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The rough set (RS) and multi-granulation rough set (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, bipolarity refers to the explicit handling of positive and negative aspects of data. In this paper, with the help of bipolar fuzzy preference relation (BFPR) and bipolar fuzzy preference δ -covering (BFP δ C), we put forward the idea of BFP δ C based optimistic multi-granulation bipolar fuzzy rough set (BFP δ C-OMG-BFRS) model and BFP δ C based pessimistic multi-granulation bipolar fuzzy rough set (BFP δ C-PMG-BFRS) model. We examine several significant structural properties of BFP δ C-OMG-BFRS and BFP δ C-PMG-BFRS models in detail. Moreover, we discuss the relationship between BFP δ C-OMG-BFRS and BFP δ C-PMG-BFRS models. Eventually, we apply the BFP δ C-OMG-BFRS model for solving multi-criteria decision-making (MCDM). Furthermore, we demonstrate the effectiveness and feasibility of our designed approach by solving a numerical example. We further conduct a detailed comparison with certain existing methods. Last but not least, theoretical studies and practical examples reveals that our suggested approach dramatically enriches the MGRS theory and offers a novel strategy for knowledge discovery, which is practical in real-world circumstances. [ABSTRACT FROM AUTHOR]

Despite the importance of divergence measures, the literature has not provided a satisfactory formulation for the case of intuitionistic fuzzy sets (IFS). This paper criticizes the existing attempts in terms of respect of the basic axioms of a divergence measure. Then, new improved, axiomatically supported divergence measures for IFSs are proposed. Additional properties of the new divergence measures are discussed to guarantee their good performance. Transformation relationships with entropy and dissimilarity measures are debated. As an application, a new intuitionistic fuzzy set theory-based ranking method for figure skaters is designed. A numerical example illustrates its applicability. It uses real data produced during the Men Single Skating Short Program performed in the Team Event in Figure Skating during the Olympic Winter Games 2018 PyeungChang held in Korea from 09.02.2018 to 25.02.2018. [ABSTRACT FROM AUTHOR]

In the modelization of frameworks for multi-attribute decision-making, hesitancy embodies a convenient attitude toward undetermined or vague knowledge provided by distinct experts from a group. Hesitant fuzzy N-soft sets are a functional improvement in hesitant fuzzy sets with the practical spirit of N-soft sets. This analytical model accommodates the hesitant situations with evaluations by grades (e.g., in terms of star ratings) and partial degrees of membership. In this article, we approach the problem of selecting alternatives that are described by this model. We advocate for the use of an adapted form of the ELECTRE-II method, that we describe under the name "hesitant fuzzy N-soft ELECTRE-II method." With the aim of designing this novel method, we first characterize the notion of hesitant fuzzy N-soft concordance and discordance sets and then construct strong and weak outranking relation, which allow us to rank the objects of the reference set. A practical example concerning the ranking of hotels based on star ratings is fully developed in order to illustrate the applicability of this method. Furthermore, an exhaustive comparison with the hesitant fuzzy N-soft ELECTRE-I and bipolar fuzzy ELECTRE-I methods is performed. [ABSTRACT FROM AUTHOR]

This paper is in the continuation of a well-established line of research in multi-criteria decision making, namely, the development of the basic ELECTRE methodology. In the present study, we make an attempt to enhance the influence of the ELECTRE III method, which was conceived in response to the deficiency that existing variants of the ELECTRE method did not provide a full ordering of the alternatives. ELECTRE III has been recently extended so as to make use of information in the form of intuitionistic fuzzy sets. Although this enhancement has been acclaimed by many scholars, it is nonetheless true that further progresses can be made in the quest of generality. In this research, we develop a two-phase Pythagorean fuzzy version of standard ELECTRE III method, called two-phase Pythagorean fuzzy ELECTRE III (PF-ELECTRE III) approach. It takes full advantage of the capabilities of Pythagorean fuzzy sets (PFS); a successful model that generalizes intuitionistic fuzzy sets in a practical and intuitive way. The group decision support system of PF-ELECTRE III first evaluates the performances of the alternatives characterized by Pythagorean fuzzy numbers. The second stage produces a complete ranking of the alternatives from their evaluations in the first phase. We formulate the approach by defining PF indifference threshold, preference threshold and veto threshold functions which provide a more reliable basis to construct outranking relations. The ranking module of the PF-ELECTRE III method is simplified by computing the concordance credibility, discordance credibility and net credibility degrees of each alternative. A diagrammatic representation of the group decision-supporting system is presented to justify the corresponding step-by-step approach for solving problems. Finally, the haze management problem is solved to verify the applicability of the approach proposed in this paper. The investigation of this model and its performance paves the way to additional generalizations by other models like q-rung orthopair fuzzy sets. [ABSTRACT FROM AUTHOR]