Data envelopment analysis

Data Envelopment Analysis
Robert M. Hayes
2005

Overview
Introduction
DEA Models
Extensions to include a priori Valuations
Strengths and Weaknesses of DEA
Implementation of DEA
The Example of Libraries
Annals of Operations Research 66

Introduction
Utility Functions
Cost/Effectiveness
Interpretation for Libraries

Utility Functions
A fundamental requirement in applying operations research
models is the identification of a "utility function" which
combines all variables relevant to a decision problem into a
single variable which is to be optimized. Underlying the
concept of a utility function is the view that it should represent
the decision-maker's perceptions of the relative importance of
the variables involved rather than being regarded as uniform
across all decision-makers or externally imposed.
The problem, of course, is that the resulting utility functions
may bear no relationship to each other and it is therefore
difficult to make comparisons from one decision context to
another. Indeed, not only may it not be possible to compare
two different decision-makers but it may not be possible to
compare the utility functions of a single decision-maker from
one context to another.

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 appear
to 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.

Cost/Effectiveness
It also must be recognized that effectiveness will usually
entail consideration of a number of products and services
and costs a number of sources of costs. Cost/effectiveness
measurement requires combining the sources of cost into
a 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 the
several measures of effectiveness. But it may also be the
case with the several measure of costs, since some costs
may be regarded as more important than others even
though they may all be measured in dollars. When some
costs cannot be measured in dollars, the problem is
compounded.

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.

This issue can be illustrated by evaluating library
performance. Effectiveness here is the extent to which
library services meet the expectations or goals set by
the organization served. It is likely to be measured by
several services which are the outputs of library
operations—making a collection available for use,
circulation or other uses of materials, answering of
information questions, instructing and consulting.
Inputs are represented by acquisitions, staff, and space,
to which evident costs can be assigned, but they are also
represented by measures of the populations served.

Efficiency measures the library’s ability to transform
its inputs (resources and demands) into production of
outputs (services). The objective in doing so is to
optimize the balance between the level of outputs and
the level of inputs. The success of the library, like that
of other organizations, depends on its ability to behave
both effectively and efficiently.
The issue at hand, then is how to combine the several
measures of input and output into a single measure of
efficiency. The method we will examine is that called
"data envelopment analysis".

Data Envelopment Analysis (DEA) measures the relative
efficiencies of organizations with multiple inputs and
multiple outputs. The organizations are called the
decision-making units, or DMUs.
DEA assigns weights to the inputs and outputs of a DMU
that give it the best possible efficiency. It thus arrives at a
weighting of the relative importance of the input and
output variables that reflects the emphasis that appears to
have been placed on them for that particular DMU.
At the same time, though, DEA then gives all the other
DMUs the same weights and compares the resulting
efficiencies with that for the DMU of focus.

If the focus DMU looks at least as good as any other
DMU, 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.

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).
0
1
2
3
4
5
6
7
8
9
0 1 2 3 4 5 6 7 8 9 10 11
Input
Output
L1
L2
L3
L4
L5
L6
L7

DEA identifies the units in the comparison set which lie
at the top and to the left, as represented by L1, L2, L3,
and L4. These units are called the efficient units, and
the line connecting them is called the "envelopment
surface" because it envelops all the cases.
DMUs L5 through L7 are not on the envelopment
surface and thus are evaluated as inefficient by the
DEA analysis. There are two ways to explain their
weakness. One is to say that, for example, L5 could
perhaps produce as much output as it does, but with
less input (comparing with L1 and L2); the other is to
say it could produce more output with the same input
(comparing with L2 and L3).

Thus, there are two possible definitions of efficiency
depending on the purpose of the evaluation. One might
be interested in possible reduction of inputs (in DEA
this is called the input orientation) or augmentation of
outputs (the output orientation) in achieving technical
efficiency. Depending on the purpose of the evaluation,
the analysis provides different sets of peer groups to
which to compare.
However, there are times when reduction of inputs or
augmentation of outputs is not sufficient. In our
example, even when L6 reduces its input from 4 units
to 2, there is still a gap between it and its peer L1 in the
amount of one unit of output. In DEA, this is called the
"slack" which means excess input or missing output
that exists even after the proportional change in the
input or the outputs.

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.

DEA Models
The Basic EDA Concept
Variations of DEA Formulation
Formulation: Primal or Dual
Orientation: Input or Output
Returns to Scale: Fixed or Variable

The Basic EDA Concept
Assume that each DMU has values for a set of inputs
and a set of outputs.
Choose non-negative weights to be applied to the inputs
and outputs for a focus DMU so as to maximize the
ratio of weighted outputs divided by weighted inputs
But do so subject to the condition that, if the same
weights are applied to each of the DMUs (including the
focus DMU), the corresponding ratios are not greater
than 1
Do that for each DMU.
The resulting value of the ratio for each DMU is its
EDA efficiency. It is 1 if the DMU is efficient and less
than 1 if it is not.

Formulation
Let (Yk,Xk) = (Yki,Xkj), k = 1 to n, i = 1 to s, j = 1 to m
Maximize µYk/νXkfor each value of k from 1 to n,
subject in each case to µYj/nXj<= 1, j= 1 to n, where
µYk means Σi µi*Yki, i = 1 to s,
νXk means Σi νi*Xki, i = 1 to m
µYj means Σi µi*Yji, i = 1 to s and j = 1 to n
νXj means Σi νi*Xji, i = 1 to m and j = 1 to n.
µi, νi >= 0
The solution is the set of maximum values for µYk/νXk
and the associated values for µ and ν

Basic Linear Programming Model
For solution, this optimization problem is transformed
into a linear programming problem, schematically
displayed as follows:
In a moment, we will interpret this display as it is
translated into alternative formulations of the
optimization target and conditional inequalities.
µ ν Min
λ Yj -Xj <= 0
a -I <= -I
b -I <= -I
>= >=
Max Yk - Xk

Variations of DEA Formulation
But first, it is necessary to identify several sources of
variation in the basic DEA formulation, leading to a
variety of different models for implementation:
We will now examine and illustrate each of those
sources 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

(1) Formulation: Primal or Dual
The first source of variation is interpretation of the
display 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 the
columns as representing the model.
Let’s examine each of those in turn.

Primal Formulation
The rows of this display are interpreted as follows:
(M) Maximize W = µYk – νXk subject to
(1) µYj – νXj <= 0, j = 1 to n
(2) -µ <= -1, or µ >= 1
(3) -ν <= -1, or ν >= 1
µ ν
(1) Yj -Xj <= 0
(2) -I <= -I
(3) -I <= -I
(M) Yk - Xk

The Dual Formulation
The Columns of this display are interpreted as follows:
(m) Minimize W = -a - b subject to
(1) λYj – a >= Yk
(2) –λXj - b >= -Xk
(1) (2) (m)
λ Yj -Xj 0
a -I -I
b -I -I
>= >=
Yk - Xk

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 and
s, the number of output variables)

Illustration
To illustrate, consider the example previously presented.
The target to be minimized in the Dual form is W = – a – b.
The values to be varied are (λ, a, b), or (µ, ν).
The following table shows the solution for both forms:
X Y W a b λ µ ν
L1 2 2 - 1.33 1.33 λ2 = 0.67 1 1.67
L2 3 5 0.00 0.00 λ2 = 1.00 1 1.67
L3 6 7 - 3.00 3.00 λ2 = 2.00 1 1.67
L4 9 8 - 7.00 7.00 λ2 = 3.00 1 1.67
L5 5 3 - 5.33 5.33 λ2 = 1.67 1 1.67
L6 4 1 - 5.67 5.67 λ2 = 1.33 1 1.67
L7 10 7 - 9.67 9.67 λ2 = 3.33 1 1.67

Illustration
Graphically, the results are as follows:
The maximum value for W, over all cases, is at L2, where
W = 0 and the ratio of Y/X is a maximum. The slack for
each other case is the vertical distance to the line which
goes from the origin (0,0) through L2 (3,5).
0
5
10
15
20
25
0 2 4 6 8 10 12

(2) Orientation: Input or Output
The second source of variation, orientation, provides
the 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?

Orientation to Input
The linear programming display for the input
orientation is as follows:
It adds one additional condition, νXk <= 1, to the
display.
µ ν Min
λ Yj -Xj <= 0
a -I <= 0
b -I <= 0
c - 1 Xk <= I
>= >=
Max Yk - Xk

The resulting Dual formulation is as follows:
(m) Minimize W = c-1 subject to
(1) λYj – a >= Yk
(2) –λXj – b + (c – 1)Xk >= -Xk or λXk + b <= cXk
(1) (2) (m)
λ Yj -Xj 0
a -I 0
b -I 0
c - 1 Xk I
>= >=
Max Yk - Xk

Continuing with the same example, the following table
shows the solutions in both formulations. The target is
W = c – 1. Values to be varied are now (λ, a, b, c) or
(µ and ν).
Note that L2 still dominates the solution, but the graph
is now quite different,
X Y W=c-1 a b λ µ ν
L1 2 2 - 0.40 λ2 = 0.40 0.30 0.50
L2 3 5 0.00 λ2 = 1.00 0.20 0.33
L3 6 7 - 0.30 λ2 = 1.40 0.10 0.17
L4 9 8 - 0.46 λ2 = 1.60 0.07 0.11
L5 5 3 - 0.64 λ2 = 0.60 0.12 0.20
L6 4 1 - 0.85 λ2 = 0.20 0.15 0.25
L7 10 7 - 0.58 λ2 = 1.40 0.06 0.10

0
2
4
6
8
10
12
0 2 4 6 8 10 12

Orientation to Output
The linear programming display for the output
orientation is as follows:
It adds one additional condition, µYk <= 1, to the
display.
µ ν Min
λ Yj -Xj <= 0
a -I <= 0
b -I <= 0
c - 1 Yk <= I
>= >=
Max Yk - Xk

(m) Minimize W = 1 – c subject to
(1) λYj – a >= cYk
(2) –λXj – b >= – Xk or λXk + b <= Xk
(1) (2) (m)
λ Yj -Xj 0
a -I 0
b -I 0
1 - c Yk I
>= >=
Max Yk - Xk

Continuing with the same example, the following table
shows the solutions in both formulations. The target is
W = 1 – c. Values to be varied are still (λ, a, b, c) or
(µ and ν).
Note that L2 still dominates the solution, but the graph
is now quite different,
X Y W=1-c a b λ µ ν
L1 2 2 - 0.67 λ2 = 0.67 0.50 0.83
L2 3 5 0.00 λ2 = 1.00 0.20 0.33
L3 6 7 - 0.43 λ2 = 2.00 0.14 0.24
L4 9 8 - 0.87 λ2 = 3.00 0.13 0.21
L5 5 3 - 1.78 λ2 = 1.67 0.33 0.56
L6 4 1 - 5.67 λ2 = 1.33 1.00 1.67
L7 10 7 - 1.38 λ2 = 3.33 0.14 0.24

Note that the graphical display is identical to that for
the general form, though the interpretation is
somewhat different (replacing efficiencies by slacks).
0
5
10
15
20
25
0 2 4 6 8 10 12

(3) Returns to Scale: Fixed or Variable
The third basis for variation among DEA models is
“returns to scale”.
The examples presented to this point have each involved
“constant returns to scale”. That is, the ratio µY/νX can
be replaced by the inequality µY – νX <= 0.
These variations of the DEA model are called CCR
models and reflect the requirement of constant returns to
scale,
But if there are “variable returns to scale”, the ratio
µY/νX must now be replaced by µY – νX + u <= 0 where
u can now vary to reflect the variable returns to scale.
The results from that change are dramatic and make the
DEA model much more interesting. The resulting models
are called BCC models.

Variable Returns to Scale, Basic Model
The linear programming display for the basic DEA
model is as follows:
It adds the variable u to the display.
µ ν u Min
λ Yj -Xj I <= 0
a -I <= - I
b -I <= - I
>= >=
Max Yk - Xk I

Variable Returns: Orientation to Input
The linear programming display for the variables
returns to scale, input orientation is as follows:
It adds one additional condition, νXk <= 1, to the
display.
µ ν u Min
λ Yj -Xj I <= 0
a -I <= 0
b -I <= 0
c - 1 Xk <= I
>= >= >=
Max Yk - Xk I

(m) Minimize W = c-1 subject to
(1) λYj – a >= Yk
(2) –λXj – b + (c – 1)Xk >= -Xk or λXk + b <= cXk
(3) λ >= 1
The new, third condition makes things interesting.
(1) (2) u (m)
λ Yj -Xj I 0
a -I 0
b -I 0
c - 1 Xk I
>= >=
Max Yk - Xk I

Continuing with the same example, the
following table shows the solutions in both
formulations. The target is W = c – 1. Values to
be varied are now (λ, a, b, c) or (µ, ν, u).
X Y W=c-1 a b λ µ ν u
L1 2 2 0.00 0.00 λ1 =1.00 0.00 0.00 0.00
L2 3 5 0.00 0.00 λ2 = 1.00 0.00 0.00 0.00
L3 6 7 0.00 0.00 λ3 = 1.00 0.00 0.00 0.00
L4 9 8 0.00 0.00 λ4 = 1.00 0.00 0.00 0.00
L5 5 3 - 4.00 2.00 λ2 = 1.00 0.00 0.00 0.00
L6 4 1 - 5.00 4.00 λ2 = 1.00 0.00 0.00 0.00
L7 10 7 - 4.00 0.00 λ3 = 1.00 0.00 0.00 0.00

Extensions to include a priori Valuations
To this point, DEA has been essentially a mathematical process in
which the data for input and output are taken as given, without
further 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 the
variations.
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

Discretionary & Non-discretionary
In identifying input and output variables, one wants to
include all that are relevant to the operation. For
example, the level of output is determined not only by
what the unit itself does but by the size of the market to
which the output is delivered.
The result, though, is that some relevant variables, such
as the size of the market, are not under the control of
management. Such variables, called non-discretionary,
are in contrast to those that are under management
control, called discretionary.
In assessing efficiency, all variables are considered, but
in determining the criterion function to be maximized or
minimized, only the discretionary variables are included.

Categorical Variables
In the DEA model as so far presented, the variables are
treated 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 of
the same type, such as public or private universities.
This is accomplished by introducing categorical
variables containing numbers for order or identifiers
for names.

A priori Restrictions on Weights
In the model as presented, the weights are limited only
by the requirements that they be non-negative.
However, there may be reason to require that weights
be further limited.
For example, it may be felt that a given variable must
be included in the assessment so its weight must have at
least a minimal value greater than zero. This might
represent an output that is essential in assessment.
As another example, a variable may be such a large
weight would over-emphasize its a priori importance so
that there should be an upper limit on the weight. This
might represent an output variable that is counter-
productive.

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.
Such a priori knowledge may be expressed as a ratio
between the weights assigned to the related variables.

A priori assessments of Efficient Units
Some DMUs may be regarded, based on a priori
knowledge, as eminently efficient or notoriously
inefficient. While one might argue about the validity of
such a priori judgments, frequently they must be
recognized.
To do so, additional conditions may be imposed upon
the choice of weights. For example, the condition
µYj/nXj<= 1 may be replaced by equality for a given
DMU which is regarded as eminently efficient.

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 each
other.
How is such substitutability to be incorporated?
Let’s explore this issue a bit further since, by doing so,
we can illuminate some additional perspectives on the
basic DEA model.

For simplicity in description, consider two input variables
and a single output variable that has the same value for
all DMUs. The graphic representation of the envelopment
surface can now best be presented not in terms of the
relationship between output and input, as previously
shown, but between the variables of input.
The two variables are “Professional Staff” and “Non-
Professional Staff”. The assumption is that they are
completely substitutable and that physicians differ only in
their “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.

0
1
2
3
4
5
6
7
8
9
10
-1 0 1 2 3 4 5
Professional Staff
Non-ProfessionalStaff
Style 1
Style 4
Style 2 Style 3

Discrimination among Efficient Units

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

Implementation of DEA
Structure
Spreadsheet implementation
Choice of Model
Spreadsheet Structure
Spreadsheet Calculations
Solver Elements in Spreadsheet
Visual Basic Program
Access to the Implementation
The data included in the spreadsheet is for ARL
libraries in 1996.

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 model
is initiated by pressing Ctrl-q.
The solution is effected by a Visual Basic program that
determines the model from the parameters and then
launches the Excel Add-In called Solver.
The program then produces the output on Sheet 3 that
shows the results.

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 row
B16:R16, a set of normalizing factors, one for each of
the variables.

The Spreadsheet calculations in D5:R14 can be
illustrated by D5:D14 and N5:N14:
C D
5 Discretionary? 1
6 Weights 0.000001
7
8
9 Comp =SUMPRODUCT(Mult,D17:D113)*D16
10Slacks 15.2073410229378
11Mod Comp =D9+D10
12=INDEX(C17:C126,MATCH($B$12,$B$17:$B$126,0),1)=INDEX(Data,MATCH($B$12,$B$17:$B$126,0),COLUMN()-3)*D16
13 =D12*$B$13
14 =IF($B$2=1,D13,D12)

The Spreadsheet calculations in D5:R14 can be
illustrated by D5:D14 and N5:N14:
C N
5 Discretionary? 1
6 Weights 9.99999999999265E-07
7
8
9 Comp =SUMPRODUCT(Mult,N17:N113)*N16
10Slacks 5.56269731722995
11Mod Comp =N9-N10
12=INDEX(C17:C126,MATCH($B$12,$B$17:$B$126,0),1) =INDEX(Data,MATCH($B$12,$B$17:$B$126,0),COLUMN()-3)*N16
13 =N12*$B$13
14 =IF($B$2=2,N13,N12)

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>= 0
1 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<= 0 $T$12= 1
7 1 0 1 B6 Max $D$6:$R$6,$S$6 $T$17:$T$113<= 0 $T$12= 1
8 1 1 0 B6 Max $D$6:$R$6 $T$17:$T$113<= 0 $T$12= 1
9 1 1 1 B6 Max $D$6:$R$6,$S$6 $T$17:$T$113<= 0 $T$12= 1
10 1 2 0 B6 Max $D$6:$R$6 $T$17:$T$113<= 0 $T$12= 1
11 1 2 1 B6 Max $D$6:$R$6,$S$6 $T$17:$T$113<= 0 $T$12= 1
B7=-Slacks = SUMPRODUCT(D5:R5,D10:R10)
B8=Inrensity-1
B9=1-Intensity
B6=SUMPRODUCT($N$12:$R$12,$N$6:$R$6)-SUMPRODUCT($D$12:$M$12,$D$6:$M$6)+IF($B$3=1,$S$6,0)
$T$17:$T$113 are illustrated by $T$17:
$T$17=SUMPRODUCT(N17:R17,$N$16:$R$16,$N$6:$R$6)-SUMPRODUCT(D17:M17,$D$16:$M$16,$D$6:$M$6)+IF($B$3=1,$S$6,0
$T$12=IF($B$2=0,1,IF($B$2=1,SUMPRODUCT($D$12:$M$12,$D$6:$M$6),SUMPRODUCT($N$12:$R$12,$N$6:$R$6)))
$B$127=SUM($A$17:$A$117)

Application.Range("B3").Select
Convex = Selection.Value
A = 6 * Form + 2 * Orient + Convex
SolverReset
'Set Target, MaxMinVal, Change
If A = 0 Or A = 1 Then 'Dual, Addition
SolverOk SetCell:="B7", MaxMinVal:=2, ValueOf:="0", _
ByChange:= "$D$10:$R$10,$A$17:$A$113"
End If
If A = 2 Or A = 3 Then 'Dual, Input
ByChange:= "$D$10:$R$10,$A$17:$A$113,$B$13"
End If
If A = 4 Or A = 5 Then 'Dual, Output
ByChange:= "$D$10:$R$10,$A$17:$A$113,$B$13"
End If
If A = 6 Or A = 8 Or A = 10 Then 'Primal, Not Convex (Constant Returns to Scale)
ByChange:= "$D$6:$R$6"
End If
If A = 7 Or A = 9 Or A = 11 Then 'Primal, Convex (Variable Returns to Scale
ByChange:= "$D$6:$R$6,$S$6"
End If

'Set Conditions
If A = 0 Or A = 1 Or A = 2 Or A = 3 Or A = 4 Or A = 5 Then 'Dual
SolverAdd CellRef:="$D$11:$R$11", Relation:=2, FormulaText:="$D$14:$R$14"
SolverAdd CellRef:="$A$17:$A$113", Relation:=3, FormulaText:="0"
SolverAdd CellRef:="$D$10:$R$10", Relation:=3, FormulaText:="0"
End If
If A = 1 Or A = 3 Or A = 5 Then 'Dual, Convex (Variable Returns to Scale)
SolverAdd CellRef:="$A$127", Relation:=2, FormulaText:="1"
End If
If A = 6 Or A = 7 Or A = 8 Or A = 9 Or A = 10 Or A = 11 Then
SolverAdd CellRef:="$T$12", Relation:=2, FormulaText:="1"
SolverAdd CellRef:="$T$17:$T$113", Relation:=1, FormulaText:="0"
End If
If A = 8 Or A = 9 Or A = 10 Or A = 11 Then 'Primal, Input or Output
SolverAdd CellRef:="$D$6:$R$6", Relation:=3, FormulaText:="$B$15"
End If
If A = 6 Or A = 7 Then 'Primal, Addition
SolverAdd CellRef:="$D$6:$R$6", Relation:=3, FormulaText:="1"
End If
SolverOptions MaxTime:=1000, Iterations:=1000, Precision:=0.000001, _
AssumeLinear:=True, StepThru:=False, Estimates:=1, Derivatives:=1, _
SearchOption:=1, IntTolerance:=5, Scaling:=False, Convergence:=0.0001, _
AssumeNonNeg:=False

For m = 1 To 97
Application.StatusBar = "Calculating Efficiency for unit " & Str(m)
' Paste unit 0's number to model worksheet
Sheets("Sheet1").Select
Application.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 sheet
Application.Range("C12").Select
Selection.Copy
Range("A2").Offset(m, 0).Select
Selection.PasteSpecial Paste:=xlValues

Application.Goto Reference:="Target1"
Selection.Copy
Selection.Value = m
Application.Goto Reference:="Results"
Selection.Copy
Selection.Value = m
Next m
Application.Goto Reference:="Start"
End Sub

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

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

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
Collection*Output

Sum of Projections
The following chart show the distribution of the sum of
the projections as a function of the Intensity.

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
SumofProjections

Distribution of Weights
The following chart shows the magnitudes of the
weights on each of the Input and Output components

-0,05
0,00
0,05
0,10
0,15
0,20
0,25
0 2 4 6 8 10 12 14 16

(1996)
Preface

Part I: DEA models, methods and
interrelations
Chapter 1.Introduction: Extensions and new developments in
W W Cooper, R.G. Thompson and R.M. Thrall
Chapter 2.A generalized data envelopment analysis model: A u
G.Yu, Q. Wet and P Binckett

Extensions in DEA
Covers (1) new measures of efficiency, (2) new models,
and (3) new implementations.
The TDT measure of “relative” efficiency takes the
criterion measure (weighted output/weighted input)
relative to the maximum for that measure
The Pareto-Koopman measure applies the Pareto
criterion (no variables can be improved without
worsening others)
The BCC model (variable returns to scale) is presented.
Congestion arises when excess inputs interfere with
outputs. It thus represents relationships among variables.

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), the
two returns to scale (fixed and variable), and three binary
parameters (1, 2, 3) in the equations
1 = 1eT
+ 12(-1)3
n+1 (for the dual)
12(-1)3
0(for the primal)
Values of (1, 2, 3) 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
models, measures and solutions (1)
Chapter 3. Translation invariance in data envelopment analys
J.T Pastor
Chapter 4. The lack of invariance of optimal dual solutions un
R.M. Thrall
Chapter 5. Duality, classification and slacks in DEA
R.M. Thrall

Translation invariance
This paper proves that several of the DEA model are
translation invariant (i.e., optimal solutions are not
changed if the original variable values are “translated”,
that is all values for a variable are replaced by some
constant minus the values).
Specifically, the primal additive model is translation
invariant.
The BCC input oriented primal model is output
translation invariant.
The CCR models are not translation invariant.

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
This paper considers the role of slacks especially in the
context of radial measures of efficiency. The effect of
alternative optima is to make slacks difficult to deal
with; 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” is
introduced and the maximizing virtual multiplier w* is
the 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

Part II: Desirable properties of
models, measures and solutions (2)
Chapter 6. On the construction of strong
complementarity
slackness solutions for DEA linear programming problems us
M.D. Gonzdlez-Ltma, R.A. Tapta and R.M. Thralt
Chapter 7. DEA multiplier analytic center sensitivity with an i
R.C. Thompson, PS. Ditarmapala, f Diaz, M.D.
Gonzdlez-Lima and R.M. Thrall

Complementarity Slackness Solutions
This paper proposes use of “primary-dual interior-point
methods” for solution of the DEA linear programming
problem (an iterative process that generates interior point
that converge to the solution).
The primary form minimizes C’x; the dual form
maximizes B’y.
The condition for solution is that C’x = B’y, called the
complementarity slackness condition.
These methods attempt to solve the primary and dual
linear programs simultaneously.
Solutions are classified as radially efficient or inefficient
using the CCT model.

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
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 of
data envelopment analysis for competitive analysis of a
high-technology manufacturing plant
K.K. Sinha
Chapter 10. Profitability and productivity changes: An
application to Swedish pharmacies
R. Aithin, R. Fare and S. Grosskopf 219

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
This paper proposes a method for assessing when some
data may not be available. It uses aggregate data on
“best practices”. It depends upon time series data

Profitability & productivity changes
It is not evident how this relates to DEA.

Part IV: Statistical and stochastic
characterizations
Chapter 11. Simulation studies of efficiency, returns to
scale and misspecification with nonlinear functions in
DEA
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 secondary
schools in Texas
VL Arnold, LR. Bardhan, WW Cooper and S.C.
Kumbhakar
Chapter 13. Satisficing DEA models under chance
constraints
W W Cooper Z Huang and S.X. Li

Simulation studies
Well, so be it.

DEA and statistical regressions
Compares the two methods. It uses a Cobb-Douglas
production model (in log form) and estimates the
parameters by a regression on the set of DMUs.
(Actually, it does a set of regressions, one for each
output variable against the uniform set of input
variables.)
It then applies DEA to the same set of input variables
(separately for each output variable in turn).
It then considers the joint outputs, taken together.

Satisficing DEA models
Introduces stochastic variables (characterized by
probability 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).

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 1988-1989
performance of the Nanjing Textiles Corporation
J. Zhu 311

China vehicle manufacturing
Evaluates the efficiency of vehicle manufacturing in
China.
It deals with the problem of zero values for some
variables.

DEA/AR analysis
Another application in China.

Annals of Operations Research 73 (1997)
Contents
Preface
Foreword

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 and
future directions
R.Allen, A. Athanassopoulos, R.O. Dyson and F.
Thanassoulis

Extending the frontiers
See earlier in this presentation

Weights restrictions & value judgments
See earlier in this presentation.

Part VII: Applications
DEA and primary care physician report cards:
Deriving preferred practice cones from managed care
service 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 RE
Byrnes
Applications of DEA to measure the efficiency of
software production at two large Canadian banks
J.C. Paradi, D.N. Reese and D. Rosen

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
Again, styles of management are identified, this time
based 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
Input to software production is taken as cost; outputs as
size (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.

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
To apply DEA to evaluation of economic performance
of 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
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 player’s standard
deviation for each variable

Efficiency of families
Family home health care is assessed using a “stepped
procedure” in DEA.
The stepped procedure involves a series of steps in
which variables are successively introduced:
Inputs Outputs
Step1 Direct Costs Medical Expense Family Income
Step2 Indirect Costs Training Patient/Caregiver
Step1 Step1
Step3 Caring Costs Hours/day Caregiver burden
Moths/caregiving Caregiver esteem
Medication
Step2

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 assessing
Information Technology impact on firm performance
C.H. Wang, R.D. Gopal and S. Zionts

Productivity & managerial performance
Examines the productivity of an organization over
time.

Information Technology impact
Examines the impact of information technology on
performance 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.

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 of
best practice regulation
P Bogetoft

Comparative advantage & disadvantage
This paper introduces a cost function into DEA
analysis as the means for calculating a comparative
advantage or disadvantage as the difference between
the 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 result
is “virtual” cost, revenue, and profit. The profit (or
loss) is then compared with the maximum profit
obtained by a best practice unit and that of the
evaluated unit.
For an efficient unit, the comparison is between the
virtual profit of the valuated unit and the maximum
profit across all other units.

Comparative disadvantage
The DEA model for determining comparative
disadvantage is:
Max R – C + w subject to Min  - 
- uY1 + R = -1 -  Y1 + Y + T0
 = 0
vX1 – C = 1 X1 – X+ T1
1
= 0
uY – vX = Iw <= 0 I= 1
uT0
<= 0, vT1
<= 0  <= 1, >= 1
R, C >= 0 >= 0

Comparative advantage
The DEA model for determining comparative
advantage is applied to the set removing the target
unit:
Max – R + C + w subject to Min  - 
- uY1 + R = – 1 -  Y1 + Y1
 + T0
 = 0
vX1 + C = 1 X1 – X1
+ T1
1
= 0
uY1
– vX1
= Iw <= 0 I= 1
uT0
<= 0, vT1
<= 0  >= 1, <= 1
R, C >= 0 >= 0

Model mis-specification
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
This paper treats DEA as a tool in game theory. One
player 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 v’y0 – u’x0 subject to v’yj – u’xj <= 0 and
v’y0 + u’x0 = 1, u, v >=0
Player 2 Minimizes a subject to Y+ ay0 >= y0, X– ax0 <= x0,
 >= 0, a unrestricted

Best practice regulation
The use of DEA in regulatory practice is discussed. The
underlying game is represented by a series of steps:
Costs and demands for service are observed or
identified
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 expected
costs of making the DMUs accept, fulfil, and minimize
costs.
The use of DEA is to determine the best practce norms.

Part VIII: Theoretical Extensions
A Data Envelopment Analysis approach to
Discriminant Analysis
D.L. Retzlaff-Roberts
Derivation of the Maximum Efficiency Ratio mode
from the maximum decisional efficiency principle
M.D. Trouft

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 membership
is already known.
This paper proposes the use of DEA as a means for
doing DA

Maximum Efficiency Ratio
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 (uryrj/ vixij), subject to uryrj/ vixij <= 1 for
all j, ur = 1, u, v >= 0

Part IX: Computational
Implementation
A Parallel and hierarchical decomposition approaches
for solving large-scale Data Envelopment Analysis
models
R.S. Barr and M.L. Durchholz

Part X: Abraham Charnes
Abraham Charnes remembered
Abraham Charnes, 1917-1992
A bibliography for Data Envelopment Analysis (1978-
1996)
LM. Setford

Data envelopment analysis

More Related Content

What's hot

Viewers also liked

Similar to Data envelopment analysis

Data envelopment analysis

Editor's Notes