Double Hierarchy Hesitant Fuzzy Linguistic Mathematical Programming Method for MAGDM Based on Shapley Values and Incomplete Preference Information

This paper investigates a double-hierarchy hesitant fuzzy linguistic mathematical programming method to multiple attribute group decision making (MAGDM) problems, where the assessment values of alternatives and the truth degrees of pairwise comparisons between alternatives are denoted by double-hierarchy hesitant fuzzy linguistic elements (DHHFLEs) and the inter-dependent or interactive characteristics among attributes and the incomplete preference information are taken into account. First, based on the distance measure between DHHFLEs, the double-hierarchy hesitant fuzzy linguistic positive ideal group consistency index (DHHFLPIGCI), double-hierarchy hesitant fuzzy linguistic positive ideal group inconsistency index (DHHFLPIGII), double-hierarchy hesitant fuzzy linguistic negative ideal group consistency index (DHHFLNIGCI), and double-hierarchy hesitant fuzzy linguistic negative ideal group inconsistency index (DHHFLNIGII) are defined, respectively. Then, to determine the positive ideal solution (PIS), negative ideal solution (NIS), and Shapley values of attributes simultaneously, a four-objective double-hierarchy hesitant fuzzy linguistic mathematical programming model is constructed by minimizing the DHHFLPIGII and DHHFLNIGII as well as maximizing the DHHFLPIGCI and DHHFLNIGCI. Subsequently, the relative closeness degrees (RCDs) of all feasible alternatives for each decision maker (DM) are obtained and the individual ranking order of alternatives for each DM is derived according to the descending order of RCDs, and thus a single-objective assignment model is established to generate the group ranking order of alternatives. Finally, a numerical example is given to illustrate the application of the proposed method and its effectiveness is demonstrated by comparison analysis.