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 x i
represents the i th input, and y j
represent the j th 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.
Virtual Input
 i
I


1 u i x i where u i
is the weight assigned to its corresponding to input x i during the aggregation. Note that u i

0.
Similarly, virtual output of a firm is obtained as the linear weighted sum of all its outputs.
Virtual Output
 j
J


1 v j y j where v j
is the weight assigned to its corresponding to output y j during the aggregation. Also v j

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.
Efficiency

Virtual Output

Virtual Input i j
J


1 v
I


1 u i j y j x i
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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.
4

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 m th DMU, and maximize its efficiency as per the definition above. Here the m th DMU is the reference DMU.
5

R. Ramanathan/ Data Envelopment Analysis/ HUT/ September-December 2000
The mathematical program now is, max E m
 j
J


1 v jm y jm i
I


1 u im x im subject to
0 v jm j
J


1 v jm y jn

, u i
I


1 u im im
 x in
0 ; i

1 ; n

1 ,

1 , 2 ,  ,
2 ,  , I ; j

N
1 , 2 ,  , J where
E m
is the efficiency of the m th DMU, y jm
is j th output of the m th DMU, v jm
is the weight of that output, x im
is i th input of the m th DMU, u im
is the weight of that input, and y jn
and x in
are j th output and i th input of the n th 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 v
VA,A be the weight associated with the only output (value added) of the Firm A. Let u
CAP,A
and u
EMP,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.
E
A

8 .
6 v
VA , A u
CAP , A

* 1 .
8
1 .
8 u
EMP , A
This efficiency is maximized subject to the following conditions. max E
A

8 .
6 u
1 .
8 v
VA , A
CAP , A

1 .
8 u
EMP , A subject to
0

E
A

8 .
6 u
CAP
1 .
8 v
VA , A
, A

1 .
8 u
EMP , A

1
0

E
B

2 .
2 u
CAP
0 .
2 v
VA , A
, A

1 .
7 u
EMP , A

1
0

E
C

2 .
8 v
VA , A
15 .
6 u
CAP , A

2 .
6 u
EMP , A

1
0
 v
VA , A
E
D
, u

31 .
6 u
CAP
4 .
1 v
VA , A
, A

12 .
3 u
EMP , A
CAP , A
, u
EMP , A

0

1
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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

2 .
2 u
0 .
2 v
VA , B
CAP , B

1 .
7 u
EMP , B subject to
0

E
A

8 .
6 u
CAP
1 .
8 v
VA , B
, B

1 .
8 u
EMP , B

1
0

E
B

2 .
2 u
CAP
0 .
2 v
VA , B
, B

1 .
7 u
EMP , B

1
0

E
C

15 .
6
2 .
8 v
VA , B u
CAP , B

2 .
6 u
EMP , B

1
0
 v
VA , B
E
D
, u

31 .
6 u
CAP
4 .
1 v
VA , B
, B

12 .
3 u
EMP , B
CAP , B
, u
EMP , B

0

1
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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 .
8 v
VA , A subject to
8 .
6 u
CAP ,
1 .
8 v
VA , A
0 .
2 v
VA , A
2 .
8 v
VA , A
4 .
1 v
VA , A
A





1 .
8 u
EMP
8 .
6 u
CAP
2 .
2 u
CAP
,
,
A
A




15 .
6 u
31 .
6 u
CAP
CAP
,
,
,
A
A
A





1
1 .
8 u
1 .
7 u
EMP
2 .
6 u
,
EMP ,
EMP
A
A





, A
12 .
3 u
EMP , A
0
0



0
0 v
VA , A
, u
CAP , 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.
11

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. min 8 .
6 u

CAP , A

1 .
8 u

EMP , A subject to
1 .
8 v

VA , A

1
1 .
8 v

VA , A
0 .
2 v

VA , A


8 .
6 u

CAP , A


2 .
2 u

CAP , A

1 .
8 u

EMP , A


0

1 .
7 u

EMP , A


0
2 .
8 v

VA , A
4 .
1 v

VA , A


15 .
6 u

CAP , A


31 .
6 u

CAP , A

2 .
6 u

EMP , A



12 .
3 u

EMP , A


0
0 v

VA , A
, u

CAP , A
, u

EMP , A

0
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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, v
VA , A
, u
CAP , A
, u
EMP , A

0 are replaced by v
VA , A
, u
CAP , A
, u
EMP , A

0 . In fact, the subsequent modification published by the authors in 1979 restricted the input and output weights such that, v
VA , A
, u
CAP , 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. max z
 j
J


1 v jm y jm subject to i
I


1 u im x im

1 j
J


1 v jm y jn
 i
I


1 u im x in

0 ; n

1 , 2 ,  , N v jm
, u im
 
; i

1 , 2 ,  , I ; j

1 , 2 ,  , J
The above formulation can be represented in matrix form as shown below. max z

V m
T
Y m subject to
T
U m
X m

1
V m
T
Y

U
T m
X

0
V m
T T
, U m
 
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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. min z
  i
I


1 u
 im x im subject to j
J


1 v
 jm y jm

1
J j

1 jm y jn
 i
I
 

1 u im x in

0 ; n

1 , 2 ,  , N v
 jm
, u
 im
 
; i

1 , 2 ,  , I ; j

1 , 2 ,  , J
The above formulation can be represented in matrix form as shown below. min z
 
U

T m
X m subject to
V m

T
Y m

1
V m
 T
Y

U
 T m
X

0
V m

T
, U

T m
 
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