Measurement of returns to scale using a non-radial DEA model: A range-adjusted measure approach

Abstract

This research theoretically explores the measurement of returns to scale (RTS), using a non-radial DEA (data envelopment analysis) model. A range-adjusted measure (RAM) is used as a representative of such non-radial models. Historically, a type of RTS has been discussed within an analytical framework of radial models. The radial-based RTS measurement is replaced by the non-radial RAM/RTS measurement in this study. When discussing the non-radial RAM/RTS measurement, this study finds a problem of multiple projections that cannot be found in the radial measurement. In this research, a new linear programming approach is proposed to identify all efficient DMUs (decision making units) on a reference set. The important feature of the proposed approach is that it can deal with a simultaneous occurrence of (a) multiple reference sets, (b) multiple supporting hyperplanes and (c) multiple projections. All of the three difficulties are handled by the proposed RAM/RTS measurement. In particular, we discuss both when the three different types of multiple solutions occur on the RAM/RTS measurement and how to deal with such difficulties. Our research results make it possible to measure not only the type of RTS but also the magnitude of RTS in the RAM measurement.

Introduction

Data envelopment analysis (DEA), first proposed by Charnes et al. (1978), is a managerial approach to evaluate performance/efficiency of various organizations in public and private sectors. The contribution of previous research efforts on DEA can be easily found in the bibliography of Tavares (2004) that lists more than 3000 works from 1951 to 2001. It is also known that DEA is now widely disseminated from USA to other industrial nations in Europe and Asia.

In DEA, each managerial entity to be evaluated is often referred to as a “decision making unit” (DMU). The performance of each DMU is characterized by a production process that uses multiple inputs to produce multiple outputs. A unique feature of DEA is that it relatively compares the achievement of a specific DMU with the remaining other DMUs in order to determine the performance level of the specific DMU (or so-called “DEA efficiency score”). In addition to the efficiency measurement, DEA can determine the type and magnitude of local RTS (returns to scale) of each DMU as an important performance measure.

Among these previous research efforts on DEA/RTS, this study pays attention to Sueyoshi (1999) as a starting point of our research. His study (1999) has discussed the analytical relationship between RTS and a supporting hyperplane(s) on a production possibility set. The research has explored both when multiple solutions occur on the supporting hyperplane (so, RTS) and how to deal with an occurrence of such multiple solutions on RTS. A problem of his study (1999) is that it does not clearly classify the type of multiple solutions from the perspective of RTS measurement.

To deal with the methodological issue, this study discusses the existence of four different types of multiple solutions occurring in the RTS measurement. The four types are referred to as (a) “Type 0: LP-proper multiple solutions”, (b) “Type 1: multiple reference sets”, (c) “Type 2: multiple supporting hyperplanes”, and (d) “Type 3: multiple projections”, respectively.

Type 0 indicates an occurrence of multiple solutions due to the fact that DEA is formulated by linear programming. This type of multiple solutions occurs on all DEA models. The occurrence of the multiple solutions is further classified into primal or dual degeneracy. The problem of Type 0 is very well known among OR researchers. Any linear programming textbook describes the issue. Hence, this study does not describe it in detail.

The occurrence of Type 0 does not imply an occurrence of multiple reference sets. The occurrence on multiple reference sets is referred to as “Type 1” in this study. This type of problem is closely linked to the RTS measurement, because RTS is determined by the position of a supporting hyperplane(s) in a data space. The supporting hyperplane is mathematically characterized by the reference set(s). Meanwhile, the problem of Type 2 implies an occurrence of multiple supporting hyperplanes on a production possibility set. The problem of Type 2 is well known among DEA researchers. However, a theoretical and practical linkage between Types 1 and 2 has not been clearly investigated in the previous DEA studies. In addition to Types 1 and 2, multiple projections may occur on an efficiency frontier. An occurrence of multiple projections is referred to as “Type 3” in this study. The issue of multiple projections has not been sufficiently explored in the previous RTS studies. [An exception may be found in Olesen and Petersen, 1996, Olesen and Petersen, 2003 who have investigated a use of multiple projections to identify a whole surface of a production possibility set.]

The purpose of this study is to explore how to measure RTS, using a non-radial model. The model shift from a conventional radial model to the non-radial model is important in terms of the RTS measurement, because the non-radial RTS measurement is not sufficiently explored in the previous studies. In this study, we discuss an existence of the three types of multiple solutions (Types 1–3) in the non-radial RTS measurement. Then, we document how to solve such difficulties. This study also discusses a quantitative RTS measurement under a simultaneous occurrence of the three types of multiple solutions.

The remaining structure of this article is organized as follows: Section 2 documents previous research efforts on the RTS measurement. This section also documents visually differences among the three types of multiple solutions: (a) multiple reference sets, (b) multiple hyperplanes and (c) multiple projections. Section 3 describes the two-stage approach proposed by Banker et al. (2004). We discuss what problems occur when their approach is applied to the RTS measurement, using a non-radial model (i.e., RAM: range-adjusted model). Section 4 proposes a new linear programming approach for the non-radial RTS measurement. The concept of scale economies (SE) is incorporated into the proposed approach in order to discuss quantitatively the non-radial RTS measurement. The research task is explored in Section 5. An illustrative example and computational results are documented in Section 6. A concluding comment and future extensions are summarized in Section 7.