Affiliations: Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran
Correspondence:
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Corresponding author. Nazila Aghayi, Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran. Tel.: +98 9143537139; Fax: +98 4533729820; E-mail: [email protected].
Abstract: Being a nonparametric method, data envelopment analysis (DEA) is confined to measuring the efficiency of a collection of decision making units (DMUs) consuming multiple crisp inputs to produce multiple crisp outputs. Since not all data in the real world have determined values, and input and output values for DMUs are often subject to fluctuation, the concept of fuzziness has been introduced to deal with such imprecise data. This study intends to evaluate the cost efficiency of DMUs in three different scenarios, with one distinct model proposed for each scenario. The main idea is to use the alpha-cut method and the extension principle to convert the fuzzy cost efficiency model into a family of conventional crisp DEA models by obtaining one lower bound and one upper bound for the cost efficiency score of a DMU for any α varying between 0 and 1. When the lower and upper bounds are invertible with respect to α, the membership function of the fuzzy efficiency of a DMU— which falls within the scope of parametric programming problems— can be obtained by finding the inverse of the lower and upper bounds as well as employing the extension principle. In this case, the value of fuzzy cost efficiency varies between 0 and 1. Otherwise, the cost efficiency scores of DMUs can be specified as intervals, which are actually α-cuts of the fuzzy membership function, by collecting results for different α values. Furthermore, we demonstrate that Farrell’s decomposition also holds for cost efficiency with fuzzy data. In addition, all DMUs are divided into three classes in each scenario, with cost-efficient and cost-inefficient DMUs falling into independent classes. In other words, units which are cost-inefficient in the upper bound for any α ranging between 0 and 1 will definitely be cost-inefficient in the lower bound, too, and the units will be cost-efficient in the upper bound if they are cost-efficient in the lower bound for a specific α varying between 0 and 1. Moreover, the upper bound of units that are cost-inefficient in the lower bound could not be judged in terms of cost efficiency. Finally, a practical example dealing with data on all branches of the National Bank of Iran across Ardabil Province, Iran, during 2012–2014 is provided to demonstrate the applicability of the proposed method.
