Document Type : Original Article
Authors
^{1} department of industrial engineering, science and research branch of Islamic Azad university, Tehran
^{2} Department of Industrial Engineering, Tehran Science and Research Branch, Islamic Azad University, Tehran, Iran.
Abstract
Keywords
Introduction
Data envelopment analysis (DEA) is a linear and nonparametric programming technique for measuring the relative efficiency of decision making units (DMUs) with multiple inputs and multiple outputs[31]. The efficiency score of a DMU is describe as the maximum of the ratio of its weighted sum of outputs to its weighted sum of inputs and it cannot be greater than 1 any DMU. A DMU is efficient if its efficiency score is equal to 1, otherwise it is inefficient. Usually, efficient DMUs are assessed to perform better than inefficient DMUs Since its institution [32], various DEA models were suggested by other scholars. One of the most popular models is the crossefficiency method, found by Sexton et al. (1986). It was developed as a DEA extension, and could be used to rank efficient DMUs by using crossefficiency scores which were related to all DMUs. When crossefficiency method is used, optimal weights computed by the DEA model are generally not unique, making crossefficiency scores generated arbitrarily [23]. This set of weights
may possibly develop some DMUs’ ranks, while worsen those of the others, which might reduce the usefulness of crossefficiency. To overcome this problem, Doyle & Green (1994) proposed two different types of models: aggressive and benevolent formulations, which represent two diametrically across from strategies. The idea of the aggressive or benevolent model is to recognize optimal weights that can not only maximize the
efficiency of the DMU under evaluation but also maximize or minimize the average efficiency of other DMUs[11]. Liang et al. (2008) suggested a series of secondary objective functions to extend Doyle and Green’s models. Each secondary objective function represents a kind of efficiency evaluation criterion, according to which the efficiency scores are compared from multiple different angles[29]. Wang & Chin (2010) proposed a neutral DEA model for cross efficiency evaluation. They belief that in evaluation of the DMU should cover whether the weights could be as favorable as possible to itself, rather than how aggressive or benevolent the weights are to the others [6]. However, neither the traditional DEA model nor the crossefficiency methods can directly handle interval data. Some theoretical research of DEA with interval data has been presented in the literature, for instance, Despotis & Smirlis (2002) and Jahanshahloo et al. (2004) suggested some new models in earlier research. These models can only classify the DMUs, but fail to rank them. In this study, A new approach is developed to address the issue of ranking all the DMUs.
The new approach, crossefficiency method applied for two stages DMUs with interval data. So, 1)we consider lower bound in first stage of whole DMUs , Forming Crossover n*n table and then Compute the average elements of each row for DMU and then calculate distance of it from one , also, 2)we consider upper bound in first stage of whole DMUs , Forming Crossover n*n table and then Compute the average elements of each row for DMU and then calculate distance of it from one. Now, Calculate the average of two above numbers. The smaller average have a better rank, so rank the first stage of DMUs. Similarly as above, we rank the second stage of DMUs.
The rest of the paper is organized as follows. Section 2introduces the literature review of interval DEA models and cross efficiency. Section 3presents the new model for the crossefficiency evaluation method with interval data for tow stage DMUs. Illustrative example for proposed method are presented in section 4. Conclusions and further remarks are given in Section 5.
Material and methods
Data envelopment analysis (DEA), originally proposed by Charnes, Cooper, and Rhodes (1978), is a nonparametric programming method for evaluating the efficiency of a group of homogenous decision making units (DMUs) with multiple inputs and multiple outputs [1, 2, 3, 4, 5] The main idea of DEA is to generate a set of optimal weights for each DMU in a set of DMUs to maximize the ratio of its sum of weighted outputs to its sum of weighted inputs while keeping all the DMU ratios at most 1. This maximum ratio is defined as the efficiency of the DMU under evaluation [6, 7].For its effectiveness in identifying the bestpractice frontier and ranking the DMUs, DEA has been widely applied in benchmarking and efficiency evaluation of schools [8], hospitals [9], bank branches [10],and so on. However, the traditional selfevaluated DEA models with total weight flexibility may evaluate many DMUs as DEA efficient and cannot make any further distinction among these efficient DMUs. Therefore, one of the main shortfalls of the traditional DEA model (CCR or BCC model) is its inability to discriminate among DMUs that are all deemed efficient [6].
To improve the power of DEA in discriminating the efficient DMUs, Sexton, Silk man, and Hogan (1986) incorporated the concept of peer evaluation into DEA, and proposed the crossefficiency evaluation method. In crossefficiency evaluation, each DMU gets a selfevaluated efficiency obtained by its own most favorable weights and n1 peerevaluated efficiencies obtained using the other DMUs’ most favorable weights. Then, all these efficiencies are aggregated into a final efficiency for the DMU under evaluation. There are at least three principal advantages of the crossefficiency evaluation. Firstly, it almost always ranks the DMUs in a unique order [11].Secondly, it eliminates unrealistic weight schemes without incorporating weight restrictions [12]. Finally, it effectively distinguishes good and poor performers among the DMUs[13].Due to these advantages, crossefficiency evaluation has been extensively applied in performance evaluation of nursing homes [14], preference ranking and project selection [15], selection of flexible manufacturing systems [16], judging suitable computer numerical control machines [17 , determining the efficient operators and measuring the labor assignment in cellular manufacturing systems [18], performance ranking of countries in the Olympic Games [19], supplier selection in public procurement [20], portfolio selection in the Korean stock market [21], energy efficiency evaluation for airlines [22], and so on.
In spite of its advantages and wide applications, there are still some shortcomings in DEA crossefficiency evaluation. For example, the nonuniqueness of the DEA optimal weights may reduce the usefulness of crossefficiency evaluation[14]. Specifically, the optimal weights generated from the traditional models (CCR or BCC model) are generally not unique. Thus, the crossefficiency scores for the DMUs are somewhat arbitrarily generated [23]. To solve this problem, Sexton, Silk man, and Hogan (1986) suggested incorporating secondary goals into crossefficiency evaluation. Based on this idea, many secondary goal models have been proposed [14]. For example, Jahanshahloo, Hosseinzadeh Lofti, Yafari, and Maddahi (2011) used selecting symmetric weights asa secondary goal in crossefficiency evaluation [24]. Wu, Sun, Zha, and Liang (2011) and Contreras (2012) proposed weights selecting models in which the secondary goal is to optimize the ranking position of the DMU under evaluation[25, 26]. Lim (2012) proposed models using the minimization (or maximization) of the best (or worst) cross efficiencies of peer DMUs as the secondary goal [27]. Maddahi, Jahanshahloo, Hosseinzadeh Lotfi, and Ebrahimnejad (2014) suggested optimizing proportional weights as a secondary goal in DEA crossefficiency evaluation[28]. Among the secondary goal models, the most commonly used are the benevolent and aggressive models [11] The main idea of the benevolent(aggressive) model is to select for each DMU a set of optimal weights that makes the other DMUs’ cross efficiencies as large (small) as possible while keeping its own efficiency at its predetermined optimal level (CCR efficiency). Liang, Wu, Cook, and Zhu (2008) extended the model of Doyle and Green (1994) by introducing various secondary objective functions. Each new secondary objective function represents an efficiency evaluation criterion and can be applied in different practical case scenarios [29]. They also proposed alternative secondary goal models, but they replaced the target efficiency of each DMU from the ideal point 1 to CCR efficiency.
Another main drawback of crossefficiency evaluation is that the generated average crossefficiency scores for the DMUs are not Pareto optimal [25], which means it may be difficult to get all the DMUs to accept these crossefficiency evaluation results. To overcome this drawback, some scholars have eliminated the average assumption for determining the ultimate crossefficiency scores by using a common weights evaluation method. For example, Wu, Liang, and Yang (2009) considered the DMUs as players in a cooperative game, in which the characteristic function values of coalitions are defined to compute the Shapley value of each DMU, and the common weights associated with the imputation of the Shapley values are used to determine the ultimate crossefficiencies scores [30].
Interval DEA Models
Assume there are n DMUs with two stages to be evaluated. Each DMU produces s outputs using min puts. Input i and output r for DMU_{j }are denoted as and , respectively. The input and output data and are not assumed to be exactly obtained because of uncertainty. Only their bounded interval and with and, are known. In order to measure the efficiencies of the DMUs with uncertain inputs and outputs data, Despotis & Smirlis (2002) proposed a pair of linear problem models to generate the lower and upper bounds of the efficiency for each DMU. However, Wang & Yang (2005) pointed out that the efficiencies calculated by the models in Despotis & Smirlis (2002) are lack of the comparability. Because different production frontiers have been adopted in the process of efficiency measurement. In order to deal with such an uncertain situation, the CCRDEA model can be defined as:
(1) 
In order to calculate the lower and upper bound of the efficiency of DMUd , Wang et al.(2005) proposed the following two linear formulations to generate the bounded interval .
(2)

In the above two models, DMU_{d} is the DMU under evaluation, and are the weights assigned to the inputs and outputs respectively. is the lower efficiency for DMU_{d}, is the upper efficiency. is the nonArchimedean infinitesimal. Considering model (3) and (4), it is clear that DMUd , can be considered as DEA efficient if its best possible upper efficiency , or it is inefficient if :
(3) 
Crossefficiency Evaluation Method with Interval Data
In fact, the DMU under evaluation heavily weighs the inputs and outputs of a few favorable DMUs and ignores those of the others in order to maximize its own efficiency ratio. Moreover, optimal weights calculated with the models (2)and (3) are generally not unique. The different calculation software may produce different optimal weights, making the generated crossefficiency scores arbitrary. Therefore, an interval crossefficiency evaluation method is used to overcome this shortcoming. The crossefficiency method in DEA uses peer evaluation instead of selfevaluation. It can define the crossefficiency scores of DMUs on their interval (Wang & Yang 2005).
In the traditional crossefficiency method, some choice of weights may lead to a lower crossefficiency for some DMUs or a higher crossefficiency for the others. So we introduce a secondary objective function to reduce the ambiguity. Model (4) can calculate the low crossefficiency values for interval data.
(4) 
Similarly, the large crossefficiency values of interval data can be computed with model (5).
(5) 
After all crossefficiency values are computed, an efficiency matrix (CEM) can be constructed as shown in Table 1.For each column, represents the lower and upper limits of the crossefficiency scores of DMUj by using the weights that DMU_{d} (j=1,…,n) has chosen. The elements on the main diagonal are the limits obtained through selfevaluation which can be calculated by using models (2) and (3).
Table 1. A generalized crossefficiency matrix (CEM)
Rating 
Rated DMU_{j} 

DMU_{d} 
1 
2 
3 
… 
n 
1 
… 

2 
… 

3 
… 

. . . 
. . . 
. . . 
. . . 
… 
. . . 
N 
… 
Results
Proposed model has shown in figure 1., according to it, crossefficiency method applied for two stages DMUs with interval data. So, 1)we consider lower bound in first stage of whole DMUs , Forming Crossover n*n table and then Compute the average elements of each row for DMU and then calculate distance of it from one , also, 2)we consider upper bound in first stage of whole DMUs , Forming Crossover n*n table and then Compute the average elements of each row for DMU and then calculate distance of it from one. Now, calculate the average of two above numbers. The smaller average has a better rank, so rank the first stage of DMUs.
Similarly as above, for second stage of DMUs 1)we consider lower bound in second stage of whole DMUs , Forming Crossover n*n table and then Compute the average elements of each row for DMU and then calculate distance of it from one , also, 2)we consider upper bound in second stage of whole DMUs , Forming Crossover n*n table and then Compute the average elements of each row for DMU and then calculate distance of it from one. Now, Calculate the average of two above numbers. The smaller average have a better rank, so rank the second stage of DMUs.
This approach can be used to comparison of DMUs with two stages. The proposed model Provide a solution to increase the efficiency of each DMU and Improving each stage of every DMU with the best performance available among DMUs in that stage. In other words, our model is a benchmark to DMUs with two stages. In order to prove the effectiveness of the proposed approach, numerical examples are illustrated finally. We considered n DMUs with two stages and interval data. Then calculated cross efficiency for lower and upper bounds of first stages separately, and for lower and upper bounds of second stages. Finally, the new approach enables us to ranking of first stage for n DMU and second stages of them. DMUs with the best rank can be used as benchmark for improving efficiency of other DMUs.
The proposed method can be used for each network that include DMUs with two stages in production process. However, this paper is the first study that examined cross efficiency of DMUs in structure framework with interval data.
Fig 1. Proposed Method
Discussion
Assume there are n DMUs with two stages to be evaluated. Each DMU produces s outputs using min puts. Input r and output k for DMUj are denoted as and , respectively. The input and output data and are not assumed to be exactly obtained because of uncertainty. Only their bounded interval , r=1,…,m and , k=1,…,s with and , are known. Efficiency score of DMUs is interval and have lower and upper bound.
Table 2. and table 3. shows lower and upper bound of each DMUs with two stages, respectively.
Table 2. Lower bound of first and second stage for DMUs
O2 
O1 
I2 
I1 


3 
2 
5 
4 
DMU1 
First stage 
2 
1 
6 
7 
DMU2 

1 
7 
1 
3 
DMU3 

8 
3 
5 
2 
DMU1 
Second stage 
2 
1 
2 
1 
DMU2 

1 
7 
5 
7 
DMU3 
Table 3. upper bound of first and second stage for DMUs
O2 
O1 
I2 
I1 


6 
4 
6 
6 
DMU1 
First stage 
4 
3 
9 
8 
DMU2 

2 
9 
4 
4 
DMU3 

9 
5 
6 
5 
DMU1 
Second stage 
3 
3 
4 
3 
DMU2 

4 
8 
6 
9 
DMU3 
First, we calculate CCR input oriented model for lower and upper bounds of first and second stage for DMUs. Results bring to table 4. and then table 5. And table 6. shows computing crossefficiency for lower and upper bounds of DMUs with two stages, respectively.
Table 4. CCR input oriented model for lower and upper bounds of first and second stage for DMUs
CCR _{Upper} 
CCR _{Lower} 


0.311355 
0.128788 
DMU1 
First stage 
0.174699 
0.055556 
DMU2 

1 
0.514706 
DMU3 

0.625 
0.333333 
DMU1 
Second stage 
0.75 
0.125 
DMU2 

0.353982 
0.243056 
DMU3 
Table 5.crossefficiency for lower bounds of first stage for DMUs
First stage 


DMU3 
DMU2 
DMU1 

First stage 
0.166667 
0.32 
 
DMU1 

0.164835 
 
0.083333 
DMU2 

 
0.2 
0.113032 
DMU3 
Table 6.crossefficiency for lower bounds of second stage for DMUs
Second stage 


DMU3 
DMU2 
DMU1 

Second stage 
0.243056 
0.083333 
 
DMU1 

0.243056 
 
0.15 
DMU2 

 
0.083333 
0.15 
DMU3 
Table 7.crossefficiency for upper bounds of first and second stage for DMUs
First stage 


DMU3 
DMU2 
DMU1 

First stage 
1 
1 
 
DMU1 

0.810811 
 
1 
DMU2 

 
1 
0.34 
DMU3 
Table 8.crossefficiency for upper bounds of second stage for DMUs
second stage 


DMU3 
DMU2 
DMU1 

Second stage 
0.353982 
0.75 
 
DMU1 

0.353982 
 
0.625 
DMU2 

 
0.75 
0.625 
DMU3 
For ranking of DMUs we Compute the average elements of lower bound and upper bound for each DMUs and then calculate distance of them from one. Now, Calculate the average of two numbers for each DMUs in first stage and then second stage. The smaller average have a better rank in each stage between all DMUs, so rank DMUs. The results are shown in table 9.
Table 9. Ranking of DMUs for first and second stages
A 
1A U 
1A L 
A U 
A L 


0.845162 
0.841399 
0.848926 
0.158601 
0.151074 
DMU1 
First stage 
0.845079 
0.841311 
0.848847 
0.158689 
0.151153 
DMU2 

0.835669 
0.831406 
0.839932 
0.168594 
0.160068 
DMU3 

0.82581 
0.821029 
0.830592 
0.178971 
0.169408 
DMU1 
second stage 
0.840764 
0.836769 
0.844759 
0.163231 
0.155241 
DMU2 

0.845162 
0.841399 
0.848926 
0.158601 
0.151074 
DMU3 
Conclusions
In many practical examples, the outputs and inputs of DMUs are not known exactly, for example, given as intervals. However, the existing classical interval DEA method is not able to rank the DMUs, but can only classify them as efficient or inefficient. In view of the drawbacks, we put forward our approach. Then new model is proposed. This approach can be used to comparison of DMUs with two stages. The proposed model Provide a solution to increase the efficiency of each DMU and Improving each stage of every DMU with the best performance available among DMUs in that stage. In other words, our model is a benchmark to DMUs with two stages. In order to prove the effectiveness of the proposed approach, numerical examples are illustrated finally. We considered n DMUs with two stages and interval data. Then calculated cross efficiency for lower and upper bounds of first stages separately, and for lower and upper bounds of second stages. Finally, the new approach enables us to ranking of first stage for n DMU and second stages of them. DMUs with the best rank can be used as benchmark for improving efficiency of other DMUs. Through the example, we can conclude that the proposed method is convenient to solve multiple attribute problems with interval data. It can make full use of the original data information, and provide complete and fair results for all DMUs. The method in this paper can be further expanded in the future.