Outsourcing the hazardous materials (HazMat) transportation is an effective way for manufacturing enterprises to avoid risks and accidents as well as to retain sustainable development in economic growth and social inclusion while not bringing negative impacts on the public and the environment. It is imperative to develop viable and effective approaches to selecting the most appropriate HazMat transportation alternatives. This paper aims at proposing an integrated multi-criteria group decision making approach that combines proportional hesitant fuzzy linguistic term set (PHFLTS) and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) to address the problem of HazMat transportation alternative evaluation and selection. PHFLTSs are adopted to represent the congregated individual evaluations in a bid to avoid information loss and increase the reliability of results. Two weight assignment models are then proposed to determine the comprehensive weights of experts and criteria. Furthermore, several novel manipulations of PHFLTS are also defined to enrich its applicability. The TOPSIS method is subsequently extended to the context of PHFLTSs to rank alternatives and choose the best one. Eventually, the feasibility and validity of the proposed approach are verified by a practical case study of a HazMat transportation alternative evaluation and selection decision and further comparison analyses.
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. INDEX TERMS Hesitant fuzzy set (HFS), proportional hesitant fuzzy set (PHFS), aggregation operators, multicriteria group decision making (MCGDM), electric vehicle battery (EVB) supplier selection. The associate editor coordinating the review of this manuscript and approving it for publication was Jerry Chun-Wei Lin .
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