Deriving the priority weights from incomplete hesitant fuzzy preference relations based on multiplicative consistency

In this paper, we define the concept of incomplete hesitant fuzzy preference relations to deal with the cases where the decision makers express their judgments by using hesitant fuzzy preference relations with incomplete information, and investigate the consistency of the incomplete hesitant fuzzypreference relations and obtain the reliable priority weights. We first establish a goal programmingmodel for deriving the priority weights from incomplete hesitant fuzzy preference relations based the ̨ -normalization. Then, we give the definition of multiplicative consistent incomplete hesitant fuzzy preference relations based on the ˇ-normalization, and develop amethod for complementing the acceptable incomplete hesitant fuzzy preference relations by using themultiplicative consistency property. Furthermore, utilizing a convex combinationmethod, a newalgorithm for obtaining thepriorityweights fromcomplete or incomplete hesitant fuzzy preference relations is presented on the basis of the ˇ-normalization. Finally, several numerical examples are provided to illustrate the validity and practicality of the proposed methods. © 2016 Elsevier B.V. All rights reserved.