Dual Interval-valued Hesitant Fuzzy Soft Set And Its Application

Many existing mathematical tools are deterministic and measurable in nature.These traditional mathematical tools can only be used to solve certain,ideal problems.However,in the complex and changeable real world,there are still many uncertain issues,and a lot of information is fuzzy,uncertain,and unpredictable.If existing mathematical methods are used to solve these problems,it will become very difficult or even impossible.For example,there are many uncertain problems in decision-making,medical diagnosis,economics,and engineering.Based on this background,a large number of uncertainty theories have been generated,including fuzzy sets,rough sets,soft sets,etc.These basic theories are the supporting points of other uncertainty theory research.In recent years,the generalization of sets and the fusion between sets are one of the hot topics in current uncertain mathematical theory.For instance,fuzzy sets are generalized as hesitant fuzzy sets,interval-valued hesitant fuzzy sets,dual hesitant fuzzy sets,etc.,fuzzy sets and rough sets are merged into fuzzy rough sets,etc.,and soft sets and fuzzy sets are merged into fuzzy soft sets.In a word,both in theory and applications have produced a lot of fruitful results.Based on this idea,this paper generalizes the interval-valued hesitant fuzzy soft sets and combines the dual hesitant-fuzzy fuzzy sets with soft sets.The specific work content is as follows:1.We study the interval-valued hesitant fuzzy soft set model when the parameter set is a fuzzy set.It is called generalized interval-valued hesitant fuzzy soft set.Based on interval-valued hesitant fuzzy soft sets,we define its basic operations such as complement,union,intersection,ring sum,and ring product,discuss the relevant properties of these operations,and prove that the operations of generalized interval-valued hesitant fuzzy soft sets satisfy the commutative law,associative law,De Morgan's laws.2.Interval-valued dual hesitant fuzzy set and soft set are fused,and an dual interval-valued hesitant fuzzy soft set model is proposed.The basic operations are defined,and the concepts of "absolute union" and "relative intersection" of dual interval-valued hesitant fuzzy soft sets are proposed.The commutative law,associative law,and De Morgan's laws of these operations are studied and detailed proofs are given.Finally,based on the distance measure of dual interval-valued hesitant fuzzy soft sets,its similarity measure is studied.Several similarity measures of dual interval-valued hesitant fuzzy soft sets are proposed and applied to practical examples.