An iterative method for determining weights in cross efficiency evaluation

https://doi.org/10.1016/j.cie.2016.08.024Get rights and content

Highlights

  • We prove that our approach produces a unique weight set for the cross efficiency evaluation.

  • Our approach selects the weight set with the least zero weights among all weight sets of the DEA model.

  • We do not impose any prior weight constraint.

  • All the mathematical programs involved in our algorithm are linear programs, which can be easily solved.

  • Our approach terminates within a finite number of iterations.

Abstract

Cross efficiency evaluation in data envelopment analysis (DEA) is a commonly used skill for ranking decision making units (DMUs). However, the presence of multiple optimal weights in traditional DEA models leads to different cross efficiency scores for a DMU, which seriously affects the validity of this skill for ranking DMUs. Although many secondary-goal techniques have been proposed to address this issue, sometimes the weight sets determined by these techniques are still not unique. On the other hand, since zero weights make much information on inputs and outputs be ignored in cross efficiency evaluation, the number of them should be reduced as many as possible. The current approaches have not fully solved the above two problems. This paper proposes an iterative method for determining weights in cross efficiency evaluation, which not only ensures a unique weight set for positive input and output data but also reduces the number of zero weights maximally without imposing any prior weight restriction. Numerical examples are used to show the validity and superiorities of the proposed method in choosing a unique weight set and reducing the number of zero weights.

Introduction

Data envelopment analysis (DEA) (Charnes, Cooper, & Rhodes, 1978) is a technique to measure the relative efficiency of a set of decision making units (DMUs) with multiple inputs and outputs. In DEA, the efficiency of each DMU is self-evaluated in an attempt to achieve the best values. As a result, the most favorable weights for the evaluated DMU would be chosen and many DMUs might be self-evaluated as the best. So the DEA technique lacks the ability to further discriminate these DMUs. To tackle this issue, several approaches have been proposed. One of them is the cross efficiency evaluation. The concept of the cross efficiency evaluation was originally proposed by Sexton, Silkman, and Hogan (1986) and developed by Doyle and Green (1994). The basic idea of the cross efficiency evaluation is to assess the overall performance of each DMU with the weights of all DMUs instead of with only its own weights. Concretely, the overall performance of a DMU is usually calculated as the arithmetic average of the efficiency scores obtained with the weights provided by all the DMUs. Different with the conventional DEA model, this approach makes each DMU not only be self-evaluated but also be peer-evaluated by other DMUs. Cross efficiency evaluation has two main advantages: (i) it usually provides a full ranking for the DMUs to be evaluated and (ii) it eliminates unrealistic weight schemes without the need to elicit weight restrictions (Anderson, Hollingsworth, & Inman, 2002). With its strong discrimination power, the cross efficiency evaluation method has been widely used in various contexts, such as R&D projects (Oral, Kettani, & Lang, 1991), preference voting (Green, Doyle, & Cook, 1996), cellular layouts (Talluri & Sarkis, 1997), electricity distribution sector (Chen, 2002), Olympic games (Wu, Liang, & Yang, 2009a), supply chains (Yu, Ting, & Chen, 2010) and public procurement (Falagario, Sciancalepore, Costantino, & Pietroforte, 2012).

There are two main research topics on the cross efficiency evaluation. One topic is the calculation of the overall performance by means of substituting a weighted average of the cross efficiencies for the usual arithmetic average. For the sake of distinction, the weights used for calculating the average are called aggregation weights. In Wu et al., 2008, Wu et al., 2009b, the DMUs are treated as players in a cooperative game and the aggregation weights are respectively defined as the nucleolus solution of game and from the Shapley value of each DMU. Wu et al., 2011, Wu et al., 2012a use the aggregation weights that reflect the entropy in the cross efficiencies provided by a given DMU. Wang, Chin, and Jiang (2011) put forward the ordered weighted averaging operator for weights, which allow one to consider the preferences of decision makers in the overall performance evaluation. Ruiz and Sirvent (2012) propose the aggregation weights reflecting the disequilibrium in the profiles of DEA weights. Yang, Yang, Liu, and Li (2013) use the evidential reasoning approach to aggregate the cross efficiencies obtained from cross evaluation through the transformation of the cross efficiency matrix to pieces of evidence.

The other topic is the selection of the weight set produced by the DEA model for the cross efficiency evaluation. Most of the existing works on cross efficiency evaluation have devoted to this topic. The possible existence of multiple optimal weight sets in a DEA model may lead to different cross efficiency scores and consequently to different overall performance rankings of DMUs. This deficiency limits the application of the cross efficiency evaluation. A potential remedy to this deficiency, as suggested by Sexton et al. (1986), is the use of alternative secondary goals to choose a weight set among multiple optimal solution sets. The benevolent and aggressive models (Doyle & Green, 1994) are two well-known examples that use a secondary goal for selecting weights. The benevolent model selects the weights that increase the cross efficiencies of other DMUs as much as possible while maintaining the self-evaluated efficiency score of the target DMU. In contrast, the aggressive model decreases the cross efficiencies of other DMUs as much as possible while keeping the self-evaluated efficiency score of the target DMU. Besides these two strategies, other secondary-goal techniques are discussed by Liang et al., 2008, Wu et al., 2009c, Lam, 2010, Lam and Bai, 2011, Wang and Chin, 2010a, Wang and Chin, 2010b, Jahanshahloo et al., 2011, Ramón et al., 2010a, Ramón et al., 2010b, Ramón et al., 2011, Wang et al., 2011, Wang et al., 2012.

It should be noted that sometimes the weight sets determined by secondary-goal techniques are still not unique. The reason is that these techniques adopt the optimal solutions of a mathematical program as weights. In general, only the optimum value of a mathematical program can be uniquely defined; variable (either primal variables or dual variables) values may not be uniquely defined since there can exist different values for primal/dual variables that lead to the same objective function value (Beasley, 2003). To the best of our knowledge, there have been no theoretical proofs that the above secondary-goal techniques always generate a unique weight set for any positive input-output data set. We will propose an iterative method which can be proved to provide a unique weight set.

On the other hand, the well-known benevolent or aggressive models produce a large number of zero weights for inputs and outputs in some cases. This shortcoming makes a large amount of information on inputs and outputs, which could affect the cross efficiency evaluation, be ignored. To avoid zero weights, some models are introduced by integrating value judgments, such as the assurance region model (Thompson, Singleton, Thrall, & Smith, 1986) and the cone ratio model (Charnes, Cooper, Huang, & Sun, 1990) with the weight restriction method proposed by Allen, Athanassopoulos, Dyson, and Thanassoulis (1997). In addition, the model proposed by Mehrabian, Jahanshahloo, Alirezaee, and Amin (2000) obtains an assurance interval for the non-archimedean epsilon and completely avoids zero weights when this epsilon takes a positive value. Ramón et al. (2010a) provide a multiplier bound approach and extend the ideas of assessing efficiency in Ramón et al. (2010b) to the cross efficiency evaluation. Recently, Wu, Sun, and Liang (2012b) introduce a weight restriction model which produces a positive minimum value for each weighted data. By constraining that each weighted data must be no less than this positive minimum value in DEA model, zero weights are avoided. One common point of the above methods is the introduction of prior restrictions on weights to achieve non-zero weights. Such a treatment might decrease the self-evaluated efficiency scores of inefficient DMUs (i.e., the efficiency scores of inefficient DMUs determined by the DEA model.) Thus, some scholars believe that the number of zero weights should be reduced without requiring any weight preference information from the decision maker, and then several neutral DEA models (Wang and Chin, 2010a, Wang et al., 2011, Wang et al., 2012) are put forward. These DEA models can reduce the number of zero weights in the cross efficiency evaluation effectively while maintaining the DEA efficiency score of each DMU unchanged. However, it is possible that the number of zero weights generated by neutral DEA models can be reduced further while keeping the DEA efficiency of each DMU. We will demonstrate this point in Section 4.

From the above literature review, we know that current studies have not fully solved the following two issues: ensuring the uniqueness of the weight set and reducing the number of zero weights maximally. Due to this, this paper aims to propose a new method for solving these two problems simultaneously. This paper differs from existing related results and contributes to the literature in the following perspectives:

  • (i)

    Our approach always produces a unique weight set for any positive input and output data in the cross efficiency evaluation. A strict theoretical proof is initially provided.

  • (ii)

    The weight set generated by our approach is the one with the least number of zero weights among all weight sets yielded by the DEA model.

  • (iii)

    We do not impose any prior weight restrictions, thus our approach does not change the original DEA efficiency of any DMU.

  • (iv)

    All the mathematical programs involved in our iterative approach are linear programs, which can be easily solved.

  • (v)

    Our approach terminates within a finite number of iterations.

The rest of the paper is organized as follows. Section 2 briefly describes the cross efficiency evaluation and several well-known secondary-goal techniques. Section 3 first proposes a primary weight determination DEA model. Based on this model, we then devise an iterative method for determining a unique weight set and reducing the number of zero weights maximally in cross efficiency evaluation. Section 4 uses three numerical examples to demonstrate the validity and superiorities of our new approach. Conclusions are shown in Section 5.

Section snippets

Theoretical backgrounds

Suppose there are n DMUs j (j=1,,n) to be evaluated and each DMU is assumed to use m different positive inputs to produce s different positive outputs, denoted by xj=(x1j,,xmj) and yj=(y1j,,ysj), respectively. The relative efficiency score of DMU p(p=1,,n) can be expressed as r=1sμrpyrpi=1mνipxip, where μrp,r=1,,s, and νip,i=1,,m, are non-negative weights assigned to s outputs and m inputs, respectively. Therefore, the relative efficiency of DMU p can be measured by the following CCR

An iterative method for weight determination

There is no doubt that the weights determined by model (Pp2) should satisfy the following system:i=1mvipxip=1,r=1surpyrp=θpp,r=1surpyrj-i=1mvipxij0,j=1,,n,jp,urp,vip0,r=1,,s,i=1,,m.

There are generally multiple groups of optimal weights meeting system (25), (26), (27), (28). To avoid this, we propose a criterion choosing a unique weight set for each DMU based on the following idea: the number of zero weights should be reduced maximally under system (25), (26), (27), (28). Such a

Numerical examples

In this section, we use three numerical examples to illustrate the significant role of our method in determining a unique weight set and reducing the number of zero weights in the cross efficiency evaluation. Considering the space limitation, all numerical results are presented with seven digits after the decimal point.

Example 1

This example is about the efficiency evaluation of seven departments in a university, which was previously studied by Wong and Beasley, 1990, Wang et al., 2011, Wang et al., 2012

Conclusions

Cross efficiency evaluation is an effective skill for ranking DMUs. It requires each DMU to determine a weight set for all the DMUs to use. The weights produced by the DEA model are usually not unique, which leads to different cross efficiency scores and consequently to different rankings of DMUs. To overcome this deficiency, many secondary-goal techniques have been proposed. However, sometimes the weights produced by these secondary-goal techniques are still not unique. On the other hand, zero

Acknowledgements

The authors are grateful to two anonymous reviewers and the editors for their constructive comments, which have helped us to improve the paper significantly in both content and style. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11301395, 11201344, 71371152 and 11571270 and the Scientific Research Foundation of Hubei Provincial Education Department (Grant No. Q20132706).

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