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 fuzzy preference relations and obtain the reliable priority weights. We first establish a goal programming model for deriving the priority weights from incomplete hesitant fuzzy preference relations based the alpha-normalization. Then, we give the definition of multiplicative consistent incomplete hesitant fuzzy preference relations based on the beta-normalization, and develop a method for complementing the acceptable incomplete hesitant fuzzy preference relations by using the multiplicative consistency property. Furthermore, utilizing a convex combination method, a new algorithm for obtaining the priority weights from complete or incomplete hesitant fuzzy preference relations is presented on the basis of the beta-normalization. Finally, several numerical examples are provided to illustrate the validity and practicality of the proposed methods. (C) 2016 Elsevier B.V. All rights reserved.