alegre (RS)

Santa Maria (RS)

In Table 4 we present the benchmarks municipalities resulting from aggregated efficiencies of the three dimensions.


Table 4- Benchmarks Municipalities

Cluster Benchmark

1 Manaus (AM)

3 São Paulo (SP)

Conclusions

This work illustrated the use of conceptual maps for the design and structuring of the Brazilian public health assessment. Knowledge mapping was used to establish a context for the qualitative and quantitative variables, to be used in Data Mining and Data Envelopment Analysis, respectively.

The approach used encompasses different major diseases that are prevalent and representative of death risks at three different age ranges. The methodology intends to evaluate three different processes and respective sets of variables that are not directly related, and thus assessed in three different DEA models, constituting a first stage.

Differently from other approaches, we restrained the analysis to health closures, represented by the worst situation, namely death occurrence. We intend to extend the method to include other minor diseases.

Other recommendations for future research consider the evolution of efficiency along the period 2009-2012, since the first coincided with the beginning of the current mandate of mayors in Brazilian municipalities. We suggest the use of Malmquist index to evaluate the actions and programs in the health municipalities during their mandate, considering the change in the health performance of municipalities between the two periods of time (2009 and 2012).

Finally, it is important to facilitate the validation of these results by qualified healthcare professionals, regarding the potential for improvements and the observation of the recommendations in the National Policy of Health Promotion, whose general objective is to promote the quality of life and reduce vulnerability and health risks related to its determinants and conditioning – ways of living, working conditions, housing, environment, education, leisure, culture and access to essential goods and services.

References

Antoun Netto, S. O. (2012). O uso de Multimetodologia para a determinação de Metas e Indicadores de Desenvolvimento Municipal na Área da Saúde. Tese (doutorado) – UFRJ/ COPPE/ Programa de Engenharia de Produção.

Lins, M. E. et al. (2007). O uso da Análise Envoltória de Dados (DEA) para avaliação de hospitais universitários brasileiros. Ciência & Saúde Coletiva, 12(4): 985-998.

Mingers, J. ; Brocklesby, J. (1997). “Multimethodology: Towards a Framework for Mixing Methodologies”, Omega, International Journal of Management Science, 25, 5 (pp. 489–509).


Ozcan, Y.A.; Estellita Lins M.P., Lobo, M.S.C.; Silva A.C.M.; Fiszman, R.; Pereira, B. B. (2010) Evaluating the Performance of Brazilian University Hospitals Annals of Operations Research; ISSN:0254-5330; Springer.

Reisman, A.; Oral, M. (2005). Soft systems methodology: A context within a 50-year retrospective of OR/MS." Interfaces 35.2, 164-78.

Rosenhead, J.; Mingers, J. (2001). Rational analysis for a problematic world: problem structuring methods for complexity, uncertainty and conflict. 375p. 2. ed. West Sussex: John Willey & Sons.

Weis S. M. e Indurkhya, N. (1999) “Predict Data Mining”; Morgan Kaufmann Publishers, Inc.


24. On the Measurement of Social Efficiency in Microfinance Institutions

Breno Sampaio Department of Economics, Federal University of Pernambuco, [email protected]*

Lúcio Silva Department of Production Engineering, Federal University of Pernambuco, [email protected]

Abstract

Do Microfinance Institutions trade-off between financial and social efficiencies? Previous evidence shows that a positive correlation exists. In this paper we argue that previous results are biased because they do not account for differences between countries. We propose a decomposition in which efficiency is broken down into a MFI and a country component. Results are quite different from the ones previously presented, given we isolate any country-specific characteristic that might bias efficiency estimates. The correlation we estimate between financial and social efficiency is about 35% of what was previously reported. Also, this correlation varies within the financial efficiency distribution. Looking at profits, we obtain that they are indeed correlated with efficiencies; the financial efficiency correlation is positive but, contrary to previous findings, the social efficiency correlation is negative. Finally, we show that countries indeed face different frontiers implying that efficiencies should be assessed by comparing units within the same country.

Keywords: DEA, microfinance institutions (MFIs), social efficiency

Introduction

Microfinance Institutions (MFIs) are nowadays present in most countries around the world. Their main purpose is to provide credit to the poor who are excluded from the traditional banking system and, for this reason, they are widely recognized for their contribution to the development of financial markets in the developing world. Contrary to what is observed in the literature about the traditional banking sector, little is known about the performance of MFIs, specially considering their mutual purpose of being financially and socially efficient (Gutiérrez-Nieto et al., 2009). Berguiga (2009), for example, concludes after an extensive review that MFIs do not trade-off between being financially and socially efficient, that is, the literature actually suggests that these two requirements are compatible and may even be complementary. This question is also addressed in Gutiérrez-Nieto et al. (2009). They analyze both the financial and social efficiency of Microfinance Institutions using data for 89 MFIs * Corresponding author


located in several countries around the world. Additionally, they also look at the relationship between these efficiencies, and how they relate to other indicators, such as profitability, and the type of institutions - Non-Governmental Organization (NGO) and non-NGO. Their main conclusions are that there is a positive, but low, correlation between social efficiency and financial efficiency, NGOs are more socially efficient than other MFIs that are run under other organizational structures, and, more importantly, the geographical area of activity of the MFI was found to be important in determining efficiency measures.

In this paper, our focus is to look at the interaction between financial and social efficiencies but taking into account differences between countries, which is a substantial improvement compared to previous studies on the subject. More specifically, we invoke one of the most basic assumptions in DEA applications, which require that the units being considered in the analysis must have the same production process. That is, when comparing MFIs in different countries one must take into account that significant differences may exist in terms of labor market conditions for women, for example, which if not correctly accounted for may introduce significant bias in the efficiency coefficients ultimately leading to misinterpretations of inefficiencies given by country-specific limitations. Hence, to account for country-specific characteristics when estimating MFIs financial and social efficiencies and to look at how different are production possibility curves in different countries, we propose a decomposition of efficiency similar to that taken by Portela and Thanassoulis (2001) and recently by Amores and Contreras (2009), in which overall efficiency is broken down into a MFI (financial and social) efficiency and a country (financial and social) efficiency. This idea is similar to estimating a regression model with country fixed effects to control for any unobserved country characteristics (see Sampaio (2012)).

After this brief introduction, in the next section we present the data. In the third section we describe the methodology and in the fourth section we present the results. Finally, in the fifth section we conclude.

Data

The data source, as in most of the recent studies looking at MFIs (Gutiérrez-Nieto et al. (2009), among others), comes from the Microfinance Information Exchange (MIX) database. The sample we use in this article was obtained by first dropping all MFIs that had missing values on any of the variables used in the analysis and then restricting our sample to include only countries that had at least 20 MFIs, given we want to construct country-specific frontiers. All the data we use consider only the year of 2008. This leaves us with 483 MFIs distributed within 14 countries. The two largest countries, according to the number of MFIs analyzed, are Russia, with 78, and India, with 57 microfinance institutions.


Methodology

The Data Envelopment Analysis (DEA) Model

We estimate DEA efficiency scores using the CCR model of constant returns to scale, which has been the model used in this specific literature. Given the main purpose of our article is to improve upon what has been done in the literature, we follow the input and output selection of Gutiérrez-Nieto et al. (2009). There are three inputs: total assets (A), represented by the total of all net asset accounts ($), taken directly from the Mixmarket dataset; Operating Cost (C), defined as expenses related to operations, such as transportation, office, and personnel expenses; and number of active employees (E) employed by the MFI. The two financial outcomes considered are: gross loan portfolio (L), defined as the outstanding balance of the all of the MFI's outstanding loans including current, delinquent and restructured loan, but not loans that have been written off (it does not include interest receivable); and financial revenue (R), defined as the revenue generated from the gross loan portfolio and from investments plus other operating revenue. The two social outcomes considered are: number of active borrowers who are female (W); and an indicator of benefit to the poorest (P), defined through the average loan balance per borrower and the per capita Gross National Income.

The number of active borrowers who are female is taken directly from the Mixmarket dataset. We interpret this as a measure of female empowerment at home or, in some degree, within her society. The indicator of benefit to the poorest is another social outcome of MFIs widely considered in the literature, since MFIs are partially designed to provide financial services to the poorest in a given country. We follow Gutierrez-Nieto et al. (2009) and define an indicator to benefit the poorest as the ratio between the average loan balance per borrower and per capita Gross National Income (pcGNI). The intuition is that wealthier individuals are able to borrow larger amounts. Thus, if the average loan balance per borrower is low, then a larger fraction of poor individuals are being financed. The division of the average loan balance per borrower by per capita Gross National Income (pcGNI) is only intended to capture differences in countries average per capita income, hence is it a normalization. This indicator is then normalized again to lie on the interval [0,1] and inverted (one minus the indicator), such that a number close to 1 represents a higher percentage of loans to the poor. Finally, this new indicator is multiplied by the number of active borrowers, such that one can construct an approximate measure of the number of poor borrowers. In this paper we consider two main empirical DEA specifications: the financial frontier, represented by inputs (Assests, Costs and Employees) and outputs (gross loan portfolio and financial revenue), named as ACE-LR; and the social frontier, represented by the same inputs but considers as outputs (number of active borrowers who are female and the indicator of benefit to the poorest) and is named as ACE-WP.

The Efficiency Decomposition

The decomposition we adopt in this paper follows closely the one proposed by Portela and Thanassoulis (2001). For our case, we will first compare MFIs within the same country, such that country characteristics are isolated from final estimates,


certainly not introducing any bias in the DEA efficiencies. Then, we will compare all MFIs in a way that differences between countries production possibility curves will be identified. Figure 1 shows how the decomposition works graphically. For simplicity assume there are only two countries A and B and that there is only one input and one output. Each triangle is a DMU, in this case a MFI, that is established in country A and each square is a MFI established in country B. Solid lines represent the DEA efficiency frontiers for each country (which we label as local efficiency - EFL), while the dashed line represent the DEA efficiency frontier considering both countries analyzed (which we label as global efficiency - EFG).

Now consider the MFI labeled as Z established in country B. Its DEA efficiency with

respect to other MFIs in the same country may be assessed by 𝑂𝑍

𝑂𝑍1, which we label as

MFI Efficiency (MFIE) and, in this case, is clearly smaller than 1 making Z an inefficient MFI. This MFI, however, is not as inefficient as it would appear if all MFIs efficiencies of all countries were assessed together, which we label as MFI Overall

Efficiency (MFIOE). In this case, Zs efficiency would be 𝑂𝑍

𝑂𝑍2 < 𝑂𝑍𝑂𝑍1. Thus, when pulling

every MFI together in a unique production frontier, previous analysis on this specific topic, such as Gutiérrez-Nieto et al. (2009), are introducing a negative bias in all MFIs efficiencies.

Figure 1: Decomposition of Overall Efficiency


Now we turn to the measurement of how efficiency frontiers vary within countries.

Note that the coefficient obtained by Gutiérrez-Nieto et al. (2009), namely 𝑂𝑍

𝑂𝑍2,

represents two effects. First, it represents the lack of effort or ability of MFI Z that prevents it from being efficient in comparison to all other MFIs in the same country. And second, the possibilities of the country to provide/produce outputs, which we label as Country Efficiency (CE). The CE, which is equal to the distance between each

country's local efficiency frontier and the global efficiency frontier, 𝑂𝑍1

𝑂𝑍2, can be

obtained by

𝑂𝑍1

𝑂𝑍2 =𝑂𝑍

𝑂𝑍2 ×𝑂𝑍1

𝑂𝑍= 𝑀𝐹𝐼𝑂𝐸 ×

1𝑀𝐹𝐼𝐸


Table 2 presents average efficiencies by country. First note that, as expected, MFIE are always greater than MFIOE. Also, MFIE are not only a shifted version of MFIOE, that is, there are substantial changes in the country ranking order if ones uses local or global efficiencies measures. This is another evidence that comparisons between financial and social efficiencies using global efficiency measures might be significantly misleading.

Table 1: Financial and Social Efficiencies by Countries

Country Financial Efficiency Social Efficiency

MFIF MFIOE MFIF MFIOF

Bangladesh 0.895 0.679 0.831 0.425

Bolivia 0.941 0.736 0.475 0.116

Brazil 0.876 0.782 0.478 0.109

Ecuador 0.907 0.752 0.521 0.167

Ghana 0.862 0.577 0.433 0.300

India 0.892 0.723 0.666 0.642

Indonesia 0.861 0.689 0.382 0.179

Mexico 0.895 0.660 0.541 0.214

Nepal 0.894 0.682 0.550 0.423

Nicaragua 0.939 0.741 0.565 0.132

Peru 0.913 0.741 0.612 0.154

Philippines 0.835 0.643 0.547 0.197

Russia 0.870 0.837 0.291 0.089

Tajikistan 0.944 0.784 0.586 0.082


To look at the correlation between financial and social efficiencies, we use a different strategy than the one adopted by Gutiérrez-Nieto et al. (2009). We estimate the correlation between efficiencies by the following fixed effects model

𝐹𝑖𝑛𝑎𝑛𝑐𝑖𝑎𝑙 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦𝑖𝑐 = 𝛽 + 𝜌. 𝑆𝑜𝑐𝑖𝑎𝑙 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦𝑖𝑐 + 𝜆𝑖𝑐 + 𝜖𝑖𝑐

where i and c stands for, respectively, the MFI and the country the institution is located at; 𝜌 is the parameter of interest; and 𝜆𝑖𝑐 and 𝜖𝑖𝑐 are, respectively, a country fixed effect and an error term. The estimated coefficients are presented in Table 2. In column 1 we estimate the correlation via equation 2 of the efficiencies presented in Table B1 of Gutiérrez-Nieto et al. (2009). Columns 2 and 3 are the estimated correlations using our data. As can be observed, the correlation between efficiencies estimated in their paper is significantly higher than the correlation we obtain considering country differences. This still implies that MFIs that are socially efficient are also financially efficient, however, the relationship between them is much weaker than what previous studies imply.

Table 2: Estimated correlations between Financial and Social Efficiencies

Variables (1) (2) (3)

Social Efficiency .252*** 0.89*** 0.90***

(.073) (.017) (.014)

Country Fixed Effects NO NO YES

Observations 89 483 483

Note: All regressions include a constant. Standard Deviation presented in parentheses. *** indicates p<0.01.

We now estimate the relationship between efficiencies and profitability. This exercise was also done by Gutiérrez-Nieto et al. (2009) in which they obtained that the correlation between social efficiency and profits was positive but never statistically different from zero. We take the return on equity (ROE) as a measure of profitability and compute correlations for financial and social efficiencies using our sample. Results are presented in Table 4. As expected, financial efficiency is positively and significantly correlated with ROE. On the other hand, the correlation between social efficiency and ROE is found to be negative.

Table 4: Estimated correlations between ROE and Financial and Social Efficiencies

Variables Financial Social

Social Efficiency . 000329** -.000524***

(.000145) (.000072)

Country Fixed Effects YES YES


Note: Standard Deviation presented in parentheses. *** indicates p<0.01 and ** indicates p<0.05.

We now look at Country Efficiencies (CE) estimated by equation 1. As one can see in Table 5, financial and social frontiers vary significantly between the countries in our sample. The correlation between efficiencies is -.474 suggesting that countries with high financial frontiers have low social frontiers. This, in our view, might be a result of two effects: (a) MFIs located in different countries may face different conditions and (b) different MFI types might be sorting themselves in different countries given some country-specific characteristic. Both effects, however, imply that DEA efficiency coefficients should be assessed by comparing units from the same country.

Table 5: Financial and Social Country Frontiers

Variables Financial Efficiency Social Efficiency Efficiency Ranking Efficiency Ranking Russia 0.962 1 0.303 9 Brazil 0.894 2 0.231 12 Ecuador 0.831 3 0.310 8 Tajikistan 0.829 4 0.132 14 Peru 0.810 5 0.240 10 India 0.808 6 0.951 1 Indonesia 0.797 7 0.419 5 Nicaragua 0.788 8 0.228 13 Bolivia 0.784 9 0.236 11 Philippines 0.769 10 0.330 7 Bangladesh 0.759 11 0.512 4 Nepal 0.748 12 0.744 3 Mexico 0.734 13 0.395 6 Ghana 0.666 14 0.800 2

Conclusions

In this paper we extend the analysis carried out by Gutiérrez-Nieto et al. (2009) by looking at the interactions between financial and social efficiencies but taking into account differences between countries. Results are quite different from the ones observed in Gutiérrez-Nieto et al. (2009). Our estimate of the correlation between financial and social efficiencies is much smaller (correlation is about 35% of what they find). Looking at profits, we show that, as expected, financial efficiency is positively and significantly correlated with ROE. On the other hand, the correlation between social efficiency and ROE is found to be negative.

References

Amores AF & Contreras I. (2009). New approach for the assignment of new European agricultural subsidies using scores from data envelopment analysis: Application to olive-growing farms in Andalusia (Spain). European Journal of Operational Research, 193: 718-729.


Berguiga I. (2009). Social Performance vs. Financial Performance of Microfinance Institutions. Mimeo, East Paris University.

Gutierrez-Nieto B, Serrano-Cinca C & Mar Molinero C. (2009). Social efficiency in microfinance institutions. Journal of the Operational Research Society, 60: 104-119.

Hallock K & Koenker R. (2001). Quantile Regression. Journal of Economic Perspectives, 15: 143-156.

Mixmarket (2012). The Microfinance Information eXchange (MIX). http://www.mixmarket.org/en/what.is.mix.asp, accessed in 8 May 2012.

Portela MCAS & Thanassoulis E. (2001). Decomposing school and school type efficiency. European Journal of Operational Research, 132: 114-130.

Sampaio B. 2012. To Generalize or Not to Generalize? Comment on Robinson and Davies. Journal of the Operational Research Society, 63: 563-565.


25. Performance Evaluation in Hospitals: a study on hospitals financed by the Brazilian Unified Health System

Antônio Artur de Souza Av. Antônio Carlos, 6627, Pampulha. Belo Horizonte-MG - Universidade Federal de Minas Gerais, [email protected]

Emerson Alves da Silva Av. Antônio Carlos, 6627, Pampulha. Belo Horizonte-MG - Universidade Federal de Minas Gerais, [email protected]

Douglas Rafael Moreira Av. Antônio Carlos, 6627, Pampulha. Belo Horizonte-MG - Universidade Federal de Minas Gerais, [email protected]

Alisson Maciel de Faria Marques Av. Antônio Carlos, 6627, Pampulha. Belo Horizonte-MG – Secretaria de Estado de Saúde de Minas Gerais, [email protected]

Ewerton Alex Avelar Av. Antônio Carlos, 6627, Pampulha. Belo Horizonte-MG - Universidade Federal de Minas Gerais, [email protected]

Bernardo Franco Tormin Av. Antônio Carlos, 6627, Pampulha. Belo Horizonte-MG - Universidade Federal de Minas Gerais, [email protected]

Abstract

Hospitals have been forced to change because of both users’ pressure for higher quality services and regulatory agencies’ pressure for better resources management. Notably, low quality health services are directly derived from poor hospital management and causes substantial user dissatisfaction. Against this background, this paper reports on a performance evaluation of hospitals financed by the Brazilian Unified Health System (SUS). It analyzes the performance of 20 hospitals (among public and voluntary organizations) in seven States in 2008. Focusing on financial management, it uses a set of operational ratios (i.e., Occupancy Rate, Average Length of Stay, and Full Time Employees per Bed) as inputs, and financial ratios (i.e., EBIT Margin, EBITDA Margin, Return on Assets, Return on Invested Capital, and Net Margin) as outputs. The assessment framework confirms the hypotheses, and shows that financial management efficiency differs between public and voluntary hospitals.

Keywords: Efficiency, Data Envelopment Analysis (DEA), Hospitals, Financial Ratios, Operational Ratios, Public and Voluntary Hospitals.


Introduction

Data Envelopment Analysis (DEA) is a non-parametric model used to assess efficiency in a set of production units, named as Decision Make Units (DMUs) (Angulo Meza et al. 2005). A DMU is a unit that is able to transform inputs into outputs (Azevedo et al., 2012).

In the face of the need to improve processes and analyze the performance of production units with a view to reaching maximum efficiency, DEA has been used as an important tool for both researchers and policy makers. Besides, the model has some characteristics that make it suitable for application in several types of analysis, especially in complex organizations (Marinho 2001).

Struett (2005) and Smet (2009), among others, contend that hospitals are usually complex organizations in which the management processes and performance assessment processes are significantly intricate. These organizations can benefit from DEA models, especially for performance evaluation and gaining insight into possible improvement actions, as shown in some exploratory studies, such as Cesconetto, Lapa & Calvo (2008), and Guerra, Souza & Moreira (2012).

This paper reports on a study aimed to develop three indicator frameworks and analyze the efficiency of Brazilian hospitals that obtain funding from the Brazilian Unified Health System (SUS). Three models were developed aiming to provide support for improved financial management in hospitals building on operational indicators. The study also included environmental variables (i.e., type of organization, number of beds, and revenues) to understand whether these factors impact the scores of the sample hospitals.

Methods

DEA Overview

DEA is a non-parametric model to assess efficiency of a set of production units, or DMUs. A DMU is a decision-make unit capable of transforming multiple inputs into multiple outputs (Azevedo et al. 2012; Moreira 2010). Unlike the parametric models, DEA aims to optimize each DMU individually, determining an empirical efficient frontier based on a set of resources (inputs) and outputs (Cesconetto, Lapa & Calvo 2008).

The model, first approached in 1978 by Charnes, Cooper & Rhodes drawing on Farrel (1957), identifies efficient DMUs based on the assumption that they operate with constant returns to scale, that is, outputs changes proportionally to changes in the inputs. The literature refers to this model as CCR, which stands for the initials: Charnes, Cooper and Rhodes. The assessment of DMUs considering variable returns to scales (VRS) bases builds on the BCC model, which stands for Banker, Charnes & Cooper (1984).

Efficiency calculation using DEA model can be oriented either to inputs or to outputs. Input-oriented DEA aims at producing the same amount of outputs despite reducing the use of inputs (Azevedo et al. 2012). On the other hand, output-oriented DEA aims


at maximizing output despite keeping inputs unchanged (Gomes, Mello & Lins 2004; Correia, Soares de Mello & Angulo-Meza 2011).

The major advantages of DEA are: (i) to classify each DMU as inefficient or efficient (the DMU that is at the efficient frontier); (ii) to be based on individual observations, instead of mean values; (iii) to incorporate inputs and outputs with different units of measures; (iv) to identify optimal production and consumption values; (v) to identify the inefficiency measure for every unit away from the frontier; (vi) to work without converting all inputs and products in monetary units; and (vii) to consider discrepant values as comparison parameters for the DMUs (Marinho 2001; Cesconetto, Lapa & Calvo 2008).

The present study applies an output-oriented DEA with Variable Returns to Scale. The analysis of hospitals’ financial performance involved separating public from voluntary hospitals. It was employed the Multiplier Method as a mathematical method of structuring the DEA.

Sample description and data collection

Financial indicators were obtained from secondary data containing hospitals’ financial statements and accounting reports. Given limited data access, data collection was restricted to the period of 2008. There was no aprioristic selection of the organizations that would be included in the sample, their inclusion thus being related to financial data availability.

The financial statements and annual reports were obtained from: (i) hospitals’ websites; (ii) official state gazettes (in the case of São Paulo and Minas Gerais); and (iii) results from GoogleTM search engine. Most data included poor specification of accounts, which were inappropriately classified in the statements or were not available. Therefore, the selection criteria also included data completeness and reliability. This led to 20 hospital organizations from seven states (UF), namely: São Paulo (SP), Minas Gerais (MG), Alagoas (AL), Ceará (CE), Pará (PA), Paraná (PR), and Rio Grande do Sul (RS).

The collected data were imported to MS ExcelTM spreadsheets. Due to the variety and amount of irrelevant accounts, a standardized Chart of Accounts was established to serve as a model. This produced 55 financial indicators, which were subsequently screened with a view to defining those that would make up a hospital efficiency assessment framework (i.e., DEA models).

The indicators deemed as relevant to assess hospital performance were: (i) EBIT Margin; (ii) EBITIDA Margin; (iii) Return on Assets (ROA); (iv) Return on Invested Capital (ROIC); and (v) Net Margin. These indicators represent, therefore, the outputs of the efficiency models formulated in the study.

The operational indicators (i.e., those related to the services provided and the organizations’ performance) were obtained from DATASUS (a databank provided by the Brazilian Unified Health System Informatics Department) and CNES (a national register of health organizations). According to Guerra, Souza & Moreira (2012), these sources provide a number of valuable data, including number of bed, medical procedures, professionals, and patients, among others.


These data were then imported to MS Excel spreadsheets in order to calculate the operational indicators: Mean time of stay (TMP); Occupancy rate (TO); e Full time workers (FTE) / Bed (FL). They were used as inputs in the efficiency models. The financial and operational indicators were subsequently standardized, but preserving their proportionality.

Finally, logarithmic transformations were performed to reduce discrepancy between financial and operational indicators. Because of null values in the sample, 1 unit was added to every indicator. The logarithm base selected was the highest value of every indicator considering all the hospitals. This produced indicators with values between 0 and 1 in a scale that does not skew weight distribution in the model, and therefore, the efficiency of each DMU.


Setting up the models

Three models were set up for data analysis. The first (M1) approaches efficiency focusing on the public nature of the hospitals and thus uses the indicators shown in Chart 1. The expectation of this model is of relative homogeneity in the way to measure profitability in public hospitals and voluntary hospitals (non-profit organizations), as public hospitals usually do not register depreciation. Depreciation can be seen as a non-financial expense that reduces the accounting profit of an organization, which could thus benefit the results of public hospitals.

Chart 1: Models 1, 2 e 3 – inputs and outputs.

Model Inputs Outputs

M1 Mean time of stay (TMP); Occupancy rate

(TO); e FTE / Bed (FL) EBITDA Margin (ME1); e Return on Invested

Capital (ROIC)


(TO); e FTE / Bed (FL) EBIT Margin (ME2); Return on Assets (ROA);

e Net Margin (ML)


(TO); eFTE / Bed (FL)

EBITDA Margin (ME1); Return on Invested Capital (ROIC); EBIT Margin (ME2); Return on

Assets (ROA); e Net Margin (ML)

Source: Created by the authors. The second model (M2) was estimated focusing on business indicators of profitability, including depreciation and amortization expenses (cf. Chart 1). This model is assumed to provide better performance results for voluntary hospitals, since their operational result include revenues from private health insurances, which usually have greater mean net margin and return on assets.

The third model (M3) is a combination of the two others (cf. Chart 1). The aim is to assess how the efficiency scores behave, assuming that this behavior will resemble one of the two other models, once the DEA will adjust the weight of the indicators that lead to better scores.


Results

Table 1 provides the mean efficiency scores of the organizations in the sample for each model. The results show that the efficiency levels were significantly high in the three models. Nevertheless, this does not mean that the organizations are efficient. As efficiency in a DEA is the comparison of organizations within the sample, this simply means that the financial results of these organizations are similar, that is, they have similar production functions.

As expected, the first model (M1) pointed to lower levels of efficiency than the other two models. As mentioned above, excluding depreciation expenses from the profitability calculation would make the organizations financially similar.

Other important result was the similarity of results in models 2 and 3. Given the increase in the indicator values in the third model the weights fit the most favorable outputs. As the M2 indicators presented better results, M3 followed suit.

Table 1 – Mean efficiency scores of the organizations in the sample for each model

Model 1 Model 2 Model 3 0.7707 0.8412 0.8421

Source: Created by the authors.

Another objective of the research is to explore some environmental factors proposed in the literature to assess the behavior of the efficiency scores. The first factor was type of organization. Hospitals were divided into two groups: voluntary and public hospitals. The results are shown in Table 2.

Table 2 – Mean efficiency scores according to type of organization, and model

Organization Public Voluntary Model 1 0.8547 0.7387 Model 2 0.8911 0.8092 Model 3 0.8946 0.8104


The efficiency scores of the public hospitals were higher than those of the voluntary hospitals in all the models under scrutiny. The largest difference is found in M1, which is the model in which the financial results are included more similarly for all organizations in the sample.

Considering the number of beds, the results show that the organizations offering from 100 through 200 beds are those with the highest mean scores in all the models (Table 3). The results suggest that this amount of beds can provide the best economies of scale to hospital organizations.


Table 3 – Mean efficiency scores according to number of beds, and model.

Beds Model 1 Model 2 Model 3 0-100 0.6766 0.7581 0.7581 101-200 0.8516 0.9145 0.9145 201-300 0.7705 0.8396 0.8396 301+ 0.7803 0.8489 0.8514


As for income (BRL million) in 2008, the results show that the hospitals earning 7-20 million Brazilian Reais had the best performance (Table 4). In order to understand this, further studies should investigate the complexity of the services provided, as well as the synergy among the procedures offered in the hospitals. This result could suggest that these organizations have economies of scope.

Table 4 – Mean efficiency scores according to revenue (million BRL), and model.

Revenue Model 1

Model 2

Model 3

0-7 0.6667 0.7433 0.7433 7-20 0.8410 0.9228 0.9228 21-40 0.7868 0.8511 0.8511 41+ 0.7776 0.8333 0.8368


Conclusions

This study aimed to analyze the financial management efficiency of Brazilian hospitals building on operational indicators as inputs and financial indicators as outputs. The results show that public hospitals have higher efficiency scores than the voluntary hospitals in all the three models applied to the sample. This result, however, should be observed with caution.

One limitation of this study is that human resources expenses are accounted differently in voluntary and public hospitals. This expense is usually not allocated to the public hospital organizations, rather to a central agency within the public administration. This can skew the results, but the accounting information of this central agency is not fully available.

Besides, DEA is a tool that is sensitive to a number of factors, including sample, inputs, outputs, and period of observation. The relative nature of the tool should always be regarded, especially in exploratory studies like this. Therefore, results vary according to DMU inputs and/or outputs, and indicators. As this study was meant to be a first attempt to design a model for hospital financial assessment, we hope that further research draws on our results to develop more robust models.


References

Angulo Meza, L.; Biondi Neto, L.; Soares de Mello, J. C. C. B.; Gomes, E. G. (2005). ISYDS - Integrated System for Decision Support (SIAD - Sistema Integrado de Apoio à Decisão): a software package for data envelopment analysis model. Pesquisa Operacional, 25(3): 493-503.

Azevedo G. H. I., Roboredo M. C., Aizemberg L., Silveira J. Q., Soares de Mello J. C. C. B. (2012) Uso de análise envoltória de dados para mensurar eficiência temporal de rodovias federais concessionadas, Journal of Transport Literature 6 (1): 37-56.

Cesconetto A., Lapa J. S., Calvo M. C. M. (2008) Avaliação de eficiência produtiva de hospitais do SUS de Santa Catarina, Brasil, Caderno de Saúde Pública 24 (10): 2407-2417.

Charnes A., Cooper W. W., Rhodes E. (1978) Measuring the Efficiency of Decision Making Units, European Journal of Operational Research 2: 429-444.

Correia T. C. V. D., Soares de Mello J. C. C. B., Ângulo-Meza L. (2011) Eficiência Técnica das companhias aéreas brasileiras: um estudo com análise envoltória de dados e conjuntos nebulosos, Produção 21 (4): 676-683.

Farrel, M. J. (1957) The Measurement of Production Efficiency, Journal of Royal Statistical Society 3: 253-290.

Gomes E. G., Mello J. C. C. B. S., Lins M. P. E. (2004) Redistribuição de Inputs e Outputs em modelos de Análise Envoltória de Dados com Ganhos de Soma Zero. Pesquisa Operacional 24 (2): 269-284.

Guerra M., Souza A. A., Moreira D. R. (2012) Performance Analysis: A Study Using Data Envelopment Analysis in 26 Brazilian Hospitals, Journal of Health Care Finance 38 (4): 19-35.

Marinho A. (2001) Estudo de Eficiência em alguns hospitais públicos e privados com a geração de rankings, Rio de Janeiro, Instituto de Pesquisa Econômica Aplicada.

Moreira D. R. (2010) Análise de Eficiência usando Data Evelopment Analysis e Composição Probabilística para Procedimentos Médicos Referentes às Doenças Isquêmicas do Coração no Estado de Minas Gerais. 111f. 2012. Dissertação (Mestrado em Engenharia de Produção) - Universidade Federal Fluminense. Niterói.

Neves A. P. T. P. (2009) Indicadores Financeiros e Operacionais para Avaliação de Desempenho em Hospitais. 71 f. 2009. Monografia (Graduação em Ciências Contábeis) – Universidade Federal de Minas Gerais, Belo Horizonte.

Struett, M. (2005) Custeio baseado em atividades em laboratórios de análises clínicas: estudo de caso em um hospital filantrópico. 165 f. 2005. Dissertação (Mestrado em Administração) – Universidade Estadual de Londrina, Londrina, 2005.


26. Performance Evaluation of expenditure in Primary Care: the Case of Brazil’s Southeastern cities.

Lucas Maia dos Santos* Instituto Federal de Minas Gerais – Campus Sabará, [email protected]

Márcio Augusto Gonçalves Universidade Federal de Minas Gerais, [email protected]

Márcia Mascarenhas Alemão Universidade Federal de Minas Gerais, [email protected]

Marco Aurélio Marques Ferreira Universidade Federal de Viçosa, [email protected]

Lucas Campos Vaz Universidade Federal de Minas Gerais, [email protected]

Heloiza Azevedo Drummond Universidade Federal de Minas Gerais, [email protected]

Abstract

This study aimed to analyze the performance of cities of Brazil’s southeastern region in resource allocation on primary care, from 2007 to 2010. In order to do the performance analysis, we used in this study the technical efficiency scores produced by the Data Envelopment Analysis (DEA) methodology. But, before starting the cities’ efficiency analysis, the cluster analysis was applied to group similar cities. This study proposes an analytical model of the performance of 1097 cities, based on the National Primary Care Policy. The efficiency scores obtained highlights the disparities in the allocation of resources and the results obtained in primary care. This fact could be justified due the absence of procedures of relative comparison between cities and the decentralization of public expenses in public healthcare. The results shed light on the possibility to improve the performance of primary care, given the current level of resource allocation.

Keywords: Brazil. Primary Care. Performance. Public Expenditure. DEA.









Introduction

According to the Ministry of Health (2008) the results which Brazil has obtained in the healthcare area, considering the amount invested are below the expected results. Little by little, the public policy makers begin to admit that money may not have been well invested. Among the main possible solutions for the problem we can state: increase the amount of financial resources – probably unviable due to the resource limitation – or increase the efficiency in resource allocation.

Focusing on the above highlighted problems and seen the increasing of efficiency as a feasible solution for the improvement of the results which Brazil has obtained in healthcare sector, this study has aimed to answer the following question: what is the performance of cities in the southeastern region of Brazil in primary care resource allocation? In this context, measuring the technical efficiency in allocating resources as a performance measure becomes a good opportunity. According to Selden and Sowa (2004) and Scott and Davis (2007), the idea of performance is focused, typically, in the outputs and outcomes of programs or policies. According to Farrel (1957) and Charnes et al. (1978) technical efficiency is the ability to produce more outputs with a given amount of inputs.

It is justified that the evaluation of efficiency in allocating public resources may serve as an instrument to support public managers’ decision making as well as identifying benchmark cities so that the inefficient cities may find the best practices in efficient cities. In performance analysis, according to Greiling (2006), benchmarking is designed to make learning easier as a continuum and systematic process of measuring products, services and practices aiming at correcting failures and improving results. This idea of a comparison between units of analysis by relative efficiency scores is found in the performance analysis literature (SELDEN; SOWA, 2004; GREILING, 2006; SCOTT; DAVIS, 2007).

Given the problems proposed and the justifications for conducting this study, the general objective was to analyze the performance of cities in the southeastern region of Brazil in primary care expenditure, from 2007 to 2010. More specifically, we separated the groups of the most similar cities according to characteristics which could have influence on efficiency scores and then applied the proposed efficiency model.

Methods

From the population of 1668 cities of Southeastern region of Brazil we chose only cities which had at least one Family Health Team – the group of physician, nurses, nurse assistants and other professionals due to that are the main strategy of the National Primary Care Policy. Afterwards, after an exploratory analysis of the data, we excluded from the sample those with inconsistent variables and with the absence of some observation for one or more analyzed years (2007-2010). It was excluded 570 cities because of that.

The final sample was comprised of about 70% of the cities in Minas Gerais, 60% of the cities in the State of Espírito Santo, 50% of the cities in Rio de Janeiro and 52% of the cities in São Paulo. In some cases, existing secondary data create an opportunity for evaluative and simplified comparisons.


For data collection and analysis, the National Primary Care Policy from 2006 was established as the reference public policy. By this definition, the analysis period was from 2007 - the first year of PNAB - to 2010, due to this being the last year with available data in the researched databanks. All of these variables were collected and tabulated during the period from July to September 2011.

Analysis procedures

In order to do the performance analysis we used in this study the technical efficiency scores produced by the Data Envelopment Analysis (DEA) methodology. But, before starting the cities’ efficiency analysis, the cluster analysis, according to Ferguson et al. (2000) was applied to group similar cities.

In the cluster analysis we used the non-hierarchical k-means method quoted in Maroco et al. (2005) and Hair et al. (2005) for partitioning the city groups.

According to Sugar and James (2003), a fundamental problem in cluster analysis is to determine the best number of groups. In order to determine this number that shall be used in the study we employed the Calinski and Harabaz index whose equations are in Milligan and Cooper (1985) and Sugar and James (2003). According to Milligan and Cooper (1985), the Calinski and Harabaz index is considered to be a robust index for defining the number of clusters.

After the definition of the clusters, the measuring of efficiency was performed through the use of the Data Envelopment Analysis (DEA) methodology, output-oriented supposing variable returns to scale (BANKER et al., 1984).

The output orientation was applied because the objective of the cities must be to increase the primary care services production and not to reduce the budget allocated in the field. Also, quoted authors have discussed the scarcity of resources for this sector, indicating that the increase in allocation efficiency is necessary (FLEURY, 2001; WORLD BANK, 2004; MINISTRY OF HEALTH, 2008).

In this study, we have the availability of a panel of efficiency scores for verifying changes in the productivity of these cities over the years in the allocation of resources in primary care. According to Banker et al. (2005) the Malmquist index was proposed by Caves, Christensen and Diewet (1982) with the objective of measuring changes in productivity between two periods of time by the distance between a DMU and the frontier of production for each period.

Analytical model for performance evaluation in Primary Care resource allocation

What will define how similar the groups are internally are the variables inserted to characterize them. In this study, the model for ranking the cities was built from four dimensions considered to be important for primary care in health and that in some way may have impact on the expenditure of distinct cities: private coverage, development, size and poverty. Figure 1 describes the four dimensions.


Figure 1 – Dimensions for characterizing the cities

Source: Elaborated by the author

After the definition of the clusters we inserted some more variables available by Ministry of Health for additional characterization the cities. These variables are related to the cities’ infrastructures as the following ones:

Proportion of houses with water piping (x102);

Proportion of houses with garbage collection (x102);

Proportion of houses with sewage piping (x102);

Proportion of houses made of bricks (x102);

Proportion of houses which have electricity (x102).

These variables were used because, according to the World Bank (2007), other factors such as the access to drinking water and sanitation may also have an influence on comparisons between expenditures and results.

With the formation of the groups it was possible to perform the relative technical efficiency analysis. In this direction, the allocation of resources for the primary care will be represented by the number of Family Health Teams and number of health establishments. These two variables according to National Primary Care Policy represent the sector’s resource allocation.

The resource allocation model for performance evaluation in Figure 2 was built by two inputs and three outputs. The first output was characterized by the number of people registered by the Family Health Team. The number of people registered by the Family Health Team, number of home visits of the Family Health Team and the ambulatory production in primary care were the outputs of the model.

Development: HDI-M. Source: United Nations Development Program.

Size: Number of city inhabitants. Source: Brazilian Institute of

Geography and Statistics

Poverty: number of beneficiaries of the government benefit Conditional

Cash Transfer program over city residing population (x102). Source:

Institute of Applied Economic Research – IPEA

Private Coverage: People with private health coverage over number of people registered by Family Health

Team (x103). Source: Ministry of Health.


Figure 2 – The model of technical efficiency in allocating resources on primary care

Source: elaborated by the authors


Comparing groups we can affirm that, in general, Group 2, shown in Table 1, presented lower averages for all the variables used to rank them, except for proportion of the population covered by the Conditional Cash Transfer program.

Group B was, after the Cluster analysis, partitioned into Groups 3 and 4. It was possible to see that, for the analyzed dimensions, groups 1, 2, 3 and 4 were formed with cities of different sizes, but similar by other dimensions.

Despite this fact, Group 2, with the lowest private coverage has the cities with the lowest population and, by analyzing the quartiles, 75% of these have less than 12 thousand inhabitants. This group also showed the smallest averages for the other variables, indicating the presence of the smallest cities and also the poorest of the sample.

Health establishments in

primary care.

People registered by the Family Health Team (x10-3)

Number of home visits of the Family Health Team (x10-3).

Ambulatory production in Primary Care (x10-3).

Inputs Outputs

FOR PRODUCING

Family Health Teams (ESF).


Table 1 Descriptive Statistics – Group A split in Group 1 and Group 2*

Gr Variable N Avg Min Max 25% Median 75%

G1

WATER 360 76.17 4.56 100.00 67.97 80.06 90.27 GARBAGE 360 81.00 23.32 100.00 71.88 83.71 92.88 SEWAGE 360 68.00 0.00 100.00 55.21 74.21 87.94 BRICK 360 95.75 46.28 100.00 95.04 98.71 99.61

ELECTRICITY 360 97.46 0.00 100.00 97.25 98.79 99.42 PRIVATE 360 47.54 0.00 100.00 24.37 46.26 71.17 TRANFER 360 5.77 1.23 10.59 4.49 5.80 7.11

POPULATION 360 27.58 1.20 830.67 4.76 10.40 22.76 HDI 360 0.76 0.67 0.85 0.74 0.75 0.78

G2



POPULATION 255 9.92 1.65 52.98 4.66 6.95 11.88 HDI 255 0.67 0.57 0.75 0.65 0.68 0.70

Variables: WATER – proportion of houses with water piping (x102); GARBAGE – proportion of cities with garbage collection (x102); SEWAGE – proportion of houses with sewage piping (x102); BRICK – proportion of houses made of bricks (x102); ELECTRICITY – proportion of houses which have electricity (x102); PRIVATE – proportion of population covered by private healthcare assistance (x10-3); TRANSFER – proportion of population benefited by the “Conditional Cash Transfer” program (x102); POPULATION – population living in the city (x10-3); HDI – Human Development index; Gr – group. Tests results: highest pseudo-F of Calinski and Harabaz: 362.71; p-value<0.01 for univariate normality asymmetry and kurtosis test and bivariate and multivariate normality of Doornik-Hansen; p-value<0.01 for differences between the groups G1 and G2 by the Kruskal-Wallis non-parametric test.

Source: research results.

Table 2 Descriptive Analysis – Group B split in Groups 3 and 4*

Gr Var N Avg Min Max 25% Median 75%

G3



POPULATION 306 98.92 1.05 2.41 13.20 34.12 94.23 HDI 306 0.80 0.73 0.92 0.78 0.80 0.82

G4



POPULATION (in 1,000) 176 23.37 1.39 356.53 5.22 9.55 19.33 HDI 176 0.74 0.57 0.80 0.73 0.75 0.76

Variables: WATER – proportion of houses with water piping (x102); GARBAGE – proportion of cities with garbage collection (x102); SEWAGE – proportion of houses with sewage piping (x102); BRICK – proportion of houses made of bricks (x102); ELECTRICITY – proportion of houses which have electricity (x102); PRIVATE – proportion of population covered by private healthcare assistance (x103); TRANSFER – proportion of population benefited by the “Conditional Cash Transfer” program (x102); POPULATION – population living in the city (x10-3); HDI – Human Development index; Gr – group.

Tests results: highest pseudo-F of Calinski and Harabaz: 314.02; p-value<0.01 for univariate normality asymmetry and kurtosis test and bivariate and multivariate normality of Doornik-Hansen; p-value<0.01 for differences between the groups G3 and G4 by the Kruskal-Wallis non-parametric test.



Group 1 and Group 4 have similar characteristics, except for the private coverage percentage. While Group 1 has 75% of the cities with less than 23 thousand inhabitants, Group 4 has 75% of the cities with less than 20 thousand inhabitants. As to development, they are close in HDI indexes, Group 1 having an HDI index average of 0.76 and Group 4 has 0.74.

Group 3 showed itself to be opposed to Group 2, presenting the highest averages for the variables analyzed. This group may be considered as the one formed by the biggest cities in the sample and, at the same time, being in average the most developed.

Placing the groups in an imaginary line which measures the dimensions stated for ranking the cities, we may say that Groups 2 and 3 are at the extremities of this imaginary line, while Groups 1 and 4 are in intermediate positions.

Technical efficiency analysis

The descriptive statistics of technical efficiency in allocating primary care resources for groups are set re set out in tables 3 to 6, for the four years which were analyzed (2007-2010).

Table 3 Technical efficiency and productivity changing for Group 1

Efficiency Malmquist Year Obs Avg Min Max 25% Median 75% Avg 25% Median 75% 2007 360 57.68 18.95 100.00 44.59 54.84 66.06 - - - - 2008 360 59.34 20.03 100.00 44.98 56.33 70.16 0,99 0,90 0,99* 1,06 2009 360 65.11 19.70 100.00 52.95 64.20 75.53 0,99 0,90 0,98* 1,05 2010 360 62.39 22.68 100.00 48.74 58.37 73.56 1,01 0,90 1,00 1,07 Obs – Observations; Min – Minimum; Max – Maximum; Avg – average Observations: p-value<0.05 for univariate normality asymmetry and kurtosis test and bivariate and multivariate normality of Doornik-Hansen; * significant at 5% by the nonparametric Wilcoxon test for the theoretical value of the median equal to 1.



Efficiency Malmquist Year Obs Avg Min Max 25% Median 75% Avg 25% Median 75% 2007 255 58.05 19.51 100 43.59 54.34 68.62 - - - - 2008 255 48.07 15.77 100 31.49 42.51 60.24 1,01 0,90 0,98* 1,06 2009 255 55.93 18.67 100 41.14 51.20 65.58 1,00 0,89 1,00 1,09 2010 255 59.70 28.34 100 43.37 54.95 72.32 0,98 0,85 0,97* 1,08

Obs – Observations; Min – Minimum; Max – Maximum; Avg - Average Observations: p-value<0.05 for univariate normality asymmetry and kurtosis test and bivariate and multivariate normality of Doornik-Hansen; * significant at 5% by the nonparametric Wilcoxon test for the theoretical value of the median equal to 1.




Efficiency Productivity changing Year Obs Avg Min Max 25% Median 75% Avg 25% Median 75% 2007 306 61.64 15.41 100 46.80 58.67 74.11 - - - - 2008 306 54.14 13.87 100 38.60 48.86 65.48 1,04 0,89 0,99 1,10 2009 306 59.53 24.22 100 42.92 54.46 74.76 1,01 0,89 0,98* 1,05 2010 306 60.06 19.64 100 43.96 56.73 72.38 0,99 0,86 0,97* 1,05 Obs – Observations; Min – Minimum; Max – Maximum; Avg - Average Observations: p-value<0.05 for univariate normality asymmetry and kurtosis test and bivariate and multivariate normality of Doornik-Hansen; * significant at 5% by the nonparametric Wilcoxon test for the theoretical value of the median equal to 1.



Efficiency Productivity changing Year Obs Avg Min Max 25% Median 75% Avg 25% Median 75% 2007 176 65.42 30.52 100.00 51.95 62.48 76.59 - - - - 2008 176 69.76 31.84 100.00 54.89 68.26 82.02 1,02 0,94 0,99 1,03 2009 176 67.33 15.63 100.00 53.36 64.31 79.73 1,03 0,96 1,01 1,05 2010 176 49.58 10.34 100.00 29.14 42.67 62.78 1,01 0,86 0,98* 1,05 Obs – Observations; Min – Minimum; Max – Maximum; Avg – Average Observations: p-value<0.05 for univariate normality asymmetry and kurtosis test and bivariate and multivariate normality of Doornik-Hansen; * significant at 5% by the nonparametric Wilcoxon test for the theoretical value of the median equal to 1.


It can be inferred that most of the municipalities analyzed showed lost in productivity in resource allocation in primary care in Brazil over the years analyzed. This result can be inferred by analyzing median of Malmquist Index of Ray and Desly (1997).

It may be possible that the reduction of this productivity is related to stabilization of Primary Care National Policy since the early years there may be greater effort by governments to promote policy and this effort would be reduced over the year. But it is not possible to tell if this is the cause of most lost in productivity.

Conclusions

One of this study’s main contributions was the proposal of the analytic model of performance in primary care expenditures. Furthermore, this study took into consideration a longitudinal analysis which is not taken into consideration in most studies which use efficiency indexes in Brazil.

Analyzing the most similar possible cities, in four different groups, the efficiency scores evidenced the disparities in resource allocation in the southeastern region, a fact which could be justified by their autonomy in allocating their resources and the absence of relative comparison procedures between them for this allocation. This study proposes a model to solve this problem of a lack of relative comparisons. Also, we may analyze through the quartiles that the median scores are far from technical efficiency. There is space for increasing the primary care service offer, given the current level of expenditures.


As a proposal for a performance analysis, this model should be applied continuously, so that it may be justified as a performance study. This study is the initial proposal of a model so that the measuring is not restricted only the time period of this study. The building of reference units, or benchmarks, may be an instrument to direct the decision-taking of city managers, as well as other higher public management levels, such as state and national.

References

BANKER, R.D.; CHARNES, A.; COOPER, W.W. (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30 (9): 1078-1092.

CHARNES, A., COOPER, W.W., RHODES, E (1978) Measuring the efficiency of decision-making units. European Journal of Operational Research, 2 (1): p. 429-444, 1978.

FERGUSON, T.D.; DEEPHOUSE, D.L.; FERGUSON, W.L (2000) Do strategic groups differ in reputation? Strategic Management Journal, 21: 1195-1214.

FLEURY, S.S.B.; BARIS, E (2001) Reshaping health care in Latin America: a comparative analysis of health care reform in Argentina, Brazil, and Mexico. International Development Research Centre (IDRC), 2001

GREILING, D. (2006) Performance measurement: a remedy for increasing the efficiency of public services? International Journal of Productivity and Performance Management, 55 (6).

HAIR, J.F.; ANDERSON, R.E.; TATHAM, R.L.; BLACK, W.C. (2005) Multivariate Data Analysis. New Jersey: Upper Saddle River, 2005.

MAROCO, J. (2003) Análise estatística. Lisboa: Sílabo, 508 p.

MILLIGAN, G.W.; COOPER, M.C. (1985) An examination of procedures for determining the number of clusters in a data set. Psychometrika, 50 (2): 159-179.

MINITRY OF HEALTH. (2008) Avaliação Econômica em Saúde. Brasília: Editora MS. 2008

MINITRY OF HEALTH. (2007) Secretaria de Atenção Básica à Saúde. Departamento de Atenção Básica. Política Nacional de Atenção Básica. 4ª Ed. Brasília: Ministério da Saúde, 2007. 68p.

RAY, S.C., DESLI, E. (1997) Productivity growth, technical progress, and efficiency change in industrialized countries: comment. The American Economic Review, 87 (5): 1033-1039.

SCOTT, W. R.; DAVIS, G. F. D. (2007) Organizations and organizing: rational, natural, and open system perspectives. Upper Saddle River, NJ: Prentice Hall, 2007.


SELDEN, S. C.; SOWA, J. E. (2004) Testing a multi-dimensional model of organizational performance: prospects and problems. Journal of Public Administration Research and Theory, 14 (3): 395-416.

SUGAR, C.A; JAMES, G.M. (2003) Finding the number of clusters in a dataset: an information-theoretic approach. Journal of the American Statistical Association, 98 (463): 750-763.

WORLD BANK. (2007) Brazil: governance in Brazil’s Unified Health System (SUS): raising the quality of public spending and resource management. Washington,DC: World Bank.

WORLD HEALTH ORGANIZATION – WHO. (2001) Brazil: health profile 2010. Available at: http://www.who.int/countries/bra/en/. Accessed on: 23 MAR. 2011.


27. Reconceptualizing the DEA Bootstrap for improved estimations in the presence of small samples

Panagiotis D. Zervopoulos China Center for Health Development Studies, Peking University,38 Xueyuan Rd, Beijing, China

Department of Business Administration of Food and Agricultural Enterprises, University of Ioannina, 2 Georgiou Seferi St, Agrinio, Greece, [email protected]

Francisco Vargas Economics Department, Universidad de Sonora,E. de Zubeldia No. 27, Hermosillo, Sonora, Mexico, [email protected]

Gang Cheng China Center for Health Development Studies, Peking University,38 Xueyuan Rd, Beijing, China, [email protected]

Abstract

This paper emphasizes the sensitivity of the Data Envelopment Analysis (DEA) efficiency scores due to sampling variations of best-practice frontier and dimensionality issues. Despite its non-parametric nature, it has been proven that DEA yields consistent estimators when large sample sizes are used. A DEA Bootstrap method is being widely applied to tackle inaccuracy of DEA estimators. The combination of DEA and a modified Bootstrap expression enhances the statistical properties of DEA estimators without overcoming the inherent limitations of each of the two methods. This paper provides a non-resampling multi-parametric methodology to deal with the sensitivity of DEA estimators when small samples are available. A comparative analysis between the DEA Bootstrap and the new method’s estimations convergence rate shows that the new method performs better than the DEA Bootstrap in the presence of small samples.

Keywords: Data Envelopment Analysis; Bootstrap; Small samples; Bias correction

Introduction

Data Envelopment Analysis (DEA) is a non-parametric technique that draws on linear programming to measure relative efficiency of decision making units (DMUs). The efficiency scores are defined against a best-practice frontier, which consists of the top-performing DMUs of the sample under evaluation. In this context, the accuracy and reliability of the obtained efficiency scores depend on the robustness of the frontier to sampling variations, dimensionality issues, and noisy data. In the presence of these uncertainties, bias is involved in the efficiency measures.


Despite the deterministic nature of DEA, Banker (1993) has proven the consistency of DEA efficiency scores under certain assumptions and when large samples are employed. However, in the case of small samples, which is not uncommon in economic and DEA literature, few studies deal with bias in non-parametric efficiency measures.

A general methodology for correcting bias in DEA efficiency estimators has been developed by Simar and Wilson (1998, 1999). The two scholars combined DEA with a smoothed expression of Bootstrap. Simar and Wilson’s approach is dominant in bias-correcting DEA efficiency estimators as more than 875 citations in academic articles are recorded. However, the DEA Bootstrap method inherits the virtues and the limitations of the two underlying methods, DEA and Bootstrap, in estimating unbiased efficiency scores.

A multi-parametric method for bias correction (MPBC) of DEA estimators is introduced in this paper in an effort to obtain more consistent estimators compared to Simar and Wilson’s DEA Bootstrap. MPBC does not utilize the original sample DEA estimators in order to make statistical inferences, but variations of the DEA estimators or of aggregate efficiency indices. The MPBC method enables classification of the new DMUs because the estimated confidence intervals of distinct DEA efficiency measures either do not overlap or overlap to a small degree. The classification property of the new method increases its managerial and decision making implications.

Method

Considering that DEA is not appropriate for measuring efficiency scores as a stand-alone method in the case of small samples because it yields upward-biased results, and also that Bootstrap has limited accuracy in approximating the true variability of population efficiency scores when it is applied in the same context, we develop a DEA-based bias-correction method that has particular applicability in the presence of small samples ( 50n < ) and complex production processes. The proposed method (MPBC) does not draw on resampling such as Bootstrap. It rather employs truncated random data generation processes to estimate the unknown population distribution Ffrom the empirical distribution F . The scope of the new method is to estimate the

population efficiency scores , 1, 2,...,p p mθΘ = = by producing an estimator F of the

population distribution F from the efficiency scores ˆˆ , 1, 2,...,i i nθΘ = = defined by

DEA. Bias-corrected efficiency scores * *, 1, 2,...,i i nθΘ = = are generated by F in the

pursuit of *Θ ≈ Θ .

By applying DEA, for instance the BCC model (Banker et al., 1984), we obtain iθ

efficiency scores 1

ˆ ˆ0 1n

i i iθ θ

=≤ ≤ for every DMUi. In the following analysis we presume

input orientation is applied.


Based on the efficiency scores ( )1ˆˆ n

i iθ

=Θ = assigned to the sample DMUs, or, the

weighted efficiency scores ( )1

n

i iθ

=Θ =

, where Θ ⊆ Θ

, a truncated random data

generation process T is utilized to produce a sequence of pseudo-numbers *

1x

ωψ

ψ =

for every DMU. Every sequence of pseudo-numbers originates from every single efficiency score or from a combination of a selected efficiency score and the average scores of the sample.

* to produce 1, 2,...,oT x ψθ θ ψ ω∀ =

or 1 *

1

ˆ ˆ(1 ) to produce

1, 2, ..., ; 1, 2, ...,

nud ud

i i i ioi

z z n xT

i n

ψθ θ θ

ψ ω

−

=

+ −

∀ = =

∑

(1), where

* * ˆmin ,io i ix xψ ψ θ=

In addition,

2*( ) ( , )T x N seθ

and * 2( ) ( ( ), ( ( ) ) )udi i iT x N cvψ θ θ

(2)

1

1

ˆ ˆ( ) (1 )n

ud udi i i

iz z nθ θ θ−

=

= + − ∑

where

z is a user-defined score that denotes the magnitude of a single efficiency score, and complementary of the sample mean efficiency scores, on the generation of a “trimmed” random sequence of data (scores). In fact, there is inherent dependency between the efficiency scores of the sample DMUs that is due to the comparative assessment procedure applied through DEA.

Moreover, *x represents the randomly generated data, the *iox ψ expresses selected

randomly generated replicas of the efficiency score for the ψ -number4 elements of the sequence, and cv stands for the coefficient of variation which is user-defined.

The bias-corrected efficiency score for every DMU is estimated as follows * *( ) 1, 2,..., ; 1, 2,...,i ios x i nψθ ψ ω= ∀ = = (3), where s is a statistic (e.g., mean, median).

It is straightforward that the bias is expressed as *ˆMPBC

i i ibias θ θ= − (4) where * [0,1)iθ ∈ .

The standard error of the proposed MPBC method is 1/2

1 * 2

1

[ ( )]MPBC

i io ise x sω

ψ

ψ

ω−

=

= −

∑ (5)

where 1 *

1

( )i ios xω

ψ

ψ

ω−

=

= ∑

Taking into account equations (3) and (5), and also a large enough sequence size (ω ) of pseudo-data so that the asymptotic normality of their distribution to be assured by

4 1, ...,ψ ω= where ω denotes a proper length for the sequence rather thanω → ∞

*

1 1

ˆ nn f f f

i ii ipθ δ θ δ

= =∴ ∀ = ∃ ≠ − , both

fδ and fp express fixed values.


the Central Limit Theorem, the confidence intervals of the bias-corrected efficiency

scores are constructed using Student’s t distribution

* (1 /2) * (1 /2)( 1) ( 1),

MPBC MPBCa at se t seω ωθ θ− +− −− +

(6)

we prove that *ˆPr , 1, 2,..., 0ubi iob i nθ θ≤ ∀ = = (7)

and 1 *

1

ˆPr , 1, 2,...,L

ubl i i

l

L ob i n eθ θ−

=

≤ ∀ = =∑ (8), where the superscript ub

stands for the upper bound of the confidence interval of the bias-corrected efficiency scores.

Acknowledging the inherit randomness in the proposed method, all the provided proofs or statements result from iterative procedures. In formula (8), the probability, that is the average of L=1000 iterations, is equal to an infinitesimal value.

The inherent randomness in the proposed method is regarded as a drawback because it is a source of instability for the obtained results when the method is applied repeatedly. To overcome this drawback, a stabilization parameter γ is introduced in

the procedure that eliminates up to 99% the variation of the bias-corrected scores. The parameter γ expresses the number of iterations for the formulas (1)-(6). The reported

results are average scores.

The proposed method5 for dealing with uncertainties in DEA is expressed by the

following formula *ˆ ( , , , , , , var )ex ex MPBCf cv z nθα ω γ θ≡ (9)

where α denotes the level of significance, cv is preferably a low-variance parameter (1cv < ), 0 1z≤ ≤ , about the interval of ω we have already discussed, and γ should be

at least equal to unity, which means that no iteration to the proposed bias-correction procedure is applied.

In formula (9), two exogenous parameters exn and varex are included; they denote the number of DMUs in the original sample and the number of input and output variables, respectively, that are utilized to define the efficiency scores through DEA.

Based on a numerical example and on results that are tested through Monte Carlo to

eliminate randomness, the proposed method yields better estimators ( *MPBCθ ) for the

population efficiency scores (θ ) than the DEA Bootstrap ( *Bootθ ) when the original

sample consists of fewer than 50 DMUs. In addition, the convergence rate of *MPBCθ s to

θ s increases against *Bootθ s when the number of input and output variables increases.

Results

From the dataset, we draw 18 samples, each sample consisting of 10, 30, 40, 50, 60, and 80 DMUs. DEA efficiency estimators are computed for each of the 6 samples when 4 (Case 1: 3 inputs & 1 output), 7 (Case 2: 5 inputs & 2 outputs), and 10 (Case 3: 7 inputs & 3 outputs) variables are employed. Additionally, DEA efficiency scores

5 The algorithm for the MPBC method has been developed in Matlab.


are calculated for the three population groups, which consist of 100 DMUs and 4 (group 1), 7 (group 2), and 10 (group 3) variables. The efficiency estimators are cross-checked with the population efficiency scores in order to test the consistency of the efficiency estimators and the maximum sample size, or the optimal combination of DMUs and variables, for which the MPBC method yields improved bias-corrected scores compared to those estimated by the smoothed Bootstrap.

The criterion for the selection of the 6 main samples (10, 30, 40, 50, 60 and 80 DMUs) was the balance below and above the minimum size set by Chernick (2008) (n≥50) to obtain consistent estimators by the Bootstrap.

The scope of this paper is to develop a method to identify consistent efficiency

estimators when small samples are available *11

n npi i ii

θ θ==

→ (10)

The efficiency estimators obtained by the proposed method (MPBC) should converge more to the true efficiency scores of the DMUs under evaluation than the efficiency scores estimated by the smoothed Bootstrap

* *MPBC BootΘ −Θ < Θ −Θ (11)

In order to test the consistency of the MPBC efficiency estimators against the efficiency estimators obtained by the smoothed Bootstrap, we compare the efficiency scores assigned to the 6 main samples (10, 30, 40, 50, 60, and 80 DMUs) with those of the population sample that consists of 100 DMUs.

In the case we employ a binary scale to distinguish the success from the failure of MPBC to yield more consistent efficiency estimators than the smoothed Bootstrap, we utilize the following criterion

1 * *

1,1 if Pr 0.5

Convergence0 otherwise

nMPBC Boot

i i ii

i zob n γθ θ θ θ−

=

− < − >= ∑

(12)

The effects of the variability of the user-defined and the exogenously-fixed parameters, illustrated in formula (9), on the consistency of the MPBC efficiency estimators in comparison with the smoothed Bootstrap efficiency estimators, are summarized in the regression models (13) and (14). The construction of the former regression model drew on efficiency convergence scores obtained for γ =[0, 2, 10], and

the latter regression model was based solely on scores estimated for γ =10.

The independent variables used in our analysis were: (1) n, (2) variables, (3) z, (4) binz, (5) binzn, and (6) binzvariables. The parametersα , cv andω are regarded as constant (α =0.05, cv =0.25,ω =100). The independent variable binz is a binary variable where binz=0 if [0.0, 0.5)z∈ , and binz=1 if [0.5, 1.0]z∈ . The regressors binzn and

binzvariables are mixed variables.

In practice, let [0, 2, 10]γ = , then

** ** **

**

% (65.219) (0.456) (0.316)(12.921)Convergence binzn

binzn= − − +

(13) **p <0.01, R2=0.448, adjR2=0.445, F=141.628, pF-stat=0.00


If the user selects a [0.0, 0.5)z∈ , the MPBC method yields more consistent efficiency

scores than the smoothed Bootstrap for samples with up to 33 DMUs, regardless of the number of variables incorporated in the identification of the DEA efficiency scores. The sample size is extended to 36 DMUs when a [0.5, 1.0]z∈ is selected for the bias-

correction procedure.

In the case of 10γ = , then

** ** **

*% (70.899) (0.537) (0.259)

(1.047)Convergence binzn

binzvariablesn= − − + (14)

**p<0.01, *p<0.05, R2=0.513, adjR2=0.505, F=68.089, pF-stat=0.00

For a [0.0, 0.5)z∈ , the MPBC efficiency estimators report better convergence than the

smoothed Bootstrap efficiencies for samples with up to 38 DMUs. In the case a [0.5, 1.0]z∈ is decided by the user, then the maximum sample size for which the

MPBC simulates at a higher degree than the smoothed Bootstrap the true efficiency varies according to the number of variables. For instance, for 4 variables, the maximum sample size is 31; for 7 and 10 variables, it is 35 and 39, respectively. In all these cases, the maximum sample size meets the criterion/rule of thumb of

max * , 3* ( )n x y x y≥ + for preventing dimensionality effects in the DEA efficiency

estimations.

Taking into account formulas (13) and (14), we develop the following roadmap to facilitate the application of MPBC towards the optimum relative convergence of the efficiency estimators to the true efficiency scores. In practice,

if variables≤ 7, and

n<31→ γ =[0, 2, 10] and z∈[0.5, 1.0]

31≤ n≤ 38→ γ =10 and z∈[0.0, 0.5)

if variables = 8, and

n<32→ γ =10 and z∈[0.5, 1.0]

32≤ n≤ 38→ γ =10 and z∈[0.0, 0.5)

If variables = 9, and

n<36→ γ =10 and z∈[0.5, 1.0]

36≤ n≤ 38→ γ =10 and z∈[0.0, 0.5)

If variables≥ 10,∀n→ γ =10 and z∈[0.5, 1.0], [max n6: adjusted to the number of

variables]

6 The MPBC method yields more consistent efficiency estimators than the smoothed Bootstrap, or relative consistent efficiency estimators, and also free of dimensionality effects, for up to 43 DMUs when 13 variables are incorporated in the DEA evaluation process.


so that, *11

n npi i ii

θ θ==

→ , and just for case 1, 1

*1

n

i

npi ii γθ θ

= =→ where

( ) 1* *# ii γθ γ θ−= and #γ denotes the number of γ parameters.

Conclusions

In this paper, we developed a multi-parametric method for bias correction (MPBC) of DEA efficiency estimators. The new method enhances the applicability and reliability of DEA as a comparative efficiency measurement technique when small or inadequate samples are available. In the presence of small or inadequate samples, DEA efficiency estimators are not regarded as consistent because they are biased by sampling variations and dimensionality.

In order to prevent any confusion to the users of the method resulting from the selection of the appropriate parameters to attain greater consistency for the efficiency estimators than the smoothed Bootstrap, we provide a detailed roadmap. Based on this roadmap, we prove that efficiency estimators obtained by DEA in conjunction with MPBC converge in higher probability to the true efficiency scores than the efficiency estimators yielded by smoothed Bootstrap, under certain circumstances. MPBC is regarded as an appropriate bias-correction method for DEA efficiency estimators when the samples consist at maximum of 43 DMUs.

References

Banker, R. D. (1993). Maximum-Likelihood, Consistency and Data Envelopment Analysis - a Statistical Foundation. Management Science, 39, 1265-1273.

Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science, 30, 1078-1092.

Chernick, M. R. (2008). Bootstrap Methods: A Guide for Practitioners and Researchers. New Jersey: John Wiley & Sons.

Cooper, W. W., Seiford, L. M., & Tone, K. (2007). Data envelopment analysis: a comprehensive text with models, applications, references and DEA-Solver software (2nd ed.). New York: Springer Science + Business Media.

Simar, L., & Wilson, P. W. (1998). Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models. Management Science, 44, 49-61.

Simar, L., & Wilson, P. W. (1999). Estimating and bootstrapping Malmquist indices. European Journal of Operational Research, 115, 459-471.


28. Relative balance as a complementary measure to relative efficiency

Heinz Ahn Technische Universität Braunschweig, Institute of Management Control and Business Accounting, Germany, [email protected]

Ludmila Neumann Technische Universität Braunschweig, Institute of Management Control and Business Accounting, Germany, [email protected]

Nadia Vazquez Novoa Technische Universität Braunschweig, Institute of Management Control and Business Accounting, Germany, [email protected]

Abstract

One of the major strengths of Data Envelopment Analysis (DEA) is the endogenous determination of the weights of performance criteria, assigning to each decision making unit (DMU) its best possible efficiency score. However, this property also leads to a significant shortcoming: it allows zero-value weights that exclude criteria from the evaluation. While many approaches that deal with this problem incorporate value judgments into analysis, our approach supports management’s efficiency analysis with a complementary performance measure that is derived from the given data set. The respective balance score evaluates the extent to which a DMU avoids concentration on only some of the crucial performance criteria. One of the possible decisions resulting from a balance analysis is to reduce the set of DMUs considered to serve as benchmarks. For this case, a modified CCR-O model is presented.

Keywords: DEA, Balance score, Inappropriate benchmarks

Introduction

Data Envelopment Analysis (DEA) is a meaningful approach to relative efficiency measurement of decision-making units (DMUs) that assesses DMUs’ performance by aggregating their input and output values into a single efficiency score. The weights of these performance criteria are endogenously determined, assigning to each DMU its best possible efficiency score. This property constitutes one of the major advantages of the basic DEA models but it also represents a source of pitfalls concerning performance assessment and performance control. Such problems result from the possibility of zero-value weights that eliminate from the analysis any input and/or output criteria in which the performance of a DMU is weak, with the aim of raising its efficiency score to its maximum level (Dyson and Thanassoulis, 1988; Allen et al., 1997; Thanassoulis et al., 2004). Unlike approaches that deal with this issue by





incorporating value judgments into the usual DEA analysis, we suggest a complementary balance score that directly refers to the given input/output values.

The balance score constitutes a source of additional information about a DMU’s performance structure. As is known, it may happen that different DMUs obtain (nearly) the same efficiency score even if they substantially differ in their pattern of criteria achievement. We suggest to cover this aspect of performance by means of the

balance score βo. A DMU will be characterized by a balance score of 100% if all of its crucial performance criteria values are – relative to the respective values of the other DMUs – achieved on a similar level. In contrast, the more the performance criteria values are not achieved on a (relative) similar level, the more the balance score decreases towards 0%.

Often, outputs can be interpreted as indicators of strategic goals and inputs as means to achieve these goals. For such cases, the pattern of the input values is estimated to be of minor relevance, whereas the pattern of output values is estimated to be of great managerial importance. For this reason, managers may prefer to focus their attention on the evaluation of the balance of the output mix. We therefore focus on an output-oriented balance score. If the analysis of the input mix would be of main interest, an input-oriented balance score could be calculated.

The rest of the paper addresses the following aspects: introduction of the balance model; benefit of the model application; numerical example and concluding remarks.

The Balance Model

According to Ahn et al. (2012) the procedure to measure relative balance is comprised of two main steps: determination of the acceptable output mix region and calculation of each DMU’s balance score. Supposing that there is no ideal output pattern or that it is at least unknown, we use the average of all DMUs as substitute and the average absolute deviation from the mean as basis for the specification of a tolerance region, called the multicone C .

While all DMUs located inside this region are considered to be balanced, the balance score for all other DMUs needs to be calculated. A reference point on the boundary of the multicone is determined for each unbalanced DMU guaranteeing that the

maximum relative variation ϖo,r of each output r necessary to obtain a balanced output

mix is minimal. The formula 1–βo reflects to what extent the extreme pattern of the output values of an unbalanced DMU differs from that of the reference point.

Balance analysis and its consequences

The complementary information provided by the balance analysis provides insights that substantially influence the DEA efficiency analysis.

Concerning performance assessment, the balance score can be characterized as a kind of secondary measure. On one hand, efficiency deficits cannot be compensated by a high balance score. On the other hand, balance deficits let the performance expressed by a high efficiency score appear in another light. For example, it may be found that the cause of a DMU’s low balance score is the pursuit of a significantly different


business strategy. Consequently, such DMUs should be excluded from the set of comparable DMUs.

Concerning performance control, the management should be cautious about using efficient but unbalanced DMUs as benchmarks. An inefficient DMU with such a benchmark would be urged to pursue a questionable performance strategy.

We suggest excluding efficient but unbalanced DMUs from the set of possible benchmarks. This can be achieved by adding the following constraint to the envelopment form of the CCR-O model:

nk ok kko ,...,1,1,0, ∈≠≠= βλ

This modification still allows all efficient DMUs to serve as their own benchmarks, while eliminating unbalanced DMUs from the efficiency frontier relevant for the inefficient DMUs. Consequently, the reference sets of the inefficient DMUs only contain balanced DMUs, and the modified efficiency frontier leads to improvements of the efficiency scores of inefficient DMUs. These improvements result from the fact that extreme targets set by the unbalanced benchmarks are replaced with more moderate targets. In special cases, it may even happen that an inefficient DMU turns into an efficient one because it becomes part of the new frontier.

The numerical example that follows in the next section illustrates the proposed balance analysis and shows the evolution in the results.

Numerical example

We refer to the example of a European pharmacy chain originally presented in Ahn et al. (2012). As Table 1 shows, 20 pharmacy stores were analyzed, based on two inputs: worked hours (WH) and store square meters (SQM) and three outputs: number of customers buying medications available only on prescription (MP), number of customers purchasing over the counter products (OTC) and total number of prescriptions (PR). The outputs represent the main goals to be pursued by the stores.

The main numerical results calculated by running the balance model are presented in Table 1. The DMUs A, E, F, O, Q, R, and T are not attain all outputs in an acceptable

proportion. For example, DMU A has a balance score βA of 0.88 since it requires a minimal output variation of 12% to be projected onto C ; an increase of 11% on MP and 12% on PR is necessary, while a reduction of 12% on OTC is allowed.

In the extreme case, where at least one output is not produced at all, the respective DMU will be characterized by a balance score of 0. The DMU R represents such an

extreme DMU because it has no OTC sales. The ϖo,r values project R (nearly) onto the origin of the coordinate system, which obviously cannot be understood as a benchmark. However, the efficiency analysis by means of traditional CCR-O model designates R as a benchmark for F and K (see Table 2), although Table 1 shows that R is specialized on two outputs, while F and K also take a third output into account. For this reason we reanalyze the efficiency of the pharmacy stores by means of a modified CCR-O model which exclude unbalanced DMUs from the reference set of other DMUs. Table 2 presents the results of the modified CCR-O model and those of the traditional CCR-O model.


Table 1. Data set of 20 pharmacy stores and the results of the balance model.

DMU Inputs Outputs Balance model (iteration 1)

WH SQM MP OTC PR βo ϖo,MP ϖo,OTC ϖo,PR

A 7,424 287 3,578 12,329 15,431 0.88 0.11 –0.12 0.12

B 14,208 168 9,115 8,765 40,034 1 0 0 0

C 8,608 146 7,644 13,340 29,453 1 0 0 0

D 8,736 281 7,866 13,055 27,304 1 0 0 0

E 10,656 333 12,244 9,307 56,743 0.89 –0.11 0.11 –0.07

F 10,272 130 8,700 10,067 52,510 0.99 0 0.01 –0.01

G 10,336 274 6,749 14,279 30,522 1 0 0 0

H 12,160 240 8,867 13,446 41,789 1 0 0 0

I 7,424 203 10,483 12,580 52,145 1 0 0 0

J 12,960 264 8,981 13,787 36,094 1 0 0 0

K 10,880 257 7,000 13,071 39,464 1 0 0 0

L 6,240 220 7,946 10,826 43,239 1 0 0 0

M 14,240 127 15,335 24,899 68,108 1 0 0 0

N 11,584 115 13,133 27,643 60,306 1 0 0 0

O 3,840 92 5,365 12,365 16,079 0.90 0 –0.10 0.10

P 9,600 266 8,119 12,908 43,039 1 0 0 0

Q 6,400 90 7,108 24,107 35,996 0.89 0.11 –0.11 0.06

R 7,040 92 7,567 1ε 44,104 0 21 ε+− 0.57 21 ε+−

S 10,176 275 8,115 12,854 26,959 1 0 0 0

T 7,040 180 2,835 11,980 18,333 0.79 0.21 –0.21 0.11

The balance and efficiency analysis concerning a period 1 is represented by the results of iteration 1. The six pharmacy stores originally identified as efficient keep this property. While the balanced DMUs I, M and N also continue to serve as benchmarks for other stores, the unbalanced DMUs O, Q and R are eliminated from the reference set of other stores. This increases the efficiency score of those inefficient stores which formerly had unbalanced DMUs in their reference sets. For example, the new benchmark of A is N, raising its efficiency score from 0.44 to 0.70. As a special case, L is now characterized as efficient, and it becomes an appropriate benchmark for K and P. It should be noted that the balance analysis indicates R as a DMU following a different business strategy. In this case, R is not comparable to the other DMUs and should be excluded from the sample.

On the basis of the data analysis of period 1, the management of the pharmacy stores should take decisions according to the respective results. If a store is unbalanced, a change of its output mix in reference to the determined reference point on the frontier of the multicone is required. Hence, the DMUs A, E, F, O, Q and T should adjust their

output values according to the proposed ϖo,r values in Table 1. Such changes in the output mix may induce a negative change in the efficiency score in the subsequent period 2 indicating the overestimation of the actual performance of the DMUs.


Table 2. Traditional versus modified efficiency scores and benchmarks.

DMU CCR-O model

Modified CCR-O model (iteration 1)

Modified CCR-O model (iteration 2)

oθ λ oθ λ

oθ λ

A 0.44 Q 0.70 N 0.61 N

B 0.55 I, N, O 0.55 I, N 0.55 I, N

C 0.70 I, N, O 0.72 I, N 0.72 I, N

D 0.64 I, O 0.72 I, N 0.72 I, N

E 0.81 I 0.81 I 0.72 I

F 0.86 N, Q, R 0.93 I, N 0.92 I, N

G 0.50 I, O, Q 0.58 N 0.58 N

H 0.56 I, N, O 0.57 I, N 0.57 I, N

I 1 I 1 I 1 I

J 0.53 I, N, O 0.54 I, N 0.54 I, N

K 0.55 I, Q, R 0.59 I, L, N 0.59 I, L, N

L 0.99 I, Q 1 L 1 L

M 1 M 1 M 1 M

N 1 N 1 N 1 N

O 1 O 1 O 1 O

P 0.66 I, Q 0.70 I, L, N 0.70 I, L, N

Q 1 Q 1 Q 1 Q

R 1 R 1 R --- ---

S 0.57 I, O 0.62 I, N 0.62 I, N

T 0.46 I, Q 0.71 N 0.56 N

Assuming that the balanced DMUs keep their balanced output mix, the pharmacy stores are again evaluated in period 2, but without the incomparable DMU R. The results of this second balance analysis are shown in Table 2, iteration 2. The DMUs A

(βA=0.99), E (βE=0.97), F (βF=0.97), Q (βQ=0.99), and T (βT=0.99) still remain unbalanced, while DMU O becomes balanced and the balance score of DMU B decreases to 0.97. The efficiency analysis by running the modified CCR-O model shows a decrease of the efficiency scores of unbalanced DMUs while the efficiency scores of balanced DMUs as well as the reference set of inefficient DMUs remain unaffected.

This iterative process of balance and efficiency analysis could be continued. Assuming that the output mix of balanced DMUs remains constant, the unbalanced DMUs will turn into balanced ones after further iterations will be conducted.


Conclusions

This paper introduces a units-invariant model for identifying DMUs with an unbalanced output mix and illustrates the method with a numerical example. Our approach supports management’s efficiency analysis with a complementary performance measure that avoids setting restrictions on the weights. Based on a typical data set for a DEA application, it allows for higher discrimination within both efficient and inefficient DMUs by quantifying their relative balance. The approach does not need additional value judgments to quantify the balance scores and bears no risk of infeasible solutions.

A thorough evaluation of the achieved balance scores provides insights that substantially influence the DEA efficiency analysis. Concerning performance assessment, the explanation for a DMU’s low balance score could be that the DMU pursues a different business strategy and therefore is not comparable to the other DMUs. Concerning performance control, the management may agree that efficient but unbalanced DMUs are not adequate benchmarks for inefficient DMUs. For this case, a modified CCR-O model is proposed, which endogenously calculates the balance score and eliminates the undesirable benchmarks. In addition, an iterative process of balance and efficiency analysis is suggested, which determines adequate efficiency scores and supports the unbalanced DMUs on their way to get balanced.

A series of alternative models can be developed by modification of the procedure to

calculate βo (Dyckhoff et al., 2012). For example, instead of using the average deviation as a measure of dispersion the less restrictive standard deviation could be

used. This would lead to a broader multicone C . Nevertheless, this alternative multicone would keep the feature of being sensitive to the distribution of the DMUs, changing its position and amplitude according to the output values of the DMUs.

References

Ahn H., L. Neumann, N. Vazquez Novoa (2012) Measuring the relative balance of DMUs. European Journal of Operational Research 221: 417-423.

Allen R., A. Athanassopoulos, R.G. Dyson, E. Thanassoulis (1997) Weights restrictions and value judgments in Data Envelopment Analysis: Evolution, development and future directions, Annals of Operations Research 73: 13-34.

Dyckhoff, H., A. Dirksen, E. Mbock (2012) Measuring balanced efficiency with DEA: New approach and case study of German business schools‘ research performance, http://ssrn.com/abstract=1990233.

Dyson R.G., E. Thanassoulis (1988) Reducing weight flexibility in Data Envelopment Analysis, Journal of the Operational Research Society 39: 563-576.

Thanassoulis E., M.C.S. Portela, R. Allen (2004) Incorporating value judgments in DEA. In Cooper, W.W., L.M. Seiford, J. Zhu (eds.), Handbook on Data Envelopment Analysis. Boston et al.

Acknowledgements

We gratefully acknowledge the financial support of the DFG (German Research Foundation) in the context of the research project “Advanced Data Envelopment Analysis”.


29. Statistical Inference and Efficient Portfolio Investment Performance

Shibo Liu Loughborough University UK, [email protected]

Tom Weyman-Jones Loughborough University UK, [email protected]

Karligash Glass

Loughborough University UK, [email protected]

Abstract

The purpose of this paper is to apply a quadratic data envelopment analysis model with bootstrap and second stage regression to estimate the efficiency of a sample of investment funds, obtain the statistical inference of the efficiency scores and detect the determinants of inefficiency. Morey and Morey (1999) developed a mutual funds efficiency measure in a traditional mean-variance model. It is derived from the standard data envelopment analysis but differs from it in having non-linear constraints in the envelopment version of the model’s structure. This paper first applies the procedures in Morey and Morey (1999) to a new modern data set comprising a multi-year sample of investment funds and then utilise Simar-Wilson (2008) bootstrapping algorithms to develop statistical inference and confidence intervals for the indexes of efficient investment fund performance. For the second stage analysis, robust-OLS regression, Tobit models and Papke-Wooldridge (PW) models are conducted and compared to evaluate contextual variables affecting the performance of investment funds. The DEA efficiency scores are regressed on potential variables to test the statistical significance of those factors. Results and inferences are drawn from an extensive new dataset of investment funds.

Keywords: nonlinear-DEA, portfolios, bootstrapping, second stage DEA

Introduction

The history of portfolio evaluation dates from the 1960th (Sharp, 1966; Treynor, 1965 and Jensen, 1968), with emphasis on both expected return and risk. Investment fund managers attempt to find efficient portfolios – those promising the greatest expected return for any given degree of risk, i.e. risk-adjusted return. However, there are many criticisms of traditional portfolio analysis which focus on their sensitivity to chosen benchmarks. Murthi et at. (1997) were the first to apply DEA methodology to fund performance evaluation. A large proportion of DEA models applied to investment funds show piecewise linear correspondence between multiple inputs and outputs.


However, according to Markowitz portfolio theory, there is correlation between different assets which should not be ignored, and these co-movements between different securities affect the relationship between expected return and risk of the combined portfolio.

Based on the unique characteristics of investment trusts, Morey and Morey (1999) developed an investment funds efficiency measure in a traditional mean-variance model. It was based on Markowitz portfolio theory and related the non-parametric methodologies to the foundations of traditional performance measurement in mean-variance space. The model is derived from the standard data envelopment analysis but differs from it in having non-linear constraints in the envelopment version of the model’s structure. Although mean and variance are considered in Morey and Morey (1999) models, they distinguish their model from traditional portfolio analysis by the fact that there is no theoretical benchmark like the market portfolio of the Capital Asset Pricing Model. Instead, the benchmarking fund in Morey and Morey (1999) consists of certain funds in the group, each with a particular weight. So rather than being compared with an idealised fund, Morey and Morey (1999) model benchmarks the funds under evaluation again themselves. This makes Morey and Morey (1999) model practically feasible and easier to test.

Therefore, this paper firstly applies the procedures in Morey and Morey (1999) to a new modern data set comprising a multi-year sample of investment funds. It then extends Morey and Morey (1999) model by adding statistical significance tests. The motivation is that the DEA efficiency scores obtained through Morey and Morey (1999) model have not hitherto been tested for statistical significance. Simar-Wilson (2008) bootstrapping algorithms are utilised to develop statistical inference and confidence intervals for the indexes of efficient investment fund performance.

After conducting the statistical test, this paper then examines the efficiency of investment trusts, analyse the factors contributing to investment trusts performance and detect the determinants of inefficiency. This framework involves two-stages. In the first stage, efficiency scores are calculated using Morey and Morey (1999) quadratic DEA model. And then in the second stage, these scores are regressed on potential explanatory variables which may have an impact on the funds’ performances. Robust-OLS regression, Tobit models and Papke-Wooldridge (PW) models are conducted and compared to evaluate contextual variables affecting the performance of investment funds. In this stage the DEA efficiency scores are regressed on potential variables including Sharpe ratio, Jensen’s alpha, expense ratio, P/E ratio, book to market ratio and market value of the investment funds to test the statistical significance of those factors.

Methods

In the essence of data envelopment analysis, Morey and Morey (1999) quadratic models use the idea of ‘funds of funds’: for each fund there is a corresponding composite benchmarking fund, which lies on the efficient frontier. These are hypothetical but potentially efficient combinations of the actual observations. DEA scores are obtained by measuring the direct distance from the position of the fund in question in mean-variance space to that of the efficient composite benchmarking fund.


Morey and Morey (1999) presented two basic quadratic programming approaches to identify those funds that are efficient. These two approaches are mean return augmentation and risk contraction. Figure1 illustrate these two quadratic models.

From Figure 1, it is in the mean-variance space with risk as input and mean return as output. These two approaches show different paths to the efficiency frontier.

Mean return augmentation method could be seen as output oriented DEA, and it represents a vertical path towards the efficient frontier, while the risk contraction model is input-oriented DEA which follows the horizontal path.

Consider N mutual funds to be evaluated, indexed j=1, 2,…,N, where 0j is the fund in

evaluation for each run. ,,...2,10 Nj = and there are N runs totally. Let T denote the

number of different time horizons, where t=1, 2,…,T. Denote )( ,tjRE as the mean

return for fund j, and 2jσ as its the variance as well as ),( ,, tjti RRCov as its covariance.

Denote jw as the weight allocated to each fund to form the benchmarking fund in

each run. Formula of mean return augmentation is as follows:

Determine ),...,,...2,1(0 0 Njjwj =≥ so that:

),...2,1(

)()(

),(

1..

,1

,

2,,

1 1 1

2,

2

1

0

0

Tt

REREw

RRCovwww

wts

Max

tj

N

jtjj

jtjti

N

j

N

jii

N

jjitjj

N

jj

=

≥

≤+

=

∑

∑ ∑∑

∑

=

=+= =

=

ϑ

σσ

θ

Where θ is the efficiency score and we have 1≥θ

.

θ is calculated by running the above programming problem once for each fund. Efficient funds will have a value of one, while inefficient ones will get a value greater than one which shows how much the actual return should be expanded for the fund to be considered technically efficient.

The second quadratic program is risk contraction, and the formula is as follows:

Mean

Return

O Graph 1

Inefficient Fund

Mean

Return Augmentation

Risk Contractio

Risk


),...2,1(

),(

)()(

1..

2,,

1 1 1

2,

2

,1

,

1

0

0

Tt

ZRRCovwww

REREw

wts

ZMin

jtjti

N

j

N

jii

N

jjitjj

tj

N

jtjj

N

jj

=

≤+

≥

=

∑ ∑∑

∑

∑

=+= =

=

=

σσ

Where 1≤Z and the efficient frontier is composed

by funds with Z equal to one.


29 funds classified by Morningstar as ‘UK equity Mid/Small cap’ are tested and the results show that both mean return augmentation and risk contraction approach identify the same 6 efficient funds and except the efficient funds, 3 funds out of 29 have the same ranking from both approaches, with other funds rank differently for different approaches. The correlation between two rankings is 0.829929, which is very high. This means that although mean return augmentation approach and risk contraction approach emphasize different aspects and have different benchmarking fund on the efficient frontier, a fund could get similar ranking based on two approaches. Also, the marginal contribution of the mean return and variance in each period to the fund’s efficiency could be obtained by solving lagrangian functions.

Then DEA scores are bootstrapped with 2000 replicates and Bandwidth h equals 0.2186 which is calculated by least-squares cross-validation method described by Silverman (1986). From the results, the initial DEA model gives an average uncorrected efficiency score of 1.2470, while the bootstrap model generates an average bias-corrected score of 1.2642. The minimum uncorrected score is 1 and the maximum is 1.9331, while the minimum bias corrected score was 1 and the maximum was 1.9736. For the most efficient funds, the 2000 bootstrap estimators are all equal to one; therefore the 95% confidence intervals for these funds become a single point. The results also reveal that all the estimated biases are negative, which is as expected, because according to Simar and Wilson (1998), the DEA estimate is upwardly biased using an input oriented model and downwardly biased for an output oriented model. The original scores have a mean bias of -0.0172. And the standard deviations for all the estimators are quite small with the maximum standard deviation equal to 0.0908.

All the funds satisfy the condition of 41

ˆ>

i

ibiasσ

, except for the most efficient funds

which have both the bias and the standard deviation equal to zero. From the results, all of the original DEA scores are within the lower and upper bounds of 95% confidence interval, with the maximum range is as small as 0.3343. Therefore the statistical test indicates that the DEA scores are reliable. This conclusion can be further proved by the comparison between rankings of the funds from original DEA scores and rankings based on bias-corrected DEA scores, which shows that except two funds, all the other funds have exactly the same rankings before and after bias correction; for the two funds which have difficult rankings, their rankings using original DEA scores and bias corrected DEA scores are very close.


Robust-OLS, two limit tobit, one limit model censoring at zero and PW model with logit function as the link function are conducted and the results indicate that firstly, Shape ratio and price/earnings ratio have positive impact on the fund performance under Robust-OLS and three tobit models, but not statistically significant, while Jensen’s alpha, net expense ratio, market value, and book to market ratio of the fund have negative impact on the fund performance, but only Jensen’s alpha is statistically significant in three tobit models. Secondly, the magnitude of all the factors are fairly close for OLS and Two limit and One limit tobit models, and PW model with the same sign, while the results from tobit model censoring at zero has the opposite sign and different magnitude. This is because in tobit model censoring at zero, the dependent variable is obtained by taking the reciprocal of DEA score minus one, therefore the positive relation between DEA scores and the dependent variable in other regressions turn to negative in this model. The p values of the marginal effects from PW model are much larger than other models except of the last factor book to market ratio, where tobit model censoring at zero has slightly larger p value.

For all the factors except the Jensen’s alpha in tobit models, the p-values are all larger than the critical value 0.05 for 95% confidence level. There could be misspecification or inclusion of irrelevant variables. Also because Sharpe ratio and Jensen’s alpha are two measurements that also in mean and variance space, a recursive model is applied which regresses DEA scores, Sharpe Ratio and Jensen’s Alpha on net expense ratio, price/earnings ratio, market value and book to market ratio respectively using robust-OLS and indicates them as model (1), model (2) and model (3) respectively. The results of model (1) indicates that the net expense ratio has a negative impact on the efficiency score indicated by the DEA score as expected, but not statistically significant, with PE ratio, Market value and book to market ratio impact the efficiency score positively, but only the coefficient of the PE ratio is statistically significant. Model (2) gives very good results, in a way that all the coefficients are statistically significant. In model (2), the PE ratio, market value, book to market ratio make significant positive contribution to explaining the efficiency indicated by Sharpe ratio. From model (3), all the factors have a positive impact on the efficiency measure indicated by Jensen’s Alpha, but none of the coefficients are statistically significant. Also, model (2) shows the highest R-square among those three models, which equals 0.7914 while the R-square in Model (1) is 0.5438.This means that both models fit well. The prob>F gives the overall significance level of the regression model. Model (1) has the smallest prob>F value, which equals 0.0006; while that in model (2) is slightly higher, but still highly significant at 1% significance level. The results imply that quadratic DEA score is better than Jensen’s Alpha and similar to Sharpe ratio as an investment fund efficiency indicator.

Conclusions

The purpose of this paper is to apply a quadratic data envelopment analysis model with bootstrap and second stage regression to estimate the efficiency of a sample of investment trusts, obtain the statistical inference of the efficiency scores and detect the determinants of inefficiency.


Firstly the relative rankings of 29 funds are obtained applying Morey and Morey (1999) quadratic DEA model, the marginal contribution of the mean return and variance in each period to the fund’s efficiency are obtained then the Simar-Wilson (2008) bootstrapping algorithms are utilized to develop statistical inference and confidence intervals for the indexes of efficient investment fund performance. Algorithms of smoothed bootstrap for this quadratic DEA model are designed. The results indicate that the DEA scores obtained from Morey and Morey (1999) quadratic DEA model are reliable. The third application in this paper applies second stage DEA models to analyse the factors contributing to investment trusts performance. Robust-OLS regression, Tobit models and Papke-Wooldridge (PW) models are conducted and compared to evaluate contextual variables affecting the performance of investment funds. The DEA scores are regressed on potential explanatory variables including Sharpe ratio, Jensen’s alpha, expense ratio, P/E ratio, book to market ratio and market value of the investment funds to test the statistical significance of those factors. Then a recursive model is applied when DEA scores, Sharpe ratio and Jensen’s alpha are used as dependent variables respectively while net expense ratio, PE ratio, market value and book to market ratio are explaining factors in all three regressions. The results show that quadratic DEA score is better than Jensen’s Alpha and similar to Sharpe ratio as an investment fund efficiency indicator. Quadratic DEA score has richest set of theoretical properties and informative estimated multipliers. There are further developments on quadratic and cubic DEA models based on Morey and Morey (1999). However, relevant empirical papers applying these methods are few. For example, Briec et al. (2004) applied a directional distance function which allowed simultaneous changes in the direction of reducing inputs and expanding outputs. They also defined an indirect mean-variance utility function, and divided overall efficiency (OE) into allocative efficiency (AE), and portfolio efficiency (PE). Briec et al. (2007) claimed that portfolio returns are generally not normally distributed, with investors preferring positive skewness so that the probability of obtaining a negative return is low. They extended the work of Briec et al. (2004) into mean-variance-skewness space using cubic programming and divided overall efficiency into portfolio efficiency, allocative efficiency, and convexity efficiency. Kerstenset al. (2010) examined different returns to scale, convexity problems and higher order moments in both quadratic and cubic optimization programming and decided that various return to scale (VRS), free disposal hull and higher moments are essential methodologies for mutual funds evaluation. Also, and none of these papers discuss the statistical properties of DEA estimators. These issues are left for future research.

References

Ayoe, H. (2007). Second stage DEA: Comparison of approaches for modelling the DEA score. European Journal of Operational Research, 181(1), 425-435.

Banker, R. D., & Natarajan, R. (2008). Evaluating contextual variables affecting productivity using data envelopment analysis. Operations Research, 56(1), 48-58.

Morey, M. R., & Morey, R. C. (1999) Mutual fund performance appraisals: A multi-horizon perspective with endogenous benchmarking. Omega, 27(2), 241-258.

Simar, L., & Wilson, P. W. (1998) Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models. Management Science, 44(1), 49-61.


30. Study of technical efficiency in product development in steel company, with application of Data Envelopment Analysis (DEA)

Regina Rocha de Morais Gonçalves FPL-MG, [email protected]

José Edson Lara, FPL-MG, [email protected]

Ana Lúcia Miranda Lopes, UFMG, [email protected]

Ronaldo Lamounier Locatelli FPL-MG, [email protected]

Abstract

This study applied the Data Envelopment Analysis (DEA) methodology in Brazilian Steel Industry. The DEA is based on non-parametric mathematical models. It evaluates the performance of each unit of observation with a multidimensional perspective. The study used a portfolio of 12 product development projects for which were obtained the scores of technical and scale efficiency. The technical efficiency scores indicated the benchmark projects, demonstrating the potential of DEA. For purposes of analysis, this research used the Data Envelopment Analysis with BCC (Banker, Charnes, Cooper) input-oriented model; content analysis and the NTCR model (News, Technology, Complexity and Rhythm). It can be concluded that the adoption of methodologies aimed at efficiency projects management is an essential tool for competitiveness of the organization, motivating employees, improving management processes and reducing time for product delivery.

Keywords: Data Envelopment Analysis (DEA); BCC, innovation and product development management; Multifunctional Integration; steel industry; technical efficiency; scale efficiency.

Introduction

In the current scenario of business in the industry stands out the capacity for innovation management and product development as a determinant of survival of organizations. Researchers and practitioners in this area have provided significant contributions of management systems and integrated processes, creating and


validating models, as a contribution to the companies in order to meet the needs of their customers efficiently and effectively.

The presented situation creates interdependence between the areas of organization. They require information and cooperation between actors from different functional departments such as Marketing, Product Development, Research and Development (R & D), Production and Quality, and a constant interaction with the customer, in order to obtain convergence to the company's strategic objectives.

This context brings solution for the industry like a management new products and process development. However, as Rozenfeld et al. (2006) pointed out the organization's success in developing new products is not guaranteed by the genius and creativity of professional R & D, or the number of resources allocated to the projects.

Clark e Wheelwright (1993) shows that one the practice essential in this discussion, it is the importance of creating a development product framework that brings a broad perspective to the process and facilitates cross-functional integration.

Shenhar and Dvir (2010) identified five dimensions of project success: project efficiency, impact on the customer; impact team, and direct sales success and preparation for the future.

In this context this research applied the Data Envelopment Analysis (DEA) in product development of a Brazilian Steel Industry in order to: measure the technical efficiency; identify the benchmark projects and measure the scale efficiency. The analysis concentrated on a portfolio of 12 product development projects, and the scores of technical and scale efficiency were calculated. The technical efficiency scores indicated the projects "benchmarks", providing the capability of the method. It can be concluded that the adoption of methodologies aimed at efficiency management project is an indispensable tool for the competitiveness of the organization.

Methods

Regarding methodological aspects, it is a case study with qualitative and quantitative approach. Scripts were used for semi-structured interviews for data collection.

The study used a portfolio of 12 product development projects for which were obtained the scores of technical and scale efficiency. The technical efficiency scores indicated the benchmark projects.

For purposes of analysis, the three techniques were applied:

• Data Envelopment Analysis (DEA) with BCC (Banker, Charnes, Cooper) input-oriented model – VRS (Variable Returns to Scale);

• Content analysis;

• Model NTCR (News, Technology, Complexity and Rhythm).

The software selected to perform the method of this research was the DEA Frontier, and it is available at www.deafrontier.com.

http://www.deafrontier.com/



The results and discussions were aggregated into two topics: the application of Data Envelopment Analysis (DEA) and the analysis of research results.

Application of Data Envelopment Analysis (DEA)

As the guidelines available for the use of DEA Frontier software were elected input and output variables, as it follows: duration and cost of the project performed (inputs) and the number of products delivered and opportunities -Tons Sold (outputs).

Table 11 presents the data used to calculate the technical efficiency and scale of projects, as well as to identify which are the benchmark projects of the portfolio analyzed.

Table 11- Project Portfolio of New Product Development

Source: Steel Industry

As shown in Table 12, technical efficiency scores of each analyzed project were obtained. It was identified two efficient DMUs, 1 and 11, because they have reached the maximum value equal to 1. Therefore, these are the benchmarks DMUs, and the others can be considered inefficient.

Program DMU Product ProjectsDuration of

Project Cost (R$)

Number of Products

Delivered Opportunities

(Tons Sold))

1 Naval 1,2,3,4,5,6 Project_Naval 1 180 473650 6 15000

2Naval 7,8,9 Project_Naval 2 180 473650 3 9000

3Naval 10 Project_Naval 3 242 622450 1 8000

4Tubo 1 Project_Tubo 1 395 736000 1 0




8 Estrutural 1 Project_Estrutural 1 152 298600 1 8000




12 Estrutural 5, 6 Projeto_Estrutural 5 183 354400 2 6000

Estr

utur

al

Product Development - TMCP Input Output

Nav

alTu

bes (

Oil a

nd G

as)


Table 12- BCC Model Execution

Source: Execution from DEA-Fontier

The investigation carried out with BCC Model oriented input indicated the DMU's 2, 3, 4, 5, 6, 7, 8, 9, 10 and 12 as inefficient. Although as Table 2 shows each one has a different degree of inefficiency.

According to the analysis of Table 3, it can be identified that two DMUs had the maximum efficiency achieved with CCR Model, since they reached the maximum value possible equal to 1. They were Projects 1 and 11, belonging to the Program Project Naval and Structural respectively. It is noteworthy that the Projects 1 and 11 also obtained maximum technical efficiency model for BCC, which brings us to prove that every project efficiently in CCR model will also be in the BCC model, FERREIRA e GOMES, (2009). However, the DMU efficiency in BCC model can´t is efficient in CCR model.

After the technical efficiencies are calculated in BCC and CCR models, it is followed by the calculation of the efficiency scale, and the results are presented in Table 4.

Inputs OutputsDuration of Project Number of Products Delivered Cost (R$) Opportunities (Tons Sold)

Input-OrientedVRS Optimal Lambdas

DMU No. DMU Name Efficiency with Benchmarks1 Project_Naval 1 1,00000 1,000 Project_Naval 12 Project_Naval 2 0,80000 0,400 Project_Naval 1 0,600 Project_Estrutural 43 Project_Naval 3 0,50107 0,021 Project_Naval 1 0,979 Project_Estrutural 44 Project_Tubo 1 0,32745 1,000 Project_Estrutural 45 Project_Tubo 2 0,32745 1,000 Project_Estrutural 46 Project_Tubo 3 0,32745 1,000 Project_Estrutural 47 Project_Tubo 4 0,32745 1,000 Project_Estrutural 48 Project_Estrutural 1 0,82345 0,021 Project_Naval 1 0,979 Project_Estrutural 49 Project_Estrutural 2 0,87793 0,091 Project_Naval 1 0,909 Project_Estrutural 4

10 Project_Estrutural 3 0,68700 1,000 Project_Estrutural 411 Project_Estrutural 4 1,00000 1,000 Project_Estrutural 412 Project_Estrutural 5 0,81131 0,200 Project_Naval 1 0,800 Project_Estrutural 4


Table 13- CCR Model Execution


As Table 4 shows the DMUs 1 and 11 are operating at optimal scale. Thus, they are considered the benchmark projects.

Table 14 - efficiency ratio scale



Input-OrientedCRS Sum of Optimal Lambdas

DMU No. DMU Name Efficiency lambdas RTS with Benchmarks1 Project_Naval 1 1,00000 1,000 Constant 1,000 Project_Naval 12 Project_Naval 2 0,60000 0,600 Increasing 0,600 Project_Naval 13 Project_Naval 3 0,40498 0,570 Increasing 0,493 Project_Naval 1 0,078 Project_Estrutural 44 Project_Tubo 1 0,10726 0,167 Increasing 0,167 Project_Naval 15 Project_Tubo 2 0,21015 0,556 Increasing 0,089 Project_Naval 1 0,467 Project_Estrutural 46 Project_Tubo 3 0,29246 0,868 Increasing 0,026 Project_Naval 1 0,841 Project_Estrutural 47 Project_Tubo 4 0,29246 0,868 Increasing 0,026 Project_Naval 1 0,841 Project_Estrutural 48 Project_Estrutural 1 0,82252 1,019 Decreasing 1,019 Project_Estrutural 49 Project_Estrutural 2 0,87393 1,083 Decreasing 1,083 Project_Estrutural 4

10 Project_Estrutural 3 0,68268 0,992 Increasing 0,002 Project_Naval 1 0,991 Project_Estrutural 411 Project_Estrutural 4 1,00000 1,000 Constant 1,000 Project_Estrutural 412 Project_Estrutural 5 0,53097 0,489 Increasing 0,302 Project_Naval 1 0,187 Project_Estrutural 4


Input-Oriented Input-OrientedCRS VRS Scale

DMU No. DMU Name Efficiency Efficiency Efficiency1 Project_Naval 1 1,00000 1,00000 1,000002 Project_Naval 2 0,60000 0,80000 0,750003 Project_Naval 3 0,40498 0,50107 0,808234 Project_Tubo 1 0,10726 0,32745 0,327565 Project_Tubo 2 0,21015 0,32745 0,641786 Project_Tubo 3 0,29246 0,32745 0,893167 Project_Tubo 4 0,29246 0,32745 0,893168 Project_Estrutural 1 0,82252 0,82345 0,998889 Project_Estrutural 2 0,87393 0,87793 0,99544

10 Project_Estrutural 3 0,68268 0,68700 0,9937211 Project_Estrutural 4 1,00000 1,00000 1,0000012 Project_Estrutural 5 0,53097 0,81131 0,65445


Results Analysis

Application of Data Envelopment Analysis (DEA) to calculate the technical efficiency, to identify benchmark projects and to calculate the scale efficiency in product development, proving the method's potential according the results of this study.

In the Project Portfolio studied, it can be noticed that 80% have technical and scale inefficiency (Table 4).

As emphasized, the simple quantification of a DMU is not sufficient to instruct on how to improve their efficiency. Ferreira and Gomes (2009) showed that it is necessary, moreover, to identify what percentage of efficiency can be improved from the elimination of excess resources as well as incorrect scale production.

To improve the efficiency of inefficient projects were identified factors that ensured the arrangements for the benchmark projects.

Respondents agree that there is a relationship between the degree of integration and performance indicators of the organization. The main characteristics observed in the benchmark projects were: increased productivity, reduced development time, anticipation of the solution, compliance dates delivery, increased value-added products, standardization of processes, integration of processes, speed of information and ease of access to information.

These results have established practices to be implemented for projects that showed technical and scale inefficiency. Among the critical success factors were identified, in the sequence described, practices that contributed to the success of Projects Naval 1 and Structural 4, which were considered the benchmarks of portfolio analysis:

- Working in cross-functional teams, including customers and suppliers,

- Development Product Process of well-defined, from concept to launching;

- Ability to capture customer requirements;

- The project leader had the technical, managerial and interpersonal skills needed to manage conflicts and problems.

By the application of the .NTCR Model, the projects are classified in: innovation, low technology, system and critical time. Except 4 projects from Tubes Program that are classified in: innovation, high technology, system and critical time. The Data Envelopment Analysis (DEA) with BCC (Banker, Charnes, Cooper) input-oriented model would apply to 8 projects instead of 12.

Conclusions

It can be concluded that the adoption of methodologies aimed at efficiency project management is an essential tool for competitiveness of the organization, motivating employees, improving management processes and reducing time for product delivery.

An attempt was made to apply DEA methodology on development product management, using a case study of Brazilian steel industry. The main results were the identification of benchmark projects, which had maximum technical efficiency and the indication that 80% of projects in the portfolio analysis showed technical and scale inefficiency.


The results are valid as a first reference for managers. However, the limited number of projects and the absence of similar studies should be considered as a limitation of this research.

References

BANKER, R. D.; CHARNES, A.; COOPER, W.; Some Models for Estimating Technical and Scale Inefficiencies. In Data Envelopment Analysis. Management Science, Vol. 30, No. 9, pp. 1078-1092, Set. 1984.

CHARNES, A.; COOPER, W.; RHODES; E.. Measuring the efficiency of decision making units. European Journal of Operational Research 2, 1978; 429-444..

CHARNES, A.; COOPER, W.; LEWIN; A.Y., SEIFORD,L. M. Data Envelopment Analysis: Theory, Methodology and Application. 3a. Ed. Massachussetts, USA, 1997. 513p

CLARK, K., WHEELRIGHT, S. C. Managing New Product and Process Development: Text and Cases. New York: Fee Press, 1993.

FERREIRA, C. M. C.; GOMES, A. P. Introdução à Análise Envoltória de Dados: Teoria, Modelos e Aplicações. Viçosa, MG: Ed. UFV, 2009.

KERZNER, Harold. Gestão de Projetos: as melhores práticas. 2ª. Edição - Porto Alegre: Bookman, 2006. 824p.

ROZENFELD, H.; FORCELLINI, F. A.; AMARAL, D.C.; TOLEDO, J.C.; SILVA, S.L.; ALLIPRANDINI, D.H; SCALICE, R.K. (2006). Gestão de Desenvolvimento de Produtos: uma referência para melhoria do processo. São Paulo: 542p, Editora Saraiva.

ROZENES, S., VITNER, G.; SPRAGGETT, S. MPCS: Multidimensional Project Control System.. International Journal Project Management, No. 22, 2004.

SHENHAR, A.; Dvir, D. Reinventando o Gerenciamento de Projetos: A abordagem diamante ao crescimento e inovação bem-sucedidos. São Paulo, M.Books do Brasil, 2010. 260p.

ZHU, J.; COOK; W. D.. Data Envelopment Analysis: Modeling Operational Processes and Measuring Productivity. York University, Canada. Wade D. Cook, 2008.


31. Technical efficiency of Burkina Faso primary public health care centers

Oumarou HEBIE Institute of Empirical Research in Political Economy, [email protected]

Simon TIENDREBEOGO Institut de Recherche en Sciences de la Santé, [email protected] (corresponding author)

Séni KOUANDA Institut de Recherche en Sciences de la Santé, [email protected] (corresponding author)

Abdel Latef ANOUZE American University of Beyrouth, [email protected] (corresponding author)

Abstract

Millennium Development Goals 4th and 5th axes aim to reduce infant mortality and enhance maternal health by the year 2015. However, in Burkina Faso (West Africa), these indicators remain worrying. To face this situation, Burkina Faso engaged the extension of Obstetrical and Neonatal Emergency Care (EmNOC) strategy since 2004. The main objective of this paper was to assess technical efficiency of the public basic health care organizations (CSPS) related to SONU. The great number of such organizations and their spatial distribution make them more accessible for women who need EmNOC. We found, by an output oriented model that the median efficiency score of the CSPS is 1.17 and less than 20% of the CSPS are fully efficient; none of the sixty three health district of the country is efficient. Rural location of CSPS and lack of electricity seems to be the main environmental factors of inefficiency according to our data.

Keywords: Data Envelopment Analysis (DEA), Two-stage estimation, Obstetrical and Neonatal Emergency Care, Burkina Faso.

Introduction

Developing countries remain the most affected by maternal mortality in the world, 536,000 women die each year worldwide due to complications of pregnancy, childbirth and the puerperium according to United Nations. Millennium Development Goals (MGDs) 4th and 5th axes aimed to reduce infant mortality and enhance maternal health by the year 2015. However, in Burkina Faso, these two indicators remain worrying. The maternal mortality ratio was successively 566, 484 and 307.3 for 100000 live births in 1993, 1998 and 2006. Other indicators show that Burkina Faso has a poor health status. The public expenditure on health is extremely low, amounting to less than 6.1% of the GDP. In addition, the distribution of health resources is highly skewed reflecting significant regional disparities in health services across the country.


In response to the increasing costs of health care, policy-makers have become increasingly concerned with improving the efficiency of health services sector. In Burkina Faso, over the last two decades, various health reform measures have been implemented to reduce the health expenditure and to improve the efficiency of health services, with the current insufficient health resources and the limited funds allotted.

Health System

The health system has two types of structures: the administrative structures and the hospitals and social-health. The health system is pyramidal and has three levels: the central level represented by the office of Minister of Health, the General Secretariat and the central departments; the intermediate or regional level (there are 13 Regional Directorates of Health); the peripheral level or operational which includes 63 health districts. The central level is the level of direction, design and policy making and national development plans in health. The intermediate or regional level is the level of monitoring of the implementation of policy and national development plans in health. The peripheral level is managed by a team headed by a district medical officer. This is the operational level where the development plans for health are implemented. The health district is the operational entity.

Methods

Data Envelopment Analysis

The literature on efficiency measurement is broadly divided into using non-parametric and parametric methods. The non-parametric methods include Data Envelopment Analysis (DEA), the most popular efficiency measurement method. In this study, DEA

is used to measure the technical efficiency of health centers because the method,

handles several outputs, does not require explicit specification of the functional forms relating inputs and outputs and does not require information on prices of inputs and outputs. DEA is a linear programming formulation for frontier analysis that defines a relationship between multiple inputs and multiple outputs. DEA was introduced by Charnes, Cooper, and Rhodes pursuing Farrell’s work. Charnes et al proposed a model that had an input orientation and assumed constant returns to scale (CRS). Subsequent works have considered alternative sets of assumptions, such as Banker, Charnes and Cooper, who proposed a variable returns to scale (VRS) model known as BCC model.

Since early 1980s, DEA has been extensively used for efficiency analysis of health care organizations. Chilingerian and Sherman noted that DEA has become the researchers’ method of choice for finding best practices and evaluated productive inefficiency in health care organizations

The DEA models are broadly divided into input-orientated and output-orientated models. The output orientated model determines by how much outputs’ quantities can be proportionally expanded without altering the quantities of inputs. The choice of model orientation depends on the extent to which the health center managers have control on its inputs or outputs. In this study, an output orientation has been chosen because CSPS managers have not total control on all inputs.


The DEA model used in this study is an output-oriented BCC which takes the following form:

Aggregated efficiency scores and Tobit regression

In this paper we also computed the aggregated efficiency scores. This is done by Simar and Zelenyuk approach (2007). Based in a previous paper of Fare and Zelenyuk (2003), they give price-independent weights to be considered for aggregate efficiency score calculation. Finally truncated tobit regression was used to regress primary health care centers efficiency scores on environment variables.

Source of data and study variables

Data for this study were mainly obtained from the Assessment Obstetrical and Neonatal Emergency Needs coupled to the mapping of the supply of care in reproductive health in Burkina Faso Report.

The variables involved in the study concerned characteristics of primary health care centers, their resources and outputs. Eleven input variables and four output variables were used (Table 1) to measure the technical relative efficiency.

Inclusion criteria

All health districts were included. In each district, all primary health care centers that met the following criteria were included: Public facility, practice of delivery, no missing data for the study variables. Outlier primary health care centers according to efficiency score were also excluded.


General description

Analysis was performed on sample of 996 primary health facilities in Burkina Faso. Most of the primary health care centers were rural (90,8%). More than one primary health care center did not receive electricity (28%) and 15% of primary health care centers was not supplied with water.

These facilities have a variation in term of equipment and resources endowment. In term of obstetric capacity, the range is between 1 and 20. More than 75% have no wise women while the maximum number found is 8 (Table1).


Table 2 – Descriptive statistics of inputs

Inputs Mean(IQR

Min Max (%)

Number of units 3(2 ;3) 1 5

Number of beds in the obstetrics 4(3 ;5) 1 20

Wise women or men number 0(0 ;0) 0 8

Number of auxiliary midwives 1(1 ;1) 0 19

Other health workers 2(2 ;3) 1 21

Number of delivery tables 1(1 ;2) 0 5

Number of category of recommended

5(5 ;5) 1 6

Others medecines

Anticonvulsive + Oxytocin 2(2 ;2) 2 2

Number of basic equipement 5(5 ;5) 1 5

Number material for

neonative care 1(1 ;1) 1 3

Number of elements in

the delivery box 8(7 ;9) 2 10

The variation is very wide in outputs compared to inputs. The range of delivery number is great than 100 (Table 3).

Table 3 – Descriptive statistics: outputs

Outputs Median

Min Max

EmNOC related units 5(5 ;6) 2 7

Total number of delivery 247(156 ;378) 1 2598

Total number of abortion

5(1 ;11) 0 254

Number of live births 244 (153 ;372) 6 3331

Efficiency estimation

Only 20% of primary health care centers are fully efficient. Most of them (80.42%)

have slacks and more than 70.5% of primary health care centers are inefficient (Table

4). Most of CSPS (75.2%) are inefficient at each step of their production process.


Table 4 – Efficiency status

Efficiency (F) range CSPS number percentage (%) F=1 292 29.5 1< F61.1 355 35.8 1.1<F61.2 232 23.4 1.2<F61.3 51 5.1 1.3<F61.5 231 23.3 1.5<F62 119 12.0 2<F6 5 3 0.3

Table 5 – description of slacks

Slack Status Fully efficient Only x slacks Only y slack x and y slack Count 194 44 8 745 Percentage 19.6 4.4 0.8 75.2

Many health centers can improve their performance. Indeed, 50% of CSPS can proportionally increase their outputs quantity by 17% with the current inputs (efficiency median index is 1.17). Moreover Burkina Faso CSPS can assess their efficiency according to outputs quantity improvement by 23%

Independent of the region, the proportion of efficient CSPS is less than 40%. Nevertheless, we notice a variation of efficiency among regions. For instance, Centre-Sud, Boucle du Mouhoun, Sahel regions have high proportions of efficiency CSPS (more than 25%). The less proportions are found in Hauts-Bassins, Centre-Est and Est regions where the proportion of efficient CSPS is less than 15%. The aggregated

efficiency score show that Boucle du Mouhoun, Centre Sud and Sahel are, by

decreasing order, the most efficient regions and Plateau Central, Est, and Cascades are the less efficient

In the same way, analyses show that Nouna, Pô health districts are the most efficient while Banfora and Bousse are the less efficient health districts.

Environmental variables and inefficiency

The multivariate analysis show that only poor women care special subsidy is significant. Nevertheless, rural location and electricity source have significant

coefficients (at level 0.1). Poor women subsidy and electricity source contribute

(respectively by 9.5% and 6.18%) to move the efficiency score toward 1. In contrast,

rural location increases inefficiency respectively by 13.8% (Table 8).


Table 8 – Tobit regression results summary

Coefficients Estimate Std. error t-value Pr(>

Signif.

Intercept 1.256 0.074 16.832 ¡2.2e-16 ***

Education level 0.05 0.059 0.850 0.395

Rural 0.139 0.070 1.989 0.047 *

Distance from the referee obstetric

-0.0006 0.0006 -0.988 0.323

Duration to the referee obstetric

0.0005 0.0003 1.529 0.126

Electrified -0.062 0.031 -1.987 0.047 *

Poor women subsidy -0.095 0.030 -3.134 0.002 **

Not modern source of electricity -0.106 0.064 -1.641 0.101

Not modern source of water -0.009 0.068 -0.139 0.889

Sigma 0.273 0.014 19.727 ¡2.2e-16 ***

Discussion

Seventy per cent of the CSPS in the sample were technically inefficient and 20% were fully efficient. A similar study of 155 primary health care clinics in in Kwazulu-Natal found 30% of health care clinics to be technically efficient. However, Our finding is quite upper than the results obtained in Nouna district in Burkina Faso (30%) by Marshall Flesha (2011), in Ghana (65%) J. AKAZILY (2008). Theses differences might be explained by a rather homogeneous structure of the health centers (cases of a single health district study), a limited number of health centers (regional study) or the strength/organization of health system and the income of the country(country level study) or the chosen inputs and outputs.

Most of primary health care centers (80%) use more inputs than need at current operational level. Theses health centers can produce the same level of outputs using 17% less of each input. It should however be noted that this does not imply the presence of excess capacity relative to needs. It might be explain by the lower utilization rate of health services due to possibly demand-side barriers of any type. Centre sud, Boucle du Mouhoun and Sahel are the most efficient regions. For Sahel and Boucle du Mouhoun regions, it might be related to the high proportion of health centers applying free cost for a normal delivery (41% and 31% respectively). Centre Sud has the highest percent of health centers (55%) with a formal system of poor women subsidy and the second low rate of homebirth (15%). These regions host the most efficient primary health centers. Nouna the best one is in Boucle du Mouhoun region while Po the second is in Centre-Sud. Marschal and Flesha found that 70% of primary health centers in Nouna are technically efficient. Our results show that health center location, power supply, and poor women care subsidy are in increasing order factors affecting health centers inefficiency. Rural location increases inefficiency. It could probably be related to, among other things, care seeking behavior of the catchment population and population density. For example, the high proportion of homebirth is found in rural location (39%). This factor is found by Tamiru Balchia in


analyzing Ethiopia health center efficiency. Others studies cited location as factor which influence efficiency. Poor women subsidy contributes to better use of health

care centers while lack of power supply makes difficult service providing in night. Power supply increase health access by improving patients sojourn in health centers.

Conclusions

The study shows that 80% of primary health care centers are inefficient. These facilities are not operating at a full technical capacity and could increase their outputs within existing budgets. The evidence indicates that capacity utilization is insufficient in rural health centers, those without power supply and system of care for indigent women. Our study recommends the adoption of policies for better management of health centers and better use of health services.

References

WW Rhodes E. Charnes A, Cooper. Measuring the efficiency of decision making units.

European Journal of Operational Research, 2 :429–444, 1978.

Rolf FÄRE and Valentin ZELENYUK. On aggregate farell efficiency. European Journal of Operational Research, 146:615–620, 2003.

Léopold SIMAR and Valentin ZELENYUK. Statistical inference for aggregates of farell-type efficiencies. Journal of Applied Econometrics, 22:1367–1394, 2007.

]Paul MARSHALL and Steffen FLESSA. Efficiency of primary care in rural burkina faso. a two-stage dea analysis. Health Economics Review, pages 1–5, 2011.

Street A Jacobs R, Smith PC. Measurin Efficiency in Health Care: Analytic Tech-niques and Health Policy. Cambridge: Cambridge University Press, 2006.


32. The efficiency of Brazilian electricity distributors during 2004 – 2009. An application using DEA corrected by environmental and stochastic factors.

Fernando Damonte Quantum7, [email protected]

Mariana De Santis Quantum8, [email protected]

Introduction

The aim of this study is to estimate the efficiency of the electricity distribution companies in Brazil during 2004 to 2009. For this purpose, a semi-parametric methodology that incorporates the impact of environmental variables and statistical noise is applied. DEA is employed supplemented with SFA methodology, following Fried et al (2002). This methodology adjusts the amounts of inputs considering environmental variables and statistical noise using a stochastic frontier approach, taking all companies to a common scenario, recalculating then the percentages of efficiency by DEA. Unlike other methodologies, the amount of each input is adjusted instead of the efficiency score, allowing environmental variables affect differently to each of the inputs. This approach is particularly advantageous when estimates of efficiency include more than one input, such as total expenditures, operating expenditures and capital expenditures.

The practice of benchmarking with price setting purposes is present in most advanced worldwide regulations. In Brazil, in the second tariff review (2007-2010), benchmarking was used in the regulation of the distribution companies to analyze the global consistency in operating costs obtained by applying the “model company” method; to relate technical losses with delinquency rates and to determine the quality of service. With an analysis of global consistency, the Brazilian regulator (ANEEL) attempted to validate the reasonableness of the costs of the reference company, obtained by "top-down" with an alternative methodology. These analyzes were not made explicit in the discussions with the agents. However, the regulator committed to explain the global consistency of analyzes to be used in the third review, where 7 Quantum – Experts in Public Utilities Regulation – www.quantumamerica.com

8 Quantum – Experts in Public Utilities Regulation – www.quantumamerica.com


benchmarking techniques will be essential. In 2011 the third review methodology was discussed and determined, in which semi-parametric benchmarking techniques were utilized to calculate optimal operating and maintenance expenditures. DEA and Corrected Ordinary Least Squares (COLS) were applied to estimate efficiency scores of each distribution company. In a second step, environmental variables were included in the analysis, such as concession area socioeconomic complexity, climatic effects, customer dispersion y regional wage levels, by employing the techniques developed by Simar and Wilson (2007) and Banker Natarajan (2008) 9.

Methodology

The methodology used in this study is a three-stage approach according to Fried et al (2002). This method combines the advantages of DEA and SFA methods. A complete analysis of the advantages of using a three stage model can be found at Fried et al (2002). In the first stage, we apply DEA to input and output data to obtain the initial efficiency scores of firms. In the second stage SFA is used to attribute variation in first stage electricity distribution companies’ performance to environmental variables, managerial inefficiency and random noise. In the third stage, firms’ inputs are adjusted to take into account the environmental effects and statistical noise uncovered in the initial DEA. Finally, DEA is applied to the adjusted inputs to obtain improved measures of managerial efficiency, net of environmental variables and random noise effects.

Data

The data used in this study are those used by the Brazilian National Electricity Agency (ANEEL) to determine the methodology that will be applied for calculating the electricity rates for the third regulatory period. This database was used by ANEEL, in the Public Hearing 40, for estimating the efficiency of the distribution companies. The input and output information was provided by the companies to the regulator. The environmental variables were surveyed by ANEEL from different sources which are mentioned below. Additionally we segregated the total number of consumers by type in small, industrial and rural with information from various public sources.

ANEEL’s database is available at its official website and it includes 61 electricity distributors for the period 2004 – 2009. Years before 2003 were not considered due to data limitations and to exclude the period of energy rationing that took place in 2001 and 2002. In this study 17 of these companies were excluded due to lack of specific data mainly related to consumer type disaggregation. Details about the definition of the variables can be found in the Technical Note 294 by ANEEL.

The variables used in this study are the following:

9 See Technical Note Number 294/2011, SRE/ANEEL


Operating and maintenance costs (OPEX)

The OPEX data comes from accounting sources (Accounting handbook of Public Service of Electricity). Costs related to services provided to third parties and those corresponding to generation and transmission activities were excluded, as well as related to not regulated activities. Figures are expressed in local currency of December 2010. To do this, we used the producer price index (IGPM), except for items Third Party Services and Personnel, which were adjusted with the consumer price index (IPCA) following identical criteria as ANEEL.

Capital Costs

The capital cost considered in this study was calculated according to an economic concept, including the Annual Depreciation of physical capital and the Opportunity Cost of Capital. Since the aim of this work is to measure the input-oriented efficiency, it was considered more appropriate to calculate the cost of capital as the annuity, because allocating the cost of investments evenly over the period does not harm the efficiency of companies whose net assets are newer neither reward with low cost (of capital) those with older assets. For the calculation of the annual rate an opportunity cost of capital of 16.07% and the average life of the assets of each company were considered, both parameters used by the regulator in its estimates of efficiency.

Electricity delivered (in MWh)

Electricity delivered is referred to the entire billed market and comes from the Tracking System Market Information for Economic Regulation of ANEEL (SAMP).

Consumers

It includes Residential, Commercial, Industrial, Rural and other categories on December of each year of the period.

Distribution network length (in kilometers)

It comprises the length of Low, Medium and High Voltage networks at the end of every year of the period.

Average wage

The salary was calculated by ANEEL for its estimates of efficiency and it is expressed in local currency of December 2010. This variable measures the labor cost a distribution company faces when hiring employees.

Socioeconomic Complexity Index (SECI)

Measures the degree of adversity faced by distributors in regards to electricity non-technical losses combat. This index, prepared by ANEEL, includes of the following


dimensions: violence, inequality, deprivation, infrastructure and commitment to income10.

Rainfall (millimeters per year)

The geocoded data regarding isohyets, which composed the calculation of the index rainfall, come from the National Water Agency - ANA.

Table 15 summarizes the main characteristics of the variables defined above for the average of the 44 companies included in the analysis.

Table 15: Descriptive statistics of the 44 electricity distributors (2004-2009)

Variable Unit Mean S. D. Minimum Maximum Cases Missing

Operating cost Million Reáis 285.5 327.5 4.5 1774.8 245 0

Capital cost Million Reáis 503.1 648.9 1.7 3161.5 245 0

Network Thousand kms.

61.3 76.4 0.4 460.6 245 0

Total Consumers Thousand 1462.3 1586.7 13 7295.2 245 0

Small Consumers Thousand 1376.8 1471.9 12.7 6685.3 245 0

Industrial Consumers

Thousand 12.1 17.2 0.1 74.2 245 0

Rural Consumers Thousand 73.4 97.6 0.2 535.6 245 0

Energy distributed GW/h 76359.9 9346.8 68.1 40260 245 0

Average wage Reáis 3285 631.7 1925.9 5904.7 240 5

SECI 0.18 0.11 0.02 0.46 240 5

Rainfall index mm/year 1457.5 333 808 2450 240 5

Source: Own elaboration from ANEEL

Estimated models

In order to obtain the efficiency of Brazilian electricity distributors during 2004-2009 three alternative models were estimated:

The first model includes as inputs, operating costs and capital costs, responding to the premise that the measurement of efficiency must take a comprehensive approach, including all inputs involved in the production process jointly. Thus, the risk of measuring the efficiency of operation costs conditioned to the quantity and quality of the distribution network, which may differ significantly among compared companies, is minimized. In other words, it excludes the problems associated with the interchangeability of remuneration to two different production factors (trade - off). The outputs considered are the number of consumers and network length per year. For the correction of the slacks of both inputs (stage 2), we considered the following environmental variables:

10 See Technical Note N. 271/2010 by SRE/ANEEL.


• Average wage,

• Socioeconomic complexity index, and

• Rain

With regard to the signs of the estimated coefficients associated to the environmental variables in the adjustment of the optimal slack by SFA is expected average wages have a positive impact, because the distributors that operate in areas where labor receives higher wages necessarily face higher costs than those located in areas where the formal labor market is more depressed. The former face higher wage levels that raise their costs, being this difference not attributed to inefficient management.

It is expected that companies located in areas with high proportion of social vulnerability face greater difficulties in reducing non-technical losses. It is therefore likely that operating costs to combat fraud and costs of anti-fraud network be in direct relation with indices of socioeconomic complexity. Accordingly, this variable is expected to be positive in the optimum adjustment of the slack.

The coefficient associated with rain is expected to be positive, since companies operating in rainy areas are exposed to higher operational costs for service restoring such as network reconnection costs, posts and line replacement, etc.

The second model includes the same inputs and outputs than the first, with the difference that the consumers are broken into three categories: small, composed by residential and commercial consumers, industrial and rural consumers.

The third model considers as input only the operation and maintenance costs. Unlike the previous models, it only focuses on the OPEX efficiency, by controlling the magnitude of invested capital with the network length. It is worth mentioning that the Brazilian regulatory benchmarking techniques used to promote efficiency in the last review conducted last year were applied only to OPEX. It was decided to consider as outputs the number of served customers, the total energy delivered and the network length. The latter is also treated as an output by the Brazilian regulator, as well as by other authors, who consider the number of kilometers of maintained network as an output. Estache et al (2010), however, show that in numerous studies of benchmarking is often used as input the length of the network as a proxy for capital input, usually in the estimates of cost functions or distance functions. (p. 143). Environmental variables used in the adjustment of the optimal slacks are the same as in the previous models.


Estimations were performed using the LIMDEP 9.0 software. In the initial stage of the estimation the efficiency of companies in the sample was calculated adopting an input orientation DEA technique under the assumption of variable returns to scale. We chose variable returns to scale, a general assumption, because there are no a priori reasons to assume that electricity distributors face constant or decreasing average costs. A summary of Stage 1 results is presented in Table.


Table 2: First stage results

Model Outputs Inputs Mean Std. Deviation Minimum

Maximum n

Model 1 Total Consumers, Network

Length, Total Energy Delivered

OPEX, CAPEX

80.44% 0.14235688 0.500967 1.000 240

Model 2

Outputs: Small Consumers, Industrial Consumers, Rural Consumers, Network Length,

Total Energy Delivered

OPEX, CAPEX 85.98% 0.133589664 0.51698 1.000 240

Model 3

Outputs: Small Consumers, Industrial Consumers, Rural Consumers, Network Length,

Total Energy Delivered

OPEX 74.05% 0.188425685 0.280055 1.000 240

From the results of the first stage, we calculated each company's slacks, which were adjusted to incorporate the effect of environmental and stochastic variables. It was run a regression for each of the slacks for each input of the model under the assumption that the error term that captures inefficiency presents a normal distribution truncated at 0. We adopted a panel data model with random effects, in which inefficiency varies in the period according to the Batese - Coelli model. The estimated coefficients presented in all cases the expected sign and are statistically significant. The variable rain only was statistically significant in Model 3. Companies operating in areas where the wage is higher face higher costs, as well as those that operate in higher social conflict areas. The parameter lambda is statistically different from zero, indicating that managerial inefficiency explains the variability of slacks between sample firms and time.

The impacts of environmental variables and the stochastic variables estimated in the second stage were incorporated into the inputs of each company, which were adjusted to carry all distributors to a common operating environment. To do so, input quantities of the producers who were benefited from a favorable environment and "good luck" were adjusted upward.

Finally, the efficiency scores were recalculated using the input adjusted in the second stage. A summary of the results is presented in Table 3, which shows the average efficiency scores for the total sample and for some groups of distributors. As expected, in all cases efficiency is higher after separating the non-managerial component of total slacks. In general, the average efficiency rises from 0.8/0.86 to 0.93/0.92 in the estimation considering OPEX and Capital Costs as inputs (Models 1 and 2), whereas in model 3, the initial mean score, 0.74 increases to 0.89 in the last stage.


Table 3: Average Efficiency scores (Stage 3)

Model 1 Model 2 Model 3 Firm group N Stage 1 Stage 3 Stage 1 Stage 3 Stage 1 Stage 3

Smallest size Less than 500000 consumers 78 0.76 0.92 0.87 0.92 0.73 0.87

Medium Size More than 500000 and less than 3

million consumers 126 0.80 0.92 0.83 0.92 0.70 0.88

Largest size More than 3 million consumers

36 0.91 0.97 0.93 0.97 0.92 0.99

All the simple 240 0.80 0.93 0.86 0.92 0.74 0.89

Note that the average score of the companies serving larger quantities of customers (more than 3 million customers) is higher than those classified as small and medium companies both in the first and third stage of the estimations in all models, while no significant disparities are observed between the smallest and medium firms. Notice that after "leveling the playing field," according to Fried et al (2002), the size of the distribution companies seems to reduce its impact in the final estimations of the companies’ efficiency. However, results suggest that larger companies present advantages with regards to smaller ones, attributable to higher density reducing the unitary cost.

It is also worth noting that model 3 shows smaller efficiency scores than models 1 and 2, suggesting that when the analysis is more complete, considering both, OPEX and Capital Costs, the existing tradeoffs between capital and labor is better captured by the model, whereas when only estimating OPEX efficiency, companies that are more labor intensive might be considered inefficient. It seems clear that when considering both inputs, inefficiency is better isolated from the other variables that affect costs.

Conclusions

In this study we have obtained estimates of the efficiency of electricity distributors in Brazil in the period 2004 - 2009, applying the DEA methodology and SFA. The results show that the impact of non-managerial variables such as level of input prices and socioeconomic complexity, as well as statistical noise, affect significantly on the efficiency of firms in the sample, being this effect more significant among small and medium firms. A notable contribution is the incorporation of the stochastic variables in the models, in contrast with the methodology used recently by the Brazilian regulator, which explicitly included environmental variables but excluded the effect of statistical noise. Another interesting aspect is the inclusion of capital costs in the models jointly with OPEX, capturing the effect of possible trade-off between these two inputs. Moreover, the efficiency scores of each electricity distributor show small variations along each of the years, unlike the results obtained by the regulator in the last revision, in some cases showing sharp fluctuations in the scores.

It is worth noting that in this study a more complete model was estimated in comparison to the common practice by considering different types of consumers, which allows taking into account the different degrees of effort required to serve diverse users. The results indicate the trade–off between OPEX and CAPEX, since the average score increases when the inputs representative of capital is explicitly


incorporated in the model. It should be noted that the frontier was recalculated after correcting the inputs by environmental variables, in clear difference with another methodologies that only correct the scores. This process is known as re-evaluation of the frontier. To conclude, we recognize it is necessary to explore other models and specifications in order to reflect with more precision the production process, i.e. to consider energy delivered by voltage level or the maximum demand.

References

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Baumol, W., Panzar, J. and Willig, R. (1988) Contestable markets and the theory of industry structure, Harcourt Brace Jovanovich, Inc.

Burns, P. and Weyman-Jones, T.G, (1996). "Cost Functions and Cost Efficiency in Electricity Distribution: A Stochastic Frontier Approach," Bulletin of Economic Research, Wiley Blackwell, vol. 48(1), pages 41-64, January.

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Farsi, M., Filippini, M. and Greene,W. (2005). "Application of Panel Data Models in Benchmarking Analysis of the Electrivity Distribution Sector," CEPE Working paper series 05-39, CEPE Center for Energy Policy and Economics, ETH Zurich.

Fried, H.O.; Lovell C.A.K.; Schnidt S.S.; Yaisawarng S. (2002), Accounting for Environmental Effects and Statistical Noise in Data Envelopment Analysis, Journal of Productivity Analysis 17, 157-174.

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Hess, B. and Cullmann, A. (2007). "Efficiency analysis of East and West German electricity distribution companies - Do the "Ossis" really beat the "Wessis"?," Utilities Policy, Elsevier, vol. 15(3), pages 206-214, September.

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33. The production efficiency in sugarcane farms

Terezinha Bezerra Albino Oliveira UFSC/UFAL. E-mail: [email protected] (Corresponding author)

Antonio Cezar Bornia UFSC. E-mail: [email protected]

Suely de Fátima Ramos Silveira (UFV. E-mail: [email protected]

Mauro Wagner de Oliveira UFAL. E-mail: [email protected]

Alexandre Matos Drumond UFV. E-mail: [email protected]

Abstract

This study was conducted to evaluate the production efficiency in sugarcane farms, by using DEA. Seventeen DMUs comprising 2010-2011 harvest were considered. The inputs were the rent of the land; raw material; and costs of the harvest, loading and transport of the sugarcane. The revenue on sugarcane sale was considered as output. The CCR addressed to inputs was used as model. The stalk productivity ranged from 76 to 114 t of stalks/ha, as the average productivity being 81 t/ha. The fertilizer costs ranged from R$14.33 to R$366.59/ha; however, the costs of harvest, loading and transport of the stalks were the highest, as ranging from R$16.43 to R$27.16. There was a relationship between crop productivity and profit in R$/ha. From 17 DMUs, six were efficient. Those six DMUs were considered as benchmark to the other inefficient DMUs. The DEA method contributed to identify efficient sugarcane farmers and to explore their knowledge in order to improve the inefficient farms.

Keywords: Data Envelopment Analysis; sugarcane; agricultural management.

Introduction

In the world scenery, Brazil stands out as the largest producer of sugarcane, with approximately 33% of the world production, and it is considered as the first one in production of sugar and ethanol, as accounting for over half of the sugar traded worldwide (MAPA, 2011). This position has been occupied mainly due to the cropped area that is relatively large, the high productivity levels achieved in the main producing regions of the country and due to the increased productive potential of new cultivars (MARTINELLI, 2011, OLIVEIRA et al., 2011).

In recent decades, has become increasable interest in the sugarcane crop, either on socioeconomic viewpoint and environmental aspect.


Among the main positive economic impacts, it may be mentioned: employment and income generations, development interiorization, increase of the foreign exchange arising from the exportations and the possibility for complementing the increasing energy demand of the Country which depends on oil and natural gas. Concerning to environmental aspect, the high fixation rate of the atmospheric CO2 by sugarcane, during an extended time period, naturally contributes to improvement of the environment, since it reduces the greenhouse effect occasioned by the burning of the fossil fuels (SOUTO, 2011, SUNDFELD; MACHADO, 2011, OLIVEIRA et al., 2007). Besides this environmental importance, this crop is distinguished in production of the ethanol, a fuel so-called "clean", since it proceeds from renewable sources which, as a complement or replacement to fossil fuels, contributes to reduce the effects of the global warming (MARTINELLI, 2011, GALLARDO; BOND, 2010, GOLDEMBERG, 2007).

The rural enterprises require an evaluation under the viewpoint of their efficiency when performing those activities. However, to evaluate the inputs and outputs in the production and efficiency improvement processes of those properties, those authors suggest the use of the techniques or methods that provide more objective and accurate information than those obtained through the analysis of either profitability and financial indicators, since there are several external factors that may affect the results (TORQUATO et al., 2009).

The managers of the organizations require operational tools that are based on theoretical and well-founded principles that evaluate and measure the efficiency. However, several scientific contributions are complex and present difficult application outside the academic ambit or technical expertise – either by the high cost, the long time to generate results and the limited flexibility to adequate to different real situations (FERREIRA; GOMES, 2009).

The Data Envelopment Analysis has been widely used on situations in which the proposal is the measurement of the performance of organizational units where the presence of multiple inputs and multiple outputs rather makes comparison difficult (REIS et al., 2011, SOUZA et al., 2009, FERREIRA; GOMES, 2009, ONUSIC et al., 2007, MELLO et al., 2005, BANKER et al., 2004, COOPER et al., 2004).

To visualize the potentialities and to help the detection of the strangulation points in the sugarcane-producing properties, it is justified the need to answer the main question: "How to evaluate the efficiency of the farms producing sugarcane, by using procedures and criteria that contemplate the main production factors that characterize the productive activity and their interrelationships".

In this scenario, the present research was carried out to evaluate the productive efficiency of 17 sugarcane properties located on northeastern Minas Gerais, through the Data Envelopment Analysis (DEA).

Methods

Concerning to objective, this research is characterized as exploratory and descriptive. In relation to results, this research is classified as applied.

The data were obtained from SEBRAE/MG, for the period 2006 to 2011 (for five sugarcane crops) and, complementarily, the technical visits were made to sugarcane


properties. To measure the relative efficiency of the sugarcane rural properties under study, the method Data Envelopment Analysis (DEA) model CCR (Constant Returns to Scale) with orientation towards inputs (CHARNES et al., 1978). The software SIAD v3.0 was used to process the inputs and the output. The inputs-oriented model was chosen due to the possibility for intervention by farmer´s managers in attempting to reduce the costs of production, which would be a possibility greater than its ability to influence the selling price of their products in the sugarcane market. From this choice, the generated results will contribute to identify the inputs and their respective proportions, therefore maintaining constant the output, the inputs can be reduced in order to reach the efficiency in the input-output relationship.

To evaluate the relative efficiency of 17 sugarcane properties (DMUs), adopted the following inputs and outputs were adopted, Figure 1:

Figure 1. Productive process in the sugarcane farms

For better understanding of the proposed model, the description of the inputs and outputs included in Figure 1 are presented as follows.

a) Input 1: The rent of the land amounted 10 tons of sugarcane at market price (R$ 45,00/t).

b) Input 2: The raw material consists of the values of the sugarcane sucker, manure, fertilizer, liming material, insecticide and herbicide (R$/ha).

c) Input 3: To meet the variable "quantity of human resources", the following factors will be considered: the remuneration of the workers in agricultural activities, as including cutting, loading and transportation of the sugarcane (R$/ha).

d) Output: Revenue (R$/ha) of the sugarcane is the result from the tons of the produced sugarcane multiplied by the price per ton of sugarcane.

The set of the inputs and outputs were chosen due to importance in evaluating the efficiency of the main components of the production system of a sugarcane stand, since according to Burnquist (2011), Macedo et al. (2004), Oliveira et al. (2011), Demattê (2005), Vitti; Mazza (2002), Fernandes (2000), the choice of efficient cropping techniques and practices should comprise from planning to implementation of the sugarcane plantation. Besides, the optimization of the use of the inputs, of the land and of the labor increases the crop productivity, reduces the production costs and increases profitability of the rural properties, besides contributing to environmental preservation when the inputs are efficiently used.



When analyzing the composition of the production costs of 17 sugarcane farms, it was identified that the the rent price and the price of the sugarcane suckers for planting were the same for all rural properties:R$450,00/ha and R$225,00/ha, respectively. The fertilization cost ranged from R$ 14,33/ha and R$ 366,59/ha. However, the highest cost were the cutting, loading and transportation of the industrialized stalks per ton, which ranged from R$ 18,26 to R$ 27,16, as being mainly affected by the distance from farm to industry as well as by the productivity of the crop.

From the data envelopment analysis, six efficient farms in the input-output relationship were identified. The results of the efficiency as well as the ranking of the Farms under study are shown in Table 1.

Table 1. Results of the efficiency score and ranking of 17 sugarcane farms, by using the CCR model, input orientation.

The DMUs 6; 8; 10; 12; 13; and 16 were considered as effective, since they obtained a score of 100% relative efficiency (Table 1). Those DMUs to the obtained production level were efficient in their use of inputs, because no losses were identified.

It is observed the differences to occur in efficiency scores of the 17 farms under analysis, as ranging from 100% to 74.2%. For farms that have not been effective, as maintaining constant the amount of the revenue obtained, it is considered that it’s possible to reduce the level of the inputs used, as the results shown in Table 2.

Among the DMUs considered as inefficient, the need for reducing the value of inputs per hectare are identified in all three variables under analysis. Table 2 presents the results for inefficient farms, and it is emphasized that they are presented in decreasing order of the efficiency score obtained.

Ranking DMUs Efficiency score Ranking DMUs Efficiency score1º DMU6 1 5º DMU17 0,9261º DMU8 1 6º DMU5 0,9221º DMU10 1 7º DMU14 0,9181º DMU12 1 8º DMU4 0,9131º DMU13 1 9º DMU11 0,8331º DMU16 1 10º DMU2 0,8192º DMU3 0,975 11º DMU9 0,8093º DMU15 0,949 12º DMU1 0,7424º DMU7 0,929 -


Table 2. Current input, target for the inputs and percent reduction of inputs in the inefficient sugarcane farms.

It should be noted among the three input variables under analysis that the total raw materials used, on average, showed a greater reduction in expenses required for the DMUs to achieve the goal of reduction (17.8%), although the indicated highest reduction refers to the variable ‘land rent’ for DMU2 (36.5%).

Those farms considered as effective are the benchmark for other units of the analysis. This reference varies according to approximation of the use levels of the inputs and outputs among farms, as ranging from 0 to 1. The importance of efficient DMU for the other is greater the closer to 1.

Table 3 presents the results of the benchmark analysis, considering the farm which is the main reference for the index and the importance of reference (lambda).

DMUs Input / ha Real Input Target Input Reductionland rent 450 439 2,5%

raw material 584 569 9,2%human resources 1516 1477 2,5%

land rent 450 427 5,1%raw material 964 914 27,5%

human resources 2399 2276 14,1%land rent 450 418 7,1%













raw material 615 457 25,8%human resources 2064 1532 30,1%Input Reduction Minimum Maximum Average

land rent/ha 2,5% 36,5% 13,8%raw material/ha 7,4% 34,2% 17,8%

human resources/ha 2,5% 30,1% 12,7%

DMU14

DMU4

DMU11

DMU2

DMU3

DMU15

DMU7

Descriptive Statistics

DMU17

DMU5

DMU9

DMU1


It is noted the DMU 6 to be the main reference for DMU 15, which is represented by lambda equal to 0.95. The DMU 8 is the main excellence partner for most number of DMUs, since this is the main reference to DMUs 7, 3, 5, 17, 14, 2 and 9. However, it is observed the reference importance index of the DMU 8 for other ones are lower than observed in the case of the DMU 6.

Table 3. Main benchmarks, inefficient farms and the importance reference index.

The DMU 12 is considered as benchmark for DMUs 11, 4 and 1. In the case of the DMU 1, however, the importance index of this reference is relatively low. Therefore, for this farm, the attempt to mirror practices of the DMU 12 becomes more distant.

It is observed that the DMUs 10, 13 and 16, although effective, have not been considered as main benchmarks for any other DMU. When analyzing the data of the DMUs 10, 13 and 16, those three were effective, but are not near the other DMUs concerning to dimension of their variables, therefore they do not stand as main benchmarks.

In this study, it was also verified a relationship between crop productivity and the profit in R$/ha, which can be described by the equation y = 24.2 x - 938.29 (R2=0.7845), Figure 2.

Figure 2. Productivity and profit from 17 farms producing sugarcane.

The sugarcane productivity varied from 76 to 114 t of industrialized stalks per hectare, as the average productivity reaching 81 t/ha.

Main benchmarks Inefficient Farms The importance reference indexDMU6 DMU15 0,949

DMU7 0,779DMU3 0,777DMU5 0,729

DMU17 0,688DMU14 0,532DMU2 0,521DMU9 0,517

DMU11 0,605DMU4 0,554DMU1 0,388

DMU8

DMU12


Conclusions

The Data Envelopment Analysis (DEA) is presented as an important tool for analyzing the efficiency, therefore it can be used in studies concerning to sugarcane farms. The application of this method for analyzing those 17 farms allowed to identify in all production factors, analyzed as input (land rent, raw material and human resources), the possibility to implement reductions in their spending in order to reach efficiently the input-output relationship.

Concerning to relative efficiency of the rural properties evaluated by DEA method, it was found that, from the total 17 DMUs, six were considered as effective - approximately 35% of the total.

The raw material is distinguished as the critical factor, for which the possible average reduction in spending was the highest. It is also identified that some farms have operated at low efficiency levels, whereas they should intervene and reduce their expenses by more than 15%, such as in the cases of the DMUs 11, 2, 9 and 1. This result indicates evident differences to occur in the performance among farms of the same activity sector.

In this study, the Data Envelopment Analysis indicates there are farms considered as efficient in input-output relationship. Those farms can be more intensively studied in order to reflect their production pattern, which could be used for the development of the other farms. Thus, further study can be conducted from the perspective of the sucroalcoholic activity development on northeastern Minas Gerais.

It is emphasized that this study can be improved under three perspectives. The first perspective refers to the analytic model adopted, which can be remade from the variant returns to scale (VRS) or BCC. The second improvement would be to detail the inputs under use, since the greater detail of the components of the raw material and human resources could facilitate the indication of intervention for improvement. The third possibility for improvement is the search for information in higher number of farms, on such a way to comprise all activity in the territorial region under analysis.

The use of the DEA method helped to identify the efficient sugarcane producers as well as to exploit their knowledge in order to improve the inefficient ones. Thus, the implementation of procedures that would increase the useful life of the reed stand, besides preserving the natural resources and increasing the efficiency of the land, of the inputs and of the human resources are recommended.

References

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BURNQUIST, H.L. (2011). Mechanical harvesting:balance between costs and opportunities. Abr-Jun. Available at: <http://www.revistaopinioes.com.br/aa/materia>. Accessed: September. 2011.


CHARNES, A., COOPER, W.W., RHODES, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2:6 (November), pp. 429–444.

COOPER, W. W.; LI, S.; SEIFORD, L. M.; ZHU, J. (2004). Sensitivity analysis in DEA. In: Handbook on data envelopment analysis. Boston: Kluwer Academic, chap. 3, p. 75-97.

DEMATTÊ, J.L.I. (2005). Recuperação e manutenção da fertilidade dos solos. Informações Agronômicas, n. 111, set.

FERNANDES, A.C. (2000). Cálculos na agroindústria da cana-de-açúcar. STAB. Piracicaba, SP, Brazil.

FERREIRA, C. M. C.; GOMES, A. P. (2009). Introdução à análise envoltória de dados: teoria, modelos e aplicações. Viçosa: Editora UFV. 389 p.

GALLARDO A.L.C.F & BOND A. (2010). Capturing the implications of land use change in Brazil through environmental assessment: Time for a strategic approach? Environ Impact Asses Rev.

GOLDEMBERG, J. (2007). Ethanol for a sustainable energy future. Science 9 February 2007: 315 (5813), 808-810. DOI:10.1126/science.1137013.

LINS, M.P.E.; ANGULO-MEZA, L. (2000). Análise Envoltória de Dados e perspectivas de integração no ambiente de Apoio à Decisão. Rio de Janeiro: COPPE/UFRJ.

MACEDO, I.C. et al. (2004). Balanço das emissões de gases do efeito estufa na produção e no uso do etanol no Brasil. São Paulo: Secretaria de Meio Ambiente do Estado de São Paulo.

MACEDO, M. A. S.; BENGIO, M. (2003). Mensurando a Eficiência da Relação Risco x Retorno em Ativos através da Análise Envoltória de Dados. XXXVIII Congresso Latino-Americano de Escolas de Administração (CLADEA), 2003, Lima, Peru. Anais do XXXVIII CLADEA.

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MARTINELLI, L. A. et al. (2011). Sugar and ethanol production as a rural development strategy in Brazil: Evidence from the state of São Paulo. Agricultural Systems [S.I.], v. 104, n. 5, p. 419-428.

MELLO, J. C. C. B. S; ÂNGULO-MEZA, L.; GOMES, E. G.; BIONDI NETO, L. (2005). Curso de Análise de Envoltória de Dados. XXXVIII Simpósio Bras. de Pesq. Operacional e Desenvolvimento Sustentável, 27 a 30/09/2005, Gramado, RS.

OLIVEIRA, T.B.A; SELIG P. M.; BARBOSA, V. M.; CAMPOS, L.M.S.; OLIVEIRA, M. W. (2011). Sustentabilidade da produção de cana-de-açúcar: um estudo de caso em uma propriedade agrícola. XII Congreso Internacional de Costos. Punta del Este. Uruguay. 27-29 nov.


ONUSIC, L. M.; CASA NOVA, S. P. C.; ALMEIDA, F. C. (2007). Modelos de previsão de insolvência utilizando a análise por envoltória de dados: aplicação a empresas brasileiras. Revista de Administração Contemporânea, v. 11, p. 77-97.

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SOUZA, M. C. S.; TANNURI-PIANTO, M. E.; ARAUJO, P.L.C.P. (2009). Residual and technical tax efficiency scores for Brazilian municipalities: a two-stage approach. 31º Meeting of the Brazilian Econometric Society, 2009, Foz do Iguaçu. Anais do XXXI Encontro Brasileiro de Econometria.

SUNDFELD, E.; MACHADO, C. (2011). Ações para o desenvolvimento de processos industriais para conversão de biomassa em biocombustíveis. Brasília, DF. Available at: <http://www.embrapa.br/imprensa/artigos>. Accessed: June.

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34. Theory of robust optimization in overall profit efficiency with data uncertainty

N. Aghayi Department of Mathematics, Ardabil Branch; Islamic Azad University, Ardabil, Iran, [email protected] (corresponding author)

M.A. Raayatpanah School of Mathematics, Statistics and Computer Science, University of Tehran, Tehran, Iran, [email protected]

Abstract

This paper presents an application of robust optimization in which data envelopment analysis (DEA) is used to measure overall profit efficiency when data are uncertain. When the inputs and outputs as well as the input and output price vectors of decision making units (DMUs) in the constraints belong to an uncertain set (a set with data uncertainty), we compute overall profit efficiency under the assumption of the worst case scenario in relation to the uncertainty and the adjustment of the level of robustness of the solution is also allowed to trade off performance with protection among these uncertain elements. We show that the maximum overall profit efficiency score may not always occur in an optimistic case and the decision maker (DM) can obtain the maximum overall profit efficiency score corresponding to a value between the optimistic and pessimistic cases. The results of the study have been exemplified by real applications.

Keywords: Data Envelopment Analysis, Robust Optimization, Overall profit efficiency, Uncertain Data.

Introduction

Data envelopment analysis (DEA) is concerned with the evaluations of performance and it particularly deals with evaluating the activities of organizations. At first, Charnes et al. (1978) introduced the CCR model under constant returns to scale (CRS) by extending linear programming production. The original CCR model, was applicable only to technologies characterized by global CRS, which was modified in the model introduced by Banker et al. (1984) assuming variable returns to scale (VRS). In traditional DEA, decision making units (DMUs) are evaluated by considering data certainty, and traditional DEA is thus not able to appraise DMUs under uncertainty conditions such as imprecision, vagueness, inconsistency, etc. Thus, the concept of uncertainty is one of the interesting subjects in DEA. In the real world, data uncertainty is shown in different ways, for instance, by interval, fuzzy, and stochastic data. Sengupta (1992) was the initial DEA study which changed the traditional view of uncertainty. Cooper et al. (1999) presented the interval approach to deal with interval


data in DEA. Entani et al. (2002) developed a DEA model where interval efficiencies were measured from both the optimistic and the pessimistic viewpoints.

In this paper, a robust overall profit efficiency score is presented for situations where the inputs and outputs as well as the input and output price vectors of DMUs involve uncertain data, which allows the degree of conservatism of the solution to be controlled. It is indicated that the maximum overall profit efficiency score may not always be obtained in the optimistic case and the DM can achieve the maximum overall profit efficiency score corresponding to a value between the optimistic and pessimistic cases. In fact, it is shown that the robust overall profit efficiency score is better than the optimistic overall profit efficiency score in both input and output uncertainty and price vector uncertainty.

Robust Overall Profit Efficiency (ROPE) Model with Input and Output Uncertainty

In this section, a model is formulated, based on robust optimization, to measure the overall profit efficiency where the inputs and the outputs of DMUs are belong to an uncertain set (a set with input and output uncertainty). We introduce numbers

njxj 1,...,=,Γ and ,1,...,=, njy

jΓ which assume values in the interval |]|[0, xjJ and

|]|[0, yjJ , where x

jJ and yjJ are the index sets of the uncertain parameters of inputs

and outputs, respectively. xjΓ and y

jΓ are to adjust the robustness of the proposed

method against the level of conservatism of the solution; indeed, they impose a budget of uncertainty in the sense that the total (scaled) variation of the parameters

cannot exceed some thresholds xjΓ and ,1,...,=, njy

jΓ and also these thresholds are

not necessarily integer-valued. The number of coefficients allowed to vary is at most

|| xjJ and || y

jJ for the inputs and the outputs respectively. Thus, our goal is to present

a model that is protected against the maximum xjΓ and y

jΓ , where only one input and

one output tjx and tjy vary by at most xtj

xj

xj ])[( Γ−Γ and y

tjyj

yj d])[( Γ−Γ , respectively.

That is, it is assumed that only a subset of the inputs and a subset of the outputs change to yeild the solution which was presented by Bertsimas and Sim (2004). The robust counterpart of Model (2) is proposed as follows:

θφ −max=ok

)(([max

)])[(..

||,|

1=1=

oyrorj

yjsr

yj

yjsy

jJyjsy

js

rll

n

oll

roorj

s

r

dp

yypts

λφ

λλφ

−+

−−

∑

∑∑

∈Γ≤⊆

≠


,,1,=0,))]

)(((])[()

1=

1=

njd

dpd

yljtl

n

oll

oy

ojtjjtyj

yj

yrll

n

oll

≤−

−Γ−Γ+−

∑

∑

≠

≠

λ

λφλ

++−+

++−

∑

∑∑

∈Γ≤⊆

≠

)(([max

)])[(

||,|

1=1=

oxioij

xjsi

xj

xjsx

jJxjsx

js

ill

n

oll

iooij

m

i

dc

xxc

λθ

λλθ

(1)

,,1,=0,))]

)(((])[()

1=

1=

njd

dcd

xljtl

n

oll

ox

ojtjjtxj

xj

xill

n

oll

≤+

+−Γ−Γ+

∑

∑

≠

≠

λ

λθλ

.,1,=0,1,=1=

njjj

n

j≥∑ λλ

It is clear that if ,1,...,=|,=||,=| nJJ yj

yj

xj

xj ΓΓ Soyster’s method is obtained and if

,1,...,=0,== njyj

xj ΓΓ we have the nominal problem. So, we have the flexibility of

adjusting the robustness of the method against the level of conservatism of the

solution by varying xjΓ and .1,...,=, njy

jΓ The above model is non-linear. We can

obtain the linear form of Model (1) by the proposition provided by Bertsimas and Sim (2004), as follows:

θφ −max=ok

,,1,=0,

])[(..1=1=

njq

zyypts

yrj

yjJr

yj

yjrll

n

oll

roorj

s

r

≤+

Γ+−−

∑

∑∑

∈

≠

λλφ

,,1,=0,

])[(1=1=

njq

zxxc

xij

xjJi

xj

xjill

n

oll

iooij

m

i

≤+

Γ+++−

∑

∑∑

∈

≠

λλθ

,,1,...,=,)( yj

yroorj

yrj

yj Jrnjdpqz ∈−≥+ λφ ,,1,...,=,

1=

yjrjl

n

oll

yrj

yj Jrnjpqz ∈≥+ ∑

≠

λ

,,1,...,=,)( xj

xiooij

xij

xj Jrnjdcqz ∈+−≥+ λθ ,,1,...,=,

1=

xjijl

n

oll

xij

xj Jinjcqz ∈≥+ ∑

≠

λ

(2)

1,=1=

j

n

jλ∑

.,,,1,=0,

0,0,0,0,xj

yj

xij

yrj

xj

yjj

JiJrnjq

qzz

∈∈≥

≥≥≥≥

λ

It should be mentioned that Model (2) can be applied in cases where the lower bound of each interval input and output is non-negative.


Model (2) is solved in two different ways for xjΓ s and y

jΓ s. In the first case, the

overall profit efficiency scores of DMUs are obtained from Model (2) for each xjΓ and

yjΓ that are randomly generated. Then, the mean values of these overall profit

efficiency scores are considered as the robust overall profit efficiency scores. In the second case, the overall profit efficiency scores of DMUs are obtained from Model (2)

by increasing xjΓ and y

jΓ , which change from zero to the number of the uncertain

inputs and outputs, respectively. The mean values of the overall profit efficiency scores are given as the robust overall profit efficiency scores.

Definition 1: DMU o is overall profit efficient if in Model (2), 0=** θφ − .

Robust Overall Profit Efficiency (ROPE) Model with the Input and Output Price Vectors Uncertainty

In this section, a model is formulated, based on robust optimization, to measure overall profit efficiency with the input and output price vector uncertainty. In real applications, it is unlikely for all coefficients of price vectors to be equal to their nominal value; it is also unlikely for them all to be equal to their worst-case value. Thus, it is desirable to adjust the level of conservativeness of the solution, so that a reasonable trade-off between robustness and performance is achieved. The numbers

njpj 1,...,=,Γ and nc

j 1,...,=,Γ are presented, which assume values in the intervals

|]|[0, pjJ and |]|[0, c

jJ , where pjJ and c

jJ are the index sets of the uncertain

parameters of the output and the input prices, respectively, and they are not

necessarily integer-valued. pjΓ and c

jΓ are to adjust the robustness of the proposed

method against the level of conservatism of the solution. In fact, they are the protection level for the j -th constraint. The number of coefficients allowed to vary is

at most pjΓ and c

jΓ for the output and input price vectors respectively. Thus, the main

purpose is to introduce a model that is protected against the maximum pjΓ and c

jΓ ,

where only one output price and only input price tjp and tjc vary by at most ptj

pj

pj ])[( Γ−Γ and c

tjcj

cj d])[( Γ−Γ , respectively. That is, it is assumed that only a subset of

the output price and a subset of the input price change to yeild the solution that was presented by Bertsimas and Sim (2004). We propose the robust counterpart of Model (5) as follows:

θφ −max=od


)([max

)(..

1=||,|

1=1=

lrl

n

lro

prj

pjsr

pj

pjsp

jJpjsp

js

lrl

n

lrorj

s

r

yyd

yypts

λφ

λφ

∑∑

∑∑

−+

−

∈Γ≤⊆

,,1,=

0,)](])[(1=

nj

yyd lljt

n

lojt

pjjt

pj

pj

≤−Γ−Γ+ ∑ λφ

)([max

)(

1=||,|

1=1=

lil

n

lio

cij

cjsi

cj

cjsc

jJcjsc

js

lil

n

lioij

m

i

xxd

xxc

λθ

λθ

∑∑

∑∑

+−+

+−

∈Γ≤⊆

(3)

,,1,=

0,)](])[(1=

nj

xxd lljt

n

lojt

cjjt

cj

cj

≤+−Γ−Γ+ ∑ λθ.,1,=0,1,=

1=njjj

n

j≥∑ λλ

The above model is non-linear. We can obtain the linear form of Model (3) using the proposition given by Bertsimas and Sim (2004), as follows:

θφ −max=od

,,1,=0,

)(..1=1=

njq

zyypts

prj

pjJr

pj

pjlrl

n

lrorj

s

r

≤+

Γ+−

∑

∑∑

∈

λφ

,,1,=0,

)(1=1=

njq

zxxc

cij

cjJi

cj

cjlil

n

lioij

m

i

≤+

Γ++−

∑

∑∑

∈

λθ

,,1,...,=, pj

pr

prj

prj

pj Jrnjfdqz ∈≥+

,,1,...,=, cj

ci

cij

cij

cj Jrnjfdqz ∈≥+

,1,...,=,1=

srfyyf prlrl

n

lro

pr ≤−≤− ∑ λφ (4)

,1,...,=,1=

mifxxf cilil

n

lio

ci ≤−≤− ∑ λθ

1,=1=

j

n

jλ∑


.,,,1,=0,

0,0,0,

0,0,0,

cj

pj

cij

prj

ci

pr

cj

pjj

JiJrnjq

qff

zz

∈∈≥

≥≥≥

≥≥≥

λ

Model (4) is solved for pjΓ s and c

jΓ s, which are once generated randomly and once

determined by being increased step by step. In both cases, the mean value of the overall profit efficiency scores are given as the robust overall profit efficiency scores.


In this section, the robust overall profit efficiency will be explained. We consider a problem with 4 DMUs, two inputs, two outputs, and two interval input and output price vectors. The input, output, and nominal price vector data are taken from Cooper et al. (2006). The data for DMUs are given in Table 1. It is assumed that all deviation

values from the nominal rjp and ijc are equal to ten, that is, 10.== cij

prj dd

We solve Model (2) for different combinations of xjΓ s and y

jΓ s by the generalized

algebraic modeling system (GAMS). Thirty random xjΓ s and y

jΓ s are generated and

the overall profit efficiency scores for each random xjΓ and y

jΓ are saved. 1jk is given

the robust overall profit efficiency score of a DMU using the mean of overall profit

efficiency scores. Model (2) is run for different xjΓ s increasing by 0.2 in each step

from zero up to 3 and different yjΓ s increasing from zero up to 5 with step length 0.2 ,

after which the values of overall profit efficiency scores are stored. The mean values of overall profit efficiency scores are given as the robust overall profit efficiency

scores, denoted by 2jk . The results are provided in Table 2.

Model (4) is run for different combinations of pjΓ s and c

jΓ s by the generalized

algebraic modeling system (GAMS). Ten random pjΓ s and c

jΓ s are generated and

overall profit efficiency scores are stored for each random pjΓ and c

jΓ . The robust

overall profit efficiency scores are determined using the mean values of overall profit

efficiency scores and are denoted by 1jd . Moreover, the values of p

jΓ s and yjΓ s are

increased from zero to 2 with step length 0.2 and the values of overall profit efficiency scores are stored. The mean values of overall profit efficiency scores are

presented as the robust overall profit efficiency scores, denoted by 2jd . The results are

given in Table 3.

Conclusions

In this paper, a deterministic methodology was proposed to address the problem of measuring overall profit efficiency subject to the input, output, and price vector parameters being within an uncertainty set. Using the robust optimization concept, an equivalent model was built without uncertainty of the same class. Specifically, the


proposed model is a linear programming problem. One of the most appealing features of this model is that it uses very little information on the input, output, and price vector distributions, and therefore it is widely applicable. If we only know the mean and the variance of the distributions, the robust policy often outperforms the nominal policy.

Acknowledgement

This research is partially supported by Ardabil Islamic Azad University.

References

Banker, R., Charnes, A., Cooper, W.W. (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis, European Journal of Operational Research 30 (9): 1078-1092.

Bertsimas, D., and Sim, M. (2004) The price of robustness, Operations Research 52 (1): 35-53.

Charnes, A., Cooper, W. W., Rhodes, E. (1978) Measuring the efficiency of decision making units, European Journal of Operational Research; 2 (6): 429-444.

Cooper, W.W., Park, K.S., Yu, G. (1999) IDEA and AR-IDEA: models for dealing with imprecise data in DEA, Management Science 45: 597-607.

Cooper, W.W., Seiford, L.M., Tone, K. Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-solver Software. 2nd ed, New York, Springer, (2006).

Entani, T., Maeda, Y., Tanaka, H. (2002) Dual models of interval DEA and its extension to interval data, European Journal of Operational Research 136: 32-45.

Sengupta, J.K. (1992) A fuzzy systems approach in Data Envelopment Analysis, Computers and Mathematics with Applications 24 (8-9): 259-266.

. 4 price nominal and output, input, :1 DMUsfordataTheTable

3500600120600721801682742200480904505516016025318004008035050150131192201055010050090100151201

21212121 jjjjj ppccOOIIDMU


.Model(2)ith ROPE :2 wofresultsTheTable .Model(4)ith ROPE :3 wofresultsTheTable

0.060.000.0041.450.610.1633.070.211.1620.290.050.191

21 Ujjjj kkkDMU

0.240.060.1340.460.150.2930.320.080.1720.120.000.001

21 Ujjjj dddDMU


35. The efficiency in Tourism Investment capture in relation to Highways Investment in Tourist Routes of Espírito Santo State, Brazil

Marta Monteiro da Costa Cruz Federal University of Espírito Santo, Brazil, Doctor in Transport Engineering, [email protected]

Josiane Baldo Federal University of Espírito Santo, Brazil, Master in Civil Engineering, [email protected]

Abstract

Tourism becomes a sector of more important time for economic and social development of a country, contributing for the reduction of disparity of the regions and favouring for its growth. The transport service, as well as road infrastructure is essential part of tourism in order to promote movement of tourists to intended destination. This paper presents the use of DEA to measure the efficiency in capture of Tourism Investment in relation to Investment in Highways. Government tourism planning of Espirito Santo State is based on eight tourist routes. Each route has an origin in Vitoria, capital of Espirito Santo State and a final destiny that define the route. For this study were analyzed 87 stretches road linking Vitória, to each urban community that compose these routes. These 87 stretch roads were defined as DMU and it was divided into two groups due to the difference in type of road, simple or double. The inputs of this problem are distance between cities, road investment, maintenance road investment and traffic volume. It was used tourism investment as product. The DEA Model used was BCC oriented by product. The results achieved shows that tourism level of investment can increase comparing with roads investments.

Keywords: Data Envelopment Analysis, Efficiency Analysis, Highways of the Espírito Santo, Tourism Investment and Highways

Introduction

According to Porter (1990), the infrastructure is essential to the promotion of systemic conditions of competitiveness in service systems - transport, energy, water, telecommunications - key to economic activity.

In the economic context, according to the Brazilian Institute of Tourism (2010), the country reached the goal of $ 5.8 billion in international currencies generated by tourism. In the Espirito Santo State, according to a survey conducted by the State Secretariat of Tourism of the Espirito Santo State - SETUR in February 2010, the


number of tourists between 2008 and 2010 in high season increased by about 47%, resulting in an increase of 109% in average spending Tourist Routes. Regarding the mode of transport used, approximately 84% of answers used the road to move around to the desired destination.

In the social context, when a region has, among other issues, a road infrastructure in perfect condition has expanded its capacity to generate employment, income distribution, providing improved quality of life in communities. Already the tourism sector provides an endless source of knowledge of historical, cultural and social development of a country, state or community.

Today, Brazil ranks seventh in the ranking of nations that hold more international events, the city of São Paulo as first place in the Americas. The country reached its target of $ 5.8 billion in international currencies generated by tourism and, little by little, distance themselves from stereotypes related to "exotic" (MINISTRY OF TOURISM, 2010c).

However, the World Tourism Organization says that 75% of global tourist flows are intra-regional, i.e., the majority of tourism takes place within the tourist region. The eminent predominance of short-distance travel and reinforces this statement demonstrates that the practice of domestic tourism has been gaining more importance in the economic context of tourism.

According to the National Council of Commerce (CNC) (2008), domestic tourism in Brazil's support base of the entire production chain of domestic tourism and presents itself also as the vector product qualification against the national tourist international travel market. Spraying of economic activities issued by domestic tourist movement houses several tourist segments and forces them to get better qualification of its products and services, preparing them well for the reception of international tourist who usually has a higher criterion of satisfaction.

In a continental country like Brazil, tourist land transport has a significant importance - both for its affordability among consumers, and by geographic accessibility near the tourist destinations and products. Moreover, it is of fundamental importance for accessibility to local natural and cultural attractions, as well as stimulating the personal interaction between tourists, travelers and tourism professionals.

The overall goal of the work is to apply Data Envelopment Analysis - DEA to evaluate the efficiency level of the Government to attract investment in the tourism sector, taking into account the investment made in road stretches that give access to Tourist Routes of Espirito Santo, Brazil. They were then identified road stretches with better technical performance.

Espirito Santo Tourist Routes

Based on the Tourism Regionalization Plan, the State Government of the Espirito Santo State, through the State Department of Tourism (SETUR), created since 2001, the eight tourist routes listed below along with the cities that make up (SETUR, 2009):

. Route Caparaó with 11 cities

. Route Immigrants with 8 cities

. Route of the Sun and Moqueca with 5 cities


. Sea Route and the Mountains with 5 cities

. Green Route and Waters with 5 cities

. Route of the Valleys and Coffee with 5 cities

. Coastal Route and Immigration with 7 cities

. Route Marble and Granite with 20 cities

Currently, the Espirito Santo State have three routes in the international market (Route of the Sun and Stew, Sea Route and the Green Mountains and Rota and Waters). Access to the tourist routes is done through federal highway (BR-262, which connects with the Espirito Santo State Minas Gerais, in the east-west, BR-101 South and North, which links the Espirito Santo State with the Rio de Janeiro and Bahia, north-south), privatized highways (ES-060 South, known as the "Highway of the Sun", which connects Vitoria with the south coast of the state) and non-privatized (ES-166, ES-185, ES -490, ES-259, ES-060 North, among others) (CRUZ, JR OLIVEIRA, 2009).

Regionalization Planning Tour of the Espirito Santo State seeks to optimize and facilitate the mobilization efforts, resources, communication and synergy of production arrangements, seeking to consolidate the sustainable development of tourism, revealing a set of differential highlighted with attractive, concentrated in a geographical space delimited. These spaces must present conditions of competition and cooperation among actors that stimulate public investment, and define market segments to work (SETUR, 2006).

Espirito Santo Roadways

Whereas the total area of the state is 46,187 km ², its road density is 14.1 km of highway per 100 km ². All seats of its 78 counties can be accessed by paved highway. The average distance within the state, to access a paved highway, federal or state, is 6.1 km. This distance varies average 2.5 km to 23.9 km on the basis of micro-planning of the state. During the surveys, it was found that 207.6 kilometers were in paving and most of the roads were unpaved natural bed (ES-DER, 2009).

Identification DMUs

The road stretches linking the capital of Espírito Santo (Vitoria) to cities that make up each route can be considered the decision-making units, homogeneous, because they use the same vectors inputs / product with the same goal, which is road transport carried by passenger vehicles, public transportation vehicles, transport vehicles and cargo motorcycles, and therefore can be evaluated by their relative efficiencies, the units identified as being able to compose the efficient production frontier.

Note that there is the existence of road stretches connecting Vitoria to a given municipality who share inputs and outputs of another passage that connects Vitoria to another city. This happens due to the composition of excerpts come from one or more cities.

The decision to take as its starting point the city of Vitoria was due to several factors: first by its geographical location, being equidistant from most cities that make up the routes except one route to which the municipality belongs. Second factor is important


because this city be cut by BR-101, the main highway Brazilian, passing through the metropolitan area of Greater Vitoria cities linking the capital to the northern region of the state, the BR101 North and south cities of the region, the South BR101 The city of Vitoria binds to the State of Minas Gerais and the cities of the region and Southwest region of Caparaó by BR262 and the Highway of the Sun, ES-060, which connects the southern coast of the state. These roads are of great importance to highway network of Tourist Routes of the Espirito Santo State, since access to Routes must be done largely through them.

Another important factor is the fact that Vitoria is the only county in the state to have an airport with regular flights, making it a great receiver for tourists throughout the state.

The road stretches analyzed vary greatly in terms of volume of traffic, flows ranging from very low to very high flows. When observations of highway sections with flow of vehicles of different intensities are used on the same sample, we expect problems regarding the interpretation of the results.

Thus, one group was performed as auxiliary support the interpretation of results. The stretches of highway linking the cities of Vitória Routes have distinct characteristics of traffic volume. You can check that the distinction traffic volume occurs in stretches of two-lane road. Thus were established two groups: the 1st group will be represented by highway sections consisting solely of single lane highways, and the 2nd group shown by road stretches composed of simple and double track trails.

DEA Variables

The initial selection was based on data availability in DER-ES on Secretariat State Tourism of the Espirito Santo State, the Undersecretary of State Budget and the literature review. Initially six variables were chosen to analyze the efficiency of road stretches. Taking into consideration the opinion of experts, the relevant variables for final analysis of efficiency of road stretches connecting Vitoria to each municipality that make up the Tourist Routes of the Espirito Santo State were:

• Inputs: Distance, investment in roads, investment in maintenance and traffic volume.

• Products: Investment in tourism

In this study we adopted the BCC model (BANKER, CHARNES, COOPER, 1984) (with variable returns to scale) product oriented, with and without restriction to weights, using the software SIAD.

The SIAD (ANGULO MESA et al, 2005) was developed to calculate all the results of DEA models, such as efficiencies of DMU's, weights for each of the targets, benchmarks and clearances. Furthermore, it also provides the option of inserting weight restrictions and the possibility of using up to 150 DMU's. Through SIAD also can see which units work with constant returns and they work with variable returns to scale.



In this study, it was found that the efficiency score of inefficient portions were very low, portraying the deficiency of tourism investment over investment in roads. In the 1st group, 82% of the stretches were analyzed efficiency below 50%. In the 2nd group this percentage was even higher, 87%. This means that the level of investment in tourism can increase large-scale investment in roads and road maintenance support this increase.

The road stretches efficient by default were Vitoria - Vila Velha (less distance, less investment in roads, lower investment in road maintenance and increased investment in tourism) and the Vitoria - Ibatiba, lower volume of vehicles.

One can see that besides the uniformity of efficiency scores of passages considered efficient with and without restriction to weights, this uniformity is also observed in the analysis of benchmarks with and without restriction to weights because practically the same passages that were benchmarks without restriction to weights were also restricted, with both the 1st group and for the 2nd group.

It is noteworthy that, despite several attempts planning and investment by the Government in the Tourist Regions of the Espirito Santo State, is still far below what is needed to ensure the efficiency of road stretches that give access to these routes. Thus, one should seek to improve investment in the tourism sector, maintaining the quality and efficiency of highways.

Conclusions

Besides the analysis of tourism investment over investment in roads, there are several other important indicators in the analysis of efficiency Tourist Routes that have not been included here. Therefore it is extremely important that prompted the sectors responsible for the inclusion of indicators such as:

• Number of tourists;

• Index accidents on highways;

• Cost of accidents on highways.

Another suggestion would be to analyze the efficiency of Tourist Routes in relation to its infrastructure (power of attraction of the municipality, number of Beds, number of Tourist attractions, number of positions of power, among other tourism products).

References

ANGULO MEZA, L;, BIONDI NETO, L.; SOARES DE MELLO, J.C.C.B.; GOMES, E. G. ISYDS (2005) Integrated System for Decision Support (SIAD – Sistema Integrado de Apoio a Decisão): a software package for data envelopment analysis model. Pesquisa Operacional, v.25, n.3, p 493-503.

BANKER, R.D.,CHARNES, A., COOPER W.W. (1984) Some models for estimating Technical and Scale inefficiencies in Data Envelopment Analysis. Management Science.


CONFEDERAÇÃO NACIONAL DO COMÉRCIO – CNC. (2008) Transporte turístico terrestre / Confederação Nacional do Comércio, Coordenação das Câmaras Brasileiras de Comércio, Câmara Brasileira de Turismo. Rio de Janeiro.

CRUZ, M. ; OLIVEIRA, R. (2009) . Rotas turísticas do Estado do Espírito Santo: atratividades versus custos envolvidos. In: XXIII ANPET - Congresso de Pesquisa e Ensino em Transportes, 2009, Vitória. Panorama nacional da pesquisa em transportes 2009. Vitória : ANPET, v. 1. p. 1-4.

DEPARTAMENTO DE ESTRADAS E RODAGENS DO ESPÍRITO SANTO- DER-ES. (2009) Plano Estratégico de Logística e Transportes do Espírito Santo: Componente Rodoviáro.Vitória. 6v.

MINISTÉRIO DO TURISMO. (2010) Turistas estrangeiros gastaram u$ 5,3 bilhões no Brasil em 2009, 22/01/10. Disponível em: http://www.turismo.gov.br/dadosefatos/. Acesso em: Mar. 2010.

MINISTÉRIO DOS TRANSPORTES. (2010) Departamento Nacional de Infra-Estrutura de Transportes. Rede Rodoviária do PNV: divisão em trechos 2010. Brasília, 2010.

ORGANIZAÇÃO MUNDIAL DO TURISMO, (1994) Banco de dados. Disponível em: <http//www.world-tourism.org>. Acesso em: Mar. 2010.

PORTER, M. (1990) The need for a new paradigm: the competitive advantage of nations. New York: The Free Press, 1990.

SECRETARIA DE ESTADO DE TURISMO DO ESPÍRITO SANTO. (2008) Pesquisa de turismo receptivo na região metropolitana da grande Vitória alta temporada. SEBRAE: FUTURA. Vitória, 2008.

http://www.turismo.gov.br/dadosefatos/


36. The investment in ports enterprises in Espirito Santo, Brazil

Karen Vassoler Martins Federal University of Espirito Santo, Brazil, Mastering in Transport Engineering, [email protected]

Marta Monteiro da Costa Cruz Federal University of Espírito Santo, Brazil, Doctor in Transport Engineering, [email protected]

Abstract

Considering the amount of resources needed for ports enterprises, it is important to evaluate the level of infrastructure utilization at ports terminals to assist managers to take decisions on improving performance or investing in increasing capacity. In this line, this paper analyses the efficiency of available elements use at public port terminals located at the state of Espírito Santo, Brazil, in order to obtain subsidies to its planning and development. For this, it was made a performance comparison between regional terminals using DEA modelling and the homogenization criteria of DMUs presented by Bertoloto (2010). The results of the study indicate the potential for increase cargo handling and can contribute to management decisions at the operational and strategic level.

Keywords: Port efficiency; Port infrastructure; DEA.

Introduction

According to the Secretary of Ports (SEP, 2011), the Brazilian port sector handles approximately 700 million tons of various goods per year, what strengthens its importance in the major trade operations (IPEA, 2010, SEP, 2011).

In the context of Brazilian international trade, the port complex located at the state of Espírito Santo has a prominent position (ESPÍRITO SANTO, 2006). In 2010, ports and terminals that belong to that complex moved around 171.46 million tons of cargo (SINDIEX, 2011), at about 24.00% of the total flow in the country.

Given its importance, the development of regional economy is closely linked to the strategies of port activities. And to support the existing demand through public terminals, it is indispensable to increase efficiency in the use of available infrastructure (ESPÍRITO SANTO, 2006).

As defined in Brazilian port sector legislation, port facilities can be classified as private or public, and public facilities can be operated by public or private companies, by prior bidding and agreement (BRAZIL, 1993; IPEA, 2009; IPEA, 2010).

Among the elements that ensure port competitiveness there are the berths and depth, both determinants of the size of ships that dock at the port, that also determine the




amount of cargo handled (BERTOLOTO, 2010; IPEA, 2010). Considering the amount of resources needed to build these elements, to know the level of use of existing structures is important at the studies for ports and terminals expansion (WANKE, 2009a).

In this sense, the analysis of efficiency, defined as the ratio between what is produced and what could be produced with the same resources (SOARES DE MELLO et al., 2005), can be useful to identify the efforts to improve ports performance. And one of the most known and used methods to get benchmarks for ports infrastructure is the Data Envelopment Analysis (FONTES AND SOARES DE MELLO, 2006; COSTA, 2007; PIRES et al., 2009; BERTOLOTO, 2010).

The Data Envelopment Analysis (DEA) is a technique for analysis and diagnosis based on mathematical programming developed by Charnes, Cooper and Rhodes (1978), that uses data input and output and the production function theory to estimate the efficiency frontier in a set of decision-making units (SOARES DE MELLO et al., 2003; BLONINGEN AND WILSON, 2008; WANKE, 2009 a; BERTOLOTO, 2010).

To analyze the efficiency of use of infrastructure utilization of public terminals located at Espírito Santo and obtain information for its planning and development, this paper presents a study of nine ports and terminals, with similar location and connectivity with other modes, based on maximum draft allowed, total length of berths and total amount of cargo handled, with application of DEA. According to the different forms of exploitation which they are subjected, these ports and terminals were classified into three groups - Direct Administration, Rented and Private - on which was used the compensation technique for non-homogeneous units presented by Bertoloto(2010).

The results indicate that there is potential to increase cargo handling at public terminals and suggest that specialization and form of operation of terminals and ports influence its efficiency.

Performance measures obtained can help to identify the level of utilization of ports and terminals, to trace the causes of inefficiencies, to define the best form of exploitation and to take decisions on berths specialization.

Methods

This study proposes a combination of theory and practice through the application of Data Envelopment Analysis model on a set of ports and terminals located at the state of Espírito Santo, Brazil, to evaluate the efficiency at local public terminals infrastructure utilization.

The DEA model chosen was BCC (BANKER et al., 1984), input-oriented.

Nine ports or terminals were analyzed at the period between the years 2008 and 2009. Were defined as inputs the maximum draft allowed and total length of berths, both measured in meters, and as output the total amount of cargo handled quarterly, measured in tons (BERTOLOTO, 2010). Data were obtained from analyzed ports and terminals and from the local Port Authority.


Ports and terminals, each quarter, were considered a decision-making unit (DMU) in order to facilitate the identification of seasonality interferences and economic situations in the amount of cargo handled.

Due the different forms of exploitation the studied ports and terminals are subjected, the set is not considered homogeneous. Because of this the units were classified in 3 distinct clusters: Direct Administration, Rented and Private, with the application of compensation method approached by Bertoloto (2010), following the script laid out by the author: 1) Running a DEA model isolated for each cluster; 2) Running a DEA model with only the efficient units of each cluster and calculate the median of the new efficiency values for each cluster; 3) Divide the outputs of DMUs of each cluster by the average efficiency found in the previous step and 4) Running a DEA model with all DMUs and calculate the efficiencies again. For this it was used SIAD (Integrated System for Decision Support), version 3.0, Meza et al (2005).


The study considered only ports and terminals with similar location and access infrastructure, in order to minimize the effects of comparisons with terminals whose hinterlands have larger markets or that are connected to more efficient modes.

It was obtained 108 DMUs, but only 104 were analyzed, since four units did not handle any amount of cargo.

Since the goal of the study is to provide subsidies to the planning and development of public terminals, the performance analysis focuses on the units belonging to clusters Direct Administration and Rented. The results can help managers to take decisions about the opportunity for specialization or diversification of cargo handling.

The average efficiencies of ports and terminals analyzed are presented in Table 1. Performance measures, beside the output targets, suggest the potential of the inefficient units to increase the volume of cargo handled.

Clusters Direct Administration and Rented concentrated 78.00% of DMUs with efficiencies lower than 50.00% - 33.33% and 44.67%, respectively. Such information can guide service level definitions of rented terminals or subsidize feasibility studies on rent.

The comparison between inefficient terminals and benchmarks contributes to the identification of decision factors for the best use of structures, assisting the operational management of terminals operated directly by Public Administration and the management of contracts for rented terminals. Such factors may be related, among other things, to the amount of equipment, labor and storage area available.

At strategic level, this performance analysis can be used to evaluate the development of port infrastructure and determine if a particular port is underused or not, before the decision to expand it, and provide information for setting the direction and scope of port activities (BERTOLOTO, 2010).


Table 1: Summary table of average efficiencies of ports and terminals

Cluster Port / Terminal Efficiency Average Individual Cluster

Direct Administration

DA1 27,57% 46,35% DA2 60,89%

DA3 50,59%

Rented R1 67,15%

39,67% R2 9,47% R3 42,20%

Private P1 78,91%

57,27% P2 9,05% P3 83,86%

Conclusions

Based on maximum draft permitted, extension of berths and total cargo handled data of a group of regional ports and terminals, it was made an analysis of efficiency in utilization of infrastructure of public terminals located at the state of Espírito Santo, Brazil.

Ports and terminals were divided into three groups, according to the form of exploitation they are subjected - Direct Administration, Rented and Private – upon which was applied the mathematical model of Data Envelopment Analysis, using the compensation method of no homogeneous units presented by Bertoloto (2010).

In the analyzed universe the average performance of private ports is higher than that of public terminals (Direct Administration and Rented units) and in this latter group, no specialized terminals presented the highest percentage of DMUs with inefficiencies lower than 50.00%. The observations suggest that specialization influences terminals efficiencies, as already stated by Wanke (2009 b).

The results of the study contribute both to the operational management, assisting in the identification of the determinants of low performance, and to strategic management, as they provide important information for decision on expansion, directing port activities and opportunity for specialization of berths.

References

Banker, R. D.; Charnes, A.; Cooper, W.W. (1984) Some models for estimating technical and scale inefficiencies in data envelopment analysis. Management Science 30 : 1078-1092.#

Bertoloto, R. F. (2010) Eficiência de portos e terminais privativos brasileiros com características distintas. 2010. 70 f. Dissertação (Mestrado em Engenharia de Produção) – Programa de Pós-graduação em Engenharia de Produção, Universidade Federal Fluminense, Niterói.


Bloningen, Beferences

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Pires, L. S.; Bertoloto, R. F.; Soares de Mello, J. C. C. B. S. (2009) Análise da eficiência de portos de carregamento de minério de ferro. Rio’s International Journal on Sciences of Industrial and Systems Engineering and Management, v.3, p. 094-01.

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Soares de Mello, J. C. C. B. S.; Meza, L. A.; Gomes, E.G.; Serapião, B. P.; Lins, M. P. E. (2003) Análise Envoltória de Dados no estudo da eficiência e dos benchmarks para companhias aéreas brasileiras. Pesquisa Operacional, v.23, n.2, p.325-345.

Soares de Mello, J. C. C. B. S.; Meza, L. A.; Gomes, E. G.; Neto, L. B. (2005) Curso de Análise Envoltória de Dados. In: XXXVII Simpósio Brasileiro de Pesquisa Operacional – SBPO 2005, Gramado, Anais do SBPO 2005.

Wanke, P. F. (2009 a) Infraestrutura Portuária. In: Wanke, P. F.; Silveira, R. V.; Barros, F. G.(eds) Introdução ao Planejamento da Infraestrutura e Operações Portuárias: Aplicações de Pesquisa Operacional. Atlas, São Paulo.

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37. Benchmarking the Efficiency of Third Party Logistics in Brazil Using Data Envelopment Analysis

Luís Filipe Azevedo de Oliveira Department of Production Engineering, Federal University of Rio Grande do Norte [email protected]

Mariana Rodrigues de Almeida Department of Production Engineering, Federal University of Rio Grande do Norte [email protected]

Abstract

This paper deals with the efficiency evaluation and analysis of 44 Third Party Logistics (3PLs) providers working in Brazil by means of the Data Envelopment Analysis (DEA). Since the analyzed companies have diversified sizes, the DEA model considered variable returns to scales and was oriented to maximize the outputs of 3PLs. The use of variable selection techniques was critical, in this research, to reach a subset of variables with greater representativeness. As the main practical result of this study, it was possible to identify the relative efficiency of 3PLs of national range and their return to scale. Furthermore, the results are of help in the decision making on investments based on the enterprises capacity, considering the expected return of each situation.

Keywords: Thrid Party Logistics, Efficiency, Variable Return to Scale

Introduction

The process of partial or total outsourcing of logistics operations is characterized by using outside companies to perform typical Third Party Logistics (3PLs) provider functions, which traditionally used to be performed by the companies themselves (Sohail and Sohal, 2003; Kayakutlu and Buyukozkan, 2011). This trend is generally found among the organizations that seek, by outsourcing nonessential activities, the reduction of internal costs and the service level increase through a raise in the supply chain management efficiency (Min and Joo, 2006; Seth, et al., 2006).

In accordance with this worldwide phenomenon, a fast growth of logistics activities has been observed in Brazil, which led companies to reach new competitive levels in several economies’ dimensions (Wanke and Affonso, 2011). However, even with an accelerated growth and with improvements in the national supply chain, Brazil still has low efficiency when compared to medium or high developed countries (Wanke and Fleury, 2006).


This paper evaluates the efficiency of 44 3PL’s working in Brazil, using Data Envelopment Analysis (DEA). DEA is a non-parametric technique that uses linear programming to calculate the relative efficiency of a set of similar organizations, which differ in the inputs and outputs arrangement of their processes. The method results in a complete ranking, beyond the dichotomized classification of efficient or inefficient, in order to refine the evaluation of the units. Furthermore, it provides a parameter to increase the performance of inefficient organizations, showing how they should evolve to achieve business performance comparable to benchmark enterprises in the market (Adler, et al., 2002; Cooper et al., 2006; Cook and Seiford, 2009).

This paper presents a framework on three pillars: i) integration of the theme of logistics and data envelopment analysis, ii) research methods, iii) results of the analysis.

Efficiency Measurement of Third Party Logistics Providers

The efficiency study of Third Party Logistics (3PLs) providers by means of the Data Envelopment Analysis (DEA) have had great prominence in current literature (Min and Joo, 2006; Ding, et al., 2008; Hamdan and Rogers, 2008; Zhou, et al., 2008; Koster, et al., 2009; Liu and Fu, 2009; Wanke and Affonso, 2011). This fact occurs due to the role of 3PLs as leverage for the industry development, facing the complex relationships that exist along the supply chain.

One of the first studies linking efficiency evaluation to 3PLs was conducted by Min and Joo (2006), developing a meaningful set of financial benchmarks that dictates best practices, though DEA analysis, applying a non-linear fractional program. The proposed DEA model helps 3PLs identify potential sources of inefficiency and provide useful hindsight for the continuous improvement of operational efficiency. Furthermore, helps 3PLs establish detailed policy guidelines in prioritizing the use of financial resources and evaluate the effects of financial investment on the profitability of 3PLs.

Ding, et al. (2008) explain that the traditional DEA model evaluates decision making units (DMU) from the most beneficial perspective to them, which easily leads to unreasonable weight assignment. To solve this problem, an improved DEA model with upper-limit and lower-limit confidence region constraint has been established based on information entropy. In this context, the paper established an Information Entropy-DEA model to evaluate 7 logistics suppliers from china.

Hamdan and Rogers (2008) introduce DEA as a tool to evaluate the efficiency of a group 19 warehouse logistics working in the United States. The relative efficiency scores for the warehouses used in the study were analyzed before and after the use of weight restrictions. As a result, it was possible to determine the impact of each input and output on the efficiency of each warehouse, and also, to examine specific warehouse characteristics and develop a set of recommendations in the improvement and design of more efficient operations.

The purpose of Zhou, et al. (2008) was to develop a benchmark for performance standards for 3PLs in the emerging market. The authors identify the factors that


significantly affect the operational efficiency of the Chinese 3PLs, proposing different ways to improve the competitiveness of them. In particular, it develops both the Constant Return to Scale (CRS) and the Variable Return to Scale (VRS) model and also uses step-wise regression analysis to identify factors influencing the performances of Chinese 3PLs.

In a different perspective, Koster, et al. (2009) applied DEA on primary data of 38 large container terminals in Europe and compared efficiency scores from the benchmarking exercise with those of previous studies, discussing the reasons behind diverging results. The authors show that DEA may be appropriate for container terminal benchmarking, but only if better quality and additional input and output data can be obtained.

Liu and Fu (2009) compared the results of CRS and VRS input oriented DEA models, calculating both the technical efficiency and scale efficiency, and also the returns to scale for each of the 16 logistics public companies from China. By comparing the efficiency of logistics companies, they could contribute to find the 3PLs’ weaknesses in order to take effective measures to improve their level of input and output so as to improve its logistics efficiency.

Wanke and Affonso (2011) studied the 3PLs industry in Brazil. Its main objective was to determine the variables that significantly impact on the 3PL scale efficiency by means of a two-stage DEA. The inputs and outputs required for this analysis were identified as well as the contextual variables that may impact on the 3PL scale efficiency. The results corroborate the evidence in the literature on the role of coordination processes in logistics performance.

Methodology

The research was conducted with data from the special issue Operadores Logísticos (2011), published annually by the brazilian magazine Tecnologística, which shows the main companies from the Third Party Logistics (3PLs) providers market in Brazil. The analysis contemplates the universe of companies working in Brazil and different sized businesses.

The analysis took into account four input variables, as follows: i) number of employees; ii) total area used for storage; iii) number of warehouses owned by the companies; and iv) number of warehouses within the client facilities. As a result of the outputs, the following variables were considered: i) number of customers under contract; ii) company’s operating revenue; iii) total managed products volume; and iv) revenue growth presented by the company as compared with the previous year.

According to Dyson, et al. (2001), the set of Decision Making Units (DMU) must be homogeneous regarding to the activities they perform, being able to run similar tasks with the same goals and also presenting the same inputs and outputs, with varying intensity of these variables. In other words, the DEA methodology requires common variables among the 3PLs to compare them, making it necessary to obtain a list of organizations, limited by data availability. In that sense, a sample of 44 3PLs was


extracted from the original set of 126 companies, taking as selection criteria those that provided the information relevant to the chosen input and output variables.

The DEA model was oriented to maximize the outputs of 3PLs and considered variable returns to scales, since the analyzed companies have different sizes and thereby have technologies with constant, increasing and decreasing returns to scale. This model was first proposed by Banker, et al. (1984) and, for such conditions, can be described as Expression 1.

Min 𝑣𝑖 ∙ 𝑥𝑖0 + 𝑣0

𝑚

𝑖=1

(1)

Subject to:

𝑢𝑟 ∙ 𝑦𝑟0

𝑠

𝑟=1

= 1

𝑢𝑟 ∙ 𝑦𝑟𝑗

𝑠

𝑟=1

− 𝑣0 − 𝑣𝑖 ∙ 𝑥𝑖𝑗

𝑚

𝑖=1≤ 0;

𝑗 = 1, … , 𝑛, 𝑣𝑖 , 𝑢𝑟 ≥ 0; 𝑟 = 1, … , 𝑠; 𝑖 = 1, … , 𝑚.

Here the yrj and xij, all positive, are the known outputs and inputs from a set of j = 1,…, n organizations. The data yrj and xij are constants and will usually be observations from past decisions on inputs and the outputs that resulted therefrom. The variables vi, ur ≥ 0 are weights to be determined by the solution of this problem, by the data on all of the DMU's which are being used as a reference set. It is worth noting the presence of variable v0, representing the return to scale presented by the analyzed unit ‘0’, determining whether the operations are conducted with increasing, constant or decreasing returns to scale. Thus, in case v > 0, returns to scale are decreasing; if v = 0, returns are constant; and if v < 0, returns will be increasing.

The efficiency measure of any DMU is obtained as the inverse of the minimum of the weighted inputs and rated relative to the others from the set that accords the most favorable weighting that the constraints allow. Thus, the weights are objectively determined to obtain a scalar measure of efficiency in any case. The constraints provide that ratios of weighted outputs to weighted inputs for every DMU be less than or equal to unity.

Efficiency of Third Party Logistics in Brazil

The quantitative analysis of this research considers the parameters to evaluate the efficiency of Third Party Logistics (3PLs) providers in Brazil, with a practical and measurable focus. The purpose of this analysis is to access and compare the measurable parameters of 3PLs by the link among the conditions offered by them – such as inputs and/or supplies – and their volume and productivity responses – as output and/or products triggered.


According to the proposed model, it was obtained an average technical efficiency of 75.74%, a minimum efficiency rate of 13.14% and the standard deviation (SD) for the sample was 28.05. The results show that 20 3PLs were classified as efficient since their efficiency score were 100%, which implies the inefficiency status for 24 3PLs, that represents 55% of the sample (Table 1).

Table 1: Performance evaluation of 3PLs

Performance Evaluation

Average 75.74%,

SD 28.05

Minimum 13.14%

100% 20

99 –70% 7

69 –40% 9

Low 39% 8

Using the company's size classification adopted by the Banco Nacional do Desenvolvimento (BNDES, which in a free translation means National Development Bank), that considers the Annual Gross Revenue, it’s possible to note that, among the successful firms, six are small (30%), three are medium-sized companies ( 15%), seven are medium to large (35%) and four are large companies (20%).

Each return to scale phase presents a specific denotation to support the results, helps in the decision making on investments based on the enterprise’s capacity, considering the expected return of each situation (Table 2). The increasing scale implies that in raising input variables, a large (disproportional) increase in output variables for the 11 companies that present this kind of return to scale. Therefore, to invest on capacity growth of these companies is attractive because it would bring a rapid growth in national performance, provided by economies of scale likely to be achieved for 25% of 3PLs examined.

Table 2: Evaluation of the scale of 3PLs

Return to Scale

Scale Evaluation

Total Efficiency

100% <100%

Increasing 11 5 6

Constant 0 0 0

Decreasing 33 15 18


Considering the efficiency score of this group, the result shows that five companies that are technically efficient presented increasing scale of production, indicating that despite the absence of inputs in excess, the production volume is below the optimal range. This means that such units can increase production by raising the amount of resources required, moving the efficiency frontier away from the analyzed group.

Noting the inefficient firms that have an increasing return to scale, they represent six organizations on the sample. These companies operate below optimal scale, and such units can also increase their production level by raising the amount of resources to achieve the optimum efficiency.

Running on a decreasing scale, with 33 3PLs from the sample, the resources used exceed the optimum level. It is not recommended to obtain more resources to increase the production capacity by 75% of firms, but to reduce the excesses. These aspects would bring benefits proportionately lower than the rates of resources increasing. The 15 units that are efficient and have decreasing returns to scale operate above the optimal scale, which implies the need to reduce its volume of production, considering that the increase in production occurs in this situation by adding a larger quantity of inputs needed, which becomes economically unattractive. Therefore, to achieve optimum efficiency, it is necessary to reduce the excessive use of input, maintaining the production level.

Finally, it is observed that 18 organizations are inefficient and have decreasing returns to scale. To increase their technical efficiency, these companies must eliminate the use of excess raw materials, while increasing the production level.

Results and Conclusions

The growth of the logistics operations complexity and the demand for efficiency gains in performing such tasks led to the Third Party Logistics (3PLs) providers’ appearance. These are companies that take a strategic role within organizations, since they allow costs to be reduced and help improve service level.

This paper focused on the efficiency evaluation and analysis of 44 3PLs in operation in Brazil by means of the Data Envelopment Analysis (DEA). As the main practical result of this study, it was possible to identify the relative efficiency of 3PLs of national range and their return to scale. Based on the constructed model results, 20 3PLs were identified as efficient and representative of national benchmarks by their optimal relationship between productivity and resource usage.

About the scale in which each 3PLs operates, it was identified that 11 work on an increasing scale and 33 on a decreasing scale, with no occurrence of constant performance. These results are helpful in making decisions on investments based on the enterprises’ capacity, considering the expected return of each situation.


When faced with these results, managers can select the best players concerning the efficiency of production processes based on a ranking that lists the best 3PLs and assess the efficiency regarding its strategic objectives, in order to drive companies to best practices and superior performance.

Therefore, these results help organizations in making decisions regarding resources allocation in order to optimize and distribute them to the appropriate channels, which justifies the use of DEA as a tool to contribute to the 3PLs pursuit of excellence.

REFERENCE

Adler, N. et al. (2002), “Review of ranking methods in the data envelopment analysis context”, European Journal of Operational Research, Vol. 140, No. 2, pp. 249-265.

Banker, R.D. et al. (1984), “Some models for estimating technical and scale inefficiencies in data envelopment analysis”, Management Science, Vol. 30, No. 9, pp. 1078-1092.

Cook, W.D., Seiford, L.M. (2009), “Data envelopment analysis (DEA): Thirty years on”, European Journal of Operational Research, Vol. 192, No. 1, pp. 1-17.

Cooper, W. et al. (2006), Introduction to Data Envelopment Analysis and Its Uses: With DEA-Solver Software and References, Springer, New York, NY.

Ding, B. et al. (2008), “Third-party Logistics Provider Efficiency Evaluation Based on Information Entropy-DEA Model”, in International Seminar on Future Information Technology and Management Engineering, Leicestershire, 2008, IEEE, Washington, pp. 166-170.

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Koster, M.B.M. et al. (2009), “On using DEA for benchmarking container terminals”, International Journal of Operations & Production Management, Vol. 29, No. 11, pp. 1140-1155.

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Wanke, P.F., Fleury, P.F. (2006), “Transporte de cargas no Brasil: estudo exploratório das principais variáveis relacionadas aos diferentes modais e às suas estruturas de custos”, in Negri, J.A., Kubota, L.C. Estrutura e dinâmica do setor de serviços no Brasil, IPEA, Brasília, pp 409-464.

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