Some distance measures for type 2 hesitant fuzzy sets and their applications to multi-criteria group decision-making problems

The fuzzy set has an important role in the modeling of uncertainties. However, the fuzzy set is not sufficient in modeling of the problems, when the decision makers do not have the same opinion about membership degree of an element. To overcome this problem, the concept of hesitant fuzzy set was defined by Torra and Narukawa. Recently, the concept of the type 2 hesitant fuzzy set was defined by Feng and a ranking method among elements of a type 2 hesitant fuzzy element was given. In this paper, firstly, we point out some shortcomings in the ranking method given by Feng and then we give a new ranking method among elements of a type 2 hesitant fuzzy element. The distance and similarity measures are the effective mathematical tools to solve the problems such as medical diagnosis, decision making, pattern recognition and marketing strategy selection. Therefore, we introduce some distance measure methods between two type 2 hesitant fuzzy sets based on Hamming, Euclidean and Hausdorff distance measures. We obtain some properties of the proposed distance measure methods. We also develop a multi-criteria group decision-making method by integrating the TOPSIS method and the proposed distance measure methods under the type 2 hesitant fuzzy environment. Furthermore, we present a numerical example of multi-criteria group decision-making problem to choose the best alternative among firms to invest in order to illustrate the process and validate of the proposed method.