Guillermo Durán Rafael Epstein Gonzalo Zamorano

Quantitative Methods for a New Configuration of Quantitative Methods for a New Configuration of Territorial Units in a Chilean Government Agency Territorial Units in a Chilean Government Agency

Tender ProcessTender Process

Guillermo DuránRafael Epstein

Gonzalo Zamorano

formerChilean Vice-Minister of Education

(Jan2008-March2010)

Universidad de Chile Universidad de Buenos Aires

Cristian Martínez

Daily reaches 2 million children, from 0 to 24 years

11,000 schools along the country (4200 Km)

Private catering firms bid on supply contracts

US$600 million a year

The largest procurement process in ChileThe largest procurement process in Chile

School Meals ProgramSchool Meals Program

Territorial Units (TUs)Territorial Units (TUs)

CHILE: 13 regions13 regions

136 TUs136 TUs

UT 1 UT

2UT 4

UT 3

UT 5

UT 6

UT 7

UT 8

UT 9

REGION 1

REGION 2

REGION 13

UT 135

CHILE

UT 136

UT 133

UT 134

Combinatorial AuctionCombinatorial Auction

In 1997 a Combinatorial Auction is implemented• Milgrom P., Putting Auction Theory to Work,

2007, Cambridge University Press. • Cramton P., Shoham Y., Steinberg R. (editors), Combinatorial

Auctions, 2006, The MIT Press.

Each year, 1/3 of all TUs is contracted

Bids provide services for 3 years

Combinatorial Auction BenefitsCombinatorial Auction Benefits

Cost synergies are reflected on the bids

Economies of scale Volume discounts in purchasing inputs

Economies of density Logistic savings when serving nearby units

US$3 billion awarded with this model

Cost reduction reported: 22%

A New Challenge: BankruptcyA New Challenge: Bankruptcy

Five firms declared bankruptcy between 2004 and Five firms declared bankruptcy between 2004 and 2007 leading to serious financial and social losses for 2007 leading to serious financial and social losses for

the governmentthe government

Bankruptcy is a big problem: Meal service is interrupted, affecting the educational process Restoring service in affected TUs is more expensive Bankruptcies eliminate an actor from the market

Bankruptcies CausesBankruptcies Causes

1° Cause: Price War among firms Tender system promotes competition Companies drop their prices to eliminate competitors After the war the prices tend to increase, higher than before

Prices Band Try to identify the abnormal low prices Eliminate these bids from the process

Bankruptcies CausesBankruptcies Causes

2° Cause: Limited liability Aggressive bids on TUs with unknown operating conditions

When the real operation is good: The firm starts a successful business

When the real operation is unsustainable: The firm assumes the private costs JUNAEB assumes the rest of the costs

• Social costs, costly small auctions, less competitive market

Uncertainty ProblemUncertainty Problem

Some TUs offer worse operating conditions

Firms don’t properly estimate its costs in these TUs

Firms offer low prices on “bad” TUs

The bidding process selects some of them

After that, they may not fulfill their contracts

Proposed SolutionProposed Solution

Redesign TUs configuration

Avoid “bad conditioned” TUs

Homogenize operating conditions of TUs

Reduce the global risk of the system

Using OR to Reconfigure TUs

Comunas The smallest administrative units 346 comunas in Chile

Chilean Geographical DivisionChilean Geographical Division

Territorial Units Groups of comunas

Firms bid over the whole TU 136 TUs

Number of meals: magnitude of the contract

Number of schools: fixed costs of supply

Area covered (in km2): transport costs

% Inaccessible schools: geographical conditions

Attractiveness Index of TUAttractiveness Index of TU

Number of Schools

Total Area

Accessibility

Number of meals

Attractiveness Category Territorial Units (TU Codes) Total TUs

Below Average 101, 1004, 1007, 1005, 1010, 302, 609, 803, 805, 1101 10

Average 102, 301, 605 3Above

Average 606, 607, 1321 3

Bankruptcy: relatively “bad” TUBankruptcy: relatively “bad” TU

TUs involved in recent bankruptcies

Central Purpose:Reconfigure the Comunas into homogeneous TUs

Based on the Analytic Hierarchy Process (Saaty, 1980)

The Attractiveness IndexThe Attractiveness Index

total attractiveness score for TU j in region r.importance of number of meals within the set of criteria weight of TU j in region r on number of meals criterionimportance of number of schools within the set of criteriaweight of TU j in region r on number of schools criterionimportance of size of area of TU within the set of criteriaweight of TU j in region r on area criterionimportance of type of access to school within the set of criteriaweight of TU j in region r on accessibility criterion

rjU ,

MarjMx ,,

rjSx ,,

ArcrjArx ,,

AcdrjAcx ,,

Sb

rjAcAcrjArArrjSSrjMMrj xdxcxbxaU ,,,,,,,,,

Importance of each characteristicImportance of each characteristic


Importance of each characteristicImportance of each characteristic


xM,j,r xS,j,r xAR,j,r xAC, j, r

calculated by using statistical data

Example: Region has 2 TUs UT1 = 40,000 meals UT2 = 60,000 meals Then xM,1,r =40 and xM,2,r=60

Method 1: Local Search HeuristicMethod 1: Local Search Heuristic

Objective: minimize the standard deviation, this is, the dispersion of TU attractiveness levels in a region

Create initial solution for every possible number of TUs

Comunas are exchanged between TUs in a Region

In each iteration, only one comuna is exchanged

The best local optimum is selected

Local Search HeuristicLocal Search Heuristic0. Initial Solution

1. First Iteration

N. After N iterations,

final solution

Local Search Heuristic, example Local Search Heuristic, example

Lower bound fixed per UT: 15,000

Upper bound fixed per UT: 40,000

This region may have 2 or 3 TUs

Table of Characteristics: 1st RegionRegion Comunas Meals Schools Area (km²) Easy access Difficult access

Arica 25,726 62 4,799 62 0Camarones 54 8 3,927 7 1

Putre 248 6 5,903 6 0General Lagos 180 9 2,244 9 0

Iquique 9,155 38 2,262 38 0Alto Hospicio 11,387 25 573 25 0

Huara 460 12 10,475 12 0Camiña 404 9 2,200 9 0

Colchane 252 5 4,016 5 0Pica 799 5 8,934 5 0

Pozo Almonte 1,685 10 13,766 10 0Total 50,350 189 59,099 188 1

1st Region

Example, starting with Example, starting with 2 UTs2 UTs TU 1: 55.9% TU 2: 44.1% TU 1: 51.4% TU 2: 48.6% TU 1: 49.7% TU 2: 50.3%

Arica A Iquique Arica A Iquique Arica A IquiqueArica B Pozo Almonte Arica B Pozo Almonte Arica B Pozo AlmonteArica C Huara Arica C Pica Arica C PicaGeneral Lagos Pica General Lagos Alto Hospicio A General Lagos Alto Hospicio APutre Alto Hospicio A Putre Alto Hospicio B Putre Alto Hospicio BCamarones Alto Hospicio B Camarones Camiña Camarones Colchane

Camiña Huara Colchane HuaraColchane Camiña

Std. Deviation 5.9261 Std. Deviation 1.3646 Std. Deviation 0.3324

Iteration 1TU 1: 55.9% TU 2: 44.1% TU 1: 51.4% TU 2: 48.6% TU 1: 49.7% TU 2: 50.3%




Iteration 2TU 1: 55.9% TU 2: 44.1% TU 1: 51.4% TU 2: 48.6% TU 1: 49.7% TU 2: 50.3%




Initial Situation

Std. Dev = 5.93 Std. Dev = 1.33 Std. Dev = 0.33

Example, starting with Example, starting with 3 UTs3 UTs

TU 1: 38.05% TU 2: 30.37% TU 3: 31.58% TU 1: 35.69% TU 2: 32.35% TU 3: 31.96% TU 1: 33.05% TU 2: 32.43% TU 3: 34.52%Arica A Arica C Alto Hospicio B Arica A Arica C Alto Hospicio B Arica A Arica C Alto Hospicio BArica B Camarones Iquique Arica B Camarones Iquique Arica B Camarones IquiqueGeneral Lagos Huara Pozo Almonte General Lagos Huara Pozo Almonte General Lagos Huara Pozo AlmontePutre Camiña Pica Putre Camiña Pica Putre Camiña

Colchane Colchane Alto Hospicio A Colchane Alto Hospicio AAlto Hospicio A Pica


Iteration 2TU 1: 38.05% TU 2: 30.37% TU 3: 31.58% TU 1: 35.69% TU 2: 32.35% TU 3: 31.96% TU 1: 33.05% TU 2: 32.43% TU 3: 34.52%

Arica A Arica C Alto Hospicio B Arica A Arica C Alto Hospicio B Arica A Arica C Alto Hospicio BArica B Camarones Iquique Arica B Camarones Iquique Arica B Camarones IquiqueGeneral Lagos Huara Pozo Almonte General Lagos Huara Pozo Almonte General Lagos Huara Pozo AlmontePutre Camiña Pica Putre Camiña Pica Putre Camiña



Iteration 1TU 1: 38.05% TU 2: 30.37% TU 3: 31.58% TU 1: 35.69% TU 2: 32.35% TU 3: 31.96% TU 1: 33.05% TU 2: 32.43% TU 3: 34.52%

Arica A Arica C Alto Hospicio B Arica A Arica C Alto Hospicio B Arica A Arica C Alto Hospicio BArica B Camarones Iquique Arica B Camarones Iquique Arica B Camarones IquiqueGeneral Lagos Huara Pozo Almonte General Lagos Huara Pozo Almonte General Lagos Huara Pozo AlmontePutre Camiña Pica Putre Camiña Pica Putre Camiña



Initial Situation

Std. Dev = 3.37 Std. Dev = 1.67 Std. Dev = 0.88

2 TUs final solutionStd. Dev = 0,33

3 TUs final solutionStd. Dev = 0,88

The 2 TUs solution was selectedThe 2 TUs solution was selected

Method 2: Integer ProgrammingMethod 2: Integer Programming

Objective: minimize the difference between the most and least attractive TU in each region

Good linearization of minimize the standard deviation

Algorithm generates all possible TUs, called clusters

Integer Programming ModelInteger Programming Model

Formulation

Integer Programming ModelInteger Programming Model

Formulation

Results Standard DeviationResults Standard Deviation

RegionStandard Deviation Improvement

Original Heuristic Model O v/s H O v/s MI 5,93 0,33 0,29 94% 95%II 0,18 0,18 0,18 0% 0%III 0,74 0,19 0,19 75% 75%IV 5,94 1,81 0,88 69% 85%V 1,47 0,68 0,99 53% 33%VI 3,27 0,47 0,16 86% 95%VII 4,05 0,41 0,53 90% 87%VIII 2,44 0,60 0,87 75% 64%IX 3,90 0,40 0,03 90% 99%X 1,04 0,84 0,88 19% 15%

Both final solutions improve standard deviation on the situation prevailing before 2007

Results Gap Results Gap (most and least attractive (most and least attractive TU )TU )

RegionDifferences (max - min) Improvement

Original Heuristic Model O v/s H O v/s MI 12,70 0,55 0,04 96% 100%II 0,20 0,20 0,20 0% 0%III 2,55 1,02 1,02 60% 60%IV 46,49 3,36 1,58 92% 97%V 12,62 4,82 1,26 62% 88%VI 40,84 2,58 0,37 94% 99%VII 63,48 1,14 0,17 98% 98%VIII 72,86 2,87 2,49 96% 96%IX 47,09 1,37 0,13 97% 100%X 3,81 2,51 2,25 34% 41%

Both final solutions improve gap (max-min) on the situation prevailing before 2007

JUNAEB used our solutionJUNAEB used our solution

The configuration was adopted by JUNAEB in 2007

Since 2007, no bankruptcies have occurred

Accepted for publication in Interfaces“Quantitative Methods for a New Configuration of Territorial Units in a Chilean Government Agency Tender Process”, Guillermo Durán, Rafael Epstein, Gonzalo Zamorano, Cristian Martínez

Improving Public Policies

Education and OpportunitiesEducation and Opportunities

Efficiency and QualityEfficiency and Quality

Quantitative Methods for a New Configuration of Quantitative Methods for a New Configuration of Territorial Units in a Chilean Government Agency Territorial Units in a Chilean Government Agency

Tender ProcessTender Process

Guillermo DuránRafael Epstein

Gonzalo Zamorano

formerChilean Vice-Minister of Education

(Jan2008-March2010)

Universidad de Chile Universidad de Buenos Aires

Cristian Martínez

Documents

Guillermo Durán Rafael Epstein Gonzalo Zamorano