An Extension of Bonferroni Mean under Cubic Pythagorean Fuzzy Environment and Its Applications in Selection-Based Problems

Cubic Pythagorean fuzzy (CPF) set (CPFS) is a hybrid set that can describe both interval-valued Pythagorean fuzzy (IVPF) sets (IVPFSs) and Pythagorean fuzzy (PF) sets (PFSs) simultaneously for addressing information ambiguities. Since an aggregation operator (AO) is a signifcant mathematical approach in decision-making (DM) problems, this article presents some novel Bonferroni mean (BM) and weighted Bonferroni mean averaging operators between CPF-numbers (CPFNs) for aggregating the diferent preferences of the decision-makers. Ten, using the proposed AOs, we develop a DM approach under the CPF environment and demonstrated it with a numerical example. Furthermore, a comparative study between the proposed and existing methods has been performed to demonstrate the practicality and efciency of the proposed DM approach.


Introduction
Decision-making is a signifcant process in order to pick the best-suited alternative from among those available. In it, a number of scholars provided a variety of theories to make the best judgments. In the past, decisions were made based on crisp numbered data sets, but this resulted in insufcient outcomes that were less applicable to real-life operational scenarios. However, as time passes and the complications of the system increase, it becomes more difcult for the decisionmaker to handle the inconsistencies in the data, and thus the traditional technique is unable to fnd the optimum alternative. Terefore, the scholars used fuzzy set (FS) theory [1], interval-valued fuzzy sets (IVFSs) [2], intuitionistic fuzzy sets (IFSs) [3], interval-valued intuitionistic fuzzy sets (IVIFSs) [4], PFS [5,6], and IVPFS [7] to describe the information. Scholars have paid increasing attention to these ideas in recent decades and have efciently implemented them in a variety of scenarios in the DM process. An aggregation operator, which generally takes the form of a mathematical formalism to accumulate all of the individual input data into a single one, is an important part of the DM process. For example, Xu and Yager [8] introduced certain geometric AOs to integrate various preferences of the decision-makers into intuitionistic fuzzy numbers (IFNs). Later, Wang and Liu [9] utilized Einstein norm operations to generalize these operators. For aggregating diferent intuitionistic fuzzy information, Garg [10] introduced generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein norm operations. Garg has presented a series of interactive AOs for IFNs in [11]. Garg [12] implemented the IFS concept to PFS and proposed generalized averaging AOs. Te symmetric Pythagorean fuzzy AOs were proposed by Ma and Xu [13]. To solve DM problems, Garg [14] presented some improved interactive aggregation operators, while Wang and Liu [15] presented some hybrid weighted aggregation operators using Einstein norm operators. Apart from that, several other authors have given other approaches to solve DM problems, such as ranking functions [16] and AOs (see [17]).
As the above, AOs have been considered by many researchers during the DM process in which they have highlighted the contribution of each factor or its ordered position but cannot represent the interrelationships of the individual information. In our real-life situation, a relationship of diferent criteria such as importance, support, and impact on each other constantly plays a signifcant role throughout the aggregation process. To deal with it, Yager [18] developed the power average (PA) AOs to address this issue and implement it into DM analysis. Xu and Yager [19] and Yu [20] proposed the prioritized averaging and geometric AOs in an IFS environment. Further, Yager [21] presented the idea of the BM aggregation operators, which has the potential to represent the interrelationship between the input arguments. To alleviate the limitation of BM, Beliakov and James [22] introduced the generalized BM. To aggregate the intuitionistic fuzzy information, Xu and Yager [23] developed an intuitionistic fuzzy BM. Tese BM operators were generalized to the interval-valued IFSs environment by Xu and Chen [24]. Te generalized intuitionistic fuzzy BMs were presented by Xia et al. [25]. Te partitioned BM operators were described by Liu et al. [26] in an IFSs environment. Shi and He [27] discussed how to optimize BMs by applying them to diferent DM processes. In an intuitionistic fuzzy soft set environment, Garg and Arora [28] proposed the BM aggregation operator. Te Pythagorean fuzzy Bonferroni mean (PFBM) is developed by Liang et al. [29], and several specifc properties and cases are described. Wang and Li proposed a Pythagorean fuzzy interaction PFBM and weighted PFBM operators [30]. Nie et al. [31] proposed a PF partitioned normalized weighted BM operator with Shapley fuzzy measure.
All of the existing research and their respective applications are mostly focused on the FS, interval-valued FS (IVFS), IFS, PFS, IVIFS, and IVPFS. Ten, Jun et al. [32] developed several logic operations of the cubic sets and familiarized the theory of cubic set (CS) and their operational laws such as P-union, P-intersection, R-union, and R-intersection of CS and investigated several related properties. CS has been employed in a variety of real-world applications. Tey use highly interconnected distinctive to solve complex issues in engineering, economics, and the environment. Because of the many uncertainty models for such situations, it is not always simple to apply standard approaches to obtain good results. Terefore, Khalil and Hassan [33] introduced the class of cubic soft algebras and their basic characteristics. Shi and Ye [34] proposed Dombi Aggregation Operators of Neutrosophic Cubic Sets for Multiple Attribute Decision-Making. Ye et al. [35] presented Multi-fuzzy Cubic Sets and Teir Correlation Coefcients for multi-criteria Group Decision-Making. Garg et al. [36] proposed Correlation Measures for Cubic m-Polar Fuzzy Sets with Applications. Khan et al. [37,38] presented some cubic AOs under this set, while Mahmood et al. [39] presented the concepts of cubic hesitant fuzzy sets and their AOs. Te above concepts only provide information in the form of membership intervals and ignore the nonmembership section of the data entities, which also perform an important role in evaluating the alternative in the DM process. In the real world, expressing the value of a membership degree by an exact value in a fuzzy set is typically challenging. In such situations, an interval value and an exact value, rather than unique interval/exact values, may be convenient to communicate uncertainty and ambiguity in the real world. Hence, the hybrid form of an interval and an exact number may be a very useful term for a person to express certainty and uncertainty as a result of his or her uncertain assessment in multifaceted DM problems. Kaur and Garg [40,41] introduced the concept of the cubic intuitionistic fuzzy set CIFS, which is characterized by two portions at the same time, one of which refects membership degrees by an IVIF value and the other by an intuitionistic fuzzy value. Each CIFS component is represented as ([ξ − , ξ + ], [] − , ] + ], ξ, ]) that satisfes the conditions ξ + + ] + ≤ 1 and ξ + υ ≤ 1. However, in some actual cases, the sum of membership and non-membership grades may be greater than 1, but their square sum is less than or equal to 1. Terefore, Abbas et al. [42] presented the concept of the CPFS, which is the generalization of CIFS. At the same time, CPFS has two phases, one of which refects the degree of membership by an IVPFS and the other by a PFS that satisfes the condition (ξ + ) 2 + (υ + ) 2 ≤ 1 and ξ 2 + υ 2 ≤ 1. Terefore, a CPFS is a hybrid set that includes both an IVPFS and a PFS. Obviously, the CIFS has the advantage of being able to contain a lot more information in order to express both the IVPFN and the PFN at the same time. Hence, CPFS has a high level of efciency and importance when used to evaluate alternatives during the DM process because the general DM process may use IVPFS or PFS data, which may loss of some important evaluation information. Tere is currently no research on AOs that refects the interrelationship between the multiple criteria of a DM process having CPF information.
According to the current communication, inspired by the BM notion and by utilizing the CPFS advantages, we suggest some new AOs called the CPF Bonferroni mean (CPFBM) and weighted cubic Pythagorean fuzzy Bonferroni mean (WCPFBM) to aggregate the preferences of decisionmakers. We have also investigated various desirable properties of these operators in detail. Te main advantage of the proposed AOs is that the interrelationships between aggregated values are taken into account. We also examine the characteristics of the proposed work and design some specifc scenarios. Te proposed AOs have been deduced from several previous studies, demonstrating that the proposed AOs are more fexible than the others. Finally, a DM method for rating the various alternatives based on the proposed AOs has been presented. Finally, a DM method for rating the various alternatives based on the proposed AOs has been presented. Te list of abbreviations used in this article is provided in Table 1.
Te rest of the article is organized as follows. Te basic concepts are briefy discussed in Section 2. Section 3 presents AOs called CPF Bonferroni mean (CPFBM) and weighted CPF Bonferroni mean (WCPFBM), as well as their applications. To address multi-criteria DM (MCDM) problems, a DM approach based on proposed operators has been developed in Section 4. In Section 5, a numerical model is provided to illustrate the proposed approach and demonstrate its practicality and applicability. Section 6 presents the conclusion including closing remarks.

Preliminaries
Some main theories associated with PFSs, IVPFSs, CSs, CIFSs and CPFSs are briefy addressed in this section.

Cubic Pythagorean Fuzzy Set
Defnition 12. (see [39]). Let S be a non-empty set. A CPFS C over S is defned as follows: where and μ ρ , ξ ρ and called as CPF number (CPFN).

Mathematical Problems in Engineering
Defnition 14 (see [39] ) be the collections of CPFNs, and φ≻0 be a real number, then

Bonferroni Mean Operators
Defnition 15 (see [44]). Let a, b ≥ 0 and q i (i � 1, 2, . . . , n) be a collection of non-negative numbers, then BM is defned as In [43], Liang et al. generalized this BM for PF environments, giving the following concepts.
Mathematical Problems in Engineering

CPF-Bonferroni Mean Aggregation Operators
In this section, a series of aggregation operators, namely, CPFBM and WCPFBM are presented.
6 Mathematical Problems in Engineering Tus, ) be the collections of CPFNs and a, b ≥ 0. For any i, j, and i ≠ j, we have Proof. By Defnition 14, we have and Mathematical Problems in Engineering Ten, 8

Mathematical Problems in Engineering
Proof. We can get the following result from Proposition 1: Mathematical Problems in Engineering 9 Tus, Proposition 2 holds.
) be the collection of CPFNs and a, b ≥ 0. We can calculate the following by using Proof. Mathematical induction on n.

□
Step 1. By adopting Proposition 1, we obtain the following result when k � 2: Tus, we have 10 Mathematical Problems in Engineering Step 2. We can obtain the following result. Eq. (14) holds for Step Hence, (31) is also true for k � k 0 + 1. As a result, Proposition 3 holds true.
Proof. Te proof is similar to that of proportion 3, so it is omitted. � 1, 2)) be the collection of CPFNs and a, b ≥ 0.
Proof. Te proof is similar to that of proportion 3, so it is omitted.
be the collection of CPFNs and a, b ≥ 0. For any i, j and i ≠ j, we Proof. Based on Eq. (29), we have Mathematical Problems in Engineering By using Defnition 14, we have 2) be the collection of CPFNs and a, b ≥ 0. By using the CPFBM operator, the aggregated value is also a CPFN, and

Mathematical Problems in Engineering
Proof. By Proposition 6 and Defnition 14, we have As a result, Teorem 1 holds true. 2) be three CPFNs. Ten, use CPFBM to aggregate these three CPFNs. Te steps are outlined below. (Supposes a � b � 1 and n � 3). By using (21), we get Mathematical Problems in Engineering Tus, Based on the results of (see [18]), we may conclude the following corollaries for Teorem 1.

Corollary 3 (monotonicity). Let ϑ
3.2. Weighted CFP Bonferroni Mean. Te CPFBM described in Teorem 1 does not take the signifcance of the aggregated arguments into account. Te weight vector of the criterion is a crucial component of the aggregate in various real-world MCDM problems. As a result, the WCPFBM operator is defned as follows. 1, 2, . . . , n) be the collection of CPFNs and a, b are any non-negative real numbers. Δ � (Δ 1 , Δ 2 , . . . , Δ n ) T is the weight vector of ϑ i such that Δ i ∈ [0, 1] and n i�1 Δ i � 1. Ten, weighted CPF Bonferroni mean (WCPFBM) operator is defned as . . , n) be the collection of CPFNs and a, b are any non-negative real numbers.
Ten, the aggregated values obtained by WCPFBM are also a CPFN, and

Mathematical Problems in Engineering
Step 7. Use eq. (4) to compute the score values of each CPFN Δ i as follows: Step 8. Arrange the alternatives A i (i1, 2, . . . , m) with respect to respective score values sc(Δ i ).

Case Study
Tese days, supply management is a major problem. From an industrial perspective, a company cannot obtain better performance levels unless its material is properly administered. Terefore, efective inventory management is the frst step to achieving high levels of production. Any scarcity of raw material in inventory might cause the entire manufacturing cycle to be afected, resulting in a signifcant loss for the company. Consider the Case [40] of a food company that has to keep track of multiple inventory materials. Te company primarily produces four diferent types of products (A i )(i � 1, 2, 3, 4).
To prepare these food products, rearranging the stock the materials in the inventory will be decided on the basis of three criteria (C j )(j � 1, 2, 3).
(i) C 1 : Cost Price (ii) C 2 : Storage Facilities (iii) C 3 : Staleness Level And Δ � (0.2, 0.38, 0.32) T is the weight vector of these features. Te given alternatives are analyzed using these three criteria and their values are rated in terms of CPFNs. In each CPFN, the IVPFNs refect the relative stock level in the inventory, and PFSs refect the estimate of agreement and disagreement towards the present stock level for the following week. Since the company does not compromise on product quality, therefore, reducing the level of stagnation is the highest priority. Te goal is then to identify the food products whose material supply must be re-ordered on a regular basis. Te phases of the proposed approach have been carried out in the following order. Te multilevel layout of the problem is shown in Figure 1.
Step 9. As indicated in Table 2, the favorites information for each alternative is signifed in CPFNs, and the group ratings are given in the decision matrix.
Step 10. Te CPFNs are normalized by using Eq. (20) and summarized in Table 3.
Step 11. We assume a � b � 1 for simplicity's convenience and then calculate the aggregated values of each alternative using Eq. Step 13. According to the score values, the ranking order of the alternatives is concluded to be Te suggested operators are symmetric with respect to the parameters a and b. Terefore, in order to examine the infuence of parameter a and b on the fnal ranking of the alternatives, an investigation has been initiated by changing them simultaneously. Te score values and raking order for diferent values of a and b are summarized in Table 3. According to Table 4, we can observe that the score values for diferent couples of the parameter a and b are diferent; however, the ranking orders of alternatives remain the same. Tis quality of the suggested operators is especially important in real-world DM problems. For example, it has been observed that as the parameters are increased, the score values of alternatives grow, giving us an optimistic impression of the decision makers. So, if the decision-makers are bullish, greater values for values for a and b can be allocated throughout the processing phase. If the decisionmakers are bearish, lower values allocated can be provided to a and b. However, the ranking of alternatives cannot be Alternative A 1 Alternative A 3 Alternative A 4 Alternative A 2 Criteria C 3 Criteria C 2 Criteria C 1 Figure 1: Te multilevel layout of the problem. changed, which indicates that the outcomes are objective and cannot be afected by decision-makers' negativity or positivity. As a consequence, the results obtained are legitimate.
By varying the parameter b, the variations of the score values of each alternative are summarized in Figures 2-4 which shows that the greatest score owing alternative remains A 4 for all cases. However, in Figure 3, by holding a � 1 and varying b from 0 to 6, it is shown that when a � 1 and b � 3.8580 then, sc(A 1 ) � sc(A 3 ) � − 1.0038, and thus, from (7), we get ac(A 1 ) � 1.0962 and ac(A 3 ) � 1.1979, as a result, the ranking order of the alternative for a � 1 and        Figure 3, by holding a � 2 and b � 5.4932 then sc(A 1 ) � sc(A 3 ) � − 0.9924; however, ac(A 1 ) � and ac(A 3 ) � and hence the ranking order of alternative for a � 2 and b � 5.4932 is also A 4 ≻A 2 ≻A 3 ≻A 1 . Te performance of score values by fxing a � 5 is shown in Figure 4.

Comparative Analysis.
Te dominance of the proposed aggregation operators with respect to the previous approaches such as interval-valued intuitionistic fuzzy aggregation operators [16,[45][46][47][48][49][50] and weighted cubic intuitionistic BM operator [40] is demonstrated. Te weight vector is Δ � (0.2, 0.38, 0.42) T . Table 5 summarizes the optimal score values and rank order of the alternatives. According to this table, we concluded that the optimum alternative corresponds with the proposed approach results that authenticate the constancy of the approach with respect to state-of-the-art. Te proposed DM approach under CPF environment covers much more estimation information on the alternatives by considering both PFSs and IVPFSs at the same time, whereas existing approaches include either PFS or IVPFS. Aa a result, the DM approaches under PFS or IVPFS may lose some valuable information, that may afect the DM results. In addition, the research concluded that the proposed computational process is diferent from the existing approaches, but the results of the proposed approach are more rational to reality in DM process. Finally, it is observed that the parameters a and b provide more ranges to the decision-makers to avail their required alternatives allocate the diferent score values of the alternatives for diferent values of a and b. Terefore, the approaches under CPS and IVPFS can be measured as a special case of the proposed approach.

Conclusion
Te CPF set is the more fexible and appropriate tool for deal with fuzziness and uncertainties by interconnecting IVPFN and PFN simultaneously in DM problems. Te main purpose of this paper is to present an aggregation operator BM whose signifcant characteristic is to capture the connections among the distinct infuences. To accomplish this, we proposed two aggregation operators, i.e., CPFBM operator and WCPFBM operator, to aggregate the various preferences of the experts over the various attributes in the CPF context. Various desired properties of these aggregation operators are also investigated. Moreover, an approach for handling DM issues has been provided by changing the values of parameters a and b. It is observed that the parameter a and b makes the proposed operators more elastic and provides numerous ranges to the decision-maker for evaluating the decisions. A comparison with several existing operators reveals that the presented operators and their accompanying approaches ofer the decision maker a more reliable, pragmatic, and optimistic character during the processing phase. As a result, we accomplish that the presented operators may be used to address the in real-world situations.

Data Availability
No data were used to support this study.

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