Download Consider the following problem

R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
Mathematical Programming Aspects of DEA
Performance evaluation for the case of two inputs and one
output was more complicated than the case of single inputoutput case. Even the graphical models cannot be used if we
consider more number of inputs and outputs. Hence, a general
formulation has to be made using the principles of mathematics
to handle the case of multiple inputs and multiple outputs.
Note that the Frontier analysis has been described by Professor
M. J. Farrel as early as in 1957 itself. But a mathematical
framework to handle the Frontier analysis could not be
established in the next thirty years!
Professors Abraham Charnes and William Cooper, along with
their doctoral student E. Rhodes, were successful in providing
the mathematical formulation in the year 1978. They published
their seminal paper in the European Journal of Operational
Research, titled, “Measuring the Efficiency of Decision Making
Units”, which provided the fundamentals of the mathematical
aspects of Frontier Analysis. These authors also coined the term,
“DATA ENVELOPMENT ANALYSIS.”
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
Let us use the subscripts i and j to represent inputs and outputs
respectively. Let x denote the inputs and y denote the outputs.
Thus xi represents the ith input, and yj represent the jth output of a
decision making unit. Let the total number of inputs and outputs
be represented by I and J respectively, where I, J > 0.
In DEA, multiple inputs and outputs are linearly aggregated
using weights. Thus a virtual input of a firm is obtained as the
linear weighted sum of all its inputs.
I
Virtual Input   ui xi
i 1
where ui is the weight assigned to its corresponding to input xi
during the aggregation. Note that ui  0.
Similarly, virtual output of a firm is obtained as the linear
weighted sum of all its outputs.
J
Virtual Output   v j y j
j 1
where vj is the weight assigned to its corresponding to output yj
during the aggregation. Also vj  0.
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
Given these virtual inputs and outputs, efficiency of the DMU in
converting the inputs to outputs can be defined as the ratio of
outputs to inputs.
J
Virtual Output
Efficiency 

Virtual Input
v j y j
j 1
I
 ui xi
i 1
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
Obviously, the most important issue at this stage is the
assessment of weights. This is a tricky issue as there is no
unique set of weights.
For example, a school that is good at arts will like to attract
higher weights to arts output. A school that has a higher
percentage of socially weaker groups in its students, would like
to emphasize this fact, giving more weight to this input
category.
Thus, these weights should be flexible and reflect the
requirement (performance) of the individual DMUs.
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
This issue of assigning weights is tackled in DEA by assigning
unique set of weights for each DMU. The weights for a DMU
are determined using mathematical programming as the weights
which will maximize its efficiency subject to the condition that
the efficiency of other DMUs (calculated using the same set of
weights) is restricted between 0 and 1. The DMU for which the
efficiency is maximized is normally termed as the reference or
base DMU.
Let there be N DMUs whose efficiencies have to be compared.
Let us take one of the DMUs, say the mth DMU, and maximize
its efficiency as per the definition above. Here the mth DMU is
the reference DMU.
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
The mathematical program now is,
J
max Em 
 v jm y jm
j 1
I
 uim xim
i 1
subject to
J
0
 v jm y jn
j 1
I
 1; n  1,2,, N
 uim xin
i 1
v jm , uim  0; i  1,2,, I ; j  1,2,, J
where
Em is the efficiency of the mth DMU,
yjm is jth output of the mth DMU,
vjm is the weight of that output,
xim is ith input of the mth DMU,
uim is the weight of that input, and
yjn and xin are jth output and ith input of the nth DMU, n = 1, 2,
…, N. Note that here n includes m.
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
Let us take our example now. Let vVA,A be the weight associated
with the only output (value added) of the Firm A. Let uCAP,A and
uEMP,A represent the weights of the two inputs, capital employed
and the number of employees respectively, of this firm. Thus, in
DEA, the efficiency of the Firm A is defined as follows.
EA 
vVA, A * 1.8
8.6uCAP , A  1.8u EMP , A
This efficiency is maximized subject to the following
conditions.
max E A 
1.8vVA, A
8.6uCAP , A  1.8u EMP , A
subject to
0  EA 
0  EB 
0  EC 
0  ED 
1.8vVA, A
8.6uCAP , A  1.8u EMP , A
0.2vVA, A
2.2uCAP , A  1.7u EMP , A
1
1
2.8vVA, A
15.6uCAP , A  2.6u EMP , A
1
4.1vVA, A
31.6uCAP , A  12.3u EMP , A
vVA, A , uCAP , A , u EMP , A  0
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1
R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
The above mathematical program, when solved, will give the
values of weights u and v, that will maximize the efficiency of
Firm A. If the efficiency is unity, then the firm is said to be
efficient, and will lie on the frontier. Otherwise, the firm is said
to be relatively inefficient.
Note that the above mathematical program gives the efficiency
of only one firm (Firm A here). To get the efficiency scores of
other firms, more such mathematical programs have to be
solved.
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
For example, to get the efficiency of Firm B, the following
mathematical program is used.
max E B 
0.2vVA, B
2.2uCAP , B  1.7u EMP , B
subject to
0  EA 
0  EB 
0  EC 
0  ED 
1.8vVA, B
8.6uCAP , B  1.8u EMP , B
0.2vVA, B
2.2uCAP , B  1.7u EMP , B
1
1
2.8vVA, B
15.6uCAP , B  2.6u EMP , B
1
4.1vVA, B
31.6uCAP , B  12.3u EMP , B
vVA, B , uCAP , B , u EMP , B  0
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1
R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
Note that the above mathematical programs are fractional
programs. It is in general quite difficult to solve fractional
programs.
If they are somehow converted to simpler formulations, such as
the linear programming formats, then they can be easily solved.
The easiest way to convert the above fractional programs to
linear programs is to normalize either the numerator or
denominator of the fractional programming objective function!
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
Let us first normalize the denominator of the objective function
of the fractional program. Then, the program for maximizing the
efficiency of Firm A becomes as follows.
max 1.8vVA, A
subject to
8.6uCAP , A  1.8u EMP , A  1
1.8vVA, A  8.6uCAP , A  1.8u EMP , A   0
0.2vVA, A  2.2uCAP , A  1.7u EMP , A   0
2.8vVA, A  15.6uCAP , A  2.6u EMP , A   0
4.1vVA, A  31.6uCAP , A  12.3u EMP , A   0
vVA, A , uCAP , A , u EMP , A  0
Thus, the weighted sum of inputs is constrained to be unity in
the above linear program. The objective function is the weighted
sum of outputs. Hence, the above formulation is generally
referred to as Output Maximization DEA program.
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
An analogous LP formulation is possible by minimizing the
weighted sum of inputs, setting the weighted sum of outputs
equal to unity. That will the Input Minimization DEA program.
The following is the Input Minimization DEA program for Firm
A.
 , A  1.8u EMP , A
min 8.6uCAP
subject to
 ,A  1
1.8vVA
 , A  8.6uCAP
 , A  1.8u EMP , A   0
1.8vVA
 , A  2.2uCAP
 , A  1.7u EMP , A   0
0.2vVA
 , A  15.6uCAP
 , A  2.6u EMP , A   0
2.8vVA
 , A  31.6uCAP
 , A  12.3u EMP , A   0
4.1vVA
 , A , uCAP
 , A , u EMP , A  0
vVA
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
These were the original models introduced by Charnes, Cooper
and Rhodes in their paper in 1978. Immediately after the
publication of this paper, the authors made a minor
modification. In a conventional LP, the decision variables are
non-negative – they can be either zero or positive. However, the
authors chose to define the decision variables of the DEA
programs (i.e. the weights) to be strictly positive. The nonnegativity constraints, vVA, A , uCAP, A , u EMP, A  0 are replaced by
vVA, A , uCAP, A , u EMP, A  0 . In fact, the subsequent modification
published by the authors in 1979 restricted the input and output
weights such that,
vVA, A , uCAP, A , u EMP, A   , where 
is an
infinitesimal or non-archimedian constant, usually of the order
of 10-5 or 10-6.
The ’s were introduced because under certain circumstances
the earlier model implied unit efficiency ratings for DMUs with
non-zero slack variables such that further improvements in
performance remained feasible.
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
The models developed so far are generally called the CCR
(Charnes, Cooper and Rhodes) Models in the DEA literature.
A general output maximization CCR DEA model can be
represented as follows.
J
max z   v jm y jm
j 1
subject to
I
 uim xim  1
i 1
J
I
j 1
i 1
 v jm y jn   uim xin  0 ; n  1,2,, N
v jm , uim   ; i  1,2,, I ; j  1,2,, J
The above formulation can be represented in matrix form as
shown below.
max z  VmT Ym
subject to
U mT X m  1
VmT Y  U mT X  0
VmT ,U mT  
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R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
Similarly, a general output maximization CCR DEA model can
be represented as follows.
I
 xim
min z    uim
i 1
subject to
J
 vjm y jm  1
j 1
J
I
j 1
i 1
 xin  0 ; n  1,2,, N
 vjm y jn   uim
   ; i  1,2,, I ; j  1,2,, J
vjm , uim
The above formulation can be represented in matrix form as
shown below.
min z   U mT X m
subject to
VmT Ym  1
VmT Y  U mT X  0
VmT ,U mT  
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