Extensions of the TOPSIS for group decision-making under fuzzy environment

The aim of this paper is to extend the TOPSIS to the fuzzy environment. Owing to vague concepts frequently represented in decision data, the crisp value are inadequate to model real-life situations. In this paper, the rating of each alternative and the weight of each criterion are described by linguistic terms which can be expressed in triangular fuzzy numbers. Then, a vertex method is proposed to calculate the distance between two triangular fuzzy numbers. According to the concept of the TOPSIS, a closeness coefficient is defined to determine the ranking order of all alternatives by calculating the distances to both the fuzzy positive-ideal solution (FPIS) and fuzzy negative-ideal solution (FNIS) simultaneously. Finally, an example is shown to highlight the procedure of the proposed method at the end of this paper.