Reverse triple I method based on the Pythagorean fuzzy inference model and its application

Abstract

This paper is devoted to discussing the reverse triple I method based on the Pythagorean fuzzy set (PFS). We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, residual Pythagorean fuzzy implication operator (RPFIO) and Pythagorean fuzzy biresiduum. The reverse triple I methods for Pythagorean fuzzy modus ponens (PFMP) and Pythagorean fuzzy modus tollens (PFMT) are also established. In addition, some interesting properties of the reverse triple I method of PFMP and PFMT inference models are analysed, including the robustness, continuity and reversibility. Finally, a practical problem is provided to illustrate the effectiveness of the reverse triple I method for PFMP in decision-making problems. The advantages of the new method over existing methods are also expounded. Overall, compared with the existing methods, the proposed methods are based on logical reasoning rather than using aggregation operators, so the novel methods are more logical, can better deal with the uncertain problems in complex decision-making environments and can completely reflect the decision-making opinions of decision-makers.

Keywords

Reverse triple I method PFS RPFIO robustness continuity

1 Introduction

To solve various types of uncertainties and complex decision-making problems, the theory of fuzzy sets was proposed by Zadeh [42]. Later, Atanassov [1] extended the concept of fuzzy sets and introduced intuitionistic fuzzy set (IFS) theory. Because decision-makers consider both membership degree and nonmembership degree in the decision-making process, this theory more accurately addresses the uncertainties and decision-making problems than fuzzy sets. However, the IFS needs to satisfy the restricted condition that the sum of the degrees of membership and the degrees of nonmembership is less than or equal to 1. Under this restricted condition, the range of IFS applications is very narrow, and there are limitations in describing uncertainty and fuzziness problems. For example, when decision-makers adopt 0.6 and 0.7 as membership degrees and nonmembership degrees to express their opinions, it is obviously beyond the range of IFS applications. Thus, Yager [35] proposed Pythagorean fuzzy sets (PFSs) with a restricted condition that the sum of the square of the membership degree and nonmembership degree is less than or equal to 1. Obviously, the range of its applications of PFSs is more accurate and sufficient than that of IFSs. Since then, many scholars have studied PFS and obtained a series of valuable research results [4 , 36]. For example, Zhang and Xu [39] defined the related operations and properties of PFSs and developed a TOPSIS multiattribute decision-making method. Zhang [40] initiated an interval-valued PFS and applied it to decision-making problems. In [15], the authors defined the concept of Pythagorean hesitant fuzzy sets by integrating hesitant fuzzy sets with PFSs. Ren [20] proposed a Pythagorean fuzzy multiattribute group decision-making method based on TODIM. By combining PFS and the N-soft set, Zhang et al. [37] initiated the Pythagorean fuzzy N-soft set theory and applied it to the decision-making problem. Wang and Garg [32] investigated the algorithm for multiple attribute decision-making with interactive Archimedean norm operations under Pythagorean fuzzy uncertainty. Additionally, Senapati and Yager [23] further proposed the concept of Fermatean fuzzy sets based on PFSs with a restricted condition that the cube sum of the membership degree and nonmembership degree is no more than 1. Since the range of its applications of Fermatean fuzzy sets is more sufficient than that of IFSs and PFSs, many researchers have investigated Fermatean fuzzy sets and obtained fruitful results [2 , 34]. For example, Rani and Mishra [21] explored some Fermatean fuzzy Einstein aggregation operators, investigated their enviable characteristics, and solved the multicriteria electric vehicle charging station location selection problem in the context of the Fermatean fuzzy set. In [22], Senapati and Yager proposed an approach for decision-making problems in light of new operators under a Fermatean fuzzy environment. Jeevaraj [9] introduced the concept of interval-valued Fermatean fuzzy sets and established mathematical operations on the class of interval-valued Fermatean fuzzy sets.

Approximate reasoning is one of the most important topics for a theory dealing with uncertainty, and fuzzy reasoning is the main component of approximate reasoning. It is known that the most fundamental models of fuzzy reasoning are fuzzy modus pones (FMP) and fuzzy modus tollens (FMT), where FMP and FMT can be expressed as follows:

FMP inference model: Suppose that $\tilde{X}$ ⟶ $\tilde{Y}$ , and given: ${\tilde{X}}^{*}$ , calculate: ${\tilde{Y}}^{*}$ .

FMT inference model: Suppose that $\tilde{X}$ ⟶ $\tilde{Y}$ , and given: ${\tilde{Y}}^{*}$ , calculate: ${\tilde{X}}^{*}$ .

where $\tilde{X}, {\tilde{X}}^{*}$ and $\tilde{Y}, {\tilde{Y}}^{*}$ are fuzzy sets over U and V, respectively. In 1973, Zadeh [43] introduced the composition rule of inference (CRI). The method is developed into the basic method of fuzzy reasoning. However, from the perspective of logical semantics, the CRI method lacks a strict logic foundation. Therefore, Wang [26] provided a new method based on the standpoint of logical semantics, the full implication reasoning method (triple I method), to establish a strict logical foundation of fuzzy reasoning. Since then, the triple I method received the widespread attention of many scholars [24 , 33] and obtained a series of valuable research results. The reversibility [19], continuity [33], and approximation properties [33] of the triple I method have also been investigated. Additionally, research concerning the reverse triple I method has received great attention. For example, Song [25] proposed the reverse triple I method based on the full implication reasoning method. Zhao and Li [41] studied the reverse triple I method of fuzzy reasoning for the implication operator R_L. Gu [6] gave the general expressions of a reverse triple I sustaining methods for fuzzy reasoning. The reverse triple I method based on pointwise sustaining degrees for the Lukasiewicz implication was studied by Liu [12]. Luo and Sang [13] discussed reverse triple I algorithms based on a class of residual implications induced by the family of Schweizer-Sklar t-norms. Luo [14] discussed the robustness of reverse triple I algorithms based on interval-valued fuzzy inference. However, regardless of whether a fuzzy set or IFS is used, the range of applications is relatively narrow compared with that of PFS, and there are limitations associated with describing uncertainty and fuzziness problems. In addition, there have been few studies on the combination of PFSs with the fuzzy reasoning method. For these reasons, to fill the research gaps in this field, we attempt to establish a reverse triple I method of the PFMP and PFMT inference model.

The innovations of this article are as follows. First, we propose the concepts of Pythagorean t-norm, Pythagore an t-conorm, and RPFIO. Furthermore, we define the degree of similarity between PFSs based on the Pythagorean fuzzy biresiduum. Then, we construct the expressions for the reverse triple I method of the PFMP and PFMT inference model and prove its robustness and continuity. Finally, we explore two decision-making methods to deal with decision-making problems based on the Pythagorean fuzzy reverse triple I method. Compared with the existing methods, we also expound on the advantages of the new methods.

The structure of this article is as follows. Section 2 reviews some basic definitions concerning t-norms, t-conorms, fuzzy implication operators and PFSs. In Section 3, the concepts of Pythagorean t-norm, Pythagorean t-conorm, and RPFIO are proposed. We construct a Pythagorean fuzzy biresiduum and define the degree of similarity between PFSs based on the Pythagorean fuzzy biresiduum in Section 4. Section 5 establishes the reverse triple I method for the PFMP and PFMT inference models, constructs the expressions for the reverse triple I method based on the PFMP and PFMT inference models and investigates its reversibility. In Section 6, the robustness and continuity of the reverse triple I method for PFMP and PFMT inference models based on the degree of similarity are explored. In Section 7, a practical example is provided to illustrate the effectiveness and practicality of the reverse triple I method based on the PFMP inference model. We conclude in Section 8.

2 Preliminaries

In this section, we shall recall several definitions that are necessary for our paper.

Definition 2.1. ([10]) Consider a function ▵ : [0, 1] ² → [0, 1] that satisfies the following three conditions: ∀α, β, γ ∈ [0, 1]:

(1) commutative: α ▵ β = β ▵ α;

(2) associative: α ▵ (β ▵ γ) = (α ▵ β) ▵ γ;

(3) monotonicity: α ≤ β ⇒ α ▵ γ ≤ β ▵ γ;

(4) 1 ▵ α = α.

Then, ▵ is said to be a t-norm. Similarly, if a function ∇ : [0, 1] ² → [0, 1] is associative, commutative, monotonic, and satisfies the condition $0 \nabla \tilde{α} = \tilde{α}$ , then ∇ is said to be a t-conorm. The functions ▵ and ∇ are referred to as dual t-norm and t-conorm if they satisfy the formulas α ▵ β = 1 - (1 - α) ∇ (1 - β) and α ∇ β = 1 - (1 - α) ▵ (1 - β).

Definition 2.2. ([10]) Suppose that ▵ is a t-norm, ∇ is a t-conorm and I is a nonempty index set. For all α, β_i ∈ [0, 1], if $α ▵ (sup_{i \in I} β_{i}) = sup_{i \in I} (α ▵ β_{i})$ , then ▵ is called a left-continuous t-norm. For all α_i, β ∈ [0, 1], if $(inf_{i \in I} α_{i}) \nabla β = inf_{i \in I} (α_{i} \nabla β)$ , then ∇ is called a right-continuous t-conorm.

Definition 2.3. ([10, 28]) Let ▵ be a left-continuous t-norm. Define an operation → : [0, 1] ² → [0, 1] as α → β =sup{ɛ ∈ [0, 1] |ɛ ▵ α ≤ β} such that (▵ , →) forms an adjoint pair, i.e., α ▵ β ≤ γ ⇔ α ≤ β → γ; then, → is said to be a residual fuzzy implication operator.

Definition 2.4. ([38]) Let ∇ be a right-continuous t-conorm. Define a function ⊖ : [0, 1] ² → [0, 1] as α → β =inf{ɛ ∈ [0, 1] |α ≤ ɛ ∇ β} such that (∇ , ⊖) forms a coadjoint pair, i.e., α ≤ β ∇ γ ⇔ α ⊖ γ ≤ β; then, ⊖ is said to be a residual fuzzy difference operator.

Definition 2.5. ([38]) Suppose that (▵ , →) and (∇ , ⊖) are the adjoint pair and coadjoint pair, respectively, where ▵, ∇ are the dual t-norm and t-conorm. Then, ∇, → , ⊖ are called the associated operators of ▵.

Theorem 2.6. ([38]) Assume that ∇, → , ⊖ are the associated operators of ▵. For all α, β ∈ [0, 1], $α ⊖ β = 1 - (1 - β) \to (1 - α) .$

Definition 2.7. ([10]) Let ▵ be a t-norm and → be a residual fuzzy implication operator. ∀α, β ∈ [0, 1], if α ↔ β = (α → β) ∧ (β → α) , then ↔ is said to be a biresiduum associated with a residual fuzzy implication operator.

Theorem 2.8. ([7, 30]) Let ▵ be a t-norm and ↔ be a biresiduum associated with the residual fuzzy implication operator; then, for ∀α, β, γ, λ ∈ [0, 1] :

(1) α↔ 1 = α ;

(2) α = β⇔ α ↔ β = 1 ;

(3) α↔ β = β ↔ α ;

(4) (α↔ β) ▵ γ ↔ λ) ≤ (α ▵ γ) ↔ β ▵ λ) ;

(5) (α↔ _Pβ) ▵ γ ↔ λ) ≤ (α → γ) ↔ β → λ) ;

(6) (α↔ β) ▵ β ↔ γ) ≤ α ↔ γ ;

(7) (α ↔ β) ∧ (γ ↔ λ) ≤ (α ∨ γ) ↔ β ∨ λ).

Lemma 2.9. ([10]) Let X, X′ : U → [0, 1] be two arbitrary functions, then

(1) ⋀_u∈U (X (u)↔ X′ (u)) ≤ (⋀ _u∈UX (u)) ↔ (⋀ _u∈UX′ (u)) ;

(2) ⋀_u∈U (X (u) ↔ X′ (u)) ≤ (⋁ _u∈UX (u)) ↔ (⋁ _u∈UX′ (u)).

In [35], Yager et al. introduced the concept of PFSs.

Definition 2.10. ([35]) Let U be a universal set; then, a PFS P on U is expressed as follows: P = {(u, μ_P (u) , η_P (u)) |u ∈ U} ,

where μ_P (u) and η_P (u) denote the membership degree and nonmembership degree, respectively, with the condition that 0 ≤ (μ_P (u)) ² + (η_P (u)) ² ≤ 1.

We call (μ_P (u) , η_P (u)) a Pythagorean fuzzy number (PFN), which can be written as $\tilde{p} = (μ, η)$ . Denote P^* = {(μ_P (u) , η_P (u)) ∈ [0, 1] ²|0 ≤ (μ_P (u)) ² + (η_P (u)) ² ≤ 1}. Assume that ${\tilde{p}}_{1} = (μ^{'}, η^{'})$ and ${\tilde{p}}_{2} = (μ^{″}, η^{″})$ are two PFNs; then, $\begin{matrix} {\tilde{p}}_{1} \lor {\tilde{p}}_{2} & = (μ^{'} \lor μ^{″}, η^{'} \land η^{″}), {\tilde{p}}_{1} \land {\tilde{p}}_{2} \\ = (μ^{'} \land μ^{″}, η^{'} \lor η^{″}) . \end{matrix}$

Definition 2.11. ([16]) Let ${\tilde{p}}_{1} = (μ^{'}, η^{'})$ and ${\tilde{p}}_{2} = (μ^{″}, η^{″})$ be two PFNs. The score function and the accuracy function of $\tilde{p}$ can be defined as $S (\tilde{p}) = μ^{2} - η^{2}$ and $H (\tilde{p}) = μ^{2} + η^{2}$ , respectively. For ${\tilde{p}}_{1} = (μ^{'}, η^{'})$ and ${\tilde{p}}_{2} = (μ^{″}, η^{″})$ , then

(1) if $S ({\tilde{p}}_{1}) > S ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} > {\tilde{p}}_{2}$ .

(2) if $S ({\tilde{p}}_{1}) = S ({\tilde{p}}_{2})$ , then

if $H ({\tilde{p}}_{1}) > H ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} > {\tilde{p}}_{2}$ ; if $H ({\tilde{p}}_{1}) = H ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} = {\tilde{p}}_{2}$ .

According to Definition 2.15, we have μ′ ≥ μ″, η′ ≤ η″. Therefore, ${\tilde{p}}_{1} \geq {\tilde{p}}_{2} .$

Theorem 2.12. ([30]) Let ▵ be a t-norm and ↔ be a biresiduum associated with a residual fuzzy implication operator; then, ∀α, β, γ, λ ∈ [0, 1] :

(1) α↔ 1 = α ;

(2) α = β⇔ α ↔ β = 1 ;

(3) α↔ β = β ↔ α ;

(4) (α↔ β) ▵ γ ↔ λ) ≤ (α ▵ γ) ↔ β ▵ λ) ;

(5) (α↔ _Pβ) ▵ γ ↔ λ) ≤ (α → γ) ↔ β → λ) ;

(6) (α↔ β) ▵ β ↔ γ) ≤ α ↔ γ ;

(7) (α ↔ β) ∧ (γ ↔ λ) ≤ (α ∨ γ) ↔ β ∨ λ).

Lemma 2.13. ([10]) Let X, X′ : U → [0, 1] be two arbitrary functions; then,

(1) ⋀_u∈U (X (u)↔ X′ (u)) ≤ (⋀ _u∈UX (u)) ↔ (⋀ _u∈UX′ (u)) ;

(2) ⋀_u∈U (X (u) ↔ X′ (u)) ≤ (⋁ _u∈UX (u)) ↔ (⋁ _u∈UX′ (u)).

In [35], Yager et al. introduced the concept of PFSs.

Definition 2.14. ([35]) Let U be a universal set; then, a PFS P on U is expressed as follows: P = {(u, μ_P (u) , η_P (u)) |u ∈ U} ,

where μ_P (u) and η_P (u) denote the membership degree and nonmembership degree, respectively, with the condition that 0 ≤ (μ_P (u)) ² + (η_P (u)) ² ≤ 1.

We call (μ_P (u) , η_P (u)) a PFN, which can be written as $\tilde{p} = (μ, η)$ . Denote P^* = {(μ_P (u) , η_P (u)) ∈ [0, 1] ²|0 ≤ (μ_P (u)) ² + (η_P (u)) ² ≤ 1}. Assume that ${\tilde{p}}_{1} = (μ^{'}, η^{'})$ and ${\tilde{p}}_{2} = (μ^{″}, η^{″})$ are two PFNs; then, $\begin{matrix} {\tilde{p}}_{1} \lor {\tilde{p}}_{2} & = (μ^{'} \lor μ^{″}, η^{'} \land η^{″}), {\tilde{p}}_{1} \land {\tilde{p}}_{2} \\ = (μ^{'} \land μ^{″}, η^{'} \lor η^{″}) . \end{matrix}$

Definition 2.15. ([16]) Let ${\tilde{p}}_{1} = (μ^{'}, η^{'})$ and ${\tilde{p}}_{2} = (μ^{″}, η^{″})$ be two PFNs. The score function and the accuracy function of $\tilde{p}$ can be defined as $S (\tilde{p}) = μ^{2} - η^{2}$ and $H (\tilde{p}) = μ^{2} + η^{2}$ , respectively. For ${\tilde{p}}_{1} = (μ^{'}, η^{'})$ and ${\tilde{p}}_{2} = (μ^{″}, η^{″})$ ,

(1) if $S ({\tilde{p}}_{1}) > S ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} > {\tilde{p}}_{2}$ .

(2) if $S ({\tilde{p}}_{1}) = S ({\tilde{p}}_{2})$ , then

if $H ({\tilde{p}}_{1}) > H ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} > {\tilde{p}}_{2}$ ; if $H ({\tilde{p}}_{1}) = H ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1} = {\tilde{p}}_{2}$ .

According to Definition 2.15, we have $μ^{'} \geq μ^{″}, η^{'} \leq η^{″} \Leftrightarrow {\tilde{p}}_{1} \geq {\tilde{p}}_{2} .$

3 Pythagorean fuzzy implication operator

In this section, we propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, and RPFIO.

Let $\tilde{α} = (μ, η), \tilde{β} = (μ^{'}, η^{'}) \in P^{*}$ . Then, we define two binary operations ▵_{P
^*} and ∇_{P
^*} on P^* as follows: $\tilde{α} ▵_{P^{*}} \tilde{β} = (\sqrt{μ^{2} ▵ μ^{' 2}}, \sqrt{η^{2} \nabla η^{' 2}}),$ $\tilde{α} \nabla_{P^{*}} \tilde{β} = (\sqrt{μ^{2} \nabla μ^{' 2}}, \sqrt{η^{2} ▵ η^{' 2}}) .$

Since 0 ≤ μ² + η² ≤ 1, 0 ≤ μ^′2 + η^′2 ≤ 1, , it follows from Definition 2.1 that $\begin{matrix} (\sqrt{μ^{2} ▵ μ^{' 2}})^{2} & + (\sqrt{η^{2} \nabla η^{' 2}})^{2} \\ = μ^{2} ▵ μ^{' 2} + η^{2} \nabla η^{' 2} \\ \leq μ^{2} ▵ μ^{' 2} + (1 - μ^{2}) \nabla (1 - μ^{' 2}) \\ = μ^{2} ▵ μ^{' 2} + 1 - μ^{2} ▵ μ^{' 2} = 1, \end{matrix}$ which implies that $\tilde{α} ▵_{P^{*}} \tilde{β} \in P^{*}$ . By the same token, we prove that $\tilde{α} \nabla_{P^{*}} \tilde{β} \in P^{*}$ .

Theorem 3.1. Let ▵_{P
^*} and ∇_{P
^*} be two binary operations on P^*. Given that $\tilde{α} = (μ, η), \tilde{β} = (μ^{'}, η^{'}), \tilde{γ} = (μ^{″}, η^{″})$ are three PFNs, then ▵_{P
^*} satisfies the following conditions:

(1) $\tilde{α} ▵_{P^{*}} \tilde{β} = \tilde{β} ▵_{P^{*}} \tilde{α}$ ;

(2) $\tilde{α} ▵_{P^{*}} (\tilde{β} ▵_{P^{*}} \tilde{γ}) = (\tilde{α} ▵_{P^{*}} \tilde{β}) ▵_{P^{*}} \tilde{γ}$ ;

(3) If $\tilde{α} \leq \tilde{β},$ then $\tilde{α} ▵_{P^{*}} \tilde{γ} \leq \tilde{β} ▵_{P^{*}} \tilde{γ}$ ;

(4) $(1, 0) ▵_{P^{*}} \tilde{α} = \tilde{α}$ .

Similarly, ∇_{P
^*} is associative, commutative and monotonic and satisfies the condition $(0, 1) \nabla_{P^{*}} \tilde{α} = \tilde{α}$ .

Proof. According to the associative law and the commutative law for ▵ and ∇, both ▵_{P
^*} and ∇_{P
^*} are associative and commutative. On the one hand, when $\tilde{α} \leq \tilde{β}$ , we have μ ≤ μ′, η ≥ η′. On the other hand, since ▵, ∇ is monotonic, we obtain $\tilde{α} ▵_{P^{*}} \tilde{γ} = (\sqrt{μ^{2} ▵ μ^{″ 2}}, \sqrt{η^{2} \nabla η^{″ 2}}) \leq (\sqrt{μ^{' 2} ▵ μ^{″ 2}}, \sqrt{η^{' 2} \nabla η^{″ 2}}) = \tilde{β} ▵_{P^{*}} \tilde{γ} .$

Thus, ▵_{P
^*} is monotonic. Furthermore, ▵_{P
^*} satisfies the boundary conditions. In fact, $(1, 0) ▵_{P^{*}} \tilde{α} = (1, 0) ▵_{P^{*}} (μ, η) = (\sqrt{μ^{2} ▵ 1}, \sqrt{η^{2} \nabla 0}) = (μ, η) = \tilde{α}$ .

Similarly, we can prove that ∇_{P
^*} is monotonic and satisfies the condition $(0, 1) \nabla_{P^{*}} \tilde{α} = \tilde{α}$ . □ Based on Theorem 3.1, we obtain the following definition.

Definition 3.2. ▵_{P
^*} is referred to as a Pythagorean t-norm induced from ▵, and ∇_{P
^*} is said to be a Pythagorean t-conorm induced from ∇.

Theorem 3.3. Let ▵ be a left-continuous t-norm and ∇ be a right-continuous t-conorm. Then, ▵_{P
^*} and ∇_{P
^*} satisfy the formulas: $\tilde{α} ▵_{P^{*}} (sup_{i \in I} {\tilde{β}}_{i}) = sup_{i \in I} (\tilde{α} ▵_{P^{*}} {\tilde{β}}_{i}),$ $(inf_{i \in I} {\tilde{α}}_{i}) \nabla_{P^{*}} \tilde{β} = inf_{i \in I} ({\tilde{α}}_{i} \nabla_{P^{*}} \tilde{β}),$

where I is a nonempty index set.

In that case, ▵_{P
^*} is called a left-continuous Pythagorean t-norm on P^*, and ∇_{P
^*} is called a right-continuous Pythagorean t-conorm on P^*.

Definition 3.4. Let ▵_{P
^*} be a left-continuous Pythagorean t-norm. If a binary operation →_{P
^*} on P^* satisfies $\tilde{α} \to_{P^{*}} \tilde{β} = sup {\tilde{ɛ} \in P^{*} | \tilde{ɛ} ▵_{P^{*}} \tilde{α} \leq \tilde{β}}$ , then →_{P
^*} is said to be an RPFIO.

Theorem 3.5. Let ▵_{P
^*} be a Pythagorean t-norm and →_{P
^*} be an RPFIO; then, the following holds:

(1) $\tilde{γ} ▵_{P^{*}} \tilde{α} \leq \tilde{β} \Leftrightarrow \tilde{γ} \leq \tilde{α} \to_{P^{*}} \tilde{β},$

(2) $\tilde{α} \to_{P^{*}} (\tilde{β} \to_{P^{*}} \tilde{γ}) = (\tilde{α} ▵_{P^{*}} \tilde{β}) \to_{P^{*}} \tilde{γ},$

(3) $\tilde{α} \to_{P^{*}} \tilde{β} = (1, 0) \Leftrightarrow \tilde{α} \leq \tilde{β},$

(4) $\tilde{γ} \leq \tilde{α} \to_{P^{*}} \tilde{β} \Leftrightarrow \tilde{α} \leq \tilde{γ} \to_{P^{*}} \tilde{β},$

(5) $(1, 0) \to_{P^{*}} \tilde{α} = \tilde{α},$

(6) $\tilde{γ} \to_{P^{*}} (\tilde{α} \to_{P^{*}} \tilde{β}) = \tilde{α} \to_{P^{*}} (\tilde{γ} \to_{P^{*}} \tilde{β}) .$

Proof. (1) From the definition of →_{P
^*}, we observe that if $\tilde{γ} ▵_{P^{*}} \tilde{α} \leq \tilde{β}$ , then $\tilde{γ} \leq \tilde{α} \to_{P^{*}} \tilde{β}$ . When $\tilde{γ} \leq \tilde{α} \to_{P^{*}} \tilde{β}$ , from the left continuity and monotonicity of ▵_{P
^*}, we can obtain $\begin{matrix} \tilde{γ} ▵_{P^{*}} \tilde{α} & \leq (\tilde{α} \to_{P^{*}} \tilde{β}) ▵_{P^{*}} \tilde{α} \\ = sup {\tilde{ɛ} \in P^{*} | \tilde{ɛ} ▵_{P^{*}} \tilde{α} \leq \tilde{β}} ▵_{P^{*}} \tilde{α} \\ = sup {(\tilde{ɛ} ▵_{P^{*}} \tilde{α}) \in P^{*} | \tilde{ɛ} ▵_{P^{*}} \tilde{α} \leq \tilde{β}} = \tilde{β} . \end{matrix}$

(2) Let $\tilde{x} = \tilde{α} \to_{P^{*}} (\tilde{β} \to_{P^{*}} \tilde{γ}) \in P^{*} .$ From the result (1), we have $\begin{matrix} \tilde{x} = \tilde{α} \to_{P^{*}} (\tilde{β} \to_{P^{*}} \tilde{γ}) & \Leftrightarrow \tilde{α} ▵_{P^{*}} \tilde{x} = \tilde{β} \to_{P^{*}} \tilde{γ} \\ \Leftrightarrow (\tilde{α} ▵_{P^{*}} \tilde{x}) ▵_{P^{*}} \tilde{β} = \tilde{γ} & \Leftrightarrow (\tilde{x} ▵_{P^{*}} \tilde{α}) ▵_{P^{*}} \tilde{β} = \tilde{γ} \\ \Leftrightarrow \tilde{x} ▵_{P^{*}} (\tilde{α} ▵_{P^{*}} \tilde{β}) = \tilde{γ} & \Leftrightarrow \tilde{x} = (\tilde{α} ▵_{P^{*}} \tilde{β}) \to_{P^{*}} \tilde{γ} . \end{matrix}$

(3) - (6) are straightforward. □

Theorem 3.6. Assume that →_{P
^*} is an RPFIO, $\tilde{α} = (μ, η), \tilde{β} = (μ^{'}, η^{'}) \in P^{*}$ . Then, $\begin{matrix} \tilde{α} \to_{P^{*}} \tilde{β} \\ = (\sqrt{μ^{2} \to μ^{' 2}} \land \sqrt{1 - (η^{' 2} ⊖ η^{2}}), \sqrt{η^{' 2} ⊖ η^{2}}) . \end{matrix}$

Proof. Let $\tilde{ɛ} = (μ^{″}, η^{″}) = \tilde{α} \to_{P^{*}} \tilde{β}$ . For all $\tilde{ɛ_{i}} = (μ_{i}^{″}, η_{i}^{″}) \in P^{*}$ , it follows from Definition 3.2 that

$\begin{matrix} \tilde{α} \to_{P^{*}} \tilde{β} = sup {\tilde{ɛ_{i}} \in P^{*} | \tilde{ɛ_{i}} ▵_{P^{*}} \tilde{α} \leq \tilde{β}} \\ = sup {(μ_{i}^{″}, η_{i}^{″}) \in P^{*} | \sqrt{{μ_{i}^{″}}^{2} ▵ μ^{2}} \leq μ^{'}, \sqrt{{η_{i}^{″}}^{2} \nabla η^{2}} \geq η^{'}} \\ = sup {(μ_{i}^{″}, η_{i}^{″}) \in P^{*} | {μ_{i}^{″}}^{2} ▵ μ^{2} \leq μ^{' 2}, {η^{″}}_{i}^{2} \nabla η^{2} \geq η^{' 2}} . \end{matrix}$

By Theorem 3.5, we have $\begin{matrix} {μ_{i}^{″}}^{2} ▵ μ^{2} \leq μ^{' 2} & {η_{i}^{″}}^{2} \nabla η^{2} \geq η^{' 2}, \\ \Rightarrow μ^{2} \to μ^{' 2} \geq {μ_{i}^{″}}^{2} & η^{' 2} ⊖ η^{2} \leq {η_{i}^{″}}^{2}, \\ \Rightarrow μ^{2} \to μ^{' 2} \geq {μ_{i}^{″}}^{2} & η^{' 2} ⊖ η^{2} \leq {η_{i}^{″}}^{2}, \\ 1 - (η^{' 2} ⊖ η^{2}) \geq 1 - {η_{i}^{″}}^{2} \geq {μ_{i}^{″}}^{2}, \\ \Rightarrow \sqrt{μ^{2} \to μ^{' 2}} \geq μ_{i}^{″} & \sqrt{η^{' 2} ⊖ η^{2}} \leq η_{i}^{″}, \\ \sqrt{1 - (η^{' 2} ⊖ η^{2})} \geq \sqrt{1 - {η_{i}^{″}}^{2}} \geq μ_{i}^{″} . \end{matrix}$ Therefore, if $\tilde{ɛ} = (μ^{″}, η^{″}) = (μ_{i}^{″}, η_{i}^{″})$ , we have $\begin{matrix} \tilde{ɛ} & = (μ^{″}, η^{″}) = \tilde{α} \to_{P^{*}} \tilde{β} \\ \leq (\sqrt{μ^{2} \to μ^{' 2}} \land \sqrt{1 - (η^{' 2} ⊖ η^{2})}, \sqrt{η^{' 2} ⊖ η^{2}}) . \end{matrix}$

However,

$\begin{matrix} (\sqrt{μ^{2} \to μ^{' 2}} & \land \sqrt{1 - (η^{' 2} ⊖ η^{2})}, \sqrt{η^{' 2} ⊖ η^{2}}) ▵_{P^{*}} \tilde{α} \\ \leq (\sqrt{μ^{2} \to μ^{' 2}}, \sqrt{η^{' 2} ⊖ η^{2}}) ▵_{P^{*}} (μ, η) \\ \leq (\sqrt{(μ^{2} \to μ^{' 2}) ▵ μ^{2}}, \sqrt{(η^{' 2} ⊖ η^{2}) \nabla η^{2}}) \\ = (\sqrt{μ^{' 2}}, \sqrt{η^{' 2}}) = (μ^{'}, η^{'}) = \tilde{β} . \end{matrix}$

By applying the result (1) of Theorem 3.5, we obtain $\begin{matrix} \tilde{α} \to_{P^{*}} \tilde{β} \\ = (\sqrt{μ^{2} \to μ^{' 2}} \land \sqrt{1 - (η^{' 2} ⊖ η^{2}}), \sqrt{η^{' 2} ⊖ η^{2}}) . \end{matrix}$ □ Example 3.7. The following three RPFIOs were induced by three residual fuzzy implication operators and residual fuzzy difference operators:

(1) When ▵ is the $G \ddot{o} del$ t-norm, α ▵ _Gβ = α ∧ β, and its associated operators are $α \nabla_{G} β = α \lor β, α \to_{G} β = {\begin{matrix} 1, & α \leq β; \\ β, & α > β . \end{matrix}$

$β \underset{G}{⊖} α = {\begin{matrix} 0, & β \leq α; \\ β, & β > α . \end{matrix}$ Then, $\tilde{α} \to_{P_{G}^{*}} \tilde{β} = {\begin{matrix} (1, 0), & μ \leq μ^{'}, η^{'} \leq η; \\ (\sqrt{1 - η^{' 2}}, η^{'}), & μ \leq μ^{'}, η^{'} > η; \\ (μ^{'}, 0), & μ > μ^{'}, η^{'} \leq η; \\ (μ^{'}, η^{'}), & μ > μ^{'}, η^{'} > η . \end{matrix}$

(2) When ▵ is Product t-norm, α ▵ _πβ = αβ, and its associated operators are $α \nabla_{π} β = α + β - α β, α \to_{π} β = {\begin{matrix} 1, & α \leq β; \\ \frac{β}{α}, & α > β . \end{matrix}$

$β \underset{π}{⊖} α = {\begin{matrix} 0, & β \leq α; \\ \frac{β - α}{1 - α}, & β > α . \end{matrix}$ Then, $\tilde{α} \to_{P_{π}^{*}} \tilde{β} =$ ${\begin{matrix} (1, 0), & μ \leq μ^{'}, η^{'} \leq η; \\ (\frac{\sqrt{1 - η^{' 2}}}{\sqrt{1 - η^{2}}}, \frac{\sqrt{η^{' 2} - η^{2}}}{\sqrt{1 - η^{2}}}), & μ \leq μ^{'}, η^{'} > η; \\ (\frac{μ^{'}}{μ}, 0), & μ > μ^{'}, η^{'} \leq η; \\ (\frac{\sqrt{1 - η^{' 2}}}{\sqrt{1 - η^{2}}} \land \frac{μ^{'}}{μ}, \frac{\sqrt{η^{' 2} - η^{2}}}{\sqrt{1 - η^{2}}}), & μ > μ^{'}, η^{'} > η . \end{matrix}$

(3) When ▵ is the R₀ t-norm, α ▵ _{R
₀}β = ${\begin{matrix} 0, & α + β \leq 1; \\ α \land β, & α + β > 1 . \end{matrix}$ and its associated operators are

$α \nabla_{R_{0}} β = {\begin{matrix} 1, & α + β \geq 1; \\ α \lor β, & α + β < 1 . \end{matrix} α \to_{R_{0}} β =$ ${\begin{matrix} 1, & α \leq β; \\ (1 - α) \lor β, & α > β . \end{matrix} β \underset{R_{0}}{⊖} α = {\begin{matrix} 0, & β \leq α; \\ β \land (1 - α), & β > α . \end{matrix}$ Then, $\tilde{α} \to_{P_{R_{0}}^{*}} \tilde{β} =$ ${\begin{matrix} (1, 0), & μ \leq μ^{'}, η^{'} \leq η; \\ (\sqrt{1 - η^{' 2}} \lor η, η^{'} \land \sqrt{1 - η^{2}}), & μ \leq μ^{'}, η^{'} > η; \\ (\sqrt{1 - μ^{2}} \lor μ^{'}, 0), & μ > μ^{'}, η^{'} \leq η; \\ ((\sqrt{1 - μ^{2}} \lor μ^{'}) \land (\sqrt{1 - η^{' 2}} \lor η), \\ η^{'} \land \sqrt{1 - η^{2}}), & μ > μ^{'}, η^{'} > η . \end{matrix}$

4 Degree of similarity between PFSs based on the Pythagorean fuzzy biresiduum

In this section, we propose the concept of the Pythagorean fuzzy biresiduum and define the degree of similarity between PFSs based on the Pythagorean fuzzy biresiduum.

Definition 4.1. Given that $\tilde{α}$ and $\tilde{β}$ are two PFNs, if $\tilde{α} \leftrightarrow_{P^{*}} \tilde{β} = (\tilde{α} \to_{P^{*}} \tilde{β}) \land (\tilde{β} \to_{P^{*}} \tilde{α}),$ then ↔_{P
^*} is said to be a Pythagorean fuzzy biresiduum associated with the RPFIO.

Theorem 4.2. Let ▵_{P
^*} be a Pythagorean t-norm, →_{P
^*} be an RPFIO and ↔_{P
^*} be a Pythagorean fuzzy biresiduum; then, the following holds:

(1) $\tilde{α} \leftrightarrow_{P^{*}} (1, 0) = \tilde{α};$

(2) $\tilde{α} = \tilde{β} \Leftrightarrow \tilde{α} \leftrightarrow_{P^{*}} \tilde{β} = (1, 0);$

(3) $(\tilde{α} \leftrightarrow_{P^{*}} \tilde{β}) ▵_{P^{*}} (\tilde{γ} \leftrightarrow_{P^{*}} \tilde{λ}) \leq (\tilde{α} ▵_{P^{*}} \tilde{γ}) \leftrightarrow_{P^{*}} (\tilde{β} ▵_{P^{*}} \tilde{λ});$

(4) $(\tilde{α} \leftrightarrow_{P^{*}} \tilde{β}) ▵_{P^{*}} (\tilde{γ} \leftrightarrow_{P^{*}} \tilde{λ}) \leq (\tilde{α} \to_{P^{*}} \tilde{γ}) \leftrightarrow_{P^{*}} (\tilde{β} \to_{P^{*}} \tilde{λ});$

Proof. Assume that $\tilde{α} = (μ, η), \tilde{β} = (μ^{'}, η^{'}), \tilde{γ} = (μ^{*}, η^{*})$ and $\tilde{λ} = (μ^{*^{'}}, η^{*^{'}}) .$ From the definition of ↔_{P
^*} and Theorem 3.5, the results from (1) to (2) are straightforward.

(3) Assume that ${\bar{η}}^{2} = 1 - η^{2}, {\bar{η}}^{'}^{2} = 1 - η^{' 2}, {\bar{η}}^{* 2} = 1 - η^{* 2}$ and ${\bar{η}}^{*^{'} 2} = 1 - η^{*^{'} 2}$ . From Definition 4.1 and the concepts of Pythagorean t-norm, we can obtain

$\begin{matrix} (\tilde{α} \leftrightarrow_{P^{*}} \tilde{β}) ▵_{P^{*}} (\tilde{γ} \leftrightarrow_{P^{*}} \tilde{λ}) \\ = ((\tilde{α} \to_{P^{*}} \tilde{β}) \land (\tilde{β} \to_{P^{*}} \tilde{α})) ▵_{P^{*}} ((\tilde{γ} \to_{P^{*}} \tilde{λ}) \land (\tilde{λ} \to_{P^{*}} \tilde{γ})) \\ = (\sqrt{μ^{2} \to μ^{' 2}} \land \sqrt{{\bar{η}}^{2} \to {\bar{η}}^{'}^{2}}, \sqrt{1 - {\bar{η}}^{2} \to {\bar{η}}^{'}^{2}}) \land (\sqrt{μ^{' 2} \to μ^{2}} \\ \land \sqrt{{\bar{η}}^{'}^{2} \to {\bar{η}}^{2}}, \sqrt{1 - {\bar{η}}^{'}^{2} \to {\bar{η}}^{2}}) ▵_{P^{*}} (\sqrt{μ^{* 2} \to μ^{*^{'} 2}} \\ \land \sqrt{{\bar{η}}^{* 2} \to {\bar{η}}^{*^{'} 2}}, \sqrt{1 - {\bar{η}}^{* 2} \to {\bar{η}}^{*^{'} 2}}) \land (\sqrt{μ^{*^{'} 2} \to μ^{* 2}} \\ \land \sqrt{{\bar{η}}^{*^{'} 2} \to {\bar{η}}^{* 2}}, \sqrt{1 - {\bar{η}}^{*^{'} 2} \to {\bar{η}}^{* 2}}) \\ = (\sqrt{((μ^{2} \leftrightarrow μ^{' 2}) \land ({\bar{η}}^{2} \leftrightarrow {\bar{η}}^{'}^{2})) ▵ ((μ^{* 2} \leftrightarrow μ^{*^{'} 2}) \land ({\bar{η}}^{* 2} \leftrightarrow {\bar{η}}^{*^{'} 2}))}, \\ \sqrt{(1 - {\bar{η}}^{'}^{2} \leftrightarrow {\bar{η}}^{2}) \nabla (1 - {\bar{η}}^{*^{'} 2} \leftrightarrow {\bar{η}}^{* 2})}) \\ \leq (\sqrt{(μ^{2} \leftrightarrow μ^{' 2}) ▵ (μ^{* 2} \leftrightarrow μ^{*^{'} 2})} \land \sqrt{({\bar{η}}^{2} \leftrightarrow {\bar{η}}^{'}^{2}) ▵ ({\bar{η}}^{* 2} \leftrightarrow {\bar{η}}^{*^{'} 2})}, \\ \sqrt{1 - (({\bar{η}}^{'}^{2} \leftrightarrow {\bar{η}}^{2}) ▵ ({\bar{η}}^{*^{'} 2} \leftrightarrow {\bar{η}}^{* 2}))}) \\ \leq (\sqrt{(μ^{2} ▵ μ^{* 2}) \leftrightarrow (μ^{' 2} ▵ μ^{*^{'} 2})} \land \sqrt{({\bar{η}}^{2} ▵ {\bar{η}}^{* 2}) \leftrightarrow ({\bar{η}}^{'}^{2} ▵ {\bar{η}}^{*^{'} 2})}, \\ \sqrt{1 - (({\bar{η}}^{'}^{2} ▵ {\bar{η}}^{*^{'} 2}) \leftrightarrow ({\bar{η}}^{2} ▵ {\bar{η}}^{* 2}))}) \\ = (\sqrt{μ^{2} ▵ μ^{* 2}}, \sqrt{1 - {\bar{η}}^{2} ▵ {\bar{η}}^{* 2}}) \leftrightarrow_{P^{*}} \sqrt{μ^{' 2} ▵ μ^{*^{'} 2}}, \sqrt{1 - {\bar{η}}^{'}^{2} ▵ {\bar{η}}^{*^{'} 2}}) \\ = (\sqrt{μ^{2} ▵ μ^{* 2}}, \sqrt{η^{2} \nabla η^{* 2}}) \leftrightarrow_{P^{*}} (\sqrt{μ^{' 2} ▵ μ^{*^{'} 2}}, \sqrt{η^{' 2} \nabla η^{*^{'} 2}}) \\ = (μ, η) ▵_{P^{*}} (μ^{*}, η^{*}) \leftrightarrow_{P^{*}} (μ^{'}, η^{'}) ▵_{P^{*}} (μ^{*^{'}}, η^{*^{'}}) \\ = (\tilde{α} ▵_{P^{*}} \tilde{γ}) \leftrightarrow_{P^{*}} (\tilde{β} ▵_{P^{*}} \tilde{λ}) . \end{matrix}$

(4) It is similar to the proof of (3). □

Lemma 4.3. Let $\tilde{X}, {\tilde{X}}^{'}$ be two PFSs over U; then,

(1) $⋀_{u \in U} (\tilde{X} (u) \leftrightarrow_{P^{*}} {\tilde{X}}^{'} (u)) \leq (⋀_{u \in U} \tilde{X} (u)) \leftrightarrow_{P^{*}} (⋀_{u \in U} {\tilde{X}}^{'} (u));$

(2) $⋀_{u \in U} (\tilde{X} (u) \leftrightarrow_{P^{*}} {\tilde{X}}^{'} (u)) \leq (⋁_{u \in U} \tilde{X} (u)) \leftrightarrow_{P^{*}} (⋁_{u \in U} {\tilde{X}}^{'} (u)) .$

Proof. This is similar to the proof of result (4) of Theorem 4.2. □ Definition 4.4. Assume that →_{P
^*} is an RPFIO and ↔_{P
^*} is a Pythagorean fuzzy biresiduum associated with RPFIO. Given that X and X′ are two PFSs over U, and δ ∈ P^*, define

$S_{P} (X, X^{'}) = ⋀_{u \in U} {\tilde{X} (u) \leftrightarrow_{P^{*}} {\tilde{X}}^{'} (u)}$ $= ⋀_{u \in U} {(\tilde{X} (u) \to_{P^{*}} {\tilde{X}}^{'} (u)) \land ({\tilde{X}}^{'} (u) \to_{P^{*}} \tilde{X} (u))};$ then, S_P (X, X′) is referred to as the degree of similarity of X and X′ induced from the RPFIO. If S_P (X, X′) ≥ δ, then we say that X and X′ are δ-equal and denote X = (δ) X′.

5 The reverse triple I method of the PFMP model and PFMT model

In [25], Song proposed the reverse triple I method of the FMP model and FMT model based on the full implication reasoning method. However, the fatal flaw of this method is that the range of its applications is relatively narrow compared with that of PFS such that there are limitations associated with describing uncertainty and fuzziness problems. For that reason, in this section, we attempt to extend the reverse triple I method of the FMP model and FMT model proposed by Song to the PFMP and PFMT environments and establish the Pythagorean fuzzy reverse triple I method.

Now, the PFMP inference model is given as follows: $\frac{\begin{matrix} Suppose that & \tilde{X} (u) & \to_{P^{*}} & \tilde{Y} (v) \\ And given & {\tilde{X}}^{*} (u) \end{matrix}}{\begin{matrix} Calculate & {\tilde{Y}}^{*} (v) \end{matrix}}$ (1)In addition, the flowchart of the PFMP inference model is shown in Fig. 1.

Fig. 1

PFMP inference model.

The principle of the reverse triple I method for the PFMP inference model is to seek the optimal ${\tilde{Y}}^{*}$ over V such that the expression $({\tilde{X}}^{*} (u) \to_{P^{*}} {\tilde{Y}}^{*} (v)) \to_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)),$ (2)takes the largest possible value for any v ∈ V. That is, the conclusion ${\tilde{Y}}^{*}$ of Equation (1) is the largest PFS over V satisfying $({\tilde{X}}^{*} (u) \to_{P^{*}} {\tilde{Y}}^{*} (v)) \to_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) = (1, 0) .$ (3)where →_{P
^*} is an RPFIO, ${\tilde{X}}^{*} (u) = (μ_{p}^{*} (u), η_{p}^{*} (u)), \tilde{X} (u) = (μ_{p} (u), η_{p} (u))$ , ${\tilde{Y}}^{*} (v) = (μ_{p}^{*^{'}} (v), η_{p}^{*^{'}} (v)),$ and $\tilde{Y} (v) = (μ_{p}^{'} (v), η_{p}^{'} (v)) .$

Theorem 5.1. Let ${\tilde{X}}^{*}, \tilde{X}$ be PFSs over U, and let ${\tilde{Y}}^{*}, \tilde{Y}$ be PFSs over V. Given that ▵_{P
^*} is a Pythagorean t-norm and →_{P
^*} is an RPFIO, the reverse triple I solution of Equation (1) can be expressed as follows: ${\tilde{Y}}^{*} (v) = ⋀_{u \in U} {{\tilde{X}}^{*} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v))} .$ (4)

Proof. First, we prove that ${\tilde{Y}}^{*} (v)$ determined by Equation (4) satisfies Equation (3). It follows from Equation (4) that ${\tilde{Y}}^{*} (v) \leq {\tilde{X}}^{*} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) .$ By Theorem 3.5, we obtain ${\tilde{X}}^{*} (u) \to_{P^{*}} {\tilde{Y}}^{*} (v) \leq \tilde{X} (u) \to_{P^{*}} \tilde{Y} (v),$ Thus, $({\tilde{X}}^{*} (u) \to_{P^{*}} {\tilde{Y}}^{*} (v)) \to_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) = (1, 0) .$

Second, we prove that ${\tilde{Y}}^{*}$ is the largest PFS over V satisfying Equation (3). Assume that ${\tilde{Z}}^{*}$ is a PFS over V satisfying Equation (3), we have $({\tilde{X}}^{*} (u) \to_{P^{*}} {\tilde{Z}}^{*} (v)) \to_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) = (1, 0) .$ By Theorem 3.5, we obtain $\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v) \geq {\tilde{X}}^{*} (u) \to_{P^{*}} {\tilde{Z}}^{*} (v) .$ Thus, ${\tilde{Z}}^{*} (v) \leq {\tilde{X}}^{*} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) .$ Therefore, it follows from Equation (4) that ${\tilde{Y}}^{*} (v) \geq {\tilde{Z}}^{*} (v) .$ □

Example 5.2. Let U = {u₁, u₂, u₃} and V = {v₁, v₂, v₃}. From the PFMP model, we consider the following form:

Assume that $\tilde{X} (u) = {(u_{1}, 0.7, 0.1), (u_{2}, 0.8, 0.5), (u_{3}, 0.8, 0.2)}$ ⟶_P $\tilde{Y} (v) = {(v_{1}, 0.8, 0.3), (v_{2}, 0.9, 0.3), (v_{3}, 0.1, 0.8)}$ And given ${\tilde{X}}^{*} (u) = {(u_{1}, 0.4, 0.6), (u_{2}, 0.8, 0.2), (u_{3}, 0.8, 0.3)}$ Calculate ${\tilde{Y}}^{*}$

If →_{P
^*} is the RPFIO induced from the Gödel t-norm ▵_G, then according to the Pythagorean fuzzy reverse triple I method, the solution Y^* can be obtained as follows:

${\tilde{Y}}^{*} (v_{1}) = ⋀_{u \in U} {{\tilde{X}}^{*} (u) ▵_{P_{G}^{*}} (\tilde{X} (u) \to_{P_{G}^{*}} \tilde{Y} (v_{1}))} = (0.4, 0.6),$ ${\tilde{Y}}^{*} (v_{2}) = ⋀_{u \in U} {{\tilde{X}}^{*} (u) ▵_{P_{G}^{*}} (\tilde{X} (u) \to_{P_{G}^{*}} \tilde{Y} (v_{2}))} = (0.4, 0.6),$ ${\tilde{Y}}^{*} (v_{3}) = ⋀_{u \in U} {{\tilde{X}}^{*} (u) ▵_{P_{G}^{*}} (\tilde{X} (u) \to_{P_{G}^{*}} \tilde{Y} (v_{3}))} = (0.1, 0.8) .$

Therefore, we obtain the following solution of the Pythagorean fuzzy reverse triple I method based on the PFMP inference model: ${\tilde{Y}}^{*} (v) = {(v_{1}, 0.4, 0.6), (v_{2}, 0.4, 0.6), (v_{3}, 0.1, 0.8)} .$

The following theorem reveals that the reverse triple I method for the PFMP inference model is reversible.

Theorem 5.3. Let ${\tilde{X}}^{*}, \tilde{X}$ be PFSs over U and ${\tilde{Y}}^{*}, \tilde{Y}$ be PFSs over V. Given that ▵_{P
^*} is a Pythagorean t-norm, →_{P
^*} is an RPFIO. If ${\tilde{X}}^{*} = \tilde{X}$ , then ${\tilde{Y}}^{*} = \tilde{Y} .$

Proof. For all u ∈ U, if ${\tilde{X}}^{*} (u) = \tilde{X} (u)$ , then it follows from Eq.(4) that ${\tilde{Y}}^{*} (v) = ⋀_{u \in U} {\tilde{X} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v))} .$ Noting that $\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v) \geq \tilde{X} (u) \to_{P^{*}} \tilde{Y} (v),$ then we have $\tilde{X} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \geq \tilde{Y} (v),$ which implies that ${\tilde{Y}}^{*} (v) \geq \tilde{Y} (v)$ .

However, assume that $\tilde{X} (u) = {\tilde{X}}^{*} (u) = (1, 0)$ for u ∈ U. From Equation (4) and Theorem 3.1, we obtain $\begin{matrix} {\tilde{Y}}^{*} (v) & \leq {\tilde{X}}^{*} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \\ = (1, 0) ▵_{P^{*}} ((1, 0) \to_{P^{*}} \tilde{Y} (v)) = \tilde{Y} (v) . \end{matrix}$ Therefore, ${\tilde{Y}}^{*} (v) = \tilde{Y} (v)$ for all v ∈ V. □ In what follows, the PFMT inference model is given as follows: $\frac{\begin{matrix} Suppose that & \tilde{X} (u) & \to_{P^{*}} & \tilde{Y} (v) \\ And given & {\tilde{Y}}^{*} (v) \end{matrix}}{\begin{matrix} Calculate & {\tilde{X}}^{*} (u) \end{matrix}}$ (5)

In addition, the PFMT inference model flowchart is shown in Fig. 2.

Fig. 2

PFMT inference model.

The principle of the reverse triple I method for the PFMT inference model is to seek the optimal ${\tilde{X}}^{*}$ over U such that the expression Equation (2) takes the largest possible value for any u ∈ U. That is, the conclusion ${\tilde{X}}^{*}$ of Equation (5) is the smallest PFS over U satisfying Equation (3) for all u ∈ U.

Theorem 5.4. Let ${\tilde{X}}^{*}, \tilde{X}$ be PFSs over U and ${\tilde{Y}}^{*}, \tilde{Y}$ be PFSs over V. Given that ▵_{P
^*} is a Pythagorean t-norm and →_{P
^*} is an RPFIO, the reverse triple I solution of Equation (5) can be expressed as follows: ${\tilde{X}}^{*} (u) = ⋁_{v \in V} {(\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} {\tilde{Y}}^{*} (v)} .$ (6)

Proof. First, we prove that ${\tilde{X}}^{*} (u)$ determined by Equation (6) satisfies Equation (3). It follows from Equation (6) that ${\tilde{X}}^{*} (u) \geq (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} {\tilde{Y}}^{*} (v) .$ By Theorem 3.5, we obtain $({\tilde{X}}^{*} (u) \to_{P^{*}} {\tilde{Y}}^{*} (v)) \to_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) = (1, 0) .$ Second, we prove that ${\tilde{X}}^{*}$ is the smallest PFS over U satisfying Equation (3). Assume that ${\tilde{H}}^{*}$ is a PFS over U satisfying Equation (3), we have $({\tilde{H}}^{*} (u) \to_{P^{*}} {\tilde{Y}}^{*} (v)) \to_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) = (1, 0) .$ By Theorem 3.5, we obtain $\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v) \leq {\tilde{H}}^{*} (u) \to_{P^{*}} {\tilde{Y}}^{*} (v) .$ Thus, ${\tilde{H}}^{*} (u) \geq (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} {\tilde{Y}}^{*} (v) .$ Therefore, it follows from Equation (6) that ${\tilde{H}}^{*} (u) \geq {\tilde{X}}^{*} (u) .$ □

Example 5.5. Let U = {u₁, u₂, u₃} and V = {v₁, v₂, v₃}. From the PFMT model, we consider the following form:

Assume that $\tilde{X} (u) = {(u_{1}, 0.6, 0.5), (u_{2}, 0.8, 0.4), (u_{3}, 0.8, 0.1)}$ ⟶_P $\tilde{Y} (v) = {(v_{1}, 0.6, 0.3), (v_{2}, 0.9, 0.2), (v_{3}, 0.8, 0.2)}$

And given ${\tilde{Y}}^{*} (v) = {(v_{1}, 0.7, 0.3), (v_{2}, 0.8, 0.3), (v_{3}, 0.6, 0.2)}$

Calculate ${\tilde{X}}^{*}$

If →_{P
^*} is the RPFIO induced from the G $\ddot{o}$ del t-norm ▵_G, then according to the Pythagorean fuzzy reverse triple I method, the solution X^* can be obtained as follows:

${\tilde{X}}^{*} (u_{1}) = ⋁_{v \in V} {(\tilde{X} (u_{1}) \to_{P_{G}^{*}} \tilde{Y} (v)) \to_{P_{G}^{*}} {\tilde{Y}}^{*} (v)}$ = (0.8, 0.2) , ${\tilde{X}}^{*} (u_{2}) = ⋁_{v \in V} {(\tilde{X} (u_{2}) \to_{P_{G}^{*}} \tilde{Y} (v)) \to_{P_{G}^{*}} {\tilde{Y}}^{*} (v)}$ = (0.95, 0.2) , ${\tilde{X}}^{*} (u_{3}) = ⋁_{v \in V} {(\tilde{X} (u_{3}) \to_{P_{G}^{*}} \tilde{Y} (v)) \to_{P_{G}^{*}} {\tilde{Y}}^{*} (v)}$ = (1, 0).

Therefore, we obtain the following solution of the Pythagorean fuzzy reverse triple I method based on the PFMT inference model: ${\tilde{X}}^{*} (u) = {(u_{1}, 0.8, 0.2), (u_{2}, 0.95, 0.2), (u_{3}, 1, 0)} .$

Theorem 5.6. Let ${\tilde{X}}^{*} \tilde{X}$ be PFSs over U and ${\tilde{Y}}^{*}, \tilde{Y}$ be PFSs over V. Given that ▵_{P
^*} is a Pythagorean t-norm, →_{P
^*} is an RPFIO, and $\tilde{α} \to_{P^{*}} (0, 1) = \tilde{α}$ . If ${\tilde{Y}}^{*} = \tilde{Y}$ , then ${\tilde{X}}^{*} = \tilde{X}$ .

Proof. For all v ∈ V, if ${\tilde{Y}}^{*} (v) = \tilde{Y} (v)$ , then it follows from Equation (4) that ${\tilde{X}}^{*} (u) = ⋁_{v \in V} {(\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} \tilde{Y} (v)} .$ Noting that $\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v) \leq \tilde{X} (u) \to_{P^{*}} \tilde{Y} (v),$ then we have $\tilde{X} (u) \geq (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} \tilde{Y} (v),$ which implies that ${\tilde{X}}^{*} (u) \leq \tilde{X} (u)$ .

In contrast, assume that $\tilde{Y} (v) = {\tilde{Y}}^{*} (v) = (0, 1)$ for v ∈ V. From Equation (6) and Theorem 3.1 we obtain $\begin{matrix} {\tilde{X}}^{*} (u) & \geq (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} \tilde{Y} (v) \\ = (\tilde{X} (u) \to_{P^{*}} (0, 1)) \to_{P^{*}} (0, 1) = \tilde{X} (u) . \end{matrix}$ Therefore ${\tilde{X}}^{*} (u) = \tilde{X} (u)$ for all u ∈ U. □

Theorem 5.6 reveals that the reverse triple I method for the PFMT inference model is also reversible.

6 The robustness and continuity properties of the reverse triple I method based on PFMP and PFMT inference models

In this section, we prove the robustness of the reverse triple I method based on PFMP and PFMT inference models.

Let ${\tilde{X}}^{*}, \tilde{X}, {\tilde{X}}^{'}$ and ${\tilde{X}}^{*^{'}}$ be PFSs over U, and let ${\tilde{Y}}^{*}, \tilde{Y}, {\tilde{Y}}^{'}$ and ${\tilde{Y}}^{*^{'}}$ be PFSs over V. Given that ▵_{P
^*} is a Pythagorean t-norm, →_{P
^*} is an RPFIO and ↔_{P
^*} is a Pythagorean fuzzy biresiduum associated with the RPFIO, δ₁, δ₂ and δ₃ are three PFNs.

Theorem 6.1. Assume that $S_{P} (\tilde{X}, {\tilde{X}}^{'}) \geq δ_{1}, S_{P} (\tilde{Y}, {\tilde{Y}}^{'})$ ≥δ₂ and $S_{P} ({\tilde{X}}^{*}, {\tilde{X}}^{*^{'}}) \geq δ_{3} .$ Given that ${\tilde{Y}}^{*}$ and ${\tilde{Y}}^{*^{'}}$ are reverse triple I solutions of PFMP $(\tilde{X}, \tilde{Y}, {\tilde{X}}^{*})$ and PFMP $({\tilde{X}}^{'}, {\tilde{Y}}^{'}, {\tilde{X}}^{*^{'}})$ given by the model (1), respectively, then $S_{P} ({\tilde{Y}}^{*}, {\tilde{Y}}^{*^{'}}) \geq δ_{1} ▵_{P^{*}} δ_{2} ▵_{P^{*}} δ_{3} .$

Proof. According to the results (4) and (5) of Theorem 4.2 and Equation (4), we have $\begin{matrix} S_{P} ( & {\tilde{Y}}^{*}, {\tilde{Y}}^{*^{'}}) \\ = ⋀_{v \in V} {{\tilde{Y}}^{*} (v) \leftrightarrow_{P} {\tilde{Y}}^{*^{'}} (v)} \\ = ⋀_{v \in V} {⋀_{u \in U} {{\tilde{X}}^{*} (u) ▵_{P} (\tilde{X} (u) \to_{P} \tilde{Y} (v))} \\ \leftrightarrow_{P} ⋀_{u \in U} {{\tilde{X}}^{*^{'}} (u) ▵_{P} ({\tilde{X}}^{'} (u) \to_{P} {\tilde{Y}}^{'} (v))}} \\ \geq ⋀_{v \in V} ⋀_{u \in U} {({\tilde{X}}^{*} (u) ▵_{P} (\tilde{X} (u) \to_{P} \tilde{Y} (v))) \\ \leftrightarrow_{P} ({\tilde{X}}^{*^{'}} (u) ▵_{P} ({\tilde{X}}^{'} (u) \to_{P} {\tilde{Y}}^{'} (v)))} \end{matrix}$

$\begin{matrix} \geq ⋀_{v \in V} ⋀_{u \in U} {({\tilde{X}}^{*} (u) \leftrightarrow_{P} {\tilde{X}}^{*^{'}} (u)) ▵_{P} ((\tilde{X} (u) \\ \to_{P} \tilde{Y} (v)) \leftrightarrow_{P} ({\tilde{X}}^{'} (u) \to_{P} {\tilde{Y}}^{'} (v)))} \\ \geq ⋀_{v \in V} ⋀_{u \in U} {({\tilde{X}}^{*} (u) \leftrightarrow_{P} {\tilde{X}}^{*^{'}} (u)) ▵_{P} (\tilde{X} (u) \\ \leftrightarrow_{P} {\tilde{X}}^{'} (u)) ▵_{P} (\tilde{Y} (v) \leftrightarrow_{P} {\tilde{Y}}^{'} (v))} \\ \geq δ_{1} ▵_{P} δ_{2} ▵_{P} δ_{3} . \end{matrix}$ □ From Theorem 6.1, we know that there exist δ₁, δ₂ and δ₃ such that $S_{P} ({\tilde{Y}}^{*}, {\tilde{Y}}^{*^{'}}) \geq δ_{1} ▵_{P^{*}} δ_{2} ▵_{P^{*}} δ_{3}$ when $S_{P} (\tilde{X}, {\tilde{X}}^{'}) \geq δ_{1}, S_{P} (\tilde{Y}, {\tilde{Y}}^{'}) \geq δ_{2}$ and $S_{P} ({\tilde{X}}^{*},$ ${\tilde{X}}^{*^{'}}) \geq δ_{3}$ , which means that the reverse triple I method for the PFMP inference model is robust.

Theorem 6.2. Assume that $S_{P} (\tilde{X}, {\tilde{X}}^{'}) \geq δ_{1}, S_{P} (\tilde{Y},$ ${\tilde{Y}}^{'}) \geq δ_{2}$ and $S_{P} ({\tilde{Y}}^{*}, {\tilde{Y}}^{*^{'}}) \geq δ_{3} .$ Given that ${\tilde{X}}^{*}$ and ${\tilde{X}}^{*^{'}}$ are reverse triple I solutions of PFMT $(\tilde{X}, \tilde{Y}, {\tilde{Y}}^{*})$ and PFMT $({\tilde{X}}^{'}, {\tilde{Y}}^{'}, {\tilde{Y}}^{*^{'}})$ given by the model (5), respectively, then $S_{P} ({\tilde{X}}^{*}, {\tilde{X}}^{*^{'}}) \geq δ_{1} ▵_{P^{*}} δ_{2} ▵_{P^{*}} δ_{3} .$

Proof. According to result (5) of Theorem 4.2 and Eq. 6, we have $\begin{matrix} S_{P} ( & {\tilde{X}}^{*}, {\tilde{X}}^{*^{'}}) \\ = ⋀_{u \in U} {{\tilde{X}}^{*} (u) \leftrightarrow_{P} {\tilde{X}}^{*^{'}} (u)} \\ = ⋀_{u \in U} {⋁_{v \in V} {(\tilde{X} (u) \to_{P} \tilde{Y} (v)) \to_{P} {\tilde{Y}}^{*} (v)} \\ \leftrightarrow_{P} ⋁_{v \in V} {({\tilde{X}}^{'} (u) \to_{P} {\tilde{Y}}^{'} (v)) \to_{P} {\tilde{Y}}^{*^{'}} (v)}} \\ \geq ⋀_{u \in U} ⋀_{v \in V} {((\tilde{X} (u) \to_{P} \tilde{Y} (v)) \to_{P} {\tilde{Y}}^{*} (v)) \\ \leftrightarrow_{P} (({\tilde{X}}^{'} (u) \to_{P} {\tilde{Y}}^{'} (v)) \to_{P} {\tilde{Y}}^{*^{'}} (v))} \\ \geq ⋀_{u \in U} ⋀_{v \in V} {((\tilde{X} (u) \to_{P} \tilde{Y} (v)) \leftrightarrow_{P} \\ ({\tilde{X}}^{'} (u) \to_{P} {\tilde{Y}}^{'} (v))) ▵_{P} ({\tilde{Y}}^{*} (v) \leftrightarrow_{P} {\tilde{Y}}^{*^{'}} (v))} \\ \geq ⋀_{u \in U} ⋀_{v \in V} {(\tilde{X} (u) \leftrightarrow_{P} {\tilde{X}}^{'} (u)) ▵_{P} (\tilde{Y} (v) \\ \leftrightarrow_{P} {\tilde{Y}}^{'} (v)) ▵_{P} ({\tilde{Y}}^{*} (v) \leftrightarrow_{P} {\tilde{Y}}^{*^{'}} (v))} \\ \geq δ_{1} ▵_{P} δ_{2} ▵_{P} δ_{3} . \end{matrix}$ □

Based on Theorem 6.2, we observe that there exist δ₁, δ₂ and δ₃ such that $S_{P} ({\tilde{X}}^{*}, {\tilde{X}}^{*^{'}}) \geq δ_{1} ▵_{P^{*}} δ_{2} ▵_{P^{*}} δ_{3}$ when $S_{P} (\tilde{X}, {\tilde{X}}^{'}) \geq δ_{1}, S_{P} (\tilde{Y}, {\tilde{Y}}^{'}) \geq δ_{2}$ and $S_{P} ({\tilde{Y}}^{*},$ ${\tilde{Y}}^{*^{'}}) \geq δ_{3}$ , which implies that the reverse triple I method for the PFMT inference model is also robust.

Remark 6.3. According to Theorems 6.1 and 6.2, we know that the reverse triple I methods based on PFMP and PFMT models have the same robustness.

In what follows, we prove the continuity of the reverse triple I method based on PFMP and PFMT inference models.

The reverse triple I method for the PFMP inference model is a mapping h : P^* (U) → P^* (V), i.e., for any input ${\tilde{X}}^{*}$ over U, there exists a corresponding output ${\tilde{Y}}^{*} = h ({\tilde{X}}^{*})$ over V, where P^* (U) and P^* (V) denote the set of all PFSs on universes U and V, respectively.

Definition 6.4. Let ${\tilde{X}}_{1}^{*}, {\tilde{X}}_{2}^{*}$ be PFSs over U. If for all $\tilde{φ} \in P^{*},$ there exists $\tilde{ω} \in P^{*}$ such that $s_{P} (h ({\tilde{X}}_{1}^{*}), h ({\tilde{X}}_{2}^{*}))$ $> \tilde{φ}$ whenever $s_{P} ({\tilde{X}}_{1}^{*}, {\tilde{X}}_{2}^{*}) > \tilde{ω},$ then method h is said to be uniformly continuous.

Theorem 6.5. Let ${\tilde{X}}_{1}^{*}, {\tilde{X}}_{2}^{*}$ and $\tilde{X}$ be PFSs over U, and let ${\tilde{Y}}_{1}^{*}, {\tilde{Y}}_{2}^{*}$ and $\tilde{Y}$ be PFSs over V. Given that ▵_{P
^*} is a Pythagorean t-norm, then the reverse triple I method for the PFMP inference model is uniformly continuous.

Proof. For all $\tilde{φ} \in P^{*},$ let $\tilde{ω} = \tilde{φ}$ . When $s_{P} ({\tilde{X}}_{1}^{*}, {\tilde{X}}_{2}^{*}) > \tilde{ω},$ according to Definition 4.4 and Theorem 4.2, we can obtain $\begin{matrix} s_{P} ( & {\tilde{Y}}_{1}^{*}, {\tilde{Y}}_{2}^{*}) \\ = ⋀_{v \in V} {{\tilde{Y}}_{1}^{*} (v) \leftrightarrow_{P^{*}} {\tilde{Y}}_{2}^{*} (v)} \\ = ⋀_{v \in V} {⋀_{u \in U} {{\tilde{X}}_{1}^{*} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v))} \\ \leftrightarrow_{P^{*}} ⋀_{u \in U} {{\tilde{X}}_{2}^{*} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v))}} \\ \geq ⋀_{v \in V} ⋀_{u \in U} {({\tilde{X}}_{1}^{*} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v))) \\ \leftrightarrow_{P^{*}} ({\tilde{X}}_{2}^{*} (u) ▵_{P^{*}} (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)))} \\ \geq ⋀_{v \in V} ⋀_{u \in U} {({\tilde{X}}_{1}^{*} (u) \leftrightarrow_{P^{*}} {\tilde{X}}_{2}^{*} (u)) ▵_{P^{*}} ((\tilde{X} (u) \\ \to_{P^{*}} \tilde{Y} (v)) \leftrightarrow (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)))} \end{matrix}$

$\begin{matrix} \geq ⋀_{v \in V} ⋀_{u \in U} {({\tilde{X}}_{1}^{*} (u) \leftrightarrow_{P^{*}} {\tilde{X}}_{2}^{*} (u))} \\ \geq ⋀_{v \in V} \tilde{ω} = \tilde{φ} . \end{matrix}$

Therefore, the reverse triple I method for the PFMP inference model is uniformly continuous. □

The reverse triple I method based on the PFMT inference model is a mapping z : P^* (V) → P^* (U), i.e., for any input ${\tilde{Y}}^{*}$ over V, there exists a corresponding output ${\tilde{X}}^{*} = z ({\tilde{Y}}^{*})$ over U.

Definition 6.6. Let ${\tilde{Y}}_{1}^{*}, {\tilde{Y}}_{2}^{*}$ be PFSs over V. If for all $\tilde{φ} \in P^{*},$ there exists $\tilde{ω} \in P^{*}$ such that $s_{P} (z ({\tilde{Y}}_{1}^{*}), z ({\tilde{Y}}_{2}^{*}))$ $> \tilde{φ}$ whenever $s_{P} ({\tilde{Y}}_{1}^{*}, {\tilde{Y}}_{2}^{*}) > \tilde{ω},$ then the method z is said to be uniformly continuous.

Theorem 6.7. Let ${\tilde{X}}_{1}^{*}, {\tilde{X}}_{2}^{*}$ and $\tilde{X}$ be PFSs over U, and let ${\tilde{Y}}_{1}^{*}, {\tilde{Y}}_{2}^{*}$ and $\tilde{Y}$ be PFSs over V. Given that ▵_{P
^*} is a Pythagorean t-norm, then the reverse triple I method for the PFMT inference model is uniformly continuous.

Proof. For all $\tilde{φ} \in P^{*},$ let $\tilde{ω} = \tilde{φ}$ . When $s_{P} ({\tilde{Y}}_{1}^{*}, {\tilde{Y}}_{2}^{*}) > \tilde{ω},$ according to Definition 4.4 and Theorem 4.2, we can obtain $\begin{matrix} s_{P} ( & {\tilde{X}}_{1}^{*}, {\tilde{X}}_{2}^{*}) \\ = ⋀_{u \in U} {{\tilde{X}}_{1}^{*} (u) \leftrightarrow_{P^{*}} {\tilde{X}}_{2}^{*} (u)} \\ = ⋀_{u \in U} {⋁_{v \in V} {(\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} {\tilde{Y}}_{1}^{*} (v)} \\ \leftrightarrow_{P^{*}} ⋁_{v \in V} {(\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} {\tilde{Y}}_{2}^{*} (v)}} \\ \geq ⋀_{u \in U} ⋀_{v \in V} {((\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} {\tilde{Y}}_{1}^{*} (v)) \\ \leftrightarrow_{P^{*}} ((\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \to_{P^{*}} {\tilde{Y}}_{2}^{*} (v))} \\ \geq ⋀_{u \in U} ⋀_{v \in V} {((\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v)) \leftrightarrow_{P^{*}} \\ (\tilde{X} (u) \to_{P^{*}} \tilde{Y} (v))) ▵_{P^{*}} ({\tilde{Y}}_{1}^{*} (v) \leftrightarrow_{P^{*}} {\tilde{Y}}_{2}^{*} (v))} \\ \geq ⋀_{u \in U} ⋀_{v \in V} {({\tilde{Y}}_{1}^{*} (v) \leftrightarrow_{P^{*}} {\tilde{Y}}_{2}^{*} (v))} \\ \geq ⋀_{u \in U} \tilde{ω} = \tilde{φ} \end{matrix}$

Therefore, the reverse triple I method for the PFMT inference model is uniformly continuous. □ Remark 6.8. According to Theorems 6.5 and 6.7, we observe that the reverse triple I methods based on PFMP and PFMT models are uniformly continuous.

7 Application example

In this section, we apply the PFMP inference model to solve the practical problem concerning how to select the best investment company.

With the improvement in living standards, a single source of income can no longer meet people’s daily expenses. Therefore, investment is becoming a channel to accumulate wealth. However, as is known to us all, any investment involves an element of risk, so it is of vital importance to choose the best way to invest. To solve this kind of problem, we take the example of choosing an investment company.

Assume that V = {v₁, v₂, v₃} is a set of three different investment companies. Let U = {u₁=investment prospects, u₂=investment income, and u₃=investment credit} be a set of parameters. Now, Investor Y wants to invest money in a company. After the primary evaluation, the investor wants to select the optimal company from the three different companies v₁, v₂ and v₃ based on the set of parameters.

To form a Pythagorean fuzzy inference rule $\tilde{X} \to_{p^{*}} \tilde{Y}$ , we can assume that the industry standard for the most basic performance of investment companies is the Pythagorean fuzzy set $\tilde{X}$ , described as

$\tilde{X} = {(u_{1}, (0.72, 0.35)), (u_{2}, (0.83, 0.17)), (u_{3}, (0.69, 0.32))}$ .

Now, assume that the evaluation requirements of Mr. Y for the investment company can be expressed as the following Pythagorean fuzzy set ${\tilde{X}}^{*}$ : ${\tilde{X}}^{*} = {(u_{1}, (1, 0)), (u_{2}, (1, 0)), (u_{3}, (1, 0))}$ .

7.1 The decision-making methods based on the Pythagorean fuzzy reverse triple I method

In the following, we adopt two methods to solve how to choose the best investment company. We first transform the Pythagorean fuzzy decision-making problem into the form of Pythagorean fuzzy reasoning rules and establish two new decision-making methods based on the reverse triple I method of the PFMP model. The decision-making steps are shown in Fig. 3.

Fig. 3

The decision-making steps based on the reverse triple I method for the PFMP model.

Table 1

The evaluation data of the three companies

$\tilde{Y}$	u ₁	u ₂	u ₃
v ₁	(0.52,0.68)	(0.72,0.32)	(0.45,0.56)
v ₂	(0.92,0.15)	(0.85,0.20)	(0.75,0.25)
v ₃	(0.67,0.45)	(0.86,0.12)	(0.68,0.30)

In Method 1, we use $\to_{P_{π}^{*}}$ as the Pythagorean fuzzy implication operator. Therefore, as per Equation (4), we can obtain an inference result based on the reverse triple I method for PFMP, which is expressed as the following Pythagorean fuzzy set ${\tilde{Y}}^{*}$ : $\begin{matrix} {\tilde{Y}}^{*} (v_{1}) \\ = & ⋀_{u \in U} {{\tilde{X}}^{*} (u) ▵_{P_{π}^{*}} (\tilde{X} (u) \to_{P_{π}^{*}} \tilde{Y} (v_{1}))} \\ = & ((1, 0) ▵_{P_{π}^{*}} ((0.72, 0.35) \to_{P_{π}^{*}} (0.52, 0.68))) \\ \land ((1, 0) ▵_{P_{π}^{*}} ((0.83, 0.17) \to_{P_{π}^{*}} (0.72, 0.32))) \\ \land ((1, 0) ▵_{P_{π}^{*}} ((0.69, 0.32) \to_{P_{π}^{*}} (0.45, 0.56))) \\ = & (0.7222, 0.6224) \land (0.8675, 0.2751) \\ \land (0.8745, 0.4851) \\ = & (0.7222, 0.6224), \end{matrix}$ $\begin{matrix} {\tilde{Y}}^{*} (v_{2}) \\ = & ⋀_{u \in U} {{\tilde{X}}^{*} (u) ▵_{P_{π}^{*}} (\tilde{X} (u) \to_{P_{π}^{*}} \tilde{Y} (v_{2}))} \\ = & ((1, 0) ▵_{P_{π}^{*}} ((0.72, 0.35) \to_{P_{π}^{*}} (0.92, 0.15))) \\ \land ((1, 0) ▵_{P_{π}^{*}} ((0.83, 0.17) \to_{P_{π}^{*}} (0.85, 0.20))) \\ \land ((1, 0) ▵_{P_{π}^{*}} ((0.69, 0.32) \to_{P_{π}^{*}} (0.75, 0.25))) \\ = & (1, 0) \land (0.9943, 0.1069) \land (1, 0) \\ = & (0.9943, 0.1069), \end{matrix}$ $\begin{matrix} {\tilde{Y}}^{*} (v_{3}) \\ = & ⋀_{u \in U} {{\tilde{X}}^{*} (u) ▵_{P_{π}^{*}} (\tilde{X} (u) \to_{P_{π}^{*}} \tilde{Y} (v_{3}))} \\ = & ((1, 0) ▵_{P_{π}^{*}} ((0.72, 0.35) \to_{P_{π}^{*}} (0.67, 0.45))) \\ \land ((1, 0) ▵_{P_{π}^{*}} ((0.83, 0.17) \to_{P_{π}^{*}} (0.86, 0.12))) \\ \land ((1, 0) ▵_{P_{π}^{*}} ((0.69, 0.32) \to_{P_{π}^{*}} (0.68, 0.30))) \\ = & (0.9306, 0.3019) \land (1, 0) \land (0.9855, 0) \\ = & (0.9306, 0.3019), \end{matrix}$

Calculate the score value for all v_i by the score function, S (v₁)=0.1342, S (v₂)=0.9772, and S (v₃)=0.7749.

Therefore, we have v₂ > v₃ > v₁.

In Method 2, by using $\to_{P_{R_{0}}^{*}}$ as the Pythagorean fuzzy implication operator, we obtain the reverse triple I solution Y^* as follows: ${\tilde{Y}}^{*} = {(v_{1}, (0.6940, 0.6800)), (v_{2}, (0.9798, 0.2000)),$ $(v_{3}, (0.6940, 0.35))} .$

Calculate the score value for all v_i by the score function, S (v₁)=0.0192, S (v₂)=0.7600, and S (v₃)=0.3591.

Therefore, we have v₂ > v₃ > v₁.

7.2 Comparative analysis with the other methods

To expand on the advantages of the developed methods, we compare them with the existing methods by solving the same example, such as the Pythagorean fuzzy weighted geometric (PFWG) operator [5], Pythagorean fuzzy weighted averaging (PFWA) operator [5], Pythagorean fuzzy Dombi weighted aggregation (PFDWA) operator [8], Pythagorean fuzzy Dombi weighted geometric (PFDWG) operator [8], q-rung orthopair fuzzy weighted geometric (q-ROFWG) operator [16] and q-rung orthopair fuzzy weighted averaging (q-ROFWA) operator [16]. By applying the abovementioned methods, we obtain the comparison results shown in Table 2 and Fig. 4.

Table 2
The comparison analysis with the different methods.( $w_{i} = \frac{1}{3}$ ,i=1,2,3.)

Methods The score function ranking

PFWG S (v₁)=-0.6404,S (v₂)=0.2237,S (v₃)= -0.1312 v₂ > v₃ > v₁

PFWA S (v₁)=0.7049,S (v₂)=0.9813,S (v₃)=0.9226 v₂ > v₃ > v₁

PFDWA(k=1) S (v₁)=0.0229,S (v₂)=0.1434,S (v₃)=0.0899 v₂ > v₃ > v₁

PFDWG(k=1) S (v₁)=-0.0059,S (v₂)=0.1188,S (v₃)=0.0653 v₂ > v₃ > v₁

q-ROFWG(q=3) S (v₁)=-0.4486,S (v₂)=0.1749,S (v₃)=-0.0570 v₂ > v₃ > v₁

q-ROFWA(q=3) S (v₁)=0.5086,S (v₂)=0.9506,S (v₃)=0.8255 v₂ > v₃ > v₁

Method 1 S (v₁)=0.1342,S (v₂)=0.9772, S (v₃)=0.7749 v₂ > v₃ > v₁

Method 2 S (v₁)=0.0192,S (v₂)=0.7600, S (v₃)=0.3591 v₂ > v₃ > v₁

Methods	The score function	ranking
PFWG	S (v₁)=-0.6404,S (v₂)=0.2237,S (v₃)= -0.1312	v₂ > v₃ > v₁
PFWA	S (v₁)=0.7049,S (v₂)=0.9813,S (v₃)=0.9226	v₂ > v₃ > v₁
PFDWA(k=1)	S (v₁)=0.0229,S (v₂)=0.1434,S (v₃)=0.0899	v₂ > v₃ > v₁
PFDWG(k=1)	S (v₁)=-0.0059,S (v₂)=0.1188,S (v₃)=0.0653	v₂ > v₃ > v₁
q-ROFWG(q=3)	S (v₁)=-0.4486,S (v₂)=0.1749,S (v₃)=-0.0570	v₂ > v₃ > v₁
q-ROFWA(q=3)	S (v₁)=0.5086,S (v₂)=0.9506,S (v₃)=0.8255	v₂ > v₃ > v₁
Method 1	S (v₁)=0.1342,S (v₂)=0.9772, S (v₃)=0.7749	v₂ > v₃ > v₁
Method 2	S (v₁)=0.0192,S (v₂)=0.7600, S (v₃)=0.3591	v₂ > v₃ > v₁

Fig. 4

Decision results by different methods.

In Table 2 and Fig. 4, we observe that the optimal ranking results by the different methods are essentially the same, even though the score functions are different in different methods. Therefore, the decision-making methods based on the Pythagorean fuzzy reverse triple I method proposed by us are reasonable and valid.

On the one hand, from Table 2 and Fig. 4, we observe that when there are three candidates, there is little difference in the score functions obtained from the existing methods, including PFWA, PFWG, PFDWA, PFDWG, q-ROFWA and q-ROFWG. Therefore, we infer that when the number of candidates increases, the effectiveness of the decision for the existing methods further weakens. That is, by applying the existing methods, we cannot describe the uncertainty and vagueness problems very well in a more complex decision-making environment. However, although the existing methods can determine the weight vector of attributes and aggregate the evaluation data of each alternative, they lack logical reasoning and cannot accurately describe the opinions of decision-makers. In general, the attribute weights must satisfy the restricted condition that the sum of the weights for each attribute must be equal to 1. The more attributes we have to consider, the smaller the role of weights to attributes will be. Compared with the existing methods, the decision-making methods we propose are based on the Pythagorean fuzzy reverse triple I method, which focuses on fuzzy logic. Additionally, in Table 2 and Fig. 4, we observe that the difference between the score values obtained by the novel methods is significant. When facing a more complex decision-making environment, the effectiveness of the decision for the novel methods significantly improves. In addition, we can give the value of ${\tilde{X}}^{*}$ according to the opinion of the decision-maker, so the best decision result can be selected based on the opinions of the decision-maker. Having said all of the above, the novel methods are more logical, better able to deal with the uncertain problems in the more complex decision-making environment and can better reflect the decision-making opinions of decision-makers.

Overall, the novel methods in this paper are more flexible and reasonable than the existing methods, including the PFWG, PFWA, PFDWA, PFDWG, q-ROFWG and q-ROFWA methods.

8 Conclusion

In this study, we attempt to establish the reverse triple I method for PFMP and PFMT inference models. We first propose the concepts of Pythagorean t-norm, Pythagorean t-conorm, RPFIO and Pythagorean fuzzy biresiduum. Furthermore, some interesting properties of the reverse triple I method of PFMP and PFMT inference models are analysed, including robustness, continuity and reversibility. Finally, the reverse triple I method of PFMP is applied to practical problems. As determined by comparing the reverse triple I method with existing methods, the advantages of the Pythagorean fuzzy reverse triple I method can be listed as follows:

The overwhelming majority of the existing decision-making methods combined with PFSs are based on aggregation operators, so these methods may lack logical reasoning, making the decision results unreasonable. However, the methods we propose focus on logical reasoning, so the novel methods in this paper are more flexible and reasonable.

The proposed methods rely on the value of ${\tilde{X}}^{*}$ provided by the decision-maker. Therefore, the proposed methods can better reflect the opinions of decision-makers than the existing decision-making methods.

The effectiveness of the decision for the novel methods can be significantly improved under a more complex decision-making environment. Therefore, the novel methods are better able to deal with uncertain problems in a more complex decision-making environment.

In the future, research on granular computing for the proposed methods is expected to be an interesting topic. In addition, triple I and reverse triple I methods based on Fermatean fuzzy sets are also a topic worthy of in-depth study.

Footnotes

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 61966032), the Natural Science Foundation of Gansu Province (No. 20JR10RA118), the Fundamental Research Funds for the Central Universities of Northwest MinZu University (No. 31920210024) and the Innovation Team for Operations Research and Cybernetics of Northwest MinZu University.

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