18 AHP Combined with Fuzzy Topsis for Evaluating a Best Alternative A.Sahaya Sudha P. Rajarajeswari Department of Mathematics Department of Mathematics Nirmala College for Women Chikkanna Government Arts College Coimbatore-641 018. Tirupur-641 602 Abstract: Fuzzy TOPSIS is one of the various models of multiple attributes Decision Making with Fuzzy values. In this paper a new method is presented for fuzzy TOPSIS with Triangular fuzzy number. In today„s world a consumer while purchasing a product looks after various criteria of a particular product and the place where it is sold. So different products are taken into account along with associated sales counters is identified and formulated. In this paper AHP and Fuzzy TOPSIS methods are proposed to determine the best alternatives using the subjective criteria. The fuzzy distance between each alternative and the ideal solution is found out with the greatest relative closeness to the ideal solution is obtained. Key Words: Fuzzy number, AHP, Fuzzy TOPSIS, Multi Criteria Decision Making 1. Introduction: The idea of fuzzy set was first proposed by Bellman and Zadeh [3], as a mean of handling uncertainty. Decision making is the process of finding the best alternative from a number of feasible alternatives which is called as Multi Criteria Decision Making. The MCDM problems may be divided into the classical MCDM in which the ratings are measured as crisp numbers and the other is the FMCDM, which is based on the vagueness of the problem and expressed in linguistic terms.In[5],[7],[13]and[15]a fuzzy version of Saaty‟s AHP method was developed by Triangular Fuzzy Numbers for Linguistic terms. The brand managers of a company wants to sell their products in all the counters of a particular city whereas there has been a lot of ambiguity between various counters and all the products are not selling in a single counter and there are a lot of variations in the sales patterns of different products in different counters. So the specific data is collected and formulated and the results are useful for the brand mangers to take a decision on further sales planning. Let P 1, P 2, P 3. ----- ,P n... be n alternatives available and C 1, C 2, C 3. ----- ,C m... be criteria involved in the expression of alternative. Let A ij be the performance of alternative P i with respect to criteria C j and w j be the relative importance of criteria. Then the decision making is the selection of the best alternative with respect to criteria 2. Literature Review: Saaty(1980) [15] introduced Analytical Hierarchy Process which is one of the approaches in Multicriteria Decision Making. This plays an important role in selecting alternatives. This is one of the widely used methods for the practical solution of MCDM problems. Traditional AHP method does not directly use fuzzy numbers or membership functions to express fuzzy information. Instead it uses the fuzziness of a multiple attribute decision making problem. Laurhoven and Pedrycz (1985)[14] proposed a method of fuzzy judgment using triangular fuzzy numbers (TFN). They applied arithmetic operation for TFN and logarithmic least square method to calculate fuzzy utilities. TOPSIS (Techniques for order Performance by similarity to ideal solution) is one of the best grading methods using multi criteria decision making (MCDM) problem was first developed by Hwang and Yoon[12]. Chen and Hwang (1992)[9] first applied fuzzy numbers to establish fuzzy TOPSIS. Later some efforts were done to expand TOPSIS model to TOPSIS fuzzy as given by Abo-Sina M.A, A.H. Amer, 2004. [1].Chen, T.C., 2000[10] made the study on extensions of the TOPSIS for group decision making under fuzzy environment. 3. Preliminaries: 3.1Fuzzy number: [13] A fuzzy set à of the real line R with membership function : [ 0, 1] () R A x    is called fuzzy number if i) A must be normal and convex fuzzy set ii) the support of Ã, must be bounded iii) α A must be a closed interval for every 0, 1];   3.2Triangular Fuzzy number: [9] A fuzzy number A is defined to be a triangular fuzzy number if its membership functions