Some Novel Interactive Hybrid Weighted Aggregation Operators with Pythagorean Fuzzy Numbers and Their Applications to Decision Making

: A Pythagorean fuzzy set (PFS) is one of the extensions of the intuitionistic fuzzy set which accommodate more uncertainties to depict the fuzzy information and hence its applications are more extensive. In the modern decision-making process, aggregation operators are regarded as a useful tool for assessing the given alternatives and whose target is to integrate all the given individual evaluation values into a collective one. Motivated by these primary characteristics, the aim of the present work is to explore a group of interactive hybrid weighted aggregation operators for assembling Pythagorean fuzzy sets to deal with the decision information. The proposed aggregation operators include interactive the hybrid weighted average, interactive hybrid weighted geometric and its generalized versions. The major advantages of the proposed operators to address the decision-making problems are (i) to consider the interaction among membership and non-membership grades of the Pythagorean fuzzy numbers, (ii) it has the property of idempotency and simple computation process, and (iii) it possess an adjust parameter value and can reﬂect the preference of decision-makers during the decision process. Furthermore, we introduce an innovative multiple attribute decision making (MADM) process under the PFS environment based on suggested operators and illustrate with numerous numerical cases to verify it. The comparative analysis as well as advantages of the proposed framework conﬁrms the supremacies of the method.


Introduction
Multiple attribute decision making (MADM) is one of the processes to find the most desirable alternative among all given alternatives in the light of finite attributes or criteria. In the decision-making process, it is commonly supposed that the evaluation information of alternatives for attributes described by decision-makers (DMs) is precise numbers. However, due to the indeterminacy of the practical environment and human cognition, DMs are usually do not find it easy to use crisp numbers to express their preferences. An appropriate way to deal with such problems is to adopt uncertain evaluations rather than crisp ones, for instance, an intuitionistic fuzzy set (IFS) [1] and fuzzy set (FS) [2]. As a successful extension of the notions of IFS and FS, the Pythagorean fuzzy set (PFS) was put forward by Yager [3,4]. Similar to the IFS, the PFS is still depicted by the membership and non-membership degrees, but their square sum within interval (0, 1). Thus, PFS is more versatile than IFS. For instance, if the membership grade is given as 0.4 by DM, while the non-membership grade is 0.8, it can be seen From these above analyses, we can see that different aggregation operators have different features and application aspects. The PFIWA and the PFIWG AOs can weigh only the significance of PFNs themselves, while the PFIOWA and the PFIOWG AOs can weigh the ordered positions of given PFNs but cannot weigh the PFNs themselves. Moreover, the PFIHA and the PFIHG operators may weigh all aggregated PFNs and correspond ordered positions of them. Therefore, the PFIHA and the PFIHG operators have some superiority over the operators described above. However, these two operators have a drawback that the aggregated value of some identical integrated PFNs relies on the weight values, that is they do not possess the property of idempotency. It was pointed out by Liao and Xu [40], hybrid aggregation operators should satisfy the basic property of idempotency, so they presented a group of hybrid operators under a hesitant fuzzy environment. However, these aggregation operators are not available to tackle the Pythagorean fuzzy MADM problems. PFS, a valuable generalization of IFSs, has been shown as a successful means to deal with the indeterminacy and fuzziness which appear in many real decision problems. The presented research concentrated on the Pythagorean fuzzy setting. Therefore, it was worth putting forward some novel Pythagorean fuzzy interactive hybrid operators. Inspired by this idea, the motivation and objective of this manuscript were to (1) explore novel Pythagorean fuzzy interactive hybrid weighted average (PFIHWA) and geometric (PFIHWG) operators, discuss some interesting properties and particular cases; (2) propose novel generalized PFIHWA (GPFIHWA) and generalized PFIHWG (GPFIHWG) operators, also study their desirable properties and special cases; (3) introduce some steps for MADM and MAGDM by using the proposed operators; (4) demonstrate the availability and flexibility of the proposed MADM and MAGDM methods through some practical examples.
The remaining paper is arranged as follows: Some fundamental notions about PFSs and the AOs are introduced in Section 2. Novel PFIHWA and PFIHWG operators along with their properties are given in Section 3. Generalized forms of the AOs are provided in Section 4. In Section 5, we use the presented AOs to tackle the MADM problems and MAGDM problems, and the availability as well as flexibility of the proposed methods is illustrated with some real examples. Section 6 summarizes the paper.

Preliminaries
We briefly review some fundamental notions about PFS and Pythagorean fuzzy AOs in this part.

Some Novel Pythagorean Fuzzy Interactive Hybrid Weighted Aggregation Operators
Although PFIHA (PFIHG) possesses both the advantages of the PFIWA (PFIWG) operator and the PFIOWA (PFIOWG) operator, have a shortcoming which is that the AOs do not meet the prominent property, that is idempotency.

New Proposed Hybrid Aggregation Operators
In this part, we have presented some novel hybrid AOs for Pythagorean fuzzy sets to settle the deficiencies of the existing AOs.

Example 3.
Let us employ the PFIHWA operator in Definition 8 to recalculate Example 1. We obtain In what follows, we shall present the novel interactive hybrid weighted geometric AOs for PFNs.

Example 4.
Let us employ PFIHWG operator to compute Example 2. According to Equation (12), we obtain

Example 5.
Let us employ the PFIHWG operator in Definition 9 to recalculate Example 1. We obtain

Generalized Pythagorean Fuzzy Interactive Hybrid Weighted Aggregation Operators
Yang and Pang [37] presented the generalized PFIWA (GPFIWA) and the generalized PFIWG (GPFIWG) operators. Next, we can extend the PFIHWA and PFIHWG operators into generalized forms.

Proof.
Based on Equation (2), we have Hence, the Theorem 5 holds.
According to Equation (15), we have Therefore, the Theorem 6 holds.
In what follows, by choosing a different parameter λ, we explored some particular cases of the GPFIHWA operator.
Proof. Analogous to Theorem 5, the proof is omitted.
In the following, by choosing a different parameter λ, we explored some particular cases of the GPFIHWG operator.

MADM Approach with PFNs by Employing the Presented AOs
In this section, we adopted our developed PFIHWA, PFIHWAG, GPFIHWA and GPFIHWG operators to handle multiple attribute single person decision making and MAGDM based on PFNs, respectively.
Step 1. Compute the normalized decision information matrix P = (p i j ) m×n of R = (r i j ) m×n . The transformation is given as follows [41]: r i j , for the benefit attribute of C j r i j c , for the cos t attribute of C j in which, r i j c = (ν r i j , µ r i j ), (i = 1, 2, 3, . . . , m; j = 1, 2, 3, . . . , n) be the complement of p i j .

Remark 3.
In MADM issues, attribute information is often divided into benefit and cost types. In order to facilitate calculation, some methods are needed to standardize the attribute information [41].

Example 7.
Consider that an organization wants to evaluate emerging technology enterprises (adapted from Reference [18]), the experts of the organization are given five potential alternatives A 1 , A 2 , A 3 , A 4 , A 5 . After careful analysis, the experts evaluate the five potential alternatives in accordance with the four attributes {C 1 , C 2 , C 3 , C 4 }. C 1 represents technical advancement; C 2 represents the likely market and market risk; C 3 represents the financial conditions and human resources; C 4 represents the science and technology development and employment creation. Suppose ε = (0.15, 0.2, 0.3, 0.35) T is the weight vector. ω = (0.2, 0.4, 0.3, 0.1) T stands for the associated weight vector of four attributes, which assigns more weight to the attribute obtained for the optimal performance. The decision values take the form of PFNs, as listed in Table 1. Table 1. Pythagorean fuzzy decision matrix R. To choose the optimal emerging technology enterprise, the following procedures are summarized: Step 1. Since every attribute is a benefit type, no transformation is needed. The evaluation matrix is P = R = (r ij ) 5×4 , described in Table 1.

Process of MADM based on the PFIHWG Operator
In order to choose the optimal one(s) based on the PFIHWG operator, the following procedures of the proposed approach are summarized as below.
Step 1. It is identical with Step 1 in Section 5.1.1.

Comparison and Discussion
To demonstrate the feasibility of the presented approach, we compare our methods with the PFIHA and PFIHG operators developed by Wei [33], the SPFWA (symmetric Pythagorean fuzzy weighted averaging) and SPFWG(symmetric Pythagorean fuzzy weighted geometric) operators developed by Ma and Xu [19], and the PFEWA(Pythagorean fuzzy Einstein weighted averaging) and PFEWG(Pythagorean fuzzy Einstein weighted geometric) operators developed by Garg [22] and Garg [23], respectively. These methods were used to solve the above example, and the aggregating values and sort outcomes are given in Table 2.
The content of Table 2 implies the aggregating results are different from each other, the ranking of alternative A 1 and A 5 is slightly different in the SPFWA, SPFWG, PFEWA and PFEWG operators, but the optimal emerging technology enterprise is still A 2 in all operators. Therefore, our methods are effective and feasible. However, comparing with the PFIHA and PFIHG operators [33] our methods are simple from the computational point of view. For instance, in the PFIHWA operator, are crisp numbers, we only compute the Pythagorean fuzzy value n ⊕ j=1 ω ( j) ε j α j n j=1 ω ( j) ε j . However, in the PFIHA operator [33], we should first compute Pythagorean fuzzy value . α j = nε j α j , then compute the Pythagorean fuzzy value From Figure 1, we observe that the Spearman correlation of SPFWA [19], SPFWG [19], PFEWA [22] and PFEWG [23] are all −0.6, whereas, the Spearman correlation of proposed operator (PFIHWA, PFIHWG), PFIHA [33] and PFIHG [33] are all 1. Comparing with PFIHA [33] and PFIHG [33], the typical characteristics of our techniques are that they possess a small amount of computation and idempotency. It further indicates that our approaches are superior. Therefore, our approach is suitable for settling some practical multiple attribute decision problems with Pythagorean fuzzy information. From Figure 1, we observe that the Spearman correlation of SPFWA [19], SPFWG [19], PFEWA [22] and PFEWG [23] are all −0.6, whereas, the Spearman correlation of proposed operator (PFIHWA, PFIHWG), PFIHA [33] and PFIHG [33] are all 1. Comparing with PFIHA [33] and PFIHG [33], the typical characteristics of our techniques are that they possess a small amount of computation and idempotency. It further indicates that our approaches are superior. Therefore, our approach is suitable for settling some practical multiple attribute decision problems with Pythagorean fuzzy information.
In what follows, we will employ GPFIHWA and GPFIHWG operators to tackle MAGDM problems with PFNs. Assume
In what follows, we will employ GPFIHWA and GPFIHWG operators to tackle MAGDM problems with PFNs. Assume A = {A 1 , A 2 , . . . , A m } and C = {C 1 , C 2 , . . . , C n } are respectively the group of alternatives and attributes. The ε = (ε 1 , ε 2 , . . . , ε n ) T is the weight vector, that meets ε j ≥ 0, n j=1 ε j = 1. Suppose ω = (ω 1 , ω 2 , . . . , ω n ) T is the associated vector for ω j ≥ 0, n j=1 ω j = 1. D = {d 1 , d 2 , . . . , d l } is the group of experts, τ = (τ 1 , τ 2 , . . . , τ l ) T is the corresponding weight vector that satisfies τ k ≥ 0, ) is a PFN offered by the expert d k ∈ D for the alternative A i ∈ A relevant to the attribute C j ∈ C.µ r (k) ij and ν r (k) ij means the grade that alternative A i meets attribute C j and doesnot meet attribute C j offered by the expert d k , 1]. Then, the procedure of the MAGDM problem (Algorithm 2) is listed below: Algorithm 2. The procedure of the MAGDM problem using GPFIHWA and GPFIHWG operators.
Step 1. It is identical with Step 1 in Algorithm 1.
Step 2. Utilize the PFIWA operator and decision matrixes P (k) = (p (k) i j ) m×n to get the group decision matrix Step 3. Utilize the assessment matrix P = (p i j ) m×n and the GPFIHWA operator to obtain the comprehensive evaluation values p i (i = 1, 2, 3, . . . , m).
Step 4. It is identical with Step 3 in Algorithm 1.
Step 5. It is identical with Step 4 in Algorithm 1.

Example 8.
Suppose a company intends to implement the ERP (Enterprise Resource Planning) system (revised from Reference [28]). Three experts {e 1 , e 2 , e 3 } from different departments form a project team to make the evaluations, including a CIO(Chief Information Officer) and two senior representatives, whose weight vector is τ = (1/3, 1/3, 1/3) T . Assume that we have five latent ERP systems {A 1 , A 2 , A 3 , A 4 , A 5 }, and four assessment attributes {C 1 , C 2 , C 3 , C 4 } were selected, C 1 stands for the technology and function; C 2 stands for the strategic adaptability; C 3 stands for competence of vendor and C 4 stands for renown of vendor. Assume ε = (0.15, 0.2, 0.3, 0.35) T is the importance degree of attributes. The associated weight vector given by the project team as ω = (0.2, 0.1, 0.3, 0.4) T , which assigns more weight to the attribute obtaining the optimal performance. The five potential ERP systems A 1 , A 2 , A 3 , A 4 , A 5 are appraised by PFNs, and are summarized in Tables 3-5. Table 3. The Pythagorean fuzzy decision matrix R (1) .  Table 4. The Pythagorean fuzzy decision matrix R (2) .

Process of MAGDM based on the GPFIHWA Operator
Step 1. Since every attribute is a benefit type, no transformation is needed. The decision matrix , is described in Tables 3-5.

Process of MAGDM based on the GPFIHWG Operator
Step 1-2. It is the same as Step 1-2 in Section 5.2.1.

Comparison and Discussion
In Step 3 of Section 5.2.2, if we employ the PFIHA and PFIHG operators [33], then the decision result is A 5 A 2 A 3 A 4 A 1 . If we use the SPFWA [19], SPFWG [19], PFEWA [22] and PFEWG [23] operators, we obtain the following result: We can obtain that the decision outcomes by the PFIHA operator [33] and PFIHG operator [33] are the same as our GPFIHWG operator, and are slightly different with our GPFIHWA operator, but the most desirable alternative by the PFIHA and PFIHG operators [33] coincide with the proposed operator results, i.e., alternative A 5 . The most desirable alternative determined by SPFWA [19], SPFWG [19], PFEWA [22] and PFEWG [23] operators is A 2 for all, the reason is that these AOs do not consider the interaction among membership and non-membership grades. Therefore, it is available and feasible in the proposed approaches. Moreover, our approaches are simple from the computational point of view compared with the PFIHA and PFIHG operators [33]. Further contrast effect can be reflected in Figure 2. and PFEWG [23] operators is 2 A for all, the reason is that these AOs do not consider the interaction among membership and non-membership grades. Therefore, it is available and feasible in the proposed approaches. Moreover, our approaches are simple from the computational point of view compared with the PFIHA and PFIHG operators [33]. Further contrast effect can be reflected in Figure2. As provided in Figure 2, the Spearman correlations of SPFWA [19], SPFWG [19], PFEWA [22] and PFEWG [23] are all 0, which shows that our methods are superior. The main features of the proposed GPFIHWA and GPFIHWG operators are that: (1) it considers the interaction among membership and non-membership grades for PFNs, and are more suitable to address actual MADM issues in some special situations; (2) it has the property of idempotency and simple computation process; (3) it possess an adjust parameter value and can reflect the preference of DMs during the decision process.

Sensitivity Analysis
Parameter λ plays a significant influence in the decision-making process; it can reflect the mentality of the DMs. For this, we chose different values of λ from 0 to 30 in Algorithm 2 to solve Example 8, so as to investigate the flexibility and sensitivity of different λ . The scores as well as decision results are listed in Tables 7 and 8.  As provided in Figure 2, the Spearman correlations of SPFWA [19], SPFWG [19], PFEWA [22] and PFEWG [23] are all 0, which shows that our methods are superior. The main features of the proposed GPFIHWA and GPFIHWG operators are that: (1) it considers the interaction among membership and non-membership grades for PFNs, and are more suitable to address actual MADM issues in some special situations; (2) it has the property of idempotency and simple computation process; (3) it possess an adjust parameter value and can reflect the preference of DMs during the decision process.

Sensitivity Analysis
Parameter λ plays a significant influence in the decision-making process; it can reflect the mentality of the DMs. For this, we chose different values of λ from 0 to 30 in Algorithm 2 to solve Example 8, so as to investigate the flexibility and sensitivity of different λ. The scores as well as decision results are listed in Tables 7 and 8.   Table 7 indicates that the scores in the GPFIHWA operator become bigger with parameter λ increasing. Therefore, the DMs with optimistic attitude should take larger values of λ. Moreover, the ranking results are different by using different values of λ, but the best alternative is always A 5 . Furthermore, we can find that (1) (1) when λ ∈ (0, 0.8765], the ranking is A 5 A 2 A 3 A 4 A 1 .
(3) when λ ∈ (4.0562, 30], the ranking is A 5 A 4 A 2 A 3 A 1 . Table 8 indicates that the scores in the GPFIHWG operator become smaller with parameter λ increasing. Therefore, the DMs with optimistic attitude should take smaller values of λ. Moreover, the ranking results are also different by employing different values of λ, and the best alternative is from A 5 to A 3 , then from A 3 to A 4 with parameter λ increasing. Furthermore, we can find that (1) when λ ∈ (0, 0.1235], the ranking is A 5 A 2 A 4 A 3 A 1 , (2) when λ ∈ (0.1235, 4.3582], the ranking is A 5 A 2 A 3 A 4 A 1 , (3) when λ ∈ (4.3582, 4.5858], the ranking is A 5 A 3 A 2 A 4 A 1 , (4) when λ ∈ (4.5858, 7.3828], the ranking is A 5 A 3 A 4 A 2 A 1 , (5) when λ ∈ (7.3828, 7.6462], the ranking is A 3 A 5 A 4 A 2 A 1 , (6) when λ ∈ (7.6462, 11.6251], the ranking is A 3 A 4 A 5 A 2 A 1 , (7) when λ ∈ (11.6251, 15.1215], the ranking is A 4 A 3 A 5 A 2 A 1 , (8) when λ ∈ (15.1215, 30], the ranking is A 4 A 3 A 2 A 5 A 1 . Therefore, the approach by using the GPFIHWA operator is relatively stable. In the actual decision environment, the DMs may select a different parameter λ in line with their preferences.
To better distinguish the presented approach with the existing approaches [19,22,23,33,[37][38][39], we summarize the differences of them in Table 9. Based on Table 9, we can obtain that the presented approaches possess the property of idempotency, and also embody the interactions among membership and non-membership during the information aggregation process. Therefore, the novel approaches can obtain more reasonable ranking results. Table 9. Characteristic comparisons of different methods.

Considers the Interactions between Membership and Non-Membership
Possesses the Property of Idempotency PFIHA [33] Yes No PFIHG [33] Yes No SPFWA [19] No Yes SPFWG [19] No Yes PFEWA [22] No Yes PFEWG [23] No Yes Liao and Xu [40] No Yes Our proposed operators Yes Yes

Conclusions
In this study, the authors have presented a group of novel Pythagorean fuzzy interactive hybrid weighted AOs, such as the PFIHWA, PFIHWG, GPFIHWA and GPFIHWG operators. It can be seen that these novel developed operators have the feature of idempotency, which indicated that the proposed AOs could overcome the shortcomings of the PFIHA and the PFIHG operators. It was also shown that some other existing AOs [33,35] were the particular cases of our presented AOs. In addition, our approaches were simple in view of computational cost and also captured the interaction over membership and non-membership grades. Afterward, two algorithms to MADM and MAGDM by the proposed operators were provided. Lastly, we verified the validity and flexibility with two practical examples.
Author Contributions: L.W. and N.L. conceived and worked together to achieve this work; H.G. studied the numerical example and investigation; L.W. and N.L. wrote the initial draft paper; L.W., H.G. revised the manuscript. Finally, all the authors read and approved the final manuscript.