Meta-frontier analysis using cross-efficiency method for performance evaluation

doi:10.1016/j.ejor.2019.06.053

European Journal of Operational Research

Volume 280, Issue 1, 1 January 2020, Pages 219-229

https://doi.org/10.1016/j.ejor.2019.06.053 Get rights and content

Highlights

•
A framework for integrating cross-efficiency and meta-frontier analysis is developed.
•
Technology gap of different frontiers is measured in the cross-evaluation perspective.
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Different inefficient reasons are obtained by aggregating and decomposing cross efficiency.
•
Cross-evaluation strategy is constructed to ensure the stability of the optimal solution.

Abstract

Efficiency overestimation and technology heterogeneity are important factors that affect the use of data envelopment analysis. This paper introduces a meta-frontier analysis framework into a cross-efficiency method to develop a new efficiency evaluation method. This method can be used to calculate, aggregate, and decompose the cross efficiencies relative to the meta-frontier and group-frontier. Then the technology gap between these frontiers can be measured and more detailed information regarding the inefficiency of decision-making units can be obtained. This enables decision makers to improve efficiency in a targeted manner. Subsequently, the non-uniqueness of the optimal solution is discussed for the new method, and the cross-evaluation strategy is introduced to ensure the stability of the optimal solution. Finally, two examples are presented to illustrate the effectiveness of this method.

Introduction

Since the innovative work by Charnes, Cooper and Rhodes (1978), data envelopment analysis (DEA) has been widely accepted as a powerful performance assessment tool. It has been applied to various tasks, including energy efficiency assessments (Mardani, Zavadskas & Streimikiene, 2017) and bank efficiency assessments (Diallo, 2018). As a nonparametric approach, DEA evaluates the relative efficiency of homogeneous decision-making units (DMU) with multiple inputs and outputs. Furthermore, evaluation results can be obtained without the influence of the subjective wishes of decision makers. However, there are two obvious theoretical deficiencies in the traditional DEA method: evaluating DMUs under a unified technical environment (Kounetas & Napolitano, 2018) and allowing each DMU to measure its efficiency with its favorable weights (Liu, Song & Yang, 2019).

The evaluation process of traditional DEA is typically performed in a unified technical environment, assuming all DMUs use the same production technology. However, owing to differences in physical, human, and other characteristics in the production process (e.g. type of machinery, size and quality to the labor force), different DMUs may have different levels of production technology (O'Donnell, Rao & Battese, 2008). Walheer (2018) noted that DMUs with different technology levels could not be directly compared, because an evaluation result ignoring technical heterogeneity could be biased. Therefore, a meta-frontier analysis framework was introduced into the DEA method to address this issue. Meta-frontier analysis splits the meta-frontier of all DMUs into several group-frontiers to help measure the technology gaps between different DMUs during the DEA evaluation process. This obtains more objective evaluation results (Carrillo & Jorge, 2018). Feng et al., 2018, Li et al., 2017, and Tian and Lin (2018) applied meta-frontier analysis to DEA with undesirable outputs, dynamic DEA, and sequential DEA, respectively. Additionally, the impacts of a technology gap on efficiency evaluation were measured.

Although meta-frontier analysis can be used to solve the deficiency of the unified technical environment in DEA, the weights of inputs/outputs in this method are still partial to measuring DMUs’ efficiency by their own favors. Thus, evaluation results may be overestimated, and many DMUs may be evaluated as efficient, and cannot be further discriminated (Li, Zhu & Liang, 2018a). The cross-efficiency method is the main method used to address this problem. It combines self-evaluation with peer-evaluation to measure the efficiency of restricting the weight flexibility of traditional DEA. Then, the efficiency overestimation is solved accordingly (Ang, Chen & Yang, 2018). As one of the main research directions of DEA (Liu, Lu & Lu, 2016), the studies on cross-efficiency can be generally divided into two categories. The first category focuses primarily on the strategy formulation of cross-evaluation. The traditional aggressive/benevolent strategy, proposed by Doyle and Green (1994), regarded the others DMUs as competitors/collaborators to construct strategy formulation. Subsequently, many cross-evaluation strategies have been formulated to ensure the relative uniqueness of the optimal cross-efficiency solution in different decision-making situations. For example, the neutral strategy proposed by Wang and Chin (2010) paid attention to the further-optimized efficiency of each DMU, while some alterative strategies proposed by Davtalab-Olyaie (2019) improved the number of satisfied units according to the attitude of decision makers. The second category focuses on the rule making of cross-efficiency aggregation. Additive average is the most commonly used aggregation method, whereas other methods are recommended to improve the information utilization ratio of the cross-efficiency matrix, such as the ordered weighted averaging method (Wang & Chin, 2011) and the entropy-weight method (Song, Zhu & Peng, 2017). Unfortunately, just as with meta-frontier analysis, the cross-efficiency method can only be used to solve one of the two theoretical deficiencies of the traditional DEA method. Thus, it can solve the efficiency overestimation caused by the weights’ flexibility. However, the technology gap between different DMUs is not considered.

In summary, the cross-efficiency method can be used to evaluate the performance of DMUs more objectively by combining self-evaluation and peer-evaluation. However, the evaluation process is performed in a unified technical environment. If different DMUs have diverse levels of production technology, then the results obtained by the cross-efficiency method could be biased. Therefore, to obtain a fair and objective result, decision makers should conduct cross-evaluations in the meta-frontier analysis framework. The efficiency overestimation and technical heterogeneity can be solved at a later time.

To the best of our knowledge, there is no research in this area. This is probably the result of the different evaluation perspectives for cross-efficiency methods causing difficulties in measuring technical gaps. Therefore, the aim of this study is to develop a new meta-frontier cross-efficiency analysis framework to solve the problem of overestimation of efficiency in the traditional meta-frontier DEA method caused by self-evaluation and the difficulty in measuring technology gaps in the cross-evaluation environment.

The rest of this paper is organized as follows. Section 2 introduces the traditional cross-efficiency and meta-frontier methods. Section 3 describes a new cross-efficiency method with the meta-frontier analysis framework. Section 4 introduces the cross-evaluation strategy into the new method. In Section 5, the classification method is discussed, and two examples are presented to illustrate the proposed method in Section 6. Finally, conclusions are provided in Section 7.

Section snippets

Traditional cross-efficiency method

Assume that there are n DMUs to be evaluated, and each DMU_j (j = 1,…,n) uses m inputs, x_ij (i = 1,…, m), to produce s outputs, y_rj (r = 1,…,s). Then, according to the study of Charnes et al. (1978), efficiency is defined as a ratio of weighted outputs to weighted inputs, and the efficiency, θ_dd, of DMU_d can be obtained using the following input-oriented model: $\begin{matrix} θ_{d d} = Max \sum_{r = 1}^{s} u_{r d} y_{r d} \\ s . t . \sum_{i = 1}^{m} v_{i d} x_{i d} = 1; \\ \sum_{r = 1}^{s} u_{r d} y_{r j} - \sum_{i = 1}^{m} v_{i d} x_{i j} \leq 0, j = 1, \dots, n; \\ \forall v_{i d}, u_{r d} \geq 0; \end{matrix}$ where v_id, u_rd denote the decision variables. Model (1)

Self-evaluation based on meta-frontier analysis (SMFA)

According to the study of Sexton et al. (1986), the process of self-evaluation in cross-efficiency is equivalent to the traditional DEA method. Thus, the traditional meta-frontier method can be regarded as SMFA. Then, the self-evaluation ME (SME), self-evaluation GE (SGE), and self-evaluation MTR (SMTR) are equivalent to traditional ME, GE, and MTR (i.e., $SME = ME$ , $SGE = GE$ , and $SMTR = MTR$ ).

To obtain better decision information about efficiency, the inefficiency of DMU_d can be further decomposed into

CMFA with cross-evaluation strategy

Just as with the traditional DEA method, the optimal solution of CMFA may be not unique, owing to two reasons. On one hand, the value of $θ_{d j}^{C M E}$ obtained by model (2) may be unstable, because model (1) may have multiple optimal solutions. On the other hand, model (9) may also have multiple optimal solutions. The non-uniqueness affects the application of CMFA method. The cross-evaluation strategy can be used here to avoid this problem (Li et al., 2018b). The aggressive strategy of

Discussion on classification method

CMFA is used to evaluate the cross-efficiency of DMUs when considering the technology gap among them. Thus, classifying different DMUs into several groups is a necessary pre-requisite for applying CMFA.

In this section, subjective experience classification method is introduced, and then all DMUs are divided into several groups based on their technical levels. Its specific classification process is that according to the characteristics of DMU production technology (e.g., policy differences,

Illustrative applications

In this section, two examples are provided to illustrate the proposed CMFA method. One is a numerical example, which shows the calculation process and method comparison. The other is an empirical example demonstrating its application in practice. All computational processes are accomplished with MATLAB R2014b.

Conclusion

As one of the most popular evaluation tools, the DEA method has attracted a great deal of attention around the world. However, efficiency overestimation and technological heterogeneity are important factors affecting the development of DEA. By introducing the meta-frontier analysis framework to the cross-efficiency method, a new CMFA method was proposed to grasp the relationship between the meta-frontier and group-frontier in the cross-evaluation environment, solving the efficiency

Acknowledgments

This research is supported by National Natural Science Foundation of China (#71801050, #71371053, #71872047, #71701050), and Social Science Planning Fund project of Fujian Province (#FJ2018C014, #FJ2017C033), Natural Science Foundation of Fujian Province (#2019J01637).

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Decision SupportMeta-frontier analysis using cross-efficiency method for performance evaluation

Highlights

Abstract

Introduction

Section snippets

Traditional cross-efficiency method

Self-evaluation based on meta-frontier analysis (SMFA)

CMFA with cross-evaluation strategy

Discussion on classification method

Illustrative applications

Conclusion

Acknowledgments

Computers & Industrial Engineering

Computers & Industrial Engineering

European Journal of Operational Research

Energy Economics

Transportation Research Part D

Economic Modelling

Resources Policy

Fuzzy Sets and Systems

Economic Analysis and Policy

Economic Modelling

Expert Systems with Applications

Expert Systems with Applications

Socio-Economic Planning Sciences

European Journal of Operational Research

Decision Support
Meta-frontier analysis using cross-efficiency method for performance evaluation