Pythagorean cubic fuzzy aggregation information based on confidence levels and its application to multi-criteria decision making process

Abstract

Pythagorean cubic fuzzy set, is an extension of the interval valued Pythagorean fuzzy set which relax the condition of the square sum of its membership and non-membership degree is less than or equal to one to supremum square sum of its membership and non-membership functions is less than one. Based on this information and by combining the idea of the confidence levels of each Pythagorean cubic fuzzy number, the proposed study investigated a new averaging and geometric operators, namely confidence Pythagorean cubic fuzzy weighted and geometric operators along with their order are presented. Some of their desired properties related to the new defined operators have been investigated. Under the given information, a multi criteria decision making process has been proposed with a numerical example for showing the effectiveness and validity of it.

Keywords

Paythagorean cubic fuzzy sets confidence levels aggregation operators decision making

1. Introduction

Multi criteria decision making (MCDM) is one of the fastest growing research active problems in these days for reaching a final decision within a reasonable time. But it is not always permissible to give the preferences in a precise manner due to various constraints and hence their corresponding results are not ideal in some circumstances. To handle it, an IFS theory (Atanassov [1]) is one of the successful and widely used by the researchers for dealing with the vagueness and impreciseness in the data. Under these environments, the various researchers pay more attention on IFSs for aggregating the different alternatives using different aggregation operators. In order to aggregate all the performance of the criteria for alternatives, weighted and ordered weighted aggregation operators (Yager [29], Yager and Kacprzyk [33]) play an important role during the information fusion process. For instance, Xu and Yager [28] presented a geometric aggregation operator while Xu [27] presented a weighted averaging operator for aggregating the different intuitionistic fuzzy numbers. Later on, Wang and Liu [21] extended these operators by using Einstein norm operations under IFS environment. Garg [7] presented a generalized improved score function to rank these numbers and applied it to the decision-making problems. Ye [35] presented a new accuracy function for interval-valued IFS. Garg [11] proposed some series of interactive aggregation operators for intuitionistic fuzzy numbers (IFNs). Garg [4] presented a generalized intuitionistic fuzzy interactive geometric interaction operators using Einstein norm operations for aggregating the different intuitionistic fuzzy information. Xu et al. [25] had presented the intuitionistic fuzzy Einstein Choquet integral based operators for decision making problems. Garg [5] presented a generalized intuitionistic fuzzy aggregation operator under the intuitionistic multiplicative preference relation instead of intuitionistic fuzzy preference relations. Apart from these, various authors have investigated the problem of the decision-making under the different environments (Nancy and Garg [13, 14] Dalman [2], Dalman et al. [3], Yu [36], Yu and Shi [38], Kumar and Garg [18], Ye [34], Garg et al. [12]) and so on. A comprehensive analysis on MCDM using different approaches under IFS environment has been summarized in Yu [15] and Xu and Zhao [26]. Zhang [40] focus on group decision making with IMPRs. First, he analyzes the flaws of the consistency, definition of an IMPR in previous work and then propose a new definition to overcome the flaws. On this basis, a linear programming-based algorithm is developed to check and improve the consistency of an IMPR. Secondly, discuss the relationships between an IMPR and a normalized intuitionistic multiplicative weight vector and develop two approaches to group decision making based on complete and incomplete IMPRs, respectively. In [22] Wu et al established the multiplicative transitivity property for IFPRs by combining the multiplication of IFSs and the extension of Tanino’s multiplicative transitivity for FPRs. Wu et al [23] presents a minimum adjustment cost feedback mechanism for higher consensus in social network group decision making under distributed linguistic trust information. In [24] Wu proposes a novel visual, interactive method for consensus in SN-GDM. This method consists of three main modules: a dual trust propagation, a trust based recommendation and a visual adoption mechanism. Liu in [19] addresses the inconsistency problem in group decision making caused by disparate opinions of multiple experts. To do so, a trust induced recommendation mechanism is investigated to generate personalised advices for the inconsistent experts to reach higher consensus level. Gong et al. [16] has explored the case when an individual opinion is an interval preference in consensus decision making. And for this purpose, they construct two multi-objective optimization models: one based on the minimum cost from the perspective of the moderator, the other the maximum return from the perspective of the individuals.

From these above studies, it has been concluded that they are valid under the restrictions that the sum of the grades of memberships is not greater than one. However, in day-today life, it is not always possible to give their preferences under this restriction. Therefore, it does not satisfy the IFS condition. Hence, under this circumstance, it is not possible for the decision maker to evaluate the performance and hence IFS theory has some drawbacks. In order to overcome this, Yager and Abbasov [32] introduced a Pythagorean fuzzy set (PFS) theory which is an extension of IFS theory. Also, it has been observed that all the intuitionistic fuzzy degrees are a part of the Pythagorean fuzzy degrees, which indicates that the PFS is more powerful to handle the uncertain problems. After their pioneer work, researchers are actively working in the field of PFS to enhance it. Yager and Abbasov [32] showed that the Pythagorean degrees are the subclasses of the complex numbers. Later on, Zhang and Xu [39] presented a technique for finding the best alternative based on its ideal solution under the Pythagorean fuzzy environment. Yager [31] developed various aggregation operators, namely, Pythagorean fuzzy weighted average (PFWA) operator, Pythagorean fuzzy weighted geometric average (PFWGA) operator, Pythagorean fuzzy weighted power average (PFWPA) operator and Pythagorean fuzzy weighted power geometric average (PFWPGA) operator to aggregate the different Pythagorean fuzzy numbers. Peng and Yang [20] defined the some new arithmetical operations and their corresponding properties for PFNs. Garg [10] defined the concepts of correlation and correlation coefficients of PFSs. Also, Garg [9] presented a novel accuracy function under the interval valued Pythagorean fuzzy set (IVPFS) for solving the decision-making problems. Garg [6 –8], further, presented a generalized averaging aggregation operator under the Pythagorean fuzzy set environment by utilizing the Einstein norm operations.

Despite the popularizes of the above work, all the above studies have investigated without considering the confidence level of the attributes. In other words, all the researchers have investigated the studies by taking the assumption that decision makers are taken to be surely familiar with the evaluated object. But in real-life situation, this type of conditions are partially fulfills. To overcome this shortcoming, the decision makers may evaluate the alternative in terms of PFNs and their corresponding confidence levels for their familiarity with the evaluation. Therefore, Garg [15] incorporated the idea of the confidence levels into the aggregation process during the evaluation of the alternative in terms of PFNs. Based on these evaluations, some series of the averaging and geometric aggregation operators are proposed, namely CPFWA, CPFOWA, CPFWG and CPFOWG along with their desired properties. Further, an MCDM method based on these operators has been proposed for solving the problems.

Khan et al. [17] presented the notion of the Pythagorean cubic fuzzy set which is the generalization of interval valued Pythagorean fuzzy set and some Pythagorean cubic fuzzy operational laws have been developed. They defined score and accuracy degree for the comparison of Pythagorean cubic fuzzy numbers, distance between Pythagorean cubic fuzzy numbers. To aggregate the Pythagorean cubic fuzzy information, they proposed (PCFWA), (PCFWG), (PCFOWA) and (PCFOWG) operator under Pythagorean cubic fuzzy environment, Moreover, he proposed a multi criteria decision-making (MCDM) approach to show the strength and effectiveness of the developed operators. All the above studies of PCFNs have investigated without considering the confidence level of the attributes, but in real-life situation, this type of conditions partially fulfills. To overcome this shortcoming, the decision makers may evaluate the alternative in terms of PCFNs and their corresponding confidence levels for their familiarity with the evaluation. Therefore, the present study incorporated the idea of the confidence levels into the aggregation process during the evaluation of the alternative in terms of PCFNs. Based on these evaluations, some series of the averaging and geometric aggregation operators are proposed, namely CPCFWA, CPCFOWA, CPCFWG and CPCFOWG along with their desired properties. Further, an MCDM method based on these operators has been proposed for solving the problems.

The rest of the paper is organized as follows. Section 2 refers to the basic definitions related to PFS and PCFSs and the operational laws on PCFNs have been defined and some desirable properties of it have been defined. Section 3, novel, series of aggregation operators, namely CPCFWA, CPCFOWA, CPCFG and CPCFOG along with their properties. Section 4 presented an algorithm for solving MCDM problems based on uncertainties on the proposed operators. Section 5 encounter a numerical example of finding the best alternatives to show the applicability and effectiveness of the proposed method. Lastly, some concluding remarks and conclusion are ends the paper.

2. Preliminaries

Definition 1. [30] A Pythagorean fuzzy set (PFS) A_P over a universal set X is defined as a set of ordered pairs of membership and non-membership is given as: $A_{P} = {〈 x, μ_{A_{P}} (x), ν_{A_{P}} (x) 〉 | x \in X},$ where μ_{A
_P} (x) and ν_{A
_P} (x) is a function from X : → [0, 1] represent the degrees of membership and non-membership of the element x ∈ XepsfboxG :/Tex/IOSPRESS/IFS/0 -181516/IF01 . eps (μ_{A
_P} (x)) ² + (ν_{A
_P} (x)) ² ≤ 1. Let $π_{A_{P}} (x) = \sqrt{1 - μ_{A_{P}}^{2} (x) - ν_{A_{P}}^{2} (x)}$ , then it is called to be the Pythagorean fuzzy index of x in X to set P, denote the indeterminacy degree of x to P [39]. We denote the Pythagorean fuzzy number (PFN) by A_P = 〈μ_{A
_P}, ν_{A
_P}〉.

Khan et al. [17] present the idea of some new concepts such as, Pythagorean cubic fuzzy set and discuss some of its properties which is not an intuitionistic cubic fuzzy set.

Definition 2. [17] A Pythagorean cubic fuzzy set A_pc is defined as a set of ordered pairs membership and non-membership over a fixed universal set X as follows: $A_{pc} = {〈 x, (μ_{A_{pc}} (x), ν_{A_{pc}} (x)) 〉 | x \in X},$ (1)

Where μ_{A
_pc} (x) = 〈A_Pc (x) , λ_Pc (x) 〉 and $ν_{A_{pc}} (x) = 〈 {\tilde{A}}_{Pc} (x), μ_{Pc} (x) 〉$ represents the membership and non-membership degree under the condition that $0 \leq (\sup (A_{Pc} (x)))^{2} + (\sup ({\tilde{A}}_{Pc} (x)))^{2} \leq 1$ , and $0 \leq λ_{Pc}^{2} (x) + μ_{Pc}^{2} (x) \leq 1$ . The degree of indeterminacy for Pythagorean cubic set can be defined as: $\begin{matrix} π_{A_{pc}} & = & 〈 \sqrt{1 - (\sup (A_{pc} (x)))^{2} - (\sup ({\tilde{A}}_{pc} (x)))^{2}}, \\ \sqrt{1 - λ_{pc}^{2} (x) - μ_{pc}^{2} (x)} 〉 . \end{matrix}$

For simplicity we call (μ_{A
_Pc}, ν_{A
_Pc}) a Pythagorean cubic fuzzy number (PCFN) denoted by A_Pc = (μ_{A
_Pc}, ν_{A
_Pc}). To motivate the Pythagorean cubic fuzzy numbers, Yager suggested that PFS is defined as a pair of degree of membership (μ_p (x)) and non-membership (ν_p (x)) of the element x ∈ X to the set P under the restriction that $μ_{p}^{2} (x) + ν_{p}^{2} (x) \leq 1$ . Here μ_p (x) represents the amount of satisfaction degree of x in P and ν_p (x) represents the degree of rejection or against of the x in P. Therefore, the term $\sqrt{1 - μ_{p}^{2} (x) - ν_{p}^{2} (x)}$ represents the degree of hesitation between the satisfaction and rejection and called the degree of hesitation associated with the membership grade of x. On the other hand, PCFSs is one of the generalization of the classic set, fuzzy set, IFS, PFS and IPFS. Many real world problems might have taken a decision in such a way that their sum of their corresponding degree of membership and non-membership is in the interval-valued cubic fuzzy numbers and is greater than one. For example, if a person expresses his preference about the degree of alternative that satisfies the criteria is 〈 [0.6, 0.7] , 0.4〉, while dissatisfies the criteria is 〈 [0.5, 0.6] , 0.5〉, therefore, this situation is not properly handled in the PFS and IPFS. To answer this shortcoming, PCFS theory is one of the most general and can handle not only incomplete information, but also the indeterminate information and inconsistent information, which exist commonly in real situations. The main difference between the IPFSand PCFS is their different constraint conditions. As in PCFSs, the constraint condition corresponding to the membership grade is $0 \leq (\sup (A_{Pc} (x)))^{2} + (\sup ({\tilde{A}}_{Pc} (x)))^{2} \leq 1,$ and $0 \leq λ_{Pc}^{2} (x) + μ_{Pc}^{2} (x) \leq 1 .$

Definition 3. [17] Let $A_{c_{1}} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ , $A_{c_{2}} = (〈 A_{2}, λ_{2} 〉, 〈 {\tilde{A}}_{2}, μ_{2} 〉)$ and $A_{c} = (〈 A, λ 〉, 〈 \tilde{A},$ μ〉) are three PCFNs and δ > 0, where A₁ =[a₁, b₁], ${\tilde{A}}_{1} = [{\tilde{a}}_{1}, {\tilde{b}}_{1}]$ , A₂ = [a₂, b₂], ${\tilde{A}}_{2} = [{\tilde{a}}_{2},$ ${\tilde{b}}_{2}]$ A = [a, b], $\tilde{A} = [\tilde{a}, \tilde{b}]$ the operation laws are:

$\begin{matrix} 1) . A_{c_{1}} \oplus A_{c_{2}} = (〈 [\sqrt{a_{1}^{2} + a_{2}^{2} - a_{1}^{2} a_{2}^{2}}, \sqrt{b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2}}], \sqrt{λ_{1}^{2} + λ_{2}^{2} - λ_{1}^{2} λ_{2}^{2}} 〉, \\ 〈 [{\tilde{a}}_{1} {\tilde{a}}_{2}, {\tilde{b}}_{1} {\tilde{b}}_{2}], μ_{1} μ_{2} 〉), \end{matrix}$ $2) . A_{c_{1}} \otimes A_{c_{2}} = (\begin{matrix} 〈 [a_{1} a_{2}, b_{1} b_{2}], λ_{1} λ_{2} 〉, \\ 〈 [\sqrt{{\tilde{a}}_{1}^{2} + {\tilde{a}}_{2}^{2} - {\tilde{a}}_{1}^{2} {\tilde{a}}_{2}^{2}}, \sqrt{{\tilde{b}}_{1}^{2} + {\tilde{b}}_{2}^{2} - {\tilde{b}}_{1}^{2} {\tilde{b}}_{2}^{2}}], \sqrt{μ_{1}^{2} + μ_{2}^{2} - μ_{1}^{2} μ_{2}^{2}} 〉 \end{matrix}),$ $3) . δ A_{c_{1}} = (\begin{matrix} 〈 [\sqrt{1 - (1 - a_{1}^{2})^{δ}}, \sqrt{1 - (1 - b_{1}^{2})^{δ}}], \sqrt{1 - (1 - λ_{1}^{2})^{δ}} 〉, \\ 〈 [({\tilde{a}}_{1} {\tilde{a}}_{2})^{δ}, ({\tilde{b}}_{1} {\tilde{b}}_{2})^{δ}], (μ_{1} μ_{2})^{δ}) 〉 \end{matrix}),$ $4) . A_{c_{1}}^{δ} = (\begin{matrix} 〈 [(a_{1} a_{2})^{δ}, (b_{1} b_{2})^{δ}], (λ_{1} λ_{2})^{δ}) 〉, \\ 〈 [\sqrt{1 - (1 - {\tilde{a}}_{1}^{2})^{δ}}, \sqrt{1 - (1 - {\tilde{b}}_{1}^{2})^{δ}}], \sqrt{1 - (1 - μ_{1}^{2})^{δ}}) 〉 \end{matrix}),$ $5) . A_{c_{1}}^{c} = 〈 {\tilde{A}}_{1}, μ_{1} 〉, 〈 A_{1}, λ_{1} 〉 .$ Definition 4. [17] Let $A_{c} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ be a PCFN, where A₁ = [a₁, b₁], ${\tilde{A}}_{1} = [{\tilde{a}}_{1}, {\tilde{b}}_{1}]$ . We can introduce the score function of A_c as: $SC (A_{c}) = {(\frac{a_{1} + b_{1} - λ_{1}}{3})}^{2} - {(\frac{{\tilde{a}}_{1} + {\tilde{b}}_{1} - μ_{1}}{3})}^{2} .$ (2)

Where SC (A_c) ∈ [-1, 1].

Definition 5. [17] Let $A_{c_{1}} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ and $A_{c_{2}} = (〈 A_{2}, λ_{2} 〉, 〈 {\tilde{A}}_{2}, μ_{2} 〉)$ be two PCFNs SC (A_{c
₁}) be the score function of A_{c
₁} and SC (A_{c
₂}) be the score function of A_{c
₂}. Then

If SC (A_c
₁) < SC (A_c
₂), then A_c
₁ < A_c
₂.

If S (p_c
₁) > S (p_c
₂), then p_c
₁ > p_c
₂.

If SC (A_c
₁) = SC (A_c
₂), then p_c
₁ ∼ p_c
₂.

Definition 6. [17] Let $A_{c} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ be a PCFN. Then we define the the accuracy degree of A_c is denoted by αc (A_c), where A₁ = [a₁, b₁], ${\tilde{A}}_{1} = [{\tilde{a}}_{1}, {\tilde{b}}_{1}]$ , can be defined as: $α c (A_{c}) = {(\frac{a_{1} + b_{1} - λ_{1}}{3})}^{2} + {(\frac{{\tilde{a}}_{1} + {\tilde{b}}_{1} - μ_{1}}{3})}^{2} .$ (3)

Where αc (A_c) ∈ [0, 1].

Definition 7. [17] Let $A_{c_{1}} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ and $A_{c_{2}} = (〈 A_{2}, λ_{2} 〉, 〈 {\tilde{A}}_{2}, μ_{2} 〉)$ be two PCFNs, αc (A_{c
₁}) be the accuracy degree of A_{c
₁} and αc (A_{c
₂}) be the accuracy degree of A_{c
₂}. Then

If αc (A_c
₁) < αc (A_c
₂), then A_c
₁ < A_c
₂.

If αc (A_c
₁) > αc (A_c
₂), then A_c
₁ > A_c
₂.

If αc (A_c
₁) = αc (A_c
₂), then A_c
₁ ∼ A_c
₂.

Definition 8. [17] Let A_{c
_i} = (μ_{c
_ij}, ν_{c
_ij}) 〉, (i = 1, 2, 3 … , n) be a collection of all PCFNs, and w = (w₁, w₂, …, w_n) ^T be the weight vector of A_{c
_i} (i = 1, 2, 3 … , n) with w_i ≥ 0 where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the aggregation result using PCFWA operator is also a PCFN and $\begin{matrix} PCFWA (A_{c_{1}}, A_{c_{2}}, \dots, A_{c_{n}}) = \\ (\begin{matrix} 〈 [\sqrt{1 - \prod_{i = 1}^{n} (1 - a_{i}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - b_{i}^{2})^{w_{i}}}]; \sqrt{1 - \prod_{i = 1}^{n} (1 - λ_{i}^{2})^{w_{i}}} 〉, \\ 〈 [\prod_{i = 1}^{n} {\tilde{a}}_{i}^{w_{i}}, \prod_{i = 1}^{n} {\tilde{b}}_{i}^{w_{i}}]; \prod_{i = 1}^{n} μ_{i}^{w_{i}} 〉 \end{matrix}) \end{matrix}$ (4)

Definition 9. [17] Let A_{c
_i} = (μ_{c
_ij}, ν_{c
_ij}) 〉, (i = 1, 2, 3, …, n) be a collection of all PCFNs, and w = (w₁, w₂, …, w_n) ^T be the weight vector of A_{c
_i} (i = 1, 2, 3, …, n) with w_i ≥ 0 where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the aggregation result using PCFWG operator is also a PCFN and $\begin{matrix} PCFWG (A_{c_{1}}, A_{c_{2}}, \dots, A_{c_{n}}) = \\ (\begin{matrix} 〈 [\prod_{i = 1}^{n} a_{i}^{w_{i}}, \prod_{i = 1}^{n} b_{i}^{w_{i}}], \prod_{i = 1}^{n} λ_{i}^{w_{i}} 〉, \\ 〈 [\sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{a}}_{i}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{b}}_{i}^{2})^{w_{i}}}], \sqrt{1 - \prod_{i = 1}^{n} (1 - μ_{i}^{2})^{w_{i}}} 〉 \end{matrix}) \end{matrix}$ (5)

Definition 10. [17] Let A_{c
_i} = (μ_{c
_ij}, ν_{c
_ij}) 〉, (i = 1, 2, 3, …, n) be a collection of all PCFNs, and w = (w₁, w₂, …, w_n) be the weight vector of A_{c
_i} (i = 1, 2, 3, …, n) with w_i ≥ 0 where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the aggregation result using PCFOWA operator is also a PCFN and $\begin{matrix} PCFOWA (A_{c_{1}}, A_{c_{2}}, \dots, A_{c_{n}}) = \\ (\begin{matrix} 〈 [\sqrt{1 - \prod_{i = 1}^{n} (1 - a_{σ (i)}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - b_{σ (i)}^{2})^{w_{i}}}]; \sqrt{1 - \prod_{i = 1}^{n} (1 - λ_{σ (i)}^{2})^{w_{i}}} 〉, \\ 〈 [\prod_{i = 1}^{n} {\tilde{a}}_{σ (i)}^{w_{i}}, \prod_{i = 1}^{n} {\tilde{b}}_{σ (i)}^{w_{i}}]; \prod_{i = 1}^{n} μ_{σ (i)}^{w_{i}} 〉 \end{matrix}) \end{matrix}$ (6)

Definition 11. [18] Let A_{c
_i} = (μ_{c
_ij}, ν_{c
_ij}) 〉, (i = 1, 2, 3, …, n) be a collection of all PCFNs, and w = (w₁, w₂, …, w_n) ^T be the weight vector of A_{c
_i} (i = 1, 2, 3, …, n) with w_i ≥ 0 where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the aggregation result using PCFOWG operator is also a PCFN and $\begin{matrix} PCFOWG (A_{c_{1}}, A_{c_{2}}, \dots, A_{c_{n}}) = \\ (\begin{matrix} 〈 [\prod_{i = 1}^{n} a_{σ (i)}^{w_{i}}, \prod_{i = 1}^{n} b_{σ (i)}^{w_{i}}], \prod_{i = 1}^{n} λ_{σ (i)}^{w_{i}} 〉, \\ 〈 \begin{matrix} [\sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{a}}_{σ (i)}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{b}}_{σ (i)}^{2})^{w_{i}}}], \\ \sqrt{1 - \prod_{i = 1}^{n} (1 - μ_{σ (i)}^{2})^{w_{i}}}) \end{matrix} 〉 \end{matrix}) \end{matrix}$ (7)

3. Pythagorean cubic fuzzy information aggregation operations with confidence levels

In this section, all the scientists have examined the studies by taking the hypothesis that decision makers are taken to be confidently aware with the calculated

objects. But in real-life situation, this type of circumstances partially fulfills. To handle this limitation, in this section, we are presenting a series of an averaging and geometric aggregation operator with different confidence levels in their understanding with an estimation.

3.1. Weighted averaging operator

Definition 12. Let κ_j be confidence levels of PCFN υ_j such that 0 ≤ κ_j ≤ 1 and Γ be the collection of PCFNs υ₁, υ₂, …, υ_n. Suppose that w = (w₁, w₂, … w_n) ^T be the weight vector of υ₁, υ₂, …, υ_n such that $\sum_{j = 1}^{n} w_{j} = 1$ and w_j = 1. Let CPCFWA : ΓⁿxrightarrowΓ if CPCFWA (〈 υ₁, κ₁ 〉 , 〈 υ₂, κ₂ 〉 , …, 〈 υ_n, κ_n 〉) = w₁ (κ₁υ₁) ⊕ w₂ (κ₂υ₂) ⊕ … ⊕ w_n (κ_nυ_n) then CPCFWA is called confidence Pythagorean cubic fuzzy weighted averaging operator.

Theorem 1. Let $υ_{j} = (〈 A_{j}, λ_{j} 〉, 〈 {\tilde{A}}_{j}, μ_{j} 〉)$ where A_j = [α_j, β_j] and ${\tilde{A}}_{j} = [{\tilde{α}}_{j}, {\tilde{β}}_{j}]$ j = 1, 2, …, n be ^′n′ PCFNs and κ_j be its confidence levels then the aggregated value be CPCFWA operator is also CPCFN and $\begin{matrix} CPCFWA (〈 υ_{1}, κ_{1} 〉, 〈 υ_{2}, κ_{2} 〉, \dots, 〈 υ_{n}, κ_{n} 〉) = \oplus_{j = 1}^{n} w_{j} (κ_{j} υ_{j}) \\ = (\begin{matrix} 〈 [\sqrt{1 - \prod_{j = 1}^{n} (1 - α_{j}^{2})^{κ_{j} w_{i}}}, \sqrt{1 - \prod_{j = 1}^{n} (1 - β_{j}^{2})^{κ_{j} w_{j}}}]; \sqrt{1 - \prod_{j = 1}^{n} (1 - λ_{j}^{2})^{κ_{j} w_{j}}} 〉, \\ 〈 [\prod_{j = 1}^{n} {\tilde{α}}_{j}^{κ_{j} w_{j}}, \prod_{j = 1}^{n} {\tilde{β}}_{j}^{κ_{j} w_{j}}]; \prod_{j = 1}^{n} μ_{j}^{κ_{j} w_{j}} 〉 \end{matrix}) \end{matrix}$ (8)

Proof. Easy to prove.

Property 1. (Idempotency) Let υ_j = υ₀ = 〈 (α₀, β₀) , κ₀〉 for all j i.e., α_j = α₀, β_j = β₀ and κ_j = κ₀ then CPCFWA 〈 (υ₁, κ₁) , (υ₂, κ₂) , …, (υ_n, κ_n) 〉 = κ₀υ₀ $\begin{matrix} CPCFWA 〈 (υ_{1}, κ_{1}), (υ_{2}, κ_{2}), . . ., (υ_{n}, κ_{n}) 〉 \\ = (\begin{matrix} 〈 [\sqrt{1 - \prod_{j = 1}^{n} (1 - α_{j}^{2})^{κ_{j} w_{i}}}, \sqrt{1 - \prod_{j = 1}^{n} (1 - β_{j}^{2})^{κ_{j} w_{j}}}]; \sqrt{1 - \prod_{j = 1}^{n} (1 - λ_{j}^{2})^{κ_{j} w_{j}}} 〉, \\ 〈 [\prod_{j = 1}^{n} {\tilde{α}}_{j}^{κ_{j} w_{j}}, \prod_{j = 1}^{n} {\tilde{β}}_{j}^{κ_{j} w_{j}}]; \prod_{j = 1}^{n} μ_{j}^{κ_{j} w_{j}} 〉 \end{matrix}) \\ = (\begin{matrix} 〈 [\sqrt{1 - (1 - α_{j}^{2})^{\sum_{j = 1}^{n} κ_{0} w_{j}}}, \sqrt{1 - (1 - β_{j}^{2})^{\sum_{j = 1}^{n} κ_{0} w_{j}}}]; \sqrt{1 - (1 - λ_{j}^{2})^{\sum_{j = 1}^{n} κ_{0} w_{j}}} 〉, \\ 〈 [{\tilde{α}}_{j}^{\sum_{j = 1}^{n} κ_{0} w_{j}} {\tilde{β}}_{j}^{\sum_{j = 1}^{n} κ_{0} w_{j}}]; μ_{j}^{\sum_{j = 1}^{n} κ_{0} w_{j}} 〉 \end{matrix}) \\ = (\begin{matrix} 〈 [\sqrt{1 - (1 - α_{j}^{2})^{κ_{0}}}, \sqrt{1 - (1 - β_{j}^{2})^{κ_{0}}}]; \sqrt{1 - (1 - λ_{j}^{2})^{κ_{0}}} 〉, \\ 〈 [{\tilde{α}}_{j}^{κ_{0}} {\tilde{β}}_{j}^{κ_{0}}]; μ_{j}^{κ_{0}} 〉 \end{matrix}) = κ_{0} υ_{0} . \end{matrix}$

Hence proof complete.

Property 2. Let $υ^{-} = 〈 min_{j} {κ_{j}, α_{j}}, max_{j} {κ_{j},$ β_j} 〉 and $υ^{+} = 〈 max_{j} {κ_{j}, α_{j}}, min_{j} {κ_{j}, β_{j}} 〉$ then υ^- ≤ CPCFWA 〈 (υ₁, κ₁) , (υ₂, κ₂) , …, (υ_t, κ_t) , (υ_t+1, κ_t+1) 〉 ≤ υ⁺

Proof. Straightforward.

3.2. Weighted averaging operator with order

Definition 13. Let Γ be a family of PFNs υ_j, κ_j be its confidence levels such that 0 ≤ κ_j ≤ 1. A confidence Pythagorean cubic fuzzy ordered weighted averaging operator is a relation Γⁿxrightarrow [] Γ $\begin{matrix} CPCFOWA 〈 (υ_{1}, κ_{1}), (υ_{2}, κ_{2}), \dots, (υ_{n}, κ_{n}) 〉 = \\ w_{1} (κ_{δ (1)} υ_{δ (1)}) \oplus w_{2} (κ_{δ (2)} υ_{δ (2)}), \dots, w_{n} (κ_{δ (n)} υ_{δ (n)}) \end{matrix}$

Where δ (1) , δ (2) , …, δ (n) is the permutation of (1, 2, … n) such that υ_δ(j-1) ≥ υ_δ(j) for all j.

Theorem 2. Let $υ_{j} = (〈 A_{j}, λ_{j} 〉, 〈 {\tilde{A}}_{j}, μ_{j} 〉)$ where A_j = [α_j, β_j] and ${\tilde{A}}_{j} = [{\tilde{α}}_{j}, {\tilde{β}}_{j}]$ j = 1, 2, …, n be ^′n′ PCFNs and κ_j be its confidence levels then the aggregated value be CPCFWA operator is also CPCFN and $\begin{matrix} CPCFOWA (〈 υ_{1}, κ_{1} 〉, 〈 υ_{2}, κ_{2} 〉, \dots, 〈 υ_{n}, κ_{n} 〉 \\ = (\begin{matrix} 〈 \begin{matrix} [\sqrt{1 - \prod_{j = 1}^{n} (1 - α_{δ (j)}^{2})^{κ_{δ (j)} w_{j}}}, \sqrt{1 - \prod_{j = 1}^{n} (1 - β_{δ (j)}^{2})^{κ_{δ (j)} w_{j}}}]; \\ \sqrt{1 - \prod_{j = 1}^{n} (1 - λ_{δ (j)}^{2})^{κ_{δ (j)} w_{j}}} \end{matrix} 〉, \\ 〈 [\prod_{j = 1}^{n} {\tilde{α}}_{δ (j)}^{κ_{δ (j)} w_{j}}, \prod_{j = 1}^{n} {\tilde{β}}_{δ (j)}^{κ_{δ (j)} w_{j}}]; \prod_{j = 1}^{n} μ_{δ (j)}^{κ_{δ (j)} w_{j}} 〉 \end{matrix}) \end{matrix} (9)$

Proof. Similar to that of theorem 1.

Property 3. Boundedness: Let $υ^{-} = 〈 min_{j}$ ${κ_{j} [α_{j}, β_{j}]}, max_{j} {κ_{j} [{\tilde{α}}_{j}, {\tilde{β}}_{j}]} 〉$ and $υ^{+} = 〈 max_{j}$ ${κ_{j} [α_{j}, β_{j}]}, min_{j} {κ_{j} [{\tilde{α}}_{j}, {\tilde{β}}_{j}]} 〉$ then υ^-≤ CPCFOWA (〈 υ₁, κ₁ 〉 , 〈 υ₂, κ₂ 〉 , …, 〈 υ_n, κ_n 〉) ≤ υ⁺

3.3. Weighted Geometric averaging operator

Definition 14. Let κ_j be confidence levels of PCFN υ_j such that 0 ≤ κ_j ≤ 1 and Γ be the collection of PCFNs υ₁, υ₂, …, υ_n. Suppose that w = (w₁, w₂, … w_n) ^T be the weight vector of υ₁, υ₂, …, υ_n such that $\sum_{j = 1}^{n} w_{j} = 1$ and w_j = 1. Let CPCFWA : ΓⁿxrightarrowΓ if CPCFWG (〈 υ₁, κ₁ 〉 , 〈 υ₂, κ₂ 〉 , …, 〈 υ_n, κ_n 〉) = (υ^{κ
₁}) ^{w
₁} ⊗ (υ^{κ
₂}) ^{w
₂} ⊗ … ⊗ (υ^{κ
_n}) ^{w
_n} then CPCFWA is called confidence Pythagorean cubic fuzzy weighted geometric operator. $\begin{matrix} CPCFWG (〈 υ_{1}, κ_{1} 〉, 〈 υ_{2}, κ_{2} 〉, \dots, 〈 υ_{n}, κ_{n} 〉) \\ = \otimes_{j = 1}^{n} w_{j} (κ_{j} υ_{j}) \\ = (υ^{κ_{1}})^{w_{1}} \otimes (υ^{κ_{2}})^{w_{2}} \otimes . . . \otimes (υ^{κ_{n}})^{w_{n}} \end{matrix}$ (9)

Where w = (w₁, w₂, … w_n) ^T be the weightvector.

Theorem 3. Let $υ_{j} = (〈 A_{j}, λ_{j} 〉, 〈 {\tilde{A}}_{j}, μ_{j} 〉)$ where A_j = [α_j, β_j] and ${\tilde{A}}_{j} = [{\tilde{α}}_{j}, {\tilde{β}}_{j}]$ j = 1, 2, …, n be n PCFNs and κ_j be its confidence levels then the aggregated value be CPCFWG operator is also CPCFN and $\begin{matrix} CPCFWG (〈 υ_{1}, κ_{1} 〉, 〈 υ_{2}, κ_{2} 〉, \dots, 〈 υ_{n}, κ n 〉) = \\ \otimes_{j = 1}^{n} w_{j} (κ_{j} υ_{j}) = (υ^{κ_{1}})^{w_{1}} \otimes (υ^{κ_{2}})^{w_{2}} \otimes . . . \otimes (υ^{κ_{n}})^{w_{n}} \\ = 〈 \begin{matrix} {[\prod_{j = 1}^{n} α_{j}, \prod_{j = 1}^{n} β_{j}}^{]} κ_{j} w_{j}; \prod_{j = 1}^{n} λ_{j}^{κ_{j} w_{j}} \\ [\sqrt{1 - \prod_{j = 1}^{n} {(1 - \tilde{α} j 2)}^{κ_{j} w_{j}}}, \sqrt{1 - \prod_{j = 1}^{n} {(1 - \tilde{β} j 2)}^{κ_{j} w_{j}}}]; \\ \sqrt{1 - \prod_{j = 1}^{n} {(1 - [μ_{j}])}^{κ_{j} w_{j}}} \end{matrix} 〉 \end{matrix} (11)$ $\begin{matrix} = (\begin{matrix} (〈 [α_{1}^{κ_{1} w_{1}} α_{2}^{κ_{2} w_{2}}, β_{1}^{κ_{1} w_{1}} β_{2}^{κ_{2} w_{2}}]; λ_{1}^{κ_{1} w_{1}} λ_{2}^{κ_{2} w_{2}}), \\ [\sqrt{1 - (1 - {\tilde{α}}_{1}^{2})^{κ_{1} w_{1}} + 1 - (1 - {\tilde{α}}_{2}^{2})^{κ_{2} w_{2}} - (1 - (1 - {\tilde{α}}_{1}^{2})^{κ_{1} w_{1}}) (1 - (1 - {\tilde{α}}_{2}^{2})^{κ_{2} w_{2}})}, \\ \sqrt{1 - (1 - {\tilde{β}}_{1}^{2})^{κ_{1} w_{1}} + 1 - (1 - {\tilde{β}}_{2}^{2})^{κ_{2} w_{2}} - (1 - (1 - {\tilde{β}}_{1}^{2})^{κ_{1} w_{1}}) (1 - (1 - {\tilde{β}}_{2}^{2})^{κ_{2} w_{2}})}]; \\ \sqrt{1 - (1 - μ_{1}^{2})^{κ_{1} w_{1}} + 1 - (1 - μ_{2}^{2})^{κ_{2} w_{2}} - (1 - (1 - μ_{1}^{2})^{κ_{1} w_{1}}) (1 - (1 - μ_{2}^{2})^{κ_{2} w_{2}})} 〉 \end{matrix}) \\ = (\begin{matrix} 〈 [\prod_{j = 1}^{2} α_{j}^{κ_{j} w_{j}}, \prod_{j = 1}^{2} β_{j}^{κ_{j} w_{j}}], \prod_{j = 1}^{2} λ_{j}^{κ_{j} w_{j}}] 〉 \\ 〈 \begin{matrix} [\sqrt{1 - \prod_{j = 1}^{2} (1 - {\tilde{α}}_{j}^{2})^{κ_{j} w_{j}}}, \sqrt{1 - \prod_{j = 1}^{2} (1 - {\tilde{β}}_{j}^{2})^{κ_{j} w_{j}}}]; \\ \sqrt{1 - \prod_{j = 1}^{2} (1 - μ_{j}^{2})^{κ_{j} w_{j}}} \end{matrix} 〉, \end{matrix}) \end{matrix} (12)$

Proof. Easy to prove.

3.4. Weighted Geometric averaging operator with order

Definition 15. Let κ_j be confidence levels of PCFN υ_j such that 0 ≤ κ_j ≤ 1 and Γ be the collection of PCFNs υ₁, υ₂, …, υ_n. Suppose that w = (w₁, w₂, … w_n) ^T be the weight vector of υ₁, υ₂, …, υ_n such that $\sum_{j = 1}^{n} w_{j} = 1$ and w_j = 1. Let CPCFWG : ΓⁿxrightarrowΓ if $CPCFWG (〈 υ_{1}, κ_{1} 〉, 〈 υ_{2}, κ_{2} 〉, . . ., 〈 υ_{n}, κ_{n} 〉) = (υ_{δ (1)}^{κ_{δ (1)}})^{w_{1}} \otimes (υ_{δ (2)}^{κ_{δ (2)}})^{w_{2}} \otimes \dots \otimes (υ_{δ (n)}^{κ_{δ (n)}})^{w_{n}}$ then CPCFWG is called confidence Pythagorean cubic fuzzy weighted geometric operator.

$\begin{matrix} CPCFWG (〈 υ_{1}, κ_{1} 〉, 〈 υ_{2}, κ_{2} 〉, . . ., 〈 υ_{n}, κ_{n} 〉) \\ = \otimes_{j = 1}^{n} w_{j} (κ_{δ (j)} υ_{δ (j)}) \\ = (υ_{δ (1)}^{κ_{δ (1)}})^{w_{1}} \otimes (υ_{δ (2)}^{κ_{δ (2)}})^{w_{2}} \otimes . . . \otimes (υ_{δ (n)}^{κ_{δ (n)}})^{w_{n}} \end{matrix}$ (10)

Where w = (w₁, w₂, … w_n) ^T be the weightvector.

Theorem 4. Let $υ_{j} = (〈 A_{j}, λ_{j} 〉, 〈 {\tilde{A}}_{j}, μ_{j} 〉)$ where A_j = [α_j, β_j] and ${\tilde{A}}_{j} = [{\tilde{α}}_{j}, {\tilde{β}}_{j}]$ j = 1, 2, …, n be ^′n′ PCFNs and κ_j be its confidence levels then the aggregated value be CPCFWG operator is also CPCFN and $\begin{matrix} CPCFWG (〈 υ_{1}, κ_{1} 〉, 〈 υ_{2}, κ_{2} 〉, \dots, 〈 υ_{n}, κ_{n} 〉) = \\ \otimes_{j = 1}^{n} w_{j} (κ_{δ (j)} υ_{δ (j)}) = (υ_{δ (1)}^{κ_{δ (1)}})^{w_{1}} \otimes (υ_{δ (2)}^{κ_{δ (2)}})^{w_{2}} \otimes \dots \otimes (υ_{δ (n)}^{κ_{δ (n)}})^{w_{n}} \\ = 〈 \begin{matrix} {[\prod_{j = 1}^{n} α_{δ (j)}, \prod_{j = 1}^{n} β_{δ (j)}]}^{κ_{δ (j)} w_{j}}; \prod_{j = 1}^{n} λ_{δ (j)}^{κ_{δ (j)} w_{j}} \\ [\sqrt{1 - \prod_{j = 1}^{n} {(1 - \tilde{α} δ (j) 2)}^{κ_{δ (j)} w_{j}}}, \sqrt{1 - \prod_{j = 1}^{n} {(1 - \tilde{β} δ (j) 2)}^{κ_{δ (j)} w_{j}}}]; \\ \sqrt{1 - \prod_{j = 1}^{n} {(1 - [μ_{δ (j)}])}^{κ_{δ (j)} w_{j}}} \end{matrix} 〉 \end{matrix} (14)$

Proof. Similar proof.

4. Group decision making approach under confidence levels

Algorithm: Consider a decision making problem in which F ={ F₁, F₂, … F_n } be the set ofalternatives, H ={ H₁, H₂, …, H_m } be the finite set ofattributes, and let us adopt that the decision-maker provids their individual decision matrix $\overset{0.5 em \leftrightarrow}{D} = (p_{c_{j}})_{m \times n}$ in which p_{c
_j} is given by the decision-maker for the alternative F with respect to H in the form of PCFN p_{c
_j} = (μ_{p
_{c
_j}}, ν_{p
_{c
_j}}). Therefore, an DM problem can be characterized in the form of the Pythagorean cubic fuzzy decision-matrix $\overset{0.5 em \leftrightarrow}{D} = (p_{c_{j}})_{m \times n}$ . Then the steps are utilize for finding the best one under the set of feasible ones.

Step 1. Collect the information related to each alternative under the different criteria from each decision maker.

Step 2. In this step we construct the normalized Pythagorean cubic fuzzy decision-making matrices. In the obtanied decision-matrix $\overset{0.5 em \leftrightarrow}{D} = (p_{c_{j}})_{m \times n}$ is classifed into two kinds of attributes, namly cost and benefit attrebutes. If all the attributes are of same type, then there is no necessity to normalized the rating values, whereas if there are benefit and cost attrebutes in MCDM, then the decision matricx $\overset{0.5 em \leftrightarrow}{D} = (p_{c_{j}})_{m \times n}$ can be transformed into the normalized Pythagorean cubic fuzzy decision matrices, where $\overset{0.5 em \leftrightarrow}{r} ij s =$ ${\begin{matrix} ɛ_{ij}^{(s)} if the attribute is of benefit type \\ {\bar{ɛ}}_{ij}^{(s)} if the attribute is of cos t type \end{matrix}$ where ${\bar{ɛ}}_{ij}^{(s)}$ is the complement of $ɛ_{ij}^{(s)}$ .

Step 3. Calculate the over-all assesments of the alternatives. Use the CPCFWA or CPCFWG operator to aggregate all the rating values ${\ddot{r}}_{ij}, (i = 1, 2, \dots, m; j = 1, 2, \dots, n)$ to each i-th alternative and get its overall rating values, that is, $\begin{matrix} {\ddot{r}}_{i} = CPCFWA ({\ddot{r}}_{i 1}, {\ddot{r}}_{i 2}, \dots, {\ddot{r}}_{in}), (i = 1, 2, \dots, n) \\ or \\ {\ddot{r}}_{i} = CPCFWG ({\ddot{r}}_{i 1}, {\ddot{r}}_{i 2}, . . ., {\ddot{r}}_{in}), (i = 1, 2, \dots, n) \end{matrix}$

Where w = (w₁, …, w_n) ^T is the weighted vector of Attribute such that w_j ∈ [0, 1] and $\sum_{j}^{n} w_{j} = 1$ .

Step 4. Rank the combined total preference values ${\ddot{r}}_{i,} (i = 1, 2, \dots, n)$ in descending order using the score function of PCFN, and, finally, select the most wanted alternative(s) between them.

5. Illustrative example

Consider a set of diagnoses $T = {\begin{matrix} T_{1} (Viral fever), T_{2} (Malaria), T_{3} (Typhoid), \\ T_{4} (Stomach problem), T_{5} (Chest problem) \end{matrix}}$ and a set of symptoms $S = {\begin{matrix} s_{1} (Temperature), s_{2} (Head Ache), \\ s_{3} (Stomach Pain), s_{4} (Cough), s_{5} (chest pain) \end{matrix}},$ suppose a patient, with respect to all the symptoms, can be represented by the following PCFS. $\begin{matrix} \tilde{P} (Patient) = \\ (\begin{matrix} 〈 [0.5, 0.6], 0.4 〉, 〈 [0.5, 0.7], 0.5 〉; 〈 [0.6, 0.7], 0.6 〉, 〈 [0.5, 0.6], 0.7 〉; \\ 〈 [0.4, 0.5], 0.7 〉, 〈 [0.7, 0.8], 0.3 〉; 〈 [0.3, 0.4], 0.5 〉, 〈 [0.6, 0.7], 0.8 〉; \\ 〈 [0.7, 0.8], 0.4 〉, 〈 [0.4, 0.5], 0.2 〉; \end{matrix}) \end{matrix}$ $T_{1} = (\begin{matrix} 〈 [0.2, 0.3], 0.4 〉, 〈 [0.1, 0.6], 0.5 〉; 〈 [0.2, 0.5], 0.4 〉, 〈 [0.2, 0.3], 0.3 〉; \\ 〈 [0.5, 0.6], 0.3 〉, 〈 [0.1, 0.4], 0.7 〉; 〈 [0.2, 0.4], 0.2 〉, 〈 [0.6, 0.7], 0.4 〉; \\ 〈 [0.1, 0.2], 0.5 〉, 〈 [0.6, 0.9], 0.3 〉 \end{matrix}),$ $T_{2} = (\begin{matrix} 〈 [0.1, 0.2], 0.3 〉, 〈 [0.2, 0.5], 0.4 〉; 〈 [0.4, 0.7], 0.3 〉, 〈 [0.3, 0.6], 0.7 〉; \\ 〈 [0.1, 0.4], 0.1 〉, 〈 [0.1, 0.3], 0.5 〉; 〈 [0.3, 0.5], 0.4 〉, 〈 [0.3, 0.4], 0.6 〉; \\ 〈 [0.5, 0.7], 0.5 〉, 〈 [0.1, 0.3], 0.7 〉 \end{matrix}),$ $T_{3} = (\begin{matrix} 〈 [0.3, 0.4], 0.5 〉, 〈 [0.2, 0.7], 0.6 〉; 〈 [0.3, 0.6], 0.5 〉, 〈 [0.4, 0.6], 0.3 〉; \\ 〈 [0.3, 0.6], 0.4 〉, 〈 [0.1, 0.5], 0.8 〉; 〈 [0.2, 0.3], 0.2 〉, 〈 [0.3, 0.4], 0.4 〉; \\ 〈 [0.1, 0.2], 0.6 〉, 〈 [0.6, 0.8], 0.7 〉 \end{matrix}),$ $T_{4} = (\begin{matrix} 〈 [0.3, 0.5], 0.4 〉, 〈 [0.5, 0.6], 0.6 〉; 〈 [0.2, 0.5], 0.4 〉, 〈 [0.3, 0.4], 0.8 〉; \\ 〈 [0.5, 0.6], 0.3 〉, 〈 [0.1, 0.4], 0.7 〉; 〈 [0.2, 0.6], 0.2 〉, 〈 [0.1, 0.3], 0.4 〉; \\ 〈 [0.2, 0.3], 0.3 〉, 〈 [0.5, 0.6], 0.5 〉 \end{matrix})$ and $T_{5} = (\begin{matrix} 〈 [0.4, 0.5], 0.4 〉, 〈 [0.1, 0.3], 0.5 〉; 〈 [0.3, 0.5], 0.3 〉, 〈 [0.5, 0.6], 0.5 〉; \\ 〈 [0.2, 0.6], 0.3 〉, 〈 [0.1, 0.4], 0.5 〉; 〈 [0.3, 0.7], 0.2 〉, 〈 [0.6, 0.7], 0.4 〉; \\ 〈 [0.1, 0.2], 0.5 〉, 〈 [0.6, 0.9], 0.6 〉 \end{matrix})$

Our target is to classify the pattern $\tilde{P}$ in the classes T₁, T₂, T₃, T₄ and T₅. For this, a developed aggregation operators as given in Equations (8 and 11) have been used to computed corresponding to it.

5.1. By CPCFWA operator

Using CPCFWA operator given in Equation (8) to aggregate and the results corresponding to it is given as: $\begin{matrix} (\tilde{p}, T_{1} (1)) = (〈 [0.8634, 0.8781]; 0.8649 〉, 〈 [0.0075, 0.8882]; 0.0375 〉) . \\ (\tilde{p}, T_{1} (2)) = (〈 [0.9558, 0.9618]; 0.9576 〉, 〈 [0.0095, 0.00175]; 0.0038 〉) . \\ (\tilde{p}, T_{1} (3)) = (〈 [0.9283, 0.9428]; 0.9443 〉, 〈 [0.0025, 0.0295]; 0.0102 〉) . \\ (\tilde{p}, T_{1} (4)) = (〈 [0.9547, 0.9628]; 0.9621 〉, 〈 [0.0144, 0.0258]; 0.0123 〉) . \\ (\tilde{p}, T_{1} (5)) = (〈 [0.8785, 0.8966]; 0.8664 〉, 〈 [0.0075, 0.0689]; 0.0046 〉) . \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{1} (1)) = 0.0850; S (\tilde{p}, T_{1} (2)) = 0.1024; S (\tilde{p},$ T₁ (3)) =0.0954; $S (\tilde{p}, T_{1} (4)) = 0.1013; S (\tilde{p}, T_{1} (5))$ = 0.0909

$\begin{matrix} (\tilde{p}, T_{2} (1)) = (〈 [0.8611, 0.8746]; 0.8589 〉, 〈 [0.0150, 0.00735]; 0.0300 〉) \\ (\tilde{p}, T_{2} (2)) = (〈 [0.9576, 0.9660]; 0.8565 〉, 〈 [0.0121, 0.00265]; 0.0396 〉) \\ (\tilde{p}, T_{2} (3)) = (〈 [0.9096, 0.9283]; 0.9403 〉, 〈 [0.0025, 0.00186]; 0.0059 〉) \\ (\tilde{p}, T_{2} (4)) = (〈 [0.9569, 0.9664]; 0.9664 〉, 〈 [0.0041, 0.00094]; 0.0255 〉) \\ (\tilde{p}, T_{2} (5)) = (〈 [0.8920, 0.9207]; 0.8664 〉, 〈 [0.0076, 0.00286]; 0.0090 〉) \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{2} (1)) = 0.0850; S (\tilde{p}, T_{2} (2)) = 0.1018; S (\tilde{p},$ T₂ (3)) =0.0995; $S (\tilde{p}, T_{2} (4)) = 0.1017; S (\tilde{p}, T_{2} (5))$ = 0.0994 $\begin{matrix} (\tilde{p}, T_{3} (1)) = (〈 [0.8673, 0.8832], 0.8729 〉, 〈 [0.0150, 0.1029], 0.0450 〉) \\ (\tilde{p}, T_{3} (2)) = (〈 [0.9565, 0.9636], 0.9590 〉, 〈 [0.0144, 0.0265], 0.0238 〉) \\ (\tilde{p}, T_{3} (3)) = (〈 [0.9158, 0.9428], 0.9479 〉, 〈 [0.0025, 0.0422], 0.0126 〉) \\ (\tilde{p}, T_{3} (4)) = (〈 [0.9547, 0.9599], 0.9621 〉, 〈 [0.0150, 0.0094], 0.0123 〉) \\ (\tilde{p}, T_{3} (5)) = (〈 [0.8785, 0.8966], 0.8752 〉, 〈 [0.0319, 0.00627], 0.0090 〉) \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{3} (1)) = 0.0850; S (\tilde{p}, T_{3} (2)) = 0.1033; S (\tilde{p},$ T₃ (3)) =0.0920; $S (\tilde{p}, T_{3} (4)) = 0.1008; S (\tilde{p}, T_{3} (5))$ = 0.0892 $\begin{matrix} (\tilde{p}, T_{4} (1)) = (〈 [0.8673, 0.8900], 0.8649 〉, 〈 [0.0375, 0.0882], 0.0450 〉) \\ (\tilde{p}, T_{4} (2)) = (〈 [0.9558, 0.9618], 0.9576 〉, 〈 [0.0121, 0.0208], 0.0429 〉) \\ (\tilde{p}, T_{4} (3)) = (〈 [0.9283, 0.9428], 0.9443 〉, 〈 [0.0025, 0.0295], 0.0102 〉) \\ (\tilde{p}, T_{4} (4)) = (〈 [0.9547, 0.9710], 0.9621 〉, 〈 [0.0006, 0.0056], 0.123 〉) \\ (\tilde{p}, T_{4} (5)) = (〈 [0.8801, 0.8989], 0.8547 〉, 〈 [0.0276, 0.0498], 0.0069 〉) \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{4} (1)) = 0.0878; S (\tilde{p}, T_{4} (2)) = 0.1024; S (\tilde{p},$ T₄ (3)) =0.0954; $S (\tilde{p}, T_{4} (4)) = 0.1022; S (\tilde{p}, T_{4} (5))$ = 0.0944 $\begin{matrix} (\tilde{p}, T_{5} (1)) = (〈 [0.8729, 0.8900], 0.8649 〉, 〈 [0.0075, 0.0441], 0.0375 〉) \\ (\tilde{p}, T_{5} (2)) = (〈 [0.9565, 0.9618], 0.9565 〉, 〈 [0.0165, 0.0265], 0.0323 〉) \\ (\tilde{p}, T_{5} (3)) = (〈 [0.9119, 0.9428], 0.9443 〉, 〈 [0.0025, 0.0295], 0.0059 〉) \\ (\tilde{p}, T_{5} (4)) = (〈 [0.9569, 0.9764], 0.9621 〉, 〈 [0.0144, 0.0258], 0.123 〉) \\ (\tilde{p}, T_{5} (5)) = (〈 [0.8785, 0.8966], 0.8664 〉, 〈 [0.0319, 0.0689], 0.0080 〉) \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{5} (1)) = 0.0896; S (\tilde{p}, T_{5} (2)) = 0.1028; S (\tilde{p},$ T₅ (3)) =0.0920; $S (\tilde{p}, T_{5} (4)) = 0.1017; S (\tilde{p}, T_{5} (5))$ = 0.0908

Since $\begin{matrix} S (\tilde{p}, T_{1} (2)) > S (\tilde{p}, T_{1} (4)) > S (\tilde{p}, T_{1} (3)) > S (\tilde{p}, T_{1} (5)) > S (\tilde{p}, T_{1} (1)) \\ S (\tilde{p}, T_{2} (2)) > S (\tilde{p}, T_{2} (4)) > S (\tilde{p}, T_{2} (3)) > S (\tilde{p}, T_{2} (5)) > S (\tilde{p}, T_{2} (1)) \\ S (\tilde{p}, T_{3} (2)) > S (\tilde{p}, T_{3} (4)) > S (\tilde{p}, T_{3} (3)) > S (\tilde{p}, T_{3} (5)) > S (\tilde{p}, T_{3} (1)) \\ S (\tilde{p}, T_{4} (2)) > S (\tilde{p}, T_{4} (4)) > S (\tilde{p}, T_{4} (3)) > S (\tilde{p}, T_{4} (5)) > S (\tilde{p}, T_{4} (1)) \\ S (\tilde{p}, T_{5} (2)) > S (\tilde{p}, T_{5} (4)) > S (\tilde{p}, T_{5} (3)) > S (\tilde{p}, T_{5} (5)) > S (\tilde{p}, T_{5} (1)) \end{matrix}$

Thus we have $\begin{matrix} T_{1} (2) ≻ T_{1} (4) ≻ T_{1} (3) ≻ T_{1} (5) ≻ T_{1} (1); T_{2} (2) ≻ T_{2} (4) ≻ T_{2} (3) ≻ T_{2} (5) ≻ T_{2} (1) \\ T_{3} (2) ≻ T_{3} (4) ≻ T_{3} (3) ≻ T_{3} (5) ≻ T_{3} (1); T_{4} (2) ≻ T_{4} (4) ≻ T_{4} (3) ≻ T_{4} (5) ≻ T_{4} (1) \\ T_{5} (2) ≻ T_{5} (4) ≻ T_{5} (3) ≻ T_{5} (5) ≻ T_{5} (1) \end{matrix}$

Hence T₁₂₃₄₅ (2) is the best pattern to be classifier with $\tilde{P}$ .

5.2. By CPCFWG operator

Using CPCFWG operator given in Equation (11) to aggregate and the results corresponding to it is given as: $\begin{matrix} (\tilde{p}, T_{1} (1)) = (〈 [0.0150, 0.0324], 0.0192 〉, 〈 [0.8611, 0.9103], 0.08803 〉) \\ (\tilde{p}, T_{1} (2)) = (〈 [0.0137, 0.0323], 0.0208 〉, 〈 [0.9536, 0.9565], 0.9595 〉) \\ (\tilde{p}, T_{1} (3)) = (〈 [0.0106, 0.0221], 0.0143 〉, 〈 [0.9403, 0.9608], 0.9443 〉) \\ (\tilde{p}, T_{1} (4)) = (〈 [0.0005, 0.0031], 0.0014 〉, 〈 [0.9974, 0.9850], 0.9828 〉) \\ (\tilde{p}, T_{1} (5)) = (〈 [0.0561, 0.0922], 0.0276 〉, 〈 [0.8513, 0.8920], 0.8678 〉) \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{1} (1)) = - 0.0881; S (\tilde{p}, T_{1} (2)) = - 0.0125;$ $S (\tilde{p}, T_{1} (3)) = - 0.2017$ ; $S (\tilde{p}, T_{1} (4)) = - 0.0166;$ $S (\tilde{p}, T_{1} (5)) = - 0.0835$ $\begin{matrix} (\tilde{p}, T_{2} (1)) = (〈 [0.0075, 0.0216], 0.0144 〉, 〈 [0.8634, 0.9025], 0.8729 〉) \\ (\tilde{p}, T_{2} (2)) = (〈 [0.0208, 0.0396], 0.0175 〉, 〈 [0.9544, 0.9610], 0.9660 〉) \\ (\tilde{p}, T_{2} (3)) = (〈 [0.0008, 0.0115], 0.0025 〉, 〈 [0.9403, 0.9582], 0.9234 〉) \\ (\tilde{p}, T_{2} (4)) = (〈 [0.0010, 0.0046], 0.0048 〉, 〈 [0.9688, 0.9764], 0.9866 〉) \\ (\tilde{p}, T_{2} (5)) = (〈 [0.0844, 0.1443], 0.0276 〉, 〈 [0.8493, 0.8617], 0.8801 〉) \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{2} (1)) = - 0.0886; S (\tilde{p}, T_{2} (2)) = - 0.0121;$ $S (\tilde{p}, T_{2} (3)) = - 0.0256$ ; $S (\tilde{p}, T_{2} (4)) = - 0.0122;$ $S (\tilde{p}, T_{2} (5)) = - 0.0722$ $\begin{matrix} (\tilde{p}, T_{3} (1)) = (〈 [0.0225, 0.0432], 0.0240 〉, 〈 [0.8634, 0.9203], 0.8900 〉) \\ (\tilde{p}, T_{3} (2)) = (〈 [0.0175, 0.0361], 0.0238 〉, 〈 [0.9555, 0.9610], 0.9595 〉) \\ (\tilde{p}, T_{3} (3)) = (〈 [0.00047, 0.0221], 0.0226 〉, 〈 [0.9403, 0.9643], 0.9582 〉) \\ (\tilde{p}, T_{3} (4)) = (〈 [0.0005, 0.0018], 0.0014 〉, 〈 [0.9688, 0.9764], 0.9828 〉) \\ (\tilde{p}, T_{3} (5)) = (〈 [0.0233, 0.0530], 0.0319 〉, 〈 [0.8752, 0.9068], 0.8801 〉) \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{3} (1)) = - 0.0886; S (\tilde{p}, T_{3} (2)) = - 0.0117; S$ $(\tilde{p}, T_{3} (3)) = - 0.0295$ ; $S (\tilde{p}, T_{3} (4)) = - 0.0129; S$ $(\tilde{p}, T_{3} (5)) = - 0.0602$

$\begin{matrix} (\tilde{p}, T_{4} (1)) = (〈 [0.0225, 0.0540], 0.0192 〉, 〈 [0.8803, 0.9103], 0.8900 〉) \\ (\tilde{p}, T_{4} (2)) = (〈 [0.0137, 0.0323], 0.0208 〉, 〈 [0.9544, 0.9576], 0.9695 〉) \\ (\tilde{p}, T_{4} (3)) = (〈 [0.0106, 0.0221], 0.0143 〉, 〈 [0.9403, 0.9608], 0.9443 〉) \\ (\tilde{p}, T_{4} (4)) = (〈 [0.0005, 0.0064], 0.0014 〉, 〈 [0.9663, 0.9746], 0.9828 〉) \\ (\tilde{p}, T_{4} (5)) = (〈 [0.0406, 0.0733], 0.0183 〉, 〈 [0.8664, 0.8811], 0.8585 〉) \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{4} (1)) = - 0.0898; S (\tilde{p}, T_{4} (2)) = - 0.0786;$ $S (\tilde{p}, T_{4} (3)) = - 0.0847$ ; $S (\tilde{p}, T_{4} (4)) = - 0.0820;$ $S (\tilde{p}, T_{4} (5)) = - 0.0868$ $\begin{matrix} (\tilde{p}, T_{5} (1)) = (〈 [0.0360, 0.0620], 0.0231 〉, 〈 [0.8566, 0.8828], 0.8727 〉) \\ (\tilde{p}, T_{5} (2)) = (〈 [0.0412 0.0844], 0.0412 〉, 〈 [0.8727, 0.8889], 0.8920 〉) \\ (\tilde{p}, T_{5} (3)) = (〈 [0.0132, 0.0498], 0.0561 〉, 〈 [0.8785, 0.9023], 0.8617 〉) \\ (\tilde{p}, T_{5} (4)) = (〈 [0.0103, 0.0361], 0.0207 〉, 〈 [0.8889, 0.9083], 0.9023 〉) \\ (\tilde{p}, T_{5} (5)) = (〈 [0.0233, 0.0530]; 0.0276 〉, 〈 [0.8752, 0.9287], 0.8678 〉) \end{matrix}$

Score values corresponding to them are $S (\tilde{p}, T_{5} (1)) = - 0.0998; S (\tilde{p}, T_{5} (2)) = - 0.0832;$ $S (\tilde{p}, T_{5} (3)) = - 0.0939$ ; $S (\tilde{p}, T_{5} (4)) = - 0.0889;$ $S (\tilde{p}, T_{5} (5)) = - 0.0971$

Hence T₁₂₃₄₅ (2) is the best pattern to be classifier with $\tilde{P}$ .

Hence, from the above defined aggregation operators, we can see that the best alternatives are the same.

To explore the developments of the suggested operators, It shows that the suggested operators are more general to the Pythagorean fuzzy and Interval-valued Pythagorean fuzzy aggregation operators. We can summarize in short, that, The CPCFWA, CPCFOWA, CPCFWG and CPCFOWG operators can more precisely express the uncertainty in the MCDM than the CPFWA, CPFOWA, CPFWG and CPFOWG. So, we can know that the PCFNs has more extensive application prospect than the other methods.

Motivation: according to the above analysis, the proposed methods have the following merits in decision making problems.

The notion of Pythagorean cubic fuzzy sets in which the membership degree and non-membership degree are cubic fuzzy numbers which hold the conditions that the square sum of its membership degree is less than or equal to $〈 \tilde{1}, 1 〉$

Since, fuzzy, intiitionistics, Pythagorean and interval valued Pythagorean fuzzy set are all the special cases of the Pythagorean cubic fuzzy set. So far, researchers have used the PFS which are characterized by the membership and non-membership degree of a particular element under the restriction that their square sum is less than or equal to 1. But, in most of the real life problem this may not be true when an expert gave their preferences towards the elements. To hold this, PCFS is the generalized form which can handle not only incomplete information, but also the indeterminate and inconsistent information, which exist commonly in real. Therefore, the current studies are more suitable than the existing ones for solving the day-today real problems.

We observed from the past studies that the results calculated by the several existing methods are under the situation without considering the confidence levels of the attributes during the evaluation. But in real-life problem, these types of conditions are partially fulfilled.

The existing operators for PFS and IVPFS are a special case of the proposed operators. Furthermore, some of the existing operators for PFS and IVPFS are also a special case of the proposed operators. Therefore, it has been concluded that the proposed aggregation operators are more generalized and suitable to solve the real-life problems more accurate than the existing ones.

The objective of this work is to present some series of new averaging and geometric aggregation operator by considering the degree of the confidence levels of each decision makers’ during evaluation. Traditionally, it has been assumed that the all the decision maker gives their preferences of the different alternative at the same level of confidence. But this shortcoming has been ruled out in the present manuscript by considering the confidence level factor of the decision maker.

Table 1
The decision results of different methods

No. Method RAnK

1 PFWA Yager [31] T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)

2 PFWG Yager [31] T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)

3 CPFWA Garg [15] T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)

4 CPFWG Garg [15] T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)

5 Our proposed algorithm with CPCFWA T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)

6 Our proposed algorithm with CPCFWG T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)

No.	Method	RAnK
1	PFWA Yager [31]	T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)
2	PFWG Yager [31]	T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)
3	CPFWA Garg [15]	T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)
4	CPFWG Garg [15]	T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)
5	Our proposed algorithm with CPCFWA	T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)
6	Our proposed algorithm with CPCFWG	T₅ (2) ≻ T₅ (4) ≻ T₅ (3) ≻ T₅ (5) ≻ T₅ (1)

6. Conclusion

The aim of this paper is to extend some novel averaging and geometric aggregation operator by taking the degree of the confidence levels of each decision makers’ during evaluation. Usually, it has been supposed that the all the decision-maker give their preferences of the different alternative at the same level of confidence. But this shortcoming has been governed out in the present work by taking the confidence level factor κ of the decision maker. Based on it, new aggregation operators, namely CPCFWA, CPCFOWA, CPCFWG and CPCFOWG are suggested under PCFS information. The desirable properties consistent to each operator has also been deliberated. Moreover, it has been detected that when κ = 1 for all the preferences, then the suggested aggregation operators reduces to the existing PCFWA and PCFWG operators. A numerical example, has been proposed which shows that the suggested operators delivers an alternative way to solve decision making process in a more actual way. In the future, we may generalize the presented study to the different applications and to solve many uncertain programming problems.

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