58 INTERNATIONAL JOURNAL OF COMPUTATIONAL COGNITION (HTTP://WWW.IJCC.US), VOL. 7, NO. 4, DECEMBER 2009 Extension of Fuzzy TOPSIS Method Based on Vague Sets Kavita Devi, Shiv Prasad Yadav and Surendra Kumar Abstract— In TOPSIS method, evaluation of alternatives based on weighted attributes play an important role in the best the alternative selection. Usually these weights are gathered by uncertain or vague resources. Thus an extension of TOPSIS to the fuzzy environment is a natural generalization of TOPSIS models. Sometimes available information is not sufficient for the exact definition of a degree of membership for certain elements. There may be some hesitation degree between membership and non-membership. In view that there are many real life situations where due to insufficiency in information availability, vague sets with ill-known membership grades are appropriate to deal with such real life situations. Vague sets have been found to be particularly useful to deal with uncertainty. In this paper, the uncertainty problems in TOPSIS are dealt with vague set theory. Copyright c  2009 Yang’s Scientific Research Institute, LLC. All rights reserved. Index Terms— TOPSIS, vague sets, decision making, fuzzy sets. I. I NTRODUCTION M ULTI-ATTRIBUTE decision making (MADM) is the most well-known branch of decision making. It is a branch of a general class of operations research models that deals with decision problems under the presence of a number of decision criteria. The MADM approach requires that the selection be made among decision alternatives described by their attributes. MADM problems are assumed to have a pre- determined, limited number of decision alternatives. Solving a MADM problem involves sorting and ranking. MADM ap- proaches can be viewed as alternative methods for combining the information in a problem’s decision matrix together with additional information from the decision maker to determine a final ranking or selection from among the alternatives. Besides the information contained in the decision matrix, all but the simplest MADM techniques require additional information from the decision maker to arrive at a final ranking or selection. MADM problems and their evaluation process usually in- volve subjective assessments, resulting with imprecise data in qualitative manner. Engineering or management decisions are generally made through available data and information that Manuscript received May 02, 2009; revised August 12, 2009. Kavita Devi and Shiv Prasad Yadav, Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, India. Surendra Kumar, Depart- ment of Electrical Engineering, Indian Institute of Technology Roorkee, Roorkee, India. Acknowledgement: The first author acknowledges the financial support given by the Council of Scientific and Industrial Research, Govt. of India, India. Publisher Item Identifier S 1542-5908(09)10408-6/$20.00 Copyright c 2009 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on December 22, 2009 at http://www.YangSky.com/ijcc/ijcc74.htm are mostly vague, imprecise, and uncertain by nature. The decision-making process in engineering schemes, developed in the concept-designing phase, is one of these typical occasions, which usually need some methods to deal with uncertain data and information that are hard to define. In designing phase, designers usually present many alternatives. However, the subjective characteristics of the alternatives are generally uncertain and need to be evaluated through decision maker’s insufficient knowledge and judgments. The nature of this kind of vagueness and uncertainty is fuzzy rather than random, especially when subjective assessments are involved in the decision-making process. Fuzzy set theory [2][14][16][15] offers a possibility for handling these sorts of data and information involving the subjective characteristics of human nature in the decision-making process. There, exist several methods to solve MADM problems, out of which the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) developed by Hwang and Yoon [8], is one of the well-known methods. The basic principle of the TOPSIS method is that the chosen alternative should have the shortest distance from the positive ideal-solution and the farthest distance from the negative ideal- solution. There exist a large amount of literature involving TOPSIS theory and applications. In classical Multi Criteria Decision Making (MCDM) methods, the ratings and the weights of the criteria are known precisely. A survey of the methods has been presented in Hwang and Yoon [8]. In the process of TOPSIS the performance ratings and the weights of the criteria are given exact values. Jahanshahloo et al [9] [10] extended the concept of TOPSIS to develop a methodology for solving MCDM problem with interval data. Triantaphyllou and Lin [12] develop a fuzzy version of the TOPSIS method based on fuzzy arithmetic operations. Chen [3] extended this method to solve group decision making problems under fuzzy environment. Wang and Elhag [13] extended TOPSIS to provide a fuzzy form of closeness co-efficient through α-cuts propagation. Guangtao Fu [5] proposed a fuzzy optimization method based on TOPSIS and demonstrated a case study of reservoir flood control operation. Most of the fuzzy versions of TOPSIS method are efficient in tackling the impreciseness and vagueness present in MADM problems, but their results are not able to include the hesitation present in the information provided by the decision maker. In real life, a person may consider that an object belongs to a set to a certain degree but it is possible that he is not sure about it. In other words the person has hesitation about the member- ship degree. In fuzzy set theory there is no means to incorpo- rate this hesitation regarding the degree of suitability to which each alternative satisfies the decision maker requirement. To