AN ε-FREE DEA AND A NEW MEASURE OF EFFICIENCY

Abstract

This study presents a DEA model without the conventional non-Archimedian infinitesimal E. This article also introduces a new DEA efficiency measure. Incorporating slacks in inputs and shortages in outputs, the new DEA measure expresses the relative efficiency of decision making units more properly than traditional one. 1 Introduction and Historical Background In their ingenious paper (lOJ, Charnes, Cooper and Rhodes introduced a fractional pro­ gramming method to measure the relative efficiency of a Decision Making Unit (DMU), which was solved by transforming the fractional programming into a linear programming problem via the Charnes-Cooper scheme (6J. The method was referred to as DEA (Da.ta En­ velopment Analysis). The DEA model proposed in (10J maintained an assumption; all the weighting values to inputs and outputs were assumed to be nonnegative. In the subsequent short communication (11), Charnes et. al. changed their DEA problems and required that the weights be strictly positive. Thus, t.he introduction of the non-Archimedian in­ finitesimal e was anticipated to distinguish bet.ween nonnegative and positive values. ( This problem was already discussed in (10J implicitly.) Although the subsequent discussions can be found in (5), (7), and (12), and the role of e has become unclear and weakened, it is still frequently used in the literature (e.g., (4),(8)) and in particular, in some cases of compu­ tational situations, values such as e = 10- 5 ,10- 6 (single precision) or e = 10- 12 (double precision) are conveniently employed to substitute for the non-Archimedian infinitesimal e. However, the approach may produce a theoretically contradicting issue. That is, we cannot uniquely determine what is the best~. Different e values yield different DEA re­ sults. Therefore, we need a completely e-free development of DEA from both theoretical and computational points of view. This article is organized as follows. Section 2 defines an input oriented DEA model based on the production possibility set. Its dual corresponds to the Charnes-Cooper­ Rhodes (CCR) model with the weights to inputs and outputs as variables. Then, we define a DMU as slackless if, for every optimal solution to the DEA model, it has no slack in inputs and no shortages in outputs. By a theorem of the alternative or the strong theorem of complemantary slackness, it will be proved that for a slackless DMU there is a strictly positive weight solution in the corresponding CCR model. Subsequently, for a DMU with non-zero slacks in an optimal solution to the DEA model, there exist no positive weight solutions in the CCR model. Section 3 defines the max-slack solution and shows a procedure to find it. The max-slack solution can be used for deciding whether the DMU is slackless or not. Then, we propose a method for finding positive weights for slackless DMUs. Thus, all jobs of the CCR model can be successfully achieved with no recourse to e. Section 4 introduces a new measure of relative efficiency, based on the max-slack solution, which