Modification to Fuzzy Extent Analysis Method and its performance Analysis (presented at the 6 th IESM Conference, October 2015, Seville, Spain) © I 4 e 2 2015 Faran Ahmed Faculty of Engineering and Natural Sciences, Sabanci university Orta Mahalle, Universite Caddesi, No:27, 34956 Tuzla, Istanbul, Turkey Email: ahmedfaran@sabanciuniv.edu Kemal Kilic Faculty of Engineering and Natural Sciences, Sabanci university Orta Mahalle, Universite Caddesi, No:27, 34956 Tuzla, Istanbul, Turkey Email: kkilic@sabanciuniv.edu Abstract—Analytic Hierarchical Process (AHP) is one of the most popular Multi-Criteria Decision Making (MCDM) tech- nique while fuzzy set theory is extensively incorporated into original AHP to address uncertainty and vagueness in human judgments. There are number of algorithms proposed in the domain of Fuzzy AHP (FAHP), however, Fuzzy Extent Analysis (FEA) is one of the most frequently used model. This study evaluates the performance of this model against a modified FEA method which utilizes centroid defuzzification. This study shows that modified FEA method performs significantly better than its original model and thus can lead to more effective decision making. I. I NTRODUCTION In a Multiple-Criteria Decision Making (MCDM) process, prioritizing and assigning weights to each criteria with ref- erence to a set of available alternatives is key to effective decision making. Analytic Hierarchy Process (AHP) proposed by Thomas L. Saaty [1] is one such technique used in MCDM through which experts provide pairwise comparisons and this information is processed in a comparison matrix to calculate priority vector. One of the major concerns in the original AHP is to trans- form human judgments, which are usually natural language phrases such as “significantly more”, “slightly more” etc, into a numerical scale due to inherent uncertainty in human observation. Disregarding this fuzziness of the human behavior in the decision analysis process may lead to wrong decisions [2]. Therefore, in order to address the issue of vagueness and uncertainty, fuzzy set theory introduced by Zadeh [3] is extensively incorporated into the original AHP in which the weighing scale is composed of fuzzy numbers and thus comparison matrices are formed in such a way that its elements are fuzzy numbers. There are number of different techniques proposed over the years which prioritize and rank the available criteria based on comparison ratios represented by fuzzy numbers [4][5][6] and a review of these techniques is provided by Buyukozkan [7]. Fuzzy Extent Analysis (FEA) proposed by Chang [8] is one of the most frequently used FAHP algorithm [9]. In this study, we propose modication to this model and show that this modication leads to more accurate weights. Rest of this paper is organized as follows. In section II, we provide an extensive overview of FEA model. In secion III, we will present the modified version of FEA model and provide an overview of research methodology. In Section IV, results of the performance analysis are given and last section contains some concluding remarks. II. FUZZY EXTENT ANALYSIS Before providing a review of FEA model, we first provide a brief overview of the fuzzy logic and its basic arithmetic. Fuzzy sets can record the imprecision arising in human judg- ments which are neither random nor stochastic [10]. Instead of a single value, fuzzy number represents a set of possible values each having its own membership function between zero and one. A triangular fuzzy number is represented by [lower value, mean value, upper value], i.e., [lmu] with membership functions μ M given by; μ M (x)= ⎧ ⎪ ⎨ ⎪ ⎩ x m−l − l m−l , x ∈ [lm] x m−u − u m−u , x ∈ [mu] 0, otherwise (1) The same is graphically illustrated in Figure 1. Fig. 1: Membership function of Triangular Fuzzy Number Let (l 1 m 1 u 1 ) and (l 2 m 2 u 2 ) then the basic fuzzy arithmetic operations are summarized as follows; • Addition: (l 1 m 1 u 1 ) ⊕ (l 2 m 2 u 2 )=(l 1 + l 2 m 1 + m 2 u 1 + u 2 )