Correlation coefficient of T-spherical type-2 hesitant fuzzy sets and their applications in clustering analysis

The fuzzy set (FS) theory has a significant role in the modelling of uncertainties. However, using the fuzzy set remain incapable in the modelling of the problems, when the decision-makers do not have the same opinion about the membership degree of an element or a decision-maker has different opinion related to membership degree of an element. To cope with this problem, the concept of a hesitant fuzzy set (HFS) was introduced by Torra and Narukawa. Recently, many generalizations of the FSs have been worked by researchers. One of them is type-2 fuzzy set (T2FS) and this set is an important role for dealing with problems involving uncertainty and linguistic variables. To take advantages of T2FSs and HFSs in the modelling of some problems into consideration, the concept of type-2 hesitant fuzzy sets (T2HFS) was defined. Another generalization of the FSs is spherical fuzzy set (SFS) and T-spherical fuzzy set (T-SFS) which is a generalization of SFS. In this research paper, to take advantages of T2HFSs and T-SFSs in the modelling of some problems into consideration, the concept of T-spherical type-2 hesitant fuzzy sets (T-ST2HFS) is defined and operations of union and intersection between two T-ST2HFSs. The basic idea behind of the set introduced in this paper is the assignment of T2HF elements for the components of a T-SFS. Then, a method is given to equalize lengths of the T2HF elements of which lengths are different from each other based on the optimistic and pessimistic approaches. The correlation coefficient is a very useful tool for dealing with some situations requiring the calculation of the relation between the two objects. With this motivation, correlation coefficients and weighted correlation coefficients are defined between proposed sets and some of their properties are obtained. Furthermore, a clustering analysis method, based on the put forward correlation coefficient formula is presented. Finally, to tackle a problem involving clustering the alternatives for a firm that wants to invest with a large amount of money, an application of the proposed method is given.