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

. 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. Tis approach has many advantages in solving real-life problems. In this study, the superiority and inferiority ranking method is illustrated and showed the efectiveness for decision makers by using multicriteria. Te 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 coefcient (PF − AA − WA − CC ) aggregation operator on the basis of the closeness coefcient 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.


Introduction
Te 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. Te (SIR) approach was frst introduced by Xu [1] in 2001. In this approach, alternatives are ranked by superiority and inferiority fows. Te 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 feld. 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.

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. Terefore, Yager [13][14][15] introduced PFS, which satisfes 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][14][15] proposed diferent kinds of aggregation operators for DM problems, where ambiguity is found in the other basis of achievement.
Many scholars worked on decision-making processes and also introduced diferent approaches. Tere 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 coefcient (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 decisionmaking 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 efectiveness 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.

Contributions of this Study.
Te rest of the study is structured as follows. In Section 2, basic concepts of IFS, PFS, ΤΝ, ΤCΝ, AA − ΤΝ, and AA − ΤCΝ are briefy 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 defned. 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.

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.
Defnition 1 (see [11]). For the universe of discourse X, an IFS "I'" is defned as where μ I (x)∈ [0, 1] denotes the degree of membership and ] I (x) ∈ [0, 1] denotes the degree of nonmembership of x in I, respectively, with the condition 0 ≤ μ Yager [13][14][15] proposed the idea of PFS as a generalization of IFS with the modifed 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.

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 frst to introduce triangular norms for statistical metric spaces. Tey have many applications in decision-making and aggregation. We shall here examine some ideas necessary for this study's development.

Proposed PF
Proof. By induction, for n � 2, which is true for n � 2. We suppose it is true for n, that is, Discrete Dynamics in Nature and Society 5 We have to show that it is true for n + 1; that is, which is true for n + 1. Tus, it is true for all values of n. □ Theorem 3. suppose _ P i � (μ _ P i , ] _ P i )(i � 1, 2, . . . , n) is an accumulation of PFNs. Let the weight vector be ϣ � (ϣ 1 , V 2 , . . . , V n ) T of _ P i (i � 1, 2, . . . , n) with ϣ i > 0, ϣ i ∈ [0, 1], and n i�1 ϣ i � 1. Ten, the following properties are satisfed.

Application of the SIR Approach for PFSs with Novel Aggregation Operators in MAGDM
In this section, we use the PF − AA − WA − CC operator to consider the MAGDM issues for PF data. Let  1, 2, . . . , l), be the PF decision matrix (DM) given by the decision maker. P ij (k) represents the attribute value such that the alternative x i satisfes the attribute c j suggested by expert e k . Ѡ �(ϣ j (k) ) l×n is the attribute weight DM, where ϣ j (k) denotes the weightage of attribute c j suggested by expert e k .
Step 3. use PF − AA − WA − CC operator to aggregate group viewpoints.
(a) Performance function: � 1, 2, . . . , m, t ≠ i; j � 1, 2, . . . , n) as preference intensity of alternative x i with alternative x t to the parallel attribute c j ; that is, here Ψ j (d) is generalized threshold functions or may defned themselves by experts. (c) determine superiority index (S-I): S � (S ij ) m×n : (d) determine inferiority index (I-I): I � (I ij ) m×n Step 5 determine superiority fow: Discrete Dynamics in Nature and Society Inferiority fow: We calculate the score function of Ψ > (x i ) and of Ψ < (x i ). Tis gives S-fow and I-fow of alternatives x i asx i (Ψ > (x i ), Ψ < (x i )). For greater Ψ > (x i ) and smaller Ψ < (x i ), alternative x i is better.
, then x i ≻x t (b) Inferiority ranking rules: Step 7. combine the rules of superiority/ inferiority ranking for the best alternative (x i ).

Numerical Example
An investment company is interested in investing in Internet stocks. So, the company employs three brands of experts: market maker (e 1 ), dealer (e 2 ), and fnder (e 3 ).
Tey select four stocks: x 1 is GAD, x 2 is FUT, x 3 is NET, and x 4 is PUM, for three attributes, c 1 (market trend), c 2 (policy direction), and c 3 (yearly performance). Te experts e k evaluate stocks x i relating to the attributes c j and form the following three PF decision matrices P(k) � (P ij (k) ) 4×3 in Table 1, weights of experts in Table 2, and attribute weights in Table 3.
(a) determine the attributes weights integration ϣ j (j � 1, 2, 3) by using equation (25): Step 4: (a) determine the performance function S ij by using equation (27): (b) determine preference intensity PI j (x i , x t ) by using (28). Setting attribute threshold function, (c) determine superiority matrix (S. Matrix) by using equation ( (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.
(a) combine superiority ranking rules with Table 4 that gives   Discrete Dynamics in Nature and Society (b) combine inferiority ranking rules with Table 4 that gives Step 7: according to the results of SIR rules, the best alternative is x 4 (PUM) for Internet stock investment.

. 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. Tus, we found another easier, authentic, and valid method for choosing the best and most attractive alternative for MAGDM for SIR approach.

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. Tis structure of AA AO based on t-norm and t-conorm with SIR techniques is more generalized that efectively 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, artifcial intelligence, social network analysis, and many other areas in uncertain scenarios [38][39][40][41][42][43][44][45][46][47][48]. Tese are all fascinating topics for future studies.

Data Availability
No data were used to support this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.