Reasoning Methods Based On Pythagorean Fuzzy Environment And Corresponding Applications

With the rapid development of modern society,we often encounter some uncertain problems in daily life.However,traditional mathematical methods cannot solve these uncertain and ambiguity problems.Therefore,to solve various types of uncertainties and complex decision-making problems,the theory of fuzzy sets was proposed by Zadeh.Later,some scholars have successively proposed uncertainty theories including intuitionistic fuzzy set theory and Pythagorean fuzzy set theory in order to extend the concept of fuzzy sets.As a mathematical tool for processing fuzzy information,Pythagorean fuzzy set deals with uncertain problems from both the degree of membership and the degree of non-membership which meets the condition that the square sum of its membership degree and non-membership degree cannot exceed 1.Fuzzy reasoning is one of the most important topics for a theory dealing with uncertainty,many scholars have provided a variety of reasoning methods with strict logical foundations for dealing with fuzzy decision problems.However,there have been few studies on the combination of Pythagorean fuzzy set with the fuzzy reasoning method.Therefore,in order to fill the research gaps in this field,we attempt to establish some fuzzy reasoning methods in the context of Pythagorean fuzzy set and discuss them in depth on theory and application.The main research contents of this paper are as follows:(1)The relationship between fuzzy implication operators and Pythagorean fuzzy implication operators is established,and the properties of the residual Pythagorean fuzzy implication operators and residual Pythagorean fuzzy implication difference operators induced by the Pythagorean t-norm and t-conorm are investigated.The unified form of Pythagorean fuzzy implication operator and residual Pythagorean fuzzy implication difference operator corresponding to several common types of left continuous t-norm is given.In addition,based on the Pythagorean fuzzy biresiduum,the degree of similarity between Pythagorean fuzzy sets is defined.(2)The full implication triple I method and reverse triple I merhod for Pythagorean fuzzy modus ponens(PFMP)and Pythagorean fuzzy modus tollens(PFMT)are established.In addition,the properties of full implication triple I method and reverse triple I method of PFMP and PFMT models including the robustness,continuity and reversibility are analyzed.Finally,a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication triple I method and Pythagorean fuzzy reverse triple I method in medical diagnosis and decision-making problem.(3)By transforming the Pythagorean fuzzy decision-making problem into the form of Pythagorean fuzzy inference rules,a new type of decision-making method based on the Pythagorean fuzzy multiple I reasoning method is established,and the reversibility and continuity properties of the full implication multiple I method of PFMP are analyzed.Finally,a practical problem is discussed to demonstrate the effectiveness of the Pythagorean fuzzy full implication multiple I method in a decision-making problem.The advantages of the new method over existing methods are also explained.