# Computing the Efficiency of a DMU with Stochastic Inputs and Outputs Using Basic DEA Models

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```					                                                               (IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 5, May 2011

Computing the Efficiency of a DMU with
Stochastic Inputs and Outputs Using Basic DEA
Models
M. Nabahat                                   F.Esmaeeli sangari                                                     S.M.Mansourfar
Sama technical and vocational                    Sama technical and vocational                                        Sama technical and vocational
university, Urmia branch,Urmia,Iran              university, Urmia branch,Urmia,Iran                                   university, Sahand branch, Sahand,
mostafa.mansourfar@gmail.com

Abstract- This paper tries to introduce essential models of                                                          2. Preliminaries
stochastic and deterministic models (DEA) using the Chance                  Consider              a    set           of          homogenous             DMUs        as
Constrained Programming model to measure the DMU, that
enter the system simultaneously with the stochastic inputs and              DMU j              j  1,2,, n                  .Each DMU Consumes                  m
outputs. Considering the fact that by adding a Decision Making              inputs           to       produce              S            outputs.       Suppose    that
                                                                
Unit to the series of DMUs the efficiency frontier may change, the
X j  x1 j ,  x mj                       and Y j  y1 j , , y sj
T                                            T
main goal is to change the available essential stochastic models of                                                                                            are the
DEA and compute the amount of efficiency of the DMU and its
position in respect to the old frontier.                                    vectors of inputs and outputs values for DMU j .
Finally an Example is shown to highlight the procedure of
changing the stochastic model to deterministic model .                      Respectively                let               X j 0&X j 0                           and
Y j  0 &Y j  0 .
Keywords-DEA; Stochastic Programming; Efficiency; frontier
By using 0-1 parameters 1 ,  2 and  3 , the production
1. Introduction                                   possibility sets can be written into its generalized form [9],
          n                n                                           
Stochastic programming is a framework of modeling                               

  j x j  x ,   j y j  y  0,                            

optimization problems that involve uncertainty. Whereas                                  j 1             j 1                                         
T  ( x. y )                                                               
deterministic optimization problems are formulated with                                        n

3
known parameters, real world problems almost invariably                                  1 (   j   2 (1) n 1 )  1 ,  j  0, j  1,..., n  1
include some unknown parameters. Stochastic programming                         
              j 1                                                     

models are similar in style but take advantage of the fact that             The following four different production possibility sets are
probability distributions governing the data are known or can               obtained by assigning different values to (1 ,  2 ,  3 ) :
be estimated.
This branch of science has been used since the late 1950s for               (i)        If         1  0 &  2  0,1 &  3  0,1,                             then
decision models where input data (coefficients in Lp problem)
have been given probability distribution. Without any attempt
T becomes the production possibility set for the                                     CCR
model.
at completeness, we might mention from the early
contributions to this field. The pioneer works were done by                                   n       n
Dantzing[1-2], Beale[3-4], Tintner[5], Simon, Charnes,Cooper                      TCCR  {( x, y) | x    j x j ,   j y j  y  0 ,  j  0, j  1,..., n}
[6-7], Avriel and Williams [8] and Since then ,a number of                                                    j 1               j 1
Stochastic programming models have been formulated in
inventory theory, system maintenance ,micro-economics
,banking and finance.
(ii)        If        1  1 &  2  0 & ,  3  0,1                                then
Among various stochastic models, we would like to introduce                 T becomes the production possibility                                   set for the BCC
the chance constrained programming, and we will use it in one               model.
of the essential models.                                                                    n       n                                              n
In this paper new DMU with stochastic inputs and outputs is                   TBCC  {( x, y) | x    j x j ,   j y j  y  0 ,   j  1,  j  0, j  1,...n}
j 1             j 1                 j 1
efficiency of this DMU and changes in frontier of production
possibility set of observed DMUs are interested.

ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 5, May 2011

n
Theorem 1. (Bon ferroni inequality):
Ai        j xij   xio                i  1,, m
j 1
If the set of arbitrary events A1 , A2 ,  An constitutes a
n
partition of the sample space S andA1c , A2 c , , An c are                                         Am  r              j yrj  yro            r  1,, s
complements of events A1 , A2 ,  An the following rule                                                                 j 1
apply:
n                  n
P( Ai )  1   P( Ai ) (1)
c                                                           Following the theorem 1:
i 1               i 1
m s                      n                      n
Pr (  Ai c )  Pr (  j xij   xio ,   j y rj  y ro )  1  
3. Adding Competitive     DMU            to the Basic DEA Models
i 1                      j 1                   j 1
In this section, we will introduce two basic models of DEA                                    (2)
with special condition, new DMU with stochastic inputs and
outputs is added to the set of observed DMUs then                                        From (1) and (2) we have:
information about efficiency of this DMU and changes in
frontier of production possibility set of observed DMUs are                                  m s            m s     
interested. Suppose          DMU n 1 with           Stochastic inputs and               Pr (  Ai c )  1   P( Ai )
i 1           i 1                     
outputs enter the previous comparative DMU system .We                                                                   Pr ( Ai )      i  1,, m  s
want to achieve results regarding the efficiency of
m s
               ms
Pr (  Ai c )  1          
DMU n 1 and impact of the amount of deterministic inputs                                    i 1                    
and outputs on the old PPS frontier.
A Chance Constrained Programming model for CCR model
We have introduced                DMU o       with normally distributed, it
with the assumption that DMU o with a stochastic inputs
means:
and outputs enters the system is defined as the following:

Min                                                                               xio ~ N ( i , i2 ) ,               yro ~ N (  r , r )
2

s.t Pr (X o , Yo )  Tc   1                                                 i  1,...m ,                          r  1,...s

 j  0,            j  1,2,, n                                   Then it can be written:

n

Where X o          and Yo are all random variables and Pr                                Pr ( Ai )  Pr (  j xij   xio )               i  1,..., m              (3)
degree of probability and follow from each constraint being                                              j 1                           ms
realized    with      a     minimum       probability    of                                                     n

1        0   1.                                                                     Pr ( Am r )  Pr (  j y rj  y ro )              r  1,..., s            (4)
j 1                      ms
For illustrative purpose, let us assume X o and Yo are
normally distributed with known means and variances. Thus
we have the following chance constrained programming                                     By (3):
problem

Min                                                                                     Pr   j 1  j xij  o xio   xio  
n
                                       i  1,...., m
n                      n                                                            j o                              ms
s.t Pr    j xij   xio ,   j y rj  y ro   1   i  1, , m r  1, , s                                                  
                                                                                Pr   j 1  j xij  (  o ) xio  
n
 j 1                 j 1                                                                                                                                 (5)
 j o                            ms
 j  0, j  1,2, , n

Suppose:

ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 5, May 2011
It is known that
X                                            Min 
if X ~ N (  ,  2 )                          Then                           ~ N (0,1)

           j xij  o [i   i (1  K            )]   [i   i (1  K
n
s.t       j 1                                                                              )]
From (5) we have:                                                                                                           j o
m s                                m s
 n  x   (   )                                    
  j 1 j ij i                                                                                                                j yrj  o [r   r ( K           1)]  [ r   r ( K
n
o                                                                                                  j 1                                                                          1)] (9)
 j o                    xio (  o )   i (  o )                                                                j o
                                 
Pr                                                             (6)                                                                                            m s                              m s
        i (  o )              i (  o )          ms
           1 ,           j  0 , j  1,2,, n
n
   
                                                                                                  j 1 j
              z                        Z               
Let         represented the cumulative distribution function (CDF)
The determination of the interval where                                          K                is
of the standard normal distribution and suppose                               K       be the                                                                                                            m s
m s                          changeable.

                        By      Y 0, yj 0, X 0, xj 0                                     j  1,..., n , we
standard normal value such that                           (K  )  1                               then
ms                           have:
m s                                                                                                        
from (6):                                                                                                          i   i (1  K  )  0  ( i   i )   i K   K   1  i (10.a)
i
m s                               m s        m s
 n  j xij  i (  o )                                                                                                                                                      
 j 1                                                                                                           r   r ( K   1)  0  ( r   r )   r K   0  K   1  r (10.b)
 j o                                                                                                                                                                          r
                             m  s  1   (K  )
m s                                         m s          m s
 i (  o )

                           

m s
And by 0    1  0  1    1:
                           
    1                    1
Finally:                                                                                                          1  0    1           1          1
ms ms        ms         ms
             j xij  i (  o )
n
1                      1
  (K  )  1        K    1 (1      ) (10.c)
j 1
j o                                                                                                                         ms                    ms
 K                 1 K                                                  m s                                      m s
 i (  o )                                 m s            m s
By (10.a), (10, b), (10.c), K  is obtained.
                                            
 j 1  j xij  o i   i (1  K  )   i   i (1  K  )                                                                            m s
n
(7)
4. Numerical example
j o
               m s                    m s 

In this section ,we work out a numerical example to illustrate
Same process can be considered for (4)                                                                            the efficiency of competing DMU with stochastic inputs and
outputs that enter our system and suppose the inputs and
Then the deterministic form of this model is:                                                                     outputs are normally distributed with known means and
variances(given by Decision-maker) deterministic model and
Min                                                                                                              its efficiency are interested .
Example: Consider the four DMUs with single inputs and
           j xij  o [i   i (1  K          )]   [ i   i (1  K
n
s.t           j 1                                                                            )]                single outputs as defined in Table 1:
j o
m s                                m s
Table 1
           j yrj  o [r   r ( K           1)]  [ r   r ( K
n
j 1                                                                      1)] (8)                                          Data set for the numerical example
j o
m s                            m s
DMU              A         B         C               D                    E
 j  0 , j  1,2,, n
Input            2          5         2               7             xE ~ N (2 ,4)
Deterministic form of BCC model for measuring the
Output           4         8          2               8             y E ~ N (8 ,9)
efficiency of            DMU o          is:

ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 5, May 2011
By adding DMU E with stochastic inputs and outputs and                               DEA. .The goal is to compute the least amount of                   in which
the discussed DMU is efficient for the first time.
using (2) for CCR model, we have:
Table 2
Min                                                                                 The Deterministic Inputs and Outputs of DMUE and Objective Function that is
measured with CCR and BCC Model.
 21  52  23  74  x155  x15  0
s.t Pr                                             1   (11)                     K           1 α          I1   O1   BCC             CCR      MPSS or
41  82  23  84  y155  y15                                                                                                           Non MPSS
2
1 , 2 ,, 5  0
0.1      0.0796     3.8      5.3      0.783       0.697       Non MPSS
The deterministic form of (11) is:                                                       0.2      0.1586     3.6      5.6      0.889       0.778       Non MPSS
0.3      0.2358     3.4      5.9        1         0.868       Non MPSS
Min                                                                                     0.4      0.3108     3.2      6.2        1         0.969       Non MPSS
0.5      0.3830     3.0      6.5        1           1         MPSS
s.t 21  52  23  74  5 [2  2(1  K  )]   [2  2(1  K  )]                   0.6      0.4514     2.8      6.8        1           1         MPSS
2                   2                 0.7      0.5160     2.6      7.1        1           1         MPSS
41  82  23  84  5 [8  3( K   1)]  [8  3( K   1)]                    0.8      0.5762     2.4      7.4        1           1         MPSS
2                  2                          0.9      0.6318     2.2      7.7        1           1         MPSS
1 , 2 ,, 5  0                                                          1.0      0.6826     2.0      8.0        1           1         MPSS
1.1      0.7286     1.8      8.3        1           1         MPSS
1.2      0.7698     1.6      8.6        1           1         MPSS
The deterministic form of BCC model for measuring the                                    1.3      0.8064     1.4      8.9        1           1         MPSS
efficiency of DMU E is:                                                                  1.4      0.8384     1.2      9.2        1           1         MPSS
1.5      0.8664     1.0      9.5        1           1         MPSS
Min                                                                                     1.6      0.8904     0.8      9.8        1           1         MPSS
1.7      0.9108     0.6     10.1        1           1         MPSS
s.t 21  5 2  23  7 4  5 [2  2(1  K  )]   [2  2(1  K  )]                 1.8      0.9282     0.4     10.4        1           1         MPSS
2                   2
1.9      0.9426     0.2     10.7        1           1         MPSS
41  8 2  23  8 4  5 [8  3( K   1)]  [8  3( K   1)]              .
2                  2

1   2  3   4  5  1                                                                                       Old CCR frontier
1 ,  2 ,  , 5  0                                                      Output

B         D   Old BCC frontier
So     from    the    intersection  of  (10.a),     (10.b),
(10.c), 0  K   2 . By Supposing Range ( K  )  0.1
direction
2                                 2                                                                        E
Now, we apply our method to the data set. The results are                                                   A
provided in Table 2. As we can see in Table 2, for DMU E
The Deterministic Inputs and Outputs of DMU E and                                    .                      C

Objective Function that is measured with CCR and BCC
Model. The last column shows that in which condition (value                                                                                    Input
of  ) DMU E is MPSS.
Fig. 1.Old Efficient and New Efficient Frontiers for the Example
Fig. 1.Old and New Efficient Frontiers for the Example
The old and new efficient frontiers to this data set are depicted                    Two cases might happen either the DMU locates on the
Fig. 1.As the figure shows, DMU E after 4 stage with special                         frontier that the other DMU's had made before the use of
value of  will be Most Product Scale Size( MPSS).                                   comparative DMU. In this case the efficiency of other DMU's
is preserved. There may be just a ranking order of the DMUs
5. Conclusion                                             (in the CCR model).or as in the second case this DMU might
not be located on the old frontier. In this case the PPS frontier
Randomness in problem data poses a serious challenge for                             changes and gains new shape. We can 'not say any thing about
solving many linear programming problems especially in                               the DMU's that were efficient before adding the competitive
DMU and their efficiency must be measured again.

ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 5, May 2011
Probability of either two cases is dependent upon the amount
of mean and variance of the DMU's available for us by the
DM.

6.Acknowledgment

Authors sincerely thank Dr. F. Hosseinzadeh Lotfi, for his
most valuable comment and encouragement, which stimulated
a significant revision on the description of research
motivation, explanation and conclusion of the paper.

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