Theory and Methodology
An enhanced DEA Russell graph efficiency measure

https://doi.org/10.1016/S0377-2217(98)00098-8Get rights and content

Abstract

The measurement of productive efficiency is an issue of great interest. Since Farrell (Farrell, M.J., 1957. Journal of Royal Statistical Society, Series A 120, 253) implemented the first measure of technical efficiency, many researchers have developed new measures or have extended the already existing ones. The beginning of Data Envelopment Analysis (DEA) meant a new way of empirically measuring productive efficiency. Under some specific technologies, Farrell's measure was implemented giving rise to the first DEA models, CCR (Charnes, A., Cooper, W.W., Rhodes, E., 1978. European Journal of Operational Research 2, 429) and BCC (Banker, R.D., Charnes, A., Cooper, W.W., 1984. Management Science, 1078). The fact that these measures only account for radial inefficiency has motivated the development of the so-called Global Efficiency Measures (GEMs) (Cooper, W.W., Pastor, J.T., 1995. Working Paper, Departamento de Estadı́stica e Investigación Operativa, Universidad de Alicante, Alicante, Spain). In this paper we propose a new GEM inspired by the Russell Graph Measure of Technical Efficiency which avoids the computational and interpretative difficulties with this latter measure. Additionally, the new measure satisfies some other desirable properties.

Introduction

The measurement of technical efficiency started with the works of Debreu (1951) and Koopmans (1951). Following them, Farrell (1957) implemented the first measure of technical efficiency. Later, Färe and Lovell (1978) pointed out some difficulties with this measure which motivated the development of new measures of technical efficiency. In their work of 1978, these authors axiomatically approached this issue by suggesting some desirable properties that an ideal technical efficiency measure should satisfy, and then proposed a measure which satisfied them (it was later, in Färe et al. (1983), when it was noted that this measure does not satisfy homogeneity of degree −1 in inputs). This measure was called the Russell Input Measure of Technical Efficiency and was extended to the multiple output case by Färe et al. (1983). An output version, the Russell Output Measure of Technical Efficiency, was similarly defined by Färe et al. (1985). They also defined the Russell Graph Measure of Technical Efficiency which extends the two previous ones in the sense that it simultaneously accounts for the inefficiency in both inputs and outputs. There are also some graph versions of the Farrell measure. Färe et al. (1985) defined two of them: the hyperbolic and the generalized hyperbolic graph efficiency measures. Recently, Briec (1997) has also proposed a new graph-type extension of the Farrell measure. The main difference between Farrell and Russell measures is that Farrell measures are radial, whereas Russell ones are not, so they do not necessarily agree in classifying the same subset of units as efficient (in the particular case of DEA, they disagree when a DMU on the frontier has nonzero slacks). A comparative study of the performance of these measures which also includes two other measures can be found in Ferrier et al. (1994) and De Borger and Kerstens (1996).

The development of measures of efficiency has also been approached from the particular perspective of DEA. Initially, Farrell's measure was implemented in the LP problems which gave rise to the first DEA models, the CCR (Charnes, Cooper and Rhodes, 1978) and the BCC (Banker, Charnes and Cooper, 1984). Due to their radial nature, the efficiency scores obtained from these models overstate efficiency when nonzero slacks are present because they do not account for the nonradial inefficiency of the slacks. In contrast to these radial models, the additive model (Charnes et al., 1985) accounts for all sources of inefficiency, i.e., radial and nonradial inefficiency, both in inputs and in outputs. However, it does not directly provide an efficiency measure. To sort out these problems, several measures which consider all types of inefficiency detected by a given DEA model have been designed in the last few years, and it is still an issue of great interest. In Cooper and Pastor (1995) a complete revision with new proposal of these measures, which they call “Global Efficiency Measures” (GEM), can be found. Besides this, the authors list four basic properties that such a measure should satisfy.

GEMs can be defined both for radial and for nonradial DEA models. In this paper, we focus on the latter possibility. Next, we refer to two GEMs of this kind existing in the literature: the “Measure of Efficiency Proportions” (MEP) developed by Banker and Cooper (1994) and the “Range Adjusted Measure” (RAM) of Cooper et al. (1998) (see Appendix Afor the expression of these measures). These two measures, together with the TDT measure (Thompson and Thrall, 1994) which is not a GEM, are the new approaches to inefficiency measurement in DEA explained in Cooper and Tone (1997).

MEP should be used after an optimal solution of the additive or the invariant additive model is obtained. Therefore, we may have different values of this measure for the different alternate optima (if any). This also happens to all GEMs not included in the DEA model from which they are computed. A way of avoiding this problem is to include these GEMs as the objective of the models used for their computation. The difficulty with this including is that it usually gives rise to nonlinear programming problems which are complicated to solve, as in the case of MEP.

With these considerations in mind, we set two main goals for the GEM we are going to develop: (1) that it is well defined and (2) easy to compute. Additionally, we want our measure to satisfy some desirable properties, like the four basic ones listed by Cooper and Pastor (1995) and, in addition, that it is readily understood. RAM is an example of a measure meeting all these requirements, so we will take it as a reference to evaluate the behavior of our measure.

Aside from the mentioned approaches, the efficiency measurement with DEA models has been extended and enhanced in other directions. Some of these developments involve incorporating judgement or prior knowledge by restricting the range for the multipliers: see, for instance, Charnes et al. (1990) for the cone ratio model, Thompson et al. (1990) for the assurance region approach and Dyson and Thanassoulis (1988) which impose bounds on individual multipliers. In other extensions stochastic elements are introduced into the DEA models: see, for example, Sengupta (1987) for efficiency measurement in the stochastic case, Banker (1993) for maximum likelihood estimation of inefficiency and hypothesis testing and Land et al. (1993), Olesen and Petersen (1995) and Cooper et al. (1996) for the chance-constrained DEA approach.

The paper unfolds as follows. In Section 2we define the new measure and show the way to compute it by means of an LP problem. Section 3contains a set of desirable properties that the new measure satisfies. In Section 4we include an example to illustrate the performance of the measure. Section 5concludes.

Section snippets

A new DEA global efficiency measure

In this section we develop a new DEA efficiency measure which is closely related to the Russell measures. Assume that we have a set of n DMUs with m inputs and s outputs,{(Xj,Yj)=(x1j,…,xmj,y1j,…,ysj),j=1,…,n},where all inputs and outputs are positive. Let us also assume that the production possibility set T={(X,Y)/Y can be produced from X} satisfies the usual postulates of convexity, free disposability, constant returns to scale and minimum extrapolation (see Banker et al., 1984), as in the

Properties of Re

Färe and Lovell (1978) were the first ones who proposed a set of desirable properties that an ideal efficiency measure should satisfy, although these were enunciated for the particular case of an input oriented measure. Recently, Cooper and Pastor (1995) listed similar requirements for the DEA context and suggested some others. Next, we study the properties which the proposed Enhanced Russell Measure satisfies.

Theorem 1. The following is true for Re:

  • (i) 0 < Re  1.

  • (ii) Re =1 ⇔ DMU0 being evaluated

Example

In order to illustrate the performance of our new GEM, we have used the data relative to the agencies engaged in supplying water and related services in the Kanto region of Japan analyzed in Aida et al. (1998). These data contain 108 observations on five inputs (Number of Employees, Operating Expenses before Depreciation, Net Plant and Equipment, Population and Length of Pipes) and two outputs (Operating Revenues and Water Billed). Variable returns to scale on the efficient frontier were

Conclusions

This paper is concerned with the measurement of efficiency from a DEA perspective. We have defined a new nonradial nonoriented efficiency measure. Because of the analogy to the Russell measures, we have called it the Enhanced Russell Measure. First of all, it represents a solution for the problem of nonzero slacks when measuring efficiency by means of DEA models. However, other interesting goals have been achieved: the measure is well defined and can be easily computed by solving an LP problem.

Acknowledgements

We are grateful to two anonymous referees for their comments and to Generalitat Valenciana (GV-C-CN-10-068-96) for its financial support.

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