Non-Discretionary Factors and Imprecise Data in DEA
F. Hosseinzadeh Lotﬁ,1 and G. R. Jahanshahloo Dept. of Math., Science and Research Branch Islamic Azad University, Tehran 14515-775, Iran M. Esmaeili Dept. of Math., Islamic Azad University, Shahrekord P.O. Box 166, Shahrekord, Iran Abstract Discretionary models of data envelopment analysis (DEA) assume that all inputs and outputs can be varied at the discretion of management or other users. In any realistic situation, however, there may exist ”exogenously ﬁxed” or non-discretionary factors that are beyond the control of a DMU’s management, which also need to be considered. Also DEA requires that the data for all discretionary inputs and outputs be known exactly. The aim of this paper is measuring the relative eﬃciency of decision making units with non-discretionary inputs and interval discretionary data.
Keywords: DEA, Interval data, Non-Discretionary inputs, Eﬃciency
Data envelopment analysis (DEA) is a mathematical programming approach for measuring and evaluating the relative eﬃciency of peer decision making units (DMUs) with multiple inputs and multiple outputs (Cooper et al., 2000). Discretionary models of DEA assume that all data are discretionary, i.e., controlled by the management of each DMU and varied at its discretion. In real world situations, however, there may exist ”exogenously ﬁxed” or nondiscretionary factors that are beyond the control of a DMU’s management, which also need to be considered. Banker and Morey (1996a) developed the ﬁrst model for evaluating DEA eﬃciency with ”exogenously ﬁxed” inputs and 1
Corresponding author:Farhad Hosseinzadeh Lotﬁ, E-mail:hosseinzadeh lotﬁ@yahoo.com
F. Hosseinzadeh Lotﬁ et al.
outputs in forms like ”age of store” in an analysis of a network of fast-food restaurants (See Ray 1991; also Roggiero 1996; Roggiero 1998; Mu˜ iz, 2006). n Some examples of non-discretionary factors in the DEA literature are the number of competitors in the branches of a restaurant chain, snowfall or weather in evaluating the eﬃciency of maintenance units, soil characteristics and topography in diﬀerent farms, age of facilities in diﬀerent universities, the populations of wards in evaluating the relative eﬃciency of public libraries. On the other hand, DEA requires that the data for all discretionary inputs and outputs be known exactly. When some discretionary data are unknown decision variables, such as interval data, ordinal data, and ratio bounded data, the DEA models become nonlinear programming problems, and this is called imprecise DEA (IDEA)(See Cooper et al. 1999; Entali et al., 2002; Zhu 2003; Zhu 2004; Despotis and Smirlis 2002). In this paper we introduce an approach for dealing with non-discretionary inputs and discretionary interval data in DEA. Our approach is to transform a non-linear DEA model to a linear programming equivalent, on the basis of the original data set, by applying transformations only to the variables. Upper and lower bounds for the relative eﬃciency scores of the DMUs are then deﬁned as natural outcomes of our formulations. The remainder of this paper is structured as follows: Section 2 introduces nondiscretionary models. In Section 3 we introduce models with non-discretionary inputs and interval discretionary data. In Section 4 we get upper and lower bounds of eﬃciency. In Section 5 we will classify units based upon their eﬃciency scores. Section 6 contains a numerical example. Some conclusions are given in Section 7.
DEA with non-discretionary factors
There are a number of modiﬁcations that have been made to the programming models to control for non-discretionary variables (See Mu˜ iz 2006). Two of n these modiﬁcations are presented below. 2.1 Banker and Morey’s model (modiﬁed CCR model) Banker and Morey (1996a) provided the ﬁrst DEA model for evaluating eﬃciency in the presence of...