Extended secondary goal models for weights selection in DEA cross-efficiency evaluation

Highlights

• Desirable targets and undesirable targets for each DMU are identified.

• The targets are reachable and more reasonable for the DMUs.

• Several secondary goal models are proposed considering both kinds of targets.

• Strong ability of discriminating the DMUs are shown in application examples.

Abstract

Data Envelopment Analysis (DEA) has been extended to cross-efficiency evaluation for ranking decision making units (DMUs) and eliminating unrealistic weighting schemes. Unfortunately, the non-unique optimal weights problem in DEA has reduced the usefulness of this extended method. Aiming at solving this problem, we first incorporate a target identification model to get reachable targets for all DMUs. Then, several secondary goal models are proposed for weights selection considering both desirable and undesirable cross-efficiency targets of all the DMUs. Compared with the traditional secondary goal models, the cross-efficiency targets are improved in that all targets are always reachable for the DMUs. In addition, the proposed models considered the DMUs’ willingness to get close to their desirable cross-efficiency targets and to avoid their undesirable cross-efficiency targets simultaneously while the traditional secondary goal models considered only the ideal targets of the DMUs. Finally, the calculation results of our proposed models are compared with those of some other traditional methods for two published examples: an efficiency evaluation of six nursing homes and an R&D project selection.

Introduction

Data Envelopment Analysis (DEA), originally proposed by Charnes, Cooper, and Rhodes (CCR) (1978), is a non-parametric method for evaluating the performance of a group of homogenous decision making units (DMUs) with multiple inputs and multiple outputs (Cook & Seiford, 2009; An, Yan, Wu, & Liang, 2015). Its main idea is to maximize the ratio of the sum of weighted outputs to the sum of weighted inputs of the DMU under evaluation, under the constraints that the ratios of all units cannot be greater than 1. The maximum ratio of each DMU is then defined as its efficiency value and a DMU is said to be DEA efficient if its efficiency score equals 1. The traditional DEA models (e.g. CCR or BCC¹ model) use a self-evaluation mode, which allows each DMU to measure its efficiency with favorable weights of its own choice. This may lead to the situation that many DMUs are evaluated as efficient, and the DEA efficient DMUs cannot be further discriminated. This lack of discriminative power is one of the main drawbacks of DEA (Wang & Chin, 2010).

For research seeking to overcome the DEA weakness in discriminating between efficient DMUs, Cooper, Seiford, and Tone (2007) proposed some guidelines. They suggested that the number of the DMUs should be no smaller than the maximum between $m*s$ and $3*(m+s)$ if a good discrimination is deemed to be achieved in the evaluation results, where m and s are the numbers of inputs and outputs of the DMUs respectively. Some scholars have succeeded in improving the traditional DEA models and have proposed some new methods to improve the discriminative power of DEA. The super-efficiency evaluation method proposed by Andersen and Petersen (1993) is a typical one. In super-efficiency evaluation, the DEA efficient DMUs may obtain efficiency scores that are larger than 1 which will further discriminate the DEA efficient DMUs. DEA supper-efficiency evaluation has been widely researched after being proposed. More detailed extensions and discussions of this method can be seen in Zhu, 1996, Zhu, 1999, Khodabakhshi, 2007, Jahanshahloo et al., 2011, Chen et al., 2013, Du et al., 2015. One more method for addressing the problem is the DEA common-weight evaluation method. Cook et al., 1990, Roll et al., 1991 firstly proposed to use common weights for efficiency evaluation in DEA in the application background of performance evaluation and ranking of highway maintenance units. Further work on DEA common-weight evaluation can be seen in Kao and Hung, 2005, Zohrehbandian et al., 2010, Sun et al., 2013. There are still other methods for enhancing the discriminative power of DEA, such as the benchmarking ranking techniques (Lu and Lo, 2009, Sinuany-Stern et al., 1994, Sueyoshi, 1999), the multi-criteria decision-making methodologies (Li and Reeves, 1999, Strassert and Prato, 2002, Wang and Jiang, 2012), and the context-dependent DEA method (Chen et al., 2005, Seiford and Zhu, 2003).

Among all the methods for addressing the problem of lack of discrimination power in DEA, the most popularly studied one is the cross-efficiency evaluation method (Sexton, Silkman, & Hogan, 1986). In cross-efficiency evaluation, each DMU is requested not only to be self-evaluated, but also to be peer-evaluated. Specifically, each DMU determines a set of weights in the traditional DEA model, resulting in n sets of weights. Then, each DMU is evaluated by the n sets of weights obtaining n efficiency values. The overall efficiency of each DMU is the average of the n efficiency values. There are at least three principal advantages of cross-efficiency evaluation: ranking the DMUs in a unique order (Doyle & Green, 1995), eliminating unrealistic weight schemes without incorporating weight restrictions (Anderson, Hollingsworth, & Inman, 2002), and effectively distinguishing good and poor performers among the DMUs (Boussofiane, Dyson, & Thanassoulis, 1991). Because of these advantages, cross-efficiency evaluation has been extensively applied in, for instance, performance ranking of the nursing homes (Sexton et al., 1986), preference voting and project ranking (Green et al., 1996, Liang et al., 2008a, Wu et al., 2015), selecting suitable computer numerical control machines (Sun, 2002), determining efficient operators and measuring the labor assignment in cellular manufacturing systems (Ertay & Ruan, 2005), analyzing the efficiencies of listed coal enterprises (Ma & Li, 2011), supplier selection in public procurement (Falagario, Sciancalepore, Costantino, & Pietroforte, 2012), portfolio selection in Korean stock market (Lim, Oh, & Zhu, 2014), and so on.

However, Doyle and Green (1994) pointed out that the optimal weights obtained by the traditional DEA model may not be unique, which will reduce the usefulness of cross-efficiency evaluation method. To solve this problem, they suggested incorporating secondary goals into cross-efficiency evaluation. Based on this suggestion, many secondary goal models have been proposed. For example, Liang et al. (2008a) proposed to use the DEA game cross-efficiency evaluation method. Their method contains the DEA game cross-efficiency model and an algorithm. The algorithm can generate for the DMUs a set of cross-efficiency scores that constitute a Nash equilibrium point. The DEA game cross-efficiency method was further extended and applied in Wu et al., 2009, Ma et al., 2014. Jahanshahloo, Hosseinzadeh Lofti, Yafari, and Maddahi (2011) incorporated a symmetric technique into DEA cross-efficiency evaluation and gave a secondary goal model which can choose symmetric weights for DMUs. Wu, Sun, and Liang (2012) proposed a weight-balanced DEA model in which the secondary goal is to reduce large differences in weighted data and decrease the number of zero weights. Contreras, 2012, Zohrehbandian and Gavgani, 2013 proposed weights selection models in which the secondary goal is to optimize the rank position of the DMU under evaluation. Maddahi, Jahanshahloo, Hosseinzadeh Lotfi, and Ebrahimnejad (2014) suggested a secondary goal model in which the goal is to optimize proportional weights for the DMUs. More recently, Wu et al. (2015) proposed a DEA cross-efficiency evaluation based on Pareto improvement. Their approach can finally generate for the DMUs a set of cross-efficiency scores that constitute a Pareto optimal solution which is claimed to be more acceptable by all the DMUs. Apart from using secondary goal models for addressing the non-unique optimal weights problem, there are still a few other methods. For instance, Cook and Zhu (2014) proposed a unit-invariant multiplicative DEA model. This model can obtain the maximum and unique cross-efficiency score for each DMU directly, which means it is not essential to identify each DMU’s unique set of optimal weights.

Among the current cross-efficiency evaluation models, the most commonly used are the aggressive and benevolent models proposed by Doyle and Green (1994). The core idea of aggressive (benevolent) model is to get a set of optimal weights by making the efficiencies of the other DMUs as small (large) as possible while keeping the efficiency of the evaluated DMU at a predetermined optimal level. A similar idea also can be seen in Liang, Wu, Cook, and Zhu (2008). They extended Doyle and Green’s (1994) models by incorporating alternative secondary objective functions based on deviations to its ideal point. However, Wang and Chin (2010) pointed out that the ideal points in the model of Liang et al. (2008) are not realizable for the DEA-inefficient DMUs. They improved the models by changing the target efficiency from the ideal point 1 to the CCR efficiency. But it could be found that the CCR efficiency may be also difficult to achieve for a DMU, if the weights selected must keep the efficiency of the DMU under evaluation at its optimal level. In addition, the traditional benevolent and aggressive models only consider the desirable targets (1 or the original efficiency scores) as the referenced efficiencies for all DMUs. However, Baumeister et al., 2001, Wang and Chin, 2011, Dotoli et al., 2015 pointed out that the undesirable targets are also important indicators that the DMUs need to consider.

The purpose of this paper is to address the deficiencies of the traditional benevolent and aggressive models discussed above and propose several new and more appropriate secondary goal models. Firstly, we give a target identification model to get desirable and undesirable cross-efficiency targets for all the DMUs. Then, several secondary goal models with different purposes are proposed considering both desirable and undesirable identified targets for all the DMUs. For each DMU, our identified cross-efficiency targets are always reachable. The proposed secondary goal models do not just consider the DMUs’ willingness to get close to their desirable cross-efficiency targets like the traditional secondary goal models. In addition, they involve the DMUs’ expectations to get away from their undesirable targets as an extension.

The rest of this paper unfolds as follows. Section 2 gives a brief introduction of the DEA cross-efficiency evaluation method. Section 3 discusses the alternative traditional secondary goal models. Section 4 describes the target identification model and our proposed weights selection models. In Section 5, a numerical example and the application of R&D project selection are given using the models. Finally, conclusions are given in Section 6.