Quintuple Implication Principle on interval-valued intuitionistic fuzzy sets

Robustness of Interval-Valued Intuitionistic Fuzzy Reasoning Quintuple Implication Method

“…Proposition 1 [23,26] Let triangular norm(t-norm) ⊕ be left continuous, then there exists a binary operation on…”

“…Lemma 1 [23,26] Let (⊗, →) be a pair of adjoint, (⊕, ) be a pair of co-adjoint, and ⊗, ⊕ be dual of each other; That is →, ⊕ and are associated operator of ⊗. ∀a, b ∈ [0, 1], we have b a = 1 − (1 − a) → (1 − b) .…”

“…∀a, b ∈ [0, 1], we have b a = 1 − (1 − a) → (1 − b) . Definition 1 [26] Let X be a non-empty universe, and an interval-valued intuitionistic fuzzy set A on X can be expressed in the following form:…”

“…) . Theorem 1 [26] Let ⊗ SL be an interval-valued intuitionistic t-norm derived from a left-continuous t-norm⊗. Then there exists an operation → SL on SL such that…”

“…Because of the diversity of intuitionistic fuzzy implication operators and triangular modules [21,22], fuzzy reasoning is further extended to intuitionistic fuzzy reasoning [23][24][25]. Quintuple implication principle(QIP) and triple implication principle(TIP) on IVIF-Ss have been presented by Jin et al [26], which is the extension of fuzzy reasoning quintuple implication algorithm on IVIFSs. Interval-valued Intuitionistic Fuzzy Modus Ponens (IVIFMP) and Interval-valued Intuitionistic Fuzzy Modus Ponens (IVIFMT) are two reasoning forms of interval-valued intuitionistic fuzzy reasoning, as illustrated in equation 1 and equation 2 respectively.…”