Programming model-based group decision making with multiplicative linguistic intuitionistic fuzzy preference relations

A new type of fuzzy sets called multiplicative linguistic intuitionistic fuzzy sets (MLIFSs) is introduced to denote the asymmetric qualitative preferred and non-preferred cognitions of decision makers. Considering the application of MLIFSs, an order relationship is offered and several operational laws are defined. Then, multiplicative linguistic intuitionistic fuzzy preference relations (MLIFPRs) are introduced. A consistency concept is defined and its several properties are researched. A programming model to judge the consistency of MLIFPRs is constructed, and an approach to derive consistent MLIFPRs is provided. Meanwhile, a programming model to ascertain missing values in incomplete MLIFPRs is built. A consensus index is defined to measure consensus and a programming model to improve the consensus level is constructed, which ensures the consistency of individual MLIFPRs. Furthermore, a programing model to determine the weights of the decision makers is established. On the basis of consistency and consensus analysis, an approach to group decision making with MLIFPRs is developed. Finally, an example about evaluating washing machine is selected to show the application of the new method.