Constructing Novel Operational Laws and Information Measures for Proportional Hesitant Fuzzy Linguistic Term Sets with Extension to PHFL-VIKOR for Group Decision Making

To obtain reliable results in a qualitative multi-attribute group decision-making (MAGDM) problem, how to retain the evaluation information as much as possible and how to determine the reasonable weights of experts and attributes are two important issues. Proportional hesitant fuzzy linguistic term set (PHFLTS) is beneficial for retaining evaluation information as it would consider the linguistic terms and corresponding proportional information simultaneously. However, PHFLTS is a relatively new concept. Some novel manipulations, such as comparison, arithmetic operations, aggregation operators, and cosine similarity and distance measures are defined in this study with the purpose of improving the completeness and applicability of PHFLTS. Furthermore, cosine similarity measure-based weight determination model and entropy measure-based weight determination model under proportional hesitant fuzzy linguistic (PHFL) environment are constructed to derive the objective weights of experts and those of attributes as well. Subsequently, an integrated weighting model is proposed to determine the comprehensive weights of experts and attributes. Based on the defined operational laws for PHFLTS and comprehensive weighting model, two MAGDM methods, PHFL aggregation operator-based method and extended PHFL-VIKOR method, are developed to deal with MAGDM problems with PHFL information. To demonstrate the applicability, efficiency, and advantages of the proposed MAGDM methods, an illustrative example and a comparison example are provided.