Correlation coefficient of r,s,t-spherical hesitant fuzzy sets and MCDM problems based on clustering algorithm and technique for order preference by similarity to ideal solution method.

The opinion of r,s,t spherical fuzzy set (r,s,t-SFS) revealed by Ali and Naeem (IEEE Access 11:46454-46475, 2023a) is one of the significant spreads of picture fuzzy set which is developed to define ambiguous, undefined information in multi criteria decision making (MCDM) problems, machine learning, data mining and medical diagnosis so on. This paper aims to present a new tool called as r,s,t spherical hesitant fuzzy set (r,s,t-SHFS) by extending to r,s,t-SFS to overcome uncertainness and impreciseness encountered in daily life scenes. This construction puts forward many advantages for experts as extract parameters, carrying more information, hosting to several clusters in its own structure, being more flexible concept. The framework of r,s,t-SHFS is a generalization of hesitant fuzzy set, spherical hesitant fuzzy set, picture hesitant fuzzy set and t-spherical hesitant fuzzy set having a gorgeous potential of overcoming with uncertain and vagueness events. When examined from this perspective, the proposed cluster may present a more flexible structure for decision makers. Further, the correlation coefficient (CC) is often used to predict how a particular factor will fluctuate relative to another. The existing work investigates CC and weighted CC for r,s,t spherical hesitant fuzzy set and some set-theoretical operations. Moreover, we build a MCDM algorithm known as clustering based on the introduced correlation coefficients. Then, we solve another example by finding ideal and non-ideal solutions based on union and intersection operations through the technique for order preference by similarity to ideal solution method. Moreover, we determine the usefulness and limited sides of the new concept by comparing with some existing instructions. Some graphical presentations are given to demonstrate the credibility and effectiveness of the defined measures. [ABSTRACT FROM AUTHOR]