# A Multiobjective Programming Method for Ranking All Units Based on Compensatory DEA Model

Authors: Haifang Cheng; Yishi Zhang; Jianhu Cai; Weilai Huang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

In order to rank all decision making units (DMUs) on the same basis, this paper proposes a multiobjective programming (MOP) model based on a compensatory data envelopment analysis (DEA) model to derive a common set of weights that can be used for the full ranking of all DMUs. We first revisit a compensatory DEA model for ranking all units, point out the existing problem for solving the model, and present an improved algorithm for which an approximate global optimal solution of the model can be obtained by solving a sequence of linear programming. Then, we applied the key idea of the compensatory DEA model to develop the MOP model in which the objectives are to simultaneously maximize all common weights under constraints that the sum of efficiency values of all DMUs is equal to unity and the sum of all common weights is also equal to unity. In order to solve the MOP model, we transform it into a single objective programming (SOP) model using a fuzzy programming method and solve the SOP model using the proposed approximation algorithm. To illustrate the ranking method using the proposed method, two numerical examples are solved.


---

## Body

## 1. Introduction

As a nonparametric method, data envelopment analysis (DEA) is proposed by [1] to evaluate the relative efficiency of a set of decision making units (DMUs) with the multiple inputs and outputs. For each DMU, the optimal weights for calculating efficiency value are obtained by solving a linear programming (LP) problem. By using DEA, DMUs can be divided into two categories: efficient DMUs and inefficient DMUs [2]. However, there are always more than one DMU to be evaluated as efficient, which would cause the problem that all DMUs cannot be fully discriminated [3]. Moreover, the efficiencies of different DMUs obtained by different sets of weights may be unable to be compared and ranked on the same basis [2–5].To deal with the above-mentioned problems, many methods have been developed to rank all DMUs under the framework of DEA. The common weights DEA introduced by [6–8] is known as one of the popular methods in which all DMUs can be evaluated by a common set of weights (CSW). Since then, DEA models with CSW have been rapidly applied in many researches. For example, Sinuany-Stern and Friedman [9] developed a DR/DEA to provide the best CSW for given inputs and outputs in order to rank all DMUs on the same scale. Jahanshahloo et al. [10] presented a method to obtain the CSW of DMUs by solving only one problem, in order to measure the efficiency and to rank the efficient DMUs. Kao and Hung [11] proposed a compromise solution approach for generating a CSW which produces the vector of efficiency scores for the DMUs. Their approach is able to not only differentiate efficient DMUs but also detect abnormal efficiency scores on a common base. Amin and Toloo [12] presented an improved integrated DEA model in order to detect the most efficient DMUs. The method in their model is able to determine a CSW for all DMUs by solving a LP problem. Liu and Peng [13] introduced a DEA method to determine a CSW for the ranking of only DEA efficient DMUs. For the decision maker (DM), this ranking is based on the optimization of the group’s efficiency. Jahanshahloo et al. [2] proposed two methods to rank DMUs concerning an ideal line or a special DMU. Their method needs to determine a CSW for efficient DMUs. Davoodi and Rezai [14] suggested a method to compute the efficiency scores of all DMUs and then rank them using a CSW determined by solving a LP problem. Ramón et al. [15] proposed a DEA approach for deriving a CSW to be used for the ranking of all DMUs. The idea of this approach is to minimize the deviations of the CSW from the DEA profiles of weights without zeros of the efficient DMUs. Lotfi et al. [16] developed the common weights DEA method to deal with total weights flexibility in order to allocate fixed resources using DEA. To see the other ranking approaches under the framework of DEA, we refer the readers to the review papers of [17, 18].Recently, Khodabakhshi and Aryavash [19] proposed a very interesting variant of the basic DEA model for ranking all DMUs. In their DEA model, the sum of efficiency values of all DMUs is supposed to be equal to unity. This assumption implies that the efficiencies of all DMUs have compensatory features, so we also call it a compensatory DEA model in this paper. In order to rank all DMUs, the minimum and maximum efficiency values of each DMU are computed. Then the rank of each DMU is determined in proportion to a combination of its minimum and maximum efficiency values. To solve the compensatory DEA model easily, they transform it into a new LP model. However, the two models are not completely equivalent; thus the optimal solution of compensatory DEA model may not be obtained by solving the transformed LP model. Moreover, their ranking method does not provide more information about the weights used for calculating the efficiency scores of each DMU. To deal with these two problems, in this paper, we first improve their solution method and propose an approximation algorithm to solve the compensatory DEA model. Then, we apply the key idea of compensatory DEA model to develop a multiobjective programming (MOP) model for determining a CSW used for calculating the efficiency scores of each DMU. In the proposed MOP model, the objectives are to simultaneously maximize all common weights assigned to each input and output under constraints that the sum of efficiency values of all DMUs is equal to unity and the sum of all weights assigned to each input and output is also equal to unity. After the CSW has been determined by solving the MOP model, all DMUs can be ranked according to the efficiency scores weighted by it.The rest of the paper is organized as follows. In Section2, we first revisit compensatory DEA model for ranking all units and present an improved algorithm to solve the model. The proposed MOP model and solution approach are presented in Section 3. A numerical example is examined in Section 4 to illustrate the ranking method using the proposed model. Conclusions are offered in Section 5.


## 2. The Improved Method for Solving Compensatory DEA Model

In this section, we first revisit the compensatory DEA model proposed by [19] for ranking all DMUs. After pointing out the existing problem of their solution method, we will propose an approximation algorithm by improving it so that the model can be solved correctly. To demonstrate the improved algorithm, a numerical example is also presented in this section.


### 2.1. A Revisit to Compensatory DEA Model

Considern DMUs that use m inputs to produce s outputs. Let x
i
j (i
=
1,2
,
…
,
m) and y
r
j (r
=
1,2
,
…
,
s) represent the input and output values of DMU
j (j
=
1,2
,
…
,
n), respectively. Suppose that all input and output elements are nonnegative deterministic numbers. For a given DMU
j, then the efficiency scores are given as follows:
(1)
θ
j
=
∑
r
=
1
s
y
r
j
u
r
∑
i
=
1
m
x
i
j
v
i
,
j
=
1,2
,
…
,
n
,
where v
i (i
=
1,2
,
…
,
m) and u
r (r
=
1,2
,
…
,
s) are the input and output weights assigned to ith input and rth output, respectively.LetDMU
o be a DMU under evaluation; then the following model is used to determine the minimum and maximum efficiency values of DMU
o:
(2)
M
1
min
⁡
and
max
⁡
θ
o
M
1
0
s
.
t
.
θ
j
=
∑
r
=
1
s
y
r
j
u
r
∑
i
=
1
m
x
i
j
v
i
,
j
=
1,2
,
…
,
n
M
1
0
s
.
t
.
∑
j
=
1
n
θ
j
=
1
M
1
0
s
.
t
.
u
r
,
v
i
,
θ
j
≥
0
,
∀
i
,
j
,
r
.Model (M1) must be run twice. The minimum/maximum value ofθ
o is determined by minimizing/maximizing the objective function of model (M1). Model (M1) is a nonlinear programming. Using the transformation w
i
j
≔
v
i
θ
j, we can replace model (M1) by the following LP problem:
(3)
M
2
min
⁡
and
max
⁡
θ
o
=
∑
r
=
1
s
y
r
o
u
r
M
2
0
s
.
t
.
∑
i
=
1
m
x
i
o
v
i
=
1
M
2
s
.
t
.
∑
i
=
1
m
x
i
j
w
i
j
-
∑
r
=
1
s
y
r
j
u
r
=
0
,
000000000000000000
j
=
1,2
,
…
,
n
M
2
s
.
t
.
∑
j
=
1
n
w
i
j
=
v
i
,
i
=
1,2
,
…
,
m
M
2
s
.
t
.
u
r
,
v
i
,
w
i
j
≥
0
,
∀
i
,
j
,
r
.Letθ
j
min
⁡ and θ
j
max
⁡ be the minimum and maximum values of θ
j, respectively, which can be obtained by solving model (M2). Then the following convex combinations are used to determine the efficiency values for each DMU:
(4)
θ
j
=
λ
θ
j
min
⁡
+
1
-
λ
θ
j
max
⁡
,
0
≤
λ
≤
1
,
j
=
1,2
,
…
,
n
,
where
(5)
λ
=
1
-
∑
j
=
1
n
θ
j
max
⁡
∑
j
=
1
n
θ
j
min
⁡
-
θ
j
max
⁡
.Considering (4) and the value of λ obtained in (5) together, the values of θ
j (j
=
1,2
,
…
,
n) are determined. Now, the DMUs are fully ranked with respect to their efficiency scores. In other words, a DMU has a better rank if it has a greater efficiency score.


### 2.2. The Existing Problem for Solving Compensatory DEA Model

In compensatory DEA model, the optimal solutions of model (M1) can be found by solving model (M2). Since constraintw
i
j
=
v
i
θ
j is not considered in model (M2), model (M2) is a relaxation problem of model (M1), and two models are not completely equivalent. Hence, we may not obtain the optimal solutions of model (M1) by solving model (M2).Letu
_
r
′ and v
_
i
′ be the optimal solutions of model (M2) with a minimum form. If they are also the optimal solutions of model (M1), then they must satisfy the constraints of model (M1); that is ∑
j
=
1
n
θ
_
j
′
=
1, where the efficiency score θ
_
j
′ is determined by (1). However the following example shows that the equation ∑
j
=
1
n
θ
_
j
′
=
1 does not hold.For example, suppose we want to evaluate DMU1 in the numerical example presented in Section 2.4. For a minimum problem, we solve model (M2) to find the optimal solutions: u
_
1
′
=
0.000678, u
_
2
′
=
0, v
_
1
′
=
0.000130, v
_
2
′
=
0.012707, and v
_
3
′
=
0.050998. Using (1) we can obtain that the efficiency scores θ
_
j
′ of twelve DMUs are 0.0454, 0.0637, 0.0631, 0.0680, 0.0928, 0.0439, 0.0604, 0.0707, 0.0828, 0.0427, 0.0276, and 0.0599, respectively. Computing the sum of those efficiency scores we have ∑
j
=
1
12
θ
_
j
′
=
0.7210
<
1, which means that u
_
r
′ and v
_
i
′ are not the optimal solutions of model (M1). Therefore, we cannot obtain the optimal solutions of model (M1) by solving model (M2).


### 2.3. The Proposed Approximation Algorithm for Solving Model (M1)

In this section, the minimum problems as an example are used to illustrate the approximation algorithm. However, the procedure can be easily extended to solve the maximum problems.Unlike the standard DEA, model (M1) involves the sum of linear-fractional functions in constraints. So model (M1) may not be considered a convex optimization problem. To prove it we first rewrite these constraints of linear-fractional functions as follows:(6)
F
j
=
θ
j
∑
i
=
1
m
x
i
j
v
i
-
∑
r
=
1
s
y
r
j
u
r
=
0
,
j
=
1,2
,
…
,
n
.The Hessian matrix of functionF
j, with the order of variables being (
θ
,
v
,
u
), can be derived as
(7)
H
F
j
=
0
x
j
0
x
j
T
0
0
0
0
0
∈
R
1
+
m
+
s
×
1
+
m
+
s
,
where x
j
=
(
x
1
j
,
x
2
j
,
…
,
x
m
j
). Zeros in the matrix are of appropriate dimension. Since principal minors of order 2 are H
(
F
j
)
2
=
-
x
1
2
<
0, that is, there exists one 2 × 2 principal minor in H
(
F
j
) which is negative, it can be concluded that these constraints function is nondefinite. Therefore, model (M1) is not a convex optimization, which implies that an efficient algorithm may be required to determine a globally optimal solution.Although the optimal solutions of model (M1) cannot be found by solving model (M2), the following shows that the optimal solutions of model (M1) can be approximated by solving a sequence of LP models on the base of the optimal solution of model (M2).Since the optimal solutions of model (M2) may not be unique, a secondary goal must be introduced in it. For the minimum problem, the secondary goal is to maximize the sum of efficiency scores of the rest DMUs; that is,(8)
max
⁡
∑
j
=
1
j
≠
o
n
∑
r
=
1
s
y
r
j
u
r
∑
i
=
1
m
x
i
j
v
i
.The fractional form of the secondary objective function can be transformed to the following linear form:(9)
max
⁡
∑
j
=
1
j
≠
o
n
n
-
1
∑
r
=
1
s
y
r
j
u
r
-
∑
i
=
1
m
x
i
j
v
i
.Therefore, introducing a sufficiently small positive numberδ, model (M2) can also be rewritten as follows:
(10)
M
3
min
⁡
θ
o
=
∑
r
=
1
s
y
r
o
u
r
M
3
min
⁡
θ
o
=
-
δ
∑
j
=
1
j
≠
o
n
n
-
1
∑
r
=
1
s
y
r
j
u
r
-
∑
i
=
1
m
x
i
j
v
i
M
3
0
s
.
t
.
u
r
,
v
i
∈
ℵ
2
u
,
v
,
where ℵ
2
(
u
,
v
) denotes the set of the feasible solution to model (M2). Since model (M2) or (M3) is a relaxation problem of model (M1), we have the following lemma.Lemma 1.
Letu
_
r and v
_
i be the optimal solutions of model (M3); then one has
(11)
∑
j
=
1
n
θ
_
j
≤
1
,
where θ
_
j is determined by using (1).Proposition 2.
If∑
j
=
1
n
θ
_
j
=
1, then the optimal solutions u
_
r and v
_
i of model (M3) are also the optimal solutions of model (M1).Proof.
If∑
j
=
1
n
θ
_
j
=
1, then the optimal solutions u
_
r and v
_
i of model (M3) are the feasible solutions of model (M1). Since the objectives of two models are equivalent, the optimal solutions u
_
r and v
_
i of model (M3) are also the optimal solutions of model (M1).If∑
j
=
1
n
θ
_
j
<
1, then we have to proceed further to find the optimal solutions on the basis of it. Let constraint w
i
j
=
v
i
θ
j be replaced by w
i
j
≥
v
i
θ
_
j, then add it into model (M3) to formulate the following LP problem:
(12)
M
4
min
⁡
θ
o
=
∑
r
=
1
s
y
r
o
u
r
M
4
min
⁡
θ
o
=
-
δ
∑
j
=
1
j
≠
o
n
n
-
1
∑
r
=
1
s
y
r
j
u
r
-
∑
i
=
1
m
x
i
j
v
i
M
4
s
.
t
.
w
i
j
≥
v
i
θ
_
j
,
i
=
1,2
,
…
,
m
,
j
=
1,2
,
…
,
n
,
M
4
s
.
t
.
u
r
,
v
i
∈
ℵ
2
u
,
v
.It is obvious that model (M4) is a relaxation problem of model (M1) and model (M3) is a relaxation problem of model (M4). Therefore, we have the following lemma.Lemma 3.
Letu
_
r
1 and v
_
i
1 be the optimal solutions of model (M4); then one has
(13)
∑
j
=
1
n
θ
_
j
≤
∑
j
=
1
n
θ
_
j
1
≤
1
,
where θ
_
j
(
1
) is determined by (1).According to Proposition2, if ∑
j
=
1
n
θ
_
j
(
1
)
=
1, then u
_
r
(
1
) and v
_
i
(
1
) are also the optimal solutions of model (M1); otherwise, we must also continue to find the optimal solutions of model (M1).Letθ
_
j
(
1
) replace θ
_
j in model (M4); then solve it to obtain the optimal solutions u
_
r
(
2
) and v
_
i
(
2
). Using (1) we will get the efficiency score θ
_
j
(
2
) of each DMU and then judge whether ∑
j
=
1
n
θ
_
j
(
2
) is equal to 1 or not. The process is repeated until the sum of efficiency scores of all DMUs is approximately equal to 1 within given error.According to the above analysis, the procedure of the proposed approximation algorithm to solve model (M1) can be described as follows.Algorithm 4.
Consider the following.
Step
1. Give permissible error ε
>
0.
Step
2. Solve model (M3) to find the optimal solutions u
_
r and v
_
i, and then calculate the efficiency score θ
_
j of each DMU using (1).
Step
3. Compute the sum of the efficiency scores of all DMUs. If ∑
j
=
1
n
θ
_
j
-
1
≤
ε, then stop with the optimal solutions u
_
r and v
_
i to model (M1); otherwise, k
≔
1; go to Step 4.
Step
4. Solve model (M4) to find optimal solutions u
_
r
(
k
) and v
_
i
(
k
), and then calculate the efficiency score θ
_
j
(
k
) of each DMU using (1).
Step
5. Compute the sum of the efficiency scores of all DMUs. If ∑
j
=
1
n
θ
_
j
(
k
)
-
1
≤
ε, then stop with the optimal solutions u
_
r
(
k
) and v
_
i
(
k
) to model (M1); otherwise go to Step 6.
Step
6. Let θ
_
j
(
k
) replace θ
_
j in model (M4); k
≔
k
+
1; go back to Step 4.Proposition 5.
The optimal solutions of model (M4) will be close to the optimal solutions of model (M1) by using Algorithm4.Proof.
Letu
_
r
(
t
) and v
_
i
(
t
), t
=
1,2
,
…
,
k, be the optimal solutions to model (M4) fortth solution; using (1) we will get the efficiency score θ
_
j
(
t
) of each DMU; then we have
(14)
∑
j
=
1
n
θ
_
j
≤
∑
j
=
1
n
θ
_
j
1
≤
∑
j
=
1
n
θ
_
j
2
≤
⋯
≤
∑
j
=
1
n
θ
_
j
k
≤
1
.
Since{
∑
j
=
1
n
θ
_
j
(
t
)
} increases monotonically and exists upper bound 1, we have ∑
j
=
1
n
θ
-
j
(
t
)
→
1, which means that the optimal solutions u
_
r
(
t
) and v
_
i
(
t
) of model (M4) are close to the optimal solutions of model (M1) by repeatedly solving model (M4).We have discussed the approximation algorithm with minimum problem. For a maximum problem, let constraintw
i
j
≥
v
i
θ
_
j be replaced by constraint w
i
j
≤
v
i
θ
_
j in model (M4). Using the above approximation algorithm, we can also obtain the optimal solutions to model (M1) with maximum form.


### 2.4. A Numerical Example

In order to compare the proposed method with the method in [19], a numerical example used by them is presented in this subsection. There are twelve DMUs with three inputs (X
1, X
2, and X
3) and two outputs (Y
1, Y
2) as shown in Table 1. The minimum and maximum efficiency scores, the integrated score, and the rank of DMUs obtained by the method in [19] are exhibited in the seventh, the eighth, and the ninth column of Table 1, respectively.Table 1
The input-output data and the rankings of DMUs.
DMU
Inputs and outputs
The method in [19]
Proposed improved method
X
1
X
2
X
3
Y
1
Y
2
[
θ
j
min
⁡, θ
j
max
⁡
]
θ
j
Rank
[
θ
j
min
⁡, θ
j
max
⁡
]
θ
j
Rank
1
350
39
9
67
751
[0.0454, 0.0989]
0.0675
11
[0.0573, 0.0862]
0.0702
11
2
298
26
8
73
611
[0.0526, 0.1199]
0.0804
7
[0.0605, 0.0993]
0.0778
7
3
422
31
7
75
584
[0.0483, 0.1076]
0.0728
9
[0.0556, 0.0893]
0.0706
10
4
281
16
9
70
665
[0.0564, 0.1468]
0.0937
4
[0.0577, 0.1371]
0.0930
4
5
301
16
6
75
445
[0.0499, 0.1679]
0.0986
2
[0.0591, 0.1435]
0.0967
3
6
360
29
17
83
1070
[0.0393, 0.1402]
0.0810
6
[0.0393, 0.1249]
0.0774
8
7
540
18
10
72
457
[0.0304, 0.1262]
0.0700
10
[0.0348, 0.1248]
0.0749
9
8
276
33
5
74
590
[0.0549, 0.1358]
0.0883
5
[0.0589, 0.1190]
0.0857
5
9
323
25
5
75
1074
[0.0800, 0.2043]
0.1313
1
[0.0975, 0.1724]
0.1308
1
10
444
64
6
74
1072
[0.0363, 0.1394]
0.0789
8
[0.0363, 0.1394]
0.0822
6
11
323
25
5
25
350
[0.0267, 0.0666]
0.0432
12
[0.0325, 0.0562]
0.0430
12
12
444
64
6
104
1199
[0.0510, 0.1560]
0.0944
3
[0.0510, 0.1560]
0.0977
2In order to rank all DMUs by the proposed method, we first use the proposed approximation Algorithm4 to calculate the minimum and maximum efficiency scores of each DMU and then use (4) and (5) to compute the integrated scores. Finally, the DMUs were ranked according to their integrated scores.Using Algorithm4, the procedure to calculate the minimum efficiency score of DMU1 is described as follows.Step
1. Give permissible error ε
=
0.001.Step
2. Let δ
=
0.0001. Solve model (M3) to find the optimal solutions: u
_
1
=
0
.
000678, u
_
2
=
0, v
_
1
=
0, v
_
2
=
0.013872, and v
_
3
=
0.050998. Using (1) we can obtain that the efficiency scores θ
_
j of twelve DMUs are 0.0454, 0.0644, 0.0646, 0.0697, 0.0964, 0.0444, 0.0643, 0.0704, 0.0845, 0.0420, 0.0282, and 0.0591, respectively.Step
3. Computing the sum of the efficiency scores of all DMUs, we have ∑
j
=
1
12
θ
_
j
=
0.7335. Since ∑
j
=
1
12
θ
_
j
-
1
=
0.2665
>
0.001, we do not obtain the optimal solutions to model (M1), and we have to proceed further to find the optimal solutions. Consider k
≔
1; go to Step 4.Steps
4–
6. Solve model (M4) repeatedly to find the optimal solutions u
_
r
(
k
) and v
_
i
(
k
), and then calculate the efficiency score θ
_
j
(
k
) of each DMU using (1), k
=
1,2
,
…. Computing the sum of the efficiency scores of all DMUs, we have {
∑
j
=
1
12
θ
_
j
(
k
)
}
=
0.9592,0.9945,0.9993
,
…. Since ∑
j
=
1
12
θ
_
j
(
3
)
-
1
=
0.0007
<
0.001, the optimal solutions of model (M1) have been found: u
1
*
=
0.000855, u
2
*
=
0, v
1
*
=
0, v
2
*
=
0.022075, v
3
*
=
0.015451, and the objective function valve is θ
1
*
=
0.0573.Hence, the minimum efficiency value of DMU1 is θ
1
min
⁡
=
0.0573. Similarly, using Algorithm 4, we can also obtain the minimum efficiency values of the rest DMUs. Let constraint w
i
j
≥
v
i
θ
_
j be replaced by constraint w
i
j
≤
v
i
θ
_
j in model (M4). Using Algorithm 4 we can also obtain the maximum efficiency values of all DMUs. The minimum and maximum efficiency scores of each DMU are shown in the tenth column of Table 1.Using (4) and (5), the integrated score θ
j is determined and the results are exhibited in the eleventh column of Table 1. The twelfth column of Table 1 shows the results of rankings according to their integrated scores.Table1 shows that the rankings of DMUs obtained by proposed method are not exactly the same as that obtained by the method in [19]. For instance, DMU6 is ranked as eighth in our method and sixth in the method of [19], whereas DMU10 is placed sixth in our method and eighth in the method of [19].


## 2.1. A Revisit to Compensatory DEA Model

Considern DMUs that use m inputs to produce s outputs. Let x
i
j (i
=
1,2
,
…
,
m) and y
r
j (r
=
1,2
,
…
,
s) represent the input and output values of DMU
j (j
=
1,2
,
…
,
n), respectively. Suppose that all input and output elements are nonnegative deterministic numbers. For a given DMU
j, then the efficiency scores are given as follows:
(1)
θ
j
=
∑
r
=
1
s
y
r
j
u
r
∑
i
=
1
m
x
i
j
v
i
,
j
=
1,2
,
…
,
n
,
where v
i (i
=
1,2
,
…
,
m) and u
r (r
=
1,2
,
…
,
s) are the input and output weights assigned to ith input and rth output, respectively.LetDMU
o be a DMU under evaluation; then the following model is used to determine the minimum and maximum efficiency values of DMU
o:
(2)
M
1
min
⁡
and
max
⁡
θ
o
M
1
0
s
.
t
.
θ
j
=
∑
r
=
1
s
y
r
j
u
r
∑
i
=
1
m
x
i
j
v
i
,
j
=
1,2
,
…
,
n
M
1
0
s
.
t
.
∑
j
=
1
n
θ
j
=
1
M
1
0
s
.
t
.
u
r
,
v
i
,
θ
j
≥
0
,
∀
i
,
j
,
r
.Model (M1) must be run twice. The minimum/maximum value ofθ
o is determined by minimizing/maximizing the objective function of model (M1). Model (M1) is a nonlinear programming. Using the transformation w
i
j
≔
v
i
θ
j, we can replace model (M1) by the following LP problem:
(3)
M
2
min
⁡
and
max
⁡
θ
o
=
∑
r
=
1
s
y
r
o
u
r
M
2
0
s
.
t
.
∑
i
=
1
m
x
i
o
v
i
=
1
M
2
s
.
t
.
∑
i
=
1
m
x
i
j
w
i
j
-
∑
r
=
1
s
y
r
j
u
r
=
0
,
000000000000000000
j
=
1,2
,
…
,
n
M
2
s
.
t
.
∑
j
=
1
n
w
i
j
=
v
i
,
i
=
1,2
,
…
,
m
M
2
s
.
t
.
u
r
,
v
i
,
w
i
j
≥
0
,
∀
i
,
j
,
r
.Letθ
j
min
⁡ and θ
j
max
⁡ be the minimum and maximum values of θ
j, respectively, which can be obtained by solving model (M2). Then the following convex combinations are used to determine the efficiency values for each DMU:
(4)
θ
j
=
λ
θ
j
min
⁡
+
1
-
λ
θ
j
max
⁡
,
0
≤
λ
≤
1
,
j
=
1,2
,
…
,
n
,
where
(5)
λ
=
1
-
∑
j
=
1
n
θ
j
max
⁡
∑
j
=
1
n
θ
j
min
⁡
-
θ
j
max
⁡
.Considering (4) and the value of λ obtained in (5) together, the values of θ
j (j
=
1,2
,
…
,
n) are determined. Now, the DMUs are fully ranked with respect to their efficiency scores. In other words, a DMU has a better rank if it has a greater efficiency score.


## 2.2. The Existing Problem for Solving Compensatory DEA Model

In compensatory DEA model, the optimal solutions of model (M1) can be found by solving model (M2). Since constraintw
i
j
=
v
i
θ
j is not considered in model (M2), model (M2) is a relaxation problem of model (M1), and two models are not completely equivalent. Hence, we may not obtain the optimal solutions of model (M1) by solving model (M2).Letu
_
r
′ and v
_
i
′ be the optimal solutions of model (M2) with a minimum form. If they are also the optimal solutions of model (M1), then they must satisfy the constraints of model (M1); that is ∑
j
=
1
n
θ
_
j
′
=
1, where the efficiency score θ
_
j
′ is determined by (1). However the following example shows that the equation ∑
j
=
1
n
θ
_
j
′
=
1 does not hold.For example, suppose we want to evaluate DMU1 in the numerical example presented in Section 2.4. For a minimum problem, we solve model (M2) to find the optimal solutions: u
_
1
′
=
0.000678, u
_
2
′
=
0, v
_
1
′
=
0.000130, v
_
2
′
=
0.012707, and v
_
3
′
=
0.050998. Using (1) we can obtain that the efficiency scores θ
_
j
′ of twelve DMUs are 0.0454, 0.0637, 0.0631, 0.0680, 0.0928, 0.0439, 0.0604, 0.0707, 0.0828, 0.0427, 0.0276, and 0.0599, respectively. Computing the sum of those efficiency scores we have ∑
j
=
1
12
θ
_
j
′
=
0.7210
<
1, which means that u
_
r
′ and v
_
i
′ are not the optimal solutions of model (M1). Therefore, we cannot obtain the optimal solutions of model (M1) by solving model (M2).


## 2.3. The Proposed Approximation Algorithm for Solving Model (M1)

In this section, the minimum problems as an example are used to illustrate the approximation algorithm. However, the procedure can be easily extended to solve the maximum problems.Unlike the standard DEA, model (M1) involves the sum of linear-fractional functions in constraints. So model (M1) may not be considered a convex optimization problem. To prove it we first rewrite these constraints of linear-fractional functions as follows:(6)
F
j
=
θ
j
∑
i
=
1
m
x
i
j
v
i
-
∑
r
=
1
s
y
r
j
u
r
=
0
,
j
=
1,2
,
…
,
n
.The Hessian matrix of functionF
j, with the order of variables being (
θ
,
v
,
u
), can be derived as
(7)
H
F
j
=
0
x
j
0
x
j
T
0
0
0
0
0
∈
R
1
+
m
+
s
×
1
+
m
+
s
,
where x
j
=
(
x
1
j
,
x
2
j
,
…
,
x
m
j
). Zeros in the matrix are of appropriate dimension. Since principal minors of order 2 are H
(
F
j
)
2
=
-
x
1
2
<
0, that is, there exists one 2 × 2 principal minor in H
(
F
j
) which is negative, it can be concluded that these constraints function is nondefinite. Therefore, model (M1) is not a convex optimization, which implies that an efficient algorithm may be required to determine a globally optimal solution.Although the optimal solutions of model (M1) cannot be found by solving model (M2), the following shows that the optimal solutions of model (M1) can be approximated by solving a sequence of LP models on the base of the optimal solution of model (M2).Since the optimal solutions of model (M2) may not be unique, a secondary goal must be introduced in it. For the minimum problem, the secondary goal is to maximize the sum of efficiency scores of the rest DMUs; that is,(8)
max
⁡
∑
j
=
1
j
≠
o
n
∑
r
=
1
s
y
r
j
u
r
∑
i
=
1
m
x
i
j
v
i
.The fractional form of the secondary objective function can be transformed to the following linear form:(9)
max
⁡
∑
j
=
1
j
≠
o
n
n
-
1
∑
r
=
1
s
y
r
j
u
r
-
∑
i
=
1
m
x
i
j
v
i
.Therefore, introducing a sufficiently small positive numberδ, model (M2) can also be rewritten as follows:
(10)
M
3
min
⁡
θ
o
=
∑
r
=
1
s
y
r
o
u
r
M
3
min
⁡
θ
o
=
-
δ
∑
j
=
1
j
≠
o
n
n
-
1
∑
r
=
1
s
y
r
j
u
r
-
∑
i
=
1
m
x
i
j
v
i
M
3
0
s
.
t
.
u
r
,
v
i
∈
ℵ
2
u
,
v
,
where ℵ
2
(
u
,
v
) denotes the set of the feasible solution to model (M2). Since model (M2) or (M3) is a relaxation problem of model (M1), we have the following lemma.Lemma 1.
Letu
_
r and v
_
i be the optimal solutions of model (M3); then one has
(11)
∑
j
=
1
n
θ
_
j
≤
1
,
where θ
_
j is determined by using (1).Proposition 2.
If∑
j
=
1
n
θ
_
j
=
1, then the optimal solutions u
_
r and v
_
i of model (M3) are also the optimal solutions of model (M1).Proof.
If∑
j
=
1
n
θ
_
j
=
1, then the optimal solutions u
_
r and v
_
i of model (M3) are the feasible solutions of model (M1). Since the objectives of two models are equivalent, the optimal solutions u
_
r and v
_
i of model (M3) are also the optimal solutions of model (M1).If∑
j
=
1
n
θ
_
j
<
1, then we have to proceed further to find the optimal solutions on the basis of it. Let constraint w
i
j
=
v
i
θ
j be replaced by w
i
j
≥
v
i
θ
_
j, then add it into model (M3) to formulate the following LP problem:
(12)
M
4
min
⁡
θ
o
=
∑
r
=
1
s
y
r
o
u
r
M
4
min
⁡
θ
o
=
-
δ
∑
j
=
1
j
≠
o
n
n
-
1
∑
r
=
1
s
y
r
j
u
r
-
∑
i
=
1
m
x
i
j
v
i
M
4
s
.
t
.
w
i
j
≥
v
i
θ
_
j
,
i
=
1,2
,
…
,
m
,
j
=
1,2
,
…
,
n
,
M
4
s
.
t
.
u
r
,
v
i
∈
ℵ
2
u
,
v
.It is obvious that model (M4) is a relaxation problem of model (M1) and model (M3) is a relaxation problem of model (M4). Therefore, we have the following lemma.Lemma 3.
Letu
_
r
1 and v
_
i
1 be the optimal solutions of model (M4); then one has
(13)
∑
j
=
1
n
θ
_
j
≤
∑
j
=
1
n
θ
_
j
1
≤
1
,
where θ
_
j
(
1
) is determined by (1).According to Proposition2, if ∑
j
=
1
n
θ
_
j
(
1
)
=
1, then u
_
r
(
1
) and v
_
i
(
1
) are also the optimal solutions of model (M1); otherwise, we must also continue to find the optimal solutions of model (M1).Letθ
_
j
(
1
) replace θ
_
j in model (M4); then solve it to obtain the optimal solutions u
_
r
(
2
) and v
_
i
(
2
). Using (1) we will get the efficiency score θ
_
j
(
2
) of each DMU and then judge whether ∑
j
=
1
n
θ
_
j
(
2
) is equal to 1 or not. The process is repeated until the sum of efficiency scores of all DMUs is approximately equal to 1 within given error.According to the above analysis, the procedure of the proposed approximation algorithm to solve model (M1) can be described as follows.Algorithm 4.
Consider the following.
Step
1. Give permissible error ε
>
0.
Step
2. Solve model (M3) to find the optimal solutions u
_
r and v
_
i, and then calculate the efficiency score θ
_
j of each DMU using (1).
Step
3. Compute the sum of the efficiency scores of all DMUs. If ∑
j
=
1
n
θ
_
j
-
1
≤
ε, then stop with the optimal solutions u
_
r and v
_
i to model (M1); otherwise, k
≔
1; go to Step 4.
Step
4. Solve model (M4) to find optimal solutions u
_
r
(
k
) and v
_
i
(
k
), and then calculate the efficiency score θ
_
j
(
k
) of each DMU using (1).
Step
5. Compute the sum of the efficiency scores of all DMUs. If ∑
j
=
1
n
θ
_
j
(
k
)
-
1
≤
ε, then stop with the optimal solutions u
_
r
(
k
) and v
_
i
(
k
) to model (M1); otherwise go to Step 6.
Step
6. Let θ
_
j
(
k
) replace θ
_
j in model (M4); k
≔
k
+
1; go back to Step 4.Proposition 5.
The optimal solutions of model (M4) will be close to the optimal solutions of model (M1) by using Algorithm4.Proof.
Letu
_
r
(
t
) and v
_
i
(
t
), t
=
1,2
,
…
,
k, be the optimal solutions to model (M4) fortth solution; using (1) we will get the efficiency score θ
_
j
(
t
) of each DMU; then we have
(14)
∑
j
=
1
n
θ
_
j
≤
∑
j
=
1
n
θ
_
j
1
≤
∑
j
=
1
n
θ
_
j
2
≤
⋯
≤
∑
j
=
1
n
θ
_
j
k
≤
1
.
Since{
∑
j
=
1
n
θ
_
j
(
t
)
} increases monotonically and exists upper bound 1, we have ∑
j
=
1
n
θ
-
j
(
t
)
→
1, which means that the optimal solutions u
_
r
(
t
) and v
_
i
(
t
) of model (M4) are close to the optimal solutions of model (M1) by repeatedly solving model (M4).We have discussed the approximation algorithm with minimum problem. For a maximum problem, let constraintw
i
j
≥
v
i
θ
_
j be replaced by constraint w
i
j
≤
v
i
θ
_
j in model (M4). Using the above approximation algorithm, we can also obtain the optimal solutions to model (M1) with maximum form.


## 2.4. A Numerical Example

In order to compare the proposed method with the method in [19], a numerical example used by them is presented in this subsection. There are twelve DMUs with three inputs (X
1, X
2, and X
3) and two outputs (Y
1, Y
2) as shown in Table 1. The minimum and maximum efficiency scores, the integrated score, and the rank of DMUs obtained by the method in [19] are exhibited in the seventh, the eighth, and the ninth column of Table 1, respectively.Table 1
The input-output data and the rankings of DMUs.
DMU
Inputs and outputs
The method in [19]
Proposed improved method
X
1
X
2
X
3
Y
1
Y
2
[
θ
j
min
⁡, θ
j
max
⁡
]
θ
j
Rank
[
θ
j
min
⁡, θ
j
max
⁡
]
θ
j
Rank
1
350
39
9
67
751
[0.0454, 0.0989]
0.0675
11
[0.0573, 0.0862]
0.0702
11
2
298
26
8
73
611
[0.0526, 0.1199]
0.0804
7
[0.0605, 0.0993]
0.0778
7
3
422
31
7
75
584
[0.0483, 0.1076]
0.0728
9
[0.0556, 0.0893]
0.0706
10
4
281
16
9
70
665
[0.0564, 0.1468]
0.0937
4
[0.0577, 0.1371]
0.0930
4
5
301
16
6
75
445
[0.0499, 0.1679]
0.0986
2
[0.0591, 0.1435]
0.0967
3
6
360
29
17
83
1070
[0.0393, 0.1402]
0.0810
6
[0.0393, 0.1249]
0.0774
8
7
540
18
10
72
457
[0.0304, 0.1262]
0.0700
10
[0.0348, 0.1248]
0.0749
9
8
276
33
5
74
590
[0.0549, 0.1358]
0.0883
5
[0.0589, 0.1190]
0.0857
5
9
323
25
5
75
1074
[0.0800, 0.2043]
0.1313
1
[0.0975, 0.1724]
0.1308
1
10
444
64
6
74
1072
[0.0363, 0.1394]
0.0789
8
[0.0363, 0.1394]
0.0822
6
11
323
25
5
25
350
[0.0267, 0.0666]
0.0432
12
[0.0325, 0.0562]
0.0430
12
12
444
64
6
104
1199
[0.0510, 0.1560]
0.0944
3
[0.0510, 0.1560]
0.0977
2In order to rank all DMUs by the proposed method, we first use the proposed approximation Algorithm4 to calculate the minimum and maximum efficiency scores of each DMU and then use (4) and (5) to compute the integrated scores. Finally, the DMUs were ranked according to their integrated scores.Using Algorithm4, the procedure to calculate the minimum efficiency score of DMU1 is described as follows.Step
1. Give permissible error ε
=
0.001.Step
2. Let δ
=
0.0001. Solve model (M3) to find the optimal solutions: u
_
1
=
0
.
000678, u
_
2
=
0, v
_
1
=
0, v
_
2
=
0.013872, and v
_
3
=
0.050998. Using (1) we can obtain that the efficiency scores θ
_
j of twelve DMUs are 0.0454, 0.0644, 0.0646, 0.0697, 0.0964, 0.0444, 0.0643, 0.0704, 0.0845, 0.0420, 0.0282, and 0.0591, respectively.Step
3. Computing the sum of the efficiency scores of all DMUs, we have ∑
j
=
1
12
θ
_
j
=
0.7335. Since ∑
j
=
1
12
θ
_
j
-
1
=
0.2665
>
0.001, we do not obtain the optimal solutions to model (M1), and we have to proceed further to find the optimal solutions. Consider k
≔
1; go to Step 4.Steps
4–
6. Solve model (M4) repeatedly to find the optimal solutions u
_
r
(
k
) and v
_
i
(
k
), and then calculate the efficiency score θ
_
j
(
k
) of each DMU using (1), k
=
1,2
,
…. Computing the sum of the efficiency scores of all DMUs, we have {
∑
j
=
1
12
θ
_
j
(
k
)
}
=
0.9592,0.9945,0.9993
,
…. Since ∑
j
=
1
12
θ
_
j
(
3
)
-
1
=
0.0007
<
0.001, the optimal solutions of model (M1) have been found: u
1
*
=
0.000855, u
2
*
=
0, v
1
*
=
0, v
2
*
=
0.022075, v
3
*
=
0.015451, and the objective function valve is θ
1
*
=
0.0573.Hence, the minimum efficiency value of DMU1 is θ
1
min
⁡
=
0.0573. Similarly, using Algorithm 4, we can also obtain the minimum efficiency values of the rest DMUs. Let constraint w
i
j
≥
v
i
θ
_
j be replaced by constraint w
i
j
≤
v
i
θ
_
j in model (M4). Using Algorithm 4 we can also obtain the maximum efficiency values of all DMUs. The minimum and maximum efficiency scores of each DMU are shown in the tenth column of Table 1.Using (4) and (5), the integrated score θ
j is determined and the results are exhibited in the eleventh column of Table 1. The twelfth column of Table 1 shows the results of rankings according to their integrated scores.Table1 shows that the rankings of DMUs obtained by proposed method are not exactly the same as that obtained by the method in [19]. For instance, DMU6 is ranked as eighth in our method and sixth in the method of [19], whereas DMU10 is placed sixth in our method and eighth in the method of [19].


## 3. Proposed Multiobjective Programming Model Based on Compensatory DEA

### 3.1. The Multiobjective Programming Model for Ranking All Units

Khodabakhshi and Aryavash [19] contribute to a very interesting variant of the basic DEA model by supposing that the sum of efficiency values of all DMUs is equal to unity. Using their method all DMUs can be fully ranked. However, their method does not provide more information about the weight used for calculating the efficiency scores of each DMU. In this section, we aim to develop a new model to find a CSW, which can be used for the full ranking of all DMUs. To do it, here we propose a MOP model as follows:
(15)
M
5
max
⁡
u
1
,
u
2
,
…
,
u
s
,
v
1
,
v
2
,
…
,
v
m
M
5
0
s
.
t
.
∑
r
=
1
s
y
^
r
1
u
r
∑
i
=
1
m
x
^
i
1
v
i
+
∑
r
=
1
s
y
^
r
2
u
r
∑
i
=
1
m
x
^
i
2
v
i
M
5
0
s
.
t
.
+
⋯
+
∑
r
=
1
s
y
^
r
n
u
r
∑
i
=
1
m
x
^
i
n
v
i
=
1
,
M
5
0
s
.
t
.
∑
r
=
1
s
u
r
+
∑
i
=
1
m
v
i
=
1
M
5
0
s
.
t
.
u
r
,
v
i
≥
0
,
∀
r
,
i
,
where v
i (i
=
1,2
…
,
m) and u
r (r
=
1,2
,
…
,
s) are the common weights assigned to ith input and rth output, respectively. x
^
i
j (i
=
1,2
,
…
,
m) and y
^
r
j (r
=
1,2
,
…
,
s) represent normalized input and output values of DMU
j (j
=
1,2
,
…
,
n), respectively. In order to eliminate the impacts of measurement units on a CSW, we normalize all inputs and outputs values by using the following equations [23]:
(16)
x
^
i
j
=
x
i
j
∑
j
=
1
n
x
i
j
,
i
=
1,2
,
…
,
m
,
j
=
1,2
,
…
,
n
,
y
^
r
j
=
y
r
j
∑
j
=
1
n
y
r
j
,
r
=
1,2
,
…
,
s
,
j
=
1,2
,
…
,
n
.In model (M5), in order to avoid a choice of zero common weights and get them as big as possible, the objectives are to simultaneously maximize the common weights assigned to each input and output. The first constraint is supposing that the sum of efficiency values of all DMUs is equal to unity, which can avoid the problem that more than one DMU is evaluated as efficient in DEA. The second constraint∑
r
=
1
s
u
r
+
∑
i
=
1
m
v
i
=
1 is used to avoid arbitrariness in determining common weights. Without it, there will be an infinite number of common weights that can meet the model (M5).


### 3.2. Fuzzy Programming Method for Solving the MOP Model (M5)

Many methods can be used to solve the MOP model, such as compromise programming method, goal programming method, and fuzzy programming method. In this paper, we use fuzzy programming methods to solve the MOP model (M5).In the MOP model (M5), it is unlikely that all objectives simultaneously achieve their optimal solutions subject to the given constraints. So in practice the DM usually chooses a satisfying solution according to the aspiration level fixed for each objective [24, 25]. Let u
r
a, r
=
1,2
,
…
,
s, v
i
a, i
=
1,2
,
…
,
m, be the aspiration level of objectives; then model (M5) can be expressed as follows [24, 26, 27]:
(17)
M
6
Find
u
,
v
M
6
To
satisfy
u
r
≥
~
u
r
a
,
r
=
1,2
,
…
,
s
,
M
6
To
satisfy
v
i
≥
~
v
i
a
,
i
=
1,2
,
…
,
m
,
M
6
To
satisfy
u
r
,
v
i
∈
ℵ
5
u
,
v
.The expressionsu
r
≥
~
u
r
a and v
i
≥
~
v
i
a are represented by a fuzzy set called fuzzy goal, which means that the DM would be satisfied even for objective value slightly less than u
r
a or v
i
a within the value of allowed deviations. The symbols “≥
~” denote the fuzzified versions of “⩾” and they can be read as “approximately greater than or equal to.” The expression (
u
r
,
v
i
)
∈
ℵ
5
(
u
,
v
) is system constraint in model (M5), and ℵ
5
(
u
,
v
) denotes the set of the feasible solution of model (M5).Model (M6) is a fuzzy multiobjective programming (FMOP) model. The procedure for finding the fuzzy efficient solution can be summarized as follows.Algorithm 6.
Consider the following.Step
1 (determine the fuzzy aspiration level of each objective). Since the DM usually knows little about the objectives, this paper uses the ideal solutions of model (M5) to determine the value of the fuzzy aspiration level of each objective. The ideal solutions are just the optimal solutions of the individual objective subject to the system constraint. That is,
(18)
M
7
max
⁡
u
r
00
or
00
max
⁡
v
i
M
7
u
r
,
v
i
∈
ℵ
5
u
,
v
.Model (M7) may not be also considered as a convex optimization problem like model (M1), and it usually has multiple local optimal solutions. Thus model (M7) is more difficult to be solved. In the following, a similar approximation algorithm with Algorithm4 is used.Let(19)
θ
j
=
∑
r
=
1
s
y
^
r
j
u
r
∑
i
=
1
m
x
^
i
j
v
i
,
j
=
1,2
,
…
,
n
;
then model (M7) can be rewritten as follows:
(20)
M
8
max
⁡
u
r
00
or
00
max
⁡
v
i
M
8
0
s
.
t
.
θ
j
∑
i
=
1
m
x
^
i
j
v
i
-
∑
r
=
1
s
y
^
r
j
u
r
=
0
,
j
=
1,2
,
…
,
n
,
M
8
0
s
.
t
.
∑
j
=
1
n
θ
j
=
1
,
M
8
0
s
.
t
.
∑
r
=
1
s
u
r
+
∑
i
=
1
m
v
i
=
1
,
M
8
0
s
.
t
.
u
r
,
v
i
,
θ
j
≥
0
,
∀
i
,
j
,
r
.Using the transformationw
i
j
=
v
i
θ
j [19], model (M8) can be rewritten by the following LP problem:
(21)
M
9
max
⁡
u
r
00
or
00
max
⁡
v
i
M
9
0
s
.
t
.
∑
i
=
1
m
x
^
i
j
w
i
j
-
∑
r
=
1
s
y
^
r
j
u
r
=
0
,
j
=
1,2
,
…
,
n
,
M
9
0
s
.
t
.
∑
j
=
1
n
w
i
j
=
v
i
,
i
=
1,2
,
…
,
m
,
M
9
0
s
.
t
.
∑
r
=
1
s
u
r
+
∑
i
=
1
m
v
i
=
1
,
M
9
0
s
.
t
.
u
r
,
v
i
,
w
i
j
≥
0
,
∀
i
,
j
,
r
.Letu
-
r and v
-
i be the optimal solutions of model (M9). Using (19), it is easy to compute the efficiency score θ
-
j of each DMU. If ∑
j
=
1
n
θ
-
j
=
1, then u
-
r and v
-
i are also the optimal solutions of model (M7); otherwise, that is, ∑
j
=
1
n
θ
-
j
≠
1, we have to proceed further to find the optimal solutions on the basis of the optimal solution of model (M9). In the following, we only discuss the case that ∑
j
=
1
n
θ
-
j
>
1 for the sake of convenient description. However, the procedure can be easily extended to the case that ∑
j
=
1
n
θ
-
j
<
1.In order to find further the optimal solutions of model (M7) when∑
j
=
1
n
θ
-
j
>
1, let constraints w
i
j
=
v
i
θ
j be replaced by constraints w
i
j
≤
v
i
θ
-
j, then add it into model (M9) to formulate the following LP problem:
(22)
M
10
max
⁡
u
r
00
or
00
max
⁡
v
i
M
10
0
s
.
t
.
w
i
j
≤
v
i
θ
-
j
,
i
=
1,2
,
…
,
m
,
j
=
1,2
,
…
,
n
,
M
10
0
s
.
t
.
u
r
,
v
i
∈
ℵ
9
u
,
v
,
where ℵ
9
(
u
,
v
) denotes the set of the feasible solutions to model (M9).Letu
-
r
(
1
) and v
-
i
(
1
) be the optimal solutions of model (M10). Using (19), we will get the efficiency score θ
-
j
(
1
) of each DMU, if ∑
j
=
1
n
θ
-
j
(
1
)
=
1. Then u
-
r
(
1
) and v
-
i
(
1
) are the optimal solutions of model (M7); otherwise, we also have to proceed further to find the optimal solutions on the basis of it.Letθ
-
j
(
1
) replace θ
-
j in model (M10), and then solve it to find the optimal solutions u
-
r
(
2
) and v
-
i
(
2
). Using (19) we will get the efficiency score θ
-
j
(
2
) of each DMU and then judge whether ∑
j
=
1
n
θ
-
j
(
2
) is equal to 1 or not. The process is repeated until that the sum of efficiency scores of all DMUs is approximately equal to 1 within given error. According to Algorithm 4, the optimal solutions of model (M10) will be close to the optimal solutions of model (M7) by repeatedly solving model (M10).We have discussed the approximation algorithm in the case that∑
j
=
1
n
θ
-
j
>
1. If ∑
j
=
1
n
θ
-
j
<
1, then let constraint w
i
j
≤
v
i
θ
-
j be replaced by constraint w
i
j
≥
v
i
θ
-
j in model (M10). According to the above analysis and Algorithm 4, we can also obtain the optimal solutions of model (M7).Step
2 (construct membership functions). In the model (M6), each fuzzy goal is represented by fuzzy sets and defined by membership functions μ
r
(
u
r
) and μ
i
(
v
i
) (r
=
1,2
,
…
,
s
,
i
=
1,2
,
…
,
m). Since linear membership functions are used more than other types of membership functions in the literature [28], we choose the linear membership functions. For each objective, the lower bound is zero, and the upper bound is the fuzzy aspiration level determined by solving model (M7). Hence, the membership functions for each objective may be defined as follows:
(23)
μ
r
u
r
=
0
u
r
≤
0
u
r
u
r
a
0
≤
u
r
≤
u
r
a
1
u
r
≥
u
r
a
,
r
=
1,2
,
…
,
s
μ
i
v
i
=
0
v
i
≤
0
v
i
v
i
a
0
≤
v
i
≤
v
i
a
1
v
i
≥
v
i
a
,
i
=
1,2
,
…
,
m
.
Step
3 (build an auxiliary crisp model). Introducing the auxiliary variable λ, the FMOP model (M6) is transformed into the following auxiliary crisp model [27, 29, 30]:
(24)
M
11
Max
⁡
λ
+
δ
∑
r
=
1
s
u
r
u
r
a
+
∑
i
=
1
m
v
i
v
i
a
M
11
0
s
.
t
.
u
r
≥
u
r
a
λ
,
r
=
1,2
,
…
,
s
,
M
11
0
s
.
t
.
v
i
≥
v
i
a
λ
,
i
=
1,2
,
…
,
m
,
M
11
0
s
.
t
.
u
r
,
v
i
∈
ℵ
5
u
,
v
,
where δ is a sufficiently small positive number. The optimal solution to model (M11) is a fuzzy efficient solution to the FMOP model (M6) and the Pareto optimal solution of MOP model (M5) [24, 29, 30].Step
4 (solve model (M11) to find the optimal CSW). The method for solving model (M11) is similar to that for solving model (M7). Using the approximation algorithm, we solve model (M11) to find the optimal solutions u
r
* (r
=
1,2
,
…
,
s) and v
i
* (i
=
1,2
,
…
,
m), which is also the optimal CSW. Then, using (19) we compute the optimal efficiency scores θ
j
* (j
=
1,2
,
…
,
n) of each DMU, which can be used to fully rank all DMUs. A DMU has a better rank if it has a greater efficiency score.


## 3.1. The Multiobjective Programming Model for Ranking All Units

Khodabakhshi and Aryavash [19] contribute to a very interesting variant of the basic DEA model by supposing that the sum of efficiency values of all DMUs is equal to unity. Using their method all DMUs can be fully ranked. However, their method does not provide more information about the weight used for calculating the efficiency scores of each DMU. In this section, we aim to develop a new model to find a CSW, which can be used for the full ranking of all DMUs. To do it, here we propose a MOP model as follows:
(15)
M
5
max
⁡
u
1
,
u
2
,
…
,
u
s
,
v
1
,
v
2
,
…
,
v
m
M
5
0
s
.
t
.
∑
r
=
1
s
y
^
r
1
u
r
∑
i
=
1
m
x
^
i
1
v
i
+
∑
r
=
1
s
y
^
r
2
u
r
∑
i
=
1
m
x
^
i
2
v
i
M
5
0
s
.
t
.
+
⋯
+
∑
r
=
1
s
y
^
r
n
u
r
∑
i
=
1
m
x
^
i
n
v
i
=
1
,
M
5
0
s
.
t
.
∑
r
=
1
s
u
r
+
∑
i
=
1
m
v
i
=
1
M
5
0
s
.
t
.
u
r
,
v
i
≥
0
,
∀
r
,
i
,
where v
i (i
=
1,2
…
,
m) and u
r (r
=
1,2
,
…
,
s) are the common weights assigned to ith input and rth output, respectively. x
^
i
j (i
=
1,2
,
…
,
m) and y
^
r
j (r
=
1,2
,
…
,
s) represent normalized input and output values of DMU
j (j
=
1,2
,
…
,
n), respectively. In order to eliminate the impacts of measurement units on a CSW, we normalize all inputs and outputs values by using the following equations [23]:
(16)
x
^
i
j
=
x
i
j
∑
j
=
1
n
x
i
j
,
i
=
1,2
,
…
,
m
,
j
=
1,2
,
…
,
n
,
y
^
r
j
=
y
r
j
∑
j
=
1
n
y
r
j
,
r
=
1,2
,
…
,
s
,
j
=
1,2
,
…
,
n
.In model (M5), in order to avoid a choice of zero common weights and get them as big as possible, the objectives are to simultaneously maximize the common weights assigned to each input and output. The first constraint is supposing that the sum of efficiency values of all DMUs is equal to unity, which can avoid the problem that more than one DMU is evaluated as efficient in DEA. The second constraint∑
r
=
1
s
u
r
+
∑
i
=
1
m
v
i
=
1 is used to avoid arbitrariness in determining common weights. Without it, there will be an infinite number of common weights that can meet the model (M5).


## 3.2. Fuzzy Programming Method for Solving the MOP Model (M5)

Many methods can be used to solve the MOP model, such as compromise programming method, goal programming method, and fuzzy programming method. In this paper, we use fuzzy programming methods to solve the MOP model (M5).In the MOP model (M5), it is unlikely that all objectives simultaneously achieve their optimal solutions subject to the given constraints. So in practice the DM usually chooses a satisfying solution according to the aspiration level fixed for each objective [24, 25]. Let u
r
a, r
=
1,2
,
…
,
s, v
i
a, i
=
1,2
,
…
,
m, be the aspiration level of objectives; then model (M5) can be expressed as follows [24, 26, 27]:
(17)
M
6
Find
u
,
v
M
6
To
satisfy
u
r
≥
~
u
r
a
,
r
=
1,2
,
…
,
s
,
M
6
To
satisfy
v
i
≥
~
v
i
a
,
i
=
1,2
,
…
,
m
,
M
6
To
satisfy
u
r
,
v
i
∈
ℵ
5
u
,
v
.The expressionsu
r
≥
~
u
r
a and v
i
≥
~
v
i
a are represented by a fuzzy set called fuzzy goal, which means that the DM would be satisfied even for objective value slightly less than u
r
a or v
i
a within the value of allowed deviations. The symbols “≥
~” denote the fuzzified versions of “⩾” and they can be read as “approximately greater than or equal to.” The expression (
u
r
,
v
i
)
∈
ℵ
5
(
u
,
v
) is system constraint in model (M5), and ℵ
5
(
u
,
v
) denotes the set of the feasible solution of model (M5).Model (M6) is a fuzzy multiobjective programming (FMOP) model. The procedure for finding the fuzzy efficient solution can be summarized as follows.Algorithm 6.
Consider the following.Step
1 (determine the fuzzy aspiration level of each objective). Since the DM usually knows little about the objectives, this paper uses the ideal solutions of model (M5) to determine the value of the fuzzy aspiration level of each objective. The ideal solutions are just the optimal solutions of the individual objective subject to the system constraint. That is,
(18)
M
7
max
⁡
u
r
00
or
00
max
⁡
v
i
M
7
u
r
,
v
i
∈
ℵ
5
u
,
v
.Model (M7) may not be also considered as a convex optimization problem like model (M1), and it usually has multiple local optimal solutions. Thus model (M7) is more difficult to be solved. In the following, a similar approximation algorithm with Algorithm4 is used.Let(19)
θ
j
=
∑
r
=
1
s
y
^
r
j
u
r
∑
i
=
1
m
x
^
i
j
v
i
,
j
=
1,2
,
…
,
n
;
then model (M7) can be rewritten as follows:
(20)
M
8
max
⁡
u
r
00
or
00
max
⁡
v
i
M
8
0
s
.
t
.
θ
j
∑
i
=
1
m
x
^
i
j
v
i
-
∑
r
=
1
s
y
^
r
j
u
r
=
0
,
j
=
1,2
,
…
,
n
,
M
8
0
s
.
t
.
∑
j
=
1
n
θ
j
=
1
,
M
8
0
s
.
t
.
∑
r
=
1
s
u
r
+
∑
i
=
1
m
v
i
=
1
,
M
8
0
s
.
t
.
u
r
,
v
i
,
θ
j
≥
0
,
∀
i
,
j
,
r
.Using the transformationw
i
j
=
v
i
θ
j [19], model (M8) can be rewritten by the following LP problem:
(21)
M
9
max
⁡
u
r
00
or
00
max
⁡
v
i
M
9
0
s
.
t
.
∑
i
=
1
m
x
^
i
j
w
i
j
-
∑
r
=
1
s
y
^
r
j
u
r
=
0
,
j
=
1,2
,
…
,
n
,
M
9
0
s
.
t
.
∑
j
=
1
n
w
i
j
=
v
i
,
i
=
1,2
,
…
,
m
,
M
9
0
s
.
t
.
∑
r
=
1
s
u
r
+
∑
i
=
1
m
v
i
=
1
,
M
9
0
s
.
t
.
u
r
,
v
i
,
w
i
j
≥
0
,
∀
i
,
j
,
r
.Letu
-
r and v
-
i be the optimal solutions of model (M9). Using (19), it is easy to compute the efficiency score θ
-
j of each DMU. If ∑
j
=
1
n
θ
-
j
=
1, then u
-
r and v
-
i are also the optimal solutions of model (M7); otherwise, that is, ∑
j
=
1
n
θ
-
j
≠
1, we have to proceed further to find the optimal solutions on the basis of the optimal solution of model (M9). In the following, we only discuss the case that ∑
j
=
1
n
θ
-
j
>
1 for the sake of convenient description. However, the procedure can be easily extended to the case that ∑
j
=
1
n
θ
-
j
<
1.In order to find further the optimal solutions of model (M7) when∑
j
=
1
n
θ
-
j
>
1, let constraints w
i
j
=
v
i
θ
j be replaced by constraints w
i
j
≤
v
i
θ
-
j, then add it into model (M9) to formulate the following LP problem:
(22)
M
10
max
⁡
u
r
00
or
00
max
⁡
v
i
M
10
0
s
.
t
.
w
i
j
≤
v
i
θ
-
j
,
i
=
1,2
,
…
,
m
,
j
=
1,2
,
…
,
n
,
M
10
0
s
.
t
.
u
r
,
v
i
∈
ℵ
9
u
,
v
,
where ℵ
9
(
u
,
v
) denotes the set of the feasible solutions to model (M9).Letu
-
r
(
1
) and v
-
i
(
1
) be the optimal solutions of model (M10). Using (19), we will get the efficiency score θ
-
j
(
1
) of each DMU, if ∑
j
=
1
n
θ
-
j
(
1
)
=
1. Then u
-
r
(
1
) and v
-
i
(
1
) are the optimal solutions of model (M7); otherwise, we also have to proceed further to find the optimal solutions on the basis of it.Letθ
-
j
(
1
) replace θ
-
j in model (M10), and then solve it to find the optimal solutions u
-
r
(
2
) and v
-
i
(
2
). Using (19) we will get the efficiency score θ
-
j
(
2
) of each DMU and then judge whether ∑
j
=
1
n
θ
-
j
(
2
) is equal to 1 or not. The process is repeated until that the sum of efficiency scores of all DMUs is approximately equal to 1 within given error. According to Algorithm 4, the optimal solutions of model (M10) will be close to the optimal solutions of model (M7) by repeatedly solving model (M10).We have discussed the approximation algorithm in the case that∑
j
=
1
n
θ
-
j
>
1. If ∑
j
=
1
n
θ
-
j
<
1, then let constraint w
i
j
≤
v
i
θ
-
j be replaced by constraint w
i
j
≥
v
i
θ
-
j in model (M10). According to the above analysis and Algorithm 4, we can also obtain the optimal solutions of model (M7).Step
2 (construct membership functions). In the model (M6), each fuzzy goal is represented by fuzzy sets and defined by membership functions μ
r
(
u
r
) and μ
i
(
v
i
) (r
=
1,2
,
…
,
s
,
i
=
1,2
,
…
,
m). Since linear membership functions are used more than other types of membership functions in the literature [28], we choose the linear membership functions. For each objective, the lower bound is zero, and the upper bound is the fuzzy aspiration level determined by solving model (M7). Hence, the membership functions for each objective may be defined as follows:
(23)
μ
r
u
r
=
0
u
r
≤
0
u
r
u
r
a
0
≤
u
r
≤
u
r
a
1
u
r
≥
u
r
a
,
r
=
1,2
,
…
,
s
μ
i
v
i
=
0
v
i
≤
0
v
i
v
i
a
0
≤
v
i
≤
v
i
a
1
v
i
≥
v
i
a
,
i
=
1,2
,
…
,
m
.
Step
3 (build an auxiliary crisp model). Introducing the auxiliary variable λ, the FMOP model (M6) is transformed into the following auxiliary crisp model [27, 29, 30]:
(24)
M
11
Max
⁡
λ
+
δ
∑
r
=
1
s
u
r
u
r
a
+
∑
i
=
1
m
v
i
v
i
a
M
11
0
s
.
t
.
u
r
≥
u
r
a
λ
,
r
=
1,2
,
…
,
s
,
M
11
0
s
.
t
.
v
i
≥
v
i
a
λ
,
i
=
1,2
,
…
,
m
,
M
11
0
s
.
t
.
u
r
,
v
i
∈
ℵ
5
u
,
v
,
where δ is a sufficiently small positive number. The optimal solution to model (M11) is a fuzzy efficient solution to the FMOP model (M6) and the Pareto optimal solution of MOP model (M5) [24, 29, 30].Step
4 (solve model (M11) to find the optimal CSW). The method for solving model (M11) is similar to that for solving model (M7). Using the approximation algorithm, we solve model (M11) to find the optimal solutions u
r
* (r
=
1,2
,
…
,
s) and v
i
* (i
=
1,2
,
…
,
m), which is also the optimal CSW. Then, using (19) we compute the optimal efficiency scores θ
j
* (j
=
1,2
,
…
,
n) of each DMU, which can be used to fully rank all DMUs. A DMU has a better rank if it has a greater efficiency score.


## 4. Numerical Examples

In this section, we provide two numerical examples to illustrate our method and then compare the results with the results of the existing models to show the potential usage of the proposed method in the full ranking of DMUs.Example 1.
Consider the numerical example used in Section2.4. There are twelve DMUs with three inputs (X
1, X
2, and X
3) and two outputs (Y
1, Y
2) as shown in Table 1. In order to rank all DMUs by using the proposed method, we first normalize all inputs and outputs values by using (16); the normalized values are shown in Table 2. The integrated score and the rank of DMUs obtained by the method in [19] are exhibited again in the seventh and the eighth column of Table 2, respectively.Table 2
The normalized input-output data and the rankings of DMUs.
DMU
The normalized values of inputs and outputs
The method in [19]
Proposed method
X
1
X
2
X
3
Y
1
Y
2
θ
j
Rank
θ
j
Rank
1
0.0802
0.1010
0.0968
0.0773
0.0847
0.0675
11
0.0703
10
2
0.0683
0.0674
0.0860
0.0842
0.0689
0.0804
7
0.0874
6
3
0.0967
0.0803
0.0753
0.0865
0.0659
0.0728
9
0.0774
7
4
0.0644
0.0415
0.0968
0.0807
0.0750
0.0937
4
0.0954
4
5
0.0690
0.0415
0.0645
0.0865
0.0502
0.0986
2
0.1042
2
6
0.0825
0.0751
0.1828
0.0957
0.1207
0.0810
6
0.0750
8
7
0.1238
0.0466
0.1075
0.0830
0.0515
0.0700
10
0.0640
11
8
0.0633
0.0855
0.0538
0.0854
0.0665
0.0883
5
0.0956
3
9
0.0740
0.0648
0.0538
0.0865
0.1211
0.1313
1
0.1250
1
10
0.1018
0.1658
0.0645
0.0854
0.1209
0.0789
8
0.0718
9
11
0.0740
0.0648
0.0538
0.0288
0.0395
0.0432
12
0.0413
12
12
0.1018
0.1658
0.0645
0.1200
0.1352
0.0944
3
0.0923
5Using the normalized values we can build a MOP model (M5), which is transformed into FMOP model (M6). We solve model (M6) to find the fuzzy efficient solution, which is also the Pareto optimal solution of MOP model (M5), using Algorithm6.The procedure to solve the model (M6) can be described as follows.Step
1 (determine the fuzzy aspiration level of each objective by solving model (M7)). The algorithm for solving model (M7) is similar to the approximation algorithm, Algorithm 4.Give permissible errorε
=
0.001. For the first objective, max
⁡
⁡
u
1, solve model (M9) to find optimal solutions: u
-
1
=
0.09333, u
-
2
=
0, v
-
1
=
0.40844, v
-
2
=
0.28013, and v
-
3
=
0.21810. Calculate the efficiencies score θ
-
j of each DMU using (19) and their sum; we have ∑
j
=
1
12
θ
-
j
=
1.2678.Since∑
j
=
1
12
θ
-
j
-
1
=
0.2678
>
ε, we do not obtain the optimal solutions to model (M7). Thus we have to proceed calculating further to find the optimal solutions. We solve model (M10) repeatedly, to find optimal solutions u
-
r
(
k
) and v
-
i
(
k
), k
=
1,2
,
…, until ∑
j
=
1
n
θ
-
j
(
k
)
-
1
≤
ε. After repeating 4 times for solving model (M10), we have ∑
j
=
1
n
θ
-
j
(
4
)
-
1
=
0.00079
<
ε, which means that the optimal solutions have been found: u
1
*
=
0.07516, u
2
*
=
0, v
1
*
=
0.32684, v
2
*
=
0.27280, and v
3
*
=
0.32521. And the objective function valve is u
1
*
=
0.07516. Therefore the fuzzy aspiration level of the first objective is u
1
a
=
0.07516.Similarly, we can also solve model (M7) to obtain the fuzzy aspiration levels of the rest of the objectives:u
2
a
=
0.07673, v
1
a
=
0.92510, v
2
a
=
0.93421, and v
3
a
=
0.93073.Step
2 (construct membership functions). The mathematical expression of membership functions is given in (23). For each objective, the lower bound is zero, and the upper bound is the fuzzy aspiration level determined by solving model (M7). According to Step 1, we have u
1
a
=
0.07516, u
2
a
=
0.07673, v
1
a
=
0.92510, v
2
a
=
0.93421, and v
3
a
=
0.93073.Step
3 (build an auxiliary crisp model (M11)). Having determining the fuzzy aspiration level of each objective and the form of membership functions, we can get the auxiliary crisp model (M11) by introducing the auxiliary variable λ.Step
4 (solve model (M11) to obtain the optimal CSW). The method for solving model (M11) is similar to that of model (M7). Set δ
=
0.01; we solve model (M11) to obtain the optimal solutions λ
*
=
0.33133, u
1
*
=
0.05014, u
2
*
=
0.02542, v
1
*
=
0.30652, v
2
*
=
0.30954, and v
3
*
=
0.30838. They are the fuzzy efficient solutions to model (M6) and the Pareto optimal solutions to model (M5). Therefore we have obtained the optimal CSW: u
1
*
=
0.05014, u
2
*
=
0.02542, v
1
*
=
0.30652, v
2
*
=
0.30954, and v
3
*
=
0.30838.Using (19) we can compute the optimal efficiency score θ
j
* of each DMU, and the results are shown in the ninth column of Table 2. The tenth column of Table 2 shows the results of ranking by our method. It shows that all DMUs are fully ranked with their optimal efficiency scores θ
j
*.Table2 shows that the rankings of DMUs using the proposed method are not entirely consistent with that of the method in [19]. The main advantage of our method compared to the method in [19] is that a CSW for fully ranking DMUs is derived and all DMUs may be able to be compared and ranked on the same basis. Moreover, we improve their solution method and propose an approximation algorithm to solve their DEA model.Example 2.
Measure the performance of nations participating in the Olympic Games. Consider the example studied by Zhang et al. [20] and Azizi and Wang [21], in which 73 countries or areas are evaluated in terms of two inputs and three outputs defined as follows:X
1: gross domestic product (GDP),
X
2: total population of the country or area,
Y
1: number of gold medals won by the country,
Y
2: number of silver medals won by the country,
Y
3: number of bronze medals won by the country.The input and output data of the Athens 2004 Summer Olympic Games [20, 21] are presented in Table 3. We use this example to compare the proposed method with Azizi and Wang’s method [21] and Ramezani-Tarkhorani et al.’s method [22].Table 3
Data of the Athens 2004 Summer Olympic Games [20, 21].
Country or area (DMU)
Inputs
Outputs
GDP (billion $)
Population (thousands)
Gold
Silver
Bronze
Argentina
151.94
38372
2
0
4
Australia
617.61
19942
17
16
16
Austria
289.72
8171
2
4
1
Azerbaijan
8.54
8355
1
0
4
Bahamas
5.5
319
1
0
1
Belarus
22.75
9811
2
6
7
Belgium
352
10400
1
0
2
Brazil
599.73
183913
5
2
3
Britain
2125.51
59479
9
9
12
Bulgaria
23.91
7780
2
1
9
Cameroon
14.43
16038
1
0
0
Canada
995.83
31958
3
6
3
Chile
93.65
16124
2
0
1
China
1649.39
1307989
32
17
14
China, Hong Kong
164.55
6963
0
1
0
Chinese Taipei
305.2
22689
2
2
1
Colombia
95.19
44915
0
0
2
Croatia
33.2
4540
1
2
2
Cuba
44.54
11245
9
7
11
Czech Republic
107.05
10229
1
3
4
Denmark
242.34
5414
2
0
6
Dominican Republic
19.44
8768
1
0
0
Egypt
77.03
72642
1
1
3
Eritrea
0.62
4232
0
0
1
Estonia
11.2
1335
0
1
2
Ethiopia
8.21
75600
2
3
2
Finland
186.18
5235
0
2
0
France
2018.08
60257
11
9
13
Georgia
4.45
4518
2
2
0
Germany
2706.67
82645
13
16
20
Greece
205.49
11098
6
6
4
Hungary
99.35
10124
8
6
3
India
661.05
1087124
0
1
0
Indonesia
257.87
220077
1
1
2
Iran
168.97
68803
2
2
2
Israel
116.34
6601
1
0
1
Italy
1680.69
58033
10
11
11
Jamaica
8.71
2639
2
1
2
Japan
4668.42
127924
16
9
12
Kazakhstan
40.75
14839
1
4
3
Kenya
15.62
33467
1
4
2
Korea, Republic
681.47
47645
9
12
9
Latvia
13.66
2318
0
4
0
Lithuania
22.17
3443
1
2
0
Mexico
676.5
105699
0
3
1
Mongolia
1.29
2614
0
0
1
Morocco
49.82
31020
2
1
0
Netherlands
577.98
16226
4
9
9
New Zealand
96.97
3989
3
2
0
Nigeria
71.33
128709
0
0
2
Norway
250.44
4598
5
0
1
Paraguay
7
6017
0
1
0
Poland
241.77
38559
3
2
5
Portugal
167.24
10441
0
2
1
Romania
71.32
21790
8
5
6
Russia
582.73
143899
27
27
38
Serbia and Montenegro
24.13
10510
0
2
0
Slovakia
41.09
5401
2
2
2
Slovenia
32.79
1967
0
1
3
South Africa
212.9
47208
1
3
2
Spain
992.99
42646
3
11
5
Sweden
346.53
9008
4
2
1
Switzerland
358
7240
1
1
3
Syria
23.74
18582
0
0
1
Thailand
163.49
63694
3
1
4
Trinidad and Tobago
12.54
1301
0
0
1
Turkey
300.09
72220
3
3
4
Ukraine
65.04
46989
9
5
9
United Arab Emirates
95.72
4284
1
0
0
United States
11733.47
295410
36
39
27
Uzbekistan
9.72
26209
2
1
2
Venezuela
107.49
26282
0
0
2
Zimbabwe
5.82
12936
1
1
1In Azizi and Wang’s method [21], a pair of bounded DEA models was proposed to measure the interval efficiencies of DMUs. The lower bound of the interval efficiency is called the worst relative efficiency or pessimistic efficiency, and its value is determined using improved pessimistic DEA model [21]. The upper bound of the interval efficiency is called the best relative efficiency or the optimistic efficiency, and its value is determined using the conventional DEA model. Using the traditional DEA model, the DEA efficiency or the optimistic efficiency of each participating country at the Athens 2004 Olympic Games is determined, and the results are shown in the second column of Table 4. The efficiency intervals of the 73 DMUs according to Azizi and Wang’s method are presented in the third column of Table 4. The fourth column of Table 4 shows the rankings based on the efficiency intervals [21].Table 4
Efficiencies of the countries participating in the Athens 2004 Olympic Games.
Country or area (DMU)
DEA efficiencies
The method in [21]
The method in [22]
Proposed method
Efficiency intervals
Rank
Efficiencies
Rank
Efficiencies
Rank
Argentina
0.08978
[0.00004, 0.08978]
52
0.03364
55
0.00429
44
Australia
1.00000
[0.00023, 1.00000]
26
0.70622
5
0.02271
15
Austria
0.42598
[0.00010, 0.42598]
34
0.31406
17
0.00746
33
Azerbaijan
1.00000
[0.00030, 1.00000]
23
0.15484
32
0.02075
16
Bahamas
1.00000
[0.00039, 1.00000]
22
1.00000
1
0.08595
1
Belarus
1.00000
[0.00212, 1.00000]
2
0.58614
7
0.05144
4
Belgium
0.11826
[0.00001, 0.11826]
67
0.06027
50
0.00222
63
Brazil
0.03770
[0.00004, 0.03770]
54
0.01161
66
0.00180
65
Britain
0.22016
[0.00004, 0.22016]
51
0.14832
34
0.00405
51
Bulgaria
1.00000
[0.00068, 1.00000]
12
0.44803
10
0.04399
6
Cameroon
0.15419
[0.00001, 0.15419]
65
0.00006
73
0.00295
59
Canada
0.17040
[0.00005, 0.17040]
47
0.13567
36
0.00351
55
Chile
0.10916
[0.00003, 0.10916]
58
0.02007
61
0.00501
41
China
1.00000
[0.00010, 1.00000]
39
0.01105
67
0.00200
64
China, Hong Kong
0.08323
[0.00004, 0.08323]
53
0.08178
43
0.00177
67
Chinese Taipei
0.10304
[0.00005, 0.10304]
48
0.06486
49
0.00388
53
Columbia
0.05115
[0.00001, 0.05115]
71
0.01437
65
0.00125
69
Croatia
0.47307
[0.00050, 0.47307]
17
0.39662
13
0.02538
12
Cuba
1.00000
[0.00155, 1.00000]
4
0.67743
6
0.07166
2
Czech Republic
0.38991
[0.00023, 0.38991]
27
0.29448
18
0.01436
22
Denmark
0.93665
[0.00003, 0.93665]
56
0.34351
14
0.00872
29
Dominican Republic
0.17635
[0.00002, 0.17635]
62
0.00011
72
0.00468
42
Egypt
0.09781
[0.00014, 0.09781]
31
0.02138
60
0.00255
61
Eritrea
1.00000
[0.00004, 1.00000]
49
0.07643
44
0.00832
30
Estonia
1.00000
[0.00069, 1.00000]
11
0.91362
3
0.04368
7
Ethiopia
1.00000
[0.00045, 1.00000]
19
0.03171
56
0.00427
45
Finland
0.22139
[0.00007, 0.22139]
43
0.21505
24
0.00340
56
France
0.24442
[0.00004, 0.24442]
50
0.15195
33
0.00465
43
Georgia
1.00000
[0.00340, 1.00000]
1
0.25823
20
0.04032
8
Germany
0.51400
[0.00005, 0.51400]
44
0.18523
28
0.00508
40
Greece
0.68020
[0.00025, 0.68020]
25
0.42443
12
0.02014
17
Hungary
0.95138
[0.00052, 0.95138]
16
0.43782
11
0.03607
9
India
0.00337
[0.00001, 0.00337]
73
0.00054
70
0.00004
73
Indonesia
0.02180
[0.00004, 0.02180]
55
0.00559
68
0.00069
71
Iran
0.05368
[0.00010, 0.05368]
35
0.02631
58
0.00306
58
Israel
0.04833
[0.00002, 0.04833]
63
0.04831
52
0.00409
49
Italy
0.23392
[0.00006, 0.23392]
46
0.16714
30
0.00534
38
Jamaica
1.00000
[0.00127, 1.00000]
6
0.46528
9
0.06032
3
Japan
0.14589
[0.00002, 0.14589]
66
0.06896
47
0.00235
62
Kazakhstan
0.35502
[0.00075, 0.35502]
10
0.22199
23
0.01790
20
Kenya
0.85474
[0.00134, 0.85474]
5
0.08899
42
0.00914
28
Korea, Republic
0.47165
[0.00015, 0.47165]
30
0.20518
25
0.01028
26
Latvia
1.00000
[0.00194, 1.00000]
3
1.00000
2
0.04885
5
Lithuania
0.49341
[0.00064, 0.49341]
14
0.33673
15
0.02417
14
Mexico
0.02075
[0.00003, 0.02075]
59
0.01948
62
0.00096
70
Mongolia
1.00000
[0.00007, 1.00000]
42
0.12370
38
0.01290
23
Morocco
0.12119
[0.00017, 0.12119]
29
0.01883
63
0.00415
47
Netherlands
0.67746
[0.00013, 0.67746]
32
0.48570
8
0.01079
25
New Zealand
0.71636
[0.00017, 0.71636]
28
0.28606
19
0.01560
21
Nigeria
0.04570
[0.00001, 0.04570]
72
0.00502
69
0.00052
72
Norway
1.00000
[0.00002, 1.00000]
60
0.06775
48
0.00809
32
Paraguay
0.33965
[0.00094, 0.33965]
7
0.09677
41
0.00721
35
Poland
0.11449
[0.00009, 0.11449]
36
0.07181
45
0.00626
37
Portugal
0.14606
[0.00008, 0.14606]
41
0.14039
35
0.00423
46
Romania
0.54421
[0.00068, 0.54421]
13
0.22247
22
0.02859
10
Russia
1.00000
[0.00043, 1.00000]
20
0.19420
27
0.01886
19
Serbia and Montenegro
0.24230
[0.00055, 0.24230]
15
0.11067
39
0.00732
34
Slovakia
0.44929
[0.00043, 0.44929]
21
0.33348
16
0.02585
11
Slovenia
1.00000
[0.00025, 1.00000]
25
0.77679
4
0.02535
13
South Africa
0.06503
[0.00011, 0.06503]
33
0.05054
51
0.00367
54
Spain
0.32958
[0.00008, 0.32958]
40
0.18407
29
0.00518
39
Sweden
0.42647
[0.00005, 0.42647]
45
0.15964
31
0.00643
36
Switzerland
0.29930
[0.00003, 0.29930]
57
0.20458
26
0.00392
52
Syria
0.08710
[0.00001, 0.08710]
68
0.01739
64
0.00166
68
Thailand
0.09048
[0.00009, 0.09048]
37
0.02944
57
0.00415
48
Trinidad and Tobago
0.37071
[0.00002, 0.37071]
61
0.24628
21
0.01227
24
Turkey
0.05986
[0.00009, 0.05986]
38
0.04200
54
0.00407
50
Ukraine
0.68481
[0.00082, 0.68481]
9
0.12401
37
0.01891
18
United Arabic Emirates
0.07446
[0.00001, 0.07446]
69
0.00023
71
0.00316
57
United States
1.00000
[0.00003, 1.00000]
64
0.10257
40
0.00265
60
Uzbekistan
0.82038
[0.00048, 0.82038]
18
0.04699
53
0.00827
31
Venezuela
0.05859
[0.00001, 0.05859]
70
0.02452
59
0.00179
66
Zimbabwe
0.59653
[0.00090, 0.59653]
8
0.07011
46
0.01012
27Liu and Peng [13] proposed a common weights analysis methodology to generate a CSW for the performance indices of only DEA efficient DMUs. All DMUs are then ranked according to the efficiency scores weighted by the CSW. Ramezani-Tarkhorani et al. [22] pointed out the problem and proposed a new approach to rank all DMUs with common weights. From the second column of Table 4, it can be seen that, among 73 DMUs, 18 DMUs are DEA efficient. Using the data of these efficient DMUs, the CSW is determined as u
1
=
0.01, u
2
=
5.829, u
3
=
3.235, v
1
=
0.01, and v
2
=
0.01. The efficiency scores of DMUs can then be calculated using the CSW and the results are shown in the fifth column of Table 4, and corresponding rankings are shown in the sixth column of Table 4.In order to rank all DMUs by using the proposed method, we first normalize all inputs and outputs values by using (16) and then build a MOP model (M5), which is transformed into FMOP model (M6). We solve model (M6) to find the fuzzy efficient solution, which is also the Pareto optimal solution of MOP model (M5), using Algorithm 6. The procedure to solve model (M6) can be described as follows.Step
1 (determine the fuzzy aspiration level of each objective by solving model (M7)). The algorithm for solving model (M7) is similar to the approximation algorithm, Algorithm 4. Give permissible error ε
=
0.01, for each objective, max
⁡
⁡
u
r or max
⁡
⁡
v
i, solve model (M7), respectively, to obtain the fuzzy aspiration levels of all objectives: u
1
a
=
0.00474, u
2
a
=
0.00442, u
3
a
=
0.00349, v
1
a
=
0.99867, and v
2
a
=
0.99688.Step
2 (construct membership functions). The mathematical expression of membership functions are given in (23). For each objective, the lower bound is zero, and the upper bound is the fuzzy aspiration level determined by solving model (M7) in Step 1.Step
3 (build an auxiliary crisp model (M11)). Having determining the fuzzy aspiration level of each objective and the form of membership functions, we can get the auxiliary crisp model (M11) by introducing the auxiliary variable λ.Step
4 (solve model (M11) to obtain the optimal CSW). The method for solving model (M11) is similar to that of model (M7). Set δ
=
0.01, we solve model (M11) to obtain the optimal solutions: λ
*
=
0.33132, u
1
*
=
0.00157, u
2
*
=
0.00146, u
3
*
=
0.00116, v
1
*
=
0.49818, and v
2
*
=
0.49763. They are the fuzzy efficient solutions to model (M6) and the Pareto optimal solutions to model (M5). Therefore we have obtained the optimal CSW: u
1
*
=
0.00157, u
2
*
=
0.00146, u
3
*
=
0.00116, v
1
*
=
0.49818, and v
2
*
=
0.49763. The efficiency scores of DMUs can then be calculated using the CSW and the results are shown in the seventh column of Table 4, and corresponding rankings are shown in the eighth column of Table 4.Table4 shows that the rankings of DMUs using three methods are not entirely consistent with each other. However, Spearman’s rank correlation coefficients for the proposed model and Azizi and Wang’s model [21] and Ramezani-Tarkhorani et al.’s model [22] are calculated as r
s
=
0.7929 and 0.8344, respectively. For the α
=
0.05 significant level, all correlation coefficients are statistically significant. This result shows that the rank that is determined by the proposed model has the same direction as that of Azizi and Wang’s model [21] and Ramezani-Tarkhorani et al.’s model [22].The main features of the proposed method compared to Azizi and Wang’s method [21] and Ramezani-Tarkhorani et al.’s method [22] are summarized as follows.(1)
In Azizi and Wang’s method [21], a pair of efficiencies, the pessimistic efficiency and the optimistic efficiency, is used to rank all DMUs; in the proposed method, the neutral efficiency, not the pessimistic efficiency or the optimistic efficiency, is used to rank all DMUs.
(2)
In Azizi and Wang’s method [21], for each DMU, the optimal weights for calculating the pessimistic efficiency and the optimistic efficiency are obtained by solving a LP problem, respectively, and the efficiencies of DMUs may be unable to be ranked on the same basis; in the proposed method, a MOP model is proposed to derive a CSW, and then all DMUs can be ranked on the same basis.
(3)
In Ramezani-Tarkhorani et al.’ model [22], only the date set of DEA efficient DMUs is used to determine a CSW, and all DMUs are then ranked according to the efficiency scores weighted by the CSW; in the proposed method, the date set of all DMUs is used to determine a CSW, and then all DMUs can be ranked with complete information of all DMUs.
(4)
In Ramezani-Tarkhorani et al.’ model [22], more than one DMU is usually evaluated as efficient DMUs, and thus new rules may be needed to rank the DMUs; in the proposed method, the efficiencies of all DMU have compensatory features, and thus all DMUs can be fully discriminated.


## 5. Conclusions

In this paper, we have developed a MOP model based on compensatory DEA model to derive a CSW for fully ranking DMUs. There are four features of the proposed ranking method as follows.(1)
In this paper, an approximation algorithm based on LP model is proposed to solve nonlinear programming models (M1), (M7), and (M11), which are not the convex optimization problem. It is desirable for the reason that optimal solution for the LP problem is global in contrast to the nonlinear one.
(2)
We suppose that the sum of efficiency values of all DMUs is equal to unity in the MOP model. It has compensatory feature and can avoid the problem that more than one DMU is evaluated as efficient. Hence, using the proposed approach one can get a full ranking of all DMUs.
(3)
In order to avoid a choice of zero common weights, the objectives of the proposed model are to simultaneously maximize all common weights assigned to each input and output. So we can get a set of nonzero common weights as big as possible.
(4)
In this paper a CSW is used for calculating the efficiency score of each DMU. Hence, all DMUs may be able to be compared and ranked on the same basis using the efficiency scores.In future research, we will develop a new MOP model for fully ranking DMUs by considering new objectives or some weights constraints.



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*Source: 807843-2014-11-05.xml*