Kazuyuki Sekitani

An occurrence of multiple projections in DEA-based measurement of technical efficiency: Theoretical comparison among DEA models from desirable properties

This study discusses nine desirable properties that a measure of technical efficiency (TE) needs to satisfy from the perspective of production economics and optimization. Seven data envelopment analysis (DEA) models are theoretically compared from a viewpoint of nine TE criteria. All the seven DEA models suffer from a problem of multiple projections even though a unique projection for efficiency comparison is one of the nine desirable properties. Furthermore, all the DEA models violate the property on aggregation of inputs and outputs. Thus, the seven DEA models do not satisfy all desirable TE properties. In addition, the comparison provides us with the following guidelines: (a) The additive model violates all desirable TE properties. (b) Russell measure and SBM (=ERGM) perform as well as RAM as a non-radial measure. If we are interested in strict monotonicity, the two models outperform the other DEA models including RAM. In contrast, if we are interested in translation invariance, RAM is better than Russell measure and SBM (=ERGM). (c) The radial measures (CCR and BCC) have the property of linear homogeneity. (d) The CCR model is useful for measuring a frontier shift among different periods. (e) If a data set contains a negative value, RAM becomes a DEA model to handle the negative value because it has the property of translation invariance. After examining the desirable TE properties, this study proposes a new approach to deal with an occurrence of multiple projections. The proposed approach includes a test to examine an occurrence of multiple projections, a mathematical expression of a projection set, and a selection process of a unique reference set as the largest one covering all the possible reference sets.

THE MEASUREMENT OF RETURNS TO SCALE UNDER A SIMULTANEOUS OCCURRENCE OF MULTIPLE SOLUTIONS IN A REFERENCE SET AND A SUPPORTING HYPERPLANE

This research theoretically explores the measurement of RTS (Returns to Scale) under a possible occurrence of multiple solutions in DEA (Data Envelopment Analysis). In this study, the occurrence of multiple solutions is classified into Type I and Type II. Type I is an occurrence of multiple solutions in a reference set. Type II is an occurrence of multiple solutions on a supporting hyperplane passing on the reference set. Both Types I and II are very well known among DEA researchers, but previous research has not sufficiently explored a simultaneous occurrence of Type I and Type II in the RTS measurement. The two types of multiple solutions influence a degree of RTS in the DEA measurement. Such a quantitative issue on RTS is examined from the perspective of the Type I and Type II problems. To deal with such difficulties, a new linear programming approach is proposed to identify all efficient DMUs (Decision Making Units) that consist of a reference set, even if multiple solutions occur on the reference set. Based upon the research result, we can identify when multiple solutions of Type I and/or Type II occur on the RTS measurement and how to deal with such difficulties. Our research result makes it possible to measure a degree of scale economies (RTS) under the simultaneous occurrence of Type I and Type II.

DEA CONGESTION AND RETURNS TO SCALE UNDER AN OCCURRENCE OF MULTIPLE OPTIMAL PROJECTIONS

Economic implications of congestion have been recently discussed in many DEA (data envelopment analysis) studies. In addition, several previous research efforts have explored a theoretical linkage between returns to scale (RTS) and the concept of congestion, because the two economic concepts are closely connected to each other. Tone and Sahoo [Tone, K., Sahoo, B.K., 2004. Degree of scale economies and congestion: A unified DEA approach. European Journal of Operational Research 158, 755-772] have published the theoretical linkage in this journal. All of the previous studies, including their research (2004), assume a unique optimal solution in the investigation on DEA-based congestion. When multiple solutions occur in DEA-based congestion measurement, the economic implications of congestion obtained from the previous research are all problematic from both theoretical and practical perspectives. To deal with the issue, this study explores how to deal with the occurrence of multiple solutions in the DEA-based congestion measurement. This study proposes a new approach for the congestion measurement and theoretically compares the proposed approach with Tone and Sahoo (2004).

Input–output substitutability and strongly monotonic p-norm least distance DEA measures

In DEA, there are two frameworks for efficiency assessment and targeting: the greatest and the least distance framework. The greatest distance framework provides us with the efficient targets that are determined by the farthest projections to the assessed decision making unit via maximization of the p-norm relative to either the strongly efficient frontier or the weakly efficient frontier. Non-radial measures belonging to the class of greatest distance measures are the slacks-based measure (SBM) and the range-adjusted measure (RAM). Whereas these greatest distance measures have traditionally been utilized because of their computational ease, least distance projections are quite often more appropriate than greatest distance projections from the perspective of managers of decision-making units because closer efficient targets may be attained with less effort. In spite of this desirable feature of the least distance framework, the least distance (in) efficiency versions of the additive measure, SBM and RAM do not even satisfy weak monotonicity. In this study, therefore, we introduce and investigate least distance p-norm inefficiency measures that satisfy strong monotonicity over the strongly efficient frontier. In order to develop these measures, we extend a free disposable set and introduce a tradeoff set that implements input–output substitutability.

Computational strategy for Russell measure in DEA: Second-order cone programming

Abstract In aggregation for data envelopment analysis (DEA), a jointly measured efficiency score among inputs and outputs is desirable in performance analysis. A separate treatment between output-oriented efficiency and input-oriented efficiency is often needed in the conventional radial DEA models. Such radial measures usually need to measure both that a current performance attains an efficiency frontier and that all the slacks are zero on optimality. In the analytical framework of the radial measure, Russell measure is proposed to deal with such a difficulty. A major difficulty associated with the Russell measure is that it is modeled by a nonlinear programming formulation. Hence, a conventional linear programming algorithm, usually applied for DEA, cannot solve the Russell measure. This study newly proposes a reformulation of the Russell measure by a second-order cone programming (SOCP) model and applies the primal–dual interior point algorithm to solve the Russell measure.

A Use of Analytic Network Process for Supply Chain Management

Strategic decision analysis on Supply Chain Management (SCM) involves tangible and intangible factors and it needs to be evaluated by interdependent multiple criteria. Analytic Hierarchy Process (AHP) is a practical and popular decision-making tool for SCM. Interdependency among criteria becomes a source of difficulty in directly applying AHP to various SCM decision problems. Hence, the managerial issue is examined by Analytic Network Process (ANP), the general form of AHP. This study addresses a new use of ANP on SCM strategic decision analysis such as a supplier selection and improvement of supply chain performance.

Decomposing the efficient frontier of the DEA production possibility set into a smallest number of convex polyhedrons by mixed integer programming

This paper extends the works by Olesen and Petersen (2003), Russell and Schworm (2006) and Cooper et al. (2007) about describing the efficient frontier of a production possibility set by the intersection of a finite number of closed halfspaces, in several ways. First, we decompose the efficient frontier into a smallest number of convex polyhedrons, or equivalently into a new class of efficient faces, called maximal efficient faces (MEFs). Second, we show how to identify all MEFs even if full dimensional efficient faces do not exist. Third, by applying the MEF decomposition to various real-world data sets, we demonstrate the validity of the MEF decomposition and how it can contribute to the DEA literature. Finally, we illustrate how to use the identified MEFs in practice.

Monotonicity of minimum distance inefficiency measures for Data Envelopment Analysis

This research explores the minimum distance inefficiency measure for the Data Envelopment Analysis (DEA) model. A critical issue is that this measure does not satisfy monotonicity, i.e., the measure may provide a better evaluation score to an inferior decision making unit (DMU) than to a superior one. To overcome this, a variant called the extended facet approach has been introduced. This approach, however, requires a certain regularity condition to be met. We discuss several special classes of the DEA model, and show that for these models, the minimum distance inefficiency measure satisfies the monotonicity property without the regularity condition. Moreover, we conducted computational experiments using real-world data sets from these special classes, and demonstrated that the extended facet approach may overestimate the performance of a DMU.