Power Bonferroni mean operators under complex pythagorean fuzzy settings and their applications in decision-making problems

Abstract

Bonferroni means (BM) operator is the extended form of the arithmetic mean operator, used for simplifying non-dominant and non-feasible problems diagnosed in genuine life scenarios. A lot of aggregation operators are the specific parts of the BM operators under the consideration of different values of parameters which are the main parts of the BM operators. In the presence of the BM operator and a very well-known conception in the scenario of fuzzy set, called complex Pythagorean fuzzy (CPF) setting, the objective of this scenario is to diagnose the CPF power BM (CPFPBM) operator and utilize their beneficial results with important properties. Moreover, a multi-attribute decision-making (MADM) technique is evaluated in the presence of invented operators for CPF settings. In the last of this study, we diagnosed the superiority and efficiency of the invented works with the help of sensitive analysis and graphical illustrations to enhance the gap of the research works.

Keywords

Complex pythagorean fuzzy sets power Bonferroni mean operators decision-making methods

1 Introduction

Decision-making strategy is a continuing technique that includes developing strategies to obtain themes and finding strategies under the consideration of given beneficial optimal. We all agreed that the MADM technique is the beneficial and dominant part of the decision-making strategy has used for determining the beneficial optimal from a lot of optimums. A huge number of implementations have been diagnosed in different places, but firstly Zadeh [1] employed the MADM technique in the region of fuzzy set (FS) theory. Further, the intuitionistic FS (IFS) was another form of achievement in FS theory, which was diagnosed by Atanassov [2]. The good looking of IFS is that they have includes the truth grade (TG) $\underset{Ξ_{IF}}{\overset{=}{M}} (X^{=})$ and falsity grade (FG) $\underset{Ξ_{IF}}{\overset{=}{ℕ}} (X^{=})$ with some new strategy $0 ⩽ \underset{Ξ_{IF}}{\overset{=}{M}} (X^{=}) + \underset{Ξ_{IF}}{\overset{=}{ℕ}} (X^{=}) ⩽ 1$ . Several implementations have been done by distinct individuals, likewise, [3] covers the conception of bipolar soft sets, [4] covers the conception of interval-valued IFSs, [5] covers the conception of measures in the presence of IFSs, [6] covers the conception of MADM technique in the presence of IFSs, [7] covers the conception of type-2 IFS, [8] covers the conception of GIS-IFS, [9] covers the conception of the three-way decision in the presence of IFSs, and [10] covers the conception of bipolar intuitionistic fuzzy softs sets.

After the development of an idea, they have gotten a lot of favor in the shape of utilization in distinct regions, but after time by time, all fields have improved by some specific intellectuals. They have diagnosed some limitations in the existing theory and tried to modify it. A similar situation was done with IFS because someone find the conception of IFS enable to handle such sort of data, whose sum cross the unit interval i.e., $\underset{Ξ_{IF}}{\overset{=}{M}} (X^{=}) + \underset{Ξ_{IF}}{\overset{=}{ℕ}} (X^{=}) > 1$ . Such a problematic scenario faced in some difficult situations, Yager [11] put forward the solution of the above problematic situation in the shape of the development of the Pythagorean FS (PFS) in 2013. The mathematical model $0 ⩽ {\overset{=}{M}}_{Ξ_{PF}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{PF}}^{2} (X^{=}) ⩽ 1$ , is the important part of the PFS. To employ the MADM technique in the region of PFS isn’t awkward, a lot of utilization have illustrated here, for example in [12] covers the conception of interval-valued PFSs, in [13] covers the conception of the ORESTE method in the presence of PFSs, in [14] covers the conception of Pythagorean m-polar FSs, in [15] covers the conception of Pythagorean m-polar fuzzy soft sets, and in [16] covers the conception of measures in the Ramot et al. [17] diagnosed the different form of FS, called the complex FS (CFS), which includes the TG in the shape of $\underset{Ξ_{CP}}{\overset{=}{M}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{M}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}))}$ , whose real and unreal parts lies between unit interval. Several applications of CFS have been diagnosed, for instance in [18] contains the interval-valued CFSs, in [19] contains the complex multi-fuzzy sets, in [20] contains the complex fuzzy soft sets, in [21] contains the interval-valued complex fuzzy soft sets, in [22] contains the complex multi-fuzzy soft expert sets, and in [23] contains the complex multi-fuzzy soft expert set. Complexity is part of a genuine life scenario, which occurred in several decision-making processes, for illustration, sometimes we faced two-dimension data and each dimension includes two different terms which expressed the behavior of the function. To control such a difficult scenario, the conception of complex IFS (CIFS) is massive necessary which was invented by Alkouri and Salleh [24]. The mathematical useful part of the CIFS is stated: $0 ⩽ \underset{Ξ_{RP}}{\overset{=}{M}} (X^{=}) + \underset{Ξ_{RP}}{\overset{=}{ℕ}} (X^{=}) ⩽ 1, 0 ⩽ \underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}) + \underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}) ⩽ 1$ , for the real and unreal part, where $\underset{Ξ_{CP}}{\overset{=}{M}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{M}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}))}$ and $\underset{Ξ_{CP}}{\overset{=}{ℕ}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{ℕ}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}))}$ , dominated as TG and FG. To employ the MADM technique in the region of CIFS isn’t awkward, a lot of utilization have illustrated here, for example in, information measures invented in [25], correlation coefficient invented in [26], geometric aggregation operators invented in [27], robust aggregation operators invented in [28], generalized aggregation operators invented in [29], power aggregation operators invented in [30] in the presence of CIFSs, and complex intuitionistic fuzzy relations invented in [31].

We have an idea that the conception of CIFS is one of the massive dominant and consistent techniques which handle complex and problematic data which exists in genuine-life scenarios, but not in all situations especially in those cases when an expert determined or faced data in the shape if we have gotten the value from outside the unit disc, which we obtained from the sum of the real parts (also for unreal parts) of the pairs. Such sort of scenario is part of genuine life troubles, for this, the conception of complex PFS (CPFS), elaborated by Ullah et al. [32]. In the presence of the previous two-term are called TG $\underset{Ξ_{CP}}{\overset{=}{M}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{M}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}))}$ and FG $\underset{Ξ_{CP}}{\overset{=}{ℕ}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{ℕ}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}))}$ , we have gotten $0 ⩽ {\overset{=}{M}}_{Ξ_{RP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{RP}}^{2} (X^{=}) ⩽ 1$ and $0 ⩽ {\overset{=}{M}}_{Ξ_{IP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{IP}}^{2} (X^{=}) ⩽ 1$ . Several implementations have been done by distinct individuals, likewise, Einstein geometric aggregation operators exposed in [33], decision-making strategy for CPFSs exposed in [34, 35], complex Pythagorean Dombi fuzzy operators exposed in [36], and prioritized weighted aggregation operators exposed in [37]. Ambiguity and vague are involved in every decision-making procedure under the consideration of several strategies:

Under the presence of appropriate measures arranged the data.

Under the presence of some operators gotten the beneficial optimal from a large number of attributes.

Under the presence of the aggregated values is to determine the beneficial optimal.

In the presence of the above illustration, the foremost influence of this scenario is to employ the MADM technique under the consideration of power BM for CPF settings, several advantages of the CPF setting are diagnosed here:

In the presence of $\underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{ℕ}} (X^{=}) = \underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}) = 0$ , in the CPF setting, then we pick the FS.

In the presence of $\underset{Ξ_{RP}}{\overset{=}{ℕ}} (X^{=}) = \underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}) = 0$ , in the CPF setting, then we pick the CFS.

In the presence of $\underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}) = \underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}) = 0$ , in the CPF setting, then we pick the IFS.

In the presence of $\underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}) = \underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}) = 0$ with $0 ⩽ {\overset{=}{M}}_{Ξ_{RP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{RP}}^{2} (X^{=}) ⩽ 1$ and $0 ⩽ {\overset{=}{M}}_{Ξ_{IP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{IP}}^{2} (X^{=}) ⩽ 1$ , in the CPF setting, then we pick the PFS.

In the presence of $0 ⩽ \underset{Ξ_{RP}}{\overset{=}{M}} (X^{=}) + \underset{Ξ_{RP}}{\overset{=}{ℕ}} (X^{=}) ⩽ 1$ and $0 ⩽ \underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}) + \underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}) ⩽ 1$ , in the CPF setting, then we pick the CIFS. The geometric form of the invented works is implemented in the presence of Fig. 1.

Fig. 1

Geometrical form of invented works.

CPF data is very well-known and most dominant due to several techniques includes in the shape: $0 ⩽ {\overset{=}{M}}_{Ξ_{RP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{RP}}^{2} (X^{=}) ⩽ 1$ and $0 ⩽ {\overset{=}{M}}_{Ξ_{IP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{IP}}^{2} (X^{=}) ⩽ 1$ . They have a lot of usage in genuine life troubles, for simplicity, we try to explain the significance of the two-dimension data which occurred in the shape of a singleton term. If the president of ABC Prof. Dr. HHA planned to implement the biometric system for teacher attendance devices (BSTAD) in all offices of the teachers and for the attending of administrations in the admin. The attending device enterprises provide data in the shape: (i) model of the BSTAD and (ii) production dates of BSTAD. The university experts have selected the latest and well-known verity device for usage in university. Here the model has stated the real part and the production date is stated as the unreal part in the complex number. For IFSs, PFSs, and CIFSs are very awkward to handle such sort of scenario. For handling such problematic and intricate situations, the conception of the CPF set is massively useful to investigate genuine life troubles. Moreover, the BM operator is the extended form of the arithmetic mean operator, used for simplifying non-dominant and non-feasible problems diagnosed in genuine life scenarios. A lot of aggregation operators are the specific parts of the BM operators under the consideration of different values of parameters which are the main parts of the BM operators. In the presence of the BM operator and a very well-known conception in the scenario of fuzzy set, called CPF setting, the objective of this scenario is to diagnose below:

To discover the very well-known theory, called CPFPBM operator.

To discuss the importance of the Power BM (PBM) operator in the presence of the well-known conception of CPF settings and discuss their important properties.

To discover a MADM technique in the presence of invented operators for CPF settings.

To diagnose the superiority and efficiency of the invented works with the help of sensitivity analysis and graphical illustrations to enhance the gap of the research works.

The organized shape of the article is illustrated here: Section 2 covers the revised version of the existing CPFSs, algebraic laws, and the original form of the PBM operator. Section 3 discovered the CPFPBM operator and considered their significant results. In section 4, a multi-attribute decision-making (MADM) technique is evaluated in the presence of invented operators for CPF settings. In section 5, we diagnosed the superiority and efficiency of the invented works with the help of sensitivity analysis and graphical illustrations to enhance the gap of the research works. The conclusion is stated in Section 6.

2 Preliminaries

This scenario stated that the mathematical shown $X^{=}, \underset{Ξ_{CP}}{\overset{=}{M}} (X^{=})$ and $\underset{Ξ_{CP}}{\overset{=}{ℕ}} (X^{=})$ , expressed the universal set, TG and FG. Further, some prevailing scenarios like CPFS, algebraic laws, and PBM operator, revised here.

Definition 1. [32] A CPFS Ξ_CP is elaborated by: $Ξ_{CP} = {(\underset{Ξ_{CP}}{\overset{=}{M}} (X^{=}), \underset{Ξ_{CP}}{\overset{=}{ℕ}} (X^{=})) : x^{=} \in {\bar{X}}^{-}}$ (1) where $\underset{Ξ_{CP}}{\overset{=}{M}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{M}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}))}$ and $\underset{Ξ_{CP}}{\overset{=}{ℕ}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{ℕ}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}))}$ , stated the TG and FG with regulations such as $0 ⩽ {\overset{=}{M}}_{Ξ_{RP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{RP}}^{2} (X^{=}) ⩽ 1$ and $0 ⩽ {\overset{=}{M}}_{Ξ_{IP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{IP}}^{2} (X^{=}) ⩽ 1$ . Further, the term $\underset{Ξ_{CP}}{\overset{=}{ℝ}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{ℝ}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{ℝ}} (X^{=}))} = {(1 - {\overset{=}{M}}_{Ξ_{RP}}^{2} (X^{=}) - {\overset{=}{ℕ}}_{Ξ_{RP}}^{2} (X^{=}))}^{\frac{1}{2}} e^{2 π i {(1 - {\overset{=}{M}}_{Ξ_{IP}}^{2} (X^{=}) - {\overset{=}{ℕ}}_{Ξ_{IP}}^{2} (X^{=}))}^{\frac{1}{2}}}$ , stated the refusal degree. In all studies, the CPFN is elaborated by: $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})})$ , $j = 1, 2, \dots, n$ .

Definition 2. [32] Suppose $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})})$ , $j = 1, 2$ , be any two CPFNs. Then

$Ξ_{CP - 1} \oplus Ξ_{CP - 2} = (\begin{matrix} {({\overset{=}{M}}_{Ξ_{RP - 1}}^{2} + {\overset{=}{M}}_{Ξ_{RP - 2}}^{2} - {\overset{=}{M}}_{Ξ_{RP - 1}}^{2} {\overset{=}{M}}_{Ξ_{RP - 2}}^{2})}^{\frac{1}{2}} e^{2 π i {({\overset{=}{M}}_{Ξ_{IP - 1}}^{2} + {\overset{=}{M}}_{Ξ_{IP - 2}}^{2} - {\overset{=}{M}}_{Ξ_{IP - 1}}^{2} {\overset{=}{M}}_{Ξ_{IP - 2}}^{2})}^{\frac{1}{2}}}, \\ \underset{Ξ_{RP - 1}}{\overset{=}{ℕ}} \underset{Ξ_{RP - 2}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - 1}}{\overset{=}{ℕ}} \underset{Ξ_{IP - 2}}{\overset{=}{ℕ}})} \end{matrix})$ (2)

$Ξ_{CP - 1} \otimes Ξ_{CP - 2} = (\begin{matrix} \underset{Ξ_{RP - 1}}{\overset{=}{M}} \underset{Ξ_{RP - 2}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - 1}}{\overset{=}{M}} \underset{Ξ_{IP - 2}}{\overset{=}{M}})}, \\ {({\overset{=}{ℕ}}_{Ξ_{RP - 1}}^{2} + {\overset{=}{ℕ}}_{Ξ_{RP - 2}}^{2} - {\overset{=}{ℕ}}_{Ξ_{RP - 1}}^{2} {\overset{=}{ℕ}}_{Ξ_{RP - 2}}^{2})}^{\frac{1}{2}} e^{2 π i {({\overset{=}{ℕ}}_{Ξ_{IP - 1}}^{2} + {\overset{=}{ℕ}}_{Ξ_{IP - 2}}^{2} - {\overset{=}{ℕ}}_{Ξ_{IP - 1}}^{2} {\overset{=}{ℕ}}_{Ξ_{IP - 2}}^{2})}^{\frac{1}{2}}} \end{matrix})$ (3)

$Ψ_{SC} Ξ_{CP - 1} = ({(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - 1}}^{2})}^{Ψ_{SC}})}^{\frac{1}{2}} e^{2 π i {(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - 1}}^{2})}^{Ψ_{SC}})}^{\frac{1}{2}}}, {\overset{=}{ℕ}}_{Ξ_{RP - 1}}^{Ψ_{SC}} e^{2 π i ({\overset{=}{ℕ}}_{Ξ_{IP - 1}}^{Ψ_{SC}})})$ (4)

$Ξ_{CP - 1}^{Ψ_{SC}} = ({\overset{=}{M}}_{Ξ_{RP - 1}}^{Ψ_{SC}} e^{2 π i ({\overset{=}{M}}_{Ξ_{IP - 1}}^{Ψ_{SC}})}, {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - 1}}^{2})}^{Ψ_{SC}})}^{\frac{1}{2}} e^{2 π i {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - 1}}^{2})}^{Ψ_{SC}})}^{\frac{1}{2}}})$ (5)

Definition 3. [32] Suppose $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})})$ , $j = 1$ , be any CPFN. Then the Score value (SV) and accuracy value (AV) is elaborated by: ${S^{=}}_{SV} (Ξ_{CP - 1}) = \frac{1}{2} ({\overset{=}{M}}_{Ξ_{RP - 1}}^{2} + {\overset{=}{M}}_{Ξ_{IP - 1}}^{2} - {\overset{=}{ℕ}}_{Ξ_{RP - 1}}^{2} - {\overset{=}{ℕ}}_{Ξ_{IP - 1}}^{2})$ (6) ${A^{=}}_{AV} (Ξ_{CP - 1}) = \frac{1}{2} ({\overset{=}{M}}_{Ξ_{RP - 1}}^{2} + {\overset{=}{M}}_{Ξ_{IP - 1}}^{2} + {\overset{=}{ℕ}}_{Ξ_{RP - 1}}^{2} + {\overset{=}{ℕ}}_{Ξ_{IP - 1}}^{2})$ (7) where ${S^{=}}_{SV} (Ξ_{CP - 1}) \in [- 1, 1]$ and ${S^{=}}_{SV} (Ξ_{CP - 1}) \in [0, 1]$ .

Definition 4. [32] Suppose $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})})$ , $j = 1$ , be any CPFN. Then, by using Equations (6) and (7), we determine the order among any number of objects under the following rules:

When ${S^{=}}_{SV} (Ξ_{CP - 1}) > {S^{=}}_{SV} (Ξ_{CP - 2}) \Rightarrow Ξ_{CP - 1} > Ξ_{CP - 2}$ ;

When ${S^{=}}_{SV} (Ξ_{CP - 1}) < {S^{=}}_{SV} (Ξ_{CP - 2}) \Rightarrow Ξ_{CP - 1} < Ξ_{CP - 2}$ ;

When ${S^{=}}_{SV} (Ξ_{CP - 1}) = {S^{=}}_{SV} (Ξ_{CP - 2}) \Rightarrow$

When ${A^{=}}_{AV} (Ξ_{CP - 1}) > {A^{=}}_{AV} (Ξ_{CP - 2}) \Rightarrow Ξ_{CP - 1} > Ξ_{CP - 2}$ ;

When ${A^{=}}_{AV} (Ξ_{CP - 1}) < {A^{=}}_{AV} (Ξ_{CP - 2}) \Rightarrow Ξ_{CP - 1} < Ξ_{CP - 2}$ ;

When ${A^{=}}_{AV} (Ξ_{CP - 1}) = {A^{=}}_{AV} (Ξ_{CP - 2}) \Rightarrow Ξ_{CP - 1} = Ξ_{CP - 2}$ .

Definition 5. [38] Suppose $Ξ_{CP - j}, j = 1, 2, \dots, n$ , be the group of non-negative integers, then the PBM operator is presented by:

$\begin{matrix} {PBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) \\ = {(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} \end{matrix}$ (8)

where, $\begin{matrix} T_{F} (Ξ_{CP - i}) = \sum_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} Sup (Ξ_{CP - i}, Ξ_{CP - k}), \\ d (Ξ_{CP - i}, Ξ_{CP - k}) = Sup (Ξ_{CP - i}, Ξ_{CP - k}) \\ = \frac{1}{2} ({\overset{=}{M}}_{Ξ_{RP - i}}^{2} + {\overset{=}{M}}_{Ξ_{IP - i}}^{2} - {\overset{=}{ℕ}}_{Ξ_{RP - k}}^{2} - {\overset{=}{ℕ}}_{Ξ_{IP - k}}^{2}) \end{matrix}$ (9)

which meet the following rules:

Sup (Ξ_CP-i, Ξ_CP-k) ∈ [0, 1];

Sup (Ξ_CP-i, Ξ_CP-k) = Sup (Ξ_CP-k, Ξ_CP-i);

Sup (Ξ_CP-i, Ξ_CP-k) ⩾ Sup (Ξ_CP-l, Ξ_CP-m) ⇔ |Ξ_CP-i - Ξ_CP-k| ⩾ |Ξ_CP-l - Ξ_CP-m|;

If d (Ξ_CP-i, Ξ_CP-k) ⩾ d (Ξ_CP-l, Ξ_CP-m), then Sup (Ξ_CP-i, Ξ_CP-k) ⩽ Sup (Ξ_CP-l, Ξ_CP-m).

3 PBM operators under CPFSs

In the presence of the BM operator and a very well-known conception in the scenario of fuzzy set, called CPF setting, the objective of this scenario is to diagnose the CPFPBM operator and utilize their beneficial results with important properties.

Definition 6. Suppose $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})}), j = 1, 2, \dots, n$ , be the group of CPFNs, then the CPFPBM operator is presented by: $\begin{matrix} {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) = \\ (\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} \end{matrix}$

$\begin{matrix} {\otimes {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} \end{matrix}$ (10)

where the support degree is initiated below,

$\begin{matrix} T_{F} (Ξ_{CP - i}) = \sum_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} Sup (Ξ_{CP - i}), d (Ξ_{CP - i}) \\ = Sup (Ξ_{CP - i}) = \\ \frac{1}{2} ({\overset{=}{M}}_{Ξ_{RP - i}}^{2} + {\overset{=}{M}}_{Ξ_{IP - i}}^{2} - {\overset{=}{ℕ}}_{Ξ_{RP - i}}^{2} - {\overset{=}{ℕ}}_{Ξ_{IP - i}}^{2}) \end{matrix}$ (11)

which meet the following rules:

Sup (Ξ_CP-i, Ξ_CP-k) ∈ [0, 1];

Sup (Ξ_CP-i, Ξ_CP-k) = Sup (Ξ_CP-k, Ξ_CP-i);

Sup (Ξ_CP-i, Ξ_CP-k) ⩾ Sup (Ξ_CP-l, Ξ_CP-m) ⇔ |Ξ_CP-i - Ξ_CP-k| ⩾ |Ξ_CP-l - Ξ_CP-m|;

If d (Ξ_CP-i, Ξ_CP-k) ⩾ d (Ξ_CP-l, Ξ_CP-m), then Sup (Ξ_CP-i, Ξ_CP-k) ⩽ Sup (Ξ_CP-l, Ξ_CP-m).

Theorem 1. Suppose $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})}), j = 1, 2, \dots, n$ , be the group of CPFNs, then by using the Equation (10), we determine $\begin{matrix} {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) \\ = (\begin{matrix} {({(1 - {(\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} \begin{matrix} (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}}) * \\ (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}}) \end{matrix})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}})}^{\frac{1}{2}} \\ e^{2 π i {({(1 - {(\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} l (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}}) * (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}}))}^{\frac{1}{n (n - 1)}})}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}})}^{\frac{1}{2}}}, \\ {(1 - {(1 - {(\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} 1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}} {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}})}^{\frac{1}{2}} \\ e^{2 π i {(1 - {(1 - {(\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} l 1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}} {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}})}^{\frac{1}{2}}} \end{matrix}) \end{matrix}$ (12)

Proof. Suppose $\begin{matrix} (\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i}) \\ = ((\begin{matrix} 1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - i}}^{2})}^{{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})}^{\frac{1}{2}}} e^{2 π i {(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{\frac{1}{2}}}, \\ {\overset{=}{ℕ}}_{Ξ_{RP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})} e^{2 π i ({\overset{=}{ℕ}}_{Ξ_{IP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})} \end{matrix})) \end{matrix}$

and $\begin{matrix} (\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k}) \\ = (\begin{matrix} {(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{\frac{1}{2}} e^{2 π i {(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{\frac{1}{2}}}, \\ {\overset{=}{ℕ}}_{Ξ_{RP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})} e^{2 π i ({\overset{=}{ℕ}}_{Ξ_{IP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})} \end{matrix}) \end{matrix}$

then $\begin{matrix} {(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} \\ = (\begin{matrix} {({(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{\frac{1}{2}})}^{𝒻_{SC}} e^{2 π i {({(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{\frac{1}{2}})}^{𝒻_{SC}}}, \\ {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}})}^{\frac{1}{2}} e^{2 π i {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}})}^{\frac{1}{2}}} \end{matrix}) \end{matrix}$ $\begin{matrix} {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}} \\ = (\begin{matrix} {({(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{\frac{1}{2}})}^{ℊ_{SC}} e^{2 π i {({(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{\frac{1}{2}})}^{ℊ_{SC}}}, \\ {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}})}^{\frac{1}{2}} e^{2 π i {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}})}^{\frac{1}{2}}} \end{matrix}) \end{matrix}$

thus, $\begin{matrix} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} \otimes {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}}) \\ = (\begin{matrix} {({(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{\frac{1}{2}})}^{𝒻_{SC}} e^{2 π i {({(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{\frac{1}{2}})}^{𝒻_{SC}}}, \\ {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}})}^{\frac{1}{2}} e^{2 π i {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}})}^{\frac{1}{2}}} \end{matrix}) \end{matrix}$ $\otimes (\begin{matrix} {({(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{\frac{1}{2}})}^{ℊ_{SC}} e^{2 π i {({(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{\frac{1}{2}})}^{ℊ_{SC}}}, \\ {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}})}^{\frac{1}{2}} e^{2 π i {(1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}})}^{\frac{1}{2}}} \end{matrix})$ $= (\begin{array}{l} {((\begin{array}{l} {(1 - {(1 - {\bar{\bar{M}}}^{2}_{Ξ R P - i})}^{(\frac{n (1 + T_{F} (Ξ_{C P - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{C P - i}))})})}^{𝒻_{S C}} * \\ {(1 - {(1 - {\bar{\bar{M}}}^{2}_{Ξ R P - k})}^{(\frac{n (1 + T_{F} (Ξ_{C P - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{C P - k}))})})}^{ℊ_{S C}} \end{array}))}^{\frac{1}{2}} \\ e^{2 π i {((\begin{array}{l} {(1 - {(1 - {\bar{\bar{M}}}^{2}_{Ξ I P - i})}^{(\frac{n (1 + T_{F} (Ξ_{C P - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{C P - i}))})})}^{𝒻_{S C}} * \\ {(1 - {(1 - {\bar{\bar{M}}}^{2}_{Ξ I P - k})}^{(\frac{n (1 + T_{F} (Ξ_{C P - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{C P - k}))})})}^{ℊ_{S C}} \end{array}))}^{\frac{1}{2}}}, \\ {(1 - ({(1 - {\bar{\bar{ℕ}}}_{Ξ R P - i}^{(\frac{2n (1 + T_{F} (Ξ_{C P - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{C P - i}))})})}^{𝒻_{S C}} {(1 - {\bar{\bar{ℕ}}}_{Ξ I P - k}^{(\frac{2n (1 + T_{F} (Ξ_{C P - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{C P - k}))})})}^{ℊ_{S C}}))}^{\frac{1}{2}} \\ e^{2 π i {(1 - ({(1 - {\bar{\bar{ℕ}}}_{Ξ R P - i}^{(\frac{2n (1 + T_{F} (Ξ_{C P - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{C P - i}))})})}^{𝒻_{S C}} {(1 - {\bar{\bar{ℕ}}}_{Ξ I P - k}^{(\frac{2n (1 + T_{F} (Ξ_{C P - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{C P - k}))})})}^{ℊ_{S C}}))}^{\frac{1}{2}}} \end{array})$ $\begin{matrix} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} \otimes {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}}) \\ = (\begin{matrix} {((\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} \begin{matrix} (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}}) * \\ (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}}) \end{matrix}))}^{\frac{1}{2}} \\ e^{2 π i {((\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} l (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}}) * (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}})))}^{\frac{1}{2}}}, \\ {(1 - (\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} 1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}} {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}}))}^{\frac{1}{2}} \\ e^{2 π i {(1 - (\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} l 1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}} {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}}))}^{\frac{1}{2}}} \end{matrix}) \end{matrix}$ $\begin{matrix} {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} \otimes {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} \\ (\begin{matrix} {({(1 - {(\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} \begin{matrix} (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}}) * \\ (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{RP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}}) \end{matrix})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}})}^{\frac{1}{2}} \\ e^{2 π i {({(1 - {(\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} l (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - i}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}}) * (1 - {(1 - {(1 - {\overset{=}{M}}_{Ξ_{IP - k}}^{2})}^{(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}}))}^{\frac{1}{n (n - 1)}})}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}})}^{\frac{1}{2}}}, \\ {(1 - {(1 - {(\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} 1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}} {(1 - {\overset{=}{ℕ}}_{Ξ_{RP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}})}^{\frac{1}{2}} \\ e^{2 π i {(1 - {(1 - {(\prod_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} l 1 - {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - i}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))})})}^{𝒻_{SC}} {(1 - {\overset{=}{ℕ}}_{Ξ_{IP - k}}^{(\frac{2 n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))})})}^{ℊ_{SC}})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}})}^{\frac{1}{2}}} \end{matrix}) \end{matrix}$

Additionally, under Equation (12), we determine the idempotency, permutation invariability, and boundedness to improve the flexibility and efficiency of the presented works.

Property 1. Suppose $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})}), j = 1, 2, \dots, n$ , be the group of CPFNs. If $Ξ_{CP - j} = Ξ_{CP} = (\underset{Ξ_{RP}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{M}})}, \underset{Ξ_{RP}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{ℕ}})})$ , then ${CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) = Ξ_{CP} .$ (13)

Proof. $\begin{matrix} {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) = \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} \otimes {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} \end{matrix}$ $= {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP}))} Ξ_{CP})}^{𝒻_{SC}} \otimes {(\frac{n (1 + T_{F} (Ξ_{CP}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP}))} Ξ_{CP})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}}$ $= {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} (Ξ_{CP}^{𝒻_{SC} + ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} = Ξ_{CP} .$

Property 2. Suppose $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})}), j = 1, 2, \dots, n$ , be the group of CPFNs. If $Ξ_{CP - j}$ is a permutation and combination of $Ξ_{CP - j}^{*}$ , then ${CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) = {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}^{*}, Ξ_{CP - 2}^{*}, \dots, Ξ_{CP - n}^{*}) .$ (14)

Proof. $\begin{matrix} {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) = \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} \otimes {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} \end{matrix}$ $= {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}^{*}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}^{*}))} Ξ_{CP - i}^{*})}^{𝒻_{SC}} \otimes {(\frac{n (1 + T_{F} (Ξ_{CP - k}^{*}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}^{*}))} Ξ_{CP - k}^{*})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}}$ $= {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}^{*}, Ξ_{CP - 2}^{*}, \dots, Ξ_{CP - n}^{*}) .$

Property 3. Suppose $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})}), j = 1, 2, \dots, n$ , be the group of CPFNs. If $Ξ_{CP - i}^{*} = \frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - i}, Ξ_{CP}^{+} = max (Ξ_{CP - i}^{*}), Ξ_{CP}^{-} = min (Ξ_{CP - i}^{*})$ , then $Ξ_{CP}^{-} ⩽ {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) ⩽ Ξ_{CP}^{+} .$ (15)

Proof. $\begin{matrix} {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) = \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} \otimes {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} \end{matrix}$ $⩽ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} (^{(Ξ_{CP}^{+})_{SC}} \otimes^{(Ξ_{CP}^{+})_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} = {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(Ξ_{CP}^{+})}^{𝒻_{SC} + ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}}$ $= {(({(Ξ_{CP}^{+})}^{𝒻_{SC} + ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} = Ξ_{CP}^{+}$

and $\begin{matrix} {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) = \\ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(\frac{n (1 + T_{F} (Ξ_{CP - i}))}{\sum_{i = 1}^{n} (1 + T_{F} (Ξ_{CP - i}))} Ξ_{CP - i})}^{𝒻_{SC}} \otimes {(\frac{n (1 + T_{F} (Ξ_{CP - k}))}{\sum_{k = 1}^{n} (1 + T_{F} (Ξ_{CP - k}))} Ξ_{CP - k})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} \end{matrix}$ $⩾ {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(Ξ_{CP}^{-})}^{𝒻_{SC}} \otimes {(Ξ_{CP}^{-})}^{ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} = {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, k \in j, \\ i \neq k = 1 \end{matrix}}^{n} ({(Ξ_{CP}^{-})}^{𝒻_{SC} + ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}}$ $= {(({(Ξ_{CP}^{-})}^{𝒻_{SC} + ℊ_{SC}}))}^{\frac{1}{𝒻_{SC} + ℊ_{SC}}} = Ξ_{CP}^{-}$

From the above analysis, we determine $Ξ_{CP}^{-} ⩽ {CPFPBM}^{𝒻_{SC}, ℊ_{SC}} (Ξ_{CP - 1}, Ξ_{CP - 2}, \dots, Ξ_{CP - n}) ⩽ Ξ_{CP}^{+} .$

4 MADM procedure

Decision-making strategy is a continuing technique that includes developing strategies to obtain themes and finding strategies under the consideration of given beneficial optimal. We all agreed that the MADM technique is the beneficial and dominant part of the decision-making strategy has used for determining the beneficial optimal from a lot of optimums. Further, the complexity involved in several life dilemmas, to find their solution with the help of the MADM procedure, we recommended some alternatives Ξ_AL-1, Ξ_AL-2, …, Ξ_AL-m and the mathematical terms Ξ_AT-1, Ξ_AT-2, …, Ξ_AT-n, stated the attributes. In the presence of the above items, we exposed the matrix with terms in the shape of CPF numbers likewise: $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})}), j = 1, 2, \dots, n$ , under the circumstances $\underset{Ξ_{CP}}{\overset{=}{M}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{M}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}))}$ and $\underset{Ξ_{CP}}{\overset{=}{ℕ}} (X^{=}) = \underset{Ξ_{RP}}{\overset{=}{ℕ}} (X^{=}) e^{2 π i (\underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}))}$ , with $0 ⩽ {\overset{=}{M}}_{Ξ_{RP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{RP}}^{2} (X^{=}) ⩽ 1$ and $0 ⩽ {\overset{=}{M}}_{Ξ_{IP}}^{2} (X^{=}) + {\overset{=}{ℕ}}_{Ξ_{IP}}^{2} (X^{=}) ⩽ 1$ . To explain the usage and importance of the MADM scenario, a lot of individuals have described the MADM technique in the scenario of distinct regions.

4.1 Decision-making processes

The major procedure of the MADM technique in this scenario are diagnosed with the help of the following stages:

Stage 1: In the presence of the CPF numbers $Ξ_{CP - j} = (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})})$ , we exposed the matrix, which includes two sorts of data like benefit or cost sorts, if an expert includes the benefit types, then no issues, but if an expert suggested cost types, then homogenize the data in the presence of Equation (53), such that $D = {\begin{matrix} (\underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}, \underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})}) & For benefit types \\ (\underset{Ξ_{RP - j}}{\overset{=}{ℕ}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{ℕ}})}, \underset{Ξ_{RP - j}}{\overset{=}{M}} e^{2 π i (\underset{Ξ_{IP - j}}{\overset{=}{M}})}) & For cost types \end{matrix}$ (16)

Stage 2: With the presence of the CPFPBM operator (Equation 12), cumulative the matrix.

Stage 3: With the presence of aggregated values, demonstrated the SVs.

Stage 4: In the presence of the SVs, find the beneficial optimal from the large numbers of alternatives.

In the existence of the above algorithm their geometrical expressions are diagnosed here, geometrical form of the invented works is implemented in the presence of Fig. 2.

Fig. 2

Geometrical form of invented works.

5 Practical illustrations

Here, we demonstrated the supremacy and effectiveness of the elaborated works under the presence of CPF settings. Several examples and their sensitivity analysis are diagnosed here.

Example 1. The mathematical five terms Ξ_AL-1, Ξ_AL-2, Ξ_AL-3, Ξ_AL-4, Ξ_AL-4 and four terms Ξ_AT-1: enterprise executive rank, Ξ_AT-2: financial benefit, Ξ_AT-3: development capability, and Ξ_AT-4: community influence, stated the alternatives and attributes in the presence of weight vectors 0.3, 0.3, 0.3, and 0.1. To explain the usage and importance of the MADM scenario, a lot of individuals have described the MADM technique in the scenario of distinct regions. The major procedure of the MADM technique in this scenario are diagnosed with the help of the following stages:

Table 1
Original decision matrix

Ξ _AT-1 Ξ _AT-2 Ξ _AT-3 Ξ _AT-4

Ξ _AL-1 $(\begin{matrix} 0.9 e^{2 π i (0.8)}, 0.1 e^{2 π i (0.1)} \end{matrix})$ $(\begin{matrix} 0.91 e^{2 π i (0.81)}, 0.11 e^{2 π i (0.11)} \end{matrix})$ $(\begin{matrix} 0.92 e^{2 π i (0.82)}, 0.12 e^{2 π i (0.12)} \end{matrix})$ $(\begin{matrix} 0.93 e^{2 π i (0.83)}, 0.13 e^{2 π i (0.13)} \end{matrix})$

Ξ_AL-2. $(\begin{matrix} 0.7 e^{2 π i (0.6)}, 0.2 e^{2 π i (0.4)} \end{matrix})$ $(\begin{matrix} 0.71 e^{2 π i (0.61)}, 0.21 e^{2 π i (0.41)} \end{matrix})$ $(\begin{matrix} 0.72 e^{2 π i (0.62)}, 0.22 e^{2 π i (0.42)} \end{matrix})$ $(\begin{matrix} 0.73 e^{2 π i (0.63)}, 0.23 e^{2 π i (0.43)} \end{matrix})$

Ξ _AL-3 $(\begin{matrix} 0.6 e^{2 π i (0.7)}, 0.4 e^{2 π i (0.3)} \end{matrix})$ $(\begin{matrix} 0.61 e^{2 π i (0.71)}, 0.41 e^{2 π i (0.31)} \end{matrix})$ $(\begin{matrix} 0.62 e^{2 π i (0.72)}, 0.42 e^{2 π i (0.32)} \end{matrix})$ $(\begin{matrix} 0.63 e^{2 π i (0.73)}, 0.43 e^{2 π i (0.33)} \end{matrix})$

Ξ _AL-4 $(\begin{matrix} 0.8 e^{2 π i (0.5)}, 0.2 e^{2 π i (0.5)} \end{matrix})$ $(\begin{matrix} 0.81 e^{2 π i (0.51)}, 0.21 e^{2 π i (0.51)} \end{matrix})$ $(\begin{matrix} 0.82 e^{2 π i (0.52)}, 0.22 e^{2 π i (0.52)} \end{matrix})$ $(\begin{matrix} 0.83 e^{2 π i (0.53)}, 0.23 e^{2 π i (0.53)} \end{matrix})$

Ξ _AL-5 $(\begin{matrix} 0.8 e^{2 π i (0.4)}, 0.1 e^{2 π i (0.3)} \end{matrix})$ $(\begin{matrix} 0.81 e^{2 π i (0.41)}, 0.11 e^{2 π i (0.31)} \end{matrix})$ $(\begin{matrix} 0.82 e^{2 π i (0.42)}, 0.12 e^{2 π i (0.32)} \end{matrix})$ $(\begin{matrix} 0.83 e^{2 π i (0.43)}, 0.13 e^{2 π i (0.33)} \end{matrix})$

	Ξ _AT-1	Ξ _AT-2	Ξ _AT-3	Ξ _AT-4
Ξ _AL-1	$(\begin{matrix} 0.9 e^{2 π i (0.8)}, 0.1 e^{2 π i (0.1)} \end{matrix})$	$(\begin{matrix} 0.91 e^{2 π i (0.81)}, 0.11 e^{2 π i (0.11)} \end{matrix})$	$(\begin{matrix} 0.92 e^{2 π i (0.82)}, 0.12 e^{2 π i (0.12)} \end{matrix})$	$(\begin{matrix} 0.93 e^{2 π i (0.83)}, 0.13 e^{2 π i (0.13)} \end{matrix})$
Ξ_AL-2.	$(\begin{matrix} 0.7 e^{2 π i (0.6)}, 0.2 e^{2 π i (0.4)} \end{matrix})$	$(\begin{matrix} 0.71 e^{2 π i (0.61)}, 0.21 e^{2 π i (0.41)} \end{matrix})$	$(\begin{matrix} 0.72 e^{2 π i (0.62)}, 0.22 e^{2 π i (0.42)} \end{matrix})$	$(\begin{matrix} 0.73 e^{2 π i (0.63)}, 0.23 e^{2 π i (0.43)} \end{matrix})$
Ξ _AL-3	$(\begin{matrix} 0.6 e^{2 π i (0.7)}, 0.4 e^{2 π i (0.3)} \end{matrix})$	$(\begin{matrix} 0.61 e^{2 π i (0.71)}, 0.41 e^{2 π i (0.31)} \end{matrix})$	$(\begin{matrix} 0.62 e^{2 π i (0.72)}, 0.42 e^{2 π i (0.32)} \end{matrix})$	$(\begin{matrix} 0.63 e^{2 π i (0.73)}, 0.43 e^{2 π i (0.33)} \end{matrix})$
Ξ _AL-4	$(\begin{matrix} 0.8 e^{2 π i (0.5)}, 0.2 e^{2 π i (0.5)} \end{matrix})$	$(\begin{matrix} 0.81 e^{2 π i (0.51)}, 0.21 e^{2 π i (0.51)} \end{matrix})$	$(\begin{matrix} 0.82 e^{2 π i (0.52)}, 0.22 e^{2 π i (0.52)} \end{matrix})$	$(\begin{matrix} 0.83 e^{2 π i (0.53)}, 0.23 e^{2 π i (0.53)} \end{matrix})$
Ξ _AL-5	$(\begin{matrix} 0.8 e^{2 π i (0.4)}, 0.1 e^{2 π i (0.3)} \end{matrix})$	$(\begin{matrix} 0.81 e^{2 π i (0.41)}, 0.11 e^{2 π i (0.31)} \end{matrix})$	$(\begin{matrix} 0.82 e^{2 π i (0.42)}, 0.12 e^{2 π i (0.32)} \end{matrix})$	$(\begin{matrix} 0.83 e^{2 π i (0.43)}, 0.13 e^{2 π i (0.33)} \end{matrix})$

Table 1 is not needed to be regulated.

Stage 2: With the presence of CPFPBM operator (Equation 12), cumulative the matrix for 𝒻_SC = 1, ℊ_SC = 1. CPFPBM^{𝒻_SC,ℊ_SC} (Ξ_AL-1) = (0.9275e^2πi(0.8626), 0.6085e^2πi(0.6085)) CPFPBM^{𝒻_SC,ℊ_SC} (Ξ_AL-2) = (0.6801e^2πi(0.6198), 0.8419e^2πi(0.9043)) CPFPBM^{𝒻_SC,ℊ_SC} (Ξ_AL-3) = (0.6081e^2πi(0.6677), 0.9113e^2πi(0.8861)) CPFPBM^{𝒻_SC,ℊ_SC} (Ξ_AL-4) = (0.7318e^2πi(0.5492), 0.8529e^2πi(0.9387)) CPFPBM^{𝒻_SC,ℊ_SC} (Ξ_AL-5) = (0.7562e^2πi(0.5074), 0.7742e^2πi(0.8687))

Stage 3: With the presence of aggregated values, demonstrated the SVs. $\begin{matrix} {S^{=}}_{SV} (Ξ_{CP - 1}) = 0.4319, {S^{=}}_{SV} (Ξ_{CP - 2}) \\ = - 0.3399, {S^{=}}_{SV} (Ξ_{CP - 3}) = - 0.4000, \\ {S^{=}}_{SV} (Ξ_{CP - 4}) = - 0.3792, {S^{=}}_{SV} (Ξ_{CP - 5}) = - 0.2623 . \end{matrix}$

Stage 4: In the presence of the SVs, find the beneficial optimal from the large numbers of alternatives. $Ξ_{CP - 1} > Ξ_{CP - 5} > Ξ_{CP - 2} > Ξ_{CP - 4} > Ξ_{CP - 3}$

In the presence of the above-ranking values, we know that the beneficial term is Ξ_CP-1. Further, several scholars have improved their invented works with the help the finding many examples. Here, we illustrated one more example, which includes CIF sort of data.

Example 2. [29] The mathematical five terms Ξ_AL-1, Ξ_AL-2, Ξ_AL-3, Ξ_AL-4, Ξ_AL-4 and four terms Ξ_AT-1: enterprise executive rank, Ξ_AT-2: financial benefit, Ξ_AT-3: development capability, and Ξ_AT-4: community influence, stated the alternatives and attributes in the presence of weight vectors 0.15, 0.25, 0.2, and 0.4. To explain the usage and importance of the MADM scenario, a lot of individuals have described the MADM technique in the scenario of distinct regions. The major procedure of the MADM technique in this scenario are diagnosed with the help of the following stages:

$\begin{matrix} {S^{=}}_{SV} (Ξ_{CP - 1}) = - 0.6652, {S^{=}}_{SV} (Ξ_{CP - 2}) \\ = - 0.4279, {S^{=}}_{SV} (Ξ_{CP - 3}) = - 0.5073, \\ {S^{=}}_{SV} (Ξ_{CP - 4}) = - 0.3792 . \end{matrix}$

In the presence of the SVs, find the beneficial optimal from the large numbers of alternatives. $Ξ_{CP - 4} > Ξ_{CP - 2} > Ξ_{CP - 3} > Ξ_{CP - 1}$

In the presence of the above-ranking values, we know that the beneficial term is Ξ_CP-4. In the consideration of the mathematical terms 𝒻_SC, and ℊ_SC, stated the parameters, we discussed their fluency in Tables 2 and 3.

Table 2

For different values of 𝒻_SC and ℊ_SC = 1

Parameters	Operators	Score values	Ranking values
𝒻_SC = 2	CPFPBM	0.4870, –0.2554, –0.3175, –0.2978, –0.1708	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3
𝒻_SC = 4	CPFPBM	0.5691, –0.0832, –0.1458, –0.1312, 0.0038	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3
𝒻_SC = 6	CPFPBM	0.6209, 0.0169, –0.0458, –0.0327, 0.1075	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3
𝒻_SC = 8	CPFPBM	0.6553, 0.0784, 0.0158, 0.0285, 0.1725	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3
𝒻_SC = 10	CPFPBM	0.6792, 0.1199, 0.05740.07, 0.2167	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3

Table 3

For different values of ℊ_SC and 𝒻_SC = 1

Parameters	Operators	Score values	Ranking values
ℊ_SC = 2	CPFPBM	0.4810, –0.2653, –0.3273, –0.3077, –0.1818	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3
ℊ_SC = 4	CPFPBM	0.5573, –0.0967, –0.1587, –0.1448, –0.0115	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3
ℊ_SC = 6	CPFPBM	0.6073, 0.0050, –0.0571, –0.045, 0.093	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3
ℊ_SC = 8	CPFPBM	0.6413, 0.0679, 0.0058, 0.0175, 0.1593	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3
ℊ_SC = 10	CPFPBM	0.6655, 0.1104, 0.0485, 0.060, 0.2045	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3

Moreover, in the consideration of the mathematical terms 𝒻_SC, and ℊ_SC, stated the parameters, we discussed their fluency in Tables 4 and 5.

Table 4

For different values of 𝒻_SC and ℊ_SC = 1

Parameters	Operators	Score values	Ranking values
𝒻_SC = 2	CPFPBM	–0.503, –0.3094, –0.4081, –0.2253	Ξ_CP-4 > Ξ_CP-2 > Ξ_CP-3 > Ξ_CP-1
𝒻_SC = 4	CPFPBM	–0.2652, –0.108, –0.2125, –0.2455	Ξ_CP-2 > Ξ_CP-3 > Ξ_CP-4 > Ξ_CP-1
𝒻_SC = 6	CPFPBM	–0.1197, 0.0099, –0.0747, 0.1234	Ξ_CP-4 > Ξ_CP-2 > Ξ_CP-3 > Ξ_CP-1
𝒻_SC = 8	CPFPBM	–0.0253, 0.0298, 0.0162, 0.2361	Ξ_CP-4 > Ξ_CP-2 > Ξ_CP-3 > Ξ_CP-1
𝒻_SC = 10	CPFPBM	0.0402, 0.0835, 0.0793, 0.4253	Ξ_CP-4 > Ξ_CP-2 > Ξ_CP-3 > Ξ_CP-1

Table 5

For different values of ℊ_SC and 𝒻_SC = 1

Parameters	Operators	Score values	Ranking values
ℊ_SC = 2	CPFPBM	–0.6125, –0.3435, –0.3344, 0.0913	Ξ_CP-4 > Ξ_CP-3 > Ξ_CP-2 > Ξ_CP-1
ℊ_SC = 4	CPFPBM	–0.4082, –0.1602, –0.1096, 0.1221	Ξ_CP-4 > Ξ_CP-3 > Ξ_CP-2 > Ξ_CP-1
ℊ_SC = 6	CPFPBM	–0.2465, –0.0399, 0.0231, 0.1368	Ξ_CP-4 > Ξ_CP-3 > Ξ_CP-2 > Ξ_CP-1
ℊ_SC = 8	CPFPBM	–0.1361, 0.0393, 0.1066, 0.1459	Ξ_CP-4 > Ξ_CP-3 > Ξ_CP-2 > Ξ_CP-1
ℊ_SC = 10	CPFPBM	–0.0575, 0.0943, 0.1637, 0.1521	Ξ_CP-4 > Ξ_CP-3 > Ξ_CP-2 > Ξ_CP-1

We have gotten the beneficial term in the shape of Ξ_CP-4, in the presence of different values of parameters. Several comparisons are diagnosed here.

5.1 Comparative analysis

Without determining the supremacy of the invented works, any research work has no depth, in the consideration of these effects, we recommended some previous principals explored in the presence of the CIFS and CPFSs and compared it with our invented ideas. Several existing data are suggested here: in [40] studied the technique of ABM operators for IFSs, in [41] studied the Bonferroni mean (BM) operators for PFSs, in [27] dominated some generalized geometric aggregation (GGA) operators for CIFSs, in [28] invented the robust averaging-geometric aggregation (RAGA) operators for CIFSs, in [39] included the prioritized weighted aggregation (PWA) operators for CPFSs, and with diagnosed works. Using the data of Example 1, Table 6, includes the comparative works of the presented and prevailing works.

Table 6
For 𝒻_SC = 1, ℊ_SC = 1, some comparison stated here

Parameters Operators Score values Ranking values

Zhao [40] WABM CannotbeCalculated Cannot be Calculated

Yang et al. [41] WBM CannotbeCalculated Cannot be Calculated

Garg and Rani [27] GGA CannotbeCalculated Cannot be Calculated

Garg and Rani [28] RAGA CannotbeCalculated Cannot be Calculated

Akram et al. [39] PWA 0.4429, –0.2288, –0.3000, –0.2681, –0.1512 Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3

Proposed work CPFPBM 0.4319, –0.3399, –0.4000, –0.3792, –0.2623 Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3

Parameters	Operators	Score values	Ranking values
Zhao [40]	WABM	CannotbeCalculated	Cannot be Calculated
Yang et al. [41]	WBM	CannotbeCalculated	Cannot be Calculated
Garg and Rani [27]	GGA	CannotbeCalculated	Cannot be Calculated
Garg and Rani [28]	RAGA	CannotbeCalculated	Cannot be Calculated
Akram et al. [39]	PWA	0.4429, –0.2288, –0.3000, –0.2681, –0.1512	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3
Proposed work	CPFPBM	0.4319, –0.3399, –0.4000, –0.3792, –0.2623	Ξ_CP-1 > Ξ_CP-5 > Ξ_CP-2 >Ξ_CP-4 > Ξ_CP-3

We know that the operators in [27 , 39– 41] are enabled to demonstrate the solution of the data given in Example 1. Using the data of Example 2, Table 7, includes the comparative works of the presented and prevailing works.

Table 7

For 𝒻_SC = 1, ℊ_SC = 1, some comparison stated here

Parameters	Operators	Score values	Ranking values
Zhao [40]	WABM	CannotbeCalculated	Cannot be Calculated
Yang et al. [41]	WBM	CannotbeCalculated	Cannot be Calculated
Garg and Rani [27]	GGA	0.3524, 0.3944, 0.4263, 0.5373	Ξ_CP-4 > Ξ_CP-3 > Ξ_CP-2 > Ξ_CP-1
Garg and Rani [28]	RAGA	0.5443, 0.5863, 0.6182, 0.7292	Ξ_CP-4 > Ξ_CP-3 > Ξ_CP-2 > Ξ_CP-1
Akram et al. [39]	PWA	0.1514, 0.1924, 0.3243, 0.4363	Ξ_CP-4 > Ξ_CP-3 > Ξ_CP-2 > Ξ_CP-1
Proposed work	CPFPBM	–0.6652, –0.4279, –0.5073, –0.3792	Ξ_CP-4 > Ξ_CP-2 > Ξ_CP-3 > Ξ_CP-1

Under the information in Tables 6 and 7, we know that the operators in [27 , 39– 41] can demonstrate the solution of the data given in Example 2. Hence, our invented theory is more flexible is compared to prevailing theories [27 , 39– 41].

6 Conclusion

One of the most dominant and flexible, called BM operator is the extended form of the arithmetic mean operator, used for simplifying non-dominant and non-feasible problems diagnosed in genuine life scenarios. A lot of aggregation operators are the specific parts of the BM operators under the consideration of different values of parameters which are the main parts of the BM operators. In the presence of the BM operator and a very well-known conception in the scenario of fuzzy set, called complex Pythagorean fuzzy (CPF) setting, the objective of this scenario is to diagnose below:

We diagnosed the CPFPBM operator and utilized their beneficial results with important properties.

A MADM technique is evaluated in the presence of invented operators for CPF settings.

We diagnosed the superiority and efficiency of the invented works with the help of the sensitivity analysis and graphical illustrations to enhance the gap of the research works

In the presence of $\underset{Ξ_{RP}}{\overset{=}{ℕ}} (X^{=}) = \underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}) = 0$ , in the CPF setting, then we pick the CFS.

In the presence of $\underset{Ξ_{IP}}{\overset{=}{M}} (X^{=}) = \underset{Ξ_{IP}}{\overset{=}{ℕ}} (X^{=}) = 0$ , in the CPF setting, then we pick the IFS.

In the future, we will use the principle of Probabilistic linguistic [42 –44], multi-granulation linguistic set [45, 46], hesitant linguistic sets [47], complex q-rung orthopair FSs [48], T-spherical FSs [49], and Complex T-spherical FSs [50, 51] to investigate the proficiency and ability of the initiated work.

Conflict of interest

The authors declare that they have no conflict of interest.

Data availability statement

The authors declare that the data used in this manuscript are hypothetical and anyone can use it without prior permission of the authors by just citing this article.

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