Data Envelopment AnalysisData Envelopment Analysis(DEA)(DEA)

Which Unit is most productive?Which Unit is most productive?

DMU = decision making unit

DMU labor hrs. #cust. 1 100 150 2 75 140 3 120 160 4 100 140 5 40 50

DEA DEA (Charnes, Coopers & Rhodes ‘78)

A multiple-input, multiple-output productivity measurement tool

Basic intuition(DMU = decision making unit)

DMU labor hrs. #cust. #cust/hr. 1 100 150 1.50 2 75 140 1.87 3 120 160 1.33 4 100 140 1.40 5 40 50 1.25

#cust.

labor hrs.

x x

50 100

100

200

x

x

x

slope = 1.87

DMU’s 1,3,4,5 are dominated by DMU 2.

Extending to multiple outputs ...Extending to multiple outputs ...Ex: Consider 8 M.D.’s working at Shouldice Hospital for the same 160 hrs. in a month. Each performs exams and surgeries.

Which ones are most “productive”?

Note: There is some “efficient” trade-off between the number of surgeries and exams that any one M.D. can do in a month, but what is it?

Doctor #Exams #Surgeries1 48 682 12 803 35 764 31 715 20 706 20 1057 36 538 15 65

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20

40

60

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120

0 10 20 30 40 50 60

#Exams

#S

urg

eri

es

Scatter plot of outputs: Efficient M.D.’s: These two M.D.’s (#1 and #6) define the most efficient trade-off between the two outputs.

efficient frontier

#6

#1

These points are dominatedby #1 and #6.

“Pareto-Koopman efficiency” along the frontier - cannot increase an output (or decrease an input) without compensating decrease in other outputs (or increase in other inputs).

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#S

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How bad are the inefficient M.D.s and where are the gaps?

73.4% of distance to frontier

Efficiency score = 73.4%

Performance “gap”#5

0

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40

60

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120

0 10 20 30 40 50 60

#Exams

#S

urg

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es #5

Reference set for #5is {1,6}

#1

#6

“Nearest” efficient points define a reference set and a linear combination of the reference set inputs and outputs defines a hypothetical composite unit (HCU)

HCU

DEA summary so far: DEA summary so far: DEA uses an efficient frontier to define multiple I/O productivity

Frontier defines the (observed) efficient trade-off among inputs and outputs within a set of DMUs.

Relative distance to the frontier defines efficiency “Nearest point” on frontier defines an efficient

comparison unit (hypothetical comparison unit (HCU)) Differences in inputs and output between DMU and

HCU define productivity “gaps” (improvement potential)

How do we do this analysis systematically?

A real-word example: A real-word example: NY Area Sporting Goods StoresNY Area Sporting Goods Stores

ProductivityProductivity

Conceptually ...

Productivity = OutputsInputs

Reality if more complex ...

Technology+

Decision Making

Inputs Outputs

equipment

facility space

server labor

mgmt. labor

#type A cust.

#type B cust.

quality index

$ oper. profit

Operating Units DifferOperating Units Differ

Mix of customers served Availability and cost of inputs Facility configuration Processes/practices used Examples

– bank branches, retail stores, clinics, schools, etc.

Questions:– How do we compare productivity of a diverse set of operating units

serving a diverse set of markets?– What are the “best practice” and under-performing units?– What are the trade-offs among inputs and outputs?– Where are the improvement opportunities and how big are they?

Some approachesSome approaches Operating ratios

– e.g. Labor-hrs/transaction, $sales/sq.-ft.– Good for highly standardized operations– Problem: Does not reflect varying mix of inputs and outputs found in more

diverse operations Financial approach: Convert everything to $$$!

Problems?– Some inputs/outputs cannot be valued in $ (non-profit)– Profitability is not the same as operating efficiency (e.g. variances in

margins and local costs matter as well)

$Inputs $Outputs

Profitability vs. effeciencyProfitability vs. effeciency

Profitability is a function of 3 elements …– Input prices (costs)– Output prices– Technical efficiency (How much input is required to generate

the firms output.)

Improving operations requires understanding technical efficiency not just overall profitability.

LP Formulation:LP Formulation:

N

iiki

M

jjkj

k

k

j

i

ik

jk

Iv

Ou

E

E

ju

iv

kiI

kjO

MjM

NiN

KkK

1

1

100%)-(0 DMU of efficiency

output on weight

input on t weigh

DMU from input of level observed

DMU from output of level observed

,...,1 outputs#

,...,1 inputs #

,...,1 (DMUs) units operting# Data

Model variables

To evaluate a give unit, e, choose nonnegative weights tosolve ...

KkE

ts

E

k

e

,...,1 ,100

..

max

Niv

Mju

KkIvOu

Iv

ts

Ou

i

j

N

iiki

M

jjkj

N

iiei

je

M

jj

,...,1 ,0

,...,1 ,0

,...,1 ,100

1

..

max

11

1

1

Which can be formulated

Normalize weighted input of

e to one

Output analysis

e setreferencek

k

k

k

DMU of in the is DMU 0

DMU with associated variabledual

NiII

MjOO

ik

jk

,...,1 ,ˆ

,...,1 ,ˆ

K

1kki

K

1kkj

These dual variables can be used to contruct an efficient hypothetical composite unit (HCU) with

Input i of HCU

Output j of HCU

Satisfying

NiII

MjOO

ie

je

,...,1 ,ˆ

,...,1 ,ˆ

i

j

HCU can be used to measure excess use of inputs and potential increase in outputs

NiII

MjOO

ie

je

,...,1 ,0ˆInput

,...,1 ,0ˆ Output

i

j

Refer to spreadsheet examples.

Using the results: Eff.-Profit MatrixUsing the results: Eff.-Profit Matrix

High Profit

Low Profit

LowEff.

HighEff.

Under-performingpotential leaders

Best practice comparison group

Under-performingpossibly profitable

Candidates forclosure

Designing DEA StudiesDesigning DEA Studies

#Inputs/OuputsK > 2(N+M)

“Ambivalence” about inputs and outputs - all should be relatively important!

“Approximate similarity” among DMUs– objectives– technology

Provides relative efficiency only– choice of units to include matters– inclusion of “global leader” unit may be desirable

Experimenting with different I/O combinations may be necessary

DEA SummaryDEA Summary

Addresses fundamental productivity measurement problems due to ...

– complexity of service outputs– variability in service outputs

Takes advantage of service operating environment– large numbers of similar facilities– diversity of practices/management/environment

Provides useful information– objective measures of productivity– reference set of comparable units– excess use of inputs measure– returns to scale measure

DEA Summary (cont.)DEA Summary (cont.)

Role of DEA– “data mining” to generate hypotheses– evaluation/measurement– benchmarking to identify “best practice” units

Caveats– “black box” - No information on root causes of inefficiency– Be aware of assumptions (e.g. linearity)– Can be sensitive to selection of inputs/outputs