finding the defining hyperplanes of production possibility set with variable returns to scale using the linear independent vectors
Subject Areas : StatisticsN. Rafati-‎Maleki 1 , M. Rostamy-Malkhalifeh 2
1 - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
2 - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Keywords: تحلیل پوششی دادهها, واحدتصمیمگیرنده, مجموعه امکان تولید, ابرصفحه سازای قوی, پاراتو کارا,
Abstract :
The Production Possibility Set (PPS) is defined as the set of all inputs and outputs of a system in which inputs can produce outputs. In Data Envelopment Analysis (DEA), it is highly important to identify the defining hyperplanes and especially the strong defining hyperplanes of the empirical PPS. Although DEA models can determine the efficiency of a Decision Making Unit (DMU), but they cannot present efficient frontiers of PPS completely. The notion of defining hyperplanes is crucial to marginal discussions, marginal rates, marginal rates of substitution, sensitivity analysis, returns to scale, and in particular, the efficiency analysis of DMUs. In this paper, we propose a new method to determine all strong efficient(Pareto-efficient) DMUs and strong defining hyperplanes of the PPS with variable returns to scale which strong efficient DMUs are located on them. Furthermore, we apply the proposed method to find the normal vectors or gradient of the strong defining hyperplanes of the PPS including strong efficient DMUs. Consequently, the equations of these hyperplanes are determined. To illustrate the ability of the proposed method, some numerical examples are finally provided. Our method can be easily implemented using existing packages for operation research, such as GAMS.
[1] A. Charnes, W. W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research 2 (1978) 429-444.
[2] R. D. Banker, A. Charens, W. W. Cooper, Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis, Management Science 30 (1984) 1078-1092.
[3] G. Yu, Q. Wei, P. Brockett, L. Zhou, Construction of all DEA efficient surfaces of the production possibility set under the GDEA, European Journal of Operational Research 95 (1996) 491-510.
[4] W. W. Cooper, S. Li, L. M. Seiford, K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software, Kluwer Academic Publisher, Norwell, Mass, 1999.
[5] G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, H. ZhianiRezai, F. Rezai Balf, Finding strong defining hyperplanes of Production Possibility Set, European Journal of Operational Research, 177 (2007) 42-54.
[6] G. R. Jahanshahloo, A. Shirzadi, S. M. Mirdehghan, Finding strong defining hyperplanes of PPS using multiplier form, European Journal of Operational Research 194 (2009) 933-938.
[7] A. Amirteimoori, S .Kordrostami, Generating strong defining hyperplanes of the production possibility set in data envelopment analysis, Applied Mathematics Letters 25 (2012) 605-609.
[8] K. Tone, M. Tsutsui, An epsilon-based measure of efficiency in DEA A third pole of technical efficiency, European Journal of Operational Research 207 (2012) 1554-1563.
[9] J. Aparicio, J. T. Pastor, A well-defined efficiency measure for dealing with closest targets in DEA, Applied Mathematics and Computation, 219, (2013) 9142-9154.
[10] ] A. Hadi.Vencheh, J. Jablonsky, A. Esmaeilzadeh, The slack-based measure model based on supporting hyperplanes of production possibility set, Expert Systems with Applications, 42 (2015) 6522-6529.
[11] M. Zohrehbandian, determination of defhning hyperplanes of DEA production possibility set, U.P.B. Sci. Bull. Series C, 78, (2016) 97-106.
[12] N. Rafati Maleki, M. Rostamy Malkhali Malkhalifeh, F. Hosseinzadeh Lotfi, Finding the defining hyperplanes of production possibility set with constant returns to scale using the linear independent vectors, Cogent Mathematics and Statistics, 5, (2018) 1447222.