Research Article  Open Access
CrossEfficiency Evaluation Method with CompeteCooperate Matrix
Abstract
Crossefficiency evaluation method is an effective and widespread adopted data envelopment analysis (DEA) method with selfassessment and peerassessment to evaluate and rank decision making units (DMUs). Extant aggressive, benevolent, and neutral crossefficiency methods are used to evaluate DMUs with competitive, cooperative, and nontendentious relationships, respectively. In this paper, a symmetric (nonsymmetric) competecooperate matrix is introduced into aggressive and benevolent crossefficiency methods and competecooperate crossefficiency method is proposed to evaluate DMUs with diverse (relative) relationships. Deviation maximization method is applied to determine the final weights of crossevaluation to enhance the differentiation ability of crossefficiency evaluation method. Numerical demonstration is provided to illustrate the reasonability and practicability of the proposed method.
1. Introduction
Data envelopment analysis (DEA) is a nonparametric programming method for evaluating the relative efficiencies of a group of decision making units (DMUs) with multiple inputs and outputs. Since Charnes et al. [1] proposed the CCR model in 1978 and Banker et al. [2] proposed the BCC model in 1984, DEA is widely used in various fields. The traditional DEA models, including the CCR and BCC model, are based on selfassessment system; the obtained input and output weights of evaluated DMUs take the aim at maximizing their own efficiency, which will cause problems in three aspects. (1) The traditional DEA models can only distinguish the efficient and inefficient DMUs but cannot rank the merits and with a lower degree of differentiation on CCRefficient DMUs. (2) The obtained efficiency weights are only beneficial to the single DMU, which is easy to exaggerate its own advantages in some inputs and outputs angles, but circumvent its disadvantages in other input and output angles, resulting in lipdeep efficient phenomena. (3) Each DMU selects its own favorable weighting scheme, lacking comparability among DMUs.
In response to these problems, scholars have proposed a number of improvements [3–5]; the typical methods include crossefficiency evaluation method [6], publicweight method [7], superefficient DEA method [8], and other DEA methods, wherein the crossefficiency evaluation method has been applied repeatedly, which is proposed by Sexton et al. [9] in 1986, as an expansion and improvement of traditional DEA model. The essence of the method is the introduction of peerassessment system, using selfassessment and peerassessment system to evaluate the efficiencies of DMUs. The crossefficiency method is possible to get complete ranks and comparable evaluated scores, which has a higher degree of differentiation on CCRefficient DMUs. Therefore, this method has been widespread and widely used to deal with specific problems in academic fields [10–13].
But the crossefficiency method may fall into a predicament of multiple solutions, so many scholars have been keen on the improvement of the traditional crossefficiency model. The typical treatments to avoid the problem include Doyle and Green’s [14] aggressive and benevolent crossefficiency evaluation methods, which introduce secondary objective functions to crossefficiency evaluation method and can select the optimal weights to minimize and maximize the sum of the outputs of other DMUs, respectively. Later Wang and Chin [15], based on the aggressive and benevolent methods, propose the neutral DEA model, which has effectively reduced the number of zero outputs’ weights. Wu et al. [16] introduce secondary goals in crossefficiency evaluation to avoid multiple solutions, they propose a novel model to determine the final crossefficiency and optimize the ranking order, they indicate that pursuing the best ranking is more important than maximizing the individual score, and this model is able to draw the best ranking. Jahanshahloo et al. [17] also introduce secondary goals to crossefficiency and select symmetric weights and propose the symmetric weight assignment technique (SWAT) method to effectively select weights from multiple optimal solutions. Wu et al. [18] propose a weightbalanced DEA method to deal with the nonunique crossefficiency scores resulting from the presence of alternate optima in traditional DEA models. This method can effectively lessen the differences in weighted data and reduce the zero weights. Wang et al. [19] introduce a virtual ideal DMU (IDMU) and a virtual antideal DMU (ADMU) to crossefficiency evaluation method, propose several new DEA models, and result in neutral crossevaluation scores, which enhance the theory and methodology of crossefficiency evaluation method and can be more neutral and logical. Wang et al. [20] introduce neutral input and output weights for each DMU, replace the aggressive or benevolent ones, thus minimize the virtual disparity in crossefficiency evaluation, and reduce the number of zero weights.
In this paper, we mainly aim for the improvement on the practicality and application of crossefficiency evaluation method. The benevolent, aggressive, and neutral crossefficiency evaluation methods suppose that the relationships of DMUs are absolutely partnership, competitive, and nontendentious, respectively. But in practical applications, the following two situations generally exist. (1) The relationships of evaluated DMUs are complex; they not only involve partnership relationships but also involve competitive relationships. (2) The relationship of a pair of DMUs is relativity. A DMU regards another DMU as friend and partner, while another DMU regards the DMUs as enemies and rivals. Focusing on the two situations, we introduce a competecooperate matrix into aggressive and benevolent crossefficiency methods and build competecooperate crossefficiency model. Our method can effectively evaluate the efficiencies of DMUs which has complex relationships compared to extant crossefficiency methods. In addition, extant crossefficiency methods obtain the final scores of DMUs by calculating the average of selfassessment scores and peerassessment scores. This method sets all DMUs on equal status, with lower degree of differentiation on selfassessment and peerassessment. In this paper, we apply the deviation maximization method [21] to calculate the weights of each model, which give different evaluated scores with different importance and can effectively widen the gap of scores of DMUs.
The paper is arranged as follows. Section 2 introduces the CCR model, aggressive model, benevolent model, and neutral model. Section 3 introduces the proposed competecooperate crossefficiency model and the deviation maximization method. Section 4 provides a numerical example. Section 5 finally shows the conclusion.
2. Traditional CrossEfficiency Models
Let there be DMUs, whereuses m kind of resources to produce s kind of outputs. The input and output vectors can be denoted as and . Then, for a given , its efficiency score of can be determined by the CCR model as follows:where and are the weights assigned to and , respectively. Let be the CCRefficiency score of and reflect its selfassessment and let be the optimal solutions to model (1). Then the crossefficiency model where we can obtain the crossefficiency scores of all the DMUs with selfassessment score and peerassessment scores can be presented as
Using the average crossefficiency scores, we can compare and rank all the DMUs. However, the crossefficiency scores may not be unique because of the existence of alternate optimal weights, which reduce the usefulness of the crossefficiency evaluation method. To resolve the problem, the most representative and most applied model and the aggressive and benevolent crossefficiency models are proposed by Doyle and Green, which can be shown as follows:where is the CCRefficiency score of obtained from model (1). The aggressive efficiency model, with a minobjective function in model (3), is given to minimize the other DMUs’ crossefficiency on the promise of unchanged CCRefficiency value, and the benevolent efficiency model, with a maxobjective function, is given to maximize the crossefficiency of other DMUs. Then Wang and Chin, based on aggressive and benevolent models, proposed a neutral DEA model for crossefficiency evaluation; the model is presented as where is the efficiency of the of the th output. Compared with the aggressive and benevolent methods, there is no difficulty for DMUs to make a subjective choice and determine the input and output weights just from their own perspective in neutral DEA method.
3. CompeteCooperate CrossEfficiency Model
3.1. CompeteCooperate Matrix and CompeteCooperate CrossEfficiency Model
In actual application of crossefficiency method, we often encounter the following two situations. (1) The relationships of DMUs are complex; there not only exist partner related DMUs, but also involve competitive related DMUs. (2) The relationship between two DMUs is relativity. Some () reckons () as a cooperative partner, while () reckons () as a competitor. In these cases, we cannot simply use aggressive or benevolent crossefficiency model to calculate the values of DMUs.
In this paper, we introduce a crossefficiency model with competecooperate matrix to resolve these problems. First of all, we should build the competecooperate matrix. For the first situation, we argue that if two DMUs are cooperative partners, set the coefficient of the matrix as 1. However, if the relationship between two DMUs is competition, then set the coefficient of the matrix as −1. The coefficient of selfassessment will be 0. For the second situation, we argue that if () reckons () as a cooperative friend, set the coefficient of the matrix as 1. If reckons as a competitor, set the coefficient of the matrix as −1 and set the coefficient of selfassessment as 0. For the first situation, the competecooperate matrix is a symmetric matrix and for the second situation, the matrix is a nonsymmetric matrix. Then the competecooperate crossefficiency model is built as follows:where is the competecooperate relationship matrix, determined by the relationship of DMUs, and is the CCRefficiency value of obtained from model (1). Obviously, the competecooperate crossefficiency model can effectively evaluate the efficiencies of DMUs with complex and relative relationship. It is the biggest advantage of the competecooperate crossefficiency model.
3.2. Deviation Maximization Method
Extant crossefficiency evaluation methods, like aggressive, benevolent, and neutral methods, generally seek the final crossefficiency scores by calculating the average after determining the self and peerassessment scores of each DMU; then each of the evaluated scores participates in the evaluation on the equal weights. Generally speaking, when we evaluate the m index of n DMUs, we usually want to widen the gap of DMUs’ efficiency values, in order to pull the grade, facilitate the sorting, and enhance the ability of differentiation. So we need to choose the best weight coefficient index to widen the gap of the efficiency values of the DMUs. Assuming is the weight coefficient vector and is m vectors of the i evaluation object, the scoring matrixThen the final scores of DMUs can be presented aswhere is the final score vector of DMUs and is the efficiency value of ; in order to widen the gap between DMUs’ efficiency values, we need to make the variance of efficiency values as large as possible, which can be presented as
Put (7) into (8), and normalize the raw data; the following equation will be obtained:where is a real symmetric matrix. is the weight vector; so . Therefore, the way to make the variance can be described aswhere is the eigenvector for the maximum eigenvalue of , and (10) gets its maximum value. Then normalize to obtain the optimal weight coefficient vector. Consider . In this paper, the deviation maximization method is applied to obtain the final weight factor for aggressive, benevolent, neutral, and proposed competecooperate crossefficiency methods.
4. An Illustrative Example
In this section, we use a specific example to illustrate our method. The example aims at evaluating the technology innovation efficiency of domestic 31 provinces (DMUs). Each province has to be evaluated in terms of three inputs and three outputs . Because of the incomplete data of the Tibet Autonomous Region, we choose the remaining 30 provinces as DMUs. We use the data of the 30 provinces in 2012, obtained from China Statistical Yearbook on Science and Technology 2013 and economy prediction system. Table 1 shows the specific description of the inputs and outputs of 30 provinces.

In this paper, we only choose the first situation as an example, since the calculating process of the two situations is only different in the data of competecooperate matrix . We apply the CCRefficiency method and the aggressive, benevolent, and neutral crossefficiency methods and report their scores in this section to be compared with the proposed competecooperate crossefficiency model for some necessary analysis. First of all, we need to give certain values to the competecooperate matrix . This paper, according to regional GDP in 2012, divides GDP amount of 30 provinces into 6 stages; Table 2 shows the specific stage of each section. We argue that the relationship of two provinces whose GDP amount belongs to the same interval is partnership; set the coefficient of the matrix as 1. The relationship of two provinces whose GDP amount belongs to different intervals is competition; set the coefficient of the matrix as −1. Set the coefficient of selfassessment as 0.

In this paper, we apply the deviation maximization method to get the optimal weights. Table 3 shows the weights for aggressive, benevolent, neutral, and proposed competecooperate crossefficiency models. Table 4 shows the final efficiency scores and ranks of the 30 provinces in CCR, aggressive, benevolent, neutral, and proposed DEA models.


It can be seen from the example scores and ranks that the traditional CCRefficiency model has lower differentiation on CCRefficient DMUs; there are a number of CCRefficient provinces, including Zhejiang, Anhui, Beijing, Guangdong, and Jiangsu provinces, while using the crossefficiency evaluation method did not get such a result. Therefore, the traditional CCR method which relies solely on selfassessment system has certain drawbacks compared with crossefficiency evaluation method.
Table 4 shows the scores and ranks of 30 provinces from CCR, aggressive, benevolent, neutral, and proposed competecooperate DEA methods, which are different in some provinces, but there are still the same parts. Compared with CCR model, the 5 CCRefficient provinces are topranked in proposed model, but there are also some provinces, such as Tianjin, which is ranked 6 in CCR model but ranked 13 in proposed model. Thus, the result of the analysis of proposed model is more flexible. Compared with aggressive, benevolent, and neutral crossefficiency methods, provinces topranked from top 1 to top 6 are the same, including Jiangsu, Zhejiang, Beijing, Anhui, Guangdong, and Heilongjiang province, but there are still subtle differences on the ranking of individual provinces, such as Shanghai and Jiangxi province.
In this paper, Spearman method is used to analyze the correlation among CCR model, aggressive, benevolent, and neutral crossefficiency model, and proposed competecooperate crossefficiency method. Table 5 shows the results of Spearman correlation analysis for the five models. Correlation coefficients are all between 0.9 and 1, which shows that the efficiency scores and ranks of the five methods are significantly correlated and highly consistent, but relatively speaking, the results of proposed method and aggressive method are highly consistent.

Figure 1 shows the efficiency scores of CCR, aggressive, benevolent, neutral, and proposed competecooperate DEA methods. It can be seen that the efficiency scores evaluated by traditional CCR model, benevolent model, and neutral model can be higher than other methods’ scores; efficiency scores calculated by aggressive crossefficiency method can be lower than other methods’ scores. However, the scores of the 30 provinces calculated by proposed method are lower than CCR, benevolent, and neutral methods but higher than aggressive model. Although the extant crossefficiency models, use selfassessment and peerassessment system, can avoid the problem of appearing multiple CCRefficient provinces in CCR model, complete competitive relationships in the aggressive model cause the lower scores, complete friendly relationships in the benevolent model cause the higher scores. The relationships of the provinces are nontendentious in neutral model, while the proposed competecooperate crossefficiency method takes the relationship of cooperation and competition of the provinces into account and produces more rational evaluated scores and more reliable ranks.
5. Conclusions
Crossefficiency evaluation method is a method for assessing and evaluating the efficiency scores of DMUs, with selfassessment and peerassessment system, avoiding low degree of differentiation of CCRefficient DMUs in traditional CCR model. However, the aggressive, benevolent, and neutral crossefficiency evaluation methods can only resolve the problem of DMUs with competitive, partnership, and nontendentious relationships, respectively. But in practice, the relationships of DMUs are not absolute; there may exist two situations: (1) Some DMUs are cooperative partnership, but others are competitive relationship. (2) There exists relative relationship between two or several DMUs. Namely, some DMU_{1} regards DMU_{2} as a partner, while DMU_{2} regards DMU_{1} as a competitor. For these cases, this paper introduced a competecooperate matrix into aggressive and benevolent crossefficiency models, built competecooperate crossefficiency model, and applied deviation maximization method to obtain the final weight factor for crossefficiency evaluation methods to widen the gap between the efficiency values of the DMUs. Furthermore, we use an example of technological innovation efficiency evaluation of the 30 provinces to analyze and interpret the proposed model. The results showed that proposed competecooperate crossefficiency model has significant consistency with CCR, aggressive, benevolent, and neutral model; it can effectively evaluate the efficiency of DMUs with complex and relative relationship issues and enhance the stability and practicality of crossefficiency evaluation. Future work may focus on determining the degree of authority of DMUs, namely, the weight of each DMU.
Conflict of Interests
There is no conflict of interests regarding the publication of this paper.
Acknowledgment
This paper is supported by Liaoning Education Department fund item “regional innovation efficiency evaluation and promotion strategy of Liaoning province” (serial no.: W2014026).
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Copyright
Copyright © 2015 Qiang Hou and Xue Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.