Super-efficiency and DEA sensitivity analysis

Abstract

This paper discusses and reviews the use of super-efficiency approach in data envelopment analysis (DEA) sensitivity analyses. It is shown that super-efficiency score can be decomposed into two data perturbation components of a particular test frontier decision making unit (DMU) and the remaining DMUs. As a result, DEA sensitivity analysis can be done in (1) a general situation where data for a test DMU and data for the remaining DMUs are allowed to vary simultaneously and unequally and (2) the worst-case scenario where the efficiency of the test DMU is deteriorating while the efficiencies of the other DMUs are improving. The sensitivity analysis approach developed in this paper can be applied to DMUs on the entire frontier and to all basic DEA models. Necessary and sufficient conditions for preserving a DMU’s efficiency classification are developed when various data changes are applied to all DMUs. Possible infeasibility of super-efficiency DEA models is only associated with extreme-efficient DMUs and indicates efficiency stability to data perturbations in all DMUs.

Introduction

During the recent years, the issue of sensitivity and stability of data envelopment analysis (DEA) results has been extensively studied. Some studies focus on the sensitivity of DEA results to the variable and model selection, e.g., Ahn and Seiford (1993) and Smith (1997). Most of the DEA sensitivity analysis studies focus on the misspecification of efficiency classification of a test decision making unit (DMU). However, we shall note that DEA is an extremal method in the sense that all extreme points are characterized as efficient. If data entry errors occur for various DMUs, the resulting isoquant may vary substantially (Sexton et al., 1986). In the current paper, as in many other DEA sensitivity studies, we say that the calculated frontiers of DEA models are stable if the frontier DMUs that determine the DEA frontier remain on the frontier after the particular data perturbations are made for all DMUs.

By updating the inverse of the basis matrix associated with a specific efficient DMU in a DEA linear programming problem, Charnes et al. (1985a) study the sensitivity of DEA model to a single output change. This is followed by a series of sensitivity analysis articles by Charnes and Neralic in which sufficient conditions preserving efficiency are determined (Charnes and Neralic, 1990).

Another type of DEA sensitivity analysis is based on super-efficiency DEA approach in which a test DMU is not included in reference set Andersen and Petersen, 1993, Seiford and Zhu, 1999. Charnes et al., 1992, Rousseau and Semple, 1995, Charnes et al., 1996 develop a super-efficiency DEA sensitivity analysis technique for the situation where simultaneous proportional change is assumed in all inputs and outputs for a specific DMU under consideration. This data variation condition is relaxed in Zhu (1996) and Seiford and Zhu (1998a) to a situation where inputs or outputs can be changed individually and the entire (largest) stability region which encompasses that of Charnes et al. (1992) is obtained. As a result, the condition for preserving efficiency of a test DMU is necessary and sufficient.

The DEA sensitivity analysis methods we have just reviewed are all developed for the situation where data variations are only applied to the test efficient DMU and the data for the remaining DMUs are assumed fixed. Obviously, this assumption may not be realistic, since possible data errors may occur in each DMU. Seiford and Zhu (1998b) generalize the technique in Zhu (1996) and Seiford and Zhu (1998a) to the worst-case scenario where the efficiency of the test DMU is deteriorating while the efficiencies of the other DMUs are improving. In their method, same maximum percentage data change of a test DMU and the remaining DMUs is assumed and sufficient conditions for preserving an extreme-efficient DMU’s efficiency are determined. Note that Thompson et al. (1994) use the strong complementary slackness condition (SCSC) multipliers to analyze the stability of the CCR model (Charnes et al., 1978) when the data for all efficient and all inefficient DMUs are simultaneously changed in opposite directions and in same percentages. Although the data variation condition is more restrictive in Seiford and Zhu (1998b) than that in Thompson et al. (1994), the super-efficiency-based approach may generate a larger stability region than the SCSC method does. Also, the SCSC method is dependent on a particular SCSC solution, among others, and therefore the resulting analysis may vary.

In this paper, we focus on the DEA sensitivity analysis methods based on super-efficiency DEA models. For the DEA sensitivity analysis based on the inverse of basis matrix, the reader is referred to Neralic (1994). It is well known that certain super-efficiency DEA models may be infeasible for some extreme-efficient DMUs. Seiford and Zhu (1999) develop the necessary and sufficient conditions for infeasibility of various super-efficiency DEA models. Although the super-efficiency DEA models employed in Charnes et al. (1992) and Charnes et al. (1996) do not encounter the infeasibility problem, the models used in Seiford and Zhu (1998a) do. Seiford and Zhu (1998a) discover the relationship between infeasibility and stability of efficiency classification. That is, infeasibility means that the CCR efficiency of the test DMU remains stable to data changes in the test DMU. Furthermore, Seiford and Zhu (1998b) show that this relationship is also true for the simultaneous data change case and other DEA models, such as BCC model of Banker et al. (1984) and additive model of Charnes et al. (1985b). This finding is critical since super-efficiency DEA models in Seiford and Zhu (1998b) are frequently infeasible for real-world data sets, indicating efficiency stability with respective to data variations in inputs/outputs associated with infeasibility.

This study attempts to generalize the results in Seiford and Zhu (1998b) to a situation where variable percentage data changes are assumed for a test DMU and for the remaining DMUs. We consider the same worst-case analysis as in Seiford and Zhu (1998b). It is shown that a particular super-efficiency score can be decomposed into two data perturbation components of a particular test DMU and the remaining DMUs. Also, necessary and sufficient conditions for preserving a DMU’s efficiency classification are developed when various data changes are applied to all DMUs. As a result, DEA sensitivity analysis can be easily applied if we employ various super-efficiency DEA models.

The current article proceeds as follows. In Section 2, we present some basic DEA models and define the data variations in a test frontier DMU and the remaining DMUs. In Section 3, we develop our sensitivity analysis method for simultaneous data changes in all DMUs. The sensitivity of the DEA constant returns to scale (CRS) model (e.g., CCR model) and of the DEA VRS (variable returns to scale) models (e.g., BCC model) is discussed. The assumption of same percentage data change of all DMUs in Seiford and Zhu (1998b) is relaxed. We discuss the efficiency stability of all frontier DMUs. Conclusions are given in Section 4.