Designing a model of fuzzy TOPSIS in multiple criteria decision making

Abstract

Decision making is the process of finding the best option among the feasible alternatives. In classical multiple attribute decision making (MADM) methods, the ratings and the weights of the criteria are known precisely. Due to vagueness of the decision data, the crisp data are inadequate for real-life situations. Since human judgments including preferences are often vague and cannot be expressed by exact numerical values, the application of fuzzy concepts in decision making is deemed to be relevant. We design a model of TOPSIS for the fuzzy environment with the introduction of appropriate negations for obtaining ideal solutions. Here, we apply a new measurement of fuzzy distance value with a lower bound of alternatives. Then similarity degree is used for ranking of alternatives. Examples are shown to demonstrate capabilities of the proposed model.

Introduction

Multiple attribute decision making (MADM) approach is often used to solve various decision making and/or selection problems. This approach often requires the decision makers to provide qualitative and/ or quantitative assessments for determining the performance of each alternative with respect to each criterion, and the relative importance of evaluation criteria with respect to the overall objective.

Technique for order preference by similarity to an ideal solution (TOPSIS), known as a classical MADM method, has been developed by Hwang and Yoon [11] for solving the MADM problem. It is based on the idea that the chosen alternative should have the shortest distance from the positive ideal solution, and, on the other side, the farthest distance from the negative ideal solution. If the assessment values are known to have various types of vagueness/imprecision or subjectiveness, then the classical decision making techniques are not useful for such problems. In the past few years, numerous attempts have been carried out to apply fuzzy set theory to multiple criteria evaluation methods [2], [3], [26]. For example, Tsaur et al. [21] first convert a fuzzy MADM problem into a crisp one via centroid defuzzification and then solve the nonfuzzy MADM problem using the TOPSIS approach. Chen and Tzeng [5] transform a fuzzy multiple criteria decision making (MCDM) problem into a nonfuzzy MADM using fuzzy integral. Instead of using distance, they employ a grey relation grade to define the relative closeness of each alternative. Chu [8], [9] also changes a fuzzy MADM problem into a crisp one and solves the problem using the TOPSIS approach. Differing from the others, he first derives the membership functions of all the weighted ratings in a weighted normalization decision matrix using interval arithmetic of fuzzy numbers and then defuzzifies them into crisp values using the ranking method of mean of removals. Chen [6] extends the TOPSIS approach to fuzzy group decision making situations by defining a crisp Euclidean distance between any two fuzzy numbers. Triantaphyllou and Lin [22] develop a fuzzy version of the TOPSIS approach based on fuzzy arithmetic operations, resulting in a fuzzy relative closeness for each alternative.

Hsu and Chen [10] discuss an aggregation of fuzzy opinions under group decision making. Li [15] proposes a simple and efficient fuzzy model to deal with multi-judges/MADM problems in a fuzzy environment. Li [16] proposes several linear programming models and methods for multi attribute decision making under “intuitionistic fuzziness”, where the concept of intuitionistic fuzzy sets is a generalization of the concept of fuzzy sets. Liang [14] incorporates fuzzy set theory and the basic concepts of positive ideal and negative ideal points, and extends MADM to a fuzzy environment. Ölçer and Odabaşi [19], while dealing with problems of ranking and selection, propose a new fuzzy multi-attribute decision making method for multiple attributive group decision making problems in a fuzzy environment. Olson and Wu [18] present a simulation of fuzzy multi-attribute models based on the concept of grey relations, reflecting either interval input or commonly used trapezoidal input. This model is a simulated fuzzy MADM being applied to multi-attribute decision making problems effectively. Yeh et al. [24] propose a fuzzy MADM method based on the concepts of positive ideal and negative ideal points to evaluate performance of bus companies. Despite the applicability of these methods to many decision making problems, typical fuzzy multi criteria analyses require the comparison of fuzzy numbers. However, the comparison process can be quite complex and produce unreliable results [1], [7], [23].

Here, we extend the approach of TOPSIS to develop a methodology for solving multi-attribute decision making problems in fuzzy environments. Considering the fuzziness in the decision data, linguistic variables are used to assess the weight of each criterion and the rating of each alternative with respect to each criterion. We convert the decision matrix into a fuzzy decision matrix and construct a weighted normalized fuzzy decision matrix once the decision makers’ fuzzy ratings have been pooled. The lower bound value of alternatives has been designed to obtain the distance value of the corresponding alternatives for detecting the fuzzy positive ideal solution (FPIS) and the fuzzy negative ideal solution (FNIS). Then, we calculate the fuzzy similarity degree of each alternative from FPIS and FNIS, respectively. Finally, a closeness coefficient is defined for each alternative to determine the rankings of all alternatives. The higher value of closeness coefficient indicates that an alternative is closer to FPIS and farther from FNIS simultaneously.

The remainder of the paper is structured as follows: The next section describes the basic definitions and notations concerning fuzzy numbers and linguistic variables. In Section 3, we present an algorithm to extend TOPSIS to deal with fuzzy data. In Section 4, the proposed method is illustrated with three examples. Finally, the conclusions are given in Section 5.