Information Measure Of Hesitant Fuzzy Sets And Structure Of Interval-valued Fuzzy Numbers In Polylines

Multi-attribute decision-making problems are an important part of decision theory.It mainly provides people with the optimal scheme or specific ranking results by comprehensively considering multiple attribute indexes and factors.However,the classic multiattribute decision-making problems are usually limited by the fuzziness of decisionmakers' cognition and the complexity of the relationship between things.Therefore,research on fuzzy decision-making theory arises at the historic moment.And the discussion about it mainly focuses on the information measure and aggregation operators of fuzzy sets,interval-valued fuzzy sets and hesitant fuzzy sets etc.At present,many excellent results have been achieved in research of different forms of information measure and aggregation operators.However,there are some defects in fuzzy sets and extended forms.On the one hand,the calculation of hesitant fuzzy sets lacks the proof from the mathematical point of view,and the information measure often shows counter intuition in describing the uncertainty of fuzzy information.On the other hand,as we all know,the operation of fuzzy sets is more complex than those of classical sets,let alone interval-valued fuzzy sets.The main reason for the complexity of the operation is that the arithmetic operation of the extended principle of Zadeh does not satisfy the closeness and linearity.So the above deficiencies limit theoretical development of interval-valued fuzzy sets and hesitant fuzzy sets to a certain extent.Therefore,the purpose of this paper is tantamount to explore the operations and related information measure of hesitant fuzzy(soft)sets and the approximate realization of linear operation of interval-valued fuzzy sets.Specifically,this paper mainly involves the following aspects:1.In this paper,we redefine he union and intersection operations of hesitant fuzzy sets based on the score function.Besides,an entropy based on(R,S)-norm is proposed to characterize the uncertainty of information hesitant fuzzy sets.Secondly,we prove that the proposed measure satisfies the axiomatic definition of entropy on hesitant fuzzy sets,and explore its related properties.Then,we offer some numerical examples to illustrate that the entropy based on parameters R and S overcomes the counterintuitive situation of the existing entropy.Finally,on the basis of the new entropy,we utilize the decisionmaking method combining prospect theory and TOPSIS to solve practical problems in business investment,so as to illustrate the practicability of the new entropy of hesitant fuzzy sets.The validity of entropy based on(R,S)-norm is verified by comparative analysis with other entropy measures.2.This study focuses on the information measure of hesitant fuzzy soft sets.Firstly,we propose the concept of the relative entropy of hesitant fuzzy soft sets which is used to measure the discrimination information,and construct the symmetric cross-entropy of hesitant fuzzy soft sets considering the relative entropy.Then,we provide the axiomatic definitions of similarity measure and entropy of hesitant fuzzy soft sets.new similarity measure and entropy are proposed based on symmetric cross entropy.Meanwhile,we develop a novel similarity measure and an entropy formula of hesitant fuzzy soft sets by using the symmetric cross-entropy.Furthermore,we present two different decisionmaking methods for multi-attribute decision making and multi-attribute group decision making where multi-attribute index information is described by hesitant fuzzy soft sets.Finally,it is shown that the new similarity measure and entropy are better than the existing ones by comparing with an specific example.3.In order to reduce the complexity of operation of interval valued fuzzy sets,we introduce the concept of n polygonal interval-valued fuzzy sets.And it is verified that n polygonal interval-valued fuzzy numbers can approximate general interval valued fuzzy numbers with any precision.Then,the arithmetic operation of n polygonal intervalvalued fuzzy sets is defined based on its good characteristics.In this paper,we also research the topological properties of n polygonal interval-valued fuzzy sets.Finally,the practical significance of this polygonal interval-valued fuzzy number is verified by a specific decision problem.4.We extend the research of n polygonal interval-valued fuzzy numbers.On the one hand,we define the generalized distance formula of n polygonal interval-valued fuzzy numbers by the distance formula of closed intervals.Whenever the completeness of closed interval space,we can obtain the complete separable metric space of n polygonal interval-valued fuzzy numbers according to the generalized distance.Finally,the approximation effect of the polygonal interval-valued fuzzy number and the general intervalvalued fuzzy number is verified by a concrete example.On the other hand,we explore the extended form of n polygonal interval-valued fuzzy numbers,which is generalized n polygonal interval-valued fuzzy numbers.We also consider its mathematical structure and related topological properties based on a given metric.