Fuzzy upper bounds and their applications M. Soleimani-damaneh * Department of Mathematics, Faculty of Mathematical Science and Computer Engineering, Teacher Training University, 599 Taleghani Avenue, Tehran 15618, Iran Accepted 7 June 2006 Communicated by Prof. B.G. Sidharth Abstract This paper considers the concept of fuzzy upper bounds and provides some relevant applications. Considering a fuzzy DEA model, the existence of a fuzzy upper bound for the objective function of the model is shown and an effective approach to solve that model is introduced. Some dual interpretations are provided, which are useful for practical purposes. Applications of the concept of fuzzy upper bounds in two physical problems are pointed out. Ó 2006 Elsevier Ltd. All rights reserved. 1. Introduction In recent years, fuzzy systems have been used in a variety of problems ranging from quantum optics and gravity [18], particle systems [19–21,26], economical systems [13,25], medicine [1,3], to bioinformatics and computational biology [5,8]. A fuzzy function is a generalization of a classical function, and different features of the classical concept of a function can be considered to be fuzzy rather than crisp. Different degrees of fuzzification of the classical concept of a function were reviewed by Zimmermann [31], while we consider a hybrid concept of these types as a fuzzifying function. Also the concept of the upper bound of a fuzzy function is a generalization of the classical concept of upper bound in crisp anal- ysis. Analogously, a fuzzy upper bound notion for fuzzy functions can be introduced, after proposing a signed distance. Nowadays evaluation of decision making units (DMUs), by using the mathematical programming-based techniques, has allocated to itself a wide variety of research in operations research (OR) field. Data envelopment analysis (DEA) is one of those techniques and was first proposed by Charnes et al. [6]. One can find several fuzzy mathematical program- ming-based approaches to evaluate DMUs in the DEA literature. Kao and Liu [14,15] use the notion of fuzziness and they transform a fuzzy DEA model to a family of crisp DEA models by applying the a-cut approach. Sengupta [23] analyzes the results of fuzzy DEA models by Zimmermann’s method [30]. Hougaard [12] suggests the application of efficiency score aggregated by value judgments or manager opinions. Guo and Tanaka [11] extend the CCR efficiency score in the crisp case to the fuzzy number and they consider the relationship between DEA and regression analysis. 0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.06.042 * Tel.: +98 21 2093947; fax: +98 21 7507772. E-mail addresses: soleimani_d@yahoo.com, m_soleimani@tmu.ac.ir. Available online at www.sciencedirect.com Chaos, Solitons and Fractals 36 (2008) 217–225 www.elsevier.com/locate/chaos