Applications of complex picture fuzzy soft power aggregation operators in multi-attribute decision making

Abstract

The major theme of this analysis is to suggest a new theory in the form of complex picture fuzzy soft (CPFS) information and to initiate their major algebraic laws, score value, and accuracy values. The mathematical form of the CPFS set includes three main functions, called supporting, abstinence, and supporting against terms with a prominent characteristic that the sum of the triplet will lie in the unit interval. Further, in the consideration of the power aggregation operator using generalized t-norm and t-conorm and CPFS information, we diagnosed the mathematical concept of CPFS power averaging (CPFSPA), CPFS weighted power averaging (CPFSWPA), CPFS ordered weighted power averaging (CPFSOWPA), CPFS power geometric (CPFSPG), CPFS weighted power geometric (CPFSWPG), CPFS ordered weighted power geometric (CPFSOWPG). Moreover, the major results and their particular investigation of the invented approaches are also deliberated. Additionally, in the consideration of diagnosed operators using CPFS information, we illustrated a MADM (“multi-attribute decision-making”) tool to find the best option from the family of decisions. Finally, we have shown the supremacy and feasibility of the diagnosed operators with the help of sensitive analysis and geometrical representations.

Introduction

Pattern recognition is a vital part of the decision-making strategy used as an application in the environment of engineering science, networking systems, and medical diagnoses. Pattern recognition is the computerized recognition of patterns and consistencies in information. It has been used in statistical information analysis, image analysis, signal procedure, bioinformatics, and machine learning. Pattern recognition has a lot of implementations in different areas, but various deficiencies were involved in the environment genuine life troubles in the consideration of crisp sets. To enhance the major theme of the research work, Atanassov¹ invented the major and well-known theory of intuitionistic fuzzy set (IFS), by generalizing the theory of fuzzy set (FS)². The main IFS is very feasible and flexible because of its shape, the prominent condition of IFS is described: $0\le {\mathfrak{M}}_{\overline{\overline{\mathfrak{L}}}}\left(\overline{\overline{\varphi }}\right)+{\mathfrak{N}}_{\overline{\overline{\mathfrak{L}}}}\left(\overline{\overline{\varphi }}\right)\le 1$. The fundamental concept of IFS has massively improved than FS for handling awkward and unreliable information. Various applications are diagnosed in the circumstances of many fields. For sample, bipolar soft sets³, geometric operators⁴, generalized operators⁵, simple operators⁶, simple measures⁷, measures using Hausdorff distance⁸, similarity measures⁹, Bonferroni operators¹⁰, generalized Heronian operators¹¹, Heronian operators¹², interval-valued IFSs¹³, cubic IFSs¹⁴ and generalized cubic IFSs¹⁵. Cuong^16,17 diagnosed the main theory of picture FS (PFS) and its applications. The main PFS is very feasible and flexible because of its shape, the prominent condition of PFS is described: $0 \le {\mathfrak{M}}_{{\overline{\overline{{\mathfrak{L}}}} }} \left( {\overline{\overline{\varphi }} } \right) + {\mathcal{A}}_{{\overline{\overline{{\mathfrak{L}}}} }} \left( {\overline{\overline{\varphi }} } \right) + {\mathfrak{N}}_{{\overline{\overline{{\mathfrak{L}}}} }} \left( {\overline{\overline{\varphi }} } \right) \le 1$. The fundamental concept of PFS has massively improved over IFS and FS for handling awkward and unreliable information. Various applications are diagnosed in the circumstances of many fields of decision-making strategy^18,19,20 and operators^21,22,23,24.

FS has a huge number of implementations in the circumstances of medical diagnoses, pattern recognition, clustering analysis, and networking systems. But in various places, the theory of FS has also faced a lot of troubles. For example, if the owner of some well-known company wants to lunch a novel sort of various software in a market, they have given two sorts of information related to each software, called name and production date of the software. For handling the above-cited information, the novel theory of complex FS (CFS)²⁵ was diagnosed as a new strategy for managing genuine life ambiguities. Further, Alkouri and Salleh²⁶ introduced the strategy of complex IFS (CIFS) as a helper for scholars, those who faced complications in selecting the best option during the decision-making process. CIFS has its level for handling awkward and ambiguous information, certain people have considered the theory of CIFS for utilizing it in the circumstances of different fields, for instance, Ali et al.²⁷ diagnosed the complex intuitionistic fuzzy soft sets. Further, the concept of the relationships among any two CIFSs was invented by Jan et al.²⁸, the mathematical shape of complex interval-valued IFSs was diagnosed by Garg and Rani²⁹ and the theory of group utilized in the region of CIFSs was explored by Gulzar et al.³⁰. Akram et al.³¹ diagnosed the main theory of complex PFS (CPFS) and its applications. The main CPFS is very feasible and flexible because of its shape, the prominent condition of CPFS is described: $0\le {\mathfrak{M}}_{\overline{\overline{{\mathfrak{L}}_{R}}}}\left(\overline{\overline{\varphi }}\right)+{\mathcal{A}}_{\overline{\overline{{\mathfrak{L}}_{R}}}}\left(\overline{\overline{\varphi }}\right)+{\mathfrak{N}}_{\overline{\overline{{\mathfrak{L}}_{R}}}}\left(\overline{\overline{\varphi }}\right)\le 1$ and $0\le {\mathfrak{M}}_{\overline{\overline{{\mathfrak{L}}_{I}}}}\left(\overline{\overline{\varphi }}\right)+{\mathcal{A}}_{\overline{\overline{{\mathfrak{L}}_{I}}}}\left(\overline{\overline{\varphi }}\right)+{\mathfrak{N}}_{\overline{\overline{{\mathfrak{L}}_{I}}}}\left(\overline{\overline{\varphi }}\right)\le 1$. The fundamental concept of CPFS has massively improved over CIFS and CFS for handling awkward and unreliable information. Various applications are diagnosed in the circumstances of many fields of decision-making strategy³². To explain the information in the above paragraph, we noticed that the prevailing information has the following major dilemmas:

• 1. How do we develop new and more effective ideas?

• 2. How do we develop a superior shape of aggregation operators, used for evaluating the collection of information?

• 3. How do we evaluate the beneficial preference from the collection of information?

Therefore, the main influence of this theory is to find the solution to the above dilemmas with the help of diagnosed power aggregation operators based on CPFS information.

Additionally, Molodtsov³³ introduced the theory of soft set (SS) by extending the theory of FS and because of their shape all scholars have employed it in different fields, for instance, the theory of fuzzy SS was invented by Maji et al.³⁴. Further, the intuitionistic fuzzy SS was discovered by Maji et al.³⁵ and the interval-valued intuitionistic fuzzy SS was explored by Jiang et al.³⁶. The generalized intuitionistic fuzzy SS was invented by Agarwal et al.³⁷. Similarly, the theory of intuitionistic fuzzy SS and its application in decision-making was discovered by Jiang et al.³⁸. The theory of power aggregation operators for IFS was discovered by Xu³⁹. Further, Rani and Garg⁴⁰ diagnosed the power aggregation operators for CIFSs and CPFSs³². In various situations, the theory of CIFS information has failed because of many complications, for instance, if someone proved information in the shape of yes, abstinence, and no against the value of parameters, where the value of yes, abstinence, and no in the shape of a complex number, then the theory of complex intuitionistic fuzzy soft sets have been invalid. For evaluating the above-complicated situations, the theory of CPFSS and its operational laws are the parts of this manuscript. From the above-cited theory, we also clear that every theory has its limitations because of its structures. Similarly, all prevailing theories have a lot of benefits, keeping the benefits of the SSs and CPFSs, the main contribution of this analysis is to explore the well-known concept of CPFS setting and their laws. The major theme of this analysis is described with the help of various points:

• 1. To pioneer the theory of CPFS information and evaluated their major algebraic laws, score value, and accuracy values.

• 2. To present the theory of CPFSPA, CPFSWPA, CPFSOWPA, CPFSPG, CPFSWPG, and CPFSOWPG operators, and diagnosed their particular cases of the invented approaches.

• 3. To diagnose some real-life situations, we evaluate a MADM tool under the consideration of diagnosed operators to find the best option from the family of decisions.

• 4. To show the supremacy and feasibility of the diagnosed operators with the help of sensitive analysis and geometrical representations.

The major consequence of this organization is the shape: In section Preliminaries, we highlighted various principles of CIFSSs and their feasible and dominant algebraic laws. Additionally, we also recall the basic idea of PAOs and the generalized t-norm (TN) and t-conorm (TCN). In section “Complex picture fuzzy soft settings”, we diagnosed the mathematical concept of CPFSPA, CPFSWPA, CPFSOWPA, CPFSPG, CPFSWPG, and CPFSOWPG. Moreover, the major results and their particular investigation of the invented approaches are also deliberated. In section “Power aggregation operators under CPSF informationPower aggregation operators under CPSF information”, the major results and their particular investigation of the invented approaches are also deliberated. Additionally, in the consideration of diagnosed operators using CPFS information, we illustrated a MADM tool to find the best option from the family of decisions. Finally, we have shown the supremacy and feasibility of the diagnosed operators with the help of sensitive analysis and geometrical representations. Section “Conclusion” contains various concluding remarks.

Preliminaries

The fundamental theme of this section is to highlight various principles of CIFSSs and their feasible and dominant algebraic laws. Additionally, we also recall the basic idea of PAOs and the generalized TN and TCN. Where the mathematical term $\overline{\overline{\mathfrak{X}}}$, proven universal sets and the truth, abstinence, and facility grades are demonstrated in the shape: ${\mathfrak{M}}_{{\mathfrak{L}}_{\mathfrak{k}}}\left(\varphi \right)={\mathfrak{M}}_{{R}_{\mathfrak{k}}}\left(\varphi \right){e}^{i2\pi \left({\mathfrak{M}}_{{I}_{\mathfrak{k}}}\left(\varphi \right)\right)},{\mathcal{A}}_{{\mathfrak{L}}_{\mathfrak{k}}}\left(\varphi \right)={\mathcal{A}}_{{R}_{\mathfrak{k}}}\left(\varphi \right){e}^{i2\pi \left({\mathcal{A}}_{{I}_{\mathfrak{k}}}\left(\varphi \right)\right)},$ and ${\mathfrak{N}}_{{\mathfrak{L}}_{\mathfrak{k}}}\left(\varphi \right)={\mathfrak{N}}_{{R}_{\mathfrak{k}}}\left(\varphi \right){e}^{i2\pi \left({\mathfrak{N}}_{{I}_{\mathfrak{k}}}\left(\varphi \right)\right)}$.

Definition 1

²⁷In the presence of the universal set $\overline{\overline{\mathfrak{X}}}$, the CIFSS $\overline{\overline{{\mathfrak{L}}_{CIFS}}}$ is organized in the structure:

$$ \overline{\overline{{{\mathfrak{L}}_{{CIFS - e_{{\mathfrak{k}}} }} }}} = \left\{ {\left( {{\mathfrak{M}}_{{{\mathfrak{L}}_{{\mathfrak{k}}} }} \left( \varphi \right),{\mathfrak{N}}_{{{\mathfrak{L}}_{{\mathfrak{k}}} }} \left( \varphi \right)} \right):\varphi \in \overline{\overline{{\mathfrak{X}}}} } \right\} $$

(1)

With $0 \le {\mathfrak{M}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathfrak{N}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right) \le 1$ and $0 \le {\mathfrak{M}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathfrak{N}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right) \le 1$. Further, ${\mathcal{R}}_{{{\mathfrak{L}}_{{\mathfrak{k}}} }} \left( \varphi \right) = {\mathcal{R}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right)e^{{i2\pi \left( {{\mathcal{R}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right)} \right)}} = 1 - \left( {{\mathfrak{M}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathfrak{N}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right)} \right){\mathfrak{N}}_{{\overline{\overline{{{\mathfrak{L}}_{{R - {\mathfrak{k}}}} }}} }} \left( {\overline{\overline{\varphi }} } \right)e^{{i2\pi \left( {1 - \left( {{\mathfrak{M}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathfrak{N}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right)} \right)} \right)}}$, represented the neutral grade. The mathematical organization $\overline{\overline{{{\mathfrak{L}}_{{CIFS - i{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{i{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{i{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{i{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{i{\mathfrak{k}}}} }} } \right)}} } \right),i = 1,2, \ldots ,n$, represented the CIFSNs.

Definition 2

²⁷ Considered any two $\overline{\overline{{{\mathfrak{L}}_{{IF - 1{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} } \right),{\mathfrak{k}} = 1,2$, then

$$ \overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} \oplus \overline{\overline{{{\mathfrak{L}}_{IF - 12} }}} = \left( {\begin{array}{*{20}c} {\left( {{\mathfrak{M}}_{{R_{11} }} + {\mathfrak{M}}_{{R_{12} }} - {\mathfrak{M}}_{{R_{11} }} {\mathfrak{M}}_{{R_{12} }} } \right)e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{11} }} + {\mathfrak{M}}_{{I_{12} }} - {\mathfrak{M}}_{{I_{11} }} {\mathfrak{M}}_{{I_{12} }} } \right)}} ,} \\ {{\mathfrak{N}}_{{R_{11} }} {\mathfrak{N}}_{{R_{12} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{11} }} {\mathfrak{N}}_{{I_{12} }} } \right)}} } \\ \end{array} } \right) $$

(2)

$$ \overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} \otimes \overline{\overline{{{\mathfrak{L}}_{IF - 12} }}} = \left( {\begin{array}{*{20}c} {{\mathfrak{M}}_{{R_{11} }} {\mathfrak{M}}_{{R_{12} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{11} }} {\mathfrak{M}}_{{I_{12} }} } \right)}} ,} \\ {\left( {{\mathfrak{N}}_{{R_{11} }} + {\mathfrak{N}}_{{R_{12} }} - {\mathfrak{N}}_{{R_{11} }} {\mathfrak{N}}_{{R_{12} }} } \right)e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{11} }} + {\mathfrak{N}}_{{I_{12} }} - {\mathfrak{N}}_{{I_{11} }} {\mathfrak{N}}_{{I_{12} }} } \right)}} } \\ \end{array} } \right) $$

(3)

$$ \overbrace {\sigma }^{{}}\overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} = \left( {\left( {1 - \left( {1 - {\mathfrak{M}}_{{R_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)e^{{i2\pi \left( {1 - \left( {1 - {\mathfrak{M}}_{{I_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)}} ,{\mathfrak{N}}_{{R_{11} }}^{{{\overbrace {\sigma }{}}}} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{11} }}^{{{\overbrace {\sigma }{}}}} } \right)}} } \right) $$

(4)

$$ \overline{\overline{{{\mathfrak{L}}_{IF - 11} }}}^{{{\overbrace {\sigma }{}}}} = \left( {{\mathfrak{M}}_{{R_{11} }}^{{{\overbrace {\sigma }{}}}} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{11} }}^{{{\overbrace {\sigma }{}}}} } \right)}} ,\left( {1 - \left( {1 - {\mathfrak{N}}_{{R_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)e^{{i2\pi \left( {1 - \left( {1 - {\mathfrak{N}}_{{I_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)}} } \right) $$

(5)

Definition 3

²⁷Considered any two $\overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} } \right),{\mathfrak{k}} = 1,2$, the score value (SV) and accuracy value (AV) are simplified by:

$$ \overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} } \right) = \left| {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} - {\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} + {\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} - {\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right|,\overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} } \right) \in \left[ { - 1,1} \right] $$

(6)

$$ \overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} } \right) = \left| {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} + {\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} + {\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} + {\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right|,\overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} } \right) \in \left[ {0,1} \right] $$

(7)

Considered any two $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} } \right)$ and $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }}^{*} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }}^{*} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }}^{*} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }}^{*} } \right)}} } \right)$, then

• 1. When $\overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) > \overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} > \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$;

• 2. When $\overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) < \overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} < \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$;

• 3. When $\overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) = \overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then.

• (i) When $\overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) > \overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} > \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$;

• (ii) When $\overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) < \overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} < \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$;

• (iii) When $\overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) = \overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} = \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$.

Definition 4

³²Considered any collection of attributes $\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} ,i = 1,2, \ldots ,n$, the PAO is simplified by:

$$ PA\left( {\overline{\overline{{{\mathfrak{L}}_{PI - 1} }}} ,\overline{\overline{{{\mathfrak{L}}_{PI - 2} }}} , \ldots ,\overline{\overline{{{\mathfrak{L}}_{PI - n} }}} } \right) = \mathop \sum \limits_{i = 1}^{n} \frac{{\left( {1 + \overline{\overline{{\mathcal{T}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} } \right)} \right)\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} }}{{\mathop \sum \nolimits_{i = 1}^{n} \left( {1 + \overline{\overline{{\mathcal{T}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} } \right)} \right)}} $$

(8)

The mathematical term $\overline{\overline{{\mathcal{T}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} } \right) = \sum\nolimits_{\begin{subarray}{l} k = 1, \\ k \ne i \end{subarray} }^{n} Sup{\left( {\overline{\overline{{\rm{\mathfrak{L}}_{{PI - i}} }}} .\overline{\overline{{\rm{\mathfrak{L}}_{{PI - k}} }}} } \right)}$, stated the support for $\overline{\overline{{{\mathfrak{L}}_{PI - i} }}}$ and $\overline{\overline{{{\mathfrak{L}}_{PI - k} }}}$, particularized by:

$$ Sup\left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - k} }}} } \right) = 1 - d\left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - k} }}} } \right) $$

(9)

where $d\left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - k} }}} } \right)$, acknowledged the measure for $\overline{\overline{{{\mathfrak{L}}_{PI - i} }}}$ and $\overline{\overline{{{\mathfrak{L}}_{PI - k} }}}$, with a technique that:

• 1. $Sup\left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - k} }}} } \right) \in \left[ {0,1} \right]$;

• 2. $Sup\left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - k} }}} } \right) = Sup\left( {\overline{\overline{{{\mathfrak{L}}_{PI - k} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} } \right)$;

• 3. If $d\left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - k} }}} } \right) \le d\left( {\overline{\overline{{{\mathfrak{L}}_{PI - l} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - q} }}} } \right)$ then $Sup\left( {\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - k} }}} } \right) \ge Sup\left( {\overline{\overline{{{\mathfrak{L}}_{PI - l} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - q} }}} } \right)$.

Definition 5

¹⁵A mapping $\overline{\overline{{\mathcal{T}}}} :\left[ {0,1} \right] \times \left[ {0,1} \right] \to \left[ {0,1} \right]$ is particularized TN when $\overline{\overline{{\mathcal{T}}}}$, justified the boundary, monotonicity, commutative, and associativity techniques. Where $\overline{\overline{{\mathfrak{S}}}} \left( {\overline{\overline{\varphi }} ,\overline{\overline{{\mathfrak{y}}}} } \right) = 1 - \overline{\overline{{\mathcal{T}}}} \left( {1 - \overline{\overline{\varphi }} ,1 - \overline{\overline{{\mathfrak{y}}}} } \right)$, diagnosed the TN and the general shape of Archimedean TN and TCN is diagnosed by: $\overline{\overline{{\mathcal{T}}}} \left( {\overline{\overline{\varphi }} ,\overline{\overline{{\mathfrak{y}}}} } \right) = \overbrace {{{\mathfrak{f}}^{ - 1} }}^{{}}\left( {\overbrace {{\mathfrak{f}}}^{{}}\left( {\overline{\overline{\varphi }} } \right) + \overbrace {{\mathfrak{f}}}^{{}}\left( {\overline{\overline{{\mathfrak{y}}}} } \right)} \right)$ and $\overline{\overline{{\mathfrak{S}}}} \left( {\overline{\overline{\varphi }} ,\overline{\overline{{\mathfrak{y}}}} } \right) = \overbrace {{{\mathfrak{g}}^{ - 1} }}^{{}}\left( {\overbrace {{\mathfrak{g}}}^{{}}\left( {\overline{\overline{\varphi }} } \right) + \overbrace {{\mathfrak{g}}}^{{}}\left( {\overline{\overline{{\mathfrak{y}}}} } \right)} \right)$, based on the continuous increasing (or decreasing) function with $\overbrace {{\mathfrak{f}}}^{{}}\left( 1 \right) = 0,\;\;\overbrace {{\mathfrak{g}}}^{{}}\left( 0 \right) = 0$ and $\overbrace {{\mathfrak{f}}}^{{}}\left( {\overline{\overline{\varphi }} } \right) = 1 - \overbrace {{\mathfrak{g}}}^{{}}\left( {1 - \overline{\overline{\varphi }} } \right)$.

Complex picture fuzzy soft settings

The major theme of this analysis is to suggest a new theory in the form of CPFS information and invented their major algebraic laws, score value, and accuracy values. The mathematical form of the CPFS set includes three main functions, called supporting, abstinence, and supporting against terms with a prominent characteristic that is the sum of the triplet will lie in the unit interval.

Definition 6

In the presence of the universal set $\overline{\overline{{\mathfrak{X}}}}$, the CPFSS $\overline{\overline{{{\mathfrak{L}}_{CIFS} }}}$ is organized in the structure:

$$ \overline{\overline{{{\mathfrak{L}}_{{CIFS - e_{{\mathfrak{k}}} }} }}} = \left\{ {\left( {{\mathfrak{M}}_{{{\mathfrak{L}}_{{\mathfrak{k}}} }} \left( \varphi \right),{\mathcal{A}}_{{{\mathfrak{L}}_{{\mathfrak{k}}} }} \left( \varphi \right),{\mathfrak{N}}_{{{\mathfrak{L}}_{{\mathfrak{k}}} }} \left( \varphi \right)} \right):\varphi \in \overline{\overline{{\mathfrak{X}}}} } \right\} $$

(10)

With $0 \le {\mathfrak{M}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathcal{A}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathfrak{N}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right) \le 1$ and $0 \le {\mathfrak{M}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathcal{A}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathfrak{N}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right) \le 1$. Further, ${\mathcal{R}}_{{{\mathfrak{L}}_{{\mathfrak{k}}} }} \left( \varphi \right) = {\mathcal{R}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right)e^{{i2\pi \left( {{\mathcal{R}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right)} \right)}} = 1 - \left( {{\mathfrak{M}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathcal{A}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathfrak{N}}_{{R_{{\mathfrak{k}}} }} \left( \varphi \right)} \right)e^{{i2\pi \left( {1 - \left( {{\mathfrak{M}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathcal{A}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right) + {\mathfrak{N}}_{{I_{{\mathfrak{k}}} }} \left( \varphi \right)} \right)} \right)}}$, represented the neutral grade. The mathematical organization $\overline{\overline{{{\mathfrak{L}}_{{CIFS - i{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{i{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{i{\mathfrak{k}}}} }} } \right)}} ,{\mathcal{A}}_{{R_{{i{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{{i{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{i{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{i{\mathfrak{k}}}} }} } \right)}} } \right),i = 1,2, \ldots ,n$, represented the CPFSNs.

Definition 7

Considered any two $\overline{\overline{{{\mathfrak{L}}_{{IF - 1{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathcal{A}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} } \right),{\mathfrak{k}} = 1,2$, then

$$ \overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} \oplus \overline{\overline{{{\mathfrak{L}}_{IF - 12} }}} = \left( {\begin{array}{*{20}c} {\left( {{\mathfrak{M}}_{{R_{11} }} + {\mathfrak{M}}_{{R_{12} }} - {\mathfrak{M}}_{{R_{11} }} {\mathfrak{M}}_{{R_{12} }} } \right)e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{11} }} + {\mathfrak{M}}_{{I_{12} }} - {\mathfrak{M}}_{{I_{11} }} {\mathfrak{M}}_{{I_{12} }} } \right)}} ,} \\ {{\mathcal{A}}_{{R_{11} }} {\mathcal{A}}_{{R_{12} }} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{11} }} {\mathcal{A}}_{{I_{12} }} } \right)}} ,{\mathfrak{N}}_{{R_{11} }} {\mathfrak{N}}_{{R_{12} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{11} }} {\mathfrak{N}}_{{I_{12} }} } \right)}} } \\ \end{array} } \right) $$

(11)

$$ \overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} \otimes \overline{\overline{{{\mathfrak{L}}_{IF - 12} }}} = \left( {\begin{array}{*{20}c} {{\mathfrak{M}}_{{R_{11} }} {\mathfrak{M}}_{{R_{12} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{11} }} {\mathfrak{M}}_{{I_{12} }} } \right)}} ,} \\ {\left( {{\mathcal{A}}_{{R_{11} }} + {\mathcal{A}}_{{R_{12} }} - {\mathcal{A}}_{{R_{11} }} {\mathcal{A}}_{{R_{12} }} } \right)e^{{i2\pi \left( {{\mathcal{A}}_{{I_{11} }} + {\mathcal{A}}_{{I_{12} }} - {\mathcal{A}}_{{I_{11} }} {\mathcal{A}}_{{I_{12} }} } \right)}} ,} \\ {\left( {{\mathfrak{N}}_{{R_{11} }} + {\mathfrak{N}}_{{R_{12} }} - {\mathfrak{N}}_{{R_{11} }} {\mathfrak{N}}_{{R_{12} }} } \right)e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{11} }} + {\mathfrak{N}}_{{I_{12} }} - {\mathfrak{N}}_{{I_{11} }} {\mathfrak{N}}_{{I_{12} }} } \right)}} } \\ \end{array} } \right) $$

(12)

$$ \overbrace {\sigma }^{{}}\overline{\overline{{{\mathfrak{L}}_{IF - 11} }}} = \left( {\begin{array}{*{20}c} {\left( {1 - \left( {1 - {\mathfrak{M}}_{{R_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)e^{{i2\pi \left( {1 - \left( {1 - {\mathfrak{M}}_{{I_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)}} ,} \\ {{\mathcal{A}}_{{R_{11} }}^{{{\overbrace {\sigma }{}}}} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{11} }}^{{{\overbrace {\sigma }{}}}} } \right)}} ,{\mathfrak{N}}_{{R_{11} }}^{{{\overbrace {\sigma }{}}}} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{11} }}^{{{\overbrace {\sigma }{}}}} } \right)}} } \\ \end{array} } \right) $$

(13)

$$ \overline{\overline{{{\mathfrak{L}}_{IF - 11} }}}^{{{\overbrace {\sigma }{}}}} = \left( {\begin{array}{*{20}c} {{\mathfrak{M}}_{{R_{11} }}^{{{\overbrace {\sigma }{}}}} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{11} }}^{{{\overbrace {\sigma }{}}}} } \right)}} ,\left( {1 - \left( {1 - {\mathcal{A}}_{{R_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)e^{{i2\pi \left( {1 - \left( {1 - {\mathcal{A}}_{{I_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)}} ,} \\ {\left( {1 - \left( {1 - {\mathfrak{N}}_{{R_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)e^{{i2\pi \left( {1 - \left( {1 - {\mathfrak{N}}_{{I_{11} }} } \right)^{{{\overbrace {\sigma }{}}}} } \right)}} } \\ \end{array} } \right) $$

(14)

Definition 8

Considered any two $\overline{\overline{{{\mathfrak{L}}_{{IF - 1{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathcal{A}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} } \right),{\mathfrak{k}} = 1,2$, the score value (SV) and accuracy value (AV) are simplified by:

$$ \overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{IF - 1{\mathfrak{k}}}} }}} } \right) = \left| {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} - {\mathcal{A}}_{{R_{{1{\mathfrak{k}}}} }} - {\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} + {\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} - {\mathcal{A}}_{{I_{{1{\mathfrak{k}}}} }} - {\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right|,\overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{IF - 1{\mathfrak{k}}}} }}} } \right) \in \left[ { - 1,1} \right] $$

(15)

$$ \overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) = \left| {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} + {\mathcal{A}}_{{R_{{1{\mathfrak{k}}}} }} + {\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} + {\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} + {\mathcal{A}}_{{I_{{1{\mathfrak{k}}}} }} + {\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right|,\overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) \in \left[ {0,1} \right] $$

(16)

Considered any two $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathcal{A}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} } \right)$ and $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }}^{*} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }}^{*} } \right)}} ,{\mathcal{A}}_{{R_{{1{\mathfrak{k}}}} }}^{*} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{{1{\mathfrak{k}}}} }}^{*} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }}^{*} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }}^{*} } \right)}} } \right)$, then

• 1. When $\overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) > \overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} > \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$;

• 2. When $\overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) < \overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} < \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$;

• 3. When $\overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) = \overline{\overline{{\mathcal{S}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then.

• 1. When $\overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) > \overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} > \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$;

• 2. When $\overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) < \overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} < \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$;

• 3. When $\overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} } \right) = \overline{\overline{{\mathcal{H}}}} \left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*} } \right),$ then $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} = \overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{*}$.

Further, we try to construct the general shape of certain algebraic laws based on CPFSSs.

Definition 9

Considered any two $\overline{\overline{{{\mathfrak{L}}_{{IF - i{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathcal{A}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} } \right),{\mathfrak{k}} = 1,2$ with ${\overbrace {\sigma }}>0$, then

$$\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\oplus \overline{\overline{{\mathfrak{L}}_{CIF-12}}}=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{12}}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{12}}\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{12}}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{12}}\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{12}}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{12}}\right)\right)\right)}\end{array}\right) $$

(17)

$$\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\otimes \overline{\overline{{\mathfrak{L}}_{CIF-12}}}=\left(\begin{array}{c} {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{M}}_{{R}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{M}}_{{R}_{12}}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{M}}_{{I}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{M}}_{{I}_{12}}\right)\right)\right)},\\ {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathcal{A}}_{{R}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathcal{A}}_{{R}_{12}}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathcal{A}}_{{I}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathcal{A}}_{{I}_{12}}\right)\right)\right)},\\ {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{N}}_{{R}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{N}}_{{R}_{12}}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{N}}_{{I}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{N}}_{{I}_{12}}\right)\right)\right)}\end{array}\right)$$

(18)

$$ {\overbrace {\sigma }}\overline{\overline{{\mathfrak{L}}_{CIF-1\mathfrak{k}}}}=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{R}_{1\mathfrak{k}}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{I}_{1\mathfrak{k}}}\right)\right)}, {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{R}_{1\mathfrak{k}}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{I}_{1\mathfrak{k}}}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{R}_{1\mathfrak{k}}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{I}_{1\mathfrak{k}}}\right)\right)}\end{array}\right) $$

(19)

$${\overline{\overline{{\mathfrak{L}}_{CIF-1\mathfrak{k}}}}}^{ {\overbrace {\sigma }}}=\left(\begin{array}{c} {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{M}}_{{R}_{1\mathfrak{k}}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{M}}_{{I}_{1\mathfrak{k}}}\right)\right)}, {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathcal{A}}_{{R}_{1\mathfrak{k}}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathcal{A}}_{{I}_{1\mathfrak{k}}}\right)\right)},\\ {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{N}}_{{R}_{1\mathfrak{k}}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{N}}_{{I}_{1\mathfrak{k}}}\right)\right)}\end{array}\right) $$

(20)

Theorem 1

Considered any two $\overline{\overline{{{\mathfrak{L}}_{{IF - 1{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathcal{A}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} } \right),{\mathfrak{k}} = 1,2$ with $\overbrace {{\sigma_{i} }}^{{}} > 0,i = 1,2$, then

• 1. $\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} \oplus \overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}} = \overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}} \oplus \overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}}$;

• 2. $\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} \otimes \overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}} = \overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}} \otimes \overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}}$;

• 3. $\overbrace {\sigma }^{{}}\left( {\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} \oplus \overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}} } \right) = \overbrace {\sigma }^{{}}\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} \oplus \overbrace {\sigma }^{{}}\overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}}$;

• 4. $\left( {\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} \otimes \overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}} } \right)^{{{\overbrace {\sigma }{}}}} = \overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}}^{{{\overbrace {\sigma }{}}}} \otimes \overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}}^{{{\overbrace {\sigma }{}}}}$;

• 5. $\overbrace {{\sigma_{1} }}^{{}}\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} \oplus \overbrace {{\sigma_{2} }}^{{}}\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} = \left( {\overbrace {{\sigma_{1} }}^{{}} + \overbrace {{\sigma_{2} }}^{{}}} \right)\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}}$;

• 6. $\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}}^{{\overbrace {{\sigma_{1} }}^{{}}}} \otimes \overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}}^{{\overbrace {{\sigma_{2} }}^{{}}}} = \overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}}^{{\left( {\overbrace {{\sigma_{1} }}^{{}} + \overbrace {{\sigma_{2} }}^{{}}} \right)}}$.

Proof

We prove Eqs. (3) and (5), because Parts (1), (2), (4), and (6) are straightforward. Assume that ${\overbrace {\sigma }}\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\oplus \overline{\overline{{\mathfrak{L}}_{CIF-12}}}\right)$, then

$$ {\overbrace {\sigma }}\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\oplus \overline{\overline{{\mathfrak{L}}_{CIF-12}}}\right)= {\overbrace {\sigma }}\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{12}}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{12}}\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{12}}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{12}}\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{12}}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{12}}\right)\right)\right)}\end{array}\right)$$

$$=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }\left( {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{12}}\right)\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }\left( {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{11}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{12}}\right)\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}\left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{12}}\right)\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}\left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{12}}\right)\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}\left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{12}}\right)\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}\left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{11}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{12}}\right)\right)\right)\right)\right)}\end{array}\right)$$

$$=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{I}_{11}}\right)\right)}, {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{I}_{11}}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{I}_{11}}\right)\right)}\end{array}\right)\oplus \left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{R}_{12}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{I}_{12}}\right)\right)}, {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{R}_{12}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{I}_{12}}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{R}_{12}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{I}_{12}}\right)\right)}\end{array}\right)$$

$$= {\overbrace {\sigma }}\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\oplus {\overbrace {\sigma }}\overline{\overline{{\mathfrak{L}}_{CIF-12}}}.$$

Hence, ${\overbrace {\sigma }}\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\oplus \overline{\overline{{\mathfrak{L}}_{CIF-12}}}\right)= {\overbrace {\sigma }}\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\oplus {\overbrace {\sigma }}\overline{\overline{{\mathfrak{L}}_{CIF-12}}}$. Assume that ${\overbrace {\sigma }}_{1}\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\oplus {\overbrace {\sigma }}_{2}\overline{\overline{{\mathfrak{L}}_{CIF-11}}}$, then

$$ {\overbrace {\sigma }}_{1}\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\oplus {\overbrace {\sigma }}_{2}\overline{\overline{{\mathfrak{L}}_{CIF-11}}}=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }}_{1} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }}_{1} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{I}_{11}}\right)\right)}, {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{1}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{1}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{I}_{11}}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{1}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{1}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{I}_{11}}\right)\right)}\end{array}\right)\oplus \left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }}_{2} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }}_{2} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{I}_{11}}\right)\right)}, {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{2}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{2}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{I}_{11}}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{2}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{2}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{I}_{11}}\right)\right)}\end{array}\right)$$

$$=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }}_{1} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{R}_{11}}+ {\overbrace {\sigma }}_{2} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }}_{1} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{I}_{11}}+ {\overbrace {\sigma }}_{2} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{I}_{11}}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{1}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{R}_{11}}+ {\overbrace {\sigma }}_{2}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{1}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{I}_{11}}+ {\overbrace {\sigma }}_{2}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{I}_{11}}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{1}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{R}_{11}}+ {\overbrace {\sigma }}_{2}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{R}_{11}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}_{1}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{I}_{11}}+ {\overbrace {\sigma }}_{2}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{I}_{11}}\right)\right)}\end{array}\right)$$

$$=\left( {\overbrace {\sigma }}_{1}+ {\overbrace {\sigma }}_{2}\right)\overline{\overline{{\mathfrak{L}}_{CIF-11}}}.$$

Hence, $\overbrace {{\sigma_{1} }}^{{}}\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} \oplus \overbrace {{\sigma_{2} }}^{{}}\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} = \left( {\overbrace {{\sigma_{1} }}^{{}} + \overbrace {{\sigma_{2} }}^{{}}} \right)\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}}$.

Power aggregation operators under CPSF information

The major theme of this analysis is to investigate the consideration of power aggregation operator using generalized t-norm and t-conorm and CPFS information, we diagnosed the mathematical concept of CPFSPA, CPFSWPA, CPFSOWPA, CPFSPG, CPFSWPG, CPFSOWPG. Moreover, the major results and their particular investigation of the invented approaches are also deliberated. Various important techniques called averaging, Einstein, and Hamacher operators are investigated using $\overbrace {{\mathfrak{g}}}^{{}}\left( \varphi \right) = - \log \left( \varphi \right),\overbrace {{\mathfrak{g}}}^{{}}\left( \varphi \right) = \log \left( {\frac{2 - \varphi }{\varphi }} \right),\varphi \ne 0$, and $\overbrace {{\mathfrak{g}}}^{{}}\left( \varphi \right) = \log \left( {\overbrace {\sigma }^{{}} + \frac{{\left( {1 - \overbrace {\sigma }^{{}}} \right)\varphi }}{\varphi }} \right),\overbrace {\sigma }^{{}} \in \left( {0,\infty } \right),\varphi \ne 0$. Throughout this manuscript, the term $\overline{\overline{{{\mathfrak{L}}_{{CIFS - i{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathcal{A}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{1{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{1{\mathfrak{k}}}} }} } \right)}} } \right),i = 1,2, \ldots ,n$, represented the CPFSNs with $\overbrace {{\sigma_{i} }}^{{}} > 0$.

Definition 10

When $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} \in \overline{\overline{{\Xi }}}$, then the CPFSPA operator is simplified by:

$$ CPFSPA:\overline{\overline{{\Xi }}}^{n} \to \overline{\overline{{\Xi }}} $$

by

$$ CPFSPA\left( {\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} ,\overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}} , \ldots ,\overline{\overline{{{\mathfrak{L}}_{CIF - nm} }}} } \right) = \oplus_{{{\mathfrak{k}} = 1}}^{m} \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{\mathfrak{k}}} \oplus_{i = 1}^{n} \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} } \right)} \right) $$

(21)

where $\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{\mathfrak{k}}} = \frac{{\left( {1 + \overline{\overline{{\mathcal{T}}}}_{{\mathfrak{k}}} } \right)}}{{\mathop \sum \nolimits_{{{\mathfrak{k}} = 1}}^{m} \left( {1 + \overline{\overline{{\mathcal{T}}}}_{{\mathfrak{k}}} } \right)}},\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} = \frac{{\left( {1 + \rm {\Re }_{i} } \right)}}{{\mathop \sum \nolimits_{i = 1}^{n} \left( {1 + \overline{\overline{\rm {\Re }}}_{i} } \right)}}$, and $\overline{\overline{{\rm {\Re }_{i} }}} = \mathop \sum \limits_{{\begin{array}{*{20}c} {k = 1} \\ {i \ne k} \\ \end{array} }}^{n} Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} .\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}} } \right),\overline{\overline{{{\mathcal{T}}_{{\mathfrak{k}}} }}} = \mathop \sum \limits_{{\begin{array}{*{20}c} {k = 1} \\ {i \ne k} \\ \end{array} }}^{m} Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - {\mathfrak{k}}}} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} } \right)$, and $Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} .\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}} } \right)$, simplified the support for $\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}}$ and $\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}}$.

Theorem 2

Using the information in Eq. (21), we get

$$CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{i\mathfrak{k}}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{i\mathfrak{k}}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{i\mathfrak{k}}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}\right)\right)\right)\right)}\end{array}\right)$$

(22)

Proof

Assume $n=1$, then using the information in Eq. (22), we get

$$CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-1m}}}\right)={\oplus }_{\mathfrak{k}=1}^{m}\left({\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\overline{\overline{{\mathfrak{L}}_{PI-1\mathfrak{k}}}}\right)$$

$$= \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{1 {\mathfrak{k}}}} }} } \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{1 {\mathfrak{k}}}} }} } \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{1 {\mathfrak{k}}}} }} } \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{1 {\mathfrak{k}}}} }} } \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{1 {\mathfrak{k}}}} }} } \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{1 {\mathfrak{k}}}} }} } \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

$$=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{1}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{1}} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{1\mathfrak{k}}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{1}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{1}} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{1\mathfrak{k}}}\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{1}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{1}}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{1\mathfrak{k}}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{1}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{1}}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{1\mathfrak{k}}}\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{1}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{1}}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{1\mathfrak{k}}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{1}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{1}}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{1\mathfrak{k}}}\right)\right)\right)\right)}\end{array}\right)$$

Similarly, assume that $m=1$, then we get

$$ \begin{gathered} CPFSPA\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - n1}} }}} } \right) = \oplus _{{i = 1}}^{n} \left( { {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overline{\overline{{ {\mathfrak{L}}_{{PI - i1}} }}} } \right) \hfill \\ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{1} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i1}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{1} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i1}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{1} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i1}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{1} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i1}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{1} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i1}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{1} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i1}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

Hence Eq. (22) is held for $m = n = 1$. Additionally, we suppose that Eq. (22) is also held for $m = k_{1} ,n = k_{2} + 1$ and $m = k_{1} + 1,n = k_{2}$, then for $m = k_{1} + 1,n = k_{2} + 1$, we have

$$ \oplus_{{{\mathfrak{k}} = 1}}^{{k_{1} + 1}} \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{\mathfrak{k}}} \oplus_{i = 1}^{{k_{2} + 1}} \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} } \right)} \right) = \oplus_{{{\mathfrak{k}} = 1}}^{{k_{1} }} \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{\mathfrak{k}}} \oplus_{i = 1}^{{k_{2} + 1}} \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} } \right)} \right) \oplus \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{k_{1} + !}} \oplus_{i = 1}^{{k_{2} + 1}} \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} } \right)} \right) $$

$$ \begin{gathered} \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} }} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} }} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} }} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} }} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} }} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} }} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} \mathop { {\mathbb{N}}_{i} }\limits^{} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \oplus \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( { {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{k_{1} + 1}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( { {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{k_{1} + 1}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( { {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{k_{1} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{k_{1} + 1}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( { {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{k_{1} + 1}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( { {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{k_{1} + 1}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( { {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{k_{1} + 1}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{{k_{1} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{{k_{2} + 1}} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

Hence Eq. (22) is held for $m={k}_{1}+1,n={k}_{2}+1$, therefore it is also true for all positive integer $m,n$.

Property 1 (Idempotency)

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}=\overline{\overline{{\mathfrak{L}}_{CIF}}}$, then

$$ CPFSPA\left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{{\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{{\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \overline{\overline{{{\mathfrak{L}}_{{CIF}} }}} ~ CPFSPA\left( {\overline{\overline{{{\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{{\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{{\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \overline{\overline{{{\mathfrak{L}}_{{CIF}} }}} ~ $$

(23)

Proof

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}=\overline{\overline{{\mathfrak{L}}_{CIF}}}$, then based on Eq. (22), we get

$$ CPFSPA\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

$$=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{R}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{I}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{R}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{I}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{R}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{I}\right)\right)}\end{array}\right)$$

$$=\left({\mathfrak{M}}_{R}{e}^{i2\pi \left({\mathfrak{M}}_{I}\right)},{\mathcal{A}}_{R}{e}^{i2\pi \left({\mathcal{A}}_{I}\right)},{\mathfrak{N}}_{R}{e}^{i2\pi \left({\mathfrak{N}}_{I}\right)}\right)=\overline{\overline{{\mathfrak{L}}_{CIF}}}.$$

Property 2

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}$ and $\overline{\overline{{\mathfrak{L}}_{CIF}}}$ be any two CPFSNs, then

$$ CPFSPA\left( {\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} \oplus \overline{\overline{{{\mathfrak{L}}_{CIF} }}} ,\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} \oplus \overline{\overline{{{\mathfrak{L}}_{CIF} }}} , \ldots ,\overline{\overline{{{\mathfrak{L}}_{CIF - nm} }}} \oplus \overline{\overline{{{\mathfrak{L}}_{CIF} }}} } \right) = CPFSPA\left( {\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} ,\overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}} , \ldots ,\overline{\overline{{{\mathfrak{L}}_{CIF - nm} }}} } \right) \oplus \overline{\overline{{{\mathfrak{L}}_{CIF - nm} }}} \overline{\overline{{{\mathfrak{L}}_{CIF} }}} $$

(24)

Proof

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}$ and $\overline{\overline{{\mathfrak{L}}_{CIF}}}$ be any two CPFSNs, then

$$\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}\oplus \overline{\overline{{\mathfrak{L}}_{CIF}}}=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{i\mathfrak{k}}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{R}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{I}\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{i\mathfrak{k}}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{R}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{I}\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{i\mathfrak{k}}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{R}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{I}\right)\right)\right)}\end{array}\right)$$

$$ \begin{gathered} CPFSPA\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} \oplus \overline{\overline{{ {\mathfrak{L}}_{{CIF}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} \oplus \overline{\overline{{ {\mathfrak{L}}_{{CIF}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} \oplus \overline{\overline{{ {\mathfrak{L}}_{{CIF}} }}} } \right) \hfill \\ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{R} } \right)} \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{I} } \right)} \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{R} } \right)} \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{I} } \right)} \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{R} } \right)} \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{I} } \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$= \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{R} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{I} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{R} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{I} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{R} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{I} } \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

$$ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{R} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{I} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{R} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{I} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{R} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{I} } \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

$$ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \oplus \left( { {\mathfrak{M}}_{R} e^{{i2\pi \left( { {\mathfrak{M}}_{I} } \right)}} , {\mathcal{A}}_{R} e^{{i2\pi \left( { {\mathcal{A}}_{I} } \right)}} , {\mathfrak{N}}_{R} e^{{i2\pi \left( { {\mathfrak{N}}_{I} } \right)}} } \right) $$

$$=CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)\oplus \overline{\overline{{\mathfrak{L}}_{CIF}}}.$$

Property 3

Prove that

$$CPFSPA\left( {\overbrace {\sigma }}\overline{\overline{{\mathfrak{L}}_{CIF-11}}}, {\overbrace {\sigma }}\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots , {\overbrace {\sigma }}\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)= {\overbrace {\sigma }}CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)$$

(25)

Property 4

We know that ${\overbrace {\sigma }}>0$, then

$$ {\overbrace {\sigma }}\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{R}_{i\mathfrak{k}}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( {\overbrace {\sigma }} \overbrace {\mathfrak{f} }{\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)\right)}, {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{R}_{i\mathfrak{k}}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathcal{A}}_{{I}_{i\mathfrak{k}}}\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{R}_{i\mathfrak{k}}}\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left( {\overbrace {\sigma }}{\overbrace {\mathfrak{g} }}{\mathfrak{N}}_{{I}_{i\mathfrak{k}}}\right)\right)}\end{array}\right)$$

then,

$$ \begin{gathered} CPFSPA\left( {\overbrace {\sigma }^{{}}\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overbrace {\sigma }^{{}}\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overbrace {\sigma }^{{}}\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) \hfill \\ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{f}}}^{{}} {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{f}}}^{{}} {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} \begin{gathered} CPFSPA\left( {\overbrace {\sigma }^{{}}\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overbrace {\sigma }^{{}}\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overbrace {\sigma }^{{}}\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) \hfill \\ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{f}}}^{{}} {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{f}}}^{{}} {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{f}}}^{{}} {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{f}}}^{{}} {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}} {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

$$ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{f}}}^{{}} {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{f}}}^{{}} {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{g}}}^{{}} {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{g}}}^{{}} {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{g}}}^{{}} {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace {\sigma }^{{}}\overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{g}}}^{{}} {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

$$= {\overbrace {\sigma }}CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right).$$

Property 4

Assume that $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}=\left({\mathfrak{M}}_{{R}_{i\mathfrak{k}}}{e}^{i2\pi \left({\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)},{\mathcal{A}}_{{R}_{i\mathfrak{k}}}{e}^{i2\pi \left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}\right)},{\mathfrak{N}}_{{R}_{i\mathfrak{k}}}{e}^{i2\pi \left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}\right)}\right),\mathfrak{k}=\mathrm{1,2},$ and $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}^{*}}}=\left({\mathfrak{M}}_{{R}_{i\mathfrak{k}}}^{*}{e}^{i2\pi \left({\mathfrak{M}}_{{I}_{i\mathfrak{k}}}^{*}\right)},{\mathcal{A}}_{{R}_{i\mathfrak{k}}}^{*}{e}^{i2\pi \left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}^{*}\right)},{\mathfrak{N}}_{{R}_{i\mathfrak{k}}}^{*}{e}^{i2\pi \left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}^{*}\right)}\right)$, then

$$CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\oplus \overline{\overline{{\mathfrak{L}}_{CIF-11}^{*}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}}\oplus \overline{\overline{{\mathfrak{L}}_{CIF-12}^{*}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\oplus \overline{\overline{{\mathfrak{L}}_{CIF-nm}^{*}}}\right)=CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)\oplus CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}^{*}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}^{*}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}^{*}}}\right) $$

(26)

Proof

Assume $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}^{*}}}=\left({\mathfrak{M}}_{{R}_{i\mathfrak{k}}}^{*}{e}^{i2\pi \left({\mathfrak{M}}_{{I}_{i\mathfrak{k}}}^{*}\right)},{\mathcal{A}}_{{R}_{i\mathfrak{k}}}^{*}{e}^{i2\pi \left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}^{*}\right)},{\mathfrak{N}}_{{R}_{i\mathfrak{k}}}^{*}{e}^{i2\pi \left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}^{*}\right)}\right)$, then

$$\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}\oplus \overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}^{*}}}=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{i\mathfrak{k}}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{i\mathfrak{k}}}^{*}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left( \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)+ \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{i\mathfrak{k}}}^{*}\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{i\mathfrak{k}}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{i\mathfrak{k}}}^{*}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}^{*}\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{i\mathfrak{k}}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{i\mathfrak{k}}}^{*}\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left({\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}\right)+{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}^{*}\right)\right)\right)}\end{array}\right)$$

then,

$$ \begin{gathered} CPFSPA\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} \oplus \overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}}^{*} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} \oplus \overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}}^{*} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} \oplus \overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}}^{*} }}} } \right) \hfill \\ \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

$$ \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right) + \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

$$ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right) + \mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right) + \mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right) + \mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right) + \mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right) + \mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right) + \mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

$$ = \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \oplus \left( \begin{gathered} \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right) \hfill \\ e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{1} \left( {\mathop \sum \limits_{{i = 1}}^{n} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} \left( {\overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }}^{*} } \right)} \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) $$

$$=CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)\oplus CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}^{*}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}^{*}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}^{*}}}\right).$$

Important cases of the invented work using the information in Eq. (22) are described here.

• 1. Assume ${\overbrace {\mathfrak{g} }}\left(\varphi \right)=-\mathrm{log}\left(\varphi \right)$ in Eq. (22), then

  $$ \begin{gathered} CPFSPA\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \hfill \\ \left( \begin{gathered} 1 - \mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( {1 - {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} e^{{i2\pi \left( {1 - \mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( {1 - {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} } \right)}} , \hfill \\ \mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} e^{{i2\pi \left( {\mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} } \right)}} , \hfill \\ \mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} e^{{i2\pi \left( {\mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} } \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

  (27)

  Stated the CPFS Archimedean weighted averaging (CPFSAWA) operator.

• 2. Assume $\overbrace {{\mathfrak{g}}}^{{}}\left( \varphi \right) = \log \left( {\frac{2 - \varphi }{\varphi }} \right),\overline{\overline{\varphi }} \ne 0$ in Eq. (22), then

  $$ \begin{gathered} CPFSPA\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \hfill \\ \left( \begin{gathered} \frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }} \hfill \\ e^{{i2\pi \left( {\frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}} \right)}} , \hfill \\ \frac{{2\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {2 - {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }} \hfill \\ e^{{i2\pi \left( {\frac{{2\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {2 - {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}} \right)}} , \hfill \\ \frac{{2\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {2 - {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{N}}_{{\overline{\overline{{R_{{i {\mathfrak{k}}}} {\mathfrak{L}}_{{R - i {\mathfrak{k}}}} }}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }} \hfill \\ e^{{i2\pi \left( {\frac{{2\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {2 - {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

  (28)

  Stated the CPFS Einstein weighted averaging (CPFSEWA) operator.

• 3. When $\overbrace {{\mathfrak{g}}}^{{}}\left( \varphi \right) = \log \left( {\overbrace {\sigma }^{{}} + \frac{{\left( {1 - \overbrace {\sigma }^{{}}} \right)\varphi }}{\varphi }} \right),\overbrace {\sigma }^{{}} \in \left( {0,\infty } \right),\varphi \ne 0$ in Eq. (22), then

  $$CPFSPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)=\left(\begin{array}{c}\frac{\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1+\left( {\overbrace {\sigma }}-1\right){\mathfrak{M}}_{{R}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}-\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1-{\mathfrak{M}}_{{R}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}{\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1+\left( {\overbrace {\sigma }}-1\right){\mathfrak{M}}_{{R}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}+\left( {\overbrace {\sigma }}-1\right)\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1-{\mathfrak{M}}_{{R}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}\\ {e}^{i2\pi \left(\frac{\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1+\left( {\overbrace {\sigma }}-1\right){\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}-\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1-{\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}{\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1+\left( {\overbrace {\sigma }}-1\right){\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}+\left( {\overbrace {\sigma }}-1\right)\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1-{\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}\right)},\\ \frac{ {\overbrace {\sigma }}\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left({\mathcal{A}}_{{R}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}{\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1+\left( {\overbrace {\sigma }}-1\right)\left(1-{\mathcal{A}}_{{R}_{i\mathfrak{k}}}\right)\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}+\left( {\overbrace {\sigma }}-1\right)\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left({\mathcal{A}}_{{R}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}\\ {e}^{i2\pi \left(\frac{ {\overbrace {\sigma }}\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}{\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1+\left( {\overbrace {\sigma }}-1\right)\left(1-{\mathcal{A}}_{{I}_{i\mathfrak{k}}}\right)\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}+\left( {\overbrace {\sigma }}-1\right)\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}\right)},\\ \frac{ {\overbrace {\sigma }}\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left({\mathfrak{N}}_{{R}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}{\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1+\left( {\overbrace {\sigma }}-1\right)\left(1-{\mathfrak{N}}_{{R}_{i\mathfrak{k}}}\right)\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}+\left( {\overbrace {\sigma }}-1\right)\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left({\mathfrak{N}}_{{R}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}\\ {e}^{i2\pi \left(\frac{ {\overbrace {\sigma }}\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}{\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left(1+\left( {\overbrace {\sigma }}-1\right)\left(1-{\mathfrak{N}}_{{I}_{i\mathfrak{k}}}\right)\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}+\left( {\overbrace {\sigma }}-1\right)\prod_{\mathfrak{k}=1}^{m}{\left(\prod_{i=1}^{n}{\left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}}}\right)}\end{array}\right) $$

  (29)

Stated the CPFS Hamacher weighted averaging (CPFSHWA) operator.

Definition 11

When $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}} \in \overline{\overline{{\Xi }}}$, the CPFSWPA operator is simplified by:

$$ CPFSWPA:\overline{\overline{{\Xi }}}^{n} \to \overline{\overline{{\Xi }}} $$

by

$$ CPFSWPA\left( {\overline{\overline{{{\mathfrak{L}}_{CIF - 11} }}} ,\overline{\overline{{{\mathfrak{L}}_{CIF - 12} }}} , \ldots ,\overline{\overline{{{\mathfrak{L}}_{CIF - nm} }}} } \right) = \oplus_{{{\mathfrak{k}} = 1}}^{m} \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{\mathfrak{k}}}^{\prime } \oplus_{i = 1}^{n} \left( {\rm {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbb{N}}^{\prime}} }_{i} \overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} } \right)} \right) $$

(30)

where ${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{{\mathfrak{k}}}^{\prime } = \frac{{\hat{\mu }_{{\mathfrak{k}}} \left( {1 + \overline{\overline{{\mathcal{T}}}}_{{\mathfrak{k}}} } \right)}}{{\mathop \sum \nolimits_{{{\mathfrak{k}} = 1}}^{m} \hat{\mu }_{{\mathfrak{k}}} \left( {1 + \overline{\overline{{\mathcal{T}}}}_{{\mathfrak{k}}} } \right)}}, {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\mathbb{N}}^{\prime}} }_{i} = \frac{{\hat{\eta }_{i} \left( {1 + {\Re }_{i} } \right)}}{{\mathop \sum \nolimits_{i = 1}^{n} \hat{\eta }_{i} \left( {1 + \overline{\overline{ {\Re }}}_{i} } \right)}}$, and $\overline{\overline{{\rm {\Re }_{i} }}} = \mathop \sum \limits_{{\begin{array}{*{20}c} {k = 1} \\ {i \ne k} \\ \end{array} }}^{n} Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} .\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}} } \right),\overline{\overline{{{\mathcal{T}}_{{\mathfrak{k}}} }}} = \mathop \sum \limits_{{\begin{array}{*{20}c} {k = 1} \\ {i \ne k} \\ \end{array} }}^{m} Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} .\overline{\overline{{{\mathfrak{L}}_{PI - i} }}} } \right)$, and $Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}} .\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}} } \right)$, simplified the support for $\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}}$ and $\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}}$, where $\hat{\mu }_{{\mathfrak{k}}}$ and $\hat{\eta }_{i}$, expressed the weight vector with $\mathop \sum \limits_{{{\mathfrak{k}} = 1}}^{m} \hat{\mu }_{{\mathfrak{k}}} = 1$ and $\mathop \sum \limits_{i = 1}^{n} \hat{\eta }_{i} = 1$.

Theorem 3

Considering the information in Eq. (30), we get

$$CPFSWPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{i\mathfrak{k}}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{i\mathfrak{k}}}\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{i\mathfrak{k}}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{i\mathfrak{k}}}\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{i\mathfrak{k}}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{i\mathfrak{k}}}\right)\right)\right)\right)}\end{array}\right) $$

(31)

Proof

Omitted.

Definition 12

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}\in \overline{\overline{\Xi }}$, then CPFSOWPA operator is simplified by:

$$CPFSOWPA:{\overline{\overline{\Xi }}}^{n}\to\overline{\overline{\Xi }}$$

by

$$CPFSOWPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)={\oplus }_{\mathfrak{k}=1}^{m}\left(\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}^{^{\prime}}}{\oplus }_{i=1}^{n}\left({\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} } \overline{\overline{{\mathfrak{L}}_{PI-o\left(i\right)o(\mathfrak{k})}}}\right)\right) $$

(32)

where $\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}^{^{\prime}}}=\frac{{\widehat{\mu }}_{\mathfrak{k}}\left(1+{\overline{\overline{\mathcal{T}}}}_{\mathfrak{k}}\right)}{\sum_{\mathfrak{k}=1}^{m}{\widehat{\mu }}_{\mathfrak{k}}\left(1+{\overline{\overline{\mathcal{T}}}}_{\mathfrak{k}}\right)},{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} } =\frac{{\widehat{\eta }}_{i}\left(1+{\mathfrak{R}}_{i}\right)}{\sum_{i=1}^{n}{\widehat{\eta }}_{i}\left(1+{\overline{\overline{\mathfrak{R}}}}_{i}\right)}$, and $\overline{\overline{{\mathfrak{R}}_{i}}}=\sum_{\begin{array}{c}k=1\\ i\ne k\end{array}}^{n}Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}\right),\overline{\overline{{\mathcal{T}}_{\mathfrak{k}}}}=\sum_{\begin{array}{c}k=1\\ i\ne k\end{array}}^{m}Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-i}}}\right)$, and $Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}\right)$, simplified the support for $\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}$ and $\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}$, where ${\widehat{\mu }}_{\mathfrak{k}}$ and ${\widehat{\eta }}_{i}$, expressed the weight vector with $\sum_{\mathfrak{k}=1}^{m}{\widehat{\mu }}_{\mathfrak{k}}=1$ and $\sum_{i=1}^{n}{\widehat{\eta }}_{i}=1$ with $o\left(i\right)\mathfrak{k}\ge o\left(i-1\right)\mathfrak{k}$ and $io\left(\mathfrak{k}\right)\ge io\left(\mathfrak{k}-1\right)$.

Theorem 4

Considering the information in Eq. (32), we get

$$CPFSOWPA\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)=\left(\begin{array}{c} {\overbrace {\mathfrak{f}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{R}_{o\left(i\right)o\left(\mathfrak{k}\right)}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{f}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}} \overbrace {\mathfrak{f} }\left({\mathfrak{M}}_{{I}_{o\left(i\right)o\left(\mathfrak{k}\right)}}\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} _{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{R}_{o\left(i\right)o\left(\mathfrak{k}\right)}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathcal{A}}_{{I}_{o\left(i\right)o\left(\mathfrak{k}\right)}}\right)\right)\right)\right)},\\ {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{R}_{o\left(i\right)o\left(\mathfrak{k}\right)}}\right)\right)\right){e}^{i2\pi \left( {\overbrace {\mathfrak{g}^{-1} }}\left(\sum_{\mathfrak{k}=1}^{m}{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}\left(\sum_{i=1}^{n}\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}{\overbrace {\mathfrak{g} }}\left({\mathfrak{N}}_{{I}_{o\left(i\right)o\left(\mathfrak{k}\right)}}\right)\right)\right)\right)}\end{array}\right) $$

(33)

Proof

Omitted.

Definition 13

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}\in \overline{\overline{\Xi }}$, the CPFSPG operator is simplified by:

$$CPFSPG:{\overline{\overline{\Xi }}}^{n}\to\overline{\overline{\Xi }}$$

by

$$CPFSPG\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)={\otimes }_{\mathfrak{k}=1}^{m}{\left({\otimes }_{i=1}^{n}{\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}}\right)}^{{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}} $$

(34)

where ${\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\mathfrak{k}}}=\frac{\left(1+{\overline{\overline{\mathcal{T}}}}_{\mathfrak{k}}\right)}{\sum_{\mathfrak{k}=1}^{m}\left(1+{\overline{\overline{\mathcal{T}}}}_{\mathfrak{k}}\right)},\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i}}=\frac{\left(1+{\mathfrak{R}}_{i}\right)}{\sum_{i=1}^{n}\left(1+{\overline{\overline{\mathfrak{R}}}}_{i}\right)}$, and $\overline{\overline{{\mathfrak{R}}_{i}}}=\sum_{\begin{array}{c}k=1\\ i\ne k\end{array}}^{n}Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}\right),\overline{\overline{{\mathcal{T}}_{\mathfrak{k}}}}=\sum_{\begin{array}{c}k=1\\ i\ne k\end{array}}^{m}Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-i}}}\right)$, and $Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}\right)$, simplified the support for $\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}$ and $\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}$.

Theorem 5

Considering the information Eq. (34), we determine

$$ \begin{gathered} CPFSPG\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \hfill \\ \left( \begin{gathered} \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} , \hfill \\ \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

(35)

Proof

Omitted.

Property 5

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}=\overline{\overline{{\mathfrak{L}}_{CIF}}}$, then

$$CPFSPG\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)=\overline{\overline{{\mathfrak{L}}_{CIF}}} (36)$$

(36)

Proof

Omitted.

Property 6

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}$ and $\overline{\overline{{\mathfrak{L}}_{CIF}}}$ be any two CIFSNs, then

$$CPFSPG\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\otimes \overline{\overline{{\mathfrak{L}}_{CIF}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}}\otimes \overline{\overline{{\mathfrak{L}}_{CIF}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\otimes \overline{\overline{{\mathfrak{L}}_{CIF}}}\right)=CPFSPG\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)\otimes \overline{\overline{{\mathfrak{L}}_{CIF}}} $$

(37)

Proof

Omitted.

Property 7

Prove that

$$CPFSPG\left({\overline{\overline{{\mathfrak{L}}_{CIF-11}}}}^{ {\overbrace {\sigma }}},{\overline{\overline{{\mathfrak{L}}_{CIF-12}}}}^{ {\overbrace {\sigma }}},\dots ,{\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}}^{ {\overbrace {\sigma }}}\right)=CPFSPG{\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)}^{ {\overbrace {\sigma }}}$$

(38)

Proof

Omitted.

Property 8

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}^{*}}}=\left({\mathfrak{M}}_{\overline{\overline{{\mathfrak{L}}_{R-i\mathfrak{k}}^{*}}}}{e}^{i2\pi \left({\mathfrak{M}}_{\overline{\overline{{\mathfrak{L}}_{I-i\mathfrak{k}}^{*}}}}\right)},{\mathfrak{M}}_{\overline{\overline{{\mathfrak{L}}_{R-i\mathfrak{k}}^{*}}}}{e}^{i2\pi \left({\mathfrak{M}}_{\overline{\overline{{\mathfrak{L}}_{I-i\mathfrak{k}}^{*}}}}\right)},{\mathfrak{N}}_{\overline{\overline{{\mathfrak{L}}_{R-i\mathfrak{k}}^{*}}}}{e}^{i2\pi \left({\mathfrak{N}}_{\overline{\overline{{\mathfrak{L}}_{I-i\mathfrak{k}}^{*}}}}\right)}\right)$, then

$$CPFSPG\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}}\otimes \overline{\overline{{\mathfrak{L}}_{CIF-11}^{*}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}}\otimes \overline{\overline{{\mathfrak{L}}_{CIF-12}^{*}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\otimes \overline{\overline{{\mathfrak{L}}_{CIF-nm}^{*}}}\right)=CPFSPG\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)\otimes CPFSPG\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}^{*}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}^{*}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}^{*}}}\right) $$

(39)

Proof

Omitted.

Important cases of the invented work using the information in Eq. (35) are described here.

• Assume ${\overbrace {\mathfrak{g} }}\left(\varphi \right)=-\mathrm{log}\left(\varphi \right)$ in Eq. (35), then

  $$ \begin{gathered} CPFSPG\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \hfill \\ \left( \begin{gathered} \mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} e^{{i2\pi \left( {\mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} } \right)}} , \hfill \\ 1 - \mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} e^{{i2\pi \left( {1 - \mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} } \right)}} , \hfill \\ 1 - \mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} e^{{i2\pi \left( {1 - \mathop \prod \limits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \limits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} } \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

  (40)

  Stated the CPFS Archimedean weighted geometric (CPFSAWG) operator.

• Assume ${\overbrace {\mathfrak{g} }}\left(\varphi \right)=\mathrm{log}\left(\frac{2-\varphi }{\varphi }\right),\varphi \ne 0$ in Eq. (35), then

  $$ \begin{gathered} CPFSPG\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \hfill \\ \left( \begin{gathered} \frac{{2\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {2 - {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }} \hfill \\ e^{{i2\pi \left( {\frac{{2\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {2 - {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}} \right)}} , \hfill \\ \frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }} \hfill \\ e^{{i2\pi \left( {\frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}} \right)}} , \hfill \\ \frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }} \hfill \\ e^{{i2\pi \left( {\frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

  (41)

  Stated the CPFS Einstein weighted geometric (CPFSEWG) operator.

• Assume ${\overbrace {\mathfrak{g} }}\left(\varphi \right)=\mathrm{log}\left( {\overbrace {\sigma }}+\frac{\left(1- {\overbrace {\sigma }}\right)\varphi }{\varphi }\right), {\overbrace {\sigma }}\in \left(0,\infty \right),\varphi \ne 0$ in Eq. (35), then

  $$ \begin{gathered} CPFSPG\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \hfill \\ \left( \begin{gathered} \frac{{\overbrace {\sigma }^{{}}\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right)\left( {1 - {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \left( {\overbrace {\sigma }^{{}} - 1} \right)\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }} \hfill \\ e^{{i2\pi \left( {\frac{{\overbrace {\sigma }^{{}}\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right)\left( {1 - {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \left( {\overbrace {\sigma }^{{}} - 1} \right)\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}} \right)}} , \hfill \\ \frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right) {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right) {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \left( {\overbrace {\sigma }^{{}} - 1} \right)\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }} \hfill \\ e^{{i2\pi \left( {\frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right) {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right) {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \left( {\overbrace {\sigma }^{{}} - 1} \right)\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}} \right)}} , \hfill \\ \frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right) {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right) {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \left( {\overbrace {\sigma }^{{}} - 1} \right)\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }} \hfill \\ e^{{i2\pi \left( {\frac{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right) {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} - \mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}{{\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 + \left( {\overbrace {\sigma }^{{}} - 1} \right) {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} + \left( {\overbrace {\sigma }^{{}} - 1} \right)\mathop \prod \nolimits_{{ {\mathfrak{k}} = 1}}^{m} \left( {\mathop \prod \nolimits_{{i = 1}}^{n} \left( {1 - {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)^{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} }} } \right)^{{ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} }} }}} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

  (42)

Stated the CPFS Hamacher weighted geometric (CPFSHWG) operator.

Definition 14

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}\in \overline{\overline{\Xi }}$, the CPFSWPG operator is simplified by:

$$CPFSWPG:{\overline{\overline{\Xi }}}^{n}\to\overline{\overline{\Xi }}$$

by

$$CPFSWPG\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)={\otimes }_{\mathfrak{k}=1}^{m}{\left({\otimes }_{i=1}^{n}{\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}\right)}^{{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} } }\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\rm{\mathfrak{k}}} } $$

(43)

where ${\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\rm{\mathfrak{k}}} } =\frac{{\widehat{\mu }}_{\mathfrak{k}}\left(1+{\overline{\overline{\mathcal{T}}}}_{\mathfrak{k}}\right)}{\sum_{\mathfrak{k}=1}^{m}{\widehat{\mu }}_{\mathfrak{k}}\left(1+{\overline{\overline{\mathcal{T}}}}_{\mathfrak{k}}\right)},{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} } =\frac{{\widehat{\eta }}_{i}\left(1+{\mathfrak{R}}_{i}\right)}{\sum_{i=1}^{n}{\widehat{\eta }}_{i}\left(1+{\overline{\overline{\mathfrak{R}}}}_{i}\right)}$, and $\overline{\overline{{\mathfrak{R}}_{i}}}=\sum_{\begin{array}{c}k=1\\ i\ne k\end{array}}^{n}Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}\right),\overline{\overline{{\mathcal{T}}_{\mathfrak{k}}}}=\sum_{\begin{array}{c}k=1\\ i\ne k\end{array}}^{m}Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-i}}}\right)$, and $Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}\right)$, stated the support for $\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}$ and $\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}$, where ${\widehat{\mu }}_{\mathfrak{k}}$ and ${\widehat{\eta }}_{i}$, expressed the weight vector with $\sum_{\mathfrak{k}=1}^{m}{\widehat{\mu }}_{\mathfrak{k}}=1$ and $\sum_{i=1}^{n}{\widehat{\eta }}_{i}=1$.

Theorem 6

Considering the information in Eq. (33), we get

$$ \begin{gathered} CPFSWPG\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \hfill \\ \left( \begin{gathered} \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathcal{A}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathcal{A}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{i {\mathfrak{k}}}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

(44)

Proof

Omitted.

Definition 14

When $\overline{\overline{{\mathfrak{L}}_{CIF-i\mathfrak{k}}}}\in \overline{\overline{\Xi }}$, the CPFSOWPG operator is simplified by:

$$CPFSOWPG:{\overline{\overline{\Xi }}}^{n}\to\overline{\overline{\Xi }}$$

by

$$CPFSOWPG\left(\overline{\overline{{\mathfrak{L}}_{CIF-11}}},\overline{\overline{{\mathfrak{L}}_{CIF-12}}},\dots ,\overline{\overline{{\mathfrak{L}}_{CIF-nm}}}\right)={\otimes }_{\mathfrak{k}=1}^{m}{\left({\otimes }_{i=1}^{n}{\left(\overline{\overline{{\mathfrak{L}}_{PI-o\left(i\right)o\left(\mathfrak{k}\right)}}}\right)}^{{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} } }\right)}^{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\rm{\mathfrak{k}}} } $$

(45)

where ${\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{\rm{\mathfrak{k}}} } =\frac{{\widehat{\mu }}_{\mathfrak{k}}\left(1+{\overline{\overline{\mathcal{T}}}}_{\mathfrak{k}}\right)}{\sum_{\mathfrak{k}=1}^{m}{\widehat{\mu }}_{\mathfrak{k}}\left(1+{\overline{\overline{\mathcal{T}}}}_{\mathfrak{k}}\right)},{\rm{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} }_{i} } =\frac{{\widehat{\eta }}_{i}\left(1+{\mathfrak{R}}_{i}\right)}{\sum_{i=1}^{n}{\widehat{\eta }}_{i}\left(1+{\overline{\overline{\mathfrak{R}}}}_{i}\right)}$, and $\overline{\overline{{\mathfrak{R}}_{i}}}=\sum_{\begin{array}{c}k=1\\ i\ne k\end{array}}^{n}Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}\right),\overline{\overline{{\mathcal{T}}_{\mathfrak{k}}}}=\sum_{\begin{array}{c}k=1\\ i\ne k\end{array}}^{m}Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-i}}}\right)$, and $Sup\left(\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}.\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}\right)$, simplified the support for $\overline{\overline{{\mathfrak{L}}_{PI-i\mathfrak{k}}}}$ and $\overline{\overline{{\mathfrak{L}}_{PI-k\mathfrak{k}}}}$, where ${\widehat{\mu }}_{\mathfrak{k}}$ and ${\widehat{\eta }}_{i}$, expressed the weight vector with $\sum_{\mathfrak{k}=1}^{m}{\widehat{\mu }}_{\mathfrak{k}}=1$ and $\sum_{i=1}^{n}{\widehat{\eta }}_{i}=1$ with $o\left(i\right)\mathfrak{k}\ge o\left(i-1\right)\mathfrak{k}$ and $io\left(\mathfrak{k}\right)\ge io\left(\mathfrak{k}-1\right)$.

Theorem 7

Considering the information in Eq. (35), we get

$$ \begin{gathered} CPFSOWPG\left( {\overline{\overline{{ {\mathfrak{L}}_{{CIF - 11}} }}} ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - 12}} }}} , \ldots ,\overline{\overline{{ {\mathfrak{L}}_{{CIF - nm}} }}} } \right) = \hfill \\ \left( \begin{gathered} \overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{M}}_{{R_{{o\left( i \right)o\left( {\mathfrak{k}} \right)}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{g}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{g}}}^{{}}\left( { {\mathfrak{M}}_{{I_{{o\left( i \right)o\left( {\mathfrak{k}} \right)}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathcal{A}}_{{R_{{o\left( i \right)o\left( {\mathfrak{k}} \right)}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathcal{A}}_{{I_{{o\left( i \right)o\left( {\mathfrak{k}} \right)}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{N}}_{{R_{{o\left( i \right)o\left( {\mathfrak{k}} \right)}} }} } \right)} \right)} \right)e^{{i2\pi \left( {\overbrace {{ {\mathfrak{f}}^{{ - 1}} }}^{{}}\left( {\mathop \sum \limits_{{ {\mathfrak{k}} = 1}}^{m} {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{M}} }_{ {\mathfrak{k}}} \left( {\mathop \sum \limits_{{i = 1}}^{n} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\mathbb{N}} _{i} \overbrace { {\mathfrak{f}}}^{{}}\left( { {\mathfrak{N}}_{{I_{{o\left( i \right)o\left( {\mathfrak{k}} \right)}} }} } \right)} \right)} \right)} \right)}} \hfill \\ \end{gathered} \right) \hfill \\ \end{gathered} $$

(46)

Proof

Omitted.

Application (“MADM Processes using Proposed Operators”)

In the consideration of diagnosed operators using CPFS information, we illustrated a MADM tool to find the best option from the family of decisions.

Strategic decision-making processes

For managing awkward and problematic information that occurred in genuine life dilemmas, the MADM tool plays an important role in the circumstance of FS theory. Here, we have discussed a procedure for resolving the above issues, we choose $m$ alternatives and $n$ attributes whose representations are described: $\overline{\overline{{{\mathfrak{L}}_{AT} }}} = \left\{ {\overline{\overline{{{\mathfrak{L}}_{AT - 1} }}} ,\overline{\overline{{{\mathfrak{L}}_{AT - 2} }}} , \ldots ,\overline{\overline{{{\mathfrak{L}}_{AT - m} }}} } \right\}$ and $\overline{\overline{{{\mathfrak{L}}_{AL} }}} = \left\{ {\overline{\overline{{{\mathfrak{L}}_{AL - 1} }}} ,\overline{\overline{{{\mathfrak{L}}_{AL - 2} }}} , \ldots ,\overline{\overline{{{\mathfrak{L}}_{AL - n} }}} } \right\}$, with weight vector $\mathop \sum \limits_{{{\mathfrak{k}} = 1}}^{m} \hat{\mu }_{{\mathfrak{k}}} = 1$ and $\mathop \sum \limits_{i = 1}^{n} \hat{\eta }_{i} = 1$, provided for attributes and parameters. Then, we give a CPFSNs $\overline{\overline{{{\mathfrak{L}}_{{CIFS - i{\mathfrak{k}}}} }}} = \left( {{\mathfrak{M}}_{{\overline{\overline{{R_{{i{\mathfrak{k}}}} }}} }} e^{{i2\pi \left( {{\mathfrak{M}}_{{I_{{i{\mathfrak{k}}}} }} } \right)}} ,{\mathcal{A}}_{{R_{{i{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathcal{A}}_{{I_{{i{\mathfrak{k}}}} }} } \right)}} ,{\mathfrak{N}}_{{R_{{i{\mathfrak{k}}}} }} e^{{i2\pi \left( {{\mathfrak{N}}_{{I_{{i{\mathfrak{k}}}} }} } \right)}} } \right),i = 1,2, \ldots ,n$, with $0 \le {\mathfrak{M}}_{{R_{{i{\mathfrak{k}}}} }} + {\mathcal{A}}_{{R_{{i{\mathfrak{k}}}} }} + {\mathfrak{N}}_{{R_{{i{\mathfrak{k}}}} }} \le 1$ and $0 \le {\mathfrak{M}}_{{I_{{i{\mathfrak{k}}}} }} + {\mathcal{A}}_{{I_{{i{\mathfrak{k}}}} }} + {\mathfrak{N}}_{{I_{{i{\mathfrak{k}}}} }} \le 1$ for each alternative in the shape of a matrix. To handle the above scenario, we described various stages for resolving the above issues.

Stage 1: Deliberated information in the form of a closed matrix that contains CPFSNs for each alternative and their attributes, the mathematical term $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{\left( b \right)} ,b = 1,2, \ldots ,z$, stated the matrix.

Stage 2: Summarized the $\rm {\Re }_{i}$, stated the support for the intellectuals, we have

$$ \overline{\overline{{\rm {\Re }_{i}^{\left( b \right)} }}} = \mathop \sum \limits_{{\begin{array}{*{20}c} {k = 1} \\ {i \ne k} \\ \end{array} }}^{n} Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}}^{\left( b \right)} .\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}}^{\left( b \right)} } \right) $$

where $Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}}^{\left( b \right)} .\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}}^{\left( b \right)} } \right) = 1 - d\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}}^{\left( b \right)} .\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}}^{\left( b \right)} } \right),d\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}}^{\left( b \right)} .\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}}^{\left( b \right)} } \right) = score of c{\mathfrak{L}}rrent entry$.

Stage 3: Considering the weighted averaging/geometric operators, we initiate $\overline{\overline{{\mathcal{T}}}}_{{\mathfrak{k}}}$, stated the support for the intellectuals, we have

$$ \overline{\overline{{\mathcal{T}}}}_{{\mathfrak{k}}}^{\left( b \right)} = \mathop \sum \limits_{{\begin{array}{*{20}c} {k = 1} \\ {i \ne k} \\ \end{array} }}^{m} Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - {\mathfrak{k}}}} }}}^{\left( b \right)} .\overline{\overline{{{\mathfrak{L}}_{PI - i} }}}^{\left( b \right)} } \right) $$

where $Sup\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - {\mathfrak{k}}}} }}}^{\left( b \right)} .\overline{\overline{{{\mathfrak{L}}_{PI - i} }}}^{\left( b \right)} } \right) = 1 - d\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - {\mathfrak{k}}}} }}}^{\left( b \right)} .\overline{\overline{{{\mathfrak{L}}_{PI - i} }}}^{\left( b \right)} } \right),d\left( {\overline{\overline{{{\mathfrak{L}}_{{PI - i{\mathfrak{k}}}} }}}^{\left( b \right)} .\overline{\overline{{{\mathfrak{L}}_{{PI - k{\mathfrak{k}}}} }}}^{\left( b \right)} } \right) = \frac{{s{\mathfrak{L}}m \;of \;all\; n{\mathfrak{L}}mber\; in}}{{order \;of \;n{\mathfrak{L}}mber \;in \;row}}$.

Stage 4: Considering the CPFSPA and CPFSPG operators, we deliberated the values from the matrices into exact values in the availability of TN and TCN ${\overbrace {\mathfrak{g} }}\left(\overline{\overline{\varphi }}\right)=-\mathrm{log}\left(\overline{\overline{\varphi }}\right)$.

Stage 5: Averaged the SV of the deliberated values.

Stage 6: Ranking all results and diagnosed the best optimal.

Illustrated example

The existing shortage of energy in various constructing countries over a decade is causing the economic growth of the country. Between the distinct energy supplies, electrical energy is the main energy supplier which is valuable in the market by the distinct area of the economy. For various years, Pakistan is suffering from various electricity issues. The intensification of the electric shortage has become a main political dilemma in Pakistan which is causing the strong construction of the economy. The problem of electricity in Pakistan does not only affect the people and the economy but also affects many other dilemmas, because of some brain-dead employees and unsuccessful planning and policy. Therefore, the management of the government of Pakistan must give special attention to these issues and try to increase the quality of the electricity demand. For this, we describe the various new source of energy that will be increased the level of electricity in the future. Here, we suggest some causes of generating energy in the form of alternatives, such that $\overline{\overline{{{\mathfrak{L}}_{AL - 1} }}}$: Hydropower, $\overline{\overline{{{\mathfrak{L}}_{AL - 2} }}} :$ Solar Energy; $\overline{\overline{{{\mathfrak{L}}_{AL - 3} }}} :$ Fuel; and $\overline{\overline{{{\mathfrak{L}}_{AL - 4} }}} :$ Coal. Assume four experts in form of attributes: $\overline{\overline{{{\mathfrak{L}}_{AT - 1} }}}$: Environmental, $\overline{\overline{{{\mathfrak{L}}_{AT - 2} }}} :$ Economic; $\overline{\overline{{{\mathfrak{L}}_{AT - 3} }}} :$ Technical and $\overline{\overline{{{\mathfrak{L}}_{AT - 4} }}}$: Social-Political with parameters, described in the form: $e_{1}$: Client Facilities, $e_{2}$: Bandwidth, $e_{3}$: Package, $e_{4}$: Total Cost, and $e_{5}$ Internet Speed. Here, we choose the values $\left( {0.3,0.2,0.3,0.2} \right)$ and $\left( {0.3,0.2,0.25,0.15,0.1} \right)$, stated the weighty vector for ${\mathfrak{L}}_{i}$. To handle the above scenario, we described various stages for resolving the above issues.

Stage 1: Deliberated information in the form of a closed matrix that contains CPFSNs for each alternative and their attributes in the form of Tables 1, 2, 3, and 4, the mathematical term $\overline{\overline{{{\mathfrak{L}}_{{CIF - i{\mathfrak{k}}}} }}}^{\left( b \right)} ,b = 1,2, \ldots ,z$, stated the matrix.

Table 1 Information is given for Scholars 1.

Table 2 Information is given for Scholars 2.

Table 3 Information is given for Scholars 3.

Table 4 Information is given for Scholars 4.

Stage 2: Summarized the ${\mathfrak{R}}_{i}$, stated the support for the intellectuals, we have

$$\overline{\overline{{\mathfrak{R}}_{i}^{\left(1\right)}}}=\left[\begin{array}{c}5.55E-17\\ 0.3\\ \begin{array}{c}0.1\\ 0.1\\ 0.4\end{array}\end{array} \begin{array}{c}0.02\\ 0.32\\ \begin{array}{c}0.12\\ 0.12\\ 0.38\end{array}\end{array} \begin{array}{c}0.04\\ 0.34\\ \begin{array}{c}0.14\\ 0.14\\ 0.36\end{array}\end{array} \begin{array}{c}0.06\\ 0.36\\ \begin{array}{c}0.16\\ 0.16\\ 0.34\end{array}\end{array} \begin{array}{c}0.08\\ 0.38\\ \begin{array}{c}0.18\\ 0.18\\ 0.32\end{array}\end{array}\right],\overline{\overline{{\mathfrak{R}}_{i}^{\left(2\right)}}}=\left[\begin{array}{c}0.1\\ 0.4\\ \begin{array}{c}5.55E-17\\ 0.3\\ 0.1\end{array}\end{array} \begin{array}{c}0.12\\ 0.38\\ \begin{array}{c}0.02\\ 0.32\\ 0.12\end{array}\end{array} \begin{array}{c}0.14\\ 0.36\\ \begin{array}{c}0.04\\ 0.34\\ 0.14\end{array}\end{array} \begin{array}{c}0.16\\ 0.34\\ \begin{array}{c}0.06\\ 0.36\\ 0.16\end{array}\end{array} \begin{array}{c}0.18\\ 0.32\\ \begin{array}{c}0.08\\ 0.38\\ 0.18\end{array}\end{array}\right],\overline{\overline{{\mathfrak{R}}_{i}^{\left(3\right)}}}=\left[\begin{array}{c}0.3\\ 0.1\\ \begin{array}{c}0.04\\ 5.55E-17\\ 0.3\end{array}\end{array} \begin{array}{c}0.32\\ 0.12\\ \begin{array}{c}0.38\\ 0.02\\ 0.32\end{array}\end{array} \begin{array}{c}0.34\\ 0.14\\ \begin{array}{c}0.36\\ 0.04\\ 0.34\end{array}\end{array} \begin{array}{c}0.36\\ 0.16\\ \begin{array}{c}0.34\\ 0.06\\ 0.36\end{array}\end{array} \begin{array}{c}0.38\\ 0.18\\ \begin{array}{c}0.32\\ 0.08\\ 0.38\end{array}\end{array}\right],\overline{\overline{{\mathfrak{R}}_{i}^{\left(4\right)}}}=\left[\begin{array}{c}0.04\\ 5.55E-17\\ \begin{array}{c}0.3\\ 0.1\\ 0.4\end{array}\end{array} \begin{array}{c}0.38\\ 0.02\\ \begin{array}{c}0.32\\ 0.12\\ 0.38\end{array}\end{array} \begin{array}{c}0.36\\ 0.04\\ \begin{array}{c}0.34\\ 0.14\\ 0.36\end{array}\end{array} \begin{array}{c}0.34\\ 0.06\\ \begin{array}{c}0.36\\ 0.16\\ 0.34\end{array}\end{array} \begin{array}{c}0.32\\ 0.08\\ \begin{array}{c}0.38\\ 0.18\\ 0.32\end{array}\end{array}\right]$$

Stage 3: Considering the weighted averaging/geometric operators, we initiate $\overline{\overline{{\mathcal{T}}}}_{{\mathfrak{k}}}$, stated the support for the intellectuals, we have$\overline{\overline{{\rm {\mathcal{T}}_{\rm {\mathfrak{k}}} }}} ^{{\left( 1 \right)}} = \left[ {\begin{array}{*{20}c} {0.04} \\ {0.34} \\ {\begin{array}{*{20}c} {0.14} \\ {0.14} \\ {0.36} \\ \end{array} } \\ \end{array} } \right],\overline{\overline{{\rm {\mathcal{T}}_{\rm {\mathfrak{k}}} }}} ^{{\left( 2 \right)}} = \left[ {\begin{array}{*{20}c} {0.14} \\ {0.36} \\ {\begin{array}{*{20}c} {0.04} \\ {0.34} \\ {0.14} \\ \end{array} } \\ \end{array} } \right],\overline{\overline{{\rm {\mathcal{T}}_{\rm {\mathfrak{k}}} }}} ^{{\left( 3 \right)}} = \left[ {\begin{array}{*{20}c} {0.34} \\ {0.14} \\ {\begin{array}{*{20}c} {0.36} \\ {0.04} \\ {0.34} \\ \end{array} } \\ \end{array} } \right],\overline{\overline{{\rm {\mathcal{T}}_{\rm {\mathfrak{k}}} }}} ^{{\left( 4 \right)}} = \left[ {\begin{array}{*{20}c} {0.36} \\ {0.04} \\ {\begin{array}{*{20}c} {0.34} \\ {0.14} \\ {0.36} \\ \end{array} } \\ \end{array} } \right]$

Stage 4: Considering the CPFSPA and CPFSPG operators, we deliberated the values from the matrices into exact values in the availability of TN and TCN ${\overbrace {\mathfrak{g} }}\left(\overline{\overline{\varphi }}\right)=-\mathrm{log}\left(\overline{\overline{\varphi }}\right)$, we have

$${\mathfrak{L}}_{1}=\left(0.4384{e}^{i2\pi \left(0.2268\right)},0.172{e}^{i2\pi \left(0.1571\right)},0.1288{e}^{i2\pi \left(0.1551\right)}\right),{\mathfrak{L}}_{2}=\left(0.4485{e}^{i2\pi \left(0.2367\right)},0.1836{e}^{i2\pi \left(0.1686\right)},0.1394{e}^{i2\pi \left(0.1666\right)}\right),{\mathfrak{L}}_{3}=\left(0.4587{e}^{i2\pi \left(0.2466\right)},0.195{e}^{i2\pi \left(0.18\right)},0.15{e}^{i2\pi \left(0.1778\right)}\right),{\mathfrak{L}}_{4}=\left(0.4689{e}^{i2\pi \left(0.2564\right)},0.2063{e}^{i2\pi \left(0.1912\right)},0.1606{e}^{i2\pi \left(0.189\right)}\right),{\mathfrak{L}}_{5}=\left(0.4791{e}^{i2\pi \left(0.2663\right)},0.2174{e}^{i2\pi \left(0.2023\right)},0.1711{e}^{i2\pi \left(0.2001\right)}\right)$$

$${\mathfrak{L}}_{1}=\left(0.4238{e}^{i2\pi \left(0.2332\right)},0.194{e}^{i2\pi \left(0.1846\right)},0.1285{e}^{i2\pi \left(0.1747\right)}\right),{\mathfrak{L}}_{2}=\left(0.4342{e}^{i2\pi \left(0.2434\right)},0.2039{e}^{i2\pi \left(0.1946\right)},0.1383{e}^{i2\pi \left(0.1846\right)}\right),{\mathfrak{L}}_{3}=\left(0.4445{e}^{i2\pi \left(0.2537\right)},0.2138{e}^{i2\pi \left(0.2046\right)},0.1481{e}^{i2\pi \left(0.1945\right)}\right),{\mathfrak{L}}_{4}=\left(0.4548{e}^{i2\pi \left(0.2639\right)},0.2237{e}^{i2\pi \left(0.2145\right)},0.1579{e}^{i2\pi \left(0.2044\right)}\right),{\mathfrak{L}}_{5}=\left(0.4651{e}^{i2\pi \left(0.2741\right)},0.2336{e}^{i2\pi \left(0.2245\right)},0.1678{e}^{i2\pi \left(0.2143\right)}\right)$$

Stage 5: Averaged the SV of the deliberated values, such that

$${\mathfrak{L}}_{1}=0.0522,{\mathfrak{L}}_{2}=0.027,{\mathfrak{L}}_{3}=0.0024,{\mathfrak{L}}_{4}=0.0217,{\mathfrak{L}}_{5}=0.0454.$$

$${\mathfrak{L}}_{1}=0.0247,{\mathfrak{L}}_{2}=0.0437,{\mathfrak{L}}_{3}=0.0628,{\mathfrak{L}}_{4}=0.0819,{\mathfrak{L}}_{5}=0.1011.$$

Stage 6: Ranking all results and diagnosed the best optimal, such that

$${\mathfrak{L}}_{1}\ge {\mathfrak{L}}_{5}\ge {\mathfrak{L}}_{2}\ge {\mathfrak{L}}_{4}\ge {\mathfrak{L}}_{3}.$$

$${\mathfrak{L}}_{5}\ge {\mathfrak{L}}_{4}\ge {\mathfrak{L}}_{3}\ge {\mathfrak{L}}_{2}\ge {\mathfrak{L}}_{1}.$$

Hence, we obtained two different sorts of ranking results in the shape of ${\mathfrak{L}}_{1}$ and ${\mathfrak{L}}_{5}$, using CPFSPA and CPFSPG operators. To further enhance the standard of the invented approaches, we discuss the sensitive analysis of diagnosed work with various suggested approaches.

Sensitive analysis

The main idea of this analysis is to prove the invented work is more beneficial and realistic than the existing operators with the help of some comparison. Many scholars have diagnosed this procedure as demonstrating the proposed approaches are more utilized than the existing operators. For this, we consider various prevailing theories are tried to compare them with our diagnosed operators. The information related to existing theories is described in Refs.^{32,40,41,42,43,44}.

• 1. Information given in Ref.⁴⁰ contained the power aggregation operators for CIFSs, the invested power aggregation operators based on CPFS information are more able to resolve intuitionistic, picture, complex intuitionistic, and complex picture fuzzy information. But one deficiency that occurred in the existing operator is that it contained two grades in the shape of truth and falsity in the form of polar coordinates, which is not suitable because the proposed work contained information in the shape of truth, abstinence, falsity grades with parameters. Therefore, the prevailing operators in Ref.⁴⁰ failed.

• 2. Information given in Ref.³² contained the power aggregation operators for CPFSs, the invested power aggregation operators based on CPFS information are more able to resolve intuitionistic, picture, complex intuitionistic, and complex picture fuzzy information. But one deficiency that occurred in the existing operator is that it contained three grades in the shape of truth, abstinence, and falsity in the form of polar coordinates, which is not suitable because the proposed work contained information in the shape of truth, abstinence, falsity grades with parameters. Therefore, the prevailing operators in Ref.³² failed.

• 3. Information given in Ref.⁴¹ contained the aggregation operators for PFSSs, the invested power aggregation operators based on CPFS information are more able to resolve intuitionistic, picture, complex intuitionistic, and complex picture fuzzy information. But one deficiency that occurred in the existing operator is that it contained three grades in the shape of truth, abstinence, and falsity with parameters in the form of one dimension, which is not suitable because the proposed work contained information in the shape of truth, abstinence, falsity grades with parameters in the shape of polar coordinates. Therefore, the prevailing operators in Ref.⁴¹ failed. The comparative analysis is described in Table 5

  Table 5 Represented the sensitivity analysis.

  .

• 4. The theory diagnosed by Garg and Arora⁴², is called generalized Maclaurin symmetric information based on IFSSs. Noticed that the theory presented by Garg and Arora⁴² based on IFSSs is the special case of the diagnosed CPFS information and because of this reason, the theory of Garg and Arora⁴² is not able to use for evaluating information discussed in section “Illustrated example”. Therefore, the theory of diagnosed information is massively powerful, and dominant compared to the prevailing theory discussed by Garg and Arora⁴².

• 5. Robust aggregation operators diagnosed in Ref.⁴³ contained the robust aggregation operators for intuitionistic hypersoft sets (IHSS), the invested robust aggregation operators based on IHSS information are more able to resolve intuitionistic information. But one deficiency that occurred in the existing operator is that it contained three grades in the shape of truth and falsity in the form of one-dimension information, which is not suitable because the proposed work contained information in the shape of truth, abstinence, falsity grades with parameters. Therefore, the prevailing operators in Ref.⁴³ failed.

• 6. The theory diagnosed by Wang et al.⁴⁴, called Hamy means information based on Pythagorean uncertain linguistic information. Noticed that the theory presented by Wang et al.⁴⁴ based on Pythagorean uncertain linguistic information is the special case of the diagnosed CPFS information and because of this reason, the theory of Wang et al.⁴⁴ is not able to use for evaluating information discussed in section “Illustrated example”. Therefore, the theory of diagnosed information is massively powerful, and dominant compared to the prevailing theory discussed by Wang et al.⁴².

From the above-cited information, we obtained the final result in the shape of ${\mathfrak{L}}_{1}$ and ${\mathfrak{L}}_{5}$. Therefore, the invented operators based on CPFSS are very effective and dominant as compared to prevailing work^32,40,41.

Conclusion

The major theme of this analysis is diagnosed below:

• 1. We invented the new theory in the form of CPFS information and invented their major algebraic laws, score value, and accuracy values. The mathematical form of the CPFS set includes three main functions, called supporting, abstinence, and supporting against terms with a prominent characteristic that is the sum of the triplet will lie in the unit interval.

• 2. In the consideration of the power aggregation operator using generalized t-norm and t-conorm and CPFS information, we diagnosed the mathematical concept of CPFSPA, CPFSWPA, CPFSOWPA, CPFSPG, CPFSWPG, CPFSOWPG.

• 3. The major results and their particular investigation of the invented approaches are also deliberated with the help of t-norm and t-conorm $\overbrace {{\mathfrak{g}}}^{{}}\left( {\overline{\overline{\varphi }} } \right) = - \log \left( {\overline{\overline{\varphi }} } \right),\overbrace {{\mathfrak{g}}}^{{}}\left( {\overline{\overline{\varphi }} } \right) = \log \left( {\frac{{2 - \overline{\overline{\varphi }} }}{{\overline{\overline{\varphi }} }}} \right),\overline{\overline{\varphi }} \ne 0$, and $\overbrace {{\mathfrak{g}}}^{{}}\left( {\overline{\overline{\varphi }} } \right) = \log \left( {\overbrace {\sigma }^{{}} + \frac{{\left( {1 - \overbrace {\sigma }^{{}}} \right)\overline{\overline{\varphi }} }}{{\overline{\overline{\varphi }} }}} \right),\overbrace {\sigma }^{{}} \in \left( {0,\infty } \right),\overline{\overline{\varphi }} \ne 0$.

• 4. In the consideration of diagnosed operators using CPFS information, we illustrated a MADM tool to find the best option from the family of decisions.

• 5. Finally, we showed the supremacy and feasibility of the diagnosed operators with the help of sensitive analysis and geometrical representations.

Limitations of the proposed approaches

No doubt, the theory of CPFS information has a lot of benefits, but in some cases, the theory of CPFS information, if someone provided such type of information whose sum is exceeded from the unit interval, then for managing with such sort of situation, we need to propose the theory of complex spherical fuzzy soft sets and complex T-spherical fuzzy soft sets.

Future work

In the upcoming times, we will try to modify the principle of complex q-rung orthopair fuzzy sets⁴⁵, complex spherical fuzzy sets^46,47, T-spherical fuzzy sets⁴⁸, and decision-making^{49,50,51,52,53,54,55,56,57} to enhance the excellence and capacity of the exploration performs.

Ethics declaration statement

The authors state that this is their original work, and it is neither submitted nor under consideration in any other journal simultaneously.