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Fuzzy sets and interval-valued fuzzy sets are two kinds of fuzzy information expression forms in real uncertain and vague environments. Their mixed multivalued information expression and operational problems are very challenging and indispensable issues in group decision-making (GDM) problems. To solve single- and interval-valued fuzzy multivalued hybrid information expression, operations, and GDM issues, this study first presents the notion of a single- and interval-valued fuzzy multivalued set/element (SIVFMS/SIVFME) with identical and/or different fuzzy values. To effectively solve operational problems for various SIVFME lengths, SIVFMS/SIVFME is converted into the interval-valued fuzzy and entropy set/element (IVFES/IVFEE) based on the mean and information entropy of SIVFME. Then, the operational relationships of IVFEEs and the expected value function and sorting rules of IVFEEs are defined. Next, the IVFEE weighted averaging and geometric operators and their mixed-weighted-averaging operation are proposed. In terms of the mixed-weighted-averaging operation and expected value function of IVFEEs, a GDM method is developed to solve multicriteria GDM problems in the environment of SIVFMSs. Finally, the proposed GDM method was utilized for a supplier selection problem in a supply chain as an actual sample to show the rationality and efficiency of SIVFMSs. Through the comparative analysis of relative decision-making methods, we found the superiority of this study in that the developed GDM method not only compensates for the defects of existing GDM methods, but also makes the GDM process more reasonable and flexible.

mathematics

Group Decision-Making Problems Based on Mixed Aggregation Operations of Interval-Valued Fuzzy and Entropy Elements in Single- and Interval-Valued Fuzzy Environments