Some models to manage additive consistency and derive priority weights from hesitant fuzzy preference relations

Consistency has a crucial influence on the rationality of preference information and even the final decision result. This paper presents two new definitions of additive consistency for hesitant fuzzy preference relations (HFPRs): completely additive consistency (CAC) and weakly additive consistency (WAC). To verify the CAC or WAC of HFPRs, some linear programming models and 0–1 mixed programming models are developed. The methods consider all the information given by the decision maker without changing the length of hesitant fuzzy elements (HFEs). Accordingly, a method of modifying an inconsistent HFPR into an additively consistent HFPR is proposed. The deviation between the original complementary matrix and the modified one is minimal. Then, several linear programming models are developed to obtain priority weights from an HFPR. From these, an integrated algorithm is designed to illustrate the process of consistency test, inconsistency modification and weights derivation for HFPRs. The proposed methods are also extended to deal with WAC and CAC of incomplete HFPRs. Finally, three numerical examples and comparative analysis are presented to illustrate the feasibility and effectiveness of the proposed method.