Abstract
Since its initiation, hesitant fuzzy sets (HFSs) have gained prominence thanks to their capability to describe the hesitation of experts to assign membership degrees to objects belonging to a concept. Proportional hesitant fuzzy sets (PHFSs) are an important extension of HFSs and are characterized by the combination of possible membership degrees and their associated proportional information. PHFSs have a huge application potential for hesitant fuzzy GDM problems, because the proportional information in PHFSs can be determined objectively and the introduction of this new information dimension can effectively reduce the uncertainty. Nevertheless, PHFSs have not yet attracted sufficient attention from researchers and practitioners, which motivates us to expand the theory of PHFSs and explore its application potential. The main work comprises the following three aspects: First, we define some basis operations on PHFSs, develop aggregation operators for PHFSs, and demonstrate their properties and interrelationships to lay the theoretical foundations for the application of PHFSs. Next, we construct two multicriteria group decision making (MCGDM) models based on the proposed PHFS-based aggregation operators to bridge between theory and practice for PHFSs. In this step, we propose a method for transforming HFSs or fuzzy sets (FSs) into PHFSs, and two methods based on the maximum entropy principle are proposed for specifying criterion weights. Finally, we investigate a practical case study of the problem of selecting an electric vehicle battery (EVB) supplier to validate the outstanding advantages of PHFSs, explore the compensation characteristics and the applicability of the PHFS-based aggregation operators, and demonstrate the effectiveness and feasibility of the proposed MCGDM models. This paper provides a useful reference for MCGDM in a hesitant fuzzy context.
Highlights
For the proposed multicriteria group decision making (MCGDM) model based on the generalized PHFOWA operator (GPHFOWA) or generalized PHFOWG operator (GPHFOWG) operator, we provide a similar method based on maximum entropy and the attitudinal character given by decision makers (DMs)
Based on a comparative analysis, we find that the aggregation operators based on Proportional hesitant fuzzy sets (PHFSs), including generalized PHFWA operator (GPHFWA), generalized PHFWG operator (GPHFWG), GPHFOWA, and GPHFOWG, outperform the aggregation operators based on Hesitant fuzzy sets (HFSs) including GHFWA, GHFWG, GHFOWA, and GHFOWG, which indicates that PHFSs perform better than HFSs for uncertain MCGDM problems
(2) Based on this case study, we find that the GPHFWA and GPHFOWA operators have a higher level of compensation than the GPHFWG and GPHFOWG operators, respectively
Summary
A fuzzy set is a class of objects with a continuum of membership degrees and is represented mathematically by a membership function that assigns a membership degree in the. PHFSs are an extension of HFSs that were introduced by [24] and are characterized by a predefined set of possible membership degrees for elements and the proportional information of each membership degree These are mainly used to address group-decision-making (GDM) problems in a fuzzy hesitant context. For the proposed MCGDM model based on the GPHFOWA or GPHFOWG operator, we provide a similar method based on maximum entropy and the attitudinal character given by DMs. we conduct a thorough practical case study of the selection of a strategic supplier of electric-vehicle batteries (EVBs), which involves multiple qualitative and quantitative criteria and necessitates a multifunctional expert team.
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