# A Novel SIR Approach to Closeness Coefficient-Based MAGDM Problems Using Pythagorean Fuzzy Aczel–Alsina Aggregation Operators for Investment Policy

Authors: Iftikhar Ul Haq; Tanzeela Shaheen; Wajid Ali; Tapan Senapati
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

In this study, a novel Pythagorean fuzzy aggregation operator is presented by combining the concepts of Aczel–Alsina (AA) T-norm and T-conorm operations for multiple attribute group decision-making (MAGDM) challenge for the superiority and inferiority ranking (SIR) approach. This approach has many advantages in solving real-life problems. In this study, the superiority and inferiority ranking method is illustrated and showed the effectiveness for decision makers by using multicriteria. The Aczel–Alsina aggregation operators on interval-valued IFSs, hesitant fuzzy sets (HFSs), Pythagorean fuzzy sets (PFSs), and T-spherical fuzzy sets (TSFSs) for multiple attribute decision-making (MADM) issues have been proposed in the literature. In addition, we propose a Pythagorean fuzzy Aczel–Alsina weighted average closeness coefficient (PF−AA−WA−CC) aggregation operator on the basis of the closeness coefficient for MAGDM challenges. To highlight the relevancy and authenticity of this approach and measure its validity, we conducted a comparative analysis with the method already in vogue.


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## Body

## 1. Introduction

The superiority and inferiority ranking (SIR) approach for MAGDM is essential to decision-making (DM) challenges. It provides the most desirable and attractive option from a set of alternatives. The (SIR) approach was first introduced by Xu [1] in 2001. In this approach, alternatives are ranked by superiority and inferiority flows. The advantages of the SIR method are that it associates the properties of other MCDM problems such as PROMETHEE, SAW, and TOPSIS. Tam et al. [2] used this method for selecting concrete pumps in 2004. Tam and Tong [3] utilized this method in development projects with a grey aggregation approach in 2008. Liu [4] introduced the SIR approach for IFSs in 2010. Ma et al. [5] continued the SIR approach with HFSs and interval-valued HFSs in 2014. Peng and Yang [6] proposed this method for PFS and showed few results in 2015. Rouhani [7] employed the fuzzy SIR method in the IT field. Chen [8] introduced a PROMETHEE-based outranking approach for PFNs in 2018. Tavana et al. [9] introduced the IF-grey SIR approach in 2018. Selvaraj and Samayan [10] extended the SIR method for HPFSs for MCDM challenges in 2020.


### 1.1. Research Gap and Motivation of the Study

Atanassov [11] suggested the idea of IFS, a successful extension of Zadeh’s fuzzy set theory [12] that deals with vagueness and uncertainties in the data. Every element in the IFS is represented by an ordered pair of the degree of membership and degree of nonmembership, whose sum ranges from zero to one. However, in some cases, the sum of membership and nonmembership degrees provided by the DM may be greater than one, but their square sum is less than or equal to one. Therefore, Yager [13–15] introduced PFS, which satisfies the condition that the sum of the square of its membership and nonmembership degrees is less than or equal to one. In addition, Yager [13–15] proposed different kinds of aggregation operators for DM problems, where ambiguity is found in the other basis of achievement.Triangular norms play an essential role in decision-making problems. Menger [16] was the first to introduce triangular norms. Deschrijver et al. [17] discussed T‐norm (ΤΝ) and T‐conorm (ΤCΝ) for IFSs. There are many ΤΝ and ΤCΝ that are used for the aggregation of data. Those are Lukasiewicz ΤΝ and ΤCΝ [18], product ΤΝ, and sum ΤCΝ [19], Archimedian ΤΝ and ΤCΝ, [20], and drastic ΤΝ and ΤCΝ [21]. These norms have an essential role in the formation of aggregation operators. Aczel–Alsina (AA) [22] proposed a new idea of ΤΝ and ΤCΝ known as Aczel–Alsina (AA) ΤΝ and ΤCΝ. Many researchers used AAΤΝ and ΤCΝ and concluded that these norms and conorms give better results due to their parameters. Senapati et al. [23–30] recognized the IF, IVIF, HF, PF, IVPF, and linguistic IF aggregation operators based on the AA-ΤΝ and AA-ΤCΝ by using the MADM challenges. Hussain et al. [31] used the AA aggregation operators on TSFSs. Hussain et al. [32] considered AA aggregation operators on PFSs using the MADM problems.Many scholars worked on decision-making processes and also introduced different approaches. There are several approaches in the literature mentioned above. By inspiring this trend of scholars, we also invented a new approach for decision-making under multiple-attribute alternatives. We developed Pythagorean fuzzy Aczel–Alsina weighted average closeness coefficient (PF−AA−WA−CC) aggregation operators, applied them to the Pythagorean environment, and ranked the superiority and inferiority of the alternatives. Specially, we evaluate the group decision-making process of multicriteria known as MAGDM and for ranking the best and worst results of the alternatives. We used the superiority and inferiority ranking method under the Pythagorean data. Furthermore, we expressed the validity and effectiveness of the proposed approach and we solved a mathematical example for selecting the best stock of Internet stocks for a valuable investment. Finally, we compared the developed approach with existing studies and showed its graphical representations.


### 1.2. Contributions of this Study

The rest of the study is structured as follows. In Section2, basic concepts of IFS, PFS, ΤΝ, ΤCΝ, AA−ΤΝ, and AA−ΤCΝ are briefly reviewed. Furthermore, in this section, we discuss AA operations on Pythagorean fuzzy numbers (PFNs), and some Pythagorean fuzzy Aczel–Alsina (PF-AA) averaging aggregation operators are defined. In Section 3, we introduce a novel PF−AA−WA−CC aggregation operator for solving MAGDM challenges for the SIR approach and with their properties. In Section 4, we developed the SIR approach to deal with MAGDM challenges for PF−AA−WA−CC operator. In Section 5, an approach is depicted by a numerical example. In Section 6, a comparison of suggested results is done with the results already present. Finally, we concluded this study with Section 7.


## 1.1. Research Gap and Motivation of the Study

Atanassov [11] suggested the idea of IFS, a successful extension of Zadeh’s fuzzy set theory [12] that deals with vagueness and uncertainties in the data. Every element in the IFS is represented by an ordered pair of the degree of membership and degree of nonmembership, whose sum ranges from zero to one. However, in some cases, the sum of membership and nonmembership degrees provided by the DM may be greater than one, but their square sum is less than or equal to one. Therefore, Yager [13–15] introduced PFS, which satisfies the condition that the sum of the square of its membership and nonmembership degrees is less than or equal to one. In addition, Yager [13–15] proposed different kinds of aggregation operators for DM problems, where ambiguity is found in the other basis of achievement.Triangular norms play an essential role in decision-making problems. Menger [16] was the first to introduce triangular norms. Deschrijver et al. [17] discussed T‐norm (ΤΝ) and T‐conorm (ΤCΝ) for IFSs. There are many ΤΝ and ΤCΝ that are used for the aggregation of data. Those are Lukasiewicz ΤΝ and ΤCΝ [18], product ΤΝ, and sum ΤCΝ [19], Archimedian ΤΝ and ΤCΝ, [20], and drastic ΤΝ and ΤCΝ [21]. These norms have an essential role in the formation of aggregation operators. Aczel–Alsina (AA) [22] proposed a new idea of ΤΝ and ΤCΝ known as Aczel–Alsina (AA) ΤΝ and ΤCΝ. Many researchers used AAΤΝ and ΤCΝ and concluded that these norms and conorms give better results due to their parameters. Senapati et al. [23–30] recognized the IF, IVIF, HF, PF, IVPF, and linguistic IF aggregation operators based on the AA-ΤΝ and AA-ΤCΝ by using the MADM challenges. Hussain et al. [31] used the AA aggregation operators on TSFSs. Hussain et al. [32] considered AA aggregation operators on PFSs using the MADM problems.Many scholars worked on decision-making processes and also introduced different approaches. There are several approaches in the literature mentioned above. By inspiring this trend of scholars, we also invented a new approach for decision-making under multiple-attribute alternatives. We developed Pythagorean fuzzy Aczel–Alsina weighted average closeness coefficient (PF−AA−WA−CC) aggregation operators, applied them to the Pythagorean environment, and ranked the superiority and inferiority of the alternatives. Specially, we evaluate the group decision-making process of multicriteria known as MAGDM and for ranking the best and worst results of the alternatives. We used the superiority and inferiority ranking method under the Pythagorean data. Furthermore, we expressed the validity and effectiveness of the proposed approach and we solved a mathematical example for selecting the best stock of Internet stocks for a valuable investment. Finally, we compared the developed approach with existing studies and showed its graphical representations.


## 1.2. Contributions of this Study

The rest of the study is structured as follows. In Section2, basic concepts of IFS, PFS, ΤΝ, ΤCΝ, AA−ΤΝ, and AA−ΤCΝ are briefly reviewed. Furthermore, in this section, we discuss AA operations on Pythagorean fuzzy numbers (PFNs), and some Pythagorean fuzzy Aczel–Alsina (PF-AA) averaging aggregation operators are defined. In Section 3, we introduce a novel PF−AA−WA−CC aggregation operator for solving MAGDM challenges for the SIR approach and with their properties. In Section 4, we developed the SIR approach to deal with MAGDM challenges for PF−AA−WA−CC operator. In Section 5, an approach is depicted by a numerical example. In Section 6, a comparison of suggested results is done with the results already present. Finally, we concluded this study with Section 7.


## 2. Preliminaries

In this section, we summarize the requisite knowledge associated with IFSs and PFSs with their operations and operators utilizing Aczel–Alsina T-Norm (AA-ΤΝ. We also discuss more familiarized ideas, which are helpful in sequential analysis.


### 2.1. IFS and PFS

Atanassov [11] introduced IFS theory, which is the extension of Fuzzy set theory (FS) [12].Definition 1.
(see [11]). For the universe of discourse X, an IFS “I^’” is defined as(1)I^=<x,μI^x,νI^x>:x∈X,where μI^x∈ [0, 1] denotes the degree of membership and νI^x∈ [0, 1] denotes the degree of nonmembership of x in I^, respectively, with the condition 0≤μI^x+νI^x≤1, for all x∈X.
Yager [13–15] proposed the idea of PFS as a generalization of IFS with the modified condition that the sum of squares of the degree of membership and degree of nonmembership is less than or equal to 1. In contrast to IFSs, PFSs include more space for the selection of grades.Definition 2.
(see [13–15]). For universe of discourse X, a PFS “P˙” is defined as(2)P˙=<x,μP˙x,νP˙x>:x∈X,where μP˙∈ [0,1] denotes the degree of membership and νP˙∈ [0,1] denotes the degree of nonmembership of x in P˙, respectively, with the condition 0μP˙x2+νP˙x2≤1. The degree of indeterminacy isπP˙x=1−μP˙x2−νP˙x2.
For convenience, Zhang and Xu [33] denoted Pythagorean fuzzy number (PFN) μP˙x,νP˙x by μP˙,νP˙.Definition 3.
(see [33]). The distance between two Pythagorean fuzzy numbers P˙1=μP˙1,νP˙1andP˙2=μP˙2,νP˙2 is defined as(3)dP˙1,P˙2=12μP˙12−μP˙22+νP˙12−νP˙22+πP˙12−πP˙22,Definition 4.
(see [33]). The score function of PFN P˙=μP˙,νP˙ is defined as(4)ScP˙=μP˙2−νP˙2,where ScP˙ ∈ [−1, 1].
For Pythagorean fuzzy numbersP˙1 and P˙2,(1)
IfSc (P˙1) < P˙c (P˙2), then P˙1 ≺ P˙2(2)
IfSc (P˙1) > P˙c (P˙2), then P˙1≻P˙2(3)
IfSc (P˙1) = P˙c (P˙2), then P˙1 ∼ P˙2Definition 5.
(see [33, 34]). the accuracy function of Pythagorean fuzzy number P˙=μP˙,νP˙ is defined as(5)A^cP˙=μP˙2+νP˙2,where′A^cP˙ ∈ [0, 1].(1)
IfSc (P˙1) < Pc˙ (P˙2), then P˙1 ≺ P˙2.(2)
IfSc (P˙1) > Sc (P˙2), then P˙1≻P˙2.(3)
IfSc (P˙1) = Sc (P˙2), then(a)
IfA^c (P˙1) > A^c (P˙2), then P˙1 ≻ P˙2(b)
IfA^c (P˙1) <A^c (P˙2), then P˙1≺P˙2(c)
IfA^c (P˙1) = A^c (P˙2), then P˙1 ∼ P˙2


### 2.2. Triangular Norm, Triangular Co-Norm, and Aczel–Alsina Triangular Norm

Triangular norms (ΤΝs) are the particular classes of functions that act as a tool for interpreting the conjunction of fuzzy logic and the intersection of fuzzy sets. Menger [16] was the first to introduce triangular norms for statistical metric spaces. They have many applications in decision-making and aggregation. We shall here examine some ideas necessary for this study’s development.Definition 6.
(see [35] [36, 37]). A function Ť: [0, 1] × [0, 1] ⟶ [0, 1] is a triangular norm ΤΝ if the following axioms are satisfied, ∀α˙,β,γ ∈ [0, 1]:(1)
Symmetry: Ť (α˙,β′) = Ť (β′,α˙)(2)
Associativity: Ť (α˙, Ť (β′ , γ)) α˙ = Ť (Ť (α˙,β′), γ)(3)
Monotonicity: Ť (α˙,β′) ≤ Ť (α˙,γ) if β′≤γ(4)
One Identity: Ť (1,α˙) = α˙
Examples ofΤΝs are ∀α˙,β,γ ∈ [0, 1]:(1)
Product triangular norm:Ťpro (α˙, β′) = α˙. β′.(2)
Minimum triangular norm:Ťmin (α˙, β′) = min (α˙, β′).(3)
Lukasiewicz triangular norm:Ťluk (α˙, β′) = max (α˙ + β′−1,0).(4)
Drastic triangular norm:(6)Ťdraα,˙β′=α˙,ifβ′=1,β′,ifα˙=1,0,otherwise.Definition 7.
(see [35] [36, 37]). A function Š: [0, 1] × [0, 1] ⟶ [0, 1] is triangular conorm ΤCΝ if the following axioms are satisfied, ∀α˙,β,γ ∈ [0, 1]:(1)
Symmetry: Š (α˙, β) = Š (β′,α˙)(2)
Associativity: Š (α˙, Š (β′, )) = Š (Š (α˙,β′), γ)(3)
Monotonicity: Š (α˙,β′) ≤ Š (α˙,γ) if β′≤γ(4)
Zero Identity: Š (0,α˙) = α˙
Examples ofΤCΝs are ∀α˙,β,γ ∈ [0, 1]:(1)
Probabilistic sum triangular co-norm:ŠPS (α˙,β′) = α˙+β′−α˙,β′.(2)
Maximum triangular co-norm:Šmax (α˙,β′) = max (α˙,β′).(3)
Lukasiewicz triangular co-norm:Šluk (α˙,β′) = min {α˙ + β′,1}.(4)
Drastic triangular co-norm:(7)Šdraα˙,β′=α˙,ifβ′=0,β′,ifα˙=0,1,otherwise.Definition 8.
(see [22]). Aczel–Alsina presented triangular norm in the circumstance of functional equations, known as Aczel–Alsina triangular norm AA−ΤΝ. For φ∈0,∞, it stated as(8)ŤAφα˙,β=Ťdraα˙,β′,ifφ=0,minα˙,β′,ifφ=∞,e−−logα˙φ+−logβ′φ1/φ,otherwise.
Forφ∈0,∞, AA−ΤCΝ is stated as(9)ŠAφα˙,β′=Šdraα˙,β′,ifφ=0,maxα˙,β′,ifφ=∞1−e−−log1−α˙φ+−log1−β′φ1/φ,otherwise.,Definition 9.
(see [32]). Let P˙ and Ṙ be two PFSs. Then, the generalizations of intersection and union are defined as(10)P˙∩Ť,ŠṘ=<x,ŤμP˙x,μP˙x,ŠνP˙x,νP˙x>:x∈X,P˙∪Ť,ŠṘ=<x,ŠμP˙x,μP˙x,ŤνP˙x,νP˙x>:x∈X,where Ť is for ΤΝ and Š is for ΤCΝ.


### 2.3. Aczel–Alsina Operations on Pythagorean Fuzzy Numbers

We supposeΤΝ Ť and ΤCΝ Š represent AA product ŤA and AA sum ŠǍ, respectively. Then, the generalization of intersection and union over two PFSs “P˙” and “Ṙ” turn out to be the AA product (represented by P˙ ⊗ Ṙ) and AA sum (represented by P˙ ⊕ Ṙ) over two PFSs “P˙” and “Ṙ,” respectively, [32]:(11)P˙⊗Ṙ=<x,ŤAμP˙x,μṘx,ŠAνP˙x,νṘx>:x∈X,P˙⊕Ṙ=<x,ŠAμP˙x,μṘx,ŤAνP˙x,νṘx>:x∈X.Definition 10.
(see [32]). Let P˙=μP˙,νP˙, P˙1=μP˙1,νP˙1,andP˙2=μP˙2,νP˙2 be three Pythagorean fuzzy numbers, with ℶ ≥ 1 and Ω > 0. Then, operations of PFNs with Definition 8 is defined as(12)P˙1⊕P˙2=1−e−−log1−μP˙12ℶ+−log1−μP˙22ℶ1/ℶ,e−−logνP˙1ℶ+−logνP˙2ℶ1/ℶ,P˙1⊕P˙2=e−−logμP˙1ℶ+−logμP˙2ℶ1/ℶ,1−e−−log1−νP˙12ℶ+−log1−νP˙22ℶ1/ℶ,ΩP˙=1−e−Ω−log1−μP˙2ℶ1/ℶ,e−Ω−logνP˙ℶ1/ℶ,P˙Ω=e−Ω−logμP˙ℶ1/ℶ,1−e−Ω−log1−νP˙2ℶ1/ℶ.


### 2.4. Pythagorean Fuzzy Aczel–Alsina (PF-AA) Average Aggregation Operators (AOs)

Hussain et al. [32] introduced some PF-AA average AOs under PFNs. Additionally, some related properties are also discussed.Definition 11.
LetP˙i=μP˙,νP˙i,i=1,2,…,n, be an accumulation of PFNs. Let the weight vector ϣ=ϣ1,ϣ2,…,ϣnT of P˙i,,i=1,2,…,n with ϣi> 0, ϣi∈0,1,and ∑i=1nϣi=1. Then, PF-AA-WA operator is a function, (PF-AA-WA): P˙n⟶P˙, defined as(13)PF−AA−WAP˙1,P˙2…,P˙n=⊕i=1nϣiP˙i=ϣ1P˙1⊕ϣ2P˙2⊕,…,⊕ϣnP˙n.Theorem 1.
LetP˙i=μP˙i,νP˙i,i=1,2,…,n, be an accumulation of PFNs. Let the weight vector ϣ=ϣ1,ϣ,…,ϣnT of P˙i,i=1,2,…,n with ϣi> 0, ϣi∈0,1, and ∑i=1nϣi=1. Then, aggregated values of PFNs by PF-AA-WA operator is defined as(14)PF−AA−WAP˙1,P˙2,…,P˙n=1−e−∑i=1nϣi−log1−μP˙i2ℶ1/ℶ,e−∑i=1nϣi−logνṖiℶ1/ℶ.


## 2.1. IFS and PFS

Atanassov [11] introduced IFS theory, which is the extension of Fuzzy set theory (FS) [12].Definition 1.
(see [11]). For the universe of discourse X, an IFS “I^’” is defined as(1)I^=<x,μI^x,νI^x>:x∈X,where μI^x∈ [0, 1] denotes the degree of membership and νI^x∈ [0, 1] denotes the degree of nonmembership of x in I^, respectively, with the condition 0≤μI^x+νI^x≤1, for all x∈X.
Yager [13–15] proposed the idea of PFS as a generalization of IFS with the modified condition that the sum of squares of the degree of membership and degree of nonmembership is less than or equal to 1. In contrast to IFSs, PFSs include more space for the selection of grades.Definition 2.
(see [13–15]). For universe of discourse X, a PFS “P˙” is defined as(2)P˙=<x,μP˙x,νP˙x>:x∈X,where μP˙∈ [0,1] denotes the degree of membership and νP˙∈ [0,1] denotes the degree of nonmembership of x in P˙, respectively, with the condition 0μP˙x2+νP˙x2≤1. The degree of indeterminacy isπP˙x=1−μP˙x2−νP˙x2.
For convenience, Zhang and Xu [33] denoted Pythagorean fuzzy number (PFN) μP˙x,νP˙x by μP˙,νP˙.Definition 3.
(see [33]). The distance between two Pythagorean fuzzy numbers P˙1=μP˙1,νP˙1andP˙2=μP˙2,νP˙2 is defined as(3)dP˙1,P˙2=12μP˙12−μP˙22+νP˙12−νP˙22+πP˙12−πP˙22,Definition 4.
(see [33]). The score function of PFN P˙=μP˙,νP˙ is defined as(4)ScP˙=μP˙2−νP˙2,where ScP˙ ∈ [−1, 1].
For Pythagorean fuzzy numbersP˙1 and P˙2,(1)
IfSc (P˙1) < P˙c (P˙2), then P˙1 ≺ P˙2(2)
IfSc (P˙1) > P˙c (P˙2), then P˙1≻P˙2(3)
IfSc (P˙1) = P˙c (P˙2), then P˙1 ∼ P˙2Definition 5.
(see [33, 34]). the accuracy function of Pythagorean fuzzy number P˙=μP˙,νP˙ is defined as(5)A^cP˙=μP˙2+νP˙2,where′A^cP˙ ∈ [0, 1].(1)
IfSc (P˙1) < Pc˙ (P˙2), then P˙1 ≺ P˙2.(2)
IfSc (P˙1) > Sc (P˙2), then P˙1≻P˙2.(3)
IfSc (P˙1) = Sc (P˙2), then(a)
IfA^c (P˙1) > A^c (P˙2), then P˙1 ≻ P˙2(b)
IfA^c (P˙1) <A^c (P˙2), then P˙1≺P˙2(c)
IfA^c (P˙1) = A^c (P˙2), then P˙1 ∼ P˙2


## 2.2. Triangular Norm, Triangular Co-Norm, and Aczel–Alsina Triangular Norm

Triangular norms (ΤΝs) are the particular classes of functions that act as a tool for interpreting the conjunction of fuzzy logic and the intersection of fuzzy sets. Menger [16] was the first to introduce triangular norms for statistical metric spaces. They have many applications in decision-making and aggregation. We shall here examine some ideas necessary for this study’s development.Definition 6.
(see [35] [36, 37]). A function Ť: [0, 1] × [0, 1] ⟶ [0, 1] is a triangular norm ΤΝ if the following axioms are satisfied, ∀α˙,β,γ ∈ [0, 1]:(1)
Symmetry: Ť (α˙,β′) = Ť (β′,α˙)(2)
Associativity: Ť (α˙, Ť (β′ , γ)) α˙ = Ť (Ť (α˙,β′), γ)(3)
Monotonicity: Ť (α˙,β′) ≤ Ť (α˙,γ) if β′≤γ(4)
One Identity: Ť (1,α˙) = α˙
Examples ofΤΝs are ∀α˙,β,γ ∈ [0, 1]:(1)
Product triangular norm:Ťpro (α˙, β′) = α˙. β′.(2)
Minimum triangular norm:Ťmin (α˙, β′) = min (α˙, β′).(3)
Lukasiewicz triangular norm:Ťluk (α˙, β′) = max (α˙ + β′−1,0).(4)
Drastic triangular norm:(6)Ťdraα,˙β′=α˙,ifβ′=1,β′,ifα˙=1,0,otherwise.Definition 7.
(see [35] [36, 37]). A function Š: [0, 1] × [0, 1] ⟶ [0, 1] is triangular conorm ΤCΝ if the following axioms are satisfied, ∀α˙,β,γ ∈ [0, 1]:(1)
Symmetry: Š (α˙, β) = Š (β′,α˙)(2)
Associativity: Š (α˙, Š (β′, )) = Š (Š (α˙,β′), γ)(3)
Monotonicity: Š (α˙,β′) ≤ Š (α˙,γ) if β′≤γ(4)
Zero Identity: Š (0,α˙) = α˙
Examples ofΤCΝs are ∀α˙,β,γ ∈ [0, 1]:(1)
Probabilistic sum triangular co-norm:ŠPS (α˙,β′) = α˙+β′−α˙,β′.(2)
Maximum triangular co-norm:Šmax (α˙,β′) = max (α˙,β′).(3)
Lukasiewicz triangular co-norm:Šluk (α˙,β′) = min {α˙ + β′,1}.(4)
Drastic triangular co-norm:(7)Šdraα˙,β′=α˙,ifβ′=0,β′,ifα˙=0,1,otherwise.Definition 8.
(see [22]). Aczel–Alsina presented triangular norm in the circumstance of functional equations, known as Aczel–Alsina triangular norm AA−ΤΝ. For φ∈0,∞, it stated as(8)ŤAφα˙,β=Ťdraα˙,β′,ifφ=0,minα˙,β′,ifφ=∞,e−−logα˙φ+−logβ′φ1/φ,otherwise.
Forφ∈0,∞, AA−ΤCΝ is stated as(9)ŠAφα˙,β′=Šdraα˙,β′,ifφ=0,maxα˙,β′,ifφ=∞1−e−−log1−α˙φ+−log1−β′φ1/φ,otherwise.,Definition 9.
(see [32]). Let P˙ and Ṙ be two PFSs. Then, the generalizations of intersection and union are defined as(10)P˙∩Ť,ŠṘ=<x,ŤμP˙x,μP˙x,ŠνP˙x,νP˙x>:x∈X,P˙∪Ť,ŠṘ=<x,ŠμP˙x,μP˙x,ŤνP˙x,νP˙x>:x∈X,where Ť is for ΤΝ and Š is for ΤCΝ.


## 2.3. Aczel–Alsina Operations on Pythagorean Fuzzy Numbers

We supposeΤΝ Ť and ΤCΝ Š represent AA product ŤA and AA sum ŠǍ, respectively. Then, the generalization of intersection and union over two PFSs “P˙” and “Ṙ” turn out to be the AA product (represented by P˙ ⊗ Ṙ) and AA sum (represented by P˙ ⊕ Ṙ) over two PFSs “P˙” and “Ṙ,” respectively, [32]:(11)P˙⊗Ṙ=<x,ŤAμP˙x,μṘx,ŠAνP˙x,νṘx>:x∈X,P˙⊕Ṙ=<x,ŠAμP˙x,μṘx,ŤAνP˙x,νṘx>:x∈X.Definition 10.
(see [32]). Let P˙=μP˙,νP˙, P˙1=μP˙1,νP˙1,andP˙2=μP˙2,νP˙2 be three Pythagorean fuzzy numbers, with ℶ ≥ 1 and Ω > 0. Then, operations of PFNs with Definition 8 is defined as(12)P˙1⊕P˙2=1−e−−log1−μP˙12ℶ+−log1−μP˙22ℶ1/ℶ,e−−logνP˙1ℶ+−logνP˙2ℶ1/ℶ,P˙1⊕P˙2=e−−logμP˙1ℶ+−logμP˙2ℶ1/ℶ,1−e−−log1−νP˙12ℶ+−log1−νP˙22ℶ1/ℶ,ΩP˙=1−e−Ω−log1−μP˙2ℶ1/ℶ,e−Ω−logνP˙ℶ1/ℶ,P˙Ω=e−Ω−logμP˙ℶ1/ℶ,1−e−Ω−log1−νP˙2ℶ1/ℶ.


## 2.4. Pythagorean Fuzzy Aczel–Alsina (PF-AA) Average Aggregation Operators (AOs)

Hussain et al. [32] introduced some PF-AA average AOs under PFNs. Additionally, some related properties are also discussed.Definition 11.
LetP˙i=μP˙,νP˙i,i=1,2,…,n, be an accumulation of PFNs. Let the weight vector ϣ=ϣ1,ϣ2,…,ϣnT of P˙i,,i=1,2,…,n with ϣi> 0, ϣi∈0,1,and ∑i=1nϣi=1. Then, PF-AA-WA operator is a function, (PF-AA-WA): P˙n⟶P˙, defined as(13)PF−AA−WAP˙1,P˙2…,P˙n=⊕i=1nϣiP˙i=ϣ1P˙1⊕ϣ2P˙2⊕,…,⊕ϣnP˙n.Theorem 1.
LetP˙i=μP˙i,νP˙i,i=1,2,…,n, be an accumulation of PFNs. Let the weight vector ϣ=ϣ1,ϣ,…,ϣnT of P˙i,i=1,2,…,n with ϣi> 0, ϣi∈0,1, and ∑i=1nϣi=1. Then, aggregated values of PFNs by PF-AA-WA operator is defined as(14)PF−AA−WAP˙1,P˙2,…,P˙n=1−e−∑i=1nϣi−log1−μP˙i2ℶ1/ℶ,e−∑i=1nϣi−logνṖiℶ1/ℶ.


## 3. ProposedPF−AA−WA−CC Operator for Solving MAGDM Problems

Now, we introducePF−AA−WA−CC operators with their properties under PFSs to deal with MAGDM challenges for decision makers.Definition 12.
supposeP˙i=μP˙i,νP˙i,i=1,2,…,n is an accumulation of PFNs. Let ξkk=1,2,…,l be the individual measure degree via weight ϣ=ϣ1,ϣ2,…,ϣl of experts. Let Sk=S1,S2,…,SlT be the normalized vector of individual measure degrees. Then, PF−AA−WA−CC operator is a function: (PF−AA−WA−CC): P˙n⟶P˙ defined as(15)PF−AA−WA−CCP˙1,P˙2,…,P˙n=⊕i,k=1nSkP˙i=S1P˙1⊕S2P˙2⊕,…,⊕SnP˙n.Theorem 2.
supposeP˙i=μP˙i,νP˙i,i=1,2,…,n, is an accumulation of PFNs. Let ξkk=1,2,…,l be the individual measure degree of experts via weight ϣ=ϣ1,ϣ2,…,ϣl. Let Sk=S1,S2,…,SlT be the normalized vector of individual measure degrees. Then, aggregated values of PFNs by PF−AA−WA−CC operator are defined as(16)PF−AA−WA−CCP˙i1,P˙i2,…,P˙il=1−e−∑k=1lSk−log1−μik2ℶ1/ℶ,e−∑k=1lSk−logνikℶ1/ℶ.Proof.
By induction, forn=2,(17)PF−AA−WA−CCP˙1,P˙2=⊕i,k=12SkP˙i=S1P˙1⊕S2P˙2=1−e−S1−log1−μṖ12ℶ1/ℶ,e−S1−logνṖ1ℶ1/ℶ⊕1−e−S2−log1−μṖ22ℶ1/ℶ,e−S2−logνṖ2ℶ1/ℶ=1−e−S1−log1−μṖ12ℶ+S2−log1−μṖ22ℶ1/ℶ,e−S1−logνṖ1ℶ+S2−logνṖ2ℶ1/ℶ=1−e−∑i,k=12Sk−log1−μṖi2ℶ1/ℶ,e−∑i,k=12Sk−logνṖiℶ1/ℶ,which is true for n=2.
We suppose it is true forn, that is,(18)PF−AA−WA−CCP˙1,P˙2,…,P˙n=⊕i,k=1nSkP˙i=S1P˙1⊕S2P˙2⊕,…,⊕SnP˙n.
We have to show that it is true forn+1; that is,(19)PF−AA−WA−CCP˙1,P˙2,…,P˙n+1=⊕i,k=1n+1SkP˙i=S1P˙1⊕S2P˙2⊕,…,⊕ζnS1P˙n+1=⊕i,k=1nζkP˙i⊕ζn+1P˙n+1=1−e−∑i,k=1nSk−log1−μṖi2ℶ1/ℶ,e−∑i,k=1nSk−logνṖiℶ1/ℶ⊕1−e−ζn+1−log1−μP˙˙n+12ℶ1/ℶ,e−ζn+1−logνP˙˙n+1ℶ1/ℶ=1−e−∑i,k=1n+1ζk−log1−μP˙i2ℶ1/ℶ,e−∑i,k=1n+1ζk−logνP˙iℶ1/ℶ,which is true for n+1. Thus, it is true for all values of n.Theorem 3.
supposeP˙i=μP˙i,νP˙ii=1,2,…,n is an accumulation of PFNs. Let the weight vector be ϣ=ϣ1,V2,…,VnT of P˙ii=1,2,…,n with ϣi> 0, ϣi∈0,1, and ∑i=1nϣi=1. Then, the following properties are satisfied.(P1) Idempotency: for equal PFNsP˙i=μP˙i,νP˙ii=1,2,…,n, that is, P˙i=P˙∀i, we have(20)PF−AA−WA−CCP˙1,P˙2,…,P˙n=P˙.(P2) Boundedness: letP˙i=μP˙i,νP˙ii=1,2,…,n, be an accumulation of PFNs, with P˙− =  minP˙1,P˙2,…,P˙n and P˙+ =  maxP˙1,P˙2,…,P˙n. Then,(21)P˙−≤PF−AA−WA−CCP˙1,P˙2,…,P˙n≤P˙+.(P3) Monotonicity: we supposeP˙i=μP˙i,νP˙i and P˙i=μP˙i,vP˙ii=1,2,…,n are any two accumulations of PFNs, and if P˙i≤P˙i and ∀i=1,2,…,n, with P˙− =  minP˙1,P˙2,…,P˙n and P˙+ =  maxP˙1,P˙2,…,P˙n, then(22)PF−AA−WA−CCP˙1,P˙2,…,P˙n≤PF−AA−WA−CCP˙1,P˙2,…,P˙n.


## 4. Application of the SIR Approach for PFSs with Novel Aggregation Operators in MAGDM

In this section, we use thePF−AA−WA−CC operator to consider the MAGDM issues for PF data. Let x = {x1,x2,…xm} be a family of alternatives and c = {c1,c2,…cn} the collection of attributes. Let e = {e1,e2,…el} be a set of experts with ϣ=ϣ1,ϣ2,…,ϣlT as weight vector. Let P˙k=P˙ijkm×n,i=1,2,…,m, (j=1,2,…,n, and (k=1,2,…,l, be the PF decision matrix (DM) given by the decision maker. Pijk represents the attribute value such that the alternative xi satisfies the attribute cj suggested by expert ek. Ѡ=ϣjkl×n is the attribute weight DM, where ϣjk denotes the weightage of attribute cj suggested by expert ek.Step 1.
find individual measure degreeξkk=1,2,…,l through weights of experts. The relative closeness coefficient is obtained as(23)ξk=dϣk,ϣ−dϣk,ϣ−+dϣk,ϣ+.
ϣ−=minμP˙i,maxνP˙i and P˙+=maxμP˙i,minνP˙i with 0≤ϣk≤1.
Ifξk⟶1, then ϣk⟶ϣ+; if ξk⟶0,then ϣk⟶ϣ−.Step 2.
normalizeξkk=1,2,…,l for sum as a unit:(24)Sk=ξk∑k=1lξk,which is the normalized vector Sk=S1,S2,…,SlT as individual measure degrees.Step 3.
usePF−AA−WA−CC operator to aggregate group viewpoints.(a)
Individual attributes’ weight integration:(25)ϣ¯j=PF−AA−WA−CCϣj1,ϣj2,…,ϣjl=1−e−∑k=1lSk−log1−μjk2ℶ1/ℶ,e−∑k=1lSk−logνjkℶ1/ℶ,(b)
Individual decision matrix integration:(26)P˙¯ij=PF−AA−WA−CCP˙ij1,P˙ij2,…,P˙ijl=1−e−∑k=1lSk−log1−μijk2ℶ1/ℶ,e−∑k=1lSk−logνijkℶ1/ℶ.
The attribute weight vectorϣ¯=ϣ¯1,ϣ¯2,…,V¯n and group integrated decision matrix P˙¯ijm×n are obtained.Step 4.
determine PF superiority matrix (SM) and inferiority matrix (IM).(a)
Performance function:(27)Sij=SP˙¯ij=dP˙¯ij,P˙¯−dP˙¯ij,Ṗ¯−+dP˙¯ij,P˙¯+,P˙¯−=cj,minμP˙ij,maxνP˙ij,P˙¯+=cj,maxμP˙,minνP˙ij,with0≤Sij≤1; if Sij⟶1, then P˙¯ij⟶P˙¯+; if S⟶0, then P˙¯ij⟶P˙¯−.(b)
Preference intensityPIj (xi,xt): w define, PIj (xi,xt) t,i=1,2,…,m,t≠i;j=1,2,…,n as preference intensity of alternative xi with alternative xt to the parallel attribute cj; that is,(28)PIjxi,xt=ΨjSij−Stj=Ψjd,hereΨjd is generalized threshold functions or may defined themselves by experts.(c)
determine superiority index (S-I):S = Sijm×n:(29)Sij=∑t=1nPIjxi,xt=∑t=1nΨjSij−S. (d) determine inferiority index (I-I): I = Iijm×n(30)Iij=∑t=1nPIjxt,xi=∑t=1nΨjStj−Sij.Step 5
determine superiority flow:(31)Ψ>xi=PF−AA−WA−CCϣ¯1,ϣ¯2,…,ϣ¯n=1−e−∑j=1nSij∑k=1lSk−log1−μjk2ℶ1/ℶ,e−∑j=1nSij∑k=1lSk−logνjkℶ1/ℶ.Inferiority flow:(32)Ψ<xi=PF−AA−WA−CCϣ¯1,ϣ¯2,…,ϣ¯n=1−e−∑j=1nIij∑k=1lSk−log1−μjk2ℶ1/ℶ,e−∑j=1nIij∑k=1lSk−logνjkℶ1/ℶ.
We calculate the score function ofΨ>xi and of Ψ<xi. This gives S-flow and I-flow of alternatives xi asxiΨ>xi,Ψ<xi. For greater Ψ>xi and smaller Ψ<xi, alternative xi is better.Step 6.
Superiority/ inferiority ranking.(a)
Superiority ranking rules.(1)
IfΨ>xi>Ψ>xt and Ψ<xi<Ψ<xt, then xi≻xt(2)
IfΨ>xi>Ψ>xt and Ψ<xi=Ψ<xt, then xi≻xt(3)
IfΨ>xi=Ψ>xt and Ψ<xi<Ψ<xt, then xi≻xt(b)
Inferiority ranking rules:(1)
IfΨ>xi<Ψ>xt and Ψ<xi>Ψ<xt, then xi≺xt(2)
IfΨ>xi<Ψ>xt and Ψ<xi=Ψ<xt, then xi≺xt(3)
IfΨ>xi=Ψ>xt and Ψ<xi>Ψ<xt, then xi≺xtStep 7.
combine the rules of superiority/ inferiority ranking for the best alternative (xi).


## 5. Numerical Example

An investment company is interested in investing in Internet stocks. So, the company employs three brands of experts: market maker (e1, dealer e2, and finder e3. They select four stocks: x1 is GAD, x2 is FUT, x3 is NET, and x4 is PUM, for three attributes, c1 (market trend), c2 (policy direction), and c3 (yearly performance). The experts ek evaluate stocks xi relating to the attributes cj and form the following three PF decision matrices Pk=Pijk4×3 in Table 1, weights of experts in Table 2, and attribute weights in Table 3.Table 1
Pythagorean fuzzy (PF) decision matrices.
c1c2c3e1x1(0.71, 0.26)(0.79, 0.35)(0.44, 0.53)x2(0.60, 0.37)(0.81, 0.38)(0.63, 0.53)x3(0.54, 0.36)(0.73, 0.11)(0.89, 0.13)x4(0.47, 0.23)(0.57, 0.25)(0.51, 0.43)e2x1(0.72, 0.43)(0.55, 0.66)(0.69, 0.22)x2(0.67, 0.29)(0.85, 0.35)(0.91, 0.23)x3(0.58, 0.39)(0.60, 0.53)(0.79, 0.45)x4(0.80, 0.47)(0.83, 0.37)(0.48, 0.49)e3x1(0.59, 0.49)(0.52, 0.63)(0.73, 0.36)x2(0.65, 0.35)(0.64, 0.62)(0.79, 0.58)x3(0.73, 0.14)(0.86, 0.51)(0.67, 0.29)x4(0.94, 0.16)(0.76, 0.45)(0.87, 0.27)Table 2
Weights of experts.
ExpertsPFEse1(0.78, 0.20)e2(0.87, 0.17)e3(0.92, 0.25)Table 3
Weights of attributes.
c1c2c3e1(0.83, 0.23)(0.79, 0.36)(0.64, 0.54)e2(0.89, 0.37)(0.87, 0.25)(0.58, 0.66)e3(0.91, 0.11)(0.78, 0.53)(0.87, 0.17)For choosing the most attractive alternate(s), we have the following steps:Step 1: we find the individual measure degreeξkk=1,2,3 through weights of experts by using equation (23):(33)ξk=0.0864,0.6239,0.8763T.Step 2: we determine the normalized vectorSkk=1,2,3, by using equation (24):(34)Sk=0.0545,0.3932,0.5523T.Step 3.(a)
determine the attributes weights integrationϣ¯jj=1,2,3 by using equation (25):(35)ϣ¯1=0.7176,0.4447,ϣ¯2=0.6320,0.6266,ϣ¯3=0.6310,0.5304.(b)
determine the aggregated decision matrix integration by using (26):(36)P¯ij4×3=0.4788,0.69760.4182,0.79570.5136,0.58030.4661,0.61250.5916,0.69840.6727,0.61870.4955,0.48610.6206,0.66220.5652,0.59490.7376,0.50960.5922,0.66680.6296,0.6141.Step 4:(a)
determine the performance functionSij by using equation (27):(37)Sij4×3=0.03670.00000.19630.25440.78040.80400.45621.00000.29500.93320.84530.7023.(b)
determine preference intensityPIj (xi,xt) by using (28).Setting attribute threshold function,(38)Ψkd=0.01,ifd>0,0.00,ifd>0.(c)
determine superiority matrix (S. Matrix) by using equation (29):(39)S=0.000.000.000.010.010.030.020.030.010.030.020.02.(d)
determine inferiority matrix (I. Matrix) by using equation (30):Step 5: we determine the S-Flow and I-Flow by equations (31) and (32) as shown in Table 4, and they are illustrated in Figure 1.Step 6.(a)
combine superiority ranking rules with Table4 that gives(40)ScΨ>x4>ScΨ>x3>ScΨ>x2>ScΨ>x1x4≻x3≻x2≻x1.(b)
combine inferiority ranking rules with Table4 that gives(41)ŞcΨ<x4<ŞcΨ<x3<ŞcΨ<x2<ŞcΨ<x1x4≻x3≻x2≻x1.Step 7: according to the results of SIR rules, the best alternative isx4 (PUM) for Internet stock investment.Figure 1
Superiority and inferiority flow.Table 4
The PF-SIR flows.
Internet stocksΨ>xiScΨ>xiΨ<xiScΨ<xix1 (GAD)(0.0000, 1.0000)− 1.0000(0.2402, 0.9531)− 0.8507x2 (FUT)(0.1786, 0.9737)− 0.9162(0.1632, 0.9788)− 0.9314x3 (NET)(0.1971, 0.9675)− 0.8972(0.1400, 0.9851)− 0.9508x4 (PUM)(0.2138, 0.9642)− 0.8840(0.1120, 0.9885)− 0.9646


## 6. Comparative Analysis

In this section, we compare the proposed results with the results already in vogue in [6] which are represented in Table 5, and they are illustrated in Figure 2. It is observed from our numerical example that aggregation operators used in [6] and aggregation operators used in the proposed method give same results. However, the accuracy and authenticity of this approach lie in the fact that AA−AOs are established on AA−ΤΝs. Hence, these operators are responsible for accurate outcomes. Thus, we found another easier, authentic, and valid method for choosing the best and most attractive alternative for MAGDM for SIR approach.Table 5
Comparison between proposed and existing work.
Proposed workExisting work [6]Internet stocksScΨ>xiScΨ<xiScΨ>xiScΨ<xix1 (GAD)− 1.0000− 0.85070.00004060.0036886x2 (FUT)− 0.9162− 0.93140.00123940.0010162x3 (NET)− 0.8972− 0.95080.00200510.0005013x4 (PUM)− 0.8840− 0.96460.00228660.0003655Figure 2
Comparison between proposed and existing work.


## 7. Conclusions

In this research article, we worked on an aggregation operator, namely, the PF-AA-WA −CC operator for PFNs with SIR techniques. Meanwhile, depending on the PF aggregation operator, we examine some properties such as idempotency, boundedness, and monotonicity. This structure of AA AO based on t-norm and t-conorm with SIR techniques is more generalized that effectively integrates the complicated problems. Mainly, we used the Pythagorean information and developed the MAGDM approach for the easiness of decision makers. A MAGDM problem for the selection of Internet stocks has been solved to demonstrate the authenticity of the proposed work and measure its validity by comparing its results with the method already in vogue. It is observed that our developed method is also feasible for intuitionistic fuzzy data and fuzzy data which are a very fruitful contribution to the literature.We will extend our developments on q-rung orthogonal pair data and cubic Pythagorean fuzzy environment in the future. Furthermore, we can spread them to other aggregation operators, such as power mean AOs, Bonferroni mean AOs, Hamacher AOs, Hamy mean AOs, and Dombi AOs with SIR techniques. In the future, there is a lot of potential in machine learning, information retrieval, data mining, artificial intelligence, social network analysis, and many other areas in uncertain scenarios [38–48]. These are all fascinating topics for future studies.



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*Source: 5172679-2022-12-13.xml*