Prioritized weighted aggregation operators under complex pythagorean fuzzy information

Abstract

Complex Pythagorean fuzzy (CPF), a worthwhile generalization of Pythagorean fuzzy set, is a powerful tool to deal with two-dimensional or periodic information. In this paper, we develop two prioritized aggregation operators (AOs) under CPF environment, namely, complex Pythagorean fuzzy prioritized weighted averaging (CPFPWA) operator and complex Pythagorean fuzzy prioritized weighted geometric (CPFPWG) operator. We consider the prioritization relationship among criteria and decision makers (DMs) to make our result more accurate as in real decision making (DM) problems, the criteria and DMs have different priority level. Further, we discuss remarkable properties of our proposed AOs. Moreover, we promote the evolution of MCDM problem by investigating an algorithm in CPF environment with its flow chart. Finally, to check the superiority and validity of proposed operators, we compare the computed results with the different existing techniques.

Keywords

Complex pythagorean fuzzy sets prioritized aggregation operators decision making

1 Introduction

MCDM is valuable research topic which involves the ranking or classification of alternatives based on the opinions provided by DMs concerning multiple criteria. MCDM approach has fascinated many researchers as it has many utilizations in various fields including operation research, engineering technology, management sciences, etc. AOs are mathematical mappings which synthesize the large and cluttered data into a single valuable form and extensively used in distinct models of classical and fuzzy set theory to aggregate the data. In order to classify distinct objects in real DM the DMs use different evaluation processes such as interval numbers or crisp numbers, etc. Due to the increasing inherent uncertainties and ambiguities of data, it became difficult for the DMs to use the exact numerical values. In order to cope with this problem, Zadeh [35] initiated the innovative conception of fuzzy set (FS) by introducing the membership function ranging from unit interval [0,1]. Zadeh’s introduction of FS made a way for the DMs to solve the DM problems comprising uncertainties and ambiguities of the data in more comprehensive way then crisp set. After that, many operators were investigated in fuzzy environment [19, 23]. In 1986, Atanassov [8] initiated the notion of intuitionistic fuzzy set (IFS), by associating a new term with membership function called non-membership function with restriction that sum of both the terms should not exceed 1. The introduction of IFS helped the DMs to solve the DM problems by the help of membership and non-membership functions. Xu [28] investigated some averaging AOs in IF environment, namely, IF weighted averaging operator, IF ordered weighted averaging operator and IF hybrid operator and utilized these operators to solve DM problems. Later on, Yager [31] initiated some AOs by assigning weights to all the values in accordance with their ranking position. In order to overcome the shortcomings of other operators, Garg [10] developed some generalized AOs by using Einstein t-norm and conorm in IF environment.

In 2019, the idea of Pythagorean fuzzy sets (PFS) primarily investigated by Yager [29], as a new generalization of IFS which can accommodate more uncertainty of human knowledge in more satisfying way with the constraint that the sum of square of membership and non-membership grades should be less then or equal to 1. After the introduction PFSs, many operators in PF environment were introduced by many researchers. Yager [32] took into account some averaging and geometric AOs in PF environment and utilize these operators in DM process. Further, Akram et al. [5] proposed a DM technique by utilizing Pythagorean Dombi fuzzy AOs. Recently, Shahzadi et al. [24] worked on PF Yager weighted averaging operators with their application in real DM. Wu [27] proposed a DM technique by investigating the concept of PF Hamacher AOs.

FS, IFS and PFS theories are effective to deal with ambiguities but in some circumstances these theories fail where the main objective is to handle the two dimensional or periodic data. To overcome the restrictions of FS and its extensions, Ramot et al. [20] put forward the conception of complex fuzzy set (CFS) which is a remarkable generalization of FS. The membership function of CFS is limited to get the values within the unit circle in the complex plane and is comprised of two valuable real valued terms, namely, amplitude and phase term. The novelty of CFS is phase term which plays pivotal role in determining the characteristics of CFS and discriminate it from all other existing models. Akram and Bashir [2] investigated ordered weighted quadratic averaging operators under CFS with their application in the field of DM. After that, Alkouri and Salleh [6] associated a non-membership grade to the membership grade of FS and developed the notion of complex intuitionitic fuzzy set (CIFS). After the conception of CFS and CIFS many AOs were introduced in CF and CIF environment. Rani and Garg [22] investigated some distance measures between the CIFS with their utilization in DM field. In 2018, they [21] also worked to introduce some power AOs based on CIFS. In 2019, Garg and Rani [11] introduced weighted averaging and geometric operator in CIF environment. To overcome the limitations of CIF, Ullah et al. [25] introduced the concept complex Pythagorean fuzzy set (CPFS) which relaxes the constraints of amplitude and phase terms. Akram et al. [1] introduced a DM strategy by introducing AOs under CPF environment. Further, Akram and Khan [3] introduced a new DM approach using CPF graphs. Moreover, Akram et al. [4] utilized ELECTRIC-I and TOPSIS method under CPFS to solve multi-criteria group decision making (MCGDM) problem.

All the above AOs were developed on the fact that the DMs and criteria are at the same priority level which in real DM are at different priority level. Therefore, to deal with the problem of prioritization, Yager [30] investigated prioritized AOs which are based on the fact that priority level of different criteria are different. Later on, Yu [33] developed some prioritized AOs, namely, IF prioritized weighted averaging operator and IF prioritized weighted geometric operator with their application in MCGDM. Further, Yu [34] also considered some IF generalized prioritized weighted averaging and geometric AOs with their utilization in MCGDM. After that, Einstein prioritized weighted averaging and Einstein prioritized weighted geometric AOs were investigated by Verma and Sharma [26] in IF environment. Khan et al. [12] analyzed some prioritized AOs based on PFSs with their application in MCGDM. Gao [9] introduced PF Hamacher prioritized AOs with their utilization in MCDM. In 2018, Arora and Garg [7] introduced some geometric and averaging operators based on IF soft set. It is clear that all these prioritized AOs are based on FS and its extensions in which the uncertainties and ambiguities in the data are handled by the membership degrees which are subsets of real numbers which may cause of lose of some useful information due to the absence of valuable phase term. In this manuscript, we propose some prioritized AOs in CPF environment.

The drawbacks and some advantages of our proposed operators are given as follows: To solve the DM problems under multiple criteria by considering that the priority level of each criteria is same may cause of some useful information because in real DM the criteria and DMs are at different priority level. The use of prioritized AOs in fuzzy environment to select the best object with distinction to the finite objects may effect the results because of the absence of phase term. The FSs and its extensions with complex membership and non-membership grades has extrapolated the existing models as they are proficient enough to manage the two dimensional or periodic data due to the existence of phase term. The CPF model as compared to CIF model is more efficient to model the uncertainty because it relaxes the constraint and broaden the space to assign membership and non-membership grade. CPF prioritized AOs overcome the deficiencies of all the other existing operators.

In this research article, we present some prioritized AOs in CPF environment, namely, CPFPWA operator and CPFPWG operator. We also discuss their useful properties, namely, idempotency, boundedness and monotonicity in detail. Moreover, we initiate an algorithm to solve the MCGDM problem with its flowchart. Further, we solve a fully developed numerical example to find the best alternative by utilizing the proposed AOs. The main contribution to this article is given below.

The prioritized AOs in CPF environment are developed with their worthwhile properties.

An algorithm with a practical example to find the best alternative with distinction to finite alternative is developed.

A comparison of proposed prioritized AOs with some existing operators are provided to check their authenticity, superiority and validity.

This paper is structured as follows; The Section 2 presents some elemental definitions which are necessary to understand before the evolution of proposed AOs. In Section 3, we introduce two prioritized AOs, namely, CPFPWA operator and CPFPWG operator with their special properties. Section 4 comprises with an algorithm of finding the best choice among the possible choices with numerical example. In Section 5, we validate our proposed AOs by providing comparison of computed results with some other existing operators available in the literature. In Section 6, we give a conclusion and future directions.

2 Preliminaries

In this section, we recall some fundamental definitions which are helpful in next evaluation.

Definition 2.1. [20] A complex fuzzy set (CFS) on a universe of discourse T, is defined as $\tilde{W} = {(g, Ψ_{\tilde{W}} (g) e^{i κ_{\tilde{W}} (g)}) : g \in T},$ where $Ψ_{\tilde{W}} (g) \in [0, 1]$ , $κ_{\tilde{W}} (g) \in [0, 2 π]$ and $i = \sqrt{- 1} .$

For every g ∈ T, $Ψ_{\tilde{W}} (g)$ is the amplitude term and $κ_{\tilde{W}} (g)$ is the phase term for the membership function of g.

Definition 2.2. [25] A complex Pythagorean fuzzy set (CPFS) on a universe of discourse T is defined as $\tilde{W} = {(g, Ψ_{\tilde{W}} (g) e^{i κ_{\tilde{W}} (g)}, Φ_{\tilde{W}} (g) e^{i χ_{\tilde{W}} (g)}) : g \in T},$ where $Ψ_{\tilde{W}} (g), Φ_{\tilde{W}} (g) \in [0, 1]$ , $κ_{\tilde{W}} (g), χ_{\tilde{W}} (g) \in [0, 2 π]$ , $i = \sqrt{- 1}$ and for all g ∈ T, $0 \leq Ψ_{\tilde{W}}^{2} (g) + Φ_{\tilde{W}}^{2} (g) \leq 1$ and $0 \leq {(\frac{κ_{\tilde{W}}}{2 π} (g))}^{2} + {(\frac{χ_{\tilde{W}}}{2 π} (g))}^{2} \leq 1 .$

For each g ∈ T, $Ψ_{\tilde{W}} (g)$ and $Φ_{\tilde{W}} (g)$ are amplitude terms for the membership and non-membership grade of g. and $κ_{\tilde{W}} (g)$ and $χ_{\tilde{W}} (g)$ are phase terms for the membership and non-membership grade of g. For simplicity, the pair (Ψe^iκ, Φe^iχ) is known as CPFN.

Definition 2.3. [1] Let ${\tilde{Y}}_{0} = (Ψ_{0} e^{i κ_{0}}, Φ_{0} e^{i χ_{0}})$ be CPFN. The score function of CPFN is given by $\overset{˘}{S} ({\tilde{Y}}_{0}) = Ψ_{0}^{2} - Φ_{0}^{2} + (\frac{κ_{0}}{2 π})^{2} - (\frac{χ_{0}}{2 π})^{2},$ (1) and the accuracy function is given as $\overset{˘}{ℍ} ({\tilde{Y}}_{0}) = Ψ_{0}^{2} + Φ_{0}^{2} + (\frac{κ_{0}}{2 π})^{2} + (\frac{χ_{0}}{2 π})^{2},$ (2) where $\overset{˘}{S} (\tilde{Y}) \in [- 2, 2]$ and $\overset{˘}{ℍ} (\tilde{Y}) \in [0, 2] .$

Definition 2.4. [1] Let ${\tilde{Y}}_{1} = (Ψ_{1} e^{i κ_{1}}, Φ_{1} e^{i χ_{1}})$ and ${\tilde{Y}}_{2} = (Ψ_{2} e^{i κ_{2}}, Φ_{2} e^{i χ_{2}})$ be two CPFNs. Then

If $\overset{˘}{S} ({\tilde{Y}}_{1}) > \overset{˘}{S} ({\tilde{Y}}_{2})$ then ${\tilde{Y}}_{1} ≻ {\tilde{Y}}_{2},$

If $\overset{˘}{S} ({\tilde{Y}}_{1}) = \overset{˘}{S} ({\tilde{Y}}_{2}),$ then accuracy function are compared as

If $\overset{˘}{ℍ} ({\tilde{Y}}_{1}) > \overset{˘}{ℍ} ({\tilde{Y}}_{2}),$ then ${\tilde{Y}}_{1} ≻ {\tilde{Y}}_{2},$

If $\overset{˘}{ℍ} ({\tilde{Y}}_{1}) = \overset{˘}{ℍ} ({\tilde{Y}}_{2}),$ then ${\tilde{Y}}_{1} \sim {\tilde{Y}}_{2} .$

Definition 2.5. [25] Let ${\tilde{Y}}_{1} = (Ψ_{1} e^{i κ_{1}}, Φ_{1} e^{i χ_{1}})$ and ${\tilde{Y}}_{2} = (Ψ_{2} e^{i κ_{2}}, Φ_{2} e^{i χ_{2}})$ be two CPFNs. Then

${\tilde{Y}}_{1} \subseteq {\tilde{Y}}_{2}$ if Ψ₁ ≤ Ψ₂, κ₁ ≤ κ₂, Φ₁ ≥ Φ₂ and χ₁ ≥ χ₂,

${\tilde{Y}}_{1} = {\tilde{Y}}_{2}$ if Ψ₁ = Ψ₂, κ₁ = κ₂, Φ₁ = Φ₂ and χ₁ = χ₂,

${\tilde{Y}}_{1}^{c} = (Φ_{1} e^{i χ_{1}}, Ψ_{1} e^{i κ_{1}}) .$

3 Complex pythagorean fuzzy prioritized aggregation operators

In this section, we investigate two CPF prioritized AOs, namely, CPFPWA operator, CPFPWG operator in CPF environment. It can be seen that the score function defined in Definition 2.3, Equation 1 belongs to the interval [-2, 2] . Therefore, to facilitate the following study, here we define another score function to score the alternatives which belongs to the interval [0, 2] .

Definition 3.1. Let ${\tilde{Y}}_{0} = (Ψ_{0} e^{i κ_{0}}, Φ_{0} e^{i χ_{0}})$ be CPFN. Then the new function of CPFN to score the alternatives is given as ${\overset{˘}{S}}^{*} ({\tilde{Y}}_{0}) = \frac{1}{2} ((1 + Ψ_{0}^{2} - Φ_{0}^{2}) + (1 + (\frac{κ_{0}}{2 π})^{2} - (\frac{χ_{0}}{2 π})^{2})),$ (3) where ${\overset{˘}{S}}^{*} ({\tilde{Y}}_{0}) \in [0, 2] .$

Definition 3.2. Let $\tilde{Y} = (Ψ e^{i κ}, Φ e^{i χ})$ , ${\tilde{Y}}_{1} =$ (Ψ₁e^{iκ
₁}, Φ₁e^{iχ
₁}) and ${\tilde{Y}}_{2} = (Ψ_{2} e^{i κ_{2}}, Φ_{2} e^{i χ_{2}})$ be three CPFNs. Then for ω > 0, the operational laws corresponding to these CPFNs are defined as

${\tilde{Y}}_{1} \oplus {\tilde{Y}}_{2} = (\sqrt{Ψ_{1}^{2} + Ψ_{2}^{2} - Ψ_{1}^{2} Ψ_{2}^{2}} e^{i 2 π \sqrt{(\frac{κ_{1}}{2 π})^{2} + (\frac{κ_{2}}{π})^{2} - (\frac{κ_{1}}{2 π})^{2} (\frac{κ_{2}}{2 π})^{2}}}, Φ_{1} Φ_{2} e^{i 2 π (\frac{χ_{1}}{2 π}) (\frac{χ_{2}}{2 π})},)$

${\tilde{Y}}_{1} ⨂ {\tilde{Y}}_{2} = (Ψ_{1} Ψ_{2} e^{i 2 π (\frac{κ_{1}}{2 π}) (\frac{κ_{2}}{2 π})}$ , $\sqrt{Φ_{1}^{2} + Φ_{2}^{2} - Φ_{1}^{2} Φ_{2}^{2}} e^{i 2 π \sqrt{(\frac{χ_{1}}{2 π})^{2} + (\frac{χ_{2}}{2 π})^{2} - (\frac{χ_{1}}{2 π})^{2} (\frac{χ_{2}}{2 π})^{2}}}),$

$ω \tilde{Y} = (\sqrt{1 - (1 - Ψ^{2})^{ω}} e^{i 2 π \sqrt{1 - (1 - (\frac{κ_{1}}{2 π})^{2})^{ω}}}, Φ^{ω} e^{i 2 π (\frac{χ}{2 π})^{ω}}),$

${\tilde{Y}}^{ω} = (Ψ^{ω} e^{i 2 π (\frac{κ}{2 π})^{ω}}, \sqrt{1 - (1 - Φ^{2})^{ω}} e^{i 2 π \sqrt{1 - (1 - (\frac{χ}{2 π})^{2})^{ω}}})$

3.1 Complex pythagorean fuzzy prioritized weighted averaging operator

This section presents complex Pythagorean fuzzy prioritized weighted AOs, namely, complex Pythagorean fuzzy prioritized weighted averaging operator and complex Pythagorean fuzzy prioritized ordered weighted averaging operator with their worthwhile properties.

Definition 3.3. Let ${\tilde{Y}}_{r} = (Ψ_{r} e^{i κ_{r}}, Φ_{r} e^{i χ_{r}})$ (r = 1, 2, ⋯ s) be a collection of CPFNs. Then complex Pythagorean fuzzy prioritized weighted averaging (CPFPWA) operator is given by $CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = \oplus_{r = 1}^{s} (\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{r})$ (4) $= (\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{1} \oplus \frac{F_{2}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{2} \oplus \dots \oplus \frac{F_{s}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{s},),$ where $F_{r} = \prod_{i = 1}^{r - 1} \overset{˘}{S^{*}} ({\tilde{Y}}_{i})$ (r = 1, 2, ⋯ s) such that 𝔽₁ = 1 and $\overset{˘}{S^{*}} ({\tilde{Y}}_{i})$ is the score of CPFN ${\tilde{Y}}_{i} = (Ψ_{i} e^{i κ_{i}}, Φ_{i} e^{i χ_{i}}) .$

Next, based on the operational laws on CPFNs as defined in Definition 3.2, we have the following theorem.

Theorem 3.1. Let ${\tilde{Y}}_{r} = (Ψ_{r} e^{i κ_{r}}, Φ_{r} e^{i χ_{r}})$ (r = 1, 2, ⋯ s) be a collection of CPFNs. Then the aggregated value obtained by utilizing the CPFPWA operators is also a CPFN, given by $CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = \oplus_{r = 1}^{s} (\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{r}) = (\begin{matrix} \sqrt{1 - \prod_{r = 1}^{s} (1 - Ψ_{r}^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{s} (1 - (\frac{κ_{r}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}}}, \\ \prod_{r = 1}^{s} (Φ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π (\prod_{r = 1}^{s} {(\frac{χ_{r}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}})} \end{matrix}),$ (5) where $F_{r} = \prod_{i = 1}^{r - 1} \overset{˘}{S^{*}} ({\tilde{Y}}_{i})$ (r = 1, 2, ⋯ s) such that 𝔽₁ = 1 and $\overset{˘}{S^{*}} ({\tilde{Y}}_{i})$ is the score of CPFN ${\tilde{Y}}_{i} = (Ψ_{i} e^{i κ_{i}}, Φ_{i} e^{i χ_{i}}) .$

Proof. We prove this theorem by mathematical induction.

First, we prove that Equation (5) is true for s = 2 . Then, since $\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{1} = (\begin{matrix} \sqrt{1 - (1 - Ψ_{1}^{2})^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - (1 - (\frac{κ_{1}}{2 π})^{2})^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}}}}, \\ (Φ_{1})^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π ({(\frac{χ_{1}}{2 π})}^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}})} \end{matrix}),$ $\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{2} = (\begin{matrix} \sqrt{1 - (1 - Ψ_{1}^{2})^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - (1 - (\frac{κ_{2}}{2 π})^{2})^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}}}}, \\ (Φ_{2})^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π ({(\frac{χ_{2}}{2 π})}^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}})} \end{matrix}) .$ Thus, we have $CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}) = \frac{F_{1}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{1} \oplus \frac{F_{2}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{2} = (\begin{matrix} \sqrt{1 - (1 - Ψ_{1}^{2})^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - (1 - (\frac{κ_{1}}{2 π})^{2})^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}}}}, \\ (Φ_{1})^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π ({(\frac{χ_{1}}{2 π})}^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}})} \end{matrix}) \oplus (\begin{matrix} \sqrt{1 - (1 - Ψ_{2}^{2})^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - (1 - (\frac{κ_{2}}{2 π})^{2})^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}}}}, \\ (Φ_{2})^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π ({(\frac{χ_{2}}{2 π})}^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}})} \end{matrix}) = (\begin{matrix} \sqrt{1 - \prod_{r = 1}^{2} (1 - Ψ_{r}^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{2} (1 - (\frac{κ_{r}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}}}, \\ \prod_{r = 1}^{2} (Φ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π (\prod_{r = 1}^{2} {(\frac{χ_{r}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}})} \end{matrix}) .$ Thus, the Equation (5) is true for s=2.

Now, assume that the Equation (5) is true for s = h . Then, we have $CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{h}) = \oplus_{r = 1}^{h} (\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}} {\tilde{Y}}_{r}) = (\begin{matrix} \sqrt{1 - \prod_{r = 1}^{h} (1 - (Ψ_{r})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{h} (1 - (\frac{κ_{r}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}}}}}, \\ \prod_{r = 1}^{h} (Φ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}}} e^{i 2 π (\prod_{r = 1}^{h} {(\frac{χ_{r}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}}})} \end{matrix}),$ Now, we show that Equation (5) is true for s=h+1. $\begin{matrix} CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{h + 1}) & = \oplus_{r = 1}^{h + 1} (\frac{F_{r}}{\sum_{r = 1}^{h + 1} F_{r}} {\tilde{Y}}_{r}) \\ = \oplus_{r = 1}^{h} (\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}} {\tilde{Y}}_{r}) \oplus (\frac{F_{h + 1}}{\sum_{r = 1}^{h + 1} F_{r}} {\tilde{Y}}_{h + 1}) \end{matrix}$ $\begin{matrix} = (\begin{matrix} \sqrt{1 - \prod_{r = 1}^{h} (1 - Ψ_{r}^{2})^{\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{h} (1 - (\frac{κ_{r}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}}}}}, \\ \prod_{r = 1}^{h} (Φ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}}} e^{i 2 π (\prod_{r = 1}^{h} {(\frac{χ_{r}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{h} F_{r}}})} \end{matrix}) \\ \oplus (\begin{matrix} \sqrt{1 - (1 - Ψ_{h + 1}^{2})^{\frac{F_{h + 1}}{\sum_{r = 1}^{h + 1} F_{r}}}} e^{i 2 π \sqrt{1 - (1 - (\frac{κ_{h + 1}}{2 π})^{2})^{\frac{F_{h + 1}}{\sum_{r = 1}^{h + 1} F_{r}}}}}, \\ (Φ_{h + 1})^{\frac{F_{r}}{\sum_{r = 1}^{h + 1} F_{h + 1}}} e^{i 2 π ({(\frac{χ_{h + 1}}{2 π})}^{\frac{F_{h + 1}}{\sum_{r = 1}^{h + 1} F_{r}}})} \end{matrix}), \\ = (\begin{matrix} \sqrt{1 - \prod_{r = 1}^{h + 1} (1 - Ψ_{r}^{2})^{\frac{F_{r}}{\sum_{r = 1}^{h + 1} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{h + 1} (1 - (\frac{κ_{r}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{h + 1} F_{r}}}}}, \\ \prod_{r = 1}^{h + 1} (Φ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{h + 1} F_{r}}} e^{i 2 π (\prod_{r = 1}^{h + 1} {(\frac{χ_{r}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{h + 1} F_{r}}})} \end{matrix}) . \end{matrix}$ Thus, the Equation (5) hold for s = h + 1 .□

Now, we discuss some desirable properties of CPFPWA operator with their proofs below.

Theorem 3.2. (Idempotency) Let ${\tilde{Y}}_{r} = (Ψ_{r} e^{i κ_{r}}, Φ_{r} e^{i χ_{r}})$ (r = 1, 2, ⋯ s) be a collection of CPFNs. If all ${\tilde{Y}}_{r}$ (r = 1, 2, ⋯ s) are equal, i.e., ${\tilde{Y}}_{r} = {\tilde{Y}}_{0}$ for all r. Then $CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = {\tilde{Y}}_{0} .$ (6)

Proof. Let ${\tilde{Y}}_{0} = (Ψ_{0} e^{i κ_{0}}, Φ_{0} e^{i χ_{0}})$ be a CPFN. Then according to Definition 2.5, we have Ψ_r = Ψ₀, κ_r = κ₀, Φ_r = Φ₀ and χ_r = χ₀, for all (r = 1, 2, ⋯ s) . Then according to Theorem 3.1, we have $\begin{matrix} CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) & = \oplus_{r = 1}^{s} (\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{r}) \\ = (\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{1} \oplus \frac{F_{2}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{2} \oplus \dots \oplus \frac{F_{s}}{\sum_{r = 1}^{s} F_{r}} {\tilde{Y}}_{s}) \\ = (\begin{matrix} \sqrt{1 - \prod_{r = 1}^{s} (1 - Ψ_{r}^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{s} (1 - (\frac{κ_{r}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}}}, \\ \prod_{r = 1}^{s} (Φ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π (\prod_{r = 1}^{s} {(\frac{χ_{r}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}})} \end{matrix}) \\ = (\begin{matrix} \sqrt{1 - \prod_{r = 1}^{s} (1 - Ψ_{0}^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{s} (1 - (\frac{κ_{0}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}}}, \\ \prod_{r = 1}^{s} (Φ_{0})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π (\prod_{r = 1}^{s} {(\frac{χ_{0}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}})} \end{matrix}) \\ = (\begin{matrix} \sqrt{1 - (1 - Ψ_{0}^{2})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - (1 - (\frac{κ_{0}}{2 π})^{2})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}}}}, \\ (Φ_{0})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π ({(\frac{χ_{0}}{2 π})}^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}})} \end{matrix}) \\ = (\begin{matrix} \sqrt{1 - (1 - Ψ_{0}^{2})} e^{i 2 π \sqrt{1 - (1 - (\frac{κ_{0}}{2 π})^{2})}}, \\ (Φ_{0}) e^{i 2 π (\frac{χ_{0}}{2 π})} \end{matrix}) = {\tilde{Y}}_{0} = (Ψ_{0} e^{i κ_{0}}, Φ_{0} e^{i χ_{0}}) . \end{matrix}$ □

Theorem 3.3. (Boundedness) Let ${\tilde{Y}}_{r} = (Ψ_{r} e^{i κ_{r}}, Φ_{r} e^{i χ_{r}})$ (r = 1, 2, ⋯ s) be a collection of CPFNs. Let ${\tilde{Y}}^{-} = (min Ψ_{r} e^{i min κ_{r}}, max Φ_{r} e^{i max χ_{r}}) = (Ψ^{-} e^{i κ^{-}}, Φ^{+} e^{i χ^{+}})$ and ${\tilde{Y}}^{+} = (max Ψ_{r} e^{i max κ_{r}}, min Φ_{r} e^{i min χ_{r}}) = (Ψ^{+} e^{i κ^{+}}, Φ^{-} e^{i χ^{-}}),$ then ${\tilde{Y}}^{-} \leq CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) \leq {\tilde{Y}}^{+} .$

Proof. Let CPFPWA $({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = (Ψ_{\tilde{Y}} e^{i κ_{\tilde{Y}}}, Φ_{\tilde{Y}} e^{i χ_{\tilde{Y}}}) .$ Since, for any CPFN ${\tilde{Y}}_{r},$ we have $\begin{matrix} min (Ψ) & \leq Ψ_{r} \leq max (Ψ) \\ \sqrt{min (Ψ_{r})^{2}} & \leq \sqrt{(Ψ_{r})^{2}} \leq \sqrt{max (Ψ_{r})^{2}} \\ \sqrt{1 - max (Ψ_{r})^{2}} & \leq \sqrt{1 - (Ψ_{r})^{2}} \leq \sqrt{1 - min (Ψ_{r})^{2}} \\ \sqrt{\prod_{r = 1}^{s} {(1 - max (Ψ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} & \leq \sqrt{\prod_{r = 1}^{s} {(1 - (Ψ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \leq \sqrt{\prod_{r = 1}^{s} {(1 - min (Ψ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \\ \sqrt{{(1 - max (Ψ_{r})^{2})}^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}}} & \leq \sqrt{\prod_{r = 1}^{s} {(1 - (Ψ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \leq \sqrt{{(1 - min (Ψ_{r})^{2})}^{\sum_{r = 1}^{s} \frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \\ \sqrt{(1 - max (Ψ_{r})^{2})} & \leq \sqrt{\prod_{r = 1}^{s} {(1 - (Ψ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \leq \sqrt{(1 - min (Ψ_{r})^{2})} \\ \sqrt{min (Ψ_{r})^{2}} & \leq \sqrt{\prod_{r = 1}^{s} {(1 - (Ψ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \leq \sqrt{max (Ψ_{r})^{2}} \\ Ψ^{-} & \leq Ψ_{\tilde{Y}} \leq Ψ^{+} . \end{matrix}$ Similarly, for the amplitude term of non-membership grade, we have $\begin{matrix} min (Φ_{r}) & \leq Φ_{r} \leq max (φ_{r}) \\ \prod_{r = 1}^{s} min (Φ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} & \leq \prod_{r = 1}^{s} Φ_{r}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} \leq \prod_{r = 1}^{s} max (φ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} \\ min (Φ_{r})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}} & \leq \prod_{r = 1}^{s} Φ_{r}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} \leq \prod_{r = 1}^{s} max (φ_{r})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}} \\ Φ^{-} & \leq \prod_{r = 1}^{s} Φ_{\tilde{Y}} \leq Φ^{+} . \end{matrix}$ In the similar manner, we can obtain the results for the phase terms of membership and non-membership garde. Therefore ${\tilde{Y}}^{-} \leq CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) \leq {\tilde{Y}}^{+} .$ □

Theorem 3.4. (Monotonicity) Let ${\tilde{Y}}_{r} = (Ψ_{{\tilde{Y}}_{r}} e^{i κ_{{\tilde{Y}}_{r}}}, Φ_{{\tilde{Y}}_{r}} e^{i χ_{{\tilde{Y}}_{r}}})$ and ${\tilde{ℤ}}_{r} = (Ψ_{{\tilde{ℤ}}_{r}} e^{i κ_{{\tilde{ℤ}}_{r}}}, Φ_{{\tilde{ℤ}}_{r}} e^{i χ_{{\tilde{ℤ}}_{r}}})$ (r = 1, 2, ⋯ s) be two collections of CPFNs. If $Ψ_{{\tilde{Y}}_{r}} \leq Ψ_{{\tilde{ℤ}}_{r}},$ $κ_{{\tilde{Y}}_{r}} \leq κ_{{\tilde{ℤ}}_{r}},$ $Φ_{{\tilde{Y}}_{r}} \geq Φ_{{\tilde{ℤ}}_{r}}$ and $χ_{{\tilde{Y}}_{r}} \geq χ_{{\tilde{ℤ}}_{r}},$ then $CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) \subseteq CPFPWA ({\tilde{ℤ}}_{1}, {\tilde{ℤ}}_{2}, \dots, {\tilde{ℤ}}_{s}) .$

Proof. Let $\begin{matrix} \frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}} = (\frac{F_{{\tilde{Y}}_{1}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{1}}}, \frac{F_{{\tilde{Y}}_{2}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{2}}}, \dots \frac{F_{{\tilde{Y}}_{s}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{s}}}), \\ \frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}} = (\frac{F_{{\tilde{ℤ}}_{1}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{1}}}, \frac{F_{{\tilde{ℤ}}_{2}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{2}}}, \dots \frac{F_{{\tilde{ℤ}}_{s}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{s}}}), \end{matrix}$ be the prioritized weight vectors of the CPFNs ${\tilde{Y}}_{r} = (Ψ_{{\tilde{Y}}_{r}} e^{i κ_{{\tilde{Y}}_{r}}}, Φ_{{\tilde{Y}}_{r}} e^{i χ_{{\tilde{Y}}_{r}}})$ and ${\tilde{ℤ}}_{r} = (Ψ_{{\tilde{ℤ}}_{r}} e^{i κ_{{\tilde{ℤ}}_{r}}}, Φ_{{\tilde{ℤ}}_{r}} e^{i χ_{{\tilde{ℤ}}_{r}}})$ , respectively such that $\frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}}, \frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}} \in [0, 1]$ with condition $\sum_{r = 1}^{s} \frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}} = 1$ and $\sum_{r = 1}^{s} \frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}} = 1 .$ Let us denote $\begin{matrix} CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = (Ψ_{\tilde{Y}} e^{i κ_{\tilde{Y}}}, Φ_{\tilde{Y}} e^{i χ_{\tilde{Y}}}) \\ CPFPWA ({\tilde{ℤ}}_{1}, {\tilde{ℤ}}_{2}, \dots, {\tilde{ℤ}}_{s}) = (Ψ_{\tilde{ℤ}} e^{i κ_{\tilde{ℤ}}}, Φ_{\tilde{ℤ}} e^{i χ_{\tilde{ℤ}}}) \end{matrix}$ Now, according to the given statement, if $Ψ_{{\tilde{Y}}_{r}} \leq Ψ_{{\tilde{ℤ}}_{r}},$ then clearly $\sqrt{(Ψ_{{\tilde{Y}}_{r}})^{2}} \leq \sqrt{(Ψ_{{\tilde{ℤ}}_{r}})^{2}} .$ Moreover, we have $\begin{matrix} \sqrt{1 - {(Ψ_{{\tilde{ℤ}}_{r}})}^{2}} & \leq & \sqrt{1 - {(Ψ_{{\tilde{Y}}_{r}})}^{2}} \\ \sqrt{\prod_{r = 1}^{s} {(1 - {(Ψ_{{\tilde{Y}}_{r}})}^{2})}^{\frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}}}} & \leq & \sqrt{\prod_{r = 1}^{s} {(1 - {(Ψ_{{\tilde{ℤ}}_{r}})}^{2})}^{\frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}}}} \\ \sqrt{1 - \prod_{r = 1}^{s} {(1 - {(Ψ_{{\tilde{Y}}_{r}})}^{2})}^{\frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}}}} & \leq & \sqrt{1 - \prod_{r = 1}^{s} {(1 - {(Ψ_{{\tilde{ℤ}}_{r}})}^{2})}^{\frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}}}} . \end{matrix}$ Thus, clearly $Ψ_{\tilde{Y}} \leq Ψ_{\tilde{ℤ}} .$ Further, since $Φ_{{\tilde{Y}}_{r}} \geq Φ_{{\tilde{ℤ}}_{r}} .$ Then, $\prod_{r = 1}^{s} (Φ_{{\tilde{Y}}_{r}})^{\frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}}} \geq \prod_{r = 1}^{s} (Φ_{{\tilde{ℤ}}_{r}})^{\frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}}} .$ Thus, clearly $Φ_{\tilde{Y}} \geq Φ_{\tilde{ℤ}} .$ Similarly, we can obtain $κ_{\tilde{Y}} \leq κ_{\tilde{ℤ}}$ and $χ_{\tilde{Y}} \geq χ_{\tilde{ℤ}}$ for the non-membership grade.□

3.2 Complex pythagorean fuzzy prioritized weighted geometric operator

In this section, we initiate CPFPWG operator with its efficacious properties below.

Definition 3.4. Let ${\tilde{Y}}_{r} = (Ψ_{r} e^{i κ_{r}}, Φ_{r} e^{i χ_{r}})$ (r = 1, 2, ⋯ s) be a collection of CPFNs. Then complex Pythagorean fuzzy prioritized weighted geometric (CPFPWG) operator is given by $CPFPWG ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = ⨂_{r = 1}^{s} {({\tilde{Y}}_{r})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}$ (7) $= ({\tilde{Y}}_{1}^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}} ⨂ {\tilde{Y}}_{2}^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}} ⨂ \dots ⨂ {\tilde{Y}}_{s}^{\frac{F_{s}}{\sum_{r = 1}^{s} F_{r}}}),$ where $F_{r} = \prod_{i = 1}^{r - 1} \overset{˘}{S^{*}} ({\tilde{Y}}_{i})$ (r = 1, 2, ⋯ s) such that 𝔽₁ = 1 and $\overset{˘}{S^{*}} ({\tilde{Y}}_{i})$ is the score of CPFN ${\tilde{Y}}_{i} = (Ψ_{i} e^{i κ_{i}}, Φ_{i} e^{i χ_{i}}) .$

Next, based on the operational laws on CPFNs as defined in Definition 3.2, we have the following theorem.

Theorem 3.5. Let ${\tilde{Y}}_{r} = (Ψ_{r} e^{i κ_{r}}, Φ_{r} e^{i χ_{r}})$ (r = 1, 2, ⋯ s) be a collection of CPFNs. Then the aggregated value obtained by utilizing the CPFPWG operators is also a CPFN, given by $CPFPWG ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = ⨂_{r = 1}^{s} {({\tilde{Y}}_{r})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} = (\begin{matrix} \prod_{r = 1}^{s} (Ψ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π (\prod_{r = 1}^{s} {(\frac{κ_{r}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}})}, \\ \sqrt{1 - \prod_{r = 1}^{s} (1 - Φ_{r}^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{s} (1 - (\frac{χ_{r}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}}} \end{matrix}),$ (8) where $F_{r} = \prod_{i = 1}^{r - 1} \overset{˘}{S^{*}} ({\tilde{Y}}_{i})$ (r = 1, 2, ⋯ s) such that 𝔽₁ = 1 and $\overset{˘}{S^{*}} ({\tilde{Y}}_{i})$ is the score of CPFN ${\tilde{Y}}_{i} = (Ψ_{i} e^{i κ_{i}}, Φ_{i} e^{i χ_{i}}) .$

Proof. Similar to Theorem 3.1.□

Now, we discuss some properties of CPFPWG operator, namely, idempotency, boundedness, and monotonicity below.

Theorem 3.6. (Idempotency) Let ${\tilde{Y}}_{r} = (Ψ_{r} e^{i κ_{r}}, Φ_{r} e^{i χ_{r}})$ (r = 1, 2, ⋯ s) be a collection of CPFNs. If all ${\tilde{Y}}_{r}$ (r = 1, 2, ⋯ s) are same, i.e., ${\tilde{Y}}_{r} = {\tilde{Y}}_{0}$ ∀ r. Then $CPFPWG ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = {\tilde{Y}}_{0} .$ (9)

Proof. Let ${\tilde{Y}}_{0} = (Ψ_{0} e^{i κ_{0}}, Φ_{0} e^{i χ_{0}})$ be a CPFN. Then according to Definition 2.5, we have Ψ_r = Ψ₀, κ_r = κ₀, Φ_r = Φ₀ and χ_r = χ₀, for all (r = 1, 2, ⋯ s) . Then according to Theorem 3.5, we have $\begin{matrix} CPFPWG ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) & = ⨂_{r = 1}^{s} ({\tilde{Y}}_{r}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}), \\ = ({\tilde{Y}}_{1}^{\frac{F_{1}}{\sum_{r = 1}^{s} F_{r}}} ⨂ {\tilde{Y}}_{2}^{\frac{F_{2}}{\sum_{r = 1}^{s} F_{r}}} ⨂ \dots ⨂ {\tilde{Y}}_{s}^{\frac{F_{s}}{\sum_{r = 1}^{s} F_{r}}}) \\ = (\begin{matrix} \prod_{r = 1}^{s} (Ψ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π (\prod_{r = 1}^{s} {(\frac{κ_{r}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}})}, \\ \sqrt{1 - \prod_{r = 1}^{s} (1 - Φ_{r}^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{s} (1 - (\frac{χ_{r}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}}} \end{matrix}) \\ = (\begin{matrix} \prod_{r = 1}^{s} (Ψ_{0})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π (\prod_{r = 1}^{s} {(\frac{κ_{0}}{2 π})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}})}, \\ \sqrt{1 - \prod_{r = 1}^{s} (1 - Φ_{0}^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - \prod_{r = 1}^{s} (1 - (\frac{χ_{0}}{2 π})^{2})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}}} \end{matrix}) \\ = (\begin{matrix} (Ψ_{0})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}} e^{i 2 π ({(\frac{κ_{0}}{2 π})}^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}})}, \\ \sqrt{1 - (1 - Φ_{0}^{2})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}}} e^{i 2 π \sqrt{1 - (1 - (\frac{χ_{0}}{2 π})^{2})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}}}} \end{matrix}) \\ = (\begin{matrix} (Ψ_{0}) e^{i 2 π (\frac{κ_{0}}{2 π})}, \\ \sqrt{1 - (1 - Φ_{0}^{2})} e^{i 2 π \sqrt{1 - (1 - (\frac{χ_{0}}{2 π})^{2})}} \end{matrix}) = {\tilde{Y}}_{0} = (Ψ_{0} e^{i κ_{0}}, Φ_{0} e^{i χ_{0}}) . \end{matrix}$ □

Theorem 3.7. (Boundedness) Let ${\tilde{Y}}_{r} = (Ψ_{r} e^{i κ_{r}}, Φ_{r} e^{i χ_{r}})$ (r = 1, 2, ⋯ s) be a collection of CPFNs. Let ${\tilde{Y}}^{-} = (min Ψ_{r} e^{i min κ_{r}}, max Φ_{r} e^{i max χ_{r}}) = (Ψ^{-} e^{i κ^{-}}, Φ^{+} e^{i χ^{+}})$ and ${\tilde{Y}}^{+} = (max Ψ_{r} e^{i max κ_{r}}, min Φ_{r} e^{i min χ_{r}}) = (Ψ^{+} e^{i κ^{+}}, Φ^{-} e^{i χ^{-}}),$ then ${\tilde{Y}}^{-} \leq CPFPWG ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) \leq {\tilde{Y}}^{+} .$

Proof. Let CPFPWG $({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = (Ψ_{\tilde{Y}} e^{i κ_{\tilde{Y}}}, Φ_{\tilde{Y}} e^{i χ_{\tilde{Y}}}) .$ Since, for any CPFN ${\tilde{Y}}_{r},$ we have $\begin{matrix} min (Ψ_{r}) & \leq Ψ_{r} \leq max (φ_{r}) \\ \prod_{r = 1}^{s} min (Ψ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} & \leq \prod_{r = 1}^{s} Ψ_{r}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} \leq \prod_{r = 1}^{s} max (φ_{r})^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} \\ min (Ψ_{r})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}} & \leq \prod_{r = 1}^{s} Ψ_{r}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}} \leq \prod_{r = 1}^{s} max (φ_{r})^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}} \\ Ψ^{-} & \leq \prod_{r = 1}^{s} Ψ_{\tilde{Y}} \leq Ψ^{+} . \end{matrix}$ Similarly, for the amplitude term of non-membership grade, we have $\begin{matrix} min (Φ) & \leq Φ_{r} \leq max (Φ) \\ \sqrt{min (Φ_{r})^{2}} & \leq \sqrt{(Φ_{r})^{2}} \leq \sqrt{max (Φ_{r})^{2}} \\ \sqrt{1 - max (Φ_{r})^{2}} & \leq \sqrt{1 - (Φ_{r})^{2}} \leq \sqrt{1 - min (Φ_{r})^{2}} \\ \sqrt{\prod_{r = 1}^{s} {(1 - max (Φ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} & \leq \sqrt{\prod_{r = 1}^{s} {(1 - (Φ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \leq \sqrt{\prod_{r = 1}^{s} {(1 - min (Φ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \\ \sqrt{{(1 - max (Φ_{r})^{2})}^{\frac{\sum_{r = 1}^{s} F_{r}}{\sum_{r = 1}^{s} F_{r}}}} & \leq \sqrt{\prod_{r = 1}^{s} {(1 - (Φ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \leq \sqrt{{(1 - min (Φ_{r})^{2})}^{\sum_{r = 1}^{s} \frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \\ \sqrt{(1 - max (Φ_{r})^{2})} & \leq \sqrt{\prod_{r = 1}^{s} {(1 - (Φ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \leq \sqrt{(1 - min (Φ_{r})^{2})} \\ \sqrt{min (Φ_{r})^{2}} & \leq \sqrt{\prod_{r = 1}^{s} {(1 - (Φ_{r})^{2})}^{\frac{F_{r}}{\sum_{r = 1}^{s} F_{r}}}} \leq \sqrt{max (Φ_{r})^{2}} \\ Φ^{-} & \leq Φ_{\tilde{Y}} \leq Φ^{+} . \end{matrix}$ In the similar manner, we can obtain the results for the phase terms of membership and non-membership garde. Therefore ${\tilde{Y}}^{-} \leq CPFPWA ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) \leq {\tilde{Y}}^{+} .$ □

Theorem 3.8. (Monotonicity) Let ${\tilde{Y}}_{r} = (Ψ_{{\tilde{Y}}_{r}} e^{i κ_{{\tilde{Y}}_{r}}}, Φ_{{\tilde{Y}}_{r}} e^{i χ_{{\tilde{Y}}_{r}}})$ and ${\tilde{ℤ}}_{r} = (Ψ_{{\tilde{ℤ}}_{r}} e^{i κ_{{\tilde{ℤ}}_{r}}}, Φ_{{\tilde{ℤ}}_{r}} e^{i χ_{{\tilde{ℤ}}_{r}}})$ (r = 1, 2, ⋯ s) be two collections of CPFNs. If $Ψ_{{\tilde{Y}}_{r}} \leq Ψ_{{\tilde{ℤ}}_{r}},$ $κ_{{\tilde{Y}}_{r}} \leq κ_{{\tilde{ℤ}}_{r}},$ $Φ_{{\tilde{Y}}_{r}} \geq Φ_{{\tilde{ℤ}}_{r}}$ and $χ_{{\tilde{Y}}_{r}} \geq χ_{{\tilde{ℤ}}_{r}},$ then $CPFPWG ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) \subseteq CPFPWG ({\tilde{ℤ}}_{1}, {\tilde{ℤ}}_{2}, \dots, {\tilde{ℤ}}_{s}) .$

Proof. Let $\begin{matrix} \frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}} = \\ (\frac{F_{{\tilde{Y}}_{1}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{1}}}, \frac{F_{{\tilde{Y}}_{2}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{2}}}, \dots \frac{F_{{\tilde{Y}}_{s}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{s}}}), \\ \frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}} = \\ (\frac{F_{{\tilde{ℤ}}_{1}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{1}}}, \frac{F_{{\tilde{ℤ}}_{2}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{2}}}, \dots \frac{F_{{\tilde{ℤ}}_{s}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{s}}}), \end{matrix}$ be the prioritized weight vector of the CPFNs ${\tilde{Y}}_{r} = (Ψ_{{\tilde{Y}}_{r}} e^{i κ_{{\tilde{Y}}_{r}}}, Φ_{{\tilde{Y}}_{r}} e^{i χ_{{\tilde{Y}}_{r}}})$ and ${\tilde{ℤ}}_{r} = (Ψ_{{\tilde{ℤ}}_{r}} e^{i κ_{{\tilde{ℤ}}_{r}}}$ , $Φ_{{\tilde{ℤ}}_{r}} e^{i χ_{{\tilde{ℤ}}_{r}}})$ , respectively such that $\frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}}$ , $\frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}} \in [0, 1]$ with condition $\sum_{r = 1}^{s} \frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}} = 1$ and $\sum_{r = 1}^{s} \frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}} .$ Let us denote $\begin{matrix} CPFPWG ({\tilde{Y}}_{1}, {\tilde{Y}}_{2}, \dots, {\tilde{Y}}_{s}) = (Ψ_{\tilde{Y}} e^{i κ_{\tilde{Y}}}, Φ_{\tilde{Y}} e^{i χ_{\tilde{Y}}}) \\ CPFPWG ({\tilde{ℤ}}_{1}, {\tilde{ℤ}}_{2}, \dots, {\tilde{ℤ}}_{s}) = (Ψ_{\tilde{ℤ}} e^{i κ_{\tilde{ℤ}}}, Φ_{\tilde{ℤ}} e^{i χ_{\tilde{ℤ}}}) \end{matrix}$ Now, according to the given statement if $Ψ_{{\tilde{Y}}_{r}} \leq Ψ_{{\tilde{ℤ}}_{r}},$ then $\prod_{r = 1}^{s} (Ψ_{{\tilde{Y}}_{{\tilde{Y}}_{r}}})^{\frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}}} \leq \prod_{r = 1}^{s} (Ψ_{{\tilde{ℤ}}_{r}})^{\frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}}} .$ Thus, clearly $Ψ_{\tilde{Y}} \leq Ψ_{\tilde{ℤ}} .$ Further, since $Φ_{{\tilde{Y}}_{r}} \geq Φ_{{\tilde{ℤ}}_{r}},$ then clearly $\sqrt{(Φ_{{\tilde{Y}}_{r}})^{2}} \geq \sqrt{(Φ_{{\tilde{ℤ}}_{r}})^{2}} .$ Moreover, we have $\begin{matrix} \sqrt{1 - {(Φ_{{\tilde{ℤ}}_{r}})}^{2}} \geq \sqrt{1 - {(Φ_{{\tilde{Y}}_{r}})}^{2}} \\ \sqrt{\prod_{r = 1}^{s} {(1 - {(Φ_{{\tilde{Y}}_{r}})}^{2})}^{\frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}}}} \geq \end{matrix}$ $\begin{matrix} \sqrt{\prod_{r = 1}^{s} {(1 - {(Φ_{{\tilde{ℤ}}_{r}})}^{2})}^{\frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}}}} \\ \sqrt{1 - \prod_{r = 1}^{s} {(1 - {(Φ_{{\tilde{Y}}_{r}})}^{2})}^{\frac{F_{{\tilde{Y}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{Y}}_{r}}}}} \geq \\ \sqrt{1 - \prod_{r = 1}^{s} {(1 - {(Φ_{{\tilde{ℤ}}_{r}})}^{2})}^{\frac{F_{{\tilde{ℤ}}_{r}}}{\sum_{r = 1}^{s} F_{{\tilde{ℤ}}_{r}}}}} . \end{matrix}$ Thus, clearly $Φ_{\tilde{Y}} \leq Φ_{\tilde{ℤ}} .$ Similarly, we can obtain $κ_{\tilde{Y}} \leq κ_{\tilde{ℤ}}$ and $χ_{\tilde{Y}} \geq χ_{\tilde{ℤ}}$ for the non-membership grade. □

4 Application

In this section, we demonstrate a MCDM problem with CPF information by utilizing the proposed AOs. Let O = {O₁, O₂, ⋯ , O_p} be the collection of objects under multiple attributes set $U = {U_{1}, U_{2}, \dots, > U_{q}}$ such that there is prioritization between attributes revealed by the linear ordering $U_{1} > U_{2} > U_{3}, \dots U_{q}$ which shows that the criteria $U_{m}$ is preferential then the attribute $U_{n}$ if m < n . Also, let the set of DMs be ℍ = {ℍ₁, ℍ₂, ⋯ , > ℍ_f} , such that there is prioritization between DMs demonstrated by the linear ordering ℍ₁ > ℍ₂ > ℍ₃, ⋯ ℍ_f which reveal that the decision maker ℍ_γ is preferential then the decision maker ℍ_ρ if γ < ρ . Let $H^{(k)} = ({\tilde{Y}}_{mn}^{(k)})_{p \times q} = (Ψ_{mn}^{(k)} e^{i κ_{mn}^{(k)}}, Φ_{mn}^{(k)} e^{i χ_{mn}^{(k)}})_{p \times q}$ be the CPF decision matrix and $({\tilde{Y}}_{mn}^{(k)}) = (Ψ_{mn}^{(k)} e^{i κ_{mn}^{(k)}}, Φ_{mn}^{(k)} e^{i χ_{mn}^{(k)}})$ be the CPFN which is given by the decision maker to the alternative, where the membership value denote the degree which the alternative satisfy the criteria and the non-membership value denote the degree which alternative does not satisfy the criteria with the condition $Ψ_{mn}^{(k)}, Φ_{mn}^{(k)} \in [0, 1]$ such that $(Ψ_{mn}^{(k)})^{2} + (Φ_{mn}^{(k)})^{2} \leq 1$ and $κ_{mn}^{(k)}, χ_{mn}^{(k)} \in [0, 2 π]$ , such that ${(\frac{κ_{mn}^{(k)}}{2 π})}^{2} + {(\frac{χ_{mn}^{(k)}}{2 π})}^{2} \leq 1$ for all (m = 1, 2, ⋯ , p) , (n = 1, 2, ⋯ , q).

In order to normalize the data, the very first step is to convert the different types of criteria in the same type because there may exist different types of criteria. There is no need of normalization if all the attributes are of same type. However, if these involve distinct scales or/and units, then it is mandatory to convert them to the similar scale or/and unit. To make this point clear, let us consider two types of attribute, namely benefit type attribute (the larger the value the better is it) and cost type attribute (the smaller the value the better is it). In such cases, the very first step is to convert the cost type attribute values into the benefit type attribute values. Therefore, transform the CPF decision matrix $H^{(k)} = ({\tilde{Y}}_{mn}^{(k)})_{p \times q}$ into normalized CPF decision matrix $H^{* (k)} = ({\tilde{ℤ}}_{mn}^{(k)})_{p \times q},$ where ${\tilde{ℤ}}_{mn}^{(k)} = {\begin{matrix} {\tilde{Y}}_{mn}^{(k)}, for benefit type attribute, \\ ({\tilde{Y}}_{mn}^{(k)})^{c} for cost type attribute . \end{matrix}$ where $({\tilde{Y}}_{mn}^{(k)})^{c}$ is the complement of ${\tilde{Y}}_{mn}^{(k)},$ such that ${({\tilde{Y}}_{mn}^{(k)})}^{c} = (Φ_{mn}^{(k)} e^{i χ_{mn}^{(k)}}, Ψ_{mn}^{(k)} e^{i κ_{mn}^{(k)}})$ for all (m = 1, 2, ⋯ , p) , (n = 1, 2, ⋯ , q) . Now, we develop an algorithm to solve the MCDM under CPF environment by utilizing the proposed aggregation operators which is given in Table 1.

Table 1
Algorithm

Algorithm Steps to solve MCDM problem by using CPFP aggregation operators

Step 1 Summarize the preference values of DMs in the form of CPF decision matrix as

$H = (\begin{matrix} (Ψ_{11} e^{i κ_{11}}, Φ_{11} e^{i χ 11}) & (Ψ_{12} e^{i κ_{12}}, Φ_{12} e^{i χ 12}) & \dots & (Ψ_{1 q} e^{i κ_{1 q}}, Φ_{1 q} e^{i χ 1 q}) \\ (Ψ_{21} e^{i κ_{21}}, Φ_{21} e^{i χ 21}) & (Ψ_{22} e^{i κ_{22}}, Φ_{22} e^{i χ 22}) & \dots & (Ψ_{2 q} e^{i κ_{2 q}}, Φ_{2 q} e^{i χ 2 q}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (Ψ_{p 1} e^{i κ_{p 1}}, Φ_{p 1} e^{i χ p 1}) & (Ψ_{p 2} e^{i κ_{p 2}}, Φ_{p 2} e^{i χ p 2}) & \dots & (Ψ_{pq} e^{i κ_{pq}}, Φ_{pq} e^{i χ pq}) \end{matrix}) .$

Step 2 Compute the values of $F_{mn}^{(k)} (k = 1, 2, \dots, f)$ as given below

$\begin{matrix} F_{mn}^{(k)} & = \prod_{s}^{k - 1} {\tilde{S}}^{} ({\tilde{ℤ}}_{mn}^{(k)}), (k = 2, 3, \dots, f), \\ such that P_{mn}^{(1)} & = 1 . \end{matrix}$

Step 3 Aggregate the CPF individual decision matrices $H^{ (k)} = ({\tilde{ℤ}}_{mn}^{(k)})_{p \times q}, (k = 1, 2, \dots, f)$ into

the collective CPF decision matrix $H = ({\tilde{ℤ}}_{mn})_{p \times q}, (m = 1, 2, \dots, p) (n = 1, 2, \dots, q)$ by

utilizing the CPFPWA or CPFPWG operators given below.

$\begin{matrix} {\tilde{ℤ}}_{mn} = (Ψ_{mn} e^{i κ_{mn}}, Ψ_{mn} e^{i χ_{mn}}) = CPFPWA ({\tilde{ℤ}}_{mn}^{(1)}, {\tilde{ℤ}}_{mn}^{(2)}, \dots {\tilde{ℤ}}_{mn}^{(f)}) \end{matrix}$

$= (\begin{matrix} \sqrt{1 - \prod_{k = 1}^{f} (1 - (Ψ_{mn}^{(k)})^{2})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}}} e^{i 2 π \sqrt{1 - \prod_{k = 1}^{f} (1 - (\frac{κ_{mn}^{(k)}}{2 π})^{2})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}}}}, \\ \prod_{k = 1}^{f} (Φ_{mn}^{(k)})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}} e^{i 2 π (\prod_{k = 1}^{f} {(\frac{χ_{mn}^{(k)}}{2 π})}^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}})} \end{matrix}),$

or by utilizing CPFPWG operator

$\begin{matrix} {\tilde{ℤ}}_{mn} = (Ψ_{mn} e^{i κ_{mn}}, Ψ_{mn} e^{i χ_{mn}}) = CPFPWG ({\tilde{ℤ}}_{mn}^{(1)}, {\tilde{ℤ}}_{mn}^{(2)}, \dots {\tilde{ℤ}}_{mn}^{(f)}) \end{matrix}$

$= (\begin{matrix} \prod_{k = 1}^{f} (Ψ_{mn}^{(k)})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}} e^{i 2 π (\prod_{k = 1}^{f} {(\frac{κ_{mn}^{(k)}}{2 π})}^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}})}, \\ \sqrt{1 - \prod_{k = 1}^{f} (1 - (Φ_{mn}^{(k)})^{2})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}}} e^{i 2 π \sqrt{1 - \prod_{k = 1}^{f} (1 - (\frac{χ_{mn}^{(k)}}{2 π})^{2})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}}}} \end{matrix}),$

Step 4 Compute the value of 𝔽_mn, (m = 1, 2, ⋯ , p) (n = 1, 2, ⋯ , q) as

$\begin{matrix} F_{mn} & = \prod_{v}^{n - 1} S^{*} ({\tilde{ℤ}}_{mv}) (m = 1, 2, \dots, p) (n = 1, 2, \dots, q), \\ such that F_{m 1} & = 1 . \end{matrix}$

Step 5 Aggregate the CPFN ${\tilde{ℤ}}_{mn}$ for each alternative O_m by using the proposed CPFPWA

or CPFPWG operator.

$\begin{matrix} {\tilde{ℤ}}_{m} = (Ψ_{m} e^{i κ_{m}}, Ψ_{m} e^{i χ_{m}}) = CPFPWA ({\tilde{ℤ}}_{m 1}, {\tilde{ℤ}}_{m 2}, \dots {\tilde{ℤ}}_{mq}) \end{matrix}$

$= (\begin{matrix} \sqrt{1 - \prod_{n = 1}^{q} (1 - (Ψ_{mn})^{2})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}}} e^{i 2 π \sqrt{1 - \prod_{n = 1}^{q} (1 - (\frac{κ_{mn}}{2 π})^{2})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}}}}, \\ \prod_{n = 1}^{q} (Φ_{mn})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}} e^{i 2 π (\prod_{n = 1}^{q} {(\frac{χ_{mn}}{2 π})}^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}})} \end{matrix}),$

or by utilizing CPFPWG operator

$\begin{matrix} {\tilde{ℤ}}_{m} = (Ψ_{m} e^{i κ_{m}}, Ψ_{m} e^{i χ_{m}}) = CPFPWG ({\tilde{ℤ}}_{m 1}, {\tilde{ℤ}}_{m 2}, \dots {\tilde{ℤ}}_{mq}) \end{matrix}$

$= (\begin{matrix} \prod_{n = 1}^{q} (Ψ_{mn})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}} e^{i 2 π (\prod_{n = 1}^{q} {(\frac{κ_{mn}}{2 π})}^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}})} \\ \sqrt{1 - \prod_{n = 1}^{q} (1 - (Φ_{mn})^{2})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}}} e^{i 2 π \sqrt{1 - \prod_{n = 1}^{q} (1 - (\frac{χ_{mn}}{2 π})^{2})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}}}}, \end{matrix}),$

Step 6 Compute the score values by using the Equation (3.1).

Step 7 Select the best alternative having highest score.

Algorithm	Steps to solve MCDM problem by using CPFP aggregation operators
Step 1	Summarize the preference values of DMs in the form of CPF decision matrix as
	$H = (\begin{matrix} (Ψ_{11} e^{i κ_{11}}, Φ_{11} e^{i χ 11}) & (Ψ_{12} e^{i κ_{12}}, Φ_{12} e^{i χ 12}) & \dots & (Ψ_{1 q} e^{i κ_{1 q}}, Φ_{1 q} e^{i χ 1 q}) \\ (Ψ_{21} e^{i κ_{21}}, Φ_{21} e^{i χ 21}) & (Ψ_{22} e^{i κ_{22}}, Φ_{22} e^{i χ 22}) & \dots & (Ψ_{2 q} e^{i κ_{2 q}}, Φ_{2 q} e^{i χ 2 q}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ (Ψ_{p 1} e^{i κ_{p 1}}, Φ_{p 1} e^{i χ p 1}) & (Ψ_{p 2} e^{i κ_{p 2}}, Φ_{p 2} e^{i χ p 2}) & \dots & (Ψ_{pq} e^{i κ_{pq}}, Φ_{pq} e^{i χ pq}) \end{matrix}) .$
Step 2	Compute the values of $F_{mn}^{(k)} (k = 1, 2, \dots, f)$ as given below
	$\begin{matrix} F_{mn}^{(k)} & = \prod_{s}^{k - 1} {\tilde{S}}^{*} ({\tilde{ℤ}}_{mn}^{(k)}), (k = 2, 3, \dots, f), \\ such that P_{mn}^{(1)} & = 1 . \end{matrix}$
Step 3	Aggregate the CPF individual decision matrices $H^{* (k)} = ({\tilde{ℤ}}_{mn}^{(k)})_{p \times q}, (k = 1, 2, \dots, f)$ into
	the collective CPF decision matrix $H = ({\tilde{ℤ}}_{mn})_{p \times q}, (m = 1, 2, \dots, p) (n = 1, 2, \dots, q)$ by
	utilizing the CPFPWA or CPFPWG operators given below.
	$\begin{matrix} {\tilde{ℤ}}_{mn} = (Ψ_{mn} e^{i κ_{mn}}, Ψ_{mn} e^{i χ_{mn}}) = CPFPWA ({\tilde{ℤ}}_{mn}^{(1)}, {\tilde{ℤ}}_{mn}^{(2)}, \dots {\tilde{ℤ}}_{mn}^{(f)}) \end{matrix}$
	$= (\begin{matrix} \sqrt{1 - \prod_{k = 1}^{f} (1 - (Ψ_{mn}^{(k)})^{2})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}}} e^{i 2 π \sqrt{1 - \prod_{k = 1}^{f} (1 - (\frac{κ_{mn}^{(k)}}{2 π})^{2})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}}}}, \\ \prod_{k = 1}^{f} (Φ_{mn}^{(k)})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}} e^{i 2 π (\prod_{k = 1}^{f} {(\frac{χ_{mn}^{(k)}}{2 π})}^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}})} \end{matrix}),$
	or by utilizing CPFPWG operator
	$\begin{matrix} {\tilde{ℤ}}_{mn} = (Ψ_{mn} e^{i κ_{mn}}, Ψ_{mn} e^{i χ_{mn}}) = CPFPWG ({\tilde{ℤ}}_{mn}^{(1)}, {\tilde{ℤ}}_{mn}^{(2)}, \dots {\tilde{ℤ}}_{mn}^{(f)}) \end{matrix}$
	$= (\begin{matrix} \prod_{k = 1}^{f} (Ψ_{mn}^{(k)})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}} e^{i 2 π (\prod_{k = 1}^{f} {(\frac{κ_{mn}^{(k)}}{2 π})}^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}})}, \\ \sqrt{1 - \prod_{k = 1}^{f} (1 - (Φ_{mn}^{(k)})^{2})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}}} e^{i 2 π \sqrt{1 - \prod_{k = 1}^{f} (1 - (\frac{χ_{mn}^{(k)}}{2 π})^{2})^{\frac{F_{mn}^{(k)}}{\sum_{k = 1}^{f} F_{mn}^{(k)}}}}} \end{matrix}),$
Step 4	Compute the value of 𝔽_mn, (m = 1, 2, ⋯ , p) (n = 1, 2, ⋯ , q) as
	$\begin{matrix} F_{mn} & = \prod_{v}^{n - 1} S^{*} ({\tilde{ℤ}}_{mv}) (m = 1, 2, \dots, p) (n = 1, 2, \dots, q), \\ such that F_{m 1} & = 1 . \end{matrix}$
Step 5	Aggregate the CPFN ${\tilde{ℤ}}_{mn}$ for each alternative O_m by using the proposed CPFPWA
	or CPFPWG operator.
	$\begin{matrix} {\tilde{ℤ}}_{m} = (Ψ_{m} e^{i κ_{m}}, Ψ_{m} e^{i χ_{m}}) = CPFPWA ({\tilde{ℤ}}_{m 1}, {\tilde{ℤ}}_{m 2}, \dots {\tilde{ℤ}}_{mq}) \end{matrix}$
	$= (\begin{matrix} \sqrt{1 - \prod_{n = 1}^{q} (1 - (Ψ_{mn})^{2})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}}} e^{i 2 π \sqrt{1 - \prod_{n = 1}^{q} (1 - (\frac{κ_{mn}}{2 π})^{2})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}}}}, \\ \prod_{n = 1}^{q} (Φ_{mn})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}} e^{i 2 π (\prod_{n = 1}^{q} {(\frac{χ_{mn}}{2 π})}^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}})} \end{matrix}),$
	or by utilizing CPFPWG operator
	$\begin{matrix} {\tilde{ℤ}}_{m} = (Ψ_{m} e^{i κ_{m}}, Ψ_{m} e^{i χ_{m}}) = CPFPWG ({\tilde{ℤ}}_{m 1}, {\tilde{ℤ}}_{m 2}, \dots {\tilde{ℤ}}_{mq}) \end{matrix}$
	$= (\begin{matrix} \prod_{n = 1}^{q} (Ψ_{mn})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}} e^{i 2 π (\prod_{n = 1}^{q} {(\frac{κ_{mn}}{2 π})}^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}})} \\ \sqrt{1 - \prod_{n = 1}^{q} (1 - (Φ_{mn})^{2})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}}} e^{i 2 π \sqrt{1 - \prod_{n = 1}^{q} (1 - (\frac{χ_{mn}}{2 π})^{2})^{\frac{F_{mn}}{\sum_{n = 1}^{q} F_{mn}}}}}, \end{matrix}),$
Step 6	Compute the score values by using the Equation (3.1).
Step 7	Select the best alternative having highest score.

The flow chart to find the best alternative is displayed in Fig. 1.

Fig. 1

Step by step procedure to find the best alternative.

4.1 Numerical example

Personnel selection evaluation is considered to be the significant part of human resources management. It has always been a big challenge to select and evaluate the best employee for any organizations as it can affect notably the performance and future competitiveness of an organization. Many organizations have their own methods to select the personal as by conducting interviews, by screening candidates, etc. For this purpose of hiring the best employee which is efficient enough to fill the goals of company, a multinational organization has invited three experts, namely, ℍ¹, ℍ² and ℍ³ to conduct interview of five short listed candidates (alternative), namely, $B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ . The expert ℍ¹ is at high priority then the other two and the expert ℍ² comes next. Thus, the prioritization relationship among experts is ℍ¹ > ℍ² > ℍ³ . Thus, it is clear that it became hard for the candidate to select if he get poor evaluation from the expert ℍ¹ . The decision maker evaluate the candidate on the basis of four criteria namely, $U_{1}, U_{2}, U_{3}$ and $U_{4},$ where

$U_{1}$ : Communication skills

$U_{2}$ : Mathematical skills

$U_{3}$ : Awareness

$U_{4}$ : Personal skill

To make a CPFN these criteria are further categorize into two characteristics given below.

Communication skills involves oral communication and written communication skills.

Mathematical skills involves numeric skills and translating information.

Awareness involves situational awareness and sustained attention

Personal skill involve flexibility and professionalism.

The prioritization relationship among criteria is

U_{1} > U_{2} > U_{3} > U_{4} .

Thus, it is clear that the attribute

U_{1}

is at high priority then the other attributes which means that it is impossible to select that candidate who has bad communication skills no matter how he has good mathematical skills, awareness and personal skills. On the other hand, the candidate having good communication skills has high chance to select then the person who is good in personal skills. Since, all the attributes are of benefit type thus there is no need of normalization.

Step 1. The CPF decision matrices $H^{(k)} = ({\tilde{Y}}^{(k)})_{5 \times 4}$ for (k = 1, 2, 3) are given in Tables 2 –4.

Table 2
CPF decision matrix $H^{1} = ({\tilde{Y}}_{mn}^{(1)})_{5 \times 4}$

$U_{1}$ $U_{2}$ $U_{3}$ $U_{4}$

$B_{1}$ (0.9e^i2π(0.60), 0.4e^i2π(0.60)) (0.6e^i2π(0.65), 0.5e^i2π(0.50)) (0.5e^i2π(0.45), 0.8e^i2π(0.45)) (0.7e^i2π(0.60), 0.6e^i2π(0.50))

$B_{2}$ (0.7e^i2π(0.60), 0.6e^i2π(0.65)) (0.3e^i2π(0.40), 0.6e^i2π(0.65)) (0.8e^i2π(0.70), 0.4e^i2π(0.50)) (0.6e^i2π(0.75), 0.5e^i2π(0.40))

$B_{3}$ (0.8e^i2π(0.65), 0.7e^i2π(0.45)) (0.7e^i2π(0.45), 0.4e^i2π(0.60)) (0.9e^i2π(0.50), 0.3e^i2π(0.45)) (0.8e^i2π(0.65), 0.5e^i2π(0.50))

$B_{4}$ (0.4e^i2π(0.50), 0.7e^i2π(0.25)) (0.8e^i2π(0.50), 0.4e^i2π(0.55)) (0.8e^i2π(0.35), 0.5e^i2π(0.60)) (0.9e^i2π(0.50), 0.4e^i2π(0.40))

$B_{5}$ (0.6e^i2π(0.45), 0.7e^i2π(0.45)) (0.3e^i2π(0.45), 0.9e^i2π(0.60)) (0.4e^i2π(0.50), 0.7e^i2π(0.45)) (0.9e^i2π(0.50), 0.4e^i2π(0.45))

	$U_{1}$	$U_{2}$	$U_{3}$	$U_{4}$
$B_{1}$	(0.9e^i2π(0.60), 0.4e^i2π(0.60))	(0.6e^i2π(0.65), 0.5e^i2π(0.50))	(0.5e^i2π(0.45), 0.8e^i2π(0.45))	(0.7e^i2π(0.60), 0.6e^i2π(0.50))
$B_{2}$	(0.7e^i2π(0.60), 0.6e^i2π(0.65))	(0.3e^i2π(0.40), 0.6e^i2π(0.65))	(0.8e^i2π(0.70), 0.4e^i2π(0.50))	(0.6e^i2π(0.75), 0.5e^i2π(0.40))
$B_{3}$	(0.8e^i2π(0.65), 0.7e^i2π(0.45))	(0.7e^i2π(0.45), 0.4e^i2π(0.60))	(0.9e^i2π(0.50), 0.3e^i2π(0.45))	(0.8e^i2π(0.65), 0.5e^i2π(0.50))
$B_{4}$	(0.4e^i2π(0.50), 0.7e^i2π(0.25))	(0.8e^i2π(0.50), 0.4e^i2π(0.55))	(0.8e^i2π(0.35), 0.5e^i2π(0.60))	(0.9e^i2π(0.50), 0.4e^i2π(0.40))
$B_{5}$	(0.6e^i2π(0.45), 0.7e^i2π(0.45))	(0.3e^i2π(0.45), 0.9e^i2π(0.60))	(0.4e^i2π(0.50), 0.7e^i2π(0.45))	(0.9e^i2π(0.50), 0.4e^i2π(0.45))

Table 3

CPF decision matrix $H^{2} = ({\tilde{Y}}_{mn}^{(2)})_{5 \times 4}$

	$U_{1}$	$U_{2}$	$U_{3}$	$U_{4}$
$B_{1}$	(0.9e^i2π(0.60), 0.2e^i2π(0.20)	(0.8e^i2π(0.85), 0.5e^i2π(0.45))	(0.7e^i2π(0.65), 0.6e^i2π(0.50))	(0.9e^i2π(0.75), 0.5e^i2π(0.5))
$B_{2}$	(0.7e^i2π(0.75), 0.5e^i2π(0.60))	(0.8e^i2π(0.75), 0.4e^i2π(0.45))	(0.6e^i2π(0.55), 0.5e^i2π(0.50))	(0.8e^i2π(0.7), 0.3e^i2π(0.3))
$B_{3}$	(0.9e^i2π(0.85), 0.4e^i2π(0.25))	(0.7e^i2π(0.65), 0.6e^i2π(0.50))	(0.3e^i2π(0.45), 0.9e^i2π(0.60))	(0.4e^i2π(0.45), 0.5e^i2π(0.5))
$B_{4}$	(0.7e^i2π(0.65), 0.4e^i2π(0.45))	(0.9e^i2π(0.75), 0.4e^i2π(0.50))	(0.8e^i2π(0.75), 0.6e^i2π(0.50))	(0.7e^i2π(0.45), 0.4e^i2π(0.4)
$B_{5}$	(0.4e^i2π(0.45), 0.9e^ii2π(0.85))	(0.8e^i2π(0.75), 0.3e^i2π(0.45))	(0.7e^i2π(0.60), 0.6e^i2π(0.50))	(0.4e^i2π(0.4), 0.6e^i2π(0.5))

Table 4

CPF decision matrix $H^{3} = ({\tilde{Y}}_{mn}^{(3)})_{5 \times 4}$

	$U_{1}$	$U_{2}$	$U_{3}$	$U_{4}$
$B_{1}$	(0.4e^i2π(0.50), 0.9e^i2π(0.75))	(0.3e^i2π(0.35), 0.9e^i2π(0.75))	(0.7e^i2π(0.65), 0.4e^i2π(0.45))	(0.8e^i2π(0.75), 0.5e^i2π(0.50))
$B_{2}$	(0.7e^i2π(0.65), 0.6e^i2π(0.55))	(0.8e^i2π(0.75), 0.4e^i2π(0.45))	(0.5e^i2π(0.55), 0.6e^i2π(0.60))	(0.7e^i2π(0.65), 0.4e^i2π(0.45))
$B_{3}$	(0.8e^i2π(0.75), 0.4e^i2π(0.45))	(0.6e^i2π(0.60), 0.5e^i2π(0.50))	(0.8e^i2π(0.75), 0.3e^i2π(0.45))	(0.9e^i2π(0.75), 0.3e^i2π(0.35))
$B_{4}$	(0.9e^i2π(0.75), 0.2e^i2π(0.25))	(0.9e^i2π(0.75), 0.2e^i2π(0.30))	(0.9e^i2π(0.80), 0.4e^i2π(0.45))	(0.4e^i2π(0.45), 0.6e^i2π(0.65))
$B_{5}$	(0.4e^i2π(0.50), 0.7e^i2π(0.65))	(0.4e^i2π(0.40), 0.7e^i2π(0.50))	(0.2e^i2π(0.25), 0.9e^i2π(0.70))	(0.7e^i2π(0.65), 0.7e^i2π(0.60))

Step 2. Calculate the values of $F_{mn}^{(k)}$ for k = 1, 2, 3 . $F_{mn}^{(1)} = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{matrix}],$ $F_{mn}^{(2)} = [\begin{matrix} 1.325 & 1.1413 & 0.8050 & 1.120 \\ 1.0338 & 0.7338 & 1.3600 & 1.2563 \\ 1.1850 & 1.0863 & 1.3838 & 1.2813 \\ 0.9288 & 1.2138 & 1.0763 & 1.3700 \\ 0.9350 & 0.5613 & 0.8588 & 1.3488 \end{matrix}],$ $F_{mn}^{(3)} = [\begin{matrix} 2.0471 & 1.6606 & 0.9268 & 1.6086 \\ 1.2625 & 1.0420 & 1.4705 & 1.8530 \\ 0.7762 & 1.2506 & 0.7767 & 1.1932 \\ 1.1842 & 1.7979 & 1.3952 & 1.6252 \\ 0.3880 & 0.8167 & 0.9619 & 1.1532 \end{matrix}] .$

Step 3 Utilize the CPFPWA operator to combine all the individual matrices into a collective single matrix (see Table 5).

Table 5

Collective CPF decision matrix $H = ({\tilde{Y}}_{mn})_{5 \times 4}$

	$U_{1}$	$U_{2}$
$B_{1}$	(0.7870e^i2π(0.5574), 0.4735e^i2π(0.4771))	(0.6099e^i2π(0.6661), 0.5198e^i2π(0.4650))
$B_{2}$	(0.7000e^i2π(0.6737), 0.5667e^i2π(0.5946))	(0.7051e^i2π(0.6683), 0.4629e^i2π(0.5137))
$B_{3}$	(0.8493e^i2π(0.7743), 0.4838e^i2π(0.3557))	(0.6669e^i2π(0.5821), 0.4963e^i2π(0.5281))
$B_{4}$	(0.7673e^i2π(0.6594), 0.3678e^i2π(0.2979))	(0.8816e^i2π(0.7068), 0.2932e^i2π(0.4073))
$B_{5}$	(0.5028e^i2π(0.4590), 0.7745e^i2π(0.6181))	(0.5373e^i2π(0.5435), 0.6370e^i2π(0.5266))
	$U_{3}$	$U_{4}$
$B_{1}$	(0.6424e^i2π(0.5917), 0.5810e^i2π(0.4642))	(0.8208e^i2π(0.7180), 0.5251e^i2π(0.5000))
$B_{2}$	(0.6440e^i2π(0.5977), 0.5367e^i2π(0.5362))	(0.7180e^i2π(0.6931), 0.3868e^i2π(0.3863))
$B_{3}$	(0.7474e^i2π(0.5702), 0.4853e^i2π(0.5104))	(0.7778e^i2π(0.6393), 0.4195e^i2π(0.4424))
$B_{4}$	(0.8494e^i2π(0.7111), 0.4837e^i2π(0.5051))	(0.7156e^i2π(0.4633), 0.4717e^i2π(0.4873))
$B_{5}$	(0.4947e^i2π(0.4784), 0.7277e^i2π(0.5402))	(0.7306e^i2π(0.5302), 0.5622e^i2π(0.5152))

Step 4. Calculate the values of 𝔽_mn (m = 1, 2, ⋯ p) , (n = 1, 2, ⋯ q) . $F_{mn} = [\begin{matrix} 1 & 1.2391 & 1.4431 & 1.5944 \\ 1 & 1.1346 & 1.3988 & 1.5362 \\ 1 & 1.4801 & 1.6713 & 1.9953 \\ 1 & 1.3998 & 2.1171 & 2.8984 \\ 1 & 0.7408 & 0.7041 & 0.5817 \end{matrix}]$

Step 5. Apply CPFPWG operator to aggregate all the preference values ${\tilde{Y}}_{mn} (m = 1, 2, \dots, 5)$ in mth lines of ℍ in Table 5 to get ${\tilde{Y}}_{m} .$ $\begin{matrix} B_{1} = (0.7347 e^{i 2 π (0.6482)}, 0.5281 e^{i 2 π (0.4774)}), \\ B_{2} = (0.6929 e^{i 2 π (0.6600)}, 0.4752 e^{i 2 π (0.4908)}), \\ B_{3} = (0.7632 e^{i 2 π (0.6395)}, 0.4652 e^{i 2 π (0.4632)}), \\ B_{4} = (0.8060 e^{i 2 π (0.6288)}, 0.4200 e^{i 2 π (0.4454)}), \\ B_{5} = (0.5692 e^{i 2 π (0.5001)}, 0.6842 e^{i 2 π (0.5562)}) . \end{matrix}$ Step 6. Compute the score values of all the CPFNs obtained in Step 5. $\begin{matrix} {\overset{˘}{S}}^{*} (B_{1}) = \frac{1}{2} ((1 + 0 . 7347^{2} - 0 . 5281^{2}) \\ + (1 + 0 . 6482^{2} - 0 . 4774^{2})) = 1.2266, \\ {\overset{˘}{S}}^{*} (B_{2}) = \frac{1}{2} ((1 + 0 . 6929^{2} - 0 . 4752^{2}) \\ + (1 + 0 . 6600^{2} - 0 . 4908^{2})) = 1.2245, \end{matrix}$ $\begin{matrix} {\overset{˘}{S}}^{*} (B_{3}) = \frac{1}{2} ((1 + 0 . 7632^{2} - 0 . 4652^{2}) \\ + (1 + 0 . 6395^{2} - 0 . 4632^{2})) = 1.2802, \\ {\overset{˘}{S}}^{*} (B_{4}) = \frac{1}{2} ((1 + 0 . 8060^{2} - 0 . 4200^{2}) \\ + (1 + 0 . 6288^{2} - 0 . 4454^{2})) = 1.3351, \\ {\overset{˘}{S}}^{*} (B_{5}) = \frac{1}{2} ((1 + 0 . 5692^{2} - 0 . 6842^{2}) \\ + (1 + 0 . 5001^{2} - 0 . 5562^{2})) = 0.8983 . \end{matrix}$

Therefore ${\overset{˘}{S}}^{*} (B_{4}) > {\overset{˘}{S}}^{*} (B_{3}) > {\overset{˘}{S}}^{*} (B_{1}) > {\overset{˘}{S}}^{*} (B_{2}) > {\overset{˘}{S}}^{*} (B_{5}) .$

Step 7. Therefore, $B_{4}$ is the best alternative.

Now, by utilizing CPFPWG operator, the main steps are given below.

Step 1*. Similar to Step 1.

Step 2*. Similar to Step 2.

Step 3*. Utilize the CPFPWG operator to combine all the individual matrices into a collective single matrix (see Table 6).

Table 6

Collective CPF decision matrix $H = ({\tilde{Y}}_{mn})_{5 \times 4}$

	$U_{1}$	$U_{2}$
$B_{1}$	(0.6157e^i2π(0.5509), 0.7509e^i2π(0.6280))	(0.4833e^i2π(0.5376), 0.7670e^i2π(0.6295))
$B_{2}$	(0.7000e^i2π(0.6635), 0.5722e^i2π(0.5991))	(0.5619e^i2π(0.5980), 0.4883e^i2π(0.5386))
$B_{3}$	(0.8386e^i2π(0.7513), 0.5388e^i2π(0.3862))	(0.6607e^i2π(0.5650), 0.4415e^i2π(0.5337))
$B_{4}$	(0.6435e^i2π(0.6309), 0.4971e^i2π(0.3284))	(0.8740e^i2π(0.6779), 0.3289e^i2π(0.4436))
$B_{5}$	(0.4763e^i2π(0.4580), 0.8107e^i2π(0.7113))	(0.4174e^i2π(0.4875), 0.7836e^i2π(0.5368))
	$U_{3}$	$U_{4}$
$B_{1}$	(0.6189e^i2π(0.5681), 0.6568e^i2π(0.4656))	(0.7997e^i2π(0.7064), 0.5303e^i2π(0.5000))
$B_{2}$	(0.6031e^i2π(0.5857), 0.5226e^i2π(0.5425))	(0.7023e^i2π(0.6885), 0.4023e^i2π(0.3989))
$B_{3}$	(0.5405e^i2π(0.5275), 0.7360e^i2π(0.5396))	(0.6451e^i2π(0.5962), 0.4455e^i2π(0.4566))
$B_{4}$	(0.8388e^i2π(0.6180), 0.5027e^i2π(0.5154))	(0.5937e^i2π(0.4437), 0.4980e^i2π(0.5280))
$B_{5}$	(0.3745e^i2π(0.4173), 0.7809e^i2π(0.5727))	(0.6063e^i2π(0.5002), 0.5985e^i2π(0.5250))

Step 4*. Calculate the values of 𝔽_mn (m = 1, 2, ⋯ p) , (n = 1, 2, ⋯ q) . $F_{mn} = [\begin{matrix} 1 & 0.8622 & 0.6630 & 0.6821 \\ 1 & 1.1219 & 1.2031 & 1.2869 \\ 1 & 1.4141 & 1.6092 & 1.3980 \\ 1 & 1.2286 & 1.7928 & 2.3012 \\ 1 & 0.6367 & 0.4806 & 0.3338 \end{matrix}]$

Step 5*. Apply CPFPWA operator to aggregate all the preference values ${\tilde{Y}}_{mn} (m = 1, 2, \dots, 5)$ in mth lines of ℍ in Table 5 to get ${\tilde{Y}}_{m} .$ $\begin{matrix} B_{1} = (0.6105 e^{i 2 π (0.5807)}, 0.7039 e^{i 2 π (0.5764)}), \\ B_{2} = (0.6388 e^{i 2 π (0.6327)}, 0.4978 e^{i 2 π (0.5226)}), \\ B_{3} = (0.7209 e^{i 2 π (0.5916)}, 0.5791 e^{i 2 π (0.4940)}), \\ B_{4} = (0.7150 e^{i 2 π (0.5596)}, 0.4732 e^{i 2 π (0.4842)}), \\ B_{5} = (0.4536 e^{i 2 π (0.4625)}, 0.7780 e^{i 2 π (0.6263)}) . \end{matrix}$

Step 6*. Compute the score values of all the CPFNs obtained in Step 5. $\begin{matrix} {\overset{˘}{S}}^{*} (B_{1}) = \frac{1}{2} ((1 + 0 . 6105^{2} - 0 . 7039^{2}) \\ + (1 + 0 . 5807^{2} - 0 . 5764^{2})) = 0.9411, \\ {\overset{˘}{S}}^{*} (B_{2}) = \frac{1}{2} ((1 + 0 . 6388^{2} - 0 . 4978^{2}) \\ + (1 + 0 . 6327^{2} - 0 . 5226^{2})) = 1.1437, \\ {\overset{˘}{S}}^{*} (B_{3}) = \frac{1}{2} ((1 + 0 . 7209^{2} - 0 . 5791^{2}) \\ + (1 + 0 . 5916^{2} - 0 . 4940^{2})) = 1.1451, \\ {\overset{˘}{S}}^{*} (B_{4}) = \frac{1}{2} ((1 + 0 . 7150^{2} - 0 . 4732^{2}) \\ + (1 + 0 . 5596^{2} - 0 . 4842^{2})) = 1.1830, \\ {\overset{˘}{S}}^{*} (B_{5}) = \frac{1}{2} ((1 + 0 . 4536^{2} - 0 . 7780^{2}) \\ + (1 + 0 . 4625^{2} - 0 . 6263^{2})) = 0.7111 . \end{matrix}$ Therefore ${\overset{˘}{S}}^{*} (B_{4}) > {\overset{˘}{S}}^{*} (B_{3}) > {\overset{˘}{S}}^{*} (B_{2}) > {\overset{˘}{S}}^{*} (B_{1}) > {\overset{˘}{S}}^{*} (B_{5}) .$

Step 7*. Therefore, $B_{4}$ is the best alternative.

The final ordering and score values are summarized in Table 7.

Table 7

Score values and ordering of alternatives

Operators	$B_{1}$	$B_{2}$	$B_{3}$	$B_{4}$	$B_{5}$
CPFPWA operator	1.2266	1.2245	1.2802	1.3351	0.8983
CPFPWG operator	0.9411	1.1437	1.1451	1.1830	0.7111
	Final ordering
CPFPWA operator	${\overset{˘}{S}}^{} (B_{4}) > {\overset{˘}{S}}^{} (B_{3}) > {\overset{˘}{S}}^{} (B_{1}) > {\overset{˘}{S}}^{} (B_{2}) > {\overset{˘}{S}}^{*} (B_{5})$
CPFPWG operator	${\overset{˘}{S}}^{} (B_{4}) > {\overset{˘}{S}}^{} (B_{3}) > {\overset{˘}{S}}^{} (B_{2}) > {\overset{˘}{S}}^{} (B_{1}) > {\overset{˘}{S}}^{*} (B_{5})$

5 Comparative analysis

In this section, to check the performance and feasibility of our proposed results and to check the validity and authenticity of our proposed operators, we aggregate the same data by utilizing the Pythagorean fuzzy prioritized weighted averaging (PFPWA) operator [12] and Pythagorean fuzzy prioritized weighted geometric (PFPWG) operator [12]. The aggregated results by utilizing these operators are summarized in Table 8.

Table 8
Aggregated results of existing operators

Operators $B_{1}$ $B_{2}$ $B_{3}$ $B_{4}$ $B_{5}$

PFPWA operator [12] (0.756, 0.507) (0.678, 0.518) (0.782, 0.495) (0.772, 0.454) (0.519, 0.744)

PFPWG operator [12] (0.649, 0.664) (0.605, 0.543) (0.712, 0.592) (0.639, 0.541) (0.470, 0.805)

Operators	$B_{1}$	$B_{2}$	$B_{3}$	$B_{4}$	$B_{5}$
PFPWA operator [12]	(0.756, 0.507)	(0.678, 0.518)	(0.782, 0.495)	(0.772, 0.454)	(0.519, 0.744)
PFPWG operator [12]	(0.649, 0.664)	(0.605, 0.543)	(0.712, 0.592)	(0.639, 0.541)	(0.470, 0.805)

The score values and ordering of alternative are displayed in Table 9.

Table 9

Score values and final ordering

Operators	$B_{1}$	$B_{2}$	$B_{3}$	$B_{4}$	$B_{5}$
PFPWA operator [12]	0.6570	0.5960	0.6830	0.6950	0.3580
PFPWG operator [12]	0.4900	0.5360	0.5780	0.5580	0.2860
The proposed CPFPWA operator	1.2266	1.2245	1.2802	1.3351	0.8983
The proposed CPFPWG operator	0.9411	1.1437	1.1451	1.1830	0.7111
	Final ordering
PFPWA operator [12]	${\overset{˘}{S}}^{} (B_{4}) > {\overset{˘}{S}}^{} (B_{3}) > {\overset{˘}{S}}^{} (B_{1}) > {\overset{˘}{S}}^{} (B_{2}) > {\overset{˘}{S}}^{*} (B_{5})$
PFPWG operator [12]	${\overset{˘}{S}}^{} (B_{3}) > {\overset{˘}{S}}^{} (B_{4}) > {\overset{˘}{S}}^{} (B_{2}) > {\overset{˘}{S}}^{} (B_{1}) > {\overset{˘}{S}}^{*} (B_{5})$
The proposed CPFPWA operator	${\overset{˘}{S}}^{} (B_{4}) > {\overset{˘}{S}}^{} (B_{3}) > {\overset{˘}{S}}^{} (B_{1}) > {\overset{˘}{S}}^{} (B_{2}) > {\overset{˘}{S}}^{*} (B_{5})$
The proposed CPFPWG operator	${\overset{˘}{S}}^{} (B_{4}) > {\overset{˘}{S}}^{} (B_{3}) > {\overset{˘}{S}}^{} (B_{2}) > {\overset{˘}{S}}^{} (B_{1}) > {\overset{˘}{S}}^{*} (B_{5})$

From Table 9 it can be seen that when we apply PFPWG operator then the optimal alternative is $B_{3}$ instead of $B_{4} .$ The main reason behind this change is the absence of phase term which cause lose of information. Therefore, our proposed AOs are superior then the existing operators. The graphical representation is displayed in Fig. 2.

Fig. 2

Comparison of proposed operators with existing operators.

6 Discussion

From the above comparison, it is clear that our proposed AOs namely, CPFPWA and CPFPWG operators are more flexible and generalized then the PFPWA and PFPWG operators because these operators are based on PFSs while our proposed operators are based on CPFS which is an effective generalization of PFS due to the existence of phase term. The optimal alternative obtained by using PFPWG operator is different form the alternative obtained from our proposed operators due to the absence of phase term. The main advantages of proposed operators are given below.

The proposed AOs are based on CPFS which is the worthwhile generalization of PFS because of the presence of phase term which helps to handle the periodic or two-dimensional data. Further, on removing the phase term we get PFSs. Thus, PFSs are the special case of CPFS.

The proposed AOs are the combination of CPFS are prioritized AOs and thus are most effective to solve the DM problems.

Thus, from the above discussion, our proposed AOs are more flexible and generalized to solve the DM problems.

7 Conclusion and future direction

The CPFS, a worthwhile generalization FS and CFS, is an indispensable tool to model the two dimensional or periodic information in logical and satisfying way. In this paper, we have investigated two prioritized AOs, namely, prioritized weighted averaging operator and prioritized weighted geometric operator in CPF environment by considering that the attributes and decision makers with different priority levels. Further, we have provided some important properties of these operators, namely, idempotency, boundedness and monotonicity. At the end, to check the superiority and validity of proposed operators we have compared the computed results with different existing techniques. Our next aim is to extend our research work to (1) Prioritized Einstein aggregation operators, and (2) Prioritized Yager aggregation operators under CPFS.

Conflict of interest

The authors declare no conflict of interest.

Footnotes

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (DF-122-130-1441). The authors, therefore, gratefully acknowledge DSR technical and financial support.

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Prioritized weighted aggregation operators under complex pythagorean fuzzy information

Abstract

Keywords

1 Introduction

2 Preliminaries

Table 8 Aggregated results of existing operators Operators B 1 B 2 B 3 B 4 B 5 PFPWA operator [12] (0.756, 0.507) (0.678, 0.518) (0.782, 0.495) (0.772, 0.454) (0.519, 0.744) PFPWG operator [12] (0.649, 0.664) (0.605, 0.543) (0.712, 0.592) (0.639, 0.541) (0.470, 0.805)

7 Conclusion and future direction

Conflict of interest

Footnotes

Acknowledgment

References

Table 8
Aggregated results of existing operators

Operators $B_{1}$ $B_{2}$ $B_{3}$ $B_{4}$ $B_{5}$

PFPWA operator [12] (0.756, 0.507) (0.678, 0.518) (0.782, 0.495) (0.772, 0.454) (0.519, 0.744)

PFPWG operator [12] (0.649, 0.664) (0.605, 0.543) (0.712, 0.592) (0.639, 0.541) (0.470, 0.805)