FLM-TOPSIS: The fuzzy linguistic multiset TOPSIS method and its application in linguistic decision making

Linguistic decision making is an important subject in decision making, many interesting and important linguistic decision making methods have been proposed, in which, alternatives-criteria decision matrix are uniformly used to express linguistic assessments of alternatives provided by decision makers with respect to criteria. Alternatives-criteria decision matrixes have some limitations when we use them to distinguish distinct, partial unknown or hesitant linguistic decision making or carry out linguistic decision making in the huge amounts of decision information and alternatives. In this paper, we propose alternatives-linguistic terms decision matrix to represent linguistic assessments of alternatives, analyze advantages of the decision matrix in representing linguistic assessments and distinguishing distinct, partial unknown or hesitant linguistic decision making. To simple and fast fuse alternatives-linguistic terms decision matrixes, we further provide linguistic multiset or fuzzy linguistic multiset to represent linguistic assessments in alternatives-linguistic terms decision matrixes, analyze the function properties of the fuzzy linguistic multiset. Motivated by fuzzy multiset and the TOPSIS method, we develop the fuzzy linguistic multiset TOPSIS method for linguistic decision making, the method is mainly consisted of transformation, aggregation and exploitation phases. In transformation phase, linguistic assessments of alternatives are transformed into fuzzy linguistic multisets by using alternatives-linguistic terms decision matrixes. In aggregation phase, we use Union, Intersection and Sum operations of multisets to obtain the positive and negative ideal solutions of linguistic decision making, which are different with the positive and negative ideal solutions of the traditional TOPSIS method, in addition, we provide a pseudo-distance between two fuzzy linguistic multisets to fast fuse linguistic assessments of alternatives. In exploitation phase, we define a new closeness degree of alternative by using pseudo-distances between the alternative and the positive and negative ideal solutions, which can be used to obtain the set of most satisfying alternatives. We also design an algorithm to carry out linguistic decision making based on the proposed method. In cases study, we use two practical examples to illustrate the practicality of the proposed method and compare it with the symbolic aggregation-based method, the hesitant fuzzy linguistic TOPSIS method, the hesitant fuzzy linguistic VIKOR method and the probabilistic linguistic term sets TOPSIS method, results indicate that alternatives-linguistic terms decision matrix and fuzzy linguistic multiset are alternative, useful and flexible tools for linguistic decision method and the fuzzy linguistic multiset TOPSIS method is suitable to deal with partial unknown or hesitant linguistic decision making.