Einstein Aggregation Operators for Pythagorean Fuzzy Soft Sets with Their Application in Multiattribute Group Decision- Making

Department of Mathematics, University of Management and Technology, Sialkot Campus Lahore, Pakistan Department of Mathematics, University of Management and Technology, Lahore, Pakistan Department of Mathematics, Cankaya University, Etimesgut, Ankara, Turkey Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan Department of Mathematics, King Abdul Aziz University, Jeddah, Saudi Arabia Department of Mathematics and Statistics, College of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Phayao 56000, Thailand


Introduction
MAGDM is the most applicable method for finding the adequate alternative from all conceivable alternatives. Conventionally, it is anticipated that all data retrieving alternatives according to attributes and their conforming weights are stated in crisp numbers. On the other hand, maximum judgments are taken in situations where the objectives are usually indefinite or ambiguous in real-life circumstances. To over-come such ambiguities and anxieties, Zadeh offered the concept of the fuzzy set (FS) [1], a prevailing tool to handle the obscurities and uncertainties in DM considering the membership values of the alternatives. Experts mostly consider a membership and a nonmembership value in the DM process that FS cannot handle. Atanassov [2] introduced the generalization of the FS, the idea of the intuitionistic fuzzy set (IFS) to overcome the inadequacy mentioned above. In 2011, Wang and Liu [3] presented numerous operations on IFS, such as Einstein product and Einstein sum, and constructed two AOs. They also discussed some essential properties of these operators and utilized their proposed operator to resolve multiattribute decision-making (MADM) for the IFS information. Atanassov [4] presented a generalized form of IFS in the light of ordinary interval values, called interval-valued IFS. Garg and Kaur [5] prolonged the impression of IFS and offered a novel idea of the cubic intuitionistic fuzzy set.
The models mentioned above have been well recognized by the specialists. Still, the existing IFS cannot handle the inappropriate and vague data. For example, if decision-makers choose membership (MD) and nonmembership (NMD) 0.9 and 0.6, respectively, then 0:9 + 0:6 ≥ 1. The IFS theory mentioned above cannot be applied to this data. To resolve the limitation described above, Yager [6,7] presented the notion of the PFS by improving the basic circumstance a + b ≤ 1 to a 2 + b 2 ≤ 1 and developed some results associated with the score function and accuracy function. Rahman et al. [8] presented the Einstein geometric aggregation operator and introduced a MAGDM methodology utilizing the proposed operator. Zhang and Xu [9] developed some basic operational laws and prolonged the technique for preference by similarity to the ideal solution (TOPSIS) method to resolve multicriteria decision-making (MCDM) complications under a PFS setting. Wei and Lu [10] proposed the power AOs for PFS and discussed their fundamental properties. They also offered a DM technique to resolve MADM complications using their presented operators. Wang and Li [11] protracted Bonferroni's mean AOs for PFS considering the interaction. IIbahar et al. [12] introduced the Pythagorean fuzzy proportional risk assessment technique to assess professional health risk. Zhang [13] proposed a novel DM approach based on similarity measures to resolve PFS information's multicriteria group decision-making (MCGDM) problems. Peng and Yang [14] introduced the division and subtraction operations for Pythagorean fuzzy numbers (PFNs), proved their basic properties, and presented a superiority and inferiority ranking approach under the PFS to overcome the MAGDM complications. Garg [15,16] introduced operational laws based on Einstein norms for PFNs, proposed weighted AOs, and ordered weighted AOs for PFS. Garg [17] introduced logarithmic operational laws for the PFS and constructed various weighted AOs based on the presented logarithm operational laws.
Gao et al. [18] settled interaction AOs under a PFS environment and gave the MADM approach to solving real-life problems. Wang et al. [19] protracted the interactive Hamacher AOs for the PFS and settled a DM method. Wang and Li [20] utilized the interval-valued PFS, presented some novel PFS operators, and offered a DM approach to resolve the MCGDM complications. Moreover, to deal with the MCDM complexities, Gao et al. [18] constructed hybrid AOs for PFS and presented a DM methodology utilizing these operators. Peng and Yuan [21] extended the AOs for PFS and introduced the generalized AOs for PFS with their desirable properties. They also constructed a MADM approach established on their advanced operators. Zulqarnain et al. [22] developed novel algorithms for multipolar neutrosophic soft sets. They utilized their established algorithms in medical diagnoses. Zulqarnain et al. [23] protracted the generalized TOPSIS method under a neutrosophic setting to solve MCDM problems. Arora and Garg [24] presented basic operational laws for linguistic IFS and suggested some AOs under the considered scenario. To examine the ranking of normal IFS and interval valued IFS, Garg [25] gave novel algorithms for solving the MADM problems. Ma and Xu [26] modified the existing score function and accuracy function for PFNs and defined novel AOs for PFNs. All of the previously mentioned methods have excessive applications in several fields, but due to their ineffectiveness, these methods have many limitations with parameterization. Molodtsov [27] presented the basic notion of soft sets (SS) and deliberated some basic operations with their belongings. Maji et al. [28] presented the idea of SS and demarcated several basic operations. In [29], the authors developed a DM approach for SS. Maji et al. [30] demonstrated the theory of IFSS and offered some basic operations with their essential properties. Zulqarnain et al. [31] presented the TOPSIS method using a correlation coefficient for interval-valued IFSS. Zulqarnain et al. [32] utilized the TOPSIS method for the prediction of diabetes patients.
Nowadays, the conception and application consequences of soft sets and the earlier-mentioned several research developments are evolving speedily. Peng et al. [33] established the concept of PFSS by merging two current models, PFS and SS. They also debated some fundamental operations with their essential possessions. Athira et al. [34] established entropy measures for the PFSS. They also offered Euclidean distance and hamming distance for the PFSS and utilized their methods for DM [35]. Naeem et al. [36] developed the TOP-SIS and VIKOR methods for PFSNs and presented an approach for the stock exchange investment problem. Zulqarnain et al. [37,38] introduced the AOs and interaction AOs under the PFSS environs. They also constructed the DM methods based on their operators and utilized them in green supplier chain management. Siddique et al. [39] settled a DM technique based on a score matrix for PFSS. Zulqarnain et al. [40] presented the correlation coefficient (CC) for PFSS and proposed the TOPSIS approach based on developed CC to resolve MADM problems. Zulqarnain et al. [41,42] introduced the Einstein-ordered weighted AOs for PFSS and settled the DM approaches using their established operators.
The PFSS can potentially disclose unconvinced and obscure information in practical applications. This article establishes a new strategy for coping with DM issues under the PFSS environs. PFSS is an innovative hybrid configuration of PFS. Enriched organization approaches captivate investigators to interpret confusing and deficient data. PFSS performs a vital part in DM by congregation various sources into a single value in terms of findings. According to the best-known familiarity, the advent of hybridization of PFS and SS is not separate from PFS's perspective. Thus, to motivate modern exploration on PFSS, we will state the AOs based on rough data, with the subsequent elementary objectives of the study: (1) PFSS is proficient in conducting complex problems competently, considering the properties in the DM progression. With this advantage in mind, we put up the Einstein AOs for PFSS 2 Journal of Function Spaces (2) In some cases, it has been noted that the prevailing AOs do not seem keen to flag precise DM techniques. To handle these specific troubles, these AOs must be amended. We presented an advanced algorithm for the Pythagorean fuzzy soft numbers (PFSNs) founded on the Einstein norm (3) The PFSEWA and PFSEWG operators are built using Einstein operational laws with some basic properties (4) An innovative MAGDM method is established based on the anticipated PFSEWA and PFSEWG operators to tackle the DM problem (5) A comparative analysis of the settled MAGDM technique and current approaches has been offered to deliberate realism and dominance The configuration of the subsequent study is prearranged as follows: in Section 2, we recalled some elementary notions such as FS, IFS, PFS, SS, FSS, IFSS, PFSS, and Einstein norms. Section 3 defined some basic operational laws for PFSNs based on Einstein norms, developed PFSEWA and PFSEWG operators, and discussed their essential properties. Section 4 settled the MAGDM methodology built on planned operators and gave a numerical illustration for finding the most delicate business to invest money in. In Section 5, a comparison with some prevailing methods has been provided.

Preliminaries
This section comprises some elementary definitions, such as SS, IFS, PFS, FSS, IFSS, and PFSS, which will deliver the basis for the structure of the following manuscript. Definition 1. (see [27]). Let X and ℕ be the universe of discourse and set of attributes, respectively. Let P ðXÞ be the power set of X and A ⊆ ℕ. A pair (Ω, A) is called a SS over X, and its mapping is expressed as follows: Also, it can be defined as follows: Definition 2. (see [6]). Let X be a collection of objects and then a PFS, A over X is defined as where a A ðxÞ, b A ðxÞ: X ⟶ ½0, 1 represents the MD and NMD such as 0 ≤ a A ðxÞ 2 + b A ðxÞ 2 ≤ 1 and I = 1 − aðxÞ 2 − b A ðxÞ 2 expressed the indeterminacy.
Definition 3. (see [30]). Let X and ℕ be the universe of discourse and set of attributes, respectively; then, a pair ðΩ, ℕÞ is called an IFSS over X.
Let Ω : ℕ ⟶ IK X be a mapping and IK X be a collection of intuitionistic fuzzy subsets. Also, it is defined as follows: where a A ðtÞ, b A ðtÞ: A ⟶ ½0, 1 are MD and NMD and 0 ≤ a A ðtÞ + b A ðtÞ ≤ 1.
The above IFSS cannot contract with the state when the combination of MD and NMD is more than one, so to contract with such circumstances, Yager [6,7] reformed the state of IFSS to MD + NMD ≤ 1 presenting a general concept with its features. ðMDÞ 2 + ðNMDÞ 2 ≤ 1.
Definition 4. (see [33]). Let X and ℕ be the universe of discourse and set of attributes, respectively; then, a pair ðΩ, ℕÞ is called a PFSS over X.
Let Ω be a mapping such that Ω : ℕ ⟶ ℘K X and ℘K X be a collection of Pythagorean fuzzy subsets. Also, it is defined as follows: where a A ðtÞ, b A ðtÞ: A ⟶ ½0, 1 are MD and NMD, respectively, and 0 ≤ a A ðtÞ 2 + b A ðtÞ 2 ≤ 1.
expressed the indeterminacy: If H ij = ha ij , b ij i is a PFSN, then to compute the alternatives, Zulqarnain et al. [37] offered the score and accuracy functions for H ij as where SðH ij Þ ∈ ½−1, 1. It is reported that the score function cannot discriminate the PFSNs in some cases. For example, if H 11 = h0:3162, 0:44720:4472i and H 12 = h0:5477, 0:6324i , then SðH 11 Þ = −0:1 and SðH 12 Þ − 0:1. In that case, the use of the score function for bargaining is incredible. An accuracy function has been developed that combines MD and NMD to handle this error.
where AðH ij Þ ∈ ½−1, 1. Thus, to compare two PFSNs H ij andR ij , the following comparison laws are defined: Journal of Function Spaces Definition 5. (see [15]). Einstein sum ⨁ ε and Einstein product ⨂ ε are good alternatives of algebraic t-norm and t -conorm, respectively, given as follows: Under the Pythagorean fuzzy environment, Einstein sum ⨁ ε and Einstein product ⨂ ε are defined as where α⨁ ε β and α⨂ ε β are known as t-norm and t -conorm, respectively, satisfying the boundary, monotonicity, commutativity, and associativity properties.

Einstein-Weighted Aggregation Operators for the Pythagorean Fuzzy Soft Set
This section will construct a couple of Einstein-weighted AOs such as PFSEWA and PFSEWG operators for PFSNs with their essential properties.

Theorem 8.
Let H ij = ða ij , b ij Þ be a collection of PFSNs; then, the aggregated value attained by equation (10) is given as where θ i , λ j denote the weight vectors such that Proof. We will employ mathematical induction. For n = 1, we get θ i = 1: Journal of Function Spaces For m = 1, we get λ j = 1: Now, for m = δ 1 + 1 and n = δ 2 + 1, So, it is true for m = δ 1 + 1 and n = δ 2 + 1. As we know that   ð17Þ Theorem 11. Let H ij = ða ij , b ij Þ be a collection of PFSNs; then, where θ i , λ j denote the weight vectors such as Proof. As we know that Again, Let Then, (22) and (26) can be converted into the forms a H ≥ a H ε If SðH Þ > SðH ε Þ, then Journal of Function Spaces So, a H = a H ε and b H = b H ε ; then, by accuracy func- From (27) and (28), we get As we know that Hence, from Examples 9 and 12, it is proven that 3.2. Properties of the PFSEWA Operator Proof. As we know that Let θ i and λ j represent the weight vectors such as

Theorem 14.
Let H ij = ða ij , b ij Þ be a collection of PFSNs; then, the aggregated value attained by equation (56) is given as where θ j , λ i denote the weight vectors such as θ j > 0, Proof. We will employ mathematical induction. For n = 1, we get θ i = 1: Journal of Function Spaces For m = 1, we get λ j = 1: So, equation (57) holds for n = 1 and m = 1. Assume for n = δ 2 and m = δ 1 + 1 and for n = δ 2 + 1 and m = δ 1 , the above equation holds.
We will apply the proposed PFSEWA and PFSEWG operators to resolve the MAGDM problem, which has the succeeding phases: Step 1. Acquire decision matrices for each alternative F = ðH ij Þ n * m in the PFSN form.
Step 2. Normalize the decision matrix to convert the rating value of cost-type parameters into benefit-type parameters by using the normalization formula.
cost-type parameter,

Journal of Function Spaces
Step 3. Use the developed PFSEWA and PFSEWG operators to aggregate the PFSNs H ij for each alternative H = fH 1 , H 2 , H 3 , ⋯, H s g.
Step 4. Compute the score values for each alternative using equation (6).
Step 5. Choose the most feasible alternative with the maximum score value.
The graphical representation of the proposed model is given in Figure 1.

Numerical
Example. Suppose a businessman desires to invest money, and he has five alternatives such as H 1 ; a restaurant, H 2 ; a filling station, H 3 ; a pharmacy, H 4 ; a leather factory, and H 5 ; a supermart. There are four considered attributes, according to which people in business must have to take decision such as t 1 ; socioeconomic impact, t 2 ; environment, t 3 ; risk of loss, and t 4 ; growth rate, with the weight vector λ = ð0:2, 0:2, 0:2, 0:4Þ T . Here, t 1 , t 3 are costtype parameters and t 2 , t 4 are benefit-type parameters. People in business hire a team of four experts O r ðr = 1, 2, 3, 4Þ for decision-making with the weight vector θ = ð0:1, 0:3, 0:3, 0:3Þ T .

By the PFSEWA Operator
Step 1. Decision-maker's opinions in the PFSN form for each alternative are prearranged in Tables 1-5.
Step 2. The normalization rule developed the normalized decision matrices for each alternative. Because t 1 and t 3 are cost-type parameters, the normalized PFS decision matrices are given in Tables 6-10.
Step 3. Using the PFSEWA operator acquired the aggregated values of each alternative in the form of PFSN such as