Equivalence in two-stage DEA approaches

Abstract

Data envelopment analysis (DEA) is a linear programming problem approach for evaluating the relative efficiency of peer decision making units (DMUs) that have multiple inputs and outputs. DMUs can have a two-stage structure where all the outputs from the first stage are the only inputs to the second stage, in addition to the inputs to the first stage and the outputs from the second stage. The outputs from the first stage to the second stage are called intermediate measures. This paper examines relations and equivalence between two existing DEA approaches that address measuring the performance of two-stage processes.

Introduction

Data envelopment analysis (DEA) is an approach for measuring the relative efficiency of peer decision making units (DMUs) that have multiple inputs and outputs. While the definition of a DMU is generic and DMUs can be in various forms such as hospitals, products, universities, cities, courts, business firms, and others, DMUs can have a two-stage structure in many cases. For example, banks use labor and assets to generate deposits which are in turn used to generate load incomes. In such a setting, a DMU represents a two-stage process and intermediate measures exist in-between the two stages. The first stage uses inputs to generate outputs which become the inputs to the second stage. The first stage outputs are therefore called intermediate measures. The second stage then uses these intermediate measures to produce outputs. A key feature here is that the first stage’s outputs are the only inputs to the second stage, i.e., in addition to the intermediate measures, the first stage does not have its own outputs and the second stage does not have its own inputs.

An usual attempt to deal with such two-stage processes is to apply the standard DEA model to each stage (see, e.g., Seiford and Zhu, 1999). However, as noted in Zhu’s, 2003, Chen and Zhu’s, 2004, such an approach may conclude that two inefficient stages lead to an overall efficient DMU with the inputs of the first stage and outputs of the second stage. Consequently, improvement to the DEA frontier can be distorted, i.e., the performance improvement of one stage affects the efficiency status of the other, because of the presence of intermediate measures.

Based upon the variable returns to scale DEA model (Banker et al., 1984), Chen and Zhu (2004) develop a linear DEA type model where each stage’s efficiency is defined on its own production possibility set. The two production possibility sets are linked with the intermediate measures which are set as decision variables for each DMU under evaluation. Chen and Zhu’s (2004) model guarantees an overall efficient two-stage process when each stage is efficient. For inefficient DMUs, Chen and Zhu (2004) model provides a DEA projection with a set of optimal intermediate measures.

Kao and Hwang (2008), on the other hand, modify the standard DEA model by taking into account the series relationship of the two stages within the whole process. Under their framework, the efficiency of the whole process can be decomposed into the product of the efficiencies of the two sub-processes. Note that such an efficiency decomposition is not available in the standard DEA approach of Seiford and Zhu (1999) and the two-stage approach of Chen and Zhu (2004).

The current paper studies the relationship between the approaches of Chen and Zhu’s, 2004, Kao and Hwang’s, 2008. Note that the approach of Kao and Hwang (2008) is developed under the assumption of constant returns to scale in the multiplier CCR DEA model of Charnes et al. (1978). We show that the CCR version of the Chen and Zhu (2004) model can be equivalent to the Kao and Hwang’s (2008) model.

The rest of the paper is organized as follows. The next section presents the Kao and Hwang (2008) model and then the Chen and Zhu (2004) model. The relation and equivalence between the two approaches are then studied. Two data sets are then used to illustrate our discussion. Section 5 are given at last.