# Gap Minimization for Peer-Evaluation in DEA Cross-Efficiency

Authors: Anrong Yang; Zigang Zhang; Yishi Zhang; Dunliang Chen
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

Cross-efficiency evaluation is an effective and widely used method for ranking decision making units (DMUs) in data envelopment analysis (DEA). Gap minimization criterion is introduced in aggressive and benevolent cross-efficiency methods to avoid possible extreme efficiency from peer-evaluation and to get equitable results. On the basis of this criterion, a weighted cross-efficiency method with similarity distance that, respectively, considers the aggressive and the benevolent formulations is proposed to determine cross-efficiency. The weights of the cross-evaluation determined by this method are positively influenced by self-evaluation and thus are propitious to resolving conflict. Numerical demonstration reveals the feasibility of the proposed method.


---

## Body

## 1. Introduction

Data envelopment analysis (DEA) is an effective and widely used nonparametric method for assessing the relative efficiency of a set of decision making units (DMUs) with structures of multiple inputs and outputs. Using the traditional DEA models like CCR model proposed by Charnes et al. [1] and BCC model by Banker et al. [2], DMUs can be divided into efficient and inefficient groups. DEA is a typical application of linear programming and has already been widely used for solving many real world problems [3–6]. While in practice, there is often a need to discriminate efficient DMUs. Besides, DEA may result in unrealistic weight distribution, thus leading to the efficiency results being not equitable and acceptable in some cases. A series of approaches have been proposed to rank DMUs and to overcome the shortcomings of traditional DEA methods [7–17].Cross-efficiency method originated by Sexton et al. [18] extends the basic DEA methods and thus can rank DMUs according to their efficiency scores. In addition, it can avoid unrealistic DEA weighting schemes by incorporating additional weight restrictions into DEA. The core idea of the cross-efficiency is to evaluate each DMU according to both its and peers’ optimal input and output weights. The efficiency scores of a DMU will thus be calculated n times and the average score of self- and peer-evaluations will be chosen as its ranking score. However, this may fall into a dilemma of multiple solutions [19]. Doyle and Green [20] propose an aggressive cross-efficiency method and a benevolent one to avoid the problem. They introduce secondary objective functions to select the optimal weights minimizing and maximizing the sum of the outputs of other DMUs, respectively. Wang and Chin [16] propose a neutral cross-efficiency method to reduce the number of zero weights of the outputs. The cross-efficiency scores are determined by the profiles of weights that are used. To effectively select weights from multiple optimal solutions, goal programming formulations are considered to be an effective method in recent literature (e.g., SWAT proposed by Jahanshahloo et al. [21] and the novel secondary objective function proposed by Hosseinzadeh-Lotfi et al. [22]). Many alternative secondary goals for the purpose of cross-efficiency are proposed in the literature, where minimizing the summation of deviations from the ideal point and minimizing the maximum quantity among all deviation variables are typically applied in [23, 24]. In addition, multicriteria method proposed by Li and Reeves [23] and its extensions like preemptive programming formulation by Bal and Örkcü [8] and weighted programming formulation by [9] have the ability to select weights with realistic weight distribution. Örkcü and Bal [14] refine goal programming models in the cross-efficiency evaluation. Recently, Wang et al. [17] propose a simultaneously input- and output-oriented weight determination cross-efficiency model for reducing the number of zero weights of the inputs.In this paper, we mainly focus on the promotion of the acceptance of the cross-efficiency ranking results for each DMU. We introduce a regularization term to minimize the gap between the upper and lower bounds of the peer-evaluation scores. In addition, we evaluate DMUs in a more neutral manner both considering the pessimistic and optimistic attitudes from the aggressive and benevolent methods and propose a weighted neutral cross-efficiency method to determine the cross-efficiency scores according to the similarity distances between DMUs. Our method has a greater ability to discriminate DEA-efficient and DEA-inefficient DMUs compared to the extant cross-efficiency methods. In addition, the gap minimization criterion and the manner of determining the weights for each cross-efficiency score in our method are propitious to resolving conflict among DMUs during cross-evaluation process.The rest of this paper is organized as follows. Section2 briefly introduces the aggressive and benevolent cross-efficiency models. Section 3 proposes a secondary goal to minimize the gap between the upper and lower bounds of the peer-evaluation scores and a new weighted method for neutral DEA cross-efficiency evaluation based on the similarity distance between pairs of DMUs. Then Section 4 demonstrates the feasibility of proposed method with a numerical example. Section 5 finally summarizes the concluding remarks.


## 2. Aggressive and Benevolent Cross-Efficiency Models

Let there ben DMUs where DMU
j
(
j
=
1,2
,
…
,
n
) uses m inputs x
i
j
(
i
=
1,2
,
…
,
m
) to produce s outputs y
r
j
(
r
=
1,2
,
…
,
s
). And suppose that all input and output indices are nonnegative constant numbers. Then for a given DMU
p
(
1
≤
p
≤
n
), the relative efficiency score under the CCR model is determined by solving the following linear programming [1]:
(1)
max
⁡
θ
p
p
:
=
∑
r
=
1
s
‍
u
r
p
y
r
p
s
.
t
.
∑
i
=
1
m
‍
v
i
p
x
i
p
=
1
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
u
r
p
≥
0
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
where v
i
p
(
i
=
1,2
,
…
,
m
) and u
r
p
(
r
=
1,2
,
…
,
s
) are the weights assigned to ith input and rth output, respectively. Let v
i
p
* and u
r
p
* be the optimal solutions to model (1); then θ
p
p
*
=
∑
r
=
1
s
‍
u
r
p
*
y
r
p is the CCR-efficiency of DMU
p and reflects the self-evaluation of this unit. DMU
p is considered to be efficient if θ
p
p
*
=
1; otherwise, it is referred to as nonefficient.As an extension to basic DEA models, cross-efficiency evaluation not only generates a ranking among the DMUs according to their efficiency but also removes unrealistic DEA weighting schemes without requiring any additional information on weight restrictions. The cross-efficiencyθ
¯
p
* of DMU
p which contains DMU
j’s peer-evaluation to DMU
p
(
j
=
1
,
…
,
n
;
j
≠
p
) is obtained as
(2)
θ
j
p
*
=
∑
r
=
1
s
u
r
j
*
y
r
p
∑
i
=
1
m
v
i
j
*
x
i
p
,
j
=
1
,
…
,
n
;
j
≠
p
,
(3)
θ
¯
p
*
=
1
n
(
∑
j
=
1
,
j
≠
p
n
‍
θ
j
p
*
+
θ
p
p
*
)
.
Using the average cross-efficiency values, all of the DMUs can be compared and ranked. However, a factor that possibly reduces the usefulness of the cross-efficiency evaluation method is that the cross-efficiency results may not be unique due to the presence of alternate optima. To resolve the problem of having multiple optimal solutions from model (1) so as to keep θ
j
p
* unchanged, Sexton et al. [18] introduce a secondary goal to select the optimal input and output weights from multiple optimal solutions while keeping the CCR efficiency θ
p
p
* unchanged. The most representative secondary goals are shown in the following models [18, 20]:
(4)
min
⁡
⁡
or
max
⁡
∑
r
=
1
s
‍
(
u
r
p
∑
j
=
1
,
j
≠
p
n
‍
y
r
j
)
s
.
t
.
∑
i
=
1
m
‍
v
i
p
(
∑
j
=
1
,
j
≠
p
n
‍
x
i
j
)
=
1
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
u
r
p
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
where θ
p
p
* is the efficiency value for DMU
p obtained from model (1). Model (4) with a min-objective function is known as the aggressive efficiency model, which minimizes the cross-efficiencies of other DMUs while preserving the efficiency of itself under evaluation. Model (4) with a max-objective function is known as the benevolent efficiency model, which maximizes the cross-efficiencies of other DMUs while preserving the efficiency of itself under evaluation.


## 3. A Weighted Neutral Cross-Efficiency Method

### 3.1. Gap Minimization Criterion for Peer-Evaluation

In the case of the benevolent model, the idea is to identify the optimal weights that not only maximize the efficiency of a particular DMU under evaluation but, at the same time, maximize the average efficiency of other DMUs. In the case of the aggressive model, one seeks weights that minimize the average efficiency of those other units. Although these two goals have been proposed as a remedy for the issue of nonuniqueness, they only focus on maximizing or minimizing the value of the efficiencies of self- and peer-evaluations but never consider the rationality of possible extreme results of peer-evaluation, for example, selecting weights making the value of peer-evaluation for a certain DMU too small or too large. This may result in dispute and impair the rationality of the selected weights and the corresponding peer-efficiency scores, especially for the DMUs in a cooperative mode. To this end, the objective function of the cross-efficiency model should not only consider the issue of nonuniqueness but also take into account the rationality of the results of peer-evaluation. For the aggressive and the benevolent models, the extreme results of peer-evaluation can be partially avoided by introducing an additional goal or a regularization term focusing on such issue. In this paper, we try to avoid the extreme positions by means of minimizing the gap between the extreme large value and the extreme small one of the peer-evaluation. This can be formed as(5)
min
⁡
(
max
⁡
j
=
1
,
…
,
n
,
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j
-
min
⁡
k
=
1
,
…
,
n
,
k
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
k
)
.
Hence, we apply (5) as the regularization term and add it as a part of the objective function for both the aggressive and the benevolent models. We show the modified aggressive and benevolent models, respectively, as
(6)
min
⁡
∑
r
=
1
s
‍
(
u
r
p
∑
j
=
1
,
j
≠
p
n
‍
y
r
j
)
+

ρ
·
(
max
⁡
j
=
1
,
…
,
n
,
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j

-

min
⁡
k
=
1
,
…
,
n
,
k
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
k
)
s
.
t
.
∑
i
=
1
m
‍
v
i
p
(
∑
j
=
1
,
j
≠
p
n
‍
x
i
j
)
=
1
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
u
r
p
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
(7)
max
⁡
∑
r
=
1
s
‍
(
u
r
p
∑
j
=
1
,
j
≠
p
n
‍
y
r
j
)
-
ρ
·
(
max
⁡
j
=
1
,
…
,
n
,
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j

-

min
⁡
k
=
1
,
…
,
n
,
k
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
k
)
s
.
t
.
∑
i
=
1
m
‍
v
i
p
(
∑
j
=
1
,
j
≠
p
n
‍
x
i
j
)
=
1
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
u
r
p
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
where ρ
∈
[
0
,
+
∞
) is the importance weight of the regularization term (5) in the objective functions of models (6) and (7). Note that ρ
=
0 makes models (6) and (7) the same as the original aggressive model and the original benevolent one, respectively. If ρ is large enough, (6) and (7) are equivalent to the model that always focuses on the gap between extremes.Models (6) and (7) can be expressed equivalently in the following forms, respectively:
(8)
min
⁡
∑
r
=
1
s
‍
(
u
r
p
∑
j
=
1
,
j
≠
p
n
‍
y
r
j
)
+
ρ
·
(
λ
¯
-
λ
_
)
s
.
t
.
∑
i
=
1
m
‍
v
i
p
(
∑
j
=
1
,
j
≠
p
n
‍
x
i
j
)
=
1
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
λ
¯
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j
-
λ
_
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
u
r
p
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
(9)
max
⁡
∑
r
=
1
s
‍
(
u
r
p
∑
j
=
1
,
j
≠
p
n
‍
y
r
j
)
-
ρ
·
(
λ
¯
-
λ
_
)
s
.
t
.
∑
i
=
1
m
‍
v
i
p
(
∑
j
=
1
,
j
≠
p
n
‍
x
i
j
)
=
1
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
λ
¯
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j
-
λ
_
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
u
r
p
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
where λ
¯ and λ
_ are the upper and lower bounds of ∑
r
=
1
s
‍
u
r
p
y
r
j, respectively.


### 3.2. A Neutral Method for Peer-Efficiency Determination

In order to eliminate bias to other DMUs brought by both of the aggressive goal and the benevolent one as well as keep the objectivity of the evaluation process, we try to evaluate DMUs in a more neutral manner and to reduce conflict and disputes by applying both of the models. That is, we can make compromised results between the aggressive and the benevolent formulations if the final peer-efficiency scores are determined in terms of such two formulations. To illustrate this, forDMU
p, we denote its self- and peer-efficiency scores γ
p
p
* and γ
j
p
*
(
j
=
1
,
…
,
n
,
j
≠
p
), respectively, by solving models (8) and (3) and self- and peer-efficiency scores δ
p
p
* and δ
j
p
*
(
j
=
1
,
…
,
n
,
j
≠
p
), respectively, by solving models (9) and (3). We hence have the following convex combinatorial expression for peer-efficiency score θ
j
p
*:
(10)
θ
j
p
*
=
α
·
γ
j
p
*
+
β
·
δ
j
p
*
,
1
≤
j
≤
n
,
where the convex combinatorial coefficients α and β satisfy α
+
β
=
1 and α
≥
0, β
≥
0. Such a combination effectively balances the two extreme attitudes from the aggressive and the benevolent formulations and thus results in a more “neutral” peer-efficiency score. Thus proposed method can be taken as a neutral cross-efficiency method. For θ
j
p
*, we have the following proposition.Proposition 1.
θ
j
p
*
≤
θ
p
p
*
(
j
=
1
,
…
,
n
).Proof.
Let the optimum solutions ofDMU
j from model (8) be v
i
j
*
(
i
=
1,2
,
…
,
m
) and u
r
j
*
(
r
=
1,2
,
…
,
s
), respectively, and let those from model (9) be ν
i
j
*
(
i
=
1,2
,
…
,
m
) and μ
r
j
*
(
r
=
1,2
,
…
,
s
), respectively. Note that both the optimum solutions from models (8) and (9) satisfy the following constraint:
(11)
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
.
Recall that θ
p
p
* denotes the optimum value of the CCR model (i.e., model (1)). This constraint can be rewrite as
(12)
∑
r
=
1
s
u
r
p
y
r
p
∑
i
=
1
m
v
i
p
x
i
p
=
θ
p
p
*
.
Hence, feasible solutions from models (8) and (9) are always optimum to the CCR model. Thus we have
(13)
γ
j
p
*
=
∑
r
=
1
s
u
r
j
*
y
r
p
∑
i
=
1
m
v
i
j
*
x
i
p
≤
∑
r
=
1
s
u
r
p
*
y
r
p
∑
i
=
1
m
v
i
p
*
x
i
p
=
θ
p
p
*
,
j
=
1
,
…
,
n
,
δ
j
p
*
=
∑
r
=
1
s
μ
r
j
*
y
r
p
∑
i
=
1
m
ν
i
j
*
x
i
p
≤
∑
r
=
1
s
μ
r
p
*
y
r
p
∑
i
=
1
m
ν
i
p
*
x
i
p
=
θ
p
p
*
,
j
=
1
,
…
,
n
.
Meanwhile, θ
j
p
*
=
α
·
γ
j
p
*
+
β
·
δ
j
p
*
(
α
+
β
=
1
) implies
(14)
min
⁡
(
γ
j
p
*
,
δ
j
p
*
)
≤
θ
j
p
*
≤
max
⁡
(
γ
j
p
*
,
δ
j
p
*
)
.
With (13) and (14), we have θ
j
p
*
≤
θ
p
p
*
(
j
=
1
,
…
,
n
).To reduce inconsistency between peer-evaluation and self-evaluation, we determineα and β from the viewpoint of efficiency distance between pairs of DMUs. From an individual’s viewpoint, DMUs with more similar efficiency scores are more important to be referred during the cross-efficiency evaluation process. More specifically, for DMU
j and DMU
p, the more γ
j
p
* (δ
j
p
*) is similar to θ
p
p
*, the more likely the profile of weights of inputs and outputs of DMU
j is to be accepted by DMU
p. Thus, we define the distance from γ
j
p
* to θ
p
p
* as
(15)
d
γ
(
j
,
p
)
=
|
γ
j
p
*
-
θ
p
p
*
|
and that from δ
j
p
* to θ
p
p
* as
(16)
d
δ
(
j
,
p
)
=
|
δ
j
p
*
-
θ
p
p
*
|
.
Note that d
γ
(
j
,
p
)
≠
d
γ
(
p
,
j
) and d
δ
(
j
,
p
)
≠
d
δ
(
p
,
j
) in most cases. Since small (large) value of d
γ
(
j
,
p
) implies strong (weak) similarity between DMU
j and DMU
p, we hence introduce a total distance d
total
(
j
,
p
) as
(17)
d
total
(
j
,
p
)
=
d
γ
(
j
,
p
)
+
d
δ
(
j
,
p
)
,
and conflict reduction may be implemented by keeping α and β in proportion to [
d
total
(
j
,
p
)
-
d
γ
(
j
,
p
)
] and [
d
total
(
j
,
p
)
-
d
δ
(
j
,
p
)
], respectively; that is,
(18)
α
∝
d
δ
(
j
,
p
)
,
β
∝
d
γ
(
j
,
p
)
.
To determine the value of α and β, normalization is needed to make them satisfy α
+
β
=
1. Hence we have
(19)
α
=
d
δ
(
j
,
p
)
d
total
(
j
,
p
)
=
|
δ
j
p
*
-
θ
p
p
*
|
|
γ
j
p
*
-
θ
p
p
*
|
+
|
δ
j
p
*
-
θ
p
p
*
|
,
β
=
d
γ
(
j
,
p
)
d
total
(
j
,
p
)
=
|
γ
j
p
*
-
θ
p
p
*
|
|
γ
j
p
*
-
θ
p
p
*
|
+
|
δ
j
p
*
-
θ
p
p
*
|
.
With Proposition 1, it is easy to find that (19) can be transferred into easy-to-solve forms as follows:
(20)
α
=
θ
p
p
*
-
δ
j
p
*
2
θ
p
p
*
-
δ
j
p
*
-
γ
j
p
*
,
β
=
θ
p
p
*
-
γ
j
p
*
2
θ
p
p
*
-
δ
j
p
*
-
γ
j
p
*
.


### 3.3. Cross-Evaluation Based on Similarity Distance

To further reduce the inconsistency between peer-evaluation and self-evaluation, we also determine the weight for cross-evaluationω
j
p from the viewpoint of efficiency distance between pairs of DMUs. We define the cross-evaluation-based distance from θ
j
p
* to θ
p
p
* as
(21)
d
(
j
,
p
)
=
|
θ
j
p
*
-
θ
p
p
*
|
,
1
≤
j
≤
n
.
Let d
max
⁡
=
max
⁡
j
d
(
j
,
p
); the cross-efficiency weight of θ
j
p
* should be proportional to (
d
max
⁡
-
d
(
j
,
p
)
). Taking normalization into consideration, we apply
(22)
ω
j
p
=
d
max
⁡
-
d
(
j
,
p
)
∑
i
=
1
n
(
d
max
⁡
-
d
(
i
,
p
)
)
as the cross-efficiency weight of θ
j
p
*. To make (22) easy to solve, we have (with Proposition 1)
(23)
d
max
⁡
=
max
⁡
j
|
θ
j
p
*
-
θ
p
p
*
|
=
θ
p
p
*
-
min
⁡
j
θ
j
p
*
.
Let θ
min
⁡
*
=
min
⁡
j
θ
j
p
*, (22) be equivalent to
(24)
ω
j
p
=
θ
j
p
*
-
θ
min
⁡
*
∑
i
=
1
n
θ
i
p
*
-
n
θ
min
⁡
*
.
And finally we have
(25)
ω
j
p
=
θ
j
p
*
-
θ
min
⁡
*
∑
i
=
1
n
θ
i
p
*
-
n
θ
min
⁡
*
,
j
=
1
,
…
,
n
,
(26)
θ
¯
p
*
=
∑
j
=
1
n
‍
ω
j
p
θ
j
p
*
.
We use (10) in neither aggressive nor benevolent manners but a more neutral way to calculate peer-evaluation and use (26) to get the weighted cross-efficiency according to the distance between the currently evaluated DMU and peers. Peers’ evaluation with different weights according to the distance can partially mitigate conflict among DMUs because the evaluation perspective of peers with short distance is more likely to be accepted by the currently evaluated DMU. We use a numeric example in the following section to illustrate the advantages of our method.


## 3.1. Gap Minimization Criterion for Peer-Evaluation

In the case of the benevolent model, the idea is to identify the optimal weights that not only maximize the efficiency of a particular DMU under evaluation but, at the same time, maximize the average efficiency of other DMUs. In the case of the aggressive model, one seeks weights that minimize the average efficiency of those other units. Although these two goals have been proposed as a remedy for the issue of nonuniqueness, they only focus on maximizing or minimizing the value of the efficiencies of self- and peer-evaluations but never consider the rationality of possible extreme results of peer-evaluation, for example, selecting weights making the value of peer-evaluation for a certain DMU too small or too large. This may result in dispute and impair the rationality of the selected weights and the corresponding peer-efficiency scores, especially for the DMUs in a cooperative mode. To this end, the objective function of the cross-efficiency model should not only consider the issue of nonuniqueness but also take into account the rationality of the results of peer-evaluation. For the aggressive and the benevolent models, the extreme results of peer-evaluation can be partially avoided by introducing an additional goal or a regularization term focusing on such issue. In this paper, we try to avoid the extreme positions by means of minimizing the gap between the extreme large value and the extreme small one of the peer-evaluation. This can be formed as(5)
min
⁡
(
max
⁡
j
=
1
,
…
,
n
,
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j
-
min
⁡
k
=
1
,
…
,
n
,
k
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
k
)
.
Hence, we apply (5) as the regularization term and add it as a part of the objective function for both the aggressive and the benevolent models. We show the modified aggressive and benevolent models, respectively, as
(6)
min
⁡
∑
r
=
1
s
‍
(
u
r
p
∑
j
=
1
,
j
≠
p
n
‍
y
r
j
)
+

ρ
·
(
max
⁡
j
=
1
,
…
,
n
,
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j

-

min
⁡
k
=
1
,
…
,
n
,
k
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
k
)
s
.
t
.
∑
i
=
1
m
‍
v
i
p
(
∑
j
=
1
,
j
≠
p
n
‍
x
i
j
)
=
1
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
u
r
p
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
(7)
max
⁡
∑
r
=
1
s
‍
(
u
r
p
∑
j
=
1
,
j
≠
p
n
‍
y
r
j
)
-
ρ
·
(
max
⁡
j
=
1
,
…
,
n
,
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j

-

min
⁡
k
=
1
,
…
,
n
,
k
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
k
)
s
.
t
.
∑
i
=
1
m
‍
v
i
p
(
∑
j
=
1
,
j
≠
p
n
‍
x
i
j
)
=
1
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
u
r
p
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
where ρ
∈
[
0
,
+
∞
) is the importance weight of the regularization term (5) in the objective functions of models (6) and (7). Note that ρ
=
0 makes models (6) and (7) the same as the original aggressive model and the original benevolent one, respectively. If ρ is large enough, (6) and (7) are equivalent to the model that always focuses on the gap between extremes.Models (6) and (7) can be expressed equivalently in the following forms, respectively:
(8)
min
⁡
∑
r
=
1
s
‍
(
u
r
p
∑
j
=
1
,
j
≠
p
n
‍
y
r
j
)
+
ρ
·
(
λ
¯
-
λ
_
)
s
.
t
.
∑
i
=
1
m
‍
v
i
p
(
∑
j
=
1
,
j
≠
p
n
‍
x
i
j
)
=
1
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
λ
¯
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j
-
λ
_
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
u
r
p
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
(9)
max
⁡
∑
r
=
1
s
‍
(
u
r
p
∑
j
=
1
,
j
≠
p
n
‍
y
r
j
)
-
ρ
·
(
λ
¯
-
λ
_
)
s
.
t
.
∑
i
=
1
m
‍
v
i
p
(
∑
j
=
1
,
j
≠
p
n
‍
x
i
j
)
=
1
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
∑
i
=
1
m
‍
v
i
p
x
i
j
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
λ
¯
-
∑
r
=
1
s
‍
u
r
p
y
r
j
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
∑
r
=
1
s
‍
u
r
p
y
r
j
-
λ
_
≥
0
,
j
=
1,2
,
…
,
n
;
j
≠
p
u
r
p
,
v
i
p
≥
0
,
r
=
1,2
,
…
,
s
;
i
=
1,2
,
…
,
m
,
where λ
¯ and λ
_ are the upper and lower bounds of ∑
r
=
1
s
‍
u
r
p
y
r
j, respectively.


## 3.2. A Neutral Method for Peer-Efficiency Determination

In order to eliminate bias to other DMUs brought by both of the aggressive goal and the benevolent one as well as keep the objectivity of the evaluation process, we try to evaluate DMUs in a more neutral manner and to reduce conflict and disputes by applying both of the models. That is, we can make compromised results between the aggressive and the benevolent formulations if the final peer-efficiency scores are determined in terms of such two formulations. To illustrate this, forDMU
p, we denote its self- and peer-efficiency scores γ
p
p
* and γ
j
p
*
(
j
=
1
,
…
,
n
,
j
≠
p
), respectively, by solving models (8) and (3) and self- and peer-efficiency scores δ
p
p
* and δ
j
p
*
(
j
=
1
,
…
,
n
,
j
≠
p
), respectively, by solving models (9) and (3). We hence have the following convex combinatorial expression for peer-efficiency score θ
j
p
*:
(10)
θ
j
p
*
=
α
·
γ
j
p
*
+
β
·
δ
j
p
*
,
1
≤
j
≤
n
,
where the convex combinatorial coefficients α and β satisfy α
+
β
=
1 and α
≥
0, β
≥
0. Such a combination effectively balances the two extreme attitudes from the aggressive and the benevolent formulations and thus results in a more “neutral” peer-efficiency score. Thus proposed method can be taken as a neutral cross-efficiency method. For θ
j
p
*, we have the following proposition.Proposition 1.
θ
j
p
*
≤
θ
p
p
*
(
j
=
1
,
…
,
n
).Proof.
Let the optimum solutions ofDMU
j from model (8) be v
i
j
*
(
i
=
1,2
,
…
,
m
) and u
r
j
*
(
r
=
1,2
,
…
,
s
), respectively, and let those from model (9) be ν
i
j
*
(
i
=
1,2
,
…
,
m
) and μ
r
j
*
(
r
=
1,2
,
…
,
s
), respectively. Note that both the optimum solutions from models (8) and (9) satisfy the following constraint:
(11)
∑
r
=
1
s
‍
u
r
p
y
r
p
-
θ
p
p
*
∑
i
=
1
m
‍
v
i
p
x
i
p
=
0
.
Recall that θ
p
p
* denotes the optimum value of the CCR model (i.e., model (1)). This constraint can be rewrite as
(12)
∑
r
=
1
s
u
r
p
y
r
p
∑
i
=
1
m
v
i
p
x
i
p
=
θ
p
p
*
.
Hence, feasible solutions from models (8) and (9) are always optimum to the CCR model. Thus we have
(13)
γ
j
p
*
=
∑
r
=
1
s
u
r
j
*
y
r
p
∑
i
=
1
m
v
i
j
*
x
i
p
≤
∑
r
=
1
s
u
r
p
*
y
r
p
∑
i
=
1
m
v
i
p
*
x
i
p
=
θ
p
p
*
,
j
=
1
,
…
,
n
,
δ
j
p
*
=
∑
r
=
1
s
μ
r
j
*
y
r
p
∑
i
=
1
m
ν
i
j
*
x
i
p
≤
∑
r
=
1
s
μ
r
p
*
y
r
p
∑
i
=
1
m
ν
i
p
*
x
i
p
=
θ
p
p
*
,
j
=
1
,
…
,
n
.
Meanwhile, θ
j
p
*
=
α
·
γ
j
p
*
+
β
·
δ
j
p
*
(
α
+
β
=
1
) implies
(14)
min
⁡
(
γ
j
p
*
,
δ
j
p
*
)
≤
θ
j
p
*
≤
max
⁡
(
γ
j
p
*
,
δ
j
p
*
)
.
With (13) and (14), we have θ
j
p
*
≤
θ
p
p
*
(
j
=
1
,
…
,
n
).To reduce inconsistency between peer-evaluation and self-evaluation, we determineα and β from the viewpoint of efficiency distance between pairs of DMUs. From an individual’s viewpoint, DMUs with more similar efficiency scores are more important to be referred during the cross-efficiency evaluation process. More specifically, for DMU
j and DMU
p, the more γ
j
p
* (δ
j
p
*) is similar to θ
p
p
*, the more likely the profile of weights of inputs and outputs of DMU
j is to be accepted by DMU
p. Thus, we define the distance from γ
j
p
* to θ
p
p
* as
(15)
d
γ
(
j
,
p
)
=
|
γ
j
p
*
-
θ
p
p
*
|
and that from δ
j
p
* to θ
p
p
* as
(16)
d
δ
(
j
,
p
)
=
|
δ
j
p
*
-
θ
p
p
*
|
.
Note that d
γ
(
j
,
p
)
≠
d
γ
(
p
,
j
) and d
δ
(
j
,
p
)
≠
d
δ
(
p
,
j
) in most cases. Since small (large) value of d
γ
(
j
,
p
) implies strong (weak) similarity between DMU
j and DMU
p, we hence introduce a total distance d
total
(
j
,
p
) as
(17)
d
total
(
j
,
p
)
=
d
γ
(
j
,
p
)
+
d
δ
(
j
,
p
)
,
and conflict reduction may be implemented by keeping α and β in proportion to [
d
total
(
j
,
p
)
-
d
γ
(
j
,
p
)
] and [
d
total
(
j
,
p
)
-
d
δ
(
j
,
p
)
], respectively; that is,
(18)
α
∝
d
δ
(
j
,
p
)
,
β
∝
d
γ
(
j
,
p
)
.
To determine the value of α and β, normalization is needed to make them satisfy α
+
β
=
1. Hence we have
(19)
α
=
d
δ
(
j
,
p
)
d
total
(
j
,
p
)
=
|
δ
j
p
*
-
θ
p
p
*
|
|
γ
j
p
*
-
θ
p
p
*
|
+
|
δ
j
p
*
-
θ
p
p
*
|
,
β
=
d
γ
(
j
,
p
)
d
total
(
j
,
p
)
=
|
γ
j
p
*
-
θ
p
p
*
|
|
γ
j
p
*
-
θ
p
p
*
|
+
|
δ
j
p
*
-
θ
p
p
*
|
.
With Proposition 1, it is easy to find that (19) can be transferred into easy-to-solve forms as follows:
(20)
α
=
θ
p
p
*
-
δ
j
p
*
2
θ
p
p
*
-
δ
j
p
*
-
γ
j
p
*
,
β
=
θ
p
p
*
-
γ
j
p
*
2
θ
p
p
*
-
δ
j
p
*
-
γ
j
p
*
.


## 3.3. Cross-Evaluation Based on Similarity Distance

To further reduce the inconsistency between peer-evaluation and self-evaluation, we also determine the weight for cross-evaluationω
j
p from the viewpoint of efficiency distance between pairs of DMUs. We define the cross-evaluation-based distance from θ
j
p
* to θ
p
p
* as
(21)
d
(
j
,
p
)
=
|
θ
j
p
*
-
θ
p
p
*
|
,
1
≤
j
≤
n
.
Let d
max
⁡
=
max
⁡
j
d
(
j
,
p
); the cross-efficiency weight of θ
j
p
* should be proportional to (
d
max
⁡
-
d
(
j
,
p
)
). Taking normalization into consideration, we apply
(22)
ω
j
p
=
d
max
⁡
-
d
(
j
,
p
)
∑
i
=
1
n
(
d
max
⁡
-
d
(
i
,
p
)
)
as the cross-efficiency weight of θ
j
p
*. To make (22) easy to solve, we have (with Proposition 1)
(23)
d
max
⁡
=
max
⁡
j
|
θ
j
p
*
-
θ
p
p
*
|
=
θ
p
p
*
-
min
⁡
j
θ
j
p
*
.
Let θ
min
⁡
*
=
min
⁡
j
θ
j
p
*, (22) be equivalent to
(24)
ω
j
p
=
θ
j
p
*
-
θ
min
⁡
*
∑
i
=
1
n
θ
i
p
*
-
n
θ
min
⁡
*
.
And finally we have
(25)
ω
j
p
=
θ
j
p
*
-
θ
min
⁡
*
∑
i
=
1
n
θ
i
p
*
-
n
θ
min
⁡
*
,
j
=
1
,
…
,
n
,
(26)
θ
¯
p
*
=
∑
j
=
1
n
‍
ω
j
p
θ
j
p
*
.
We use (10) in neither aggressive nor benevolent manners but a more neutral way to calculate peer-evaluation and use (26) to get the weighted cross-efficiency according to the distance between the currently evaluated DMU and peers. Peers’ evaluation with different weights according to the distance can partially mitigate conflict among DMUs because the evaluation perspective of peers with short distance is more likely to be accepted by the currently evaluated DMU. We use a numeric example in the following section to illustrate the advantages of our method.


## 4. An Illustrative Example

In this section, our method is illustrated with a numerical example. In the example, fourteen major international passenger airlines (DMUs) are needed to be evaluated in terms of three inputs (x
1
,
x
2
,
x
3) and two outputs (y
1
,
y
2) [25]. Description of the inputs and the outputs is shown in Table 1. We apply in this section the aggressive and benevolent models (shown in model (4)) as the comparison models. We also apply CCR model as the basic DEA model and report their CCR-efficiency scores for some necessary analysis. Recall that the importance weight ρ in models (8) and (9) reflects the magnitude of the gap minimization goal (see (5)) and thus may result in various solutions if its value changes. To balance the importance of the gap minimization goal, ρ should be neither too large nor too small. In this example, we heuristically set ρ
=
0.2.Table 1
Description of the inputs and the outputs of 14 passenger airlines.
Indices
Description
Inputs
x
1
Aircraft capacity in ton kilometers
x
2
Operating cost
x
3
Nonflight assets
Outputs
y
1
Passenger kilometers
y
2
Nonpassenger revenueTable2 shows the input and output data, proposed neutral weighted cross-efficiency results, the aggressive and benevolent cross-efficiency results (shown in “θ
¯
*” and “#” columns), and the CCR-efficiency results (shown in “θ
CCR
*” column) of the fourteen passenger airlines. Efficiency scores from proposed method, aggressive and benevolent methods, and CCR are also shown in Figure 1. It can be seen that seven DMUs are CCR-efficient (i.e., the 5th, 7th, and 10–14th DMUs). For these CCR-efficient DMUs, three of the cross-efficiency methods all fully rank them with peer- and self-evaluation scores. Compared with the aggressive and benevolent methods, proposed method produces different efficiency rankings for 14 international passenger airlines. Specifically speaking, all CCR-efficient DMUs are top-ranked from top 1 to top 7, respectively, using proposed method, whereas neither aggressive nor benevolent methods can achieve this. For example, at least one CCR-inefficient DMU (e.g., DMU
4 evaluated by both aggressive and benevolent methods) is ranked as one of the top 7 DMUs while at least one CCR-efficient DMU (e.g., DMU
7 evaluated by aggressive method and DMU
10 evaluated by benevolent method) is out of them. For CCR-efficient DMUs, proposed method is flexible in balancing the different attitudes from aggressive and benevolent methods, for example, the results of DMU
10. For CCR-inefficient DMUs, it is less likely that proposed method regards them to be more efficient than CCR-efficient ones. This illustrates that proposed method is effective to discriminate CCR-efficient from CCR-inefficient DMUs and to evaluate and rank DMUs from a more neutral perspective. We believe that our method is more equitable and reliable than the methods that rank DMUs according to either the aggressive or the benevolent cross-efficiency scores, for the gap minimization goal to avoid extreme efficiency scores is more acceptable and an integration of both pessimistic and optimistic attitudes is more effective to mitigate conflict especially for DMUs in a cooperative mode.Table 2
Data, efficiency scores, and ranks for 14 passenger airlines using different cross-efficiency models.
DMU
x
1
x
2
x
3
y
1
y
2
θ
CCR
*
Aggressive
Benevolent
Proposed
θ
-
*
#
θ
-
*
#
θ
-
*
#
1
5723
3239
2003
26667
697
0.8681
0.5990
12
0.7541
12
0.7175
13
2
5895
4225
4557
3081
539
0.3379
0.1652
14
0.1894
14
0.2328
14
3
24099
9560
6267
124055
1266
0.9475
0.6226
11
0.7678
9
0.7549
11
4
13565
7499
3213
64734
1563
0.9581
0.6734
7
0.8222
6
0.7848
9
5
5183
1880
783
23604
513
1
0.7983
1
0.8912
3
0.8831
4
6
19080
8032
3272
95011
572
0.9766
0.6385
9
0.7554
11
0.7909
8
7
4603
3457
2360
22112
969
1
0.6478
8
0.8214
7
0.8442
6
8
12097
6779
6474
52363
2001
0.8588
0.5855
13
0.7242
13
0.7356
12
9
6587
3341
3581
26504
1297
0.9477
0.6309
10
0.7590
10
0.7818
10
10
5654
1878
1916
19277
972
1
0.6813
6
0.7803
8
0.8176
7
11
12559
8098
3310
41925
3398
1
0.7742
2
0.9193
1
0.9391
1
12
5728
2481
2254
27754
982
1
0.7314
5
0.8850
4
0.8906
3
13
4715
1792
2485
31332
543
1
0.7503
3
0.9190
2
0.9076
2
14
22793
9874
4145
122528
1404
1
0.7316
4
0.8659
5
0.8795
5Figure 1
Efficiency scores from proposed, aggressive, benevolent model and CCR methods.


## 5. Conclusions

Cross-efficiency evaluation is an important method for comparing and ranking DMUs. Extant research has made efforts to tackle the shortcomings of cross-efficiency but seldom considers how to effectively discriminate DEA-efficient from DEA-inefficient DMUs and to reduce conflict during the cross-evaluation process. Based on two typical cross-efficiency methods, we propose a secondary goal to minimize the gap between the upper and lower bounds of the peer-evaluation scores and a weighted neutral cross-efficiency method to determine cross-efficiency of each DMU with a combination of its aggressive and benevolent cross-efficiency. A numerical example is conducted to illustrate the proposed method. The results show that the proposed method has the ability to discriminate DEA-efficient from DEA-inefficient DMUs and rank DMUs from a more neutral perspective. In addition, the gap minimization criterion to reduce extreme peer-evaluation results and the weights of the cross-efficiency positively influenced by the DEA-efficiency scores of the currently evaluated DMU are also beneficial to conflict mitigation. The results also indicate that the proposed method is compatible with typical cross-efficiency models and makes valid contributions to cross-efficiency evaluation. Future work may focus on how to determine the importance weightρ of the gap minimization goal in different application contexts.



---

*Source: 453749-2014-08-06.xml*