Generalised operations on hesitant fuzzy values in the framework of Dempster–Shafer theory

The hesitant fuzzy sets theory (HFS) is probably the latest generalisation of fuzzy sets theory and seems to be especially useful in the solution of multiple criteria group decision making (MCGDM) problems, where it enables us to avoid some specific problems concerned with the aggregation of expert’s opinions. Currently there are only few different definitions and generalisations of HFS proposed in the literature. The key issue of hesitant fuzzy sets theory is the formulation of operation laws on the hesitant fuzzy elements (HFE), as they make it possible to use HFS for the solution of real-world problems. This paper presents a critical analysis of conventional operations on HFE and their applicability to the solution of multiple criteria decision making (MCDM) problems. It is shown that the known approaches to the definitions of HFS and corresponding operation laws have some important limitations and drawbacks. Therefore, a new generalised definition of HFS and operation laws based on the interpretation of intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory of evidence (DST) are proposed and analysed. With the use of corresponding theorems it is proved that the proposed approach is free of limitations and drawbacks of known methods.The corresponding methods for aggregation of local criteria presented by HFEs in the framework of DST are proposed and analysed. The proposed approach allows us to solve MCDM and MCGDM problems without intermediate defuzzification when not only criteria, but their weights are HFEs. The advantages of the proposed approach are illustrated with numerical examples and the case study.