Methods For Group Decision Making With Interval-valued Fuzzy Preference Relations

Under the rapid impacts of economic globalization,the decision-making environment faced by human beings is becoming more and more complicated.Group decision making(GDM),as a decision theory which can effectively take advantage of the wisdom of the group,has been widely applied during the decision-making process of governments and enterprises.In practical GDM procedures,considering the pressure of time,the vagueness of available information,and the limited knowledge of decision makers,the previous processing methods for decision information will be inapplicable.For this reason,scholars put forward the interval-valued fuzzy sets and fuzzy preference relations,which are used to accurately describe the uncertainty and fuzziness in the decision-making process,and provided a fundamental tool for decision makers to describe their own reasonable judgments through pair-wise comparisons of alternatives.Based on the interval-valued intuitionistic fuzzy preference relations(IVIFPRs),the interval-valued Pythagorean fuzzy preference relations(IVPFPRs)and interval-valued q-rung orthopair fuzzy preference relations(IVq-ROFPRs),in this paper,we research the GDM methods under interval-valued fuzzy environment from the aspects of the consistency,expert weights,group consensus and the priority weights of alternatives,respectively,and then apply them to the practical decision-making problems.The major work and innovations are mainly summarized as follows:(1)Based on the newly defined multiplicative consistency,we investigate an algorithm for the GDM method with IVIFPRs.Firstly,a novel multiplicative consistency concept is proposed,which is proved to satisfy an important property: robustness.A conversion formula is devised to accomplish the multiplicative consistent IVIFPRs by utilizing the normalized priority weights.Subsequently,a consistency measure and inconsistent repairing process are put forward to ensure that every individual IVIFPR is of acceptable multiplicative consistency.Afterward,in the context of minimizing the deviations between the given IVIFPRs and their corresponding consistent ones,two fractional programming models are constructed to generate the normalized individual interval-valued intuitionistic fuzzy weights and collective ones,where the experts are considered as individuals and a group,respectively.Finally,the applicability and reliability of this method are illustrated by an example of virtual enterprise partner selection.(2)We research a new method to solve GDM problems with IVPFPRs.Firstly,novel multiplicative consistency and consensus measures are proposed.Subsequently,the procedure for improving consistency and consensus levels are put forward to ensure that every individual IVPFPR is of acceptable multiplicative consistency and consensus simultaneously.In the context of minimizing the deviations between the individual and collective IVPFPRs,the objective experts weights are decided according to the optimization model and the aggregated IVPFPR is derived.Afterwards,a programming model is built to derive the normalized Pythagorean fuzzy priority weights,then the priority weights of alternatives are identified as well.Finally,an example of the emergency plan selection is analyzed to illustrate the effectiveness and superiority of this method.(3)We focus on GDM with incomplete IVq-ROFPRs.Based on interval-valued q-rung orthopair fuzzy sets,the novel concepts of IVq-ROFPRs and related definitions are proposed.Subsequently,two optimization models are established for deriving the complete IVq-ROFPRs and acceptably additive consistent IVq-ROFPRs,respectively.Inspired by the Markov model,the weight-generating method is designed to obtain the weights of decision makers.For the purpose of improving the consensus level,a modification process is constructed under the consideration of minimizing the deviations between the adjusted IVq-ROFPRs and original ones.Afterward,a goal programming model is constructed to derive the normalized q-rung orthopair fuzzy priority weights.The interval-valued q-rung orthopair fuzzy priority weights of alternatives are further determined.Finally,the feasibility and practicability of this method are verified by an example of civilized city selection.