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Data Envelopment Analysis

Robert M. Hayes

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Overview

Introduction


DEA Models

Extensions to includea priori Valuations

Strengths and Weaknesses of DEA

Implementation of DEA

The Example of Libraries

Annals of Operations Research 66


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Introduction

Utility Functions

Cost/Effectiveness

Interpretation for Libraries

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Utility Functions

A fundamental requirement in applying operations research

models is the identification of a "utility function" whichcombines all variables relevant to a decision problem into asingle variable which is to be optimized. Underlying theconcept of a utility function is the view that it should representthe decision-maker's perceptions of the relative importance ofthe variables involved rather than being regarded as uniformacross all decision-makers or externally imposed.

The problem, of course, is that the resulting utility functionsmay bear no relationship to each other and it is therefore

difficult to make comparisons from one decision context toanother. Indeed, not only may it not be possible to comparetwo different decision-makers but it may not be possible tocompare the utility functions of a single decision-maker fromone context to another.

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Cost/Effectiveness

A traditional way to combine variables in a utility

function is to use a cost/effectiveness ratio, called an

"efficiency" measure. It measures utility by the "cost

per unit produced". On the surface, that would appearto make comparison between two contexts possible by

comparing the two cost/effectiveness ratios. The

problem, though, is that two different decision-makers

may place different emphases on the two variables.

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Cost/Effectiveness

It also must be recognized that effectiveness will usuallyentail consideration of a number of products and servicesand costs a number of sources of costs. Cost/effectivenessmeasurement requires combining the sources of cost intoa single measure of cost and the products and services

into a single measure of effectiveness. Again, the problem of different emphases of importance

must be recognized. This is especially the case for theseveral measures of effectiveness. But it may also be thecase with the several measure of costs, since some costs

may be regarded as more important than others eventhough they may all be measured in dollars. When somecosts cannot be measured in dollars, the problem iscompounded.

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Cost/Effectiveness

More generally, instead of costs and effectiveness, the

variables may be identified as "input" and "output".

The efficiency ratio is then no long characterized as

cost/effectiveness but as "output/input", but the issues

identified above are the same.

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This issue can be illustrated by evaluating libraryperformance. Effectiveness here is the extent to whichlibrary services meet the expectations or goals set bythe organization served. It is likely to be measured by

several services which are the outputs of libraryoperationsmaking a collection available for use,circulation or other uses of materials, answering ofinformation questions, instructing and consulting.

Inputs are represented by acquisitions, staff, and space,to which evident costs can be assigned, but they are alsorepresented by measures of the populations served.

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Efficiency measures the librarys ability to transformits inputs (resources and demands) into production ofoutputs (services). The objective in doing so is tooptimize the balance between the level of outputs and

the level of inputs. The success of the library, like thatof other organizations, depends on its ability to behaveboth effectively and efficiently.

The issue at hand, then is how to combine the several

measures of input and output into a single measure ofefficiency. The method we will examine is that called"data envelopment analysis".

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Data Envelopment Analysis (DEA) measures the relativeefficiencies of organizations with multiple inputs andmultiple outputs. The organizations are called thedecision-making units, or DMUs.

DEA assigns weights to the inputs and outputs of a DMUthat give it the best possible efficiency. It thus arrives at aweighting of the relative importance of the input andoutput variables that reflects the emphasis that appears tohave been placed on them for that particular DMU.

At the same time, though, DEA then gives all the otherDMUs the same weights and compares the resultingefficiencies with that for the DMU of focus.

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If the focus DMU looks at least as good as any otherDMU, it receives a maximum efficiency score. But if

some other DMU looks better than the focus DMU, the

weights having been calculated to be most favorable to

the focus DMU, then it will receive an efficiency score

less than maximum.

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Graphical Illustration To illustrate, consider seven DMUs which each have

one input and one output: L1 = (2,2), L2 = (3,5), L3 =(6,7), L4 = (9,8), L5 = (5,3), L6 = (4,1), L7 = (10,7).

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7

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9

0 1 2 3 4 5 6 7 8 9 10 11

Input

Output

L1

L2

L3

L4

L5

L6

L7

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Graphical Illustration

DEA identifies the units in the comparison set which lieat the top and to the left, as represented by L1, L2, L3,and L4. These units are called the efficient units, andthe line connecting them is called the "envelopmentsurface" because it envelops all the cases.

DMUs L5 through L7 are not on the envelopmentsurface and thus are evaluated as inefficient by theDEA analysis. There are two ways to explain theirweakness. One is to say that, for example, L5 could

perhaps produce as much output as it does, but withless input (comparing with L1 and L2); the other is tosay it could produce more output with the same input(comparing with L2 and L3).

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Thus, there are two possible definitions of efficiencydepending on the purpose of the evaluation. One mightbe interested in possible reduction of inputs (in DEAthis is called the input orientation) or augmentation ofoutputs (the output orientation) in achieving technicalefficiency. Depending on the purpose of the evaluation,the analysis provides different sets of peer groups towhich to compare.

However, there are times when reduction of inputs oraugmentation of outputs is not sufficient. In our

example, even when L6 reduces its input from 4 unitsto 2, there is still a gap between it and its peer L1 in theamount of one unit of output. In DEA, this is called the"slack" which means excess input or missing outputthat exists even after the proportional change in the

input or the outputs.

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This example will be used to illustrate the several forms

that the DEA model can take.

In each case, the results presented are based on the

implementation of DEA that will be discussed later in

this presentation. It is an Excel spreadsheet using the

add-in Solver capability.

The spreadsheet is identical for all of the forms, but the

application of Solver differs in the target optimized and

in the values to be varied, so for each form the target

and the values to be varied will be identified.

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DEA Models

The Basic EDA Concept

Variations of DEA Formulation

Formulation: Primal or Dual Orientation: Input or Output

Returns to Scale: Fixed or Variable

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The Basic EDA Concept

Assume that each DMU has values for a set of inputsand a set of outputs.

Choose non-negative weights to be applied to the inputsand outputs for a focus DMU so as to maximize theratio of weighted outputs divided by weighted inputs

But do so subject to the condition that, if the sameweights are applied to each of the DMUs (including thefocus DMU), the corresponding ratios are not greaterthan 1

Do that for each DMU.

The resulting value of the ratio for each DMU is itsEDA efficiency. It is 1 if the DMU is efficient and lessthan 1 if it is not.

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Formulation

Let (Yk,Xk) = (Yki,Xkj), k = 1 to n, i = 1 to s, j = 1 to m

Maximize Yk/Xk for each value of k from 1 to n,subject in each case to Yj/nXj = 0

The solution is the set of maximum values for Yk/Xkand the associated values for and

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Basic Linear Programming Model

For solution, this optimization problem is transformedinto a linear programming problem, schematicallydisplayed as follows:

In a moment, we will interpret this display as it istranslated into alternative formulations of theoptimization target and conditional inequalities.

Min Yj -Xj

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Variations of DEA Formulation

But first, it is necessary to identify several sources ofvariation in the basic DEA formulation, leading to avariety of different models for implementation:

We will now examine and illustrate each of thosesources of variation.

(1) Formulation Primal Form Dual Form

(2) Orientation Input Minimization Output Maximization(3) Returns to Scale Fixed Returns Variable Returns

(4) Discretionary? Discretionary Variables Non-discretionary Variables

(5) Models Additive Multiplicative

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(1) Formulation: Primal or Dual

The first source of variation is interpretation of thedisplay for the linear programming model.

One interpretation, called the Primal, treats the rows of

the display as representing the model.

The other interpretation, called the Dual, treats thecolumns as representing the model.

Lets examine each of those in turn.

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Primal Formulation

The rows of this display are interpreted as follows:

(M) Maximize W = Yk

Xk

subject to

(1) YjXj

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The Dual Formulation

The Columns of this display are interpreted as follows:

(m) Minimize W = -a - b subject to

(1) Yja >= Yk (2) Xj - b >= -Xk

(m) Yj -Xj 0a -I -I

b -I -I

>= >=

Yk - Xk

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The Choice of Formulation

Since the results from the two formulation are equal,though expressed differently, the choice between them

is based on computational efficiency or, perhaps, ease

of interpretation.

The Dual form is more efficient in computation if the

number of DMUs is large compared to the number of

input and output variables. Note that the Primal form

entails n conditions (n being the number of DMUs)

which, in the Dual form, are replaced by just m + s

conditions (m being the number of input variables ands, the number of output variables)

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Illustration

To illustrate, consider the example previously presented. Thetarget to be minimized in the Dual form is W =ab. The

values to be varied are (, a, b), or (,. The following table shows the solution for both forms:

X Y W a b L1 2 2 - 1.33 1.33 = 0.67 1 1.67L2 3 5 0.00 0.00 = 00 1 1.67L3 6 7 - 3.00 3.00 = 00 1 1.67L4 9 8 - 7.00 7.00 = 00 1 1.67L5 5 3 - 5.33 5.33 = 1 1.67L6 4 1 - 5.67 5.67 = 1 1.67L7 10 7 - 9.67 9.67 = 1 1.67

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Illustration

Graphically, the results are as follows:

The maximum value for W, over all cases, is at L2, whereW = 0 and the ratio of Y/X is a maximum. The slack foreach other case is the vertical distance to the line which

goes from the origin (0,0) through L2 (3,5).

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(2) Orientation: Input or Output

The second source of variation, orientation, providesthe means for focusing on minimizing input or on

maximizing output.

These represent two quite different objectives in

making assessments of efficiency. Is the objective to be

minimally expensive (e.g., to save money) or is it to be

maximally effective?

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Orientation to Input

The linear programming display for the inputorientation is as follows:

It adds one additional condition, Xk

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The resulting Dual formulation is as follows: (m) Minimize W = c-1 subject to

(1) Yja >= Yk (2)Xjb + (c1)Xk >= -Xk or Xk + b = >=

Max Yk - Xk

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Continuing with the same example, the following table

shows the solutions in both formulations. The target isW = c1. Values to be varied are now (, a, b, c) or( and .

Note that L2 still dominates the solution, but the graph

is now quite different,

X Y W=c-1 a b L1 2 2 - 0.40 = 0.40 0.30 0.50L2 3 5 0.00 = 00 0.20 0.33L3 6 7 - 0.30 = 0 0.10 0.17L4 9 8 - 0.46 = 0 0.07 0.11L5 5 3 - 0.64 = 00 0.12 0.20L6 4 1 - 0.85 = 00 0.15 0.25L7 10 7 - 0.58 = 0 0.06 0.10

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Orientation to Output

The linear programming display for the outputorientation is as follows:

It adds one additional condition, Yk

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The resulting Dual formulation is as follows:

(m) Minimize W = 1c subject to (1) Yja >= cYk (2)Xjb >=Xk or Xk + b = >=

Max Yk - Xk

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Continuing with the same example, the following table

shows the solutions in both formulations. The target isW = 1c. Values to be varied are still (, a, b, c) or( and .



X Y W=1-c a b L1 2 2 - 0.67 = 0.67 0.50 0.83L2 3 5 0.00 = 00 0.20 0.33L3 6 7 - 0.43 = 00 0.14 0.24L4 9 8 - 0.87 = 00 0.13 0.21L5 5 3 - 1.78 = 0.33 0.56L6 4 1 - 5.67 = 1.00 1.67L7 10 7 - 1.38 = 0.14 0.24

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Note that the graphical display is identical to that forthe general form, though the interpretation is

somewhat different (replacing efficiencies by slacks).

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5

10

15

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25

0 2 4 6 8 10 12

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(3) Returns to Scale: Fixed or Variable

The third basis for variation among DEA models isreturns to scale.

The examples presented to this point have each involvedconstant returns to scale. That is, the ratio Y/X canbe replaced by the inequality YX

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Variable Returns to Scale, Basic Model

The linear programming display for the basic DEAmodel is as follows:

It adds the variable u to the display.

u Min Yj -Xj I

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Variable Returns: Orientation to Input

The linear programming display for the variablesreturns to scale, input orientation is as follows:

It adds one additional condition, Xk

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Orientation to Input The resulting Dual formulation is as follows:

(m) Minimize W = c-1 subject to (1) Yja >= Yk (2)Xjb + (c1)Xk >= -Xk or Xk + b = 1

The new, third condition makes things interesting.

u (m) Yj -Xj I 0a -I 0

b -I 0

c - 1 Xk I>= >=

Max Yk - Xk I

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Continuing with the same example, the

following table shows the solutions in bothformulations. The target is W = c1. Values to

be varied are now (, a, b, c) or (, , u.X Y W=c-1 a b u

L1 2 2 0.00 0.00 =1.00 0.00 0.00 0.00L2 3 5 0.00 0.00 = 1.00 0.00 0.00 0.00L3 6 7 0.00 0.00 = 1.00 0.00 0.00 0.00L4 9 8 0.00 0.00 = 1.00 0.00 0.00 0.00L5 5 3 - 4.00 2.00 = 1.00 0.00 0.00 0.00L6 4 1 - 5.00 4.00 = 1.00 0.00 0.00 0.00L7 10 7 - 4.00 0.00 = 1.00 0.00 0.00 0.00

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The linear programming display for the outputorientation is as follows:

It adds one additional condition, Yk

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The resulting Dual formulation is as follows:

(m) Minimize W = 1c subject to

(1) Yja >= cYk (2)Xjb >=Xk or Xk + b = >=

Max Yk - Xk

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Continuing with the same example, the following tableshows the solutions in both formulations. The target isW = 1c. Values to be varied are still (, a, b, c) or( and .



X Y W=1-c a b L1 2 2 - 0.67 = 0.67 0.50 0.83L2 3 5 0.00 = 00 0.20 0.33L3 6 7 - 0.43 = 00 0.14 0.24L4 9 8 - 0.87 = 00 0.13 0.21L5 5 3 - 1.78 = 0.33 0.56L6 4 1 - 5.67 = 1.00 1.67L7 10 7 - 1.38 = 0.14 0.24

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Note that the graphical display is identical to that forthe general form, though the interpretation is

somewhat different (replacing efficiencies by slacks).

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0 2 4 6 8 10 12

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Extensions to includea priori Valuations

To this point, DEA has been essentially a mathematical process inwhich the data for input and output are taken as given, withoutfurther interpretation with respect to the reality of operations.

But reality needs to be recognized, so there are several extensions

that can be made to the basic DEA model, applicable to any of thevariations.

They fall into seven categories:

(1) Discretionary and Non-discretionary Variables

(2) Categorical Variables

(3)A priori restrictions on Weights (4) Relationships between Weights on Variables

(5)A priori assessments of Efficient Units

(6) Substitutability of Variables

(7) Discrimination among Efficient Units

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Discretionary & Non-discretionary

In identifying input and output variables, one wants toinclude all that are relevant to the operation. Forexample, the level of output is determined not only bywhat the unit itself does but by the size of the market towhich the output is delivered.

The result, though, is that some relevant variables, suchas the size of the market, are not under the control ofmanagement. Such variables, called non-discretionary,are in contrast to those that are under managementcontrol, called discretionary.

In assessing efficiency, all variables are considered, butin determining the criterion function to be maximized orminimized, only the discretionary variables are included.

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Categorical Variables

In the DEA model as so far presented, the variables aretreated as essentially quantitative, but sometimes one

would like to identify non-quantitative variables, such

as ordinal or nominal variables.

For example, one might like to compare institutions ofthe same type, such as public or private universities.

This is accomplished by introducing categorical

variables containing numbers for order or identifiers

for names.

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A priori Restrictions on Weights

In the model as presented, the weights are limited onlyby the requirements that they be non-negative.

However, there may be reason to require that weightsbe further limited.

For example, it may be felt that a given variablemustbe included in the assessment so its weight must have atleast a minimal value greater than zero. This mightrepresent an output that is essential in assessment.

As another example, a variable may be such a large

weight would over-emphasize itsa priori importance sothat there should be an upper limit on the weight. Thismight represent an output variable that is counter-productive.

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Relationships between Weights

Sometimes,a priori knowledge may imply that there is

a necessary relationship among variables. For example,

an output variable may absolutely require some level of

an input variable. Sucha priori knowledge may be expressed as a ratio

between the weights assigned to the related variables.

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A priori assessments of Efficient Units

Some DMUs may be regarded, based ona priori

knowledge, as eminently efficient or notoriously

inefficient. While one might argue about the validity of

such a priori judgments, frequently they must berecognized.

To do so, additional conditions may be imposed upon

the choice of weights. For example, the condition

Yj/nXj

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Substitutability of Variables

A still unresolved issue is the means for representing

substitutability of variables. For example, two input

variables may represent two different type of labor

which may be, to some extent, substitutable for eachother.

How is such substitutability to be incorporated?

Lets explore this issue a bit further since, by doing so,

we can illuminate some additional perspectives on thebasic DEA model.

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For simplicity in description, consider two input variables

and a single output variable that has the same value forall DMUs. The graphic representation of the envelopmentsurface can now best be presented not in terms of therelationship between output and input, as previouslyshown, but between the variables of input.

The two variables are Professional Staff and Non-Professional Staff. The assumption is that they arecompletely substitutable and that physicians differ only intheir styles of providing service, represented by the mix

of the two means for doing so. The efficient DMUs are located on the red envelopment

surface, which shows the minimums in use of variables.

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0

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7

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-1 0 1 2 3 4 5

Professional Staff

Non-ProfessionalStaff

Style 1

Style 4

Style 2 Style 3

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Discrimination among Efficient Units

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Strengths & Weaknesses of DEA

Strengths

DEA can handle multiple inputs and multiple outputs

DEA doesn't require relating inputs to outputs.

Comparisons are directly against peers

Inputs and outputs can have very different units

Weaknesses

Measurement error can cause significant problems

DEA does not measure"absolute" efficiency

Statistical tests are not applicable

Large problems can be computationally intensive

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Implementation of DEA

Structure

Spreadsheet implementation

Choice of Model

Spreadsheet Structure

Spreadsheet Calculations

Solver Elements in Spreadsheet

Visual Basic Program

Access to the Implementation X

The data included in the spreadsheet is for ARLlibraries in 1996.
http://f/External%20Disk%20Backup/Fundamental%20Backup/My%20Writings/Class%20CD/PowerPoint%20Files/Economics%20PPT/DEA-ARL-Complete.xlshttp://dea-arl-complete.xls/http://dea-arl-complete.xls/http://f/External%20Disk%20Backup/Fundamental%20Backup/My%20Writings/Class%20CD/PowerPoint%20Files/Economics%20PPT/DEA-ARL-Complete.xls

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Choice of Model

The spreadsheet includes means to identify the choice

of model by means of three parameters: Form: Dual represented by 0 and Primal by 1

Orientation: Addition by 0, Input by 1, Output by 2

Convexity: No by 0, Yes by 1

Given the specification, solution of the resulting modelis initiated by pressing Ctrl-q.

The solution is effected by a Visual Basic program thatdetermines the model from the parameters and thenlaunches the Excel Add-In called Solver.

The program then produces the output on Sheet 3 thatshows the results.

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Spreadsheet Structure

The DEA Spreadsheet for application to ARL libraries

consists of three main parts:

(1) The source data, stored in cells B16:R117

(2) The spreadsheet calculations, stored in cells D5:R15

(3) The Solver related calculations, stored in cells

B1:B15, A7:A117, T12:T117

The source data consists of the 10 input and 5 output

variables for each of the ARL institutions plus, in rowB16:R16, a set of normalizing factors, one for each of

the variables.

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The Spreadsheet calculations in D5:R14 can be

illustrated by D5:D14 and N5:N14:

C D5 Discretionary? 16 Weights 0.0000017

8

9 Comp =SUMPRODUCT(Mult,D17:D113)*D1610Slacks 15.207341022937811Mod Comp =D9+D10

12=INDEX(C17:C126,MATCH($B$12,$B$17:$B$126,0),1) =INDEX(Data,MATCH($B$12,$B$17:$B$126,0),COLUMN()-3)*D1613 =D12*$B$1314 =IF($B$2=1,D13,D12)

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The Spreadsheet calculations in D5:R14 can be

illustrated by D5:D14 and N5:N14:

C N5 Discretionary? 16 Weights 9.99999999999265E-077

8

9 Comp =SUMPRODUCT(Mult,N17:N113)*N1610Slacks 5.5626973172299511Mod Comp =N9-N10

12=INDEX(C17:C126,MATCH($B$12,$B$17:$B$126,0),1) =INDEX(Data,MATCH($B$12,$B$17:$B$126,0),COLUMN()-3)*N1613 =N12*$B$1314 =IF($B$2=2,N13,N12)

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Solver Elements in Spreadsheet

# B1 B2 B3 Target Vary Conditions

0 0 0 0 B7 Min $D$10:$R$10,$A$17:$A$113 $D$11:$R$11= $D$14:$R$14 $A$17:$A$113>= 01 0 0 1 B7 Min $D$10:$R$10,$A$17:$A$113 $D$11:$R$11= $D$14:$R$14 $A$17:$A$113>= 0 $B$127= 1

2 0 1 0 B8 Min $D$10:$R$10,$A$17:$A$113,$B$13 $D$11:$R$11= $D$14:$R$14 $A$17:$A$113>= 0

3 0 1 1 B8 Min $D$10:$R$10,$A$17:$A$113,$B$13 $D$11:$R$11= $D$14:$R$14 $A$17:$A$113>= 0 $B$127= 1

4 0 2 0 B9 Min $D$10:$R$10,$A$17:$A$113,$B$13 $D$11:$R$11= $D$14:$R$14 $A$17:$A$113>= 0

5 0 2 1 B9 Min $D$10:$R$10,$A$17:$A$113,$B$13 $D$11:$R$11= $D$14:$R$14 $A$17:$A$113>= 0 $B$127= 1

6 1 0 0 B6 Max $D$6:$R$6 $T$17:$T$113

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Visual Basic ProgramApplication.Range("B3").SelectConvex = Selection.Value

A = 6 * Form + 2 * Orient + ConvexSolverReset

'Set Target, MaxMinVal, ChangeIf A = 0 Or A = 1 Then 'Dual, Addition

SolverOk SetCell:="B7", MaxMinVal:=2, ValueOf:="0", _ByChange:= "$D$10:$R$10,$A$17:$A$113"

End IfIf A = 2 Or A = 3 Then 'Dual, Input

SolverOk SetCell:="B8", MaxMinVal:=2, ValueOf:="0", _ByChange:= "$D$10:$R$10,$A$17:$A$113,$B$13"

End IfIf A = 4 Or A = 5 Then 'Dual, Output

SolverOk SetCell:="B9", MaxMinVal:=2, ValueOf:="0", _ByChange:= "$D$10:$R$10,$A$17:$A$113,$B$13"

End If

If A = 6 Or A = 8 Or A = 10 Then 'Primal, Not Convex (Constant Returns to Scale)SolverOk SetCell:="B6", MaxMinVal:=1, ValueOf:="0", _

ByChange:= "$D$6:$R$6"End IfIf A = 7 Or A = 9 Or A = 11 Then 'Primal, Convex (Variable Returns to Scale

SolverOk SetCell:="B6", MaxMinVal:=1, ValueOf:="0", _ByChange:= "$D$6:$R$6,$S$6"

End If

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Vi l B i P

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For m = 1 To 97Application.StatusBar = "Calculating Efficiency for unit " & Str(m)

' Paste unit 0's number to model worksheetSheets("Sheet1").SelectApplication.Goto Reference:="unit"Selection.Value = m

' Run Solver (with the dialog box turned off)SolverSolve (True)

' Paste unit's number and name to All Results sheetSheets("Sheet1").Select

Application.Range("C12").SelectSelection.Copy

Sheets("Sheet3").SelectRange("A2").Offset(m, 0).SelectSelection.PasteSpecial Paste:=xlValues

Vi l B i P

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Sheets("Sheet1").Select

Application.Goto Reference:="Target1"Selection.CopySheets("Sheet3").SelectRange("A2").Offset(m, 1).SelectSelection.Value = mRange("A2").Offset(m, 2).SelectSelection.PasteSpecial Paste:=xlValues

Sheets("Sheet1").SelectApplication.Goto Reference:="Results"Selection.CopySheets("Sheet3").SelectRange("A2").Offset(m, 1).SelectSelection.Value = mRange("A2").Offset(m, 3).SelectSelection.PasteSpecial Paste:=xlValues

Next mApplication.Goto Reference:="Start"

End Sub

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The Example of Libraries

Selection of Data

Input Variables (10):

Collection Characteristics (Discretionary)

Staff Characteristics (Discretionary) University Characteristics (Non-discretionary)

Output Variables (5):

Scaling of Data

Constraints on Weights

Results

Effects of the several Variables

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Selection of Data

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The Variables

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Scaling of the Variables

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Constraints on Weights

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Results

Efficiency Distribution

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Efficiency Distribution

The following chart display the efficiency distribution

for the 97 U.S. ARL libraries.

The input and output components for each institution

have been multiplied by the size of the collection.

Note the cluster of inefficient institutions below the

3,000,000 volumes of holdings.

There appear to be three groups of institutions:

The efficient ones, lying on the red line

The seven that are more then 4 million and mildly inefficient

Those that are less than 4 million and range in efficiency

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1.00

3.00

5.00

7.00

9.00

11.00

13.00

1.00 3.00 5.00 7.00 9.00 11.00 13.00

Collection*Input

Collec

tion*Output

S f P j ti

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Sum of Projections

The following chart show the distribution of the sum of

the projections as a function of the Intensity.

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0,00

1,00

2,00

3,00

4,00

5,00

6,00

7,00

8,00

9,00

0,00 0,20 0,40 0,60 0,80 1,00 1,20

Intensity

Sumof

Projections

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Distribution of Weights

The following chart shows the magnitudes of the

weights on each of the Input and Output components

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-0,05

0,00

0,05

0,10

0,15

0,20

0,25

0 2 4 6 8 10 12 14 16

A l f O ti R h 66

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(1996)

Preface

P t I DEA d l th d d

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Part I: DEA models, methods and

interrelations

Chapter 1. Introduction: Extensions and new

developments in DEA

W W Cooper, R.G. Thompson and R.M. Thrall

Chapter 2. A generalized data envelopment analysis

model: A unification and extension of existing methods

for efficiency analysis of decision making units

G.Yu, Q. Wet and P Binckett

E t i i DEA

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Extensions in DEA

Covers (1) new measures of efficiency, (2) new models,

and (3) new implementations.

The TDT measure of relative efficiency takes thecriterion measure (weighted output/weighted input)relative to the maximum for that measure

The Pareto-Koopman measure applies the Paretocriterion (no variables can be improved without worseningothers)

The BCC model (variable returns to scale) is presented.

Congestion arises when excess inputs interfere withoutputs. It thus represents relationships among variables.

G li d DEA d l

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Generalized DEA model

Essentially, this paper does what I have been trying to do in

implementation of DEA.

It does so by identifying the primal and dual (P and D), thetwo returns to scale (fixed and variable), and three binaryparameters (d1, d2, d3) in the equations

d1 = d1eT + d1d2(-1)d3n+1 (for the dual)d1d2(-1)d30(for the primal)Values of (d1, d2, d3) include:(0,-,-) the CCR model

(1,0,-) the BCC model(1,1,0) the FG model

(1,1,1) the ST model

The relationships among the several models are discussed.

Part II: Desirable properties of

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models, measures and solutions (1)

Chapter 3. Translation invariance in data envelopment

analysis: A generalization

J.T Pastor

Chapter 4. The lack of invariance of optimal dual

solutions under translation

R.M. Thrall

Chapter 5. Duality, classification and slacks in DEA

R.M. Thrall

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Lack of invariance

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Lack of invariance

This paper supplements the prior one. It shows that in

neither the BCC model nor the additive model are the

optimal solutions for the dual (i.e., multipler)

formulation invariant under translation.

Duality classification and slack

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Duality, classification and slack

This paper considers the role of slacks especially in the

context of radial measures of efficiency. The effect ofalternative optima is to make slacks difficult to dealwith; the theory presented resolves the difficulties.

The CCT model presented eliminates the need for non-Archimedean models and permits dealing with zero

values for the variables.

The concept of an admissible virtual multiplier isintroduced and the maximizing virtual multiplier w* isthe basis for categorizing efficient DMUs into 3 groups:

Extreme Efficient: all variables are included in w* Efficient: the variables in w* are all positive

Weak efficient: w* has at least one zero variable

Similarly for non-efficient DMUs


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models, measures and solutions (2)

Chapter 6. On the construction of strong

complementarity slackness solutions for DEA linear

programming problems using a primal-dual interior-

point method M.D. Gonzdlez-Ltma, R.A. Tapta and R.M. Thralt

Chapter 7. DEA multiplier analytic center sensitivity

with an illustrative application to independent oil

companies R.C. Thompson, PS. Ditarmapala, f Diaz, M.D.

Gonzdlez-Lima and R.M. Thrall

Complementarity Slackness Solutions

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Complementarity Slackness Solutions

This paper proposes use of primary-dual interior-point

methods for solution of the DEA linear programmingproblem (an iterative process that generates interior pointthat converge to the solution).

The primary form minimizes Cx; the dual formmaximizes By.

The condition for solution is that Cx = By, called thecomplementarity slackness condition.

These methods attempt to solve the primary and duallinear programs simultaneously.

Solutions are classified as radially efficient or inefficientusing the CCT model.

Multiplier Sensitivity

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Multiplier Sensitivity

The stability of the set E of extreme efficient DMUs is

examined to determine the sensitivity to changes in the

data,

Part III: Frontier shifts and efficiency

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Part III: Frontier shifts and efficiency

evaluations

Chapter 8. Estimating production frontier shifts: An

application of DEA to technology assessment

R.D. Banker and R.C. Morey

Chapter 9. Moving frontier analysis: An application ofdata envelopment analysis for competitive analysis of a

high-technology manufacturing plant

K.K. Sinha

Chapter 10. Profitability and productivity changes: Anapplication to Swedish pharmacies

R. Aithin, R. Fare and S. Grosskopf 219

Production Frontier Shifts

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Production Frontier Shifts

This paper divides the set of DMUs into two categories(representing the use or non-use of a technology). For a

DMU without the technology, comparison is made only

with others without the technology; for those with the

technology, comparison is made with all DMUs.

The result is a basis for assessment of the impact of the

technology.

Moving Frontier Analysis

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Moving Frontier Analysis

This paper proposes a method for assessing when somedata may not be available. It uses aggregate data on

best practices. It depends upon time series data

P fit bilit & d ti it h

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Profitability & productivity changes

It is not evident how this relates to DEA.

Part IV: Statistical and stochastic

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Part IV: Statistical and stochastic

characterizations

Chapter 11. Simulation studies of efficiency, returns toscale and misspecification with nonlinear functions inDEA

RD. Banker; H. Chang and WW Cooper Chapter 12. New uses of DEA and statistical

regressions for efficiency evaluation and estimation -with an illustrative application to public secondaryschools in Texas

VL Arnold, LR. Bardhan, WW Cooper and S.C.Kumbhakar

Chapter 13. Satisficing DEA models under chanceconstraints

W W Cooper Z Huang and S.X. Li

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Simulation studies

Well, so be it.

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Satisficing DEA models

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Satisficing DEA models

Introduces stochastic variables (characterized byprobability distributions) and the concept of

stochastic efficiency.

It distinguishes between a rule (which has a

probability of 1) and a policy (which has a

probability between 0.5 and 1).

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Part V: Some new applications

Chapter 14. Evaluating the efficiency of vehicle

manufacturing with different products

G. Zeng

Chapter 15. DEA/AR analysis of the 1988-1989performance of the Nanjing Textiles Corporation

J.Zhu 311

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DEA/AR analysis

Another application in China.


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(1997)

Contents

Preface

Foreword

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Part VI: Extending Frontiers

Extending the frontiers of Data Envelopment Analysis

A.Y Lewin and LM. Seijord

Weights restrictions and value judgements in Data

Envelopment Analysis: Evolution, development andfuture directions

R.Allen, A. Athanassopoulos, R.O. Dyson and F.

Thanassoulis

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Extending the frontiers

See earlier in this presentation

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Weights restrictions & value judgments

See earlier in this presentation.

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Part VII: Applications

DEA and primary care physician report cards:Deriving preferred practice cones from managed careservice concepts and operating strategies

IA. Chilingerian and H.D. Sherman

An analysis of staffing efficiency in U.S.manufacturing: 1983 and 1989

PT Ward, J.E. Storbeck, S.L. Mangum and REByrnes

Applications of DEA to measure the efficiency ofsoftware production at two large Canadian banks

J.C. Paradi, D.N. Reese and D. Rosen

Primary care physician

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Primary care physician

This papers identifies styles of management based on

ratios of input variables aimed at input cost minimizing.

The example used is comparison of hospital days versus

office visits

Staffing efficiency

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g y

Again, styles of management are identified, this timebased on ratios of types of staffing (e.g., professional vs.

non-professional). Industries are divided into types (batch

vs. line processing industries) and best practices for

each type are identified by DEA.

software production

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software production

Input to software production is taken as cost; outputs assize (measure by function points), quality (measured

by defects or rework hours), and time to market.

The DEA is compared to performance ratio analyses,

such as Cost/Function, Defects/Function, Days/Function. Then, constraints on the weights are introduced. One set

of constraints consisted of bounds on ratios of weights. A

second set of constraints consisted of tradeoffs between

variables, again represented by bounds on ratios.

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Restricted best practice selection in DEA: An overview

with a case study evaluating the socio-economic

performance of nations

B.Golany and S. Thore A new measure of baseball batters using DEA

T.R. Anderson and G.P Sharp

Efficiency of families managing home health care

CE. Smith, S. VM. Kiembeck, K. Fernengel and L.S.

Mayer

Economic performance of nations

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p

To apply DEA to evaluation of economic performanceof nations, it is necessary to recognize some constraints:

International requirements (treaties, bilateral agreements)

Externalities (e.g., mandated quotas)

Issues of equity These constraints are then incorporated into DEA

Baseball batters

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Baseball batters

Traditional methods for evaluating batters include

fixed and variable weight statistics (homers, batting

average, slugging average, RBI, etc.). The point in this

article is that use of DEA allows one to determine the

effect of changes over time. Another effect of interest is noise. To correct for

noise, the DEA model derates the data for each

player by a factor based on the players standard

deviation for each variable

Efficiency of families

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Family home health care is assessed using a steppedprocedure in DEA.

The stepped procedure involves a series of steps in

which variables are successively introduced:

Inputs OutputsStep 1 Direct Costs Medical Expense Family Income

Step 2 Indirect Costs Training Patient/Caregiver

Step 1 Step 1

Step 3 Caring Costs Hours /day Caregiver burden

Moths/caregiving Caregiver esteemMedication

Step 2

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A DEA-based analysis of productivity change and

intertemporal managerial performance

E.Grifell-Tatje and C.A.K. LoveII

Use of Data Envelopment Analysis in assessingInformation Technology impact on firm performance

C.H. Wang, R.D. Gopal and S. Zionts

Productivity & managerial performance

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Examines the productivity of an organization overtime.

Information Technology impact

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gy p

Examines the impact of information technology onperformance of firms. It divides operations into two

stages: (1) Accumulation of resources and (2) Use of

resources. (These are illustrated in banking by (1) the

collection of funds from depositors and (2) use of those

funds for generating income).

It examines separately the effect of information

technology (represented by ATM machines) on the two

stages.

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Part VIII: Theoretical Extensions

Comparative advantage and disadvantage in DEA

A.I. Alt and CS. LeTine

Model misspecification in Data Envelopment Analysis

P Smith Dominant Competitive Factors for evaluating program

efficiency in grouped data

J.J.Rousseau and J.H. Semple

DEA-based yardstick competition. The optimality ofbest practice regulation

P Bogetoft

Comparative advantage & disadvantage

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p g g

This paper introduces a cost function into DEAanalysis as the means for calculating a comparativeadvantage or disadvantage as the difference betweenthe costs of input and the income from output.

It interprets the weights in each DMUs optimum as

prices for the respective inputs and outputs. The resultis virtual cost, revenue, and profit. The profit (orloss) is then compared with the maximum profitobtained by a best practice unit and that of theevaluated unit.

For an efficient unit, the comparison is between thevirtual profit of the valuated unit and the maximumprofit across all other units.

Comparative disadvantage

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p g

The DEA model for determining comparativedisadvantage is:

Max RC + w subject to Min h - w - uY1 + R = -1 - w Y1 + Y + T0r = 0 vX1C = 1 hX1X + T1r1 = 0 uYvX = Iw

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The DEA model for determining comparativeadvantage is applied to the set removing the target

unit:

MaxR + C + w subject to Min h - w - uY1 + R =1 - w Y1 + Y1 + T0r = 0 vX1 + C = 1 hX1X1 + T1r1 = 0 uY1vX1 = Iw

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This paper examines the effects of various types of mis-

specifications of the DEA model. They include:

Omission of a necessary input

Inclusion of an extraneous variable Erroneous assumption about returns to scale

Dominant Competitive Factors

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p

This paper treats DEA as a tool in game theory. Oneplayer has control over the weights applied to the

variables, the other over the weights applied to the

DMUs. Each tries to optimize against the other.

The solution is of the pair of prime-dual problems:

Player 1 Maximizes vy0ux0subject to vyjuxj =0

Player 2 Minimizes a subject to Y + ay0 >= y0, Xax0 = 0, a unrestricted

Best practice regulation

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The use of DEA in regulatory practice is discussed. Theunderlying game is represented by a series of steps:

Costs and demands for service are observed oridentified

Schemes are proposed by the regulator

The schemes are rejected or accepted by the DMUs

Costs are selected by the DMUs

Data on performance are observed

Compensations are paid

The aim of the regulator is to minimize the expectedcosts of making the DMUs accept, fulfil, and minimizecosts.

The use of DEA is to determine the best practce norms.

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Part VIII: Theoretical Extensions

A Data Envelopment Analysis approach to

Discriminant Analysis

D.L. Retzlaff-Roberts

Derivation of the Maximum Efficiency Ratio modefrom the maximum decisional efficiency principle

M.D. Trouft

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Discriminant Analysis

Discriminant analysis is a means for determining group

classification for a set of similar units or observations.

It determines a set of factor weights which best

separate the groups, given units for which membershipis already known.

This paper proposes the use of DEA as a means for

doing DA

Maximum Efficiency Ratio

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Maximum efficiency ratio (MER) is intended to

prioritize the DEA efficient DMUs by defining common

weights. This paper supposes the existence of a ratio

form criterion common to all the DMUs but not

necessarily frontier oriented.

Maxu,v (Minj (Suryrj/Svixij), subject to Suryrj/Svixij = 0

Part IX: Computational

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Implementation

A Parallel and hierarchical decomposition approaches

for solving large-scale Data Envelopment Analysis

models

R.S. Barr and M.L. Durchholz

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Part X: Abraham Charnes

Abraham Charnes remembered

Abraham Charnes, 1917-1992

A bibliography for Data Envelopment Analysis (1978-

1996) LM. Setford

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The End

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