Algorithmic Strategy for Assurance Region Analysis in Dea

An algorithm stral:egy is proposed for use with the assurance region (AR) approach in data envelopment analysis (DEA). The strategy addressed in this study characterizes and cla.5sifies all decision making units (DMUs) into several subsets, using the revised simplex method of linear programming. Then, each DMU subset is solved by a different algorithm. Experimental studies consisting of randomly generated data sets have confirmed that be proposed algorithm outperforms the conventional DEA use of the revised simplex method. An important feature related to the DEAjAR algorithm is that it can deal effectively with large data sets. L Introduction Charnes et al. [1 J have opened up a new nonparametric approach, referred to data envelopment analysis (DEA), tha,t can empirically determine the efficiency level of many organizations in public and privated sectors. The DEA development not only provides practitioners with an opportunity to enhance productive efficiency, but also provides researchers with numerous research issues related to efficiency measurement. As such example, Seiford [9J reports more than 400 DEA contributions in the last decade. In Japan, Tone [16J introduced the DEA technique and its related underlying concepts in terms of production economics. In order to describe the research objective of this study, this article starts with describing two research issues on which many DEA researchers have been paying recent attention. First, a series of research works have investigated DEA algorithms and these related computation theory. The need for DEA algorithm development has been first argued by Charnes et al. [2, 3J because DEA applications require tremendous computation efforts. An analytical way of classifying DMUs is also proposed in [2, 3J so as to improve DEA algorithmic efficiency. Although [2, 3J did not present any computational results concerning the proposed algorithm, these studies described explicitly the importance and need for further algorithmic development. The con1ributions of [2, 3J are summarized in the forms of representation and classification theorems. Following the theorems; Chang and Sueyoshi [5], Sueyoshi and Chang [13], and Sueyoshi [10, 12J have presented efficient algorithms applied to different DEA models, incorporating several different algorithmic strategies. The other important research area is to develop a new D EA approach (often referred to as "post-D EA approach") that is designed to measure the efficiency of each decision making unit (DMU) by restricting dual variables of DEA to acceptable ranges. The restriction on dual variables in the DEA method is important because it can incorporate a priori information concerning DEA efficiency :11easurement. As a consequence of such restriction, the following new perspectives are added to DEA applications: a. First, the restriction of dual variables reduces the number of efficient DMUs and 62 © 1992 The Operations Research Society of Japan A igorithm for DEA 63 more sharply delineates the best DMU(s) for a decision maker. [See [I1J for such an example representing how the number of efficient DMUs is reduced.J b. Second, the restriction provides DEA with an analytical scheme of measuring allocative efficiency (AE). [The conventional use of DEA focuses upon only technical efficiency (TE). The concept indicates the achievement of an efficient frontier. Meanwhile, the AE denotes the measure as to how a DMU operates, using input (output) quantities along the vector indicating by given input (output) prices. See [11 J for a graphical description regarding the difference between TE and AE.J In the DEA literature two approaches are proposed to restrict dual variables to previously specified ranges. That is, DEAl assurance region (DEAl AR) analysis is proposed in [14J and cone-ratio DEA method is proposed in [4J. [A detailed description concerning theoretical differences between the two post DEA approaches lies beyond the scope of this study.J The purpose of this study is to develop an efficient algorithm for measuring TE and AE within the framework of the DEAl AR approach. In this sense, this study may be considered as an important extension of previous studies [5, 10, 11, 12, 13J. Furthermore, this research will document several results of experimental simulation studies applied to the newly developed DEA algorithm. The remainder of this article is organized as follows. Section 2 describes DEAl AR approach and describes its unique features in terms of algorithmic development. Section 3 states the DEA algorithm for AR that incorporates several computational strategies to enhance its algorithmic efficiency. Section 4 exhibits the results of a simulation study in which the proposed algorithm is compared with the conventional use of the revised simplex method for DEAl AR. Conclusion and future extensions are summarized in the last section. 2. DEAl AR Approach 2.1 DEA ratio form This section reviews the original DEA ratio form and then extends its discussion into DEAl AR approach. Suppose that there are n DMUs denoted j E J, each of which yields a nonzero output vector }j = (Y1j, Y2j, ... , Ysj f ~ 0, using a nonzero input vector Xj = (X1j, X2j, ... , Xmj f ~ O. Here, the term "nonezero" indicates that at least one vector component is positive. The superscript "T" indicates the transpose of a vector. It i~ also assumed that there is no DMUs in J whose data domain is propotional to that of another DMU. [See Charnes et al. [3J for a discussion of this assumption.J Admitting that there are many different DEA models as presented in [9], this study focuses upon only the original DEA ratio form in which a specific DMUo uses Xo to produce Yo: minimize () n subject to L Xij)..j + ()Xio ~ 0, j=l i = 1,···,m ~ Yro, r = 1,··· ,8, ).. J ~ 0, j = 1,··· ,n. (1) As shown in [1], (1) is a linear programming form equivalent to the original nonlinear ratio form. The optimal () indicates a DEA efficiency score on the closed interval [O,lJ. The vector ).. = ()..1, )..2, ... ,)..nf is used to construct a convex hull covering all data points. Copyright © by ORSJ. Unauthorized reproduction of this article is prohibited.