Hesitant Fuzzy Multiple Integrals for Information Aggregation

Abstract

Hesitant fuzzy element (HFE), membership degree of which consists of a set of crisp values, is used as an expression form when the decision makers encounter hesitance in providing their judgements. To aggregate large-scale group decision-making information under hesitant fuzzy environment, in this paper, we construct the multiple integrals for the HFEs. First, we propose the linear operations and limitation approaches of the HFEs. Next, we construct the hesitant fuzzy additive integral and hesitant fuzzy multiplicative integral and discuss their properties. Based on the two hesitant fuzzy integrals, we derive the hesitant fuzzy additive integral operator and hesitant fuzzy multiplicative integral operator to aggregate the continuous HFEs. Moreover, the useful properties of these operators are also investigated. Finally, we apply the hesitant fuzzy additive integral and hesitant fuzzy multiplicative integral operators to fuse the massive discrete HFEs which will be turned into the continuous form in advance to validate the effectiveness of the proposed multiple integrals operators. Compared with the existing methods which can only process a finite number of discrete HFEs, our methods can deal with infinitely large number of HFEs and decrease the loss of information.