Fuzzy TOPSIS method based on alpha level sets with an application to bridge risk assessment * Ying-Ming Wang * , Taha M.S. Elhag School of Mechanical, Aerospace & Civil Engineering, The University of Manchester, PO Box 88, Manchester M60 1QD, UK Abstract This paper proposes a fuzzy TOPSIS method based on alpha level sets and presents a nonlinear programming (NLP) solution procedure. The relationship between the fuzzy TOPSIS method and fuzzy weighted average (FWA) is also discussed. Three numerical examples including an application to bridge risk assessment are investigated using the proposed fuzzy TOPSIS method to illustrate its applications and the differences from the other procedures. It is shown that the proposed fuzzy TOPSIS method performs better than the other fuzzy versions of the TOPSIS method. q 2005 Elsevier Ltd. All rights reserved. Keywords: TOPSIS; Fuzzy multiple criteria decision making; Fuzzy weighted average; Alpha level sets; Bridge risk assessment 1. Introduction TOPSIS method is a popular approach to multiple criteria decision making (MCDM) and has been widely used in the literature (Abo-Sinna & Amer, 2005; Agrawal, Kohli & Gupta, 1991; Cheng, Chan & Huang, 2003; Deng, Yeh & Willis, 2000; Feng & Wang, 2000, 2001; Hwang & Yoon, 1981; Jee & Kang, 2000; Kim, Park & Yoon, 1997; Lai, Liu & Hwang, 1994; Liao, 2003; Olson, 2004; Opricovic & Tzeng, 2004; Parkan & Wu, 1997, 1999; Tong & Su, 1997; Tzeng, Lin & Opricovic, 2005; Zanakis, Solomon, Wishart & Dublish, 1998). The method has also been extended to deal with fuzzy MCDM problems. For example, Tsaur, Chang and Yen (2002) first convert a fuzzy MCDM problem into a crisp one via centroid defuzzification and then solve the nonfuzzy MCDM problem using the TOPSIS method. Chen and Tzeng (2004) transform a fuzzy MCDM problem into a nonfuzzy MCDM using fuzzy integral. Instead of using distance, they employ grey relation grade to define the relative closeness of each alternative. Chu (2002a; 2002b) and Chu and Lin (2003) also change a fuzzy MCDM problem into a crisp one and solve the crisp MCDM problem using the TOPSIS method. Differing from the others, they first derive the membership functions of all the weighted ratings in a weighted normalization decision matrix using interval arithmetics of fuzzy numbers and then defuzzify them into crisp values using the ranking method of mean of removals (Kaufmann & Gupta, 1991). Chen (2000) extends the TOPSIS method to fuzzy group decision making situations by defining a crisp Euclidean distance between any two fuzzy numbers. Triantaphyllou and Lin (1996) develop a fuzzy version of the TOPSIS method based on fuzzy arithmetic operations, which leads to a fuzzy relative closeness for each alternative. Our literature review clearly shows that except for Triantaphyllou and Lin’s fuzzy TOPSIS method, all the others mentioned above lead to a crisp relative closeness for each alternative. It is argued that fuzzy weights and fuzzy ratings should result in fuzzy relative closenesses. Crisp relative closeness provides only one possible solution to a fuzzy MCDM problem, but cannot reflect the whole picture of its all possible solutions. In spite of the fact that Triantaphyllou and Lin’s fuzzy TOPSIS method offers a fuzzy relative closeness for each alternative, the closeness is badly distorted and over exaggerated because of the reason of fuzzy arithmetic operations. This will be shown in Section 5 through the examination of a numerical example. So, there is a need to develop an exact fuzzy TOPSIS method for fuzzy MCDM problems. Motivated by such a need, this paper proposes a fuzzy TOPSIS method based on alpha level sets and the fuzzy extension principle, which turns out to be a nonlinear programming (NLP) problem and can be solved by Microsoft Excel Solver or LINGO software package. Expert Systems with Applications 31 (2006) 309–319 www.elsevier.com/locate/eswa 0957-4174/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2005.09.040 * This research was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) under Grant No. GR/S66770/01. * Corresponding author. Tel.: C44 161 2005974; fax: C44 161 2004646. E-mail addresses: msymwang@hotmail.com (Y.-M. Wang), yingming. wang@manchester.ac.uk (Y.-M. Wang).