Robustness of Interval-Valued Intuitionistic Fuzzy Reasoning Quintuple Implication Method

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The interval-valued intuitionistic fuzzy quintuple implication algorithm, as an extension of the fuzzy reasoning algorithm, may better characterize and deal with uncertainty in the reasoning, but how to select distance measure and analyze the algorithm’s robustness is an important and unsolved topic. In this paper, a novel distance measure of interval-valued intuitionistic fuzzy sets is constructed based on interval-valued intuitionistic fuzzy biresiduum similarity. The unified form of the conclusion about the robustness of interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm for interval-valued intuitionistic fuzzy modus ponens(IVIFMP) and interval-valued intuitionistic fuzzy modus tollens(IVIFMT) is obtained. Especially, the robustness of the interval-valued intuitionistic fuzzy reasoning quintuple implication algorithm based on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$G\ddot {o}del$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Lukasiewicz$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Goguen$ </tex-math></inline-formula> implication operators is presented. An application example and experiment are offered to demonstrate the validity of the obtained conclusion. Furthermore, the new distance metric is compared to traditional distances, and its benefits and limits are discussed. The results show that our approach to research the robustness is simpler and more representative, and the robustness of the algorithm based on other implication operators can be obtained by simple substitution.