Hybrid monotonic inclusion measures for interval-valued hesitant fuzzy sets
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The inclusion measure indicates the degree to which a given partially ordered set is contained in another one. Hybrid monotonic inclusion measure is the rational generalization of inclusion measure. So, the main purpose of this paper is to introduce the hybrid monotonic inclusion measure for interval-valued hesitant fuzzy sets (IVHFSs). Firstly, we present the construction approaches to hybrid monotonic inclusion measures between any two interval-valued hesitant fuzzy elements (IVHFEs) under different order relationships in extended environment. In addition, a new partial order relationship $\leq_{\vee}$ for IVHFEs is also defined, and then some hybrid monotonic inclusion measures under the new partial order $\leq_{\vee}$ for IVHFEs and IVHFSs will be constructed and be proved in detail. Finally, we present some new formulas to calculate the similarity measure and fuzzy entropy derived from the hybrid monotonic inclusion measure for IVHFSs.
Abstract
The inclusion measure indicates the degree to which a given partially ordered set is contained in another one. Hybrid monotonic inclusion measure is the rational generalization of inclusion measure. So, the main purpose of this paper is to introduce the hybrid monotonic inclusion measure for interval-valued hesitant fuzzy sets (IVHFSs). Firstly, we present the construction approaches to hybrid monotonic inclusion measures between any two interval-valued hesitant fuzzy elements (IVHFEs) under different order relationships in extended environment. In addition, a new partial order relationship $\leq_{\vee}$ for IVHFEs is also defined, and then some hybrid monotonic inclusion measures under the new partial order $\leq_{\vee}$ for IVHFEs and IVHFSs will be constructed and be proved in detail. Finally, we present some new formulas to calculate the similarity measure and fuzzy entropy derived from the hybrid monotonic inclusion measure for IVHFSs.
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