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

Authors: Muhammad Rahim; Fazli Amin; Asad Ali; Kamal Shah
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

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 significant 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 different preferences of the decision-makers. Then, 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 efficiency of the proposed DM approach.


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## Body

## 1. Introduction

Decision-making is a significant 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 insufficient outcomes that were less applicable to real-life operational scenarios. However, as time passes and the complications of the system increase, it becomes more difficult for the decision-maker to handle the inconsistencies in the data, and thus the traditional technique is unable to find the optimum alternative. Therefore, 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 efficiently 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 different 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. The 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 different criteria such as importance, support, and impact on each other constantly plays a significant 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. These BM operators were generalized to the interval-valued IFSs environment by Xu and Chen [24]. The generalized intuitionistic fuzzy BMs were presented by Xia et al. [25]. The 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 different DM processes. In an intuitionistic fuzzy soft set environment, Garg and Arora [28] proposed the BM aggregation operator. The Pythagorean fuzzy Bonferroni mean (PFBM) is developed by Liang et al. [29], and several specific 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. Then, 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. They 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. Therefore, 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 Their Correlation Coefficients 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. The 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 reflects membership degrees by an IVIF value and the other by an intuitionistic fuzzy value. Each CIFS component is represented as ξ−,ξ+,ν−,ν+,ξ,ν that satisfies the conditions ξ++ν+≤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. Therefore, 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 reflects the degree of membership by an IVPFS and the other by a PFS that satisfies the condition ξ+2+υ+2≤1 and ξ2+υ2≤1. Therefore, 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 efficiency 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. There is currently no research on AOs that reflects 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 decision-makers. We have also investigated various desirable properties of these operators in detail. The 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 specific scenarios. The proposed AOs have been deduced from several previous studies, demonstrating that the proposed AOs are more flexible 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. The list of abbreviations used in this article is provided in Table1.Table 1
List of abbreviations.
SymbolsAbbreviationsCPFCubic Pythagorean fuzzyCPFSCubic Pythagorean fuzzy setPFPythagorean fuzzyIVPFSInterval-valued Pythagorean fuzzy setPFSPythagorean fuzzy setAOsAggregation operatorsDMDecision-makingCPFNsCubic Pythagorean fuzzy numbersFSFuzzy setIVFSsInterval-valued fuzzy setsIFSsIntuitionistic fuzzy setsIVIFSsInterval-valued intuitionistic fuzzy setsIFNsIntuitionistic fuzzy numbersPAPower averageBMBonferroni meanPFBMPythagorean fuzzy Bonferroni meanCSCubic setCIFSCubic intuitionistic fuzzy setCPFBMCubic Pythagorean fuzzy Bonferroni meanWCPFBMWeighted cubic Pythagorean fuzzy Bonferroni meanThe rest of the article is organized as follows. The basic concepts are briefly discussed in Section2. 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.


## 2. Preliminaries

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


### 2.1. Pythagorean Fuzzy Set

Definition 1 (see [5, 6]).
LetS be a universal set. A PFS p over S is defined as(1)P=x,ξPx,υPx|xεS,where 0≤ξPx,υPx≤1 and ξPx2+υPx2≤1. This couple is denoted as P=ξP,υP and is referred to as a PF number (PFN).Definition 2 (see [43]).
Letα1=ξP1,υP1, α2=ξP2,υP2 and α=ξP,υP be three PFNs and φ≥0 then(i)
α1⊕α2=ξP12+ξP22−ξP12ξP22,υP1υP2;(ii)
α1⊗α2=ξP1ξP2,υP12+υP22−υP12υP22;(iii)
φα=1−1−ξ2φ,υφ;(iv)
αφ=ξφ,1−1−υ2φ.


### 2.2. Interval-Valued Pythagorean Fuzzy Set

Definition 3 (see [7]).
LetS be a fixed set. An IVPFS Q over S is defined as(2)Q=x,ξQx,υQx|xεS,where ξQx=ξQ−x,ξQ+x, υQx=υQ−x,υQ+x such that ξQ+2+υQ+2≤1. For the sake of simplicity, we denote this pair as β=ξQ−,ξQ+,υQ−,υQ+ and called as IVPF number (IVPFN).Definition 4 (see [7]).
Letβ1=ξ1−,ξ1+,υ1−,υ1+, β2=ξ2−,ξ2+,υ2−,υ2+ and β=ξ−,ξ+,υ−,υ+ be three IVPFNs, then(i)
β1⊕β2=ξ1−2+ξ2−2−ξ1−2ξ2−2,ξ1+2+ξ2+2−ξ1+2ξ2+2,υ1−υ2−,υ1+υ2+;(ii)
β1⊗β2=ξ1−ξ2−,ξ1+ξ2+,υ1−2+υ2−2−υ1−2υ2−2,υ1+2+υ2+2−υ1+2υ2+2;(iii)
φβ=1−1−ξ−2φ,1−1−υ+2φ,υ−φ,υ+φ;(iv)
βφ=ξ−φ,ξ+φ,1−1−υ−2φ,1−1−υ+2φ.


### 2.3. Cubic Set

Definition 5 (see [33]).
LetS be a universal set. A CS over S is defined as(3)ℛ=x,ℛAx,λA|xεSwhere ℛAx=ℛ−x,ℛ+x is IVF set in X and ϑ is a FS. A CS ℛ is said to be an internal cubic set if ℛ−x≼x≼ℛ+x and a CS ℛ an external cubic set if ϑx∉ℛ−x,ℛ+x. A CS ℛ=x,ℛAx,λAx|x∈X is simply denoted by η=ℛ,λ.Definition 6.
(see [32]). Let η1=ℛ1,λ1 and η2=ℛ2,λ2 be CSs in X. Then,(i)
Equality. If ℛ1=ℛ2 and λ1=λ2 then η1=η2;(ii)
P-Order. If ℛ1⊆ℛ2 and λ1≼λ2 then η1⊆Pη2;(iii)
R-Order. If ℛ1⊆ℛ2 and λ1≽λ2 then η1⊆Rη2.Definition 7 (see [32]).
Letℛi=x,ℛAix,λAix|x∈S be a collection CSs where i∈Δ, then(i)
P-Union.Δi∈ΔPℛi=x,Δi∈ΔℛAix,Δi∈ΔλAix|x∈S;(ii)
P-Intersection. Δi∈ΔPℛi=x,Δi∈ΔℛAix,Δi∈ΔλAix|x∈S;(iii)
R-Union. Δi∈ΔPℛi=x,Δi∈ΔℛAix,Δi∈ΔλAix|x∈S;(iv)
R-Intersection.Δi∈ΔrRi=x,Δi∈ΔRAix,Δi∈ΔλAix|x∈S.


### 2.4. Cubic Intuitionistic Fuzzy Set

Definition 8 (see [40, 41]).
LetX be a non-empty set. A CIFS C¨ over x∈S is defined as follows:(4)B=x,Bx,λx|x∈S,where Bx=x,ξB−x,ξB+x,νB−x,νB+x|x∈S is IVIFS while λx=x,ξλx,νλx|x∈X represents IFS such that 0°ξB−x°ξB+x°1, 0≼νB−x≼νB+x≼1 and 0≺¯νB+x+νB+x≺¯1. Also, 0≺¯ξλx,νλx≺¯1 and 0≼ξλx+νλx≼1. In order to keep it simple, the pair ℬ=B,λ, where ξB−,ξB+,νB−,νB+ and ξλ,νλ and called as CIF number (CIFN).Definition 9 (see [41]).
For a family of CIFSℬi,i∈Δ, then(i)
P-Union. Ui∈ΔPℬi=maxi∈Δξi−,maxi∈Δξi+,mini∈Δνi−,mini∈Δνi+,maxi∈Δξi,mini∈Δνi.(ii)
P-Intersection. Ui∈ΔRℬi=mini∈Δξi−,mini∈Δξi+,maxi∈Δνi−,maxi∈Δνi+,mini∈Δξi,maxi∈Δνi.(iii)
R-Union. Ui∈ΔRℬi=maxi∈Δξi−,maxi∈Δξi+,mini∈Δνi−,mini∈Δνi+,mini∈Δξi,maxi∈Δνi.(iv)
R-Intersection. Ui∈ΔPℬi=maxi∈Δξi−,maxi∈Δξi+,mini∈Δνi−,mini∈Δνi+,mini∈Δξi,maxi∈Δνi.Definition 10 (see [40]).
Letℬ1=ξ1−,ξ1+,ν1−,ν1+,ξ1,ν1 and ℬ2=ξ2−,ξ2+,ν2−,ν2+,ξ2,ν2 be two CIFSs in S. Then,(i)
Equality. ℬ1=ℬ2, if and only if ξ1−,ξ1+=ξ2−,ξ2+, ν1−,ν1+=ν2−,ν2+, ξ1=ξ2 and ν1=ν2.(ii)
P-Order. B1⊆PB2 if ξ1−,ξ1+⊆ξ2−,ξ2+, ν1−,ν1+⊇ν2−,ν2+, ξ1≼ξ2 and ν1≽ν2.(iii)
R-Order. ℬ1⊆Rℬ2 if ξ1−,ξ1+⊆ξ2−,ξ2+, ν1−,ν1+⊇ν2−,ν2+, ξ1≽ξ2 and ν1≼ν2.Definition 11 (see [40]).
Letℬ=ξ−,ξ+,ν−,ν+,ξ,ν, ℬ1=ξi−,ξi+,νi−,νi+,ξi,νii=1,2 be the collections of CIFNs, and ψ≻0 be a real number, then(5)ℬ1⊕ℬ2=1−∏i=121−ξi−,1−∏i=121−ξi+,∏i=12νi−,∏i2νi+,∏i=12ξi,1−∏i=121−υi,,ℬ1⊗ℬ2=∏i=12ξi−,∏i2ξi+,1−∏i=121−νi−,1−∏i=121−νi+,1−∏i=121−ξi,∏i=12υi,φℬ=1−1−ξ−ψ,1−1−ξ+ψ,ν−ψ,ν+ψ,ξψ,1−1−νψℬψ=ξ−ψ,ξ+ψ,1−1−ν−ψ,1−1−ν+ψ,1−1−ξψ,νψ,


### 2.5. Cubic Pythagorean Fuzzy Set

Definition 12.
(see [39]). Let S be a non-empty set. A CPFS C over S is defined as follows:(6)C=x,Dx,ρx|x∈S,where Dx=x,ξD−x,ξD+x,νD−x,νD+x represents IVPFS while ρx=x,ξρx,νρx represents PFS for all x∈S such that 0≼ξD−x≼ξD+x≼1, 0≼νD−x≼νD+x≼1 and 0≼ξD+x2+νD+x2≼1. Also, 0≼ξρx,νρx≼1 and 0≼ξρx2+νρx2≼1. In order to keep it simple, the pair C=D,ρ, where ξD−,ξD+,νD−,νD+ and μρ,ξρ and called as CPF number (CPFN).Definition 13 (see [39]).
Letϑ=μϑ−,μϑ+,νϑ−,νϑ+,μϑ,νϑ be a CPFN then score sc and accuracy function ac is defined as follows:(7)scϑ=12μϑ−2+μϑ+2−νϑ−2−νϑ+22−μϑ2−νϑ2,where −1≼scβ≼1.(8)acβ=12μϑ−2+μϑ+2+νϑ−2+νϑ+22+μϑ2+νϑ2where 0≼scϑ≼1.Definition 14 (see [39]).
Letϑ=ξ−,ξ+,ν−,ν+,ξ,ν, ϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collections of CPFNs, and φ≻0 be a real number, then(9)ϑ1∪ϑ2=maxξϑ1−,ξϑ2−,maxξϑ1+,ξϑ2+,minνϑ1−,νϑ2−,minνϑ1+,νϑ2+;maxξϑ1,ξϑ2,minνϑ1,νϑ2ϑ1∩ϑ2=minξϑ1−,ξϑ2−,minξϑ1+,ξϑ2+,maxνϑ1−,νϑ2−,maxνϑ1+,νϑ2+;minξϑ1,ξϑ2,maxυϑ1,υϑ2δϑ=1−1−ξϑ−2φ,1−1−ξϑ+2φ,νϑ−φ,νϑ+φ;ξϑφ,1−1−νϑ2φϑδ=ξϑ−φ,ξϑ+φ,1−1−νϑ−2φ,1−1−νϑ+2φ;1−1−ξϑ2φ,νϑφϑ1⊕ϑ2=ξϑ1−2+ξϑ2−2−ξϑ1−2ξϑ2−2,ξϑ1+2+ξϑ2+2−ξϑ1+2ξϑ2+2,νϑ1−νϑ2−,νϑ1+νϑ2+;ξϑ1ξϑ2,νϑ12+νϑ22−νϑ12νϑ22ϑ1⊗ϑ2=ξϑ1−ξϑ2−,ξϑ1+ξϑ2+,νϑ1−2+νϑ2−2−νϑ1−2νϑ2−2,νϑ1+2+νϑ2+2−νϑ1+2νϑ2+2;ξϑ12+ξϑ22−ξϑ12ξϑ22,νϑ1νϑ2.


### 2.6. Bonferroni Mean Operators

Definition 15 (see [44]).
Leta,b≥0 and qii=1,2,…,n be a collection of non-negative numbers, then BM is defined as(10)BMa,bq1,q2,…,qn=1nn+1∑i,j=1i≠jnqiaqjb1/a+b.
In [43], Liang et al. generalized this BM for PF environments, giving the following concepts.Definition 16 (see [43]).
Letαi=ξi,υii=1,2,…,n be the collection of PFNs. For any a,b≥0, the PF Bonferroni mean (PFBM) is defined as(11)PFBMa,bα1,α2,…,αn=1nn+1⊕i,j=1i≠jnαia⊕αjb1/a+b.Definition 17 (see [43]).
Letαi=ξi,υii=1,2,…,n be the collection of PFNs and a,b≥0. Δ=Δ1,Δ2,…,ΔnT is the weight vector of αi such that Δiϵ0,1 and ∑i=1nΔi=1, then weighted PF Bonferroni mean (WPFBM) is defined as(12)WPFBMa,bα1,α2,…,αn=1nn+1⊕i,j=1i≠jnΔIαiaΔΔJαjb1/a+b


## 2.1. Pythagorean Fuzzy Set

Definition 1 (see [5, 6]).
LetS be a universal set. A PFS p over S is defined as(1)P=x,ξPx,υPx|xεS,where 0≤ξPx,υPx≤1 and ξPx2+υPx2≤1. This couple is denoted as P=ξP,υP and is referred to as a PF number (PFN).Definition 2 (see [43]).
Letα1=ξP1,υP1, α2=ξP2,υP2 and α=ξP,υP be three PFNs and φ≥0 then(i)
α1⊕α2=ξP12+ξP22−ξP12ξP22,υP1υP2;(ii)
α1⊗α2=ξP1ξP2,υP12+υP22−υP12υP22;(iii)
φα=1−1−ξ2φ,υφ;(iv)
αφ=ξφ,1−1−υ2φ.


## 2.2. Interval-Valued Pythagorean Fuzzy Set

Definition 3 (see [7]).
LetS be a fixed set. An IVPFS Q over S is defined as(2)Q=x,ξQx,υQx|xεS,where ξQx=ξQ−x,ξQ+x, υQx=υQ−x,υQ+x such that ξQ+2+υQ+2≤1. For the sake of simplicity, we denote this pair as β=ξQ−,ξQ+,υQ−,υQ+ and called as IVPF number (IVPFN).Definition 4 (see [7]).
Letβ1=ξ1−,ξ1+,υ1−,υ1+, β2=ξ2−,ξ2+,υ2−,υ2+ and β=ξ−,ξ+,υ−,υ+ be three IVPFNs, then(i)
β1⊕β2=ξ1−2+ξ2−2−ξ1−2ξ2−2,ξ1+2+ξ2+2−ξ1+2ξ2+2,υ1−υ2−,υ1+υ2+;(ii)
β1⊗β2=ξ1−ξ2−,ξ1+ξ2+,υ1−2+υ2−2−υ1−2υ2−2,υ1+2+υ2+2−υ1+2υ2+2;(iii)
φβ=1−1−ξ−2φ,1−1−υ+2φ,υ−φ,υ+φ;(iv)
βφ=ξ−φ,ξ+φ,1−1−υ−2φ,1−1−υ+2φ.


## 2.3. Cubic Set

Definition 5 (see [33]).
LetS be a universal set. A CS over S is defined as(3)ℛ=x,ℛAx,λA|xεSwhere ℛAx=ℛ−x,ℛ+x is IVF set in X and ϑ is a FS. A CS ℛ is said to be an internal cubic set if ℛ−x≼x≼ℛ+x and a CS ℛ an external cubic set if ϑx∉ℛ−x,ℛ+x. A CS ℛ=x,ℛAx,λAx|x∈X is simply denoted by η=ℛ,λ.Definition 6.
(see [32]). Let η1=ℛ1,λ1 and η2=ℛ2,λ2 be CSs in X. Then,(i)
Equality. If ℛ1=ℛ2 and λ1=λ2 then η1=η2;(ii)
P-Order. If ℛ1⊆ℛ2 and λ1≼λ2 then η1⊆Pη2;(iii)
R-Order. If ℛ1⊆ℛ2 and λ1≽λ2 then η1⊆Rη2.Definition 7 (see [32]).
Letℛi=x,ℛAix,λAix|x∈S be a collection CSs where i∈Δ, then(i)
P-Union.Δi∈ΔPℛi=x,Δi∈ΔℛAix,Δi∈ΔλAix|x∈S;(ii)
P-Intersection. Δi∈ΔPℛi=x,Δi∈ΔℛAix,Δi∈ΔλAix|x∈S;(iii)
R-Union. Δi∈ΔPℛi=x,Δi∈ΔℛAix,Δi∈ΔλAix|x∈S;(iv)
R-Intersection.Δi∈ΔrRi=x,Δi∈ΔRAix,Δi∈ΔλAix|x∈S.


## 2.4. Cubic Intuitionistic Fuzzy Set

Definition 8 (see [40, 41]).
LetX be a non-empty set. A CIFS C¨ over x∈S is defined as follows:(4)B=x,Bx,λx|x∈S,where Bx=x,ξB−x,ξB+x,νB−x,νB+x|x∈S is IVIFS while λx=x,ξλx,νλx|x∈X represents IFS such that 0°ξB−x°ξB+x°1, 0≼νB−x≼νB+x≼1 and 0≺¯νB+x+νB+x≺¯1. Also, 0≺¯ξλx,νλx≺¯1 and 0≼ξλx+νλx≼1. In order to keep it simple, the pair ℬ=B,λ, where ξB−,ξB+,νB−,νB+ and ξλ,νλ and called as CIF number (CIFN).Definition 9 (see [41]).
For a family of CIFSℬi,i∈Δ, then(i)
P-Union. Ui∈ΔPℬi=maxi∈Δξi−,maxi∈Δξi+,mini∈Δνi−,mini∈Δνi+,maxi∈Δξi,mini∈Δνi.(ii)
P-Intersection. Ui∈ΔRℬi=mini∈Δξi−,mini∈Δξi+,maxi∈Δνi−,maxi∈Δνi+,mini∈Δξi,maxi∈Δνi.(iii)
R-Union. Ui∈ΔRℬi=maxi∈Δξi−,maxi∈Δξi+,mini∈Δνi−,mini∈Δνi+,mini∈Δξi,maxi∈Δνi.(iv)
R-Intersection. Ui∈ΔPℬi=maxi∈Δξi−,maxi∈Δξi+,mini∈Δνi−,mini∈Δνi+,mini∈Δξi,maxi∈Δνi.Definition 10 (see [40]).
Letℬ1=ξ1−,ξ1+,ν1−,ν1+,ξ1,ν1 and ℬ2=ξ2−,ξ2+,ν2−,ν2+,ξ2,ν2 be two CIFSs in S. Then,(i)
Equality. ℬ1=ℬ2, if and only if ξ1−,ξ1+=ξ2−,ξ2+, ν1−,ν1+=ν2−,ν2+, ξ1=ξ2 and ν1=ν2.(ii)
P-Order. B1⊆PB2 if ξ1−,ξ1+⊆ξ2−,ξ2+, ν1−,ν1+⊇ν2−,ν2+, ξ1≼ξ2 and ν1≽ν2.(iii)
R-Order. ℬ1⊆Rℬ2 if ξ1−,ξ1+⊆ξ2−,ξ2+, ν1−,ν1+⊇ν2−,ν2+, ξ1≽ξ2 and ν1≼ν2.Definition 11 (see [40]).
Letℬ=ξ−,ξ+,ν−,ν+,ξ,ν, ℬ1=ξi−,ξi+,νi−,νi+,ξi,νii=1,2 be the collections of CIFNs, and ψ≻0 be a real number, then(5)ℬ1⊕ℬ2=1−∏i=121−ξi−,1−∏i=121−ξi+,∏i=12νi−,∏i2νi+,∏i=12ξi,1−∏i=121−υi,,ℬ1⊗ℬ2=∏i=12ξi−,∏i2ξi+,1−∏i=121−νi−,1−∏i=121−νi+,1−∏i=121−ξi,∏i=12υi,φℬ=1−1−ξ−ψ,1−1−ξ+ψ,ν−ψ,ν+ψ,ξψ,1−1−νψℬψ=ξ−ψ,ξ+ψ,1−1−ν−ψ,1−1−ν+ψ,1−1−ξψ,νψ,


## 2.5. Cubic Pythagorean Fuzzy Set

Definition 12.
(see [39]). Let S be a non-empty set. A CPFS C over S is defined as follows:(6)C=x,Dx,ρx|x∈S,where Dx=x,ξD−x,ξD+x,νD−x,νD+x represents IVPFS while ρx=x,ξρx,νρx represents PFS for all x∈S such that 0≼ξD−x≼ξD+x≼1, 0≼νD−x≼νD+x≼1 and 0≼ξD+x2+νD+x2≼1. Also, 0≼ξρx,νρx≼1 and 0≼ξρx2+νρx2≼1. In order to keep it simple, the pair C=D,ρ, where ξD−,ξD+,νD−,νD+ and μρ,ξρ and called as CPF number (CPFN).Definition 13 (see [39]).
Letϑ=μϑ−,μϑ+,νϑ−,νϑ+,μϑ,νϑ be a CPFN then score sc and accuracy function ac is defined as follows:(7)scϑ=12μϑ−2+μϑ+2−νϑ−2−νϑ+22−μϑ2−νϑ2,where −1≼scβ≼1.(8)acβ=12μϑ−2+μϑ+2+νϑ−2+νϑ+22+μϑ2+νϑ2where 0≼scϑ≼1.Definition 14 (see [39]).
Letϑ=ξ−,ξ+,ν−,ν+,ξ,ν, ϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collections of CPFNs, and φ≻0 be a real number, then(9)ϑ1∪ϑ2=maxξϑ1−,ξϑ2−,maxξϑ1+,ξϑ2+,minνϑ1−,νϑ2−,minνϑ1+,νϑ2+;maxξϑ1,ξϑ2,minνϑ1,νϑ2ϑ1∩ϑ2=minξϑ1−,ξϑ2−,minξϑ1+,ξϑ2+,maxνϑ1−,νϑ2−,maxνϑ1+,νϑ2+;minξϑ1,ξϑ2,maxυϑ1,υϑ2δϑ=1−1−ξϑ−2φ,1−1−ξϑ+2φ,νϑ−φ,νϑ+φ;ξϑφ,1−1−νϑ2φϑδ=ξϑ−φ,ξϑ+φ,1−1−νϑ−2φ,1−1−νϑ+2φ;1−1−ξϑ2φ,νϑφϑ1⊕ϑ2=ξϑ1−2+ξϑ2−2−ξϑ1−2ξϑ2−2,ξϑ1+2+ξϑ2+2−ξϑ1+2ξϑ2+2,νϑ1−νϑ2−,νϑ1+νϑ2+;ξϑ1ξϑ2,νϑ12+νϑ22−νϑ12νϑ22ϑ1⊗ϑ2=ξϑ1−ξϑ2−,ξϑ1+ξϑ2+,νϑ1−2+νϑ2−2−νϑ1−2νϑ2−2,νϑ1+2+νϑ2+2−νϑ1+2νϑ2+2;ξϑ12+ξϑ22−ξϑ12ξϑ22,νϑ1νϑ2.


## 2.6. Bonferroni Mean Operators

Definition 15 (see [44]).
Leta,b≥0 and qii=1,2,…,n be a collection of non-negative numbers, then BM is defined as(10)BMa,bq1,q2,…,qn=1nn+1∑i,j=1i≠jnqiaqjb1/a+b.
In [43], Liang et al. generalized this BM for PF environments, giving the following concepts.Definition 16 (see [43]).
Letαi=ξi,υii=1,2,…,n be the collection of PFNs. For any a,b≥0, the PF Bonferroni mean (PFBM) is defined as(11)PFBMa,bα1,α2,…,αn=1nn+1⊕i,j=1i≠jnαia⊕αjb1/a+b.Definition 17 (see [43]).
Letαi=ξi,υii=1,2,…,n be the collection of PFNs and a,b≥0. Δ=Δ1,Δ2,…,ΔnT is the weight vector of αi such that Δiϵ0,1 and ∑i=1nΔi=1, then weighted PF Bonferroni mean (WPFBM) is defined as(12)WPFBMa,bα1,α2,…,αn=1nn+1⊕i,j=1i≠jnΔIαiaΔΔJαjb1/a+b


## 3. CPF-Bonferroni Mean Aggregation Operators

In this section, a series of aggregation operators, namely, CPFBM and WCPFBM are presented.


### 3.1. CPF Bonferroni Mean Operators

Definition 18.
A CPF Bonferroni mean (CPFBM) operator is a mappingCPFBM:Φ⟶Φ defined on a set of CPFNs ϑii=1,2,…,n and is given by(13)CPFBMa,bϑ1,ϑ2,…,ϑn=1nn+1⊕i,j=1i≠jnϑia⊕ϑjb1/a+b,where ϑ1,ϑ2,…,ϑn are CPFNs, and a and b are positive real numbers.Example 1.
Letϑ1=0.4,0.5,0.5,0.7,0.4,0.7, and ϑ2=0.2,0.4,0.7,0.8,0.3,0.6 be two CPFNs. Then, use CPFBM to aggregate these three CPFNs. The steps are outlined below. (Supposes a=b=2 and n=3).
By using (11), we get(14)1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+11/a+b=1−1−0.42a×0.22b×1−0.22a×0.42b161/a+b=0.5231,(15)1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+11/a+b=1−1−0.52a×0.42b×1−0.52a×0.42b1/61/a+b=0.6314,(16)1−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b=1−1−1−1−0.52a1−0.7b×1−1−0.72a1−0.52b1/61/a+b=0.4789,(17)1−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+11/a+b=1−1−1−1−0.72a1−0.82b×1−1−0.82a1−0.72b1/61/a+b=0.6476,(18)1−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b=1−1−1−1−0.52a1−0.32b×1−1−0.32a1−0.52b1/61/a+b=4150,(19)1−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b=1−1−0.72a×0.62b×1−0.62a×0.72b161/a+b=0.6246.
Thus,(20)CPFBMa,bϑ1,ϑ2,ϑ3=1/22+1⊕i,j=1i≠j2ϑi1⊕ϑj11/2+2=0.5231,0.6314,0.4789,0.6476,0.4150,0.6246.Proposition 1.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collections of CPFNs and a,b≥0. For any i,j, and i≠j, we have(21)ϑia⊗ϑjb=ξϑi−aξϑj−b,ξϑi+bξϑj+b,1−1−νϑi−2a1−νϑj−2b,1−1−νϑi+2a1−νϑj+2b;1−1−ξϑi2a1−ξϑj2b,νϑiaνϑjb.Proof.
By Definition14, we have(22)ϑia=ξϑi−a,ξϑi+a,1−1−νϑi−2a,1−1−νϑi+2a;1−1−ξϑi2a,νϑiaand(23)ϑjb=ξϑj−b,ξϑj+b,1−1−νϑj−2b,1−1−νϑj+2b;1−1−ξϑj2b,νϑjb
Then,(24)ξϑi−a,ξϑi+a,1−1−νϑi−2a,1−1−νϑi+2a;1−1−ξϑi2a,νϑia⊗ξϑj−b,ξϑj+b,1−1−νϑj−2b,1−1−νϑj+2b;1−1−ξϑj2b,νϑjb=ξϑi−aξϑj−b,ξϑi+bξϑj+b,1−1−νϑi−2a1−νϑj−2b,1−1−νϑi+2a1−νϑj+2b;1−1−ξϑi2a1−ξϑj2b,νϑiaνϑjbProposition 2.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collections of CPFNs and a,b≥0. For any i,j and i≠j, we have(25)=ϑia⊗ϑjb⊕ϑia⊗ϑjb1−1−ξϑi−2aξϑi−2b1−ξϑj−2aξϑj−2b,1−1−ξϑi+2aξϑi+2b1−ξϑj+2aξϑj+2b,1−1−νϑi−2a1−νϑj−2b1−1−νϑj−2a1−νϑi−2b,1−1−νϑi+2a1−νϑj+2b1−1−νϑj+2a1−νϑi+2b,1−1−ξϑi2a1−ξϑj2b1−1−ξϑj2a1−ξϑi2b,1−1−νϑi2aνϑj2b1−νϑj2aνϑi2b.Proof.
We can get the following result from Proposition1:ϑia⊗ϑjb=ξϑi−aξϑj−b,ξϑi+bξϑj+b,1−1−νϑi−2a1−νϑj−2b,1−1−νϑi+2a1−νϑj+2b;1−1−ξϑi2a1−ξϑj2b,νϑiaνϑjb,ϑja⊗ϑib=ξϑj−aξϑi−b,ξϑj+bξϑi+b,1−1−νϑj−2a1−νϑi−2b,1−1−νϑj+2a1−νϑi+2b;1−1−ξϑj2a1−ξϑi2b,νϑjaνϑib
By Definition14, we have ϑia⊗ϑjb⊕ϑia⊗ϑjb(26)=ξϑi−aξϑj−b,ξϑi+bξϑj+b,1−1−νϑi−2a1−νϑj−2b,1−1−νϑi+2a1−νϑj+2b;1−1−ξϑi2a1−ξϑj2b,νϑiaνϑjb⊕ξϑj−aξϑi−b,ξϑj+bξϑi+b,1−1−νϑj−2a1−νϑi−2b,1−1−νϑj+2a1−νϑi+2b;1−1−ξϑj2a1−ξϑi2b,νϑjaνϑib=ξϑi−aξϑj−b2+ξϑj−aξϑi−b2−ξϑi−aξϑj−b2ξϑj−aξϑi−b2,ξϑi+aξϑj+b2+ξϑj+aξϑi+b2−ξϑi+aξϑj+b2ξϑj+aξϑi+b2,1−1−νϑi−2a1−νϑj−2b1−1−νϑj−2a1−νϑi−2b1−1−νϑi−2a1−νϑj−2b1−1−νϑj−2a1−νϑi−2b1−1−ξϑi2a1−ξϑj2b1−1−ξϑj2a1−ξϑi2bνϑiaνϑjb2+νϑjaνϑib2−νϑiaνϑjb2νϑjaνϑib2=1−1−ξϑi−2aξϑi−2b1−ξϑj−2aξϑj−2b,1−1−ξϑi+2aξϑi+2b1−ξϑj+2aξϑj+2b,1−1−νϑi−2a1−νϑj−2b1−1−νϑj−2a1−νϑi−2b,1−1−νϑi+2a1−νϑj+2b1−1−νϑj+2a1−νϑi+2b,1−1−ξϑi2a1−ξϑj2b1−1−ξϑj2a1−ξϑi2b,1−1−νϑi2aνϑj2b1−νϑj2aνϑi2b.
Thus, Proposition2 holds.Proposition 3.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collection of CPFNs and a,b≥0. We can calculate the following by using the value of k⊕i=1kϑia⊗ϑk+1b.(27)=1−∏i=1k1−ξϑi−2aξϑk+1−2b,1−∏i=1k1−ξϑi+2aξϑk+1+2b,∏i=1k1−1−νϑi−2a1−νϑk+1−2b,∏i=1k1−1−νϑi+2a1−νϑk+1+2b;∏i=1k1−1−ξϑi2a1−ξϑk+12b,1−∏i=1k1−νϑi2aνϑk+12b,where 1≤k≤n.Proof.
Mathematical induction onn.Step 1.
By adopting Proposition1, we obtain the following result when k = 2:(28)ϑia⊗ϑ2+1b=ξϑi−aξϑ2+1−b,ξϑi+bξϑ2+1+b,1−1−νϑi−2a1−νϑ2+1−2b,1−1−νϑi+2a1−νϑ2+1+2b;1−1−ξϑi2a1−ξϑ2+12b,νϑiaνϑ2+1b,
Thus, we have(29)⊕i=12ϑia⊗ϑ3b=ϑ1a⊗ϑ3b⊕ϑ1a⊗ϑ3b=ξϑ1−aξϑ3−b,ξϑ1+bξϑ3+b,1−1−νϑ1−2a1−νϑ3−2b,1−1−νϑi+2a1−νϑ3+2b;1−1−ξϑ12a1−ξϑ32b,νϑ1aνϑ3b⊗ξϑ1−aξϑ3−b,ξϑ1+bξϑ3+b,1−1−νϑ1−2a1−νϑ3−2b,1−1−νϑi+2a1−νϑ3+2b;1−1−ξϑ12a1−ξϑ32b,νϑ1aνϑ3b=1−1−ξϑ1−2aξϑ3−2b1−ξϑ1−2aξϑ3−2b,1−1−ξϑ1+2aξϑ3+2b1−ξϑ1+2aξϑ3+2b,1−1−νϑ1−2a1−νϑ3−2b1−1−νϑ1−2a1−νϑ3−2b,1−1−νϑ1+2a1−νϑ3+2b1−1−νϑ1+2a1−νϑ3+2b,1−1−ξϑ12a1−ξϑ32b1−1−ξϑ12a1−ξϑ32b,1−1−νϑ12aνϑ32b1−νϑ12aνϑ32b,Step 2.
We can obtain the following result. Eq. (14) holds for k=k0,⊕i=1k0ϑia⊗ϑk+1b(30)=1−∏i=1k01−ξϑi−2aξϑk0+1−2b,1−∏i=1k01−ξϑi+2aξϑk0+1+2b,∏i=1k01−1−νϑi−2a1−νϑk0+1−2b,∏i=1k01−1−νϑi+2a1−νϑk0+1+2b;∏i=1k01−1−ξϑi2a1−ξϑk0+12b,1−∏i=1k01−νϑi2aνϑk0+12b,Step 3.
Whenk=k0+1, then we have ,,⊕i=1k0+1ϑia⊗ϑk0+2b=⊕i=1k0ϑia⊗ϑk0+2b⊕ϑia⊗ϑk0+2b,(31)=1−∏i=1k01−ξϑi−2aξϑk0+1−2b,1−∏i=1k01−ξϑi+2aξϑk0+1+2b,∏i=1k01−1−νϑi−2a1−νϑk0+1−2b,∏i=1k01−1−νϑi+2a1−νϑk0+1+2b;∏i=1k01−1−ξϑi2a1−ξϑk0+12b,1−∏i=1k01−νϑi2aνϑk0+12b⊕ξϑk0+1−aξϑk0+2−b,ξϑk0+1+bξϑk0+2+b,1−1−νϑk0+1−2a1−νϑk0+2−2b,1−1−νϑk0+1+2a1−νϑk0+2+2b;1−1−ξϑk0+12a1−ξϑk0+22b,νϑk0+1aνϑk0+2b=1−∏i=1k0+11−ξϑi−2aξϑk0+2−2b,1−∏i=1k0+11−ξϑi+2aξϑk0+2+2b,∏i=1k0+11−1−νϑi−2a1−νϑk0+2−2b,∏i=1k0+11−1−νϑi+2a1−νϑk0+2+2b;∏i=1k0+11−1−ξϑi2a1−ξϑk0+22b,1−∏i=1k0+11−νϑi2aνϑk0+22b,
Hence, (31) is also true for k=k0+1. As a result, Proposition 3 holds true.Proposition 4.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2 be the collections of CPFNs and a,b≥0. Given the value of k, we have ⊕j=1kϑk+1a⊗ϑib.(32)=1−∏j=1k1−ξϑk+1−2aξϑj−2b,1−∏j=1k1−ξϑk+1+2aξϑj+2b,∏j=1k1−1−νϑk+1−2a1−νϑj−2b,∏j=1k1−1−νϑk+1+2a1−νϑj+2b;∏j=1k1−1−ξϑk+12a1−ξϑj2b,1−∏j=1k1−νϑk+12aνϑj2b.Proof.
The proof is similar to that of proportion 3, so it is omitted.Proposition 5.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collection of CPFNs and a,b≥0. For any i,j and i≠j, we have ⊕i,j=1i≠jnϑia⊕ϑjb.(33)=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b,1−∏i,j=1i≠jn1−νϑi2aνϑj2b,Proof.
The proof is similar to that of proportion 3, so it is omitted.Proposition 6.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collection of CPFNs and a,b≥0. For any i,j and i≠j, we have 1/nn+1⊕i,j=1i≠jnϑia⊕ϑjb.(34)=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+1,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+1,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+1,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+1;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+1,1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+1Proof.
Based on Eq. (29), we have(35)⊕i,j=1i≠jnϑia⊕ϑjb=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b,1−∏i,j=1i≠jn1−νϑi2aνϑj2b,
By using Definition14, we have 1/nn+1⊕i,j=1i≠jnϑia⊕ϑjb=1/nn+11−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b,1−∏i,j=1i≠jn1−νϑi2aνϑj2b=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+1,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+1,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+1,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+1;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+1,1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+1.Theorem 1.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collection of CPFNs and a,b≥0. By using the CPFBM operator, the aggregated value is also a CPFN, and(36)CPFBMa,bϑ1,ϑ2,…,ϑn=1nn+1⊕i,j=1i≠jnϑia⊕ϑjb1/a+b=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+11/a+b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+11/a+b;1−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b,Proof.
By Proposition6 and Definition 14, we have(37)CPFBMa,bϑ1,ϑ2,…,ϑn=1nn+1⊕i,j=1i≠jnϑia⊕ϑjb1/a+b,=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+1,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+1,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+1,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+1;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+1,1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+1=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+11/a+b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+11/a+b;1−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b,
As a result, Theorem1 holds true.Example 2.
Letϑ1=0.4,0.5,0.5,0.7,0.4,0.7, ϑ2=0.2,0.4,0.7,0.8,0.3,0.6 and ϑ1=0.5,0.6,0.4,0.5,0.3,0.2 be three CPFNs. Then, use CPFBM to aggregate these three CPFNs. The steps are outlined below. (Supposes a=b=1 and n=3).
By using (21), we get(38)1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+11/a+b=1−1−0.42a×0.22b×1−0.22a×0.42b×1−0.22a×0.52b×1−0.52a×0.22b×1−0.42a×0.52b×1−0.52a×0.42b1/121/a+b=0.37091−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+11/a+b=1−1−0.52a×0.42b×1−0.52a×0.42b×1−0.42a×0.62b×1−0.62a×0.42b×1−0.52a×0.62b×1−0.62a×0.52b1/121/a+b=0.50051−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b=1−1−1−1−0.52a1−0.7b×1−1−0.72a1−0.52b×1−1−0.72a1−0.42b×1−1−0.42a1−0.72b×1−1−0.52a1−0.42b×1−1−0.42a1−0.52b1/121/a+b=0.54601−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+11/a+b=1−1−1−1−0.72a1−0.82b×1−1−0.82a1−0.72b×1−1−0.82a1−0.52b×1−1−0.52a1−0.82b×1−1−0.72a1−0.52b×1−1−0.52a1−0.72b1/121/a+b=0.68391−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b=1−1−1−1−0.52a1−0.32b×1−1−0.32a1−0.52b×1−1−0.32a1−0.32b×1−1−0.32a1−0.32b×1−1−0.52a1−0.32b×1−1−0.32a1−0.52b1/121/a+b=33521−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b=1−1−0.72a×0.62b×1−0.62a×0.72b×1−0.62a×0.22b×1−0.22a×0.62b×1−0.72a×0.22b×1−0.22a×0.72b1/121/a+b=0.5203=1−1−0.52a×0.42b×1−0.52a×0.42b×1−0.42a×0.62b×1−0.62a×0.42b×1−0.52a×0.62b×1−0.62a×0.52b1/121/a+b(39)1−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b=1−1−1−1−0.52a1−0.7b×1−1−0.72a1−0.52b×1−1−0.72a1−0.42b×1−1−0.42a1−0.72b×1−1−0.52a1−0.42b×1−1−0.42a1−0.52b1/121/a+b=0.5460,(40)1−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/121/a+b=1−1−1−1−0.72a1−0.82b×1−1−0.82a1−0.72b×1−1−0.82a1−0.52b×1−1−0.52a1−0.82b×1−1−0.72a1−0.52b×1−1−0.52a1−0.72b1/121/a+b=0.6839(41)1−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b=1−1−1−1−0.52a1−0.32b×1−1−0.32a1−0.52b×1−1−0.32a1−0.32b×1−1−0.32a1−0.32b×1−1−0.52a1−0.32b×1−1−0.32a1−0.52b1/121/a+b=3352(42)1−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b,
Thus,(43)CPFBMa,bϑ1,ϑ2,ϑ3=1/33+1⊕i,j=1i≠j3ϑi1⊕ϑj11/1+1=0.3709,0.5005,0.5460,0.6839,0.3352,0.5203.
Based on the results of (see [18]), we may conclude the following corollaries for Theorem 1.Corollary 1 (idempotency).
If any PCFNsϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2,…,n are equal, i.e., ϑi=ϑ=ξ−,ξ+,ν−,ν+,ξ,ν then CPFBMa,bϑ1,ϑ2,…,ϑn=ϑ=ξ−,ξ+,ν−,ν+,ξ,ν.Corollary 2 (commutativity).
Ifβi′ is any permutation of βii1,2,…,n then we have the following relationship CPFBMa,bβ1′,β2′,…,βn′=CPFBMa,bβ1,β2,…,βn.Corollary 3 (monotonicity).
Letϑi=ξϑi−,ξϑi+,νϑi−,νϑi+,ξϑi,νϑi and ϑ˜i=ξϑ˜i−,ξϑ˜i+,νϑ˜i−,νϑ˜i+,ξϑ˜i,νϑ˜i be the collections of CPFNs. Assisted by CPFBM operator, we assume CPFBMa,bϑ1,ϑ2,…,ϑn=ξϑ−,ξϑ+,νϑ−,νϑ+,ξϑ,νϑ and CPFBMa,bϑ˜1,ϑ˜2,…,ϑ˜n=ξϑ˜−,ξϑ˜+,νϑ˜−,νϑ˜+,ξϑ˜,νϑ˜. If ξϑi−≤ξϑ˜i−, ξϑi+≤ξϑ˜i+, νϑi−≥νϑ˜i−, νϑi+≥νϑ˜i+, ξϑi≥ξϑ˜i and νϑi≤νϑ˜i holds, then ξϑ−≤ξϑ˜−, ξϑ+≤ξϑ˜+, νϑ−≥νϑ˜−, νϑ+≥νϑ˜+, ξϑ≥ξϑ˜ and νϑ≤νϑ˜.


### 3.2. Weighted CFP Bonferroni Mean

The CPFBM described in Theorem1 does not take the significance of the aggregated arguments into account. The 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 defined as follows.Definition 19.
Letϑi=ξϑi−,ξϑi+,νϑi−,νϑi+,ξϑi,νϑii=1,2,…,n be the collection of CPFNs and a,b are any non-negative real numbers. Δ=Δ1,Δ2,…,ΔnT is the weight vector of ϑi such that Δi∈0,1 and ∑i=1nΔi=1. Then, weighted CPF Bonferroni mean (WCPFBM) operator is defined as(44)WCPFBMa,bϑ1,ϑ2,…,ϑn=1n+1⊕i,j=1i≠jnΔiϑia⊗Δjϑjb1/a+b,Theorem 2.
Letϑi=ξϑi−,ξϑi+,νϑi−,νϑi+,ξϑi,νϑii=1,2,…,n be the collection of CPFNs and a,b are any non-negative real numbers. Δ=Δ1,Δ2,…,ΔnT is the weight vector of ϑi such that Δi∈0,1 and ∑i=1nΔi=1. Then, the aggregated values obtained by WCPFBM are also a CPFN, and (45)WCPFBMa,bϑ1,ϑ2,…,ϑn=1−∏i,j=1i≠jn1−1−1−ξϑi−2ωia1−1−1−ξϑj−2ωjb1/nn+11/a+b,1−∏i,j=1i≠jn1−1−1−ξϑi+2ωia1−1−1−ξϑj+2ωjb1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi−2ωia1−νϑj−2ωib1nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi+2ωia1−νϑj+2ωib1/nn+11/a+b;1−1−∏i,j=1i≠jn1−1−ξϑi2ωia1−ξϑj2bωi1/nn+11/a+b,1−∏i,j=1i≠jn1−1−1−vϑi−2ωia1−1−1−vϑj−2ωjb1/nn+1.Proof.
We omitted this proof since it is similar to Theorem2.
The outcomes of the CPFBM and WCPFBM operators are CPFNs according to Theorems1 and 2. WCPFBM increases the significance of the aggregated arguments in addition to CPFBM. We present a technique for applying the CPFBM and WCPFBM operator to MCDM with the help of the abovementioned conclusions. The new approach is designed as follows.


## 3.1. CPF Bonferroni Mean Operators

Definition 18.
A CPF Bonferroni mean (CPFBM) operator is a mappingCPFBM:Φ⟶Φ defined on a set of CPFNs ϑii=1,2,…,n and is given by(13)CPFBMa,bϑ1,ϑ2,…,ϑn=1nn+1⊕i,j=1i≠jnϑia⊕ϑjb1/a+b,where ϑ1,ϑ2,…,ϑn are CPFNs, and a and b are positive real numbers.Example 1.
Letϑ1=0.4,0.5,0.5,0.7,0.4,0.7, and ϑ2=0.2,0.4,0.7,0.8,0.3,0.6 be two CPFNs. Then, use CPFBM to aggregate these three CPFNs. The steps are outlined below. (Supposes a=b=2 and n=3).
By using (11), we get(14)1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+11/a+b=1−1−0.42a×0.22b×1−0.22a×0.42b161/a+b=0.5231,(15)1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+11/a+b=1−1−0.52a×0.42b×1−0.52a×0.42b1/61/a+b=0.6314,(16)1−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b=1−1−1−1−0.52a1−0.7b×1−1−0.72a1−0.52b1/61/a+b=0.4789,(17)1−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+11/a+b=1−1−1−1−0.72a1−0.82b×1−1−0.82a1−0.72b1/61/a+b=0.6476,(18)1−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b=1−1−1−1−0.52a1−0.32b×1−1−0.32a1−0.52b1/61/a+b=4150,(19)1−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b=1−1−0.72a×0.62b×1−0.62a×0.72b161/a+b=0.6246.
Thus,(20)CPFBMa,bϑ1,ϑ2,ϑ3=1/22+1⊕i,j=1i≠j2ϑi1⊕ϑj11/2+2=0.5231,0.6314,0.4789,0.6476,0.4150,0.6246.Proposition 1.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collections of CPFNs and a,b≥0. For any i,j, and i≠j, we have(21)ϑia⊗ϑjb=ξϑi−aξϑj−b,ξϑi+bξϑj+b,1−1−νϑi−2a1−νϑj−2b,1−1−νϑi+2a1−νϑj+2b;1−1−ξϑi2a1−ξϑj2b,νϑiaνϑjb.Proof.
By Definition14, we have(22)ϑia=ξϑi−a,ξϑi+a,1−1−νϑi−2a,1−1−νϑi+2a;1−1−ξϑi2a,νϑiaand(23)ϑjb=ξϑj−b,ξϑj+b,1−1−νϑj−2b,1−1−νϑj+2b;1−1−ξϑj2b,νϑjb
Then,(24)ξϑi−a,ξϑi+a,1−1−νϑi−2a,1−1−νϑi+2a;1−1−ξϑi2a,νϑia⊗ξϑj−b,ξϑj+b,1−1−νϑj−2b,1−1−νϑj+2b;1−1−ξϑj2b,νϑjb=ξϑi−aξϑj−b,ξϑi+bξϑj+b,1−1−νϑi−2a1−νϑj−2b,1−1−νϑi+2a1−νϑj+2b;1−1−ξϑi2a1−ξϑj2b,νϑiaνϑjbProposition 2.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collections of CPFNs and a,b≥0. For any i,j and i≠j, we have(25)=ϑia⊗ϑjb⊕ϑia⊗ϑjb1−1−ξϑi−2aξϑi−2b1−ξϑj−2aξϑj−2b,1−1−ξϑi+2aξϑi+2b1−ξϑj+2aξϑj+2b,1−1−νϑi−2a1−νϑj−2b1−1−νϑj−2a1−νϑi−2b,1−1−νϑi+2a1−νϑj+2b1−1−νϑj+2a1−νϑi+2b,1−1−ξϑi2a1−ξϑj2b1−1−ξϑj2a1−ξϑi2b,1−1−νϑi2aνϑj2b1−νϑj2aνϑi2b.Proof.
We can get the following result from Proposition1:ϑia⊗ϑjb=ξϑi−aξϑj−b,ξϑi+bξϑj+b,1−1−νϑi−2a1−νϑj−2b,1−1−νϑi+2a1−νϑj+2b;1−1−ξϑi2a1−ξϑj2b,νϑiaνϑjb,ϑja⊗ϑib=ξϑj−aξϑi−b,ξϑj+bξϑi+b,1−1−νϑj−2a1−νϑi−2b,1−1−νϑj+2a1−νϑi+2b;1−1−ξϑj2a1−ξϑi2b,νϑjaνϑib
By Definition14, we have ϑia⊗ϑjb⊕ϑia⊗ϑjb(26)=ξϑi−aξϑj−b,ξϑi+bξϑj+b,1−1−νϑi−2a1−νϑj−2b,1−1−νϑi+2a1−νϑj+2b;1−1−ξϑi2a1−ξϑj2b,νϑiaνϑjb⊕ξϑj−aξϑi−b,ξϑj+bξϑi+b,1−1−νϑj−2a1−νϑi−2b,1−1−νϑj+2a1−νϑi+2b;1−1−ξϑj2a1−ξϑi2b,νϑjaνϑib=ξϑi−aξϑj−b2+ξϑj−aξϑi−b2−ξϑi−aξϑj−b2ξϑj−aξϑi−b2,ξϑi+aξϑj+b2+ξϑj+aξϑi+b2−ξϑi+aξϑj+b2ξϑj+aξϑi+b2,1−1−νϑi−2a1−νϑj−2b1−1−νϑj−2a1−νϑi−2b1−1−νϑi−2a1−νϑj−2b1−1−νϑj−2a1−νϑi−2b1−1−ξϑi2a1−ξϑj2b1−1−ξϑj2a1−ξϑi2bνϑiaνϑjb2+νϑjaνϑib2−νϑiaνϑjb2νϑjaνϑib2=1−1−ξϑi−2aξϑi−2b1−ξϑj−2aξϑj−2b,1−1−ξϑi+2aξϑi+2b1−ξϑj+2aξϑj+2b,1−1−νϑi−2a1−νϑj−2b1−1−νϑj−2a1−νϑi−2b,1−1−νϑi+2a1−νϑj+2b1−1−νϑj+2a1−νϑi+2b,1−1−ξϑi2a1−ξϑj2b1−1−ξϑj2a1−ξϑi2b,1−1−νϑi2aνϑj2b1−νϑj2aνϑi2b.
Thus, Proposition2 holds.Proposition 3.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collection of CPFNs and a,b≥0. We can calculate the following by using the value of k⊕i=1kϑia⊗ϑk+1b.(27)=1−∏i=1k1−ξϑi−2aξϑk+1−2b,1−∏i=1k1−ξϑi+2aξϑk+1+2b,∏i=1k1−1−νϑi−2a1−νϑk+1−2b,∏i=1k1−1−νϑi+2a1−νϑk+1+2b;∏i=1k1−1−ξϑi2a1−ξϑk+12b,1−∏i=1k1−νϑi2aνϑk+12b,where 1≤k≤n.Proof.
Mathematical induction onn.Step 1.
By adopting Proposition1, we obtain the following result when k = 2:(28)ϑia⊗ϑ2+1b=ξϑi−aξϑ2+1−b,ξϑi+bξϑ2+1+b,1−1−νϑi−2a1−νϑ2+1−2b,1−1−νϑi+2a1−νϑ2+1+2b;1−1−ξϑi2a1−ξϑ2+12b,νϑiaνϑ2+1b,
Thus, we have(29)⊕i=12ϑia⊗ϑ3b=ϑ1a⊗ϑ3b⊕ϑ1a⊗ϑ3b=ξϑ1−aξϑ3−b,ξϑ1+bξϑ3+b,1−1−νϑ1−2a1−νϑ3−2b,1−1−νϑi+2a1−νϑ3+2b;1−1−ξϑ12a1−ξϑ32b,νϑ1aνϑ3b⊗ξϑ1−aξϑ3−b,ξϑ1+bξϑ3+b,1−1−νϑ1−2a1−νϑ3−2b,1−1−νϑi+2a1−νϑ3+2b;1−1−ξϑ12a1−ξϑ32b,νϑ1aνϑ3b=1−1−ξϑ1−2aξϑ3−2b1−ξϑ1−2aξϑ3−2b,1−1−ξϑ1+2aξϑ3+2b1−ξϑ1+2aξϑ3+2b,1−1−νϑ1−2a1−νϑ3−2b1−1−νϑ1−2a1−νϑ3−2b,1−1−νϑ1+2a1−νϑ3+2b1−1−νϑ1+2a1−νϑ3+2b,1−1−ξϑ12a1−ξϑ32b1−1−ξϑ12a1−ξϑ32b,1−1−νϑ12aνϑ32b1−νϑ12aνϑ32b,Step 2.
We can obtain the following result. Eq. (14) holds for k=k0,⊕i=1k0ϑia⊗ϑk+1b(30)=1−∏i=1k01−ξϑi−2aξϑk0+1−2b,1−∏i=1k01−ξϑi+2aξϑk0+1+2b,∏i=1k01−1−νϑi−2a1−νϑk0+1−2b,∏i=1k01−1−νϑi+2a1−νϑk0+1+2b;∏i=1k01−1−ξϑi2a1−ξϑk0+12b,1−∏i=1k01−νϑi2aνϑk0+12b,Step 3.
Whenk=k0+1, then we have ,,⊕i=1k0+1ϑia⊗ϑk0+2b=⊕i=1k0ϑia⊗ϑk0+2b⊕ϑia⊗ϑk0+2b,(31)=1−∏i=1k01−ξϑi−2aξϑk0+1−2b,1−∏i=1k01−ξϑi+2aξϑk0+1+2b,∏i=1k01−1−νϑi−2a1−νϑk0+1−2b,∏i=1k01−1−νϑi+2a1−νϑk0+1+2b;∏i=1k01−1−ξϑi2a1−ξϑk0+12b,1−∏i=1k01−νϑi2aνϑk0+12b⊕ξϑk0+1−aξϑk0+2−b,ξϑk0+1+bξϑk0+2+b,1−1−νϑk0+1−2a1−νϑk0+2−2b,1−1−νϑk0+1+2a1−νϑk0+2+2b;1−1−ξϑk0+12a1−ξϑk0+22b,νϑk0+1aνϑk0+2b=1−∏i=1k0+11−ξϑi−2aξϑk0+2−2b,1−∏i=1k0+11−ξϑi+2aξϑk0+2+2b,∏i=1k0+11−1−νϑi−2a1−νϑk0+2−2b,∏i=1k0+11−1−νϑi+2a1−νϑk0+2+2b;∏i=1k0+11−1−ξϑi2a1−ξϑk0+22b,1−∏i=1k0+11−νϑi2aνϑk0+22b,
Hence, (31) is also true for k=k0+1. As a result, Proposition 3 holds true.Proposition 4.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2 be the collections of CPFNs and a,b≥0. Given the value of k, we have ⊕j=1kϑk+1a⊗ϑib.(32)=1−∏j=1k1−ξϑk+1−2aξϑj−2b,1−∏j=1k1−ξϑk+1+2aξϑj+2b,∏j=1k1−1−νϑk+1−2a1−νϑj−2b,∏j=1k1−1−νϑk+1+2a1−νϑj+2b;∏j=1k1−1−ξϑk+12a1−ξϑj2b,1−∏j=1k1−νϑk+12aνϑj2b.Proof.
The proof is similar to that of proportion 3, so it is omitted.Proposition 5.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collection of CPFNs and a,b≥0. For any i,j and i≠j, we have ⊕i,j=1i≠jnϑia⊕ϑjb.(33)=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b,1−∏i,j=1i≠jn1−νϑi2aνϑj2b,Proof.
The proof is similar to that of proportion 3, so it is omitted.Proposition 6.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collection of CPFNs and a,b≥0. For any i,j and i≠j, we have 1/nn+1⊕i,j=1i≠jnϑia⊕ϑjb.(34)=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+1,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+1,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+1,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+1;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+1,1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+1Proof.
Based on Eq. (29), we have(35)⊕i,j=1i≠jnϑia⊕ϑjb=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b,1−∏i,j=1i≠jn1−νϑi2aνϑj2b,
By using Definition14, we have 1/nn+1⊕i,j=1i≠jnϑia⊕ϑjb=1/nn+11−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b,1−∏i,j=1i≠jn1−νϑi2aνϑj2b=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+1,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+1,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+1,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+1;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+1,1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+1.Theorem 1.
Letϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2) be the collection of CPFNs and a,b≥0. By using the CPFBM operator, the aggregated value is also a CPFN, and(36)CPFBMa,bϑ1,ϑ2,…,ϑn=1nn+1⊕i,j=1i≠jnϑia⊕ϑjb1/a+b=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+11/a+b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+11/a+b;1−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b,Proof.
By Proposition6 and Definition 14, we have(37)CPFBMa,bϑ1,ϑ2,…,ϑn=1nn+1⊕i,j=1i≠jnϑia⊕ϑjb1/a+b,=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+1,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+1,∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+1,∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+1;∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+1,1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+1=1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+11/a+b,1−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+11/a+b;1−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b,1−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b,
As a result, Theorem1 holds true.Example 2.
Letϑ1=0.4,0.5,0.5,0.7,0.4,0.7, ϑ2=0.2,0.4,0.7,0.8,0.3,0.6 and ϑ1=0.5,0.6,0.4,0.5,0.3,0.2 be three CPFNs. Then, use CPFBM to aggregate these three CPFNs. The steps are outlined below. (Supposes a=b=1 and n=3).
By using (21), we get(38)1−∏i,j=1i≠jn1−ξϑi−2aξϑj−2b1/nn+11/a+b=1−1−0.42a×0.22b×1−0.22a×0.42b×1−0.22a×0.52b×1−0.52a×0.22b×1−0.42a×0.52b×1−0.52a×0.42b1/121/a+b=0.37091−∏i,j=1i≠jn1−ξϑi+2aξϑj+2b1/nn+11/a+b=1−1−0.52a×0.42b×1−0.52a×0.42b×1−0.42a×0.62b×1−0.62a×0.42b×1−0.52a×0.62b×1−0.62a×0.52b1/121/a+b=0.50051−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b=1−1−1−1−0.52a1−0.7b×1−1−0.72a1−0.52b×1−1−0.72a1−0.42b×1−1−0.42a1−0.72b×1−1−0.52a1−0.42b×1−1−0.42a1−0.52b1/121/a+b=0.54601−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/nn+11/a+b=1−1−1−1−0.72a1−0.82b×1−1−0.82a1−0.72b×1−1−0.82a1−0.52b×1−1−0.52a1−0.82b×1−1−0.72a1−0.52b×1−1−0.52a1−0.72b1/121/a+b=0.68391−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b=1−1−1−1−0.52a1−0.32b×1−1−0.32a1−0.52b×1−1−0.32a1−0.32b×1−1−0.32a1−0.32b×1−1−0.52a1−0.32b×1−1−0.32a1−0.52b1/121/a+b=33521−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b=1−1−0.72a×0.62b×1−0.62a×0.72b×1−0.62a×0.22b×1−0.22a×0.62b×1−0.72a×0.22b×1−0.22a×0.72b1/121/a+b=0.5203=1−1−0.52a×0.42b×1−0.52a×0.42b×1−0.42a×0.62b×1−0.62a×0.42b×1−0.52a×0.62b×1−0.62a×0.52b1/121/a+b(39)1−1−∏i,j=1i≠jn1−1−νϑi−2a1−νϑj−2b1/nn+11/a+b=1−1−1−1−0.52a1−0.7b×1−1−0.72a1−0.52b×1−1−0.72a1−0.42b×1−1−0.42a1−0.72b×1−1−0.52a1−0.42b×1−1−0.42a1−0.52b1/121/a+b=0.5460,(40)1−1−∏i,j=1i≠jn1−1−νϑi+2a1−νϑj+2b1/121/a+b=1−1−1−1−0.72a1−0.82b×1−1−0.82a1−0.72b×1−1−0.82a1−0.52b×1−1−0.52a1−0.82b×1−1−0.72a1−0.52b×1−1−0.52a1−0.72b1/121/a+b=0.6839(41)1−1−∏i,j=1i≠jn1−1−ξϑi2a1−ξϑj2b1/nn+11/a+b=1−1−1−1−0.52a1−0.32b×1−1−0.32a1−0.52b×1−1−0.32a1−0.32b×1−1−0.32a1−0.32b×1−1−0.52a1−0.32b×1−1−0.32a1−0.52b1/121/a+b=3352(42)1−1−∏i,j=1i≠jn1−νϑi2aνϑj2b1/nn+11/a+b,
Thus,(43)CPFBMa,bϑ1,ϑ2,ϑ3=1/33+1⊕i,j=1i≠j3ϑi1⊕ϑj11/1+1=0.3709,0.5005,0.5460,0.6839,0.3352,0.5203.
Based on the results of (see [18]), we may conclude the following corollaries for Theorem 1.Corollary 1 (idempotency).
If any PCFNsϑi=ξi−,ξi+,νi−,νi+,ξi,νii=1,2,…,n are equal, i.e., ϑi=ϑ=ξ−,ξ+,ν−,ν+,ξ,ν then CPFBMa,bϑ1,ϑ2,…,ϑn=ϑ=ξ−,ξ+,ν−,ν+,ξ,ν.Corollary 2 (commutativity).
Ifβi′ is any permutation of βii1,2,…,n then we have the following relationship CPFBMa,bβ1′,β2′,…,βn′=CPFBMa,bβ1,β2,…,βn.Corollary 3 (monotonicity).
Letϑi=ξϑi−,ξϑi+,νϑi−,νϑi+,ξϑi,νϑi and ϑ˜i=ξϑ˜i−,ξϑ˜i+,νϑ˜i−,νϑ˜i+,ξϑ˜i,νϑ˜i be the collections of CPFNs. Assisted by CPFBM operator, we assume CPFBMa,bϑ1,ϑ2,…,ϑn=ξϑ−,ξϑ+,νϑ−,νϑ+,ξϑ,νϑ and CPFBMa,bϑ˜1,ϑ˜2,…,ϑ˜n=ξϑ˜−,ξϑ˜+,νϑ˜−,νϑ˜+,ξϑ˜,νϑ˜. If ξϑi−≤ξϑ˜i−, ξϑi+≤ξϑ˜i+, νϑi−≥νϑ˜i−, νϑi+≥νϑ˜i+, ξϑi≥ξϑ˜i and νϑi≤νϑ˜i holds, then ξϑ−≤ξϑ˜−, ξϑ+≤ξϑ˜+, νϑ−≥νϑ˜−, νϑ+≥νϑ˜+, ξϑ≥ξϑ˜ and νϑ≤νϑ˜.


## 3.2. Weighted CFP Bonferroni Mean

The CPFBM described in Theorem1 does not take the significance of the aggregated arguments into account. The 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 defined as follows.Definition 19.
Letϑi=ξϑi−,ξϑi+,νϑi−,νϑi+,ξϑi,νϑii=1,2,…,n be the collection of CPFNs and a,b are any non-negative real numbers. Δ=Δ1,Δ2,…,ΔnT is the weight vector of ϑi such that Δi∈0,1 and ∑i=1nΔi=1. Then, weighted CPF Bonferroni mean (WCPFBM) operator is defined as(44)WCPFBMa,bϑ1,ϑ2,…,ϑn=1n+1⊕i,j=1i≠jnΔiϑia⊗Δjϑjb1/a+b,Theorem 2.
Letϑi=ξϑi−,ξϑi+,νϑi−,νϑi+,ξϑi,νϑii=1,2,…,n be the collection of CPFNs and a,b are any non-negative real numbers. Δ=Δ1,Δ2,…,ΔnT is the weight vector of ϑi such that Δi∈0,1 and ∑i=1nΔi=1. Then, the aggregated values obtained by WCPFBM are also a CPFN, and (45)WCPFBMa,bϑ1,ϑ2,…,ϑn=1−∏i,j=1i≠jn1−1−1−ξϑi−2ωia1−1−1−ξϑj−2ωjb1/nn+11/a+b,1−∏i,j=1i≠jn1−1−1−ξϑi+2ωia1−1−1−ξϑj+2ωjb1/nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi−2ωia1−νϑj−2ωib1nn+11/a+b,1−1−∏i,j=1i≠jn1−1−νϑi+2ωia1−νϑj+2ωib1/nn+11/a+b;1−1−∏i,j=1i≠jn1−1−ξϑi2ωia1−ξϑj2bωi1/nn+11/a+b,1−∏i,j=1i≠jn1−1−1−vϑi−2ωia1−1−1−vϑj−2ωjb1/nn+1.Proof.
We omitted this proof since it is similar to Theorem2.
The outcomes of the CPFBM and WCPFBM operators are CPFNs according to Theorems1 and 2. WCPFBM increases the significance of the aggregated arguments in addition to CPFBM. We present a technique for applying the CPFBM and WCPFBM operator to MCDM with the help of the abovementioned conclusions. The new approach is designed as follows.


## 4. DM Approach Based on Proposed CPF Bonferroni Mean Operators

A new technique of processing MCDM is developed based on the abovementioned discussion, which featured the notion of MB operators in a CPF environment, as well as its core algorithms and the CPF-matrix collection approach.Step 4.
In this step, the finite set of alternativesAii=1,2,…,m is determined. We select the set of criteria ℂjj=1,2,…,n defending on the real DM problem and produce the weight vector Δi such that Δi∈0,1 and ∑i=1nΔi=1. The CPF decision matrix is then formed as follows: (46)D=ϑijm×n=ξij−,ξij+,νij−,νij+,ξij,νijm×n=ξ11−,ξ11+,ν11−,ν11+,ξ11,ν11⋯ξ1n−,ξ1n+,ν1n−,ν1n+,ξ1n,ν1n⋮⋱⋮ξm1−,ξm1+,νm1−,νm1+,ξm1,νm1⋯ξmn−,ξmn+,νmn−,νmn+,ξmn,νmn.Step 5.
Use the normalization procedure given in Eq. (16) to change the confidence and confidence and capability of cost type into benefit type, in each of these collective information decision-matrixes.(47)Δij=ξij−,ξ11+,νij−,νij+,ξij,νij;given  the criteria of benefit typeνij−,νij+,ξij−,ξij+,νij,ξij;given  the criteria of cost type,and hence integrate it into the decision-matrix ℛij=Δijm×n.Step 6.
Use the proposed AOs to aggregate the different preference valuesΔijj=1,2,…,n of the alternative Aii=1,2,…m as follows:(48)CPFBMa,bWCPFBMa,bΔi1,Δi2,…,Δin=1n+1⊕i=1i≠jnΔiΔia⊗ΔjΔjb1/a+bΔi1,Δi2,…,Δin=1nn+1⊕i=1i≠jnΔiaΔΔjb1/a+b,Step 7.
Use eq. (4) to compute the score values of each CPFN Δi as follows:(49)scΔi=12ξi−2+ξi+2−νi−2−νi+22−νi2−ξi2.Step 8.
Arrange the alternativesAii1,2,…,m with respect to respective score values scΔi.


## 5. Case Study

These days, supply management is a major problem. From an industrial perspective, a company cannot obtain better performance levels unless its material is properly administered. Therefore, effective inventory management is the first step to achieving high levels of production. Any scarcity of raw material in inventory might cause the entire manufacturing cycle to be affected, resulting in a significant loss for the company. Consider the Case [40] of a food company that has to keep track of multiple inventory materials. The company primarily produces four different types of productsAii=1,2,3,4.(i)
A1: Beverages(ii)
A2: Edible Oils(iii)
A3: Pickles(iv).
A4: Bakery Items.To prepare these food products, rearranging the stock the materials in the inventory will be decided on the basis of three criteriaℂjj=1,2,3.(i)
ℂ1: Cost Price(ii)
ℂ2: Storage Facilities(iii)
ℂ3: Staleness LevelAndΔ=0.2,0.38,0.32T is the weight vector of these features. The given alternatives are analyzed using these three criteria and their values are rated in terms of CPFNs. In each CPFN, the IVPFNs reflect the relative stock level in the inventory, and PFSs reflect 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. The goal is then to identify the food products whose material supply must be re-ordered on a regular basis. The phases of the proposed approach have been carried out in the following order. The multilevel layout of the problem is shown in Figure 1.Figure 1
The multilevel layout of the problem.Step 9.
As indicated in Table2, the favorites information for each alternative is signified in CPFNs, and the group ratings are given in the decision matrix.Table 2
Summarized rating values of each alternative in terms of CPFNs.
AlternativesC1A10.5,0.6,0.1,0.2,0.4,0.20.2,0.3,0.4,0.5,0.3,0.20.4,0.6,0.2,0.3,0.2,0.35A20.2,0.3,0.4,0.5,0.4,0.60.15,0.25,0.3,0.35,0.2,0.30.2,0.4,0.1,0.2,0.4,0.5A30.5,0.6,0.2,0.3,0.2,0.40.4,0.6,0.25,0.35,0.3,0.40.5,0.7,0.1,0.15,0.3,0.5A40.3,0.5,0.1,0.3,0.1,0.30.4,0.55,0.15,0.2,0.3,0.40.4,0.5,0.2,0.3,0.45,0.35Step 10.
The CPFNs are normalized by using Eq. (20) and summarized in Table 3.Table 3
Decision matrix with normalization.
Alternativesℂ1ℂ2ℂ3A10.1,0.2,0.5,0.6,0.2,0.40.2,0.3,0.4,0.5,0.3,0.20.2,0.3,0.4,0.6,0.35,0.2A20.4,0.5,0.2,0.3,0.6,0.40.15,0.25,0.3,0.35,0.2,0.30.1,0.2,0.2,0.4,0.5,0.4A30.2,0.3,0.5,0.6,0.4,0.20.4,0.6,0.25,0.35,0.3,0.40.1,0.15,0.5,0.7,0.5,0.3A40.1,0.3,0.3,0.5,0.3,0.10.4,0.55,0.15,0.2,0.3,0.40.2,0.3,0.4,0.5,0.35,0.45Step 11.
We assumea=b=1 for simplicity’s convenience and then calculate the aggregated values of each alternative using Eq. (21) as follows: Δ1=0.1022,0.1601,0.7615,0.83170.6688,0.1462, Δ2=0.1122,0.1735,0.6300,0.7157,0.7582,0.2125Δ3=0.1288,0.1976,0.7528,0.8288,0.7462,0.1825 and Δ4=0.1430,0.2279,0.6675,0.7427,0.6927,0.2126.Step 12.
. The score values of each alternative are calculated using Eq. (22) as follows: scA1=−1.0437, scA2=−0.9630, scA3=−1.1224 and scA4=−0.8970Step 13.
According to the score values, the ranking order of the alternatives is concluded to beA4>A2>A1>A3.
The suggested operators are symmetric with respect to the parametersa and b. Therefore, in order to examine the influence of parameter a and b on the final ranking of the alternatives, an investigation has been initiated by changing them simultaneously. The score values and raking order for different values of a and b are summarized in Table 3. According to Table 4, we can observe that the score values for different couples of the parameter a and b are different; however, the ranking orders of alternatives remain the same. This 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 decision-makers are bearish, lower values allocated can be provided to a and b. However, the ranking of alternatives cannot be changed, which indicates that the outcomes are objective and cannot be affected by decision-makers’ negativity or positivity. As a consequence, the results obtained are legitimate.
By varying the parameterb, the variations of the score values of each alternative are summarized in Figures 2–4 which shows that the greatest score owing alternative remains A4 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, scA1=scA3=−1.0038, and thus, from (7), we get acA1=1.0962 and acA3=1.1979, as a result, the ranking order of the alternative for a=1 and b=3.8580 is A4≻A2≻A3≻A1. Similarly, in Figure 3, by holding a=2 and b=5.4932 then scA1=scA3=−0.9924; however, acA1= and acA3= and hence the ranking order of alternative for a=2 and b=5.4932 is also A4≻A2≻A3≻A1. The performance of score values by fixing a=5 is shown in Figure 4.Table 4
Influence on the score values and ranking with the variation ofa and b.
abscA1scA2scA3scA4 Rankinga=1b=1b=2b=3b=4b=5−1.0437−1.0311−1.0158−1.0018−0.9897−0.9630−0.9321−0.8927−0.8561−0.8243−1.1224−1.0882−1.0418−0.9978−0.9595−0.8970−0.8639−0.8252−0.7895−0.7582A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A3≻A1A4≻A2≻A3≻A1a=2b=1b=2b=3b=4b=5−1.0311−1.0286−1.0190−1.0081−0.9977−0.9321−0.9324−0.9111−0.8844−0.8577−1.0882−1.0950−1.0720−1.0400−1.0073−0.8639−0.8626−0.8421−0.8160−0.7896A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3a=3b=1b=2b=3b=4b=5−1.0158−1.0190−1.0140−1.0063−0.9982−0.8927−0.9111−0.9052−0.8896−0.8702−1.0418−1.0720−1.0720−1.0547−1.0308−0.8252−0.8421−0.8371−0.8218−0.8024A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3a=4b=1b=2b=3b=4b=5−1.0018−1.0081−1.0063−1.0014−1.0024−0.8561−0.8844−0.8896−0.8837−0.8797−1.0078−1.0400−1.0547−1.0530−1.0397−0.7895−0.8160−0.8218−0.8164−0.8044A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3a=5b=1b=2b=3b=4b=5−0.9897−0.9977−0.9982−1.0024−1.0210−0.8243−0.8577−0.8702−0.8797−0.8748−0.9593−1.0073−1.0308−1.0397−1.0483−0.7582−0.7896−0.8024−0.8044−0.7993A4≻A2≻A3≻A1A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3A4≻A2≻A1≻A3Figure 2
The influence of parameterb on the score values by fixing a=1.Figure 3
The influence of parameterb on the score values by fixing a=2.Figure 4
The influence of parameterb on the score values by fixing a=5.


### 5.1. Comparative Analysis

The dominance of the proposed aggregation operators with respect to the previous approaches such as interval-valued intuitionistic fuzzy aggregation operators [16, 45–50] and weighted cubic intuitionistic BM operator [40] is demonstrated. The weight vector is Δ=0.2,0.38,0.42T. 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. The 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 affect the DM results. In addition, the research concluded that the proposed computational process is different 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 different score values of the alternatives for different values of a and b. Therefore, the approaches under CPS and IVPFS can be measured as a special case of the proposed approach.Table 5
Alternatives with various methods are scored and ranked in order.
ApproachesScore valuesRankingA1A2A3A4Garg [16]0.10820.11010.12300.1649A4≻A3≻A2≻A1Wan et al. [45]−0.2610−0.0705−0.1835−0.0100A4≻A2≻A3≻A1Chen et al. [46]−0.2608−0.0613−0.2154−0.0315A4≻A2≻A2≻A1Sivaraman el al. [47]−0.2120−0.0792−0.19100.0214A4≻A2≻A2≻A1Dugenci [48]0.71060.76930.73160.7940A4≻A2≻A2≻A1Chen et al. [49]−0.2633−0.0630−0.14250.0096A4≻A2≻A2≻A1Xu and chen [24]−0.7152−0.58470.6880−0.5830A4≻A2≻A2≻A1Shi and hi [27]−0.2680−0.0685−0.2244−0.0301A4≻A2≻A2≻A1Wang and liu [50]−0.2593−0.0635−0.15520.0194A4≻A2≻A2≻A1Kaur and garg [40]−1.2942−1.1989−1.3185−1.1474A4≻A2≻A2≻A1a=b=1


## 5.1. Comparative Analysis

The dominance of the proposed aggregation operators with respect to the previous approaches such as interval-valued intuitionistic fuzzy aggregation operators [16, 45–50] and weighted cubic intuitionistic BM operator [40] is demonstrated. The weight vector is Δ=0.2,0.38,0.42T. 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. The 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 affect the DM results. In addition, the research concluded that the proposed computational process is different 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 different score values of the alternatives for different values of a and b. Therefore, the approaches under CPS and IVPFS can be measured as a special case of the proposed approach.Table 5
Alternatives with various methods are scored and ranked in order.
ApproachesScore valuesRankingA1A2A3A4Garg [16]0.10820.11010.12300.1649A4≻A3≻A2≻A1Wan et al. [45]−0.2610−0.0705−0.1835−0.0100A4≻A2≻A3≻A1Chen et al. [46]−0.2608−0.0613−0.2154−0.0315A4≻A2≻A2≻A1Sivaraman el al. [47]−0.2120−0.0792−0.19100.0214A4≻A2≻A2≻A1Dugenci [48]0.71060.76930.73160.7940A4≻A2≻A2≻A1Chen et al. [49]−0.2633−0.0630−0.14250.0096A4≻A2≻A2≻A1Xu and chen [24]−0.7152−0.58470.6880−0.5830A4≻A2≻A2≻A1Shi and hi [27]−0.2680−0.0685−0.2244−0.0301A4≻A2≻A2≻A1Wang and liu [50]−0.2593−0.0635−0.15520.0194A4≻A2≻A2≻A1Kaur and garg [40]−1.2942−1.1989−1.3185−1.1474A4≻A2≻A2≻A1a=b=1


## 6. Conclusion

The CPF set is the more flexible and appropriate tool for deal with fuzziness and uncertainties by interconnecting IVPFN and PFN simultaneously in DM problems. The main purpose of this paper is to present an aggregation operator BM whose significant characteristic is to capture the connections among the distinct influences. 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 parametersa 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 offer 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.



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*Source: 9735100-2022-11-01.xml*