Cross efficiency evaluation method based on weight-balanced data envelopment analysis model

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Abstract

In this paper, the cross efficiency evaluation method, regarded as a DEA extension tool, is firstly reviewed for its utilization in identifying the Decision Making Unit (DMU) with the best practice and ranking the DMUs by their respective cross-efficiency scores. However, we then point out that the main drawback of the method lies in non-uniqueness of cross-efficiency scores resulted from the presence of alternate optima in traditional DEA models, obviously making it become less effective. Aiming at the research gap, a weight-balanced DEA model is proposed to lessen large differences in weighted data (weighted inputs and weighted outputs) and to effectively reduce the number of zero weights for inputs and outputs. Finally, we use two examples of the literature to illustrate the performance of this approach and discuss some issues of interest regarding the choosing of weights in cross-efficiency evaluations.

Highlights

► The main drawback of the cross-efficiency evaluation method is considered. ► A weight-balanced DEA model is proposed to determine the ultimate cross efficiency. ► Two examples are illustrated to examine the validity of the proposed method.

Introduction

Data envelopment analysis (DEA) has been proven to be an effective approach in identifying the best practice frontiers and ranking the Decision Making Units (DMUs) since it was first introduced by Charnes, Cooper, and Rhodes (1978). DEA, a non-parametric programming technique, is used to evaluate efficiency of a set of homogenous DMUs with multiple inputs and multiple outputs. Typical applications of DEA have involved researches of performance evaluation and benchmarking in such settings as schools, hospitals, bank branches, production plants, and so on (Bruni et al., 2009, Charnes et al., 1994, Liu and Wang, 2008, Lozano et al., 2011, Reichmann and Sommersguter-Reichmann, 2006, Wang et al., 2008, Yang et al., 2009).

However, traditional DEA models, such as CCR model in Charnes et al. (1978) can simply classify the DMUs into two groups, namely efficient DMUs and inefficient DMUs. The DMU under evaluation heavily weighs few favorable measures and ignores other inputs and outputs in order to maximize its own DEA efficiency so that many DMUs are often evaluated as DEA efficient and cannot be discriminated any further. Moreover, it is sometimes possible in traditional DEA models that some inefficient DMUs are in fact better overall performers than some efficient ones as the unrestricted weight flexibility problem in DEA leads to an unreasonable self-rated scheme (Dyson and Thannassoulis, 1988, Wong and Beasley, 1990).

In order to increase the power of discriminating efficient DMUs, the cross-efficiency evaluation method was developed as a DEA extension technique to rank DMUs according to cross-efficiency scores which are linked to all DMUs (Sexton, Silkman, & Hogan, 1986). The main idea of the cross-evaluation method is to use DEA in a peer evaluation mode instead of a self-evaluation mode and the change brings about at least three advantages. Firstly, it provides a unique ordering among the DMUs (Sexton et al., 1986). Secondly, it eliminates unrealistic weight schemes without requiring the elicitation of weight restrictions from application area experts (Anderson, Hollingsworth, & Inman, 2002). Finally, the cross-evaluation method can effectively differentiate between good and poor performers (Boussofiane, Dyson, & Thanassoulis, 1991). All these advantages make the method be widely used for ranking performance of DMUs, for example, efficiency evaluation of nursing homes (Sexton et al., 1986), selection of a flexible manufacturing system (Shang & Sueyoshi, 1995), preference voting and project ranking (Green, Doyle, & Cook, 1996), industrial robot selection (Baker & Talluri, 1997), justification of advanced manufacturing technology (Talluri & Yoon, 2000), determination of the best labor assignment in a cellular manufacturing system (Ertay & Ruan, 2005), and measuring the performance of the nations participating in Summer Olympic Games (Wu et al., 2008, Wu et al., 2009).

In spite of wide applications, the method also has several disadvantages, such as the non-uniqueness of the DEA optimal weights which may reduce the usefulness of cross-efficiency evaluation (Doyle & Green, 1994). Actually, the optimal weights calculated by the original DEA model are generally not unique. Moreover, different calculation software may have different optimal weights (Despotis, 2002). Consequently, cross-efficiency scores may be generated arbitrarily. To solve this problem, several different secondary goals are proposed by Sexton et al. (1986) and Doyle and Green (1994) to deal with the issue of non-uniqueness. They present aggressive and benevolent formulations, representing two diametrically opposite strategies. The idea of the benevolent model is to identify optimal weights that can maximize not only the efficiency of the DMU under evaluation but also the average efficiency of other DMUs. In the case of the aggressive strategy, the optimal weights should maximize the efficiency of the evaluated DMU, whereas minimize the average efficiency of other units. See also Liang, Wu, Cook, and Zhu (2008) for extensions of these models which use series of secondary goals to provide each DMU with a set of optimal weights obtained by imposing some conditions on the resulting cross-efficiencies for all the DMUs. Similar thoughts also appeared in the article of Lim (2012), a minimax or a maximin objective is incorporated into cross-efficiency to choose the optimal weights that maximize the efficiency of a DMU under evaluation and subsequently improve (or reduce) the efficiency of the worst (or best) peer DMU as much as possible. A different idea can be found in Wang and Chin (2010), where they propose a “neutral” DEA model with which each DMU determines the weights only from its own point of view, neglecting their impact on the other DMUs. To be specific, the choice of weights made in their paper seeks to maximize the relative contribution of outputs by using a max–min formulation, which can effectively reduce the number of zero weights for outputs as well. Because Wang and Chin’s model just considered the output weights, Wang et al., 2011, Wang et al., 2011 extended their model to be an input and output-oriented weight determination DEA model, which can determine input and output weights simultaneously for the cross-efficiency evaluation. Aimed at approximating each weighted output or input component to weighted output or input sum, Örkcü and Bal (2012) suggested a new technique in order to coordinate the contribution of each input and output component in evaluating the efficiency. They also introduced goal programming method into cross efficiency (Örkcü & Bal, 2011), the new models depending on multiple criteria DEA model have three different efficiency concepts: classical DEA, minmax and minsum efficiency criteria. In the paper of Lam (2010), discriminate analysis, super-efficiency DEA model and mixed-integer linear programming were applied to determine the optimal weight sets, and the character of this new method is that each set of obtained weights can reflect the relative strengths of the efficient DMU. Interested readers may refer to Jahanshahloo et al., 2011, Jahanshahloo et al., 2011, Wang et al., 2011, Wang et al., 2011 and so on for more details.

The approach proposed in our paper actually accords with that by Wang and Chin (2010) and Örkcü and Bal (2012) in the sense that each DMU makes its own choice of weights without considering the effects on the other DMUs. In our particular case, the profile of weights to be used in the cross-efficiency evaluation is chosen by each DMU itself so that large differences in the weights attached to both inputs and outputs are avoided. In addition, non-zero weights are ensured.

So, the purpose of this study is to present a weight-balanced model to solve the non-uniqueness of the optimal weights in DEA models. The proposed model cannot only guarantee the maximum self-assessment efficiency of DMU under evaluation, but also reduce the differences in the weighted inputs and weighted outputs during the evaluation process. Another significant advantage of the weight-balanced DEA model is that it can effectively reduce the number of zero weights for inputs and outputs. In other words, all the resources of inputs and outputs in this newly proposed model can be made full use as much as possible. Similar to the issues considered in the current study, the ideas of using the multiplier bound approach for the assessment of efficiency without slacks in Ramón, Ruiz, and Sirvent (2010a) was further extended by them (2010b) to be used in cross-efficiency evaluations. Unlike our study herein, the models they used looked for the profiles with the least dissimilar weights (not the weighted data, i.e., weighted inputs and weighted outputs), however, they guaranteed non-zero weights. Besides, the first procedure in their methods was a nonlinear programming, which may make model calculation become more difficult.

The rest of this paper is organized as follows. Section 2 presents the cross-efficiency evaluation approach. Weight-balanced model is introduced in Section 3. The weight restriction model is shown in Section 4. Section 5 gives two illustrative examples, and conclusion and remarks are shown in Section 6.

Section snippets

Cross-efficiency evaluation

Using the traditional denotations in DEA, we assume that there is a set of n DMUs, and each DMUj (j = 1, 2,  , n) produces s different outputs using m different inputs which are denoted as xij(i = 1, 2,  , m) and yrj (r = 1, 2,  , s), respectively. For any evaluated DMUd (d = 1, 2,  , n), the efficiency score Edd can be calculated by the following CCR model.maxr=1sμrdyrd=Edds.t.i=1mωidxij-r=1sμrdyrj0,j=1,2,,n.i=1mωidxid=1.ωid0,i=1,2,,m.μrd0,r=1,2,,s.

For each DMUd (d = 1, 2,  , n), we can obtain a group of

A weight-balanced model for cross-efficiency evaluation

It is noticed that the selection of input and output weights only cares how to maximize or minimize the efficiencies of other DMUs under evaluation in the cross-efficiency method with most commonly used secondary goals. This approach usually generates the case that the weights at very small values (or even zero) are assigned to some inputs or outputs and very large values to other inputs or outputs. With this scenario, in the forms of weighted inputs and weighted outputs, some input or output

Avoiding zero weights

As can be seen in CCR model or the traditional cross-efficiency model (e.g. benevolent and aggressive models), these models cannot avoid the appearance of zero weights, leading to the situation that DEA assesses DMUs only with some weights of inputs and outputs, and ignores the other variables by giving them a zero weight. For this problem, earlier scholars suggested some models that integrated value judgments, such as the Assurance Regions (ARs) models (Thompson, Singleton, Thrall, & Smith,

Illustrations

To illustrate the proposed methods above, we consider two numerical examples with the data presented in Table 1, Table 3, respectively.

Conclusions

In most practical cases, the standard DEA models assess the efficiency units by using reference points on the frontier of the production possibility set (PPS), which are usually not Pareto-efficient, especially for the inefficient units. The reason is that these models generally give zero weights to inputs or outputs with lower values and very large weights to other inputs or outputs with higher values. In other words, strictly positive values for the optimal slacks are always got by their

Acknowledgments

The research is supported by National Natural Science Funds of China for Innovative Research Groups (No. 71121061), National Natural Science Funds of China (No. 70901069), Research Fund for the Doctoral Program of Higher Education of China for New Teachers (No. 20093402120013), Research Fund for the Excellent Youth Scholars of Higher School of Anhui Province of China (No. 2010SQRW001ZD), Social Science Foundation of Anhui, China (AHSKF09-10D116), the Fundamental Research Funds for the Central

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