5. Conclusions
In the process of group decision making, the decision makers may uses incomplete hesitant fuzzy preference relations to express their preference information due to the complexity, uncertainty, and hesitancy involved in real decision problems. In order to correctly operation on incomplete HFPRs, we have two principles to normalize them, i.e., the ˛-normalization and the ˇ-normalization. The ˛-normalization reduces a HFPR to a crisp FPR by removing some preference values of some elements of the HFPR, while the ˇ-normalization adds some preference values to some elements which have less lengths than the other ones until all the elements of the HFPR have the same length. Based on the ˛-normalization, we have developed a goal programming model to derive the priority weights from incomplete HFPRs under GDM situations. Then, on the basis of the ˇ-normalization, a multiplicative consistent hesitant fuzzy preference relation, an acceptable incomplete hesitant fuzzy preference relation, and a multiplicative consistent incomplete hesitant fuzzy preference relation have been defined. An algorithm for estimating the missing preference values in an acceptable incomplete HFPR has been proposed. Furthermore, a practical algorithm for obtaining the collective priority weight vector of the GDM problem with several complete or incomplete HFPRs has been proposed. Several numerical examples have been given to verify the proposed goal programming model and these algorithms. The main advantages ofthe proposed methods are summarized as follows:(1) The proposed methods are applicable to the GDM problem in which allthe DMs use the incomplete HFPRs to express their preferences. (2) According to the ˛-normalization, a new hesitantfuzzy goal programming model is developed to derive the priority weights from incomplete HFPRs based on multiplicative consistency. Meanwhile, by solving this model we can select the optimal preference value from all possible ones for each paired comparison of alternatives, and then obtain a corresponding FPR consisting of these optimal preferences as the reduction of the original HFPR. (3) According to the ˇ- normalization, a family of crisp FPRs with acceptable consistency can be obtained from an acceptably consistent HFPR by using a convex combination method, whose weights are found to be a form of convex combination. An interval weight generation method is also used to obtain interval weights from acceptably consistent HFPR. The obtained results show that the proposed methods are simple, effective and applicable to both consistent HFPRs and inconsistent ones generated from concrete decision-making problems, without solving any mathematical programming and complicated algebraic operation.
