Cross-efficiency evaluation in data envelopment analysis based on the perspective of fairness utility

Highlights

• The cross-efficiency based on a fairness utility is proposed for the first time.

• The selection on optimal weights is more reasonable and acceptable for DMUs.

• One algorithm is proposed to solve the nonlinear fairness utility model.

• Numerical examples are given to verify our proposed models.

• One algorithm is proposed to guarantee the unique optimal weights.

Abstract

Data envelopment analysis (DEA) has been widely applied as an effective data-driven tool in evaluating the efficiency of decision making units (DMUs). However, traditional DEA models evaluate DMU efficiencies in a self-evaluation mode. As an extension of traditional DEA, cross-efficiency evaluation links one DMU’s performance with others and has the appeal that scores arise from peer evaluation. Unfortunately, the problem of non-unique optimal weights has reduced the usefulness of this extended method. Although, some current secondary goal approaches have reduced the non-uniqueness of optimal weights, they are mostly based on the perspectives of improving or worsening the evaluated DMU’s or other DMUs’ position or efficiency, which fails to consider DMU’s fairness mentality that plays an important role in guiding human interaction in behavioral economics. To fill this gap, we propose the concept of a fairness utility to construct our secondary goal model. Specifically, the secondary goal is to maximize the minimum fairness of the other DMUs when keeping the evaluated DMU’s optimal efficiency. Two algorithms are proposed. One is used to solve the nonlinear fairness utility model. The other is used to guarantee the unique optimal weights. Finally, a numerical example is presented, and an empirical application is given to verify the proposed method.

Introduction

Developed by Charnes, Cooper, and Rhodes (1978), data envelopment analysis (DEA) is data-driven tool which has been widely used to measure the relative efficiency of a group of homogenous decision making units (DMUs) (Banker et al., 1984, Cook and Seiford, 2009, Dulá, 2011, Zhou et al., 2016). It is because DEA has its special advantages mentioned in many studies (e.g., Tone, 2001, Zheng and Padmanabhan, 2007, Zhu et al., 2020a). As we know, DEA has been extensively applied in the performance evaluations for hospitals (Du, Wang, Chen, Chou, & Zhu, 2014), universities (Rahimian & Soltanifar, 2013), banks (Wang et al., 2014, Li et al., 2020), and many other entities (Wu et al., 2016d, Li et al., 2019).

The traditional DEA models allow each DMU to choose its own optimal input/output weights to maximize its own efficiency value. In other words, all DMUs are self-evaluated and may use different input/output weights when determining efficiency scores. This principle may, however, have some disadvantages, since the self-evaluated weights may result in any advantages being overstated and the disadvantages being overlooked (Wang & Chin, 2010). Also, the traditional DEA models may evaluate many DMUs as fully efficient such that those efficient DMUs cannot be further ranked or distinguished.

Many methods have been proposed to solve the disadvantages mentioned above. Among them, the cross-efficiency evaluation is the best-known approach. Proposed by Sexton, Silkman, and Hogan (1986), the DEA cross-efficiency evaluation method evaluates efficiency using peer-evaluation instead of pure self-evaluation. A peer-evaluation system can yield a better sequencing for all DMUs (Doyle and Green, 1995, Anderson et al., 2002, Wang and Wang, 2013). Hence, DEA cross-efficiency has been extensively applied in efficiency evaluation (e.g., Shang and Sueyoshi, 1995, Ertay and Ruan, 2005, Wu et al., 2009, Li et al., 2018, Shiraz et al., 2016).

Notwithstanding the advantages, the DEA cross-efficiency evaluation method still has one main shortfall, which has to do with the non-uniqueness of optimal weights for each DMU (Sexton et al., 1986, Doyle and Green, 1994, Shi et al., 2019). Specifically, one DMU’s optimal weights may not be unique, which in turn will produce different efficiency values for the other n − 1 DMUs when using this DMU’s optimal weights (Despotis, 2002, Cui and Li, 2015, Kao and Liu, 2019, Wang et al., 2011). Many approaches have been proposed to address this problem. For example, Liang, Wu, Cook, and Zhu (2008b) generalized the traditional DEA cross-efficiency concept to develop game cross-efficiency. More precisely, each DMU is viewed as a player that seeks to maximize its own efficiency, under the condition that the cross-efficiency values of each of the other DMUs do not deteriorate based on the weights of each of the other DMUs.

Further, Wu, Chu, Sun, and Zhu (2016) proposed a cross-efficiency evaluation method based on Pareto improvement. The main idea behind their model is the intention to maximize the efficiency of a DMU under evaluation while keeping all DMUs’ cross-efficiency values no worse than their present cross-efficiency values based on the weights of the evaluated DMU. Cook and Zhu (2013) proposed a unit-invariant multiplicative DEA model to obtain the optimal cross-efficiency value for each DMU directly.

In addition, the perspective of meeting secondary goals when keeping the DMU’s optimal efficiency is commonly used (Sexton et al., 1986). For example, Doyle and Green (1994) first began to construct secondary goal models from two different perspectives: benevolent and aggressive. The benevolent strategy aims to maximize the sum efficiencies of the other n − 1 DMUs while maintaining the optimal efficiency of the DMU under evaluation. The aggressive strategy aims to minimize the sum efficiency of the other n − 1 DMUs under the same condition. Based on the two different strategies, Liang et al., 2008a, Wang and Chin, 2010, Lim, 2012; and Wu, Chu, Sun, Zhu, and Liang (2016) introduced a series of benevolent and aggressive models for different specific scenarios. There are also other perspectives about secondary goal models. For example, the secondary goal in Jahanshahloo, Lotfi, Jafari, and Maddahi (2011) is to select a symmetric set of weights. The secondary goal in Contreras (2012) is to optimize the ranking position of the DMU under evaluation. The secondary goal in Wu, Sun, and Liang (2012) is to select a balanced weight, that is, to lessen the large differences among weights and reduce the number of zero-weights. The secondary goal in Wu, Chu, Zhu et al. (2016) is to maximize the least satisfaction degrees among all the other DMUs. Recently, Davtalab-Olyaie (2019) proposed the secondary goal by considering the cardinality of satisfied set. The proposed secondary goal approach of Liu et al., 2019, Chen et al., 2019 is mainly based on prospect theory.

Surveying the previous studies concerning the cross-efficiency evaluation method, we find that they are based mostly on the perspectives of improving or worsening the position or efficiency of the evaluated DMUs or other DMUs. When an evaluated DMU selects its own weights to maximize its efficiency, it may also consider the fairness of what those choices impose on the other DMUs since its weights have helped to determine the efficiency values of other DMUs. From behavioral economics, the perspective of fairness is that players care not only about their own profit but also about how profits are obtained by all the other players (Cui, Raju, & Zhang, 2007). Fairness has been long recognized as one of the most important factors guiding human interactions in everyday life (Adams, 1965, Scheer et al., 2003). It is closely related to other-regarding preferences, such as status, altruism, reciprocity, and other common factors in the everyday lives of individuals, and these preferences also play an important role in the enterprise environment (Katok and Pavlov, 2013, An et al., 2020). Fairness has been widely used in behavioral operations management. For example, Cui et al. (2007) incorporated the concept of fairness in a conventional dyadic channel to investigate how fairness may affect channel coordination. De Bruyn and Bolton (2008) investigated whether fairness considerations are stable enough across bargaining situations to be quantified and used to forecast bargaining behavior accurately. Ho, Su, and Wu (2014) investigated how both types of fairness (peer-induced and distributional) might interact and influence economic outcomes in a supply chain.

In this paper, we use the concept of fairness to construct our models. The main contributions can be summarized as follows. First, the fairness mentality is firstly introduced into the secondary goal of cross-efficiency evaluation in DEA. Specifically, the secondary goal of the evaluated DMU is to maximize the minimum fairness corresponding to the other n − 1 DMUs when keeping its optimal efficiency. The fairness concept can further encourage the DMUs participating in the efficiency evaluation. Second, to handle the nonlinear nature of our proposed model, we propose a simple algorithm to solve it. Also, we further propose an algorithm to get a unique set of weights for each DMU. The results from the numerical studies show that our proposed model can further aggregate the efficiency values of all DMUs.

The rest of the paper is organized as follows. In Section 2, we briefly introduce some traditional DEA cross-efficiency evaluation methods. In Section 3, the DEA cross-efficiency model based on a Fairness utility is proposed, including the model construction, an algorithm for solving the nonlinear model, and an algorithm for guaranteeing the uniqueness of the input/output weights. In Section 4, the technique is applied to two examples to illustrate the proposed approach. Finally, conclusions are given in Section 5.