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Some Models for Estimating Technical and Scale Inefficiencies in Data Envelopment Analysis

Author(s): R. D. Banker, A. Charnes, W. W. Cooper

Source:

Management Science,

Vol. 30, No. 9 (Sep., 1984), pp. 1078-1092

Published by: INFORMS

Stable URL: http://www.jstor.org/stable/2631725

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MANAGEMENT SCIENCE

Vol. 30, No. 9, September 1984

Printed

U.S.A.

SOME MODELS FOR ESTIMATING TECHNICAL

AND

SCALE INEFFICIENCIES

IN DATA

ENVELOPMENT

ANALYSIS*

R. D.

BANKER,

CHARNES

AND W. W.

COOPER

School

of Urban

and Public

Affairs, Carnegie-Mellon University,

Pittsburgh, Pennsylvania 15213

Graduate School

of Business, University of Texas, Austin,

Texas 78712

management contexts, mathematical programming is usually used to evaluate a collec-

tion

possible

alternative courses

action en route

to selecting

one which is best.

this

capacity, mathematical programming serves

as a

planning aid to management. Data

Envelop-

ment Analysis reverses this role and employs mathematical programming to obtain ex

post

facto evaluations

the relative efficiency

management accomplishments, however

they

may

have been

planned

executed. Mathematical programming

thereby extended

for use

as a

tool

for

control and evaluation of past accomplishments as well as a tool to aid in

planning

future activities. The

CCR ratio

form

introduced by Charnes, Cooper and Rhodes,

part of

their Data

Envelopment Analysis approach, comprehends

both

technical and scale inefficien-

cies

via the

optimal

value

the ratio

form,

obtained directly

from

the data without

requiring

priori specification of weights and/or explicit delineation of assumed functional

forms

of relations

between inputs

and

outputs.

separation into technical and

scale

efficien-

cies

accomplished by

the methods

developed

this paper

without

altering the

latter

conditions

for

use

of DEA

directly

observational data.

Technical

inefficiencies

are

identified

with

failures to achieve best

possible output levels and/or usage

of excessive

amounts

inputs.

Methods

for

identifying and correcting

the

magnitudes

these

inefficien-

cies,

supplied

prior work,

are illustrated. In the

present paper,

new separate

variable

introduced

which

makes it

possible

determine whether operations were conducted

regions

increasing,

constant or

decreasing returns to

scale

(in multiple input

and

multiple

output

situations).

The

results

are discussed and related not

only

to classical

(single output)

econom-

ics

but

also

to more

modern versions

economics which are identified with "contestable

market theories."

(EFFICIENCY;

TECHNICAL

INEFFICIENCY; RETURNS

SCALE;

MATHEMATI-

CAL

PROGRAMMING;

LINEAR

PROGRAMMING)

1. Background

Charnes, Cooper and Rhodes (CCR) (1978a, 1979) introduced a ratio definition

efficiency,

also called the CCR

ratio

definition,

which

generalizes

the

single-output

single-input

classical

engineering-science ratio definition

multiple

outputs

and

inputs

without

requiring preassigned weights.

This is

done via

the

extremal

principle

incorporated

the following model:

s2=rYO

iUrYrj

maxh =h

subject

, j =

..., n,

with

(1)

Ur, Vi

O,1

..=

1....

Here the

yrj,

xii

represent output

and

input

data

for

decision

making

unit

(DMU)

*Accepted by

Arie

Lewin; received December 11, 1981. This paper has been with the

authors

months for

revisions.

'See

Charnes, Cooper, Lewin, Morey,

and

Rousseau

for an exact non-Archimedean

expression

of this

"positivity"

with

necessary algebraic closure.

1078

0025-1 909/84/3009/

078$O

.25

1984, The Institute of

Management Sciences

INEFFICIENCIES IN DATA

ENVELOPMENT ANALYSIS 1079

Output

Production function

x1=X3

Input

FIGURE

with the ranges for i, r and j

indicated in (1). The data may be in the form of

theoretically prescribed values or they

may be

the

form of

observations. The unit to

rated is included in the functional

with an index

as well as in the constraints, with

the latter

ensuring

that

an optimal ho

maxho

will always satisfy

with

optimal

solution

values u,*, v* > 0.

The

main uses

these ideas have

been in evaluations of "management" and

"program"

efficiencies2 of

decision

making units (DMUs) of a not-for-profit variety

such as

schools,3 hospitals,4 etc. The

ability to deal directly with multiple outputs and

inputs

forms

one part

the appeal

offered by these models and methods for uses such

as these. Another

part

of its

appeal

comes

from

the

development

Charnes, Cooper

and Rhodes

(1978a)

which

showed

how

the

theory

fractional

programming,

provided

Charnes and Cooper

(1962), could be used to obtain access to a linear

programming equivalent. This,

turn,

yields an implementable form for securing

solutions to

(1) and

also yields a

variety

duality

relations for

interpreting and

utilizing

the

resulting u*, v*" >

Strong (and sharp)

theoretical

underpinnings

physics

and

engineering

are not

available

applications

such

as we are

considering.

These

must be

replaced by

weaker

support-such

as can be

obtained

from

other

disciplines

like economics. It

is,

fact,

one

purpose

of the

present paper

sharpen

some of the latter

contacts,

but even after

this has

been

accomplished,

one must

generally

be satisfied with

weaker

results. For

instance,

one

must be satisfied with a

measure

only relative efficiency

based

the

available observations

without recourse to what a

stronger theory might provide.

now

try

clarify

what

has

already

been covered

reference

the illustration

Figure

Here we have

portrayed

the situation to be

considered

in terms of

a single

output,

amounts, y, and

single

input,

amounts

x. Three decision

making

units

are to be rated for

managerial

efficiency.

The

production

function

represents

the

maximum output that can

produced for

any specified input.

The

DMUs associated

with

P2 and

both achieve

the maximum

possible outputs for their input levels,

while

the DMU associated with

falls

short of

the

output

level

which

attainable from

its

x3 input

value.

evaluate

the

efficiency

PI,

utilize

(1), which,

for

this one

output-one input

2See Charnes, Cooper and

Rhodes (1981),

for

further

discussion of differences in

"management" and

"program" efficiency.

3See Bessent,

Bessent, Kennington

and

Reagan (1982).

4See

D. Sherman

(1982)

which also

contains

interesting comparison

that

highlights

deficiencies of

statistical

regressions

(including translog and Cobb-Douglas

regressions)

and econometric estimation and

similar

approaches

that have been addressed to these multiple

output situations

the past.

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