Pythagorean cubic fuzzy aggregation operators and their application to multi-criteria decision making problems

Abstract

Since Pythagorean fuzzy sets and interval-valued Pythagorean fuzzy sets are better tools to deal with fuzziness and vagueness. Therefore, in this paper, we present 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 $〈 \tilde{1}, 1 〉$ . We define some basic operators and to compare two Pythagorean cubic fuzzy numbers we develop score and accuracy function. We also define the distance between two Pythagorean cubic fuzzy numbers. Based on the defined operators, we propose Pythagorean cubic fuzzy weighted averaging (PCFWA), Pythagorean cubic fuzzy weighted geometric (PCFWG), Pythagorean cubic fuzzy ordered weighted averaging (PCFOWA) and Pythagorean cubic fuzzy ordered weighted geometric (PCFOWG) operators. We discuss some of its operational laws of the established operators and suggest a multi criteria decision-making (MCDM) approach based on the developed operators. Moreover, the methods and operators proposed in this paper are providing more general, more accurate and precise results as compared to the existing methods because these methods and operators are the generalization of their existing methods. Furthermore, the method for multi-criteria decision-making problems based on these proposed operators was developed, and the operational processes were also illustrated in detail. Finally, an illustrative example is given to show the decision-making steps in detail of these proposed methods and operators to show the validity, practicality and effectiveness.

Keywords

Interval valued pythagorean fuzzy set pythagorean cubic fuzzy set PCFWA operator PCFWG operator PCFOWA operator PCFOWG operator multi criteria decision-making

1 Introduction

Multiple-criteria decision making (MCDM) means that the best alternative is selected from the limited alternatives set according to the multiple criteria, which can be regarded as cognitive processing. Multi criteria decision-making (MCDM) is an important branch of the decision-making theory which has been widely used in human activities [9]. Because the real decision making problems were frequently produced from the complicated environment, the evaluation information is usually fuzzy. In general, the fuzzy information takes two forms: one quantitative and one qualitative. The quantitative fuzzy information can be expressed by fuzzy set (FS) [39], intuitionistic fuzzy set (IFS) [1], Pythagorean fuzzy set (PFS) [32] and so on. FS theory proposed by Zadeh [39] has been used to describe fuzzy quantitative information which contains only a membership degree. Based on this, Atanassov [1] presented IFS, which consists of a member-ship and a non-membership degrees that fulfills the restriction form that the sum of two degrees is ⩽1. However, sometimes the two degrees don’t meet the restriction, but the sum of squares of the two degrees is ⩽1. Yager [32] introduced the PFS in which the square sum of membership and non-membership degrees is equal to or less than 1. In some situation, the PFS has the greater ability to express the fuzzy information than the IFS. For example, if an expert gives the membership about the support to a matter is 0.8 and the non-membership for against is 0.6. Obviously, IFS is unable to describe this decision information, but it can be effectively described by PFS. Peng et al. [20], presented some new properties in the Pythagorean fuzzy set which are division, subtraction and other important properties. Writers deal with the supremacy and dependency ranking methods to explain the multi-criteria decision-making problems in the Pythagorean fuzzy environment. In [11], khan et al. Developed prioritized aggregation operators for multi criteria decision-making established on Pythagorean fuzzy sets. Peng et al. [21] proposed Pythagorean fuzzy Linguistic sets (PFLSs) and there operating laws and score function of Pythagorean fuzzy linguistic numbers. Wei et al. [14] introduced a maximizing deviation method to explain decision-making problems based on interval-valued Pythagorean fuzzy environments. The subtraction and division concept of Pythagorean fuzzy numbers (PFNS) were proposed by Gou et al. [8]. Pend et al. [22], presented the notion of the obvious definition of Pythagorean fuzzy distance degree, which is communicated by a Pythagorean fuzzy number that will decrease the data loss and continue further creative evidence. Also the well-defined notion of novel score function. Liang et al. [15], presented the notation of Pythagorean fuzzy geometric Bonferroni mean and weighted Pythagorean fuzzy geometric Bonferroni (WPFGBM) operator. In [3], Garg proposed interval-valued Pythagorean fuzzy geometric (IVPFWA) operator, interval-valued Pythagorean fuzzy geometric (IVPFWG) operator and presented the notation of new accuracy function based on interval-valued Pythagorean fuzzy environment. Khan et al extended to concept of TOPSIS method for multi criteria decision-making and developed Choquet integral TOPSIS method based on IVPFNs [12]. Khan and Abdullah in [13], proposed gray relational analysis (GRA) scheme for multi criteria decision-making under interval valued Pythagorean fuzzy situation. The authors first developed interval valued Pythagorean fuzzy Choquet integral average (IVPFCI) operator, then advanced a method to multi criteria decision-making built on GRA method. Peng et al. [23], propose two algorithms to solve stochastic multi criteria decision-making problem under Interval-valued fuzzy soft set approaches. Peng and Liu in [24], presents three novel single-valued neutrosophic soft set methods and propose three algorithms to solve single-valued neutrosophic soft decision making problem. Peng and Selvachandran in [25], present an overview on Pythagorean fuzzy set with aim of offering a clear perspective on the different concepts, tools and trends related to their extension and algorithms in decision making. Peng [26], initiate some new interval-valued Pythagorean fuzzy operators and discuss their properties in relation with some existing operators in detail. It will promote the development of interval-valued Pythagorean fuzzy operators and propose an algorithm to deal with multi-criteria decision making (MCDM) problem based on proposed operator. Wei et al. [28], presents the generalized Heronian mean operator and geometric Heronian mean operator under the q-rung orthopair fuzzy sets is studied. Wei et al. [29], proposed Pythagorean hesitant fuzzy aggregation operators Pythagorean hesitant fuzzy Hamacher weighted average (PHFHWA) operator, Pythagorean hesitant fuzzy Hamacher weighted geometric (PHFHWG) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted average (PHFHOWA) operator, Pythagorean hesitant fuzzy Hamacher ordered weighted geometric (PHFHOWG) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted average (PHFHIOWA) operator, Pythagorean hesitant fuzzy Hamacher induced ordered weighted geometric (PHFHIOWG) operator, Pythagorean hesitant fuzzy Hamacher induced correlated aggregation operators, Pythagorean hesitant fuzzy Hamacher prioritized aggregation operators, and Pythagorean hesitant fuzzy Hamacher power aggregation operators based on Hamacher t-norm and t-conorm. Wu et al. in [30], developed 2-tuple linguistic neutrosophic Hamy mean (2TLNHM) operator, 2-tuple linguistic neutrosophic weighted Hamy mean (2TLNWHM) operator, 2-tuple linguistic neutrosophic dual Hamy mean (2TLNDHM) operator, and 2-tuple linguistic neutrosophic weighted dual Hamy mean (2TLNWDHM) operator under 2-tuple linguistic neutrosophic environment.

Wang et al. in [31], presented 2-tuple linguistic neutrosophic numbers weighted Bonferroni mean (2TLNNWBM) operator, 2-tuple linguistic neutrosophic numbers weighted geometric Bonferroni mean (2TLNNWGBM) operator, generalized 2-tuple linguistic neutrosophic numbers weighted Bonferroni mean (G2TLNNWBM) operator, generalized 2-tuple linguistic neutrosophic numbers weighted geometric Bonferroni mean (G2TLNNWGBM) operator, dual generalized 2-tuple linguistic neutrosophic numbers weighted Bonferroni mean (DG2TLNNWBM) operator, and dual generalized 2-tuple linguistic neutrosophic numbers weighted geometric Bonferroni mean (DG2TLNNWGBM) operator to deal with MCDM problems. Gao et al. in [7], investigated the dual hesitant bipolar fuzzy MCDM problems in which there exists a prioritization relationship over attributes and proposed dual hesitant bipolar fuzzy Hamacher prioritized geometric (DHBFHPG) operator, dual hesitant bipolar fuzzy Hamacher prioritized weighted average (DHBFHPWA) operator, dual hesiant bipolar fuzzy Hamacher prioritized weighted geometric (DHBFHPWG) operator.

In this article, we present the notion of the Pythagorean cubic fuzzy set (PCFS) which is the simplification of interval-valued Pythagorean fuzzy set based on the constraint that the square sum of supremum to its membership degrees is ⩽1. Here, we present the notion of the Pythagorean cubic fuzzy set (PCFS) which is the generality of the notion of interval valued Pythagorean fuzzy set. We study some properties of PCFS. We define score and deviation degree of the Pythagorean cubic fuzzy numbers (PCFNs) for the comparison of Pythagorean cubic fuzzy numbers. We also define distance measure between Pythagorean cubic fuzzy numbers. Pythagorean cubic fuzzy set hold the condition that the sum of square of supremum to its membership’s degrees is ⩽ 1. Based on this information, aggregation operators, namely, Pythagorean cubic fuzzy weighted averaging (PCFWA), Pythagorean cubic fuzzy weighted geometric (PCFWG), Pythagorean cubic fuzzy ordered weighted averaging (PCFOWA) and Pythagorean cubic fuzzy ordered weighted geometric (PCFOWG). Furthermore, these operators are then used for decision-making problems in which specialists offer their preferences in the Pythagorean cubic fuzzy information to show the practicality and effectiveness of the new approach.

2 Preliminaries

In this section, we introduce some elementary definition and their important properties.

Definition 1. [21] Let X be a universe of discourse, then a FS set F is an object having the following formulation: $F = {〈 x, μ_{\tilde{F}} (x) 〉 | x \in X},$ (1)

where $μ_{\tilde{F}}$ is a function from X to [0, 1], and $μ_{\tilde{F}} (x)$ is called the membership degree of x in X.

Definition 2. [1] Let X be a universe of discourse, then a IFS set I is an object having the following form: $I = {〈 x, μ_{\tilde{I}} (x), ν_{\tilde{I}} (x) 〉 | x \in X},$ (2)

where $μ_{\tilde{I}} (x)$ and $ν_{\tilde{I}} (x)$ is a function from X : → [0, 1], also hold $0 ⩽ μ_{\tilde{I}} (x) ⩽ 1, 0 ⩽ ν_{\tilde{I}} (x) ⩽ 1$ for all x in X, and denote the membership and nonmembership degree of element x in X to set I.

Definition 3. [42] Let U be a universal set. Then a cubic set can be defined as: $C = {〈 x, {\bar{μ}}_{c} (x), ν_{c} (x) 〉 | x \in X} .$

Where ${\bar{μ}}_{c}$ is an interval-valued fuzzy set in U and ν_c is a fuzzy set in U.

Definition 4. [37] Let X be a universe of discourse, then a PFS set P is an object having the following form: $P = {〈 x, μ_{\tilde{P}} (x), ν_{\tilde{P}} (x) 〉 | x \in X},$ (3)

where $μ_{\tilde{P}} (x)$ and $ν_{\tilde{P}} (x)$ is a function from X : → [0, 1], such that $0 ⩽ μ_{\tilde{P}} (x) ⩽ 1, 0 ⩽ ν_{\tilde{P}} (x) ⩽ 1$ , and $0 ⩽ μ_{\tilde{P}}^{2} (x) ⩽ 1, 0 ⩽ ν_{\tilde{P}}^{2} (x) ⩽ 1$ for all x in X, and denote the membership and nonmembership degree of element x in X to set P. Let $π_{\tilde{P}} (x) = \sqrt{1 - μ_{\tilde{P}}^{2} (x) - ν_{\tilde{P}}^{2} (x)}$ , 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. Also $0 ⩽ π_{\tilde{P}} (x) ⩽ 1$ for every x ∈ X. We denote the Pythagorean fuzzy number (PFN) by P = 〈Λ_P, Γ_P〉. To compare two Pythagorean fuzzy numbers, the authors presented the notion of score function, accuracy degree for them.

Definition 5. [37] Let P = 〈μ_P, λ_P〉 P₁ = 〈μ_{P
₁}, λ_{P
₁}〉, P₂ = 〈μ_{P
₂}, λ_{P
₂}〉 be three (PFNs), then we have

$P_{1} \oplus P_{2} = 〈 \sqrt{μ_{P_{1}}^{2} + μ_{P_{2}}^{2} - μ_{P_{1}}^{2} μ_{P_{2}}^{2}}, ν_{P_{1}} ν_{P_{2}} 〉$ ;

$P_{1} \otimes P_{2} = 〈 μ_{P_{1}} μ_{P_{2}}, \sqrt{ν_{P_{1}}^{2} + ν_{P_{2}}^{2} - ν_{P_{1}}^{2} ν_{P_{2}}^{2}} 〉;$

$λ P = 〈 \sqrt{1 - (1 - μ_{P}^{2})^{λ}}, (ν_{P})^{λ} 〉 (λ > 0)$ ;

$P^{λ} = 〈 (μ_{P})^{λ}, \sqrt{1 - (1 - ν_{P}^{2})^{λ}} 〉, (λ > 0);$

P^c = (ν_P, μ_P) .

Definition 6. [38] Let X be a universe of discourse, then a IVPFS set R can be defined as: $A = {〈 x, a_{A} (x), b_{A} (x) 〉 | x \in X},$ (4)

where $a_{\tilde{R}} (x) = [a_{\tilde{R}}^{L} (x), a_{\tilde{R}}^{U} (x)] \subset [0, 1],$ $b_{A} (x) = [b_{A}^{L} (x), b_{A}^{U} (x)] \subset [0, 1]$ are the intervals, and $a_{A}^{L} (x) = \inf a_{A} (x)$ and $a_{A}^{U} (x) = \sup a_{A} (x)$ , similarly $b_{A}^{L} (x) = \inf b_{A} (x)$ and $b_{A}^{U} (x) = \sup b_{A} (x)$ for all x in X. And also $0 ⩽ (a_{A}^{U} (x))^{2} + (b_{A}^{U} (x))^{2} ⩽ 1$ .

Let $π_{A} (x) = [π_{A}^{L} (x), π_{A}^{U} (x)]$ , for all x ∈ X, then it is called the interval-valued Pythagorean fuzzy index of x to A, where $π_{A}^{L} (x) = \sqrt{1 - (μ_{A}^{U} (x))^{2} - (ν_{A}^{U} (x))^{2}}$ , and $π_{A}^{U} (x) = \sqrt{1 - (μ_{A}^{L} (x))^{2} - (ν_{A}^{L} (x))^{2}}$ .

Which satisfy the following relation.

If $a_{A}^{L} (x) = a_{A}^{U} (x)$ and $b_{A}^{L} (x) = b_{A}^{U} (x)$ , then an IVPFS set reduces to PFS.

If $a_{A}^{U} (x) + b_{A}^{U} (x) ⩽ 1$ , then an IVPFS reduces to an IVIFS.

Definition 7. [19] Let $A = ([a_{A}^{L}, a_{A}^{U}], [b_{A}^{L}, b_{A}^{U}])$ are the IVPFN, we can defined the score functionof A in the following way: $S (A) = \frac{1}{2} [(a_{A}^{L})^{2} + (a_{A}^{U})^{2} - (b_{A}^{L})^{2} - (b_{A}^{U})^{2}]$

where S(A) ∈ [0,1].

Definition 8. [19] Let $A = 〈 [a_{A}^{L}, a_{A}^{U}], [b_{A}^{L}, b_{A}^{U}] 〉$ are the IVPFN, we can defined the accurracy function of A in the following way: $H (A) = \frac{1}{2} [(a_{A}^{L})^{2} + (a_{A}^{U})^{2} + (b_{A}^{L})^{2} + (b_{A}^{U})^{2}]$

where H(A) ∈ [0,1].

Definition 9. [19] Let $A = 〈 [a_{A}^{L}, a_{A}^{U}], [b_{A}^{L}, b_{A}^{U}] 〉$ and $A_{1} = 〈 [a_{A_{1}}^{L}, a_{A_{1}}^{U}], [b_{A_{1}}^{L}, b_{A_{1}}^{U}] 〉$ be two IVPFNs, then $\begin{matrix} S (A) = \frac{1}{2} [(a_{A}^{L})^{2} + (a_{A}^{U})^{2} - (b_{A}^{L})^{2} - (b_{A}^{U})^{2}], \\ S (A_{1}) = \frac{1}{2} [(a_{A_{1}}^{L})^{2} + (a_{A_{1}}^{U})^{2} - (b_{A_{1}}^{L})^{2} - (b_{A_{1}}^{U})^{2}] \end{matrix}$

Are the score of A and A₁, separately, while $\begin{matrix} H (A) = \frac{1}{2} [(a_{A}^{L})^{2} + (a_{A}^{U})^{2} + (b_{A}^{L})^{2} + (b_{A}^{U})^{2}], \\ H (A_{1}) = \frac{1}{2} [(a_{A_{1}}^{L})^{2} + (a_{A_{1}}^{U})^{2} + (b_{A_{1}}^{L})^{2} + (b_{A_{1}}^{U})^{2}] \end{matrix}$

Are the accuracy of A and A₁, separately, which satisfy the following properties:

If S (A) < S (A₁) then A < A₁

If S (A) = S (A₁), we have.

If H (A) = H (A₁), then A = A₁,

If H (A) < H (A₁), then A < A₁,

If H (A) > H (A₁), then A > A₁,

Definition 10. [4, 5] Let A_i = 〈 [a_i, b_i] , [c_i, d_i] 〉 (i = 1, 2) be the collection of IVPFNs, and δ > 0, then the following oprational las are satisfied: $\begin{matrix} \begin{matrix} δ A_{1} = 〈 [\begin{matrix} \sqrt{1 - (1 - (a_{1})^{2})^{δ},} \\ \sqrt{1 - (1 - (b_{1})^{2})^{δ}} \end{matrix}], [(c_{1})^{δ}, (d_{1})^{δ}] 〉, \\ (A_{1})^{δ} = 〈 \begin{matrix} [(a_{1})^{δ}, (b_{1})^{δ}], \\ [\sqrt{1 - (1 - (c_{1})^{2})^{δ},} \sqrt{1 - (1 - (d_{1})^{2})^{δ}}] \end{matrix} 〉 \\ A_{1} \otimes A_{2} = 〈 \begin{matrix} [a_{1} a_{2}, b_{1} b_{2}], \\ [\begin{matrix} \sqrt{(c_{1})^{2} + (c_{2})^{2} - (c_{1})^{2} (c_{2})^{2},} \\ \sqrt{(d_{1})^{2} + (d_{2})^{2} - (d_{1})^{2} (d_{2})^{2}} \end{matrix}] \end{matrix} 〉, \\ A_{1} \oplus A_{2} = 〈 \begin{matrix} [\begin{matrix} \sqrt{(a_{1})^{2} + (a_{2})^{2} - (a_{1})^{2} (a_{2})^{2},} \\ \sqrt{(b_{1})^{2} + (b_{2})^{2} - (b_{1})^{2} (b_{2})^{2}} \end{matrix}], \\ [c_{1} c_{2}, d_{1} d_{2}] \end{matrix} 〉 . \end{matrix} \end{matrix}$

Definition 11. [27] Let A_i = 〈 [a_i, b_i] , [c_i, d_i] 〉 (i = 1, 2, …, n) be the collection of IVPFNs, then IVPFWA oprator is defined as: $\begin{matrix} {IVPFWA}_{w} (A_{1}, A_{2}, \dots, A_{n}) \\ = 〈 \begin{matrix} [\sqrt{1 - \prod_{i = 1}^{n} (1 - (a_{i})^{2})^{w_{i}},} \sqrt{1 - \prod_{i = 1}^{n} (1 - (b_{i})^{2})^{w_{i}}}], \\ [\prod_{i = 1}^{n} (c_{i})^{w_{i}}, \prod_{i = 1}^{n} (d_{i})^{w_{i}}] \end{matrix} 〉 \end{matrix}$

where w = (w₁, w₂, …, w_n) ^T be the weight vector of p_i (i = 1, 2, 3 … , n) and w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ .

Definition 12. [27] Let A_i = 〈 [a_i, b_i] , [c_i, d_i] 〉 (i = 1, 2, …, n) be the collection of IVPFNs, then IVPFWG oprator is defined as: $\begin{matrix} {IVPFWG}_{w} (A_{1}, A_{2}, \dots, A_{n}) \\ = 〈 \begin{matrix} [\prod_{i = 1}^{n} (a_{i})^{w_{i}}, \prod_{i = 1}^{n} (b_{i})^{w_{i}}], \\ [\sqrt{1 - \prod_{i = 1}^{n} (1 - (c_{i})^{2})^{w_{i}},} \sqrt{1 - \prod_{i = 1}^{n} (1 - (d_{i})^{2})^{w_{i}}}] \end{matrix} 〉 \end{matrix}$

where w = (w₁, w₂, …, w_n) ^T be the weight vector of A_i (i = 1, 2, 3 … , n) and w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ .

3 Pythagorean cubic fuzzy numbers and there properties

In this section, we present the idea of some new concepts such as, Pythagorean cubic fuzzy set and discuss some properties with the help of an example for Pythagorean cubic fuzzy set which is not an intuitionistic cubic fuzzy set. Throughout in this paper a Pythagorean cubic fuzzy set will be denoted by P_C.

Definition 13. Let X be a fixed set, then a Pythagorean cubic fuzzy set can be defined as: $P_{C} = {〈 x, (Λ_{P_{C}} (x), Γ_{P_{C}} (x)) 〉 | x \in X},$ (5)

Where Λ_{P
_C} (x) =〈A_{P
_C} (x) ; λ_{P
_C} (x) 〉 = 〈 [a_{P
_C} (x) , |b_{P
_C} (x)] ; λ_{P
_C} (x) 〉 $Γ_{P_{C}} (x) = 〈 {\tilde{A}}_{P_{C}} (x); μ_{P_{C}} (x) 〉 = 〈 [{\tilde{a}}_{P_{C}}, {\tilde{b}}_{P_{C}}]; μ_{P_{C}} (x) 〉$ and $0 ⩽ (Λ_{P_{C}} (x))^{2} + (Γ_{P_{C}} (x))^{2} ⩽ [\tilde{1}, 1] .$

The above condition can be also written in the following way: $\begin{matrix} 0 ⩽ (\sup (A_{P_{C}} (x)))^{2} + (\sup ({\tilde{A}}_{P_{C}} (x)))^{2} ⩽ 1, \\ 0 ⩽ λ_{P_{C}}^{2} (x) + μ_{P_{C}}^{2} (x) ⩽ 1 . \end{matrix}$

The degree of indeterminacy for Pythagorean cubic set can be defined as: $\begin{matrix} π_{\hat{C}} = 〈 \sqrt{1 - (\sup (A_{P_{C}} (x)))^{2} - (\sup ({\tilde{A}}_{P_{C}} (x)))^{2}}; \\ \sqrt{1 - λ_{P_{C}}^{2} (x) - μ_{P_{C}}^{2} (x)} 〉 . \end{matrix}$

For simplicity we call $(〈 A; λ 〉, 〈 \tilde{A}; μ 〉)$ a Pythagorean cubic fuzzy number (PCFN) denoted by $p_{c} = (〈 A; λ 〉, 〈 \tilde{A}; μ 〉)$ .

Example 1. Let X = {x₁, x₂, x₃} be a fixed set and consider a set in X by $P_{C} = {\begin{matrix} (x_{1}, 〈 [0.5, 0.6]; 0.7), ([0.6, 0.7]; 0.5 〉), \\ (x_{2}, 〈 [0.4, 0.7]; 0.6 〉, 〈 [0.5, 0.8]; 0.6 〉), \\ (x_{3}, 〈 [0.4, 0.6], 0.7 〉, 〈 [0.5, 0.9]; 0.6 〉) \end{matrix}} .$

Then, $\hat{C}$ is a PCFS.

Definition 14. Let $p_{c} = (〈 A; λ 〉, 〈 \tilde{A}; μ 〉)$ , $p_{c_{1}} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ and $p_{c_{2}} = (〈 A_{2}; λ_{2} 〉, 〈 {\tilde{A}}_{2}; μ_{2} 〉)$ be 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], $\tilde{A} = [\tilde{a}, \tilde{b}]$ . Then the operations laws are: $\begin{matrix} 1) p_{c_{1}} \oplus p_{c_{2}} = (\begin{matrix} 〈 \begin{matrix} [\sqrt{a_{1}^{2} + a_{2}^{2} - a_{1}^{2} a_{2}^{2}}, \sqrt{b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2}}]; \\ \sqrt{λ_{1}^{2} + λ_{2}^{2} - λ_{1}^{2} λ_{2}^{2}} \end{matrix} 〉, \\ 〈 [{\tilde{a}}_{1} {\tilde{a}}_{2}, {\tilde{b}}_{1} {\tilde{b}}_{2}]; μ_{1} μ_{2} 〉 \end{matrix}), \\ 2) p_{c_{1}} \otimes p_{c_{2}} = (\begin{matrix} 〈 [a_{1} a_{2}, b_{1} b_{2}]; λ_{1} λ_{2} 〉, \\ 〈 \begin{matrix} [\sqrt{{\tilde{a}}_{1}^{2} + {\tilde{a}}_{2}^{2} - {\tilde{a}}_{1}^{2} {\tilde{a}}_{2}^{2}}, \sqrt{{\tilde{b}}_{1}^{2} + {\tilde{b}}_{2}^{2} - {\tilde{b}}_{1}^{2} {\tilde{b}}_{2}^{2}}]; \\ \sqrt{μ_{1}^{2} + μ_{2}^{2} - μ_{1}^{2} μ_{2}^{2}} \end{matrix} 〉 \end{matrix}), \\ 3) δ p_{c_{1}} = (\begin{matrix} 〈 \begin{matrix} [\sqrt{1 - (1 - a_{1}^{2})^{δ}}, \sqrt{1 - (1 - b_{1}^{2})^{δ}}]; \\ \sqrt{1 - (1 - λ_{1}^{2})^{δ}} \end{matrix} 〉, \\ 〈 [({\tilde{a}}_{1} {\tilde{a}}_{2})^{δ}, ({\tilde{b}}_{1} {\tilde{b}}_{2})^{δ}]; (μ_{1} μ_{2})^{δ} 〉 \end{matrix}), \\ 4) p_{c_{1}}^{δ} = (\begin{matrix} 〈 [(a_{1} a_{2})^{δ}, (b_{1} b_{2})^{δ}]; (λ_{1} λ_{2})^{δ} 〉, \\ 〈 \begin{matrix} [\sqrt{1 - (1 - {\tilde{a}}_{1}^{2})^{δ}}, \sqrt{1 - (1 - {\tilde{b}}_{1}^{2})^{δ}}]; \\ \sqrt{1 - (1 - μ_{1}^{2})^{δ}} \end{matrix} 〉 \end{matrix}), \\ 5) p_{c}^{'} = 〈 \tilde{A}; μ 〉, 〈 A; λ 〉 . \end{matrix}$

Theorem 1. Let $p_{c} = (〈 A; λ 〉, 〈 \tilde{A}; μ 〉)$ , $p_{c_{1}} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ and $p_{c_{2}} = (〈 A_{2}; λ_{2} 〉, 〈 {\tilde{A}}_{2}; μ_{2} 〉)$ be three PCFNs and δ > 0, δ₁ > 0andδ₂ > 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], $\tilde{A} = [\tilde{a}, \tilde{b}]$ then the following will hold:

p_c
₁ ⊕ p_c
₂ = p_c
₂ ⊕ p_c
₁

p_c
₁ ⊗ p_c
₂ = p_c
₂ ⊗ p_c
₁

δ (p_c
₁ ⊕ p_c
₂) = δ (p_c
₁) ⊕ δ (p_c
₂)

(δ₁ + δ₂) p_c = δ₁p_c ⊕ δ₂p_c

(p_c
₁ ⊗ p_c
₂) ^δ = (p_c
₁) ^δ ⊗ (p_c
₂) ^δ

$p_{c}^{(δ_{1} + δ_{2})} = p_{c}^{δ_{1}} \otimes p_{c}^{δ_{2}}$

Proof. Proof is very simple.

To compare two PCFNs, we define score function and there basic properties.

Definition 15. Let $p_{c} = (〈 A_{c}; λ_{c} 〉, 〈 {\tilde{A}}_{c}; μ_{c} 〉)$ be a PCFN, where A_c = [a_c, b_c], ${\tilde{A}}_{c} = [{\tilde{a}}_{c}, {\tilde{b}}_{c}]$ . We can introduce the score function of p_c as: $S (p_{c}) = {(\frac{a + b - λ}{3})}^{2} - {(\frac{\tilde{a} + \tilde{b} - μ}{3})}^{2} .$ (6)

Where S (p_c) ∈ [-1, 1].

Definition 16. Let $p_{c_{1}} = (〈 A_{1}; λ_{1} 〉, 〈 {\tilde{A}}_{1}; μ_{1} 〉)$ and $p_{c_{2}} = (〈 A_{2}; λ_{2} 〉, 〈 {\tilde{A}}_{2}; μ_{2} 〉)$ be two PCFNs, S (p_{c
₁}) be the score function of p_{c
₁} and S (p_{c
₂}) be the score function of p_{c
₂}. Then

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

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

If S (p_c
₁) = S (p_c
₂), then p_c
₁ ∼ p_c
₂.

Example 2. Letp_{c
₁} = (〈 [0.5, 0.7] ;0.9 〉 , 〈 [0.1, 0.5] ; 0.6 〉), p_{c
₂} = (〈 [0.4, 0.7] ;0.6 〉 , 〈 [0.2, 0.4] ;0.6 〉) and p_{c
₃} = (〈 [0.03, 0.8] ;0.9 〉 , 〈 [0.0, 0.3] ;0.7 〉) be three PCFNs. Then by Definition 20, we have, S (p_{c
₁}) =0.01, S (p_{c
₂}) =0.027 and S (p_{c
₃}) = -0.0173. Thus, S (p_{c
₂}) > S (p_{c
₁}) > S (p_{c
₃}).

Let p_{c
₁} = (〈 [0.5, 0.7] ;0.9 〉 , 〈 [0.1, 0.5] ;0.6 〉) and p_{c
₂} = (〈 [0.4, 0.7] ;0.6 〉 , 〈 [0.2, 0.4] ;0.7 〉) be two PCFNs. Then by Definition (15), we have, S (p_{c
₁}) =0.01 and S (p_{c
₂}) =0.01 Thus, S (p_{c
₁}) = S (p_{c
₂}). Therefore, by Definition (16) we cannot get information from p_{c
₁} and p_{c
₂}. Essentially, such a case has typically risen in preparation. From Definition (16), It is clear that it we cannot take the condition that two PCFNs in equal score. While the deviancy may be changed. The accuracy function of all the components to the average number in a PCFNs returns that they may be agree with each other. To improve this matter, we present the concept of accuracy degree for the compression of two PCFNs.

Definition 17. Let $p_{c} = (〈 A; λ 〉, 〈 \tilde{A}; μ 〉)$ be a PCFN. Then we define the accuracy degree of p_c is denoted by α (p_c), where A = [a, b], $\tilde{A} = [\tilde{a}, \tilde{b}]$ , can be defined as: $α (p_{c}) = {(\frac{a + b - λ}{3})}^{2} + {(\frac{\tilde{a} + \tilde{b} - μ}{3})}^{2} .$ (7)

Where α (p_c) ∈ [0, 1].

Definition 18. Let $p_{c_{1}} = (〈 A_{1}; λ_{1} 〉, 〈 {\tilde{A}}_{1}; μ_{1} 〉)$ and $p_{c_{2}} = (〈 A_{2}; λ_{2} 〉, 〈 {\tilde{A}}_{2}; μ_{2} 〉)$ be two PCFNs, α (p_{c
₁}) be the accuracy degree of p_{c
₁} and α (p_{c
₂}) be the accuracy degree of p_{c
₂}. Then

If α (p_c
₁) < α (p_c
₂), then p_c
₁ < p_c
₂.

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

If α (p_c
₁) = α (p_c
₂), then p_c
₁ ∼ p_c
₂.

Example 3. From Example 2, since S (p_{c
₁}) =0.01 and S (p_{c
₂}) =0.01 Thus, S (p_{c
₁}) = S (p_{c
₂}). So we have α (p_{c
₁}) =0.01 and α (p_{c
₂}) =0.044. Thus α (p_{c
₁}) > α (p_{c
₂}). Hence p_{c
₁} > p_{c
₂}. Which clearing the situation that two PCFNs have the same score.

Definition 19. Let p_{c
₁} and p_{c
₂} be any two PCFNs on a set X. The distance measure between p_{c
₁} and p_{c
₂} can be defined in the following way: $d (p_{c_{1}}, p_{c_{2}}) = \frac{1}{6} [\begin{matrix} | a_{1}^{2} - a_{2}^{2} | + | b_{1}^{2} - b_{2}^{2} | + \\ | {\tilde{a}}_{1}^{2} - {\tilde{a}}_{2}^{2} | + | {\tilde{b}}_{1}^{2} - {\tilde{b}}_{2}^{2} | + \\ | λ_{1}^{2} - λ_{2}^{2} | + | μ_{1}^{2} - μ_{2}^{2} | \end{matrix}]$ (8)

Example 4. Let p_{c
₁} = (〈[0.6, 0.7] ;0.3〉, 〈[0.5, 0.7] ; 0.8〉) and p_{c
₂} (〈[0.5, 0.6] ;0.4〉, 〈[0.4, 0.7] ;0.5〉) be two PCFNs. Then $\begin{matrix} D (p_{c_{1}}, p_{c_{2}}) = \frac{1}{6} [\begin{matrix} | 0.36 - 0.25 | + | 0.49 - 0.36 | + \\ | 0.25 - 0.16 | + | 0.49 - 0.49 | + \\ | 0.9 - 0.16 | + | 0.64 - 0.25 | \end{matrix}] \\ D (p_{c_{1}}, p_{c_{2}}) = \frac{1}{6} [0.11 + 0.13 + 0.09 + 0.74 + 0.39] \\ = 0.2433 . \end{matrix}$

4 Aggregation operators for Pythagorean cubic fuzzy environment

In this section, we suggest some aggregation operators for Pythagorean cubic fuzzy numbers, and examine some of its properties.

Definition 20. Let $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ (i = 1, 2, …, n) be a collection of all PCFNs, while w = (w₁, w₂, …, w_n) ^T be the weight vector of p_{c
_i} (i = 1, 2, …, n) with w_i ⩾ 0 (i = 1, 2, 3 … , n) where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then Pythagorean cubic fuzzy weighted averaging (PCFWA) operator is a mapping PCFWA : PCFNⁿ → PCFN defined by $\begin{matrix} PCFWA (p_{c_{1}}, p_{c_{2}}, . . ., p_{c_{n}}) \\ = w_{1} p_{c_{1}} \oplus w_{2} p_{c_{2}} \oplus, . . ., \oplus w_{n} p_{c_{n}} \end{matrix}$ (9)

where PCFWA is called Pythagorean cubic fuzzy weighted averaging (PCFWA) operator.

Definition 21. Let $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ (i = 1, 2, …, n) be a collection of all PCFNs, while w = (w₁, w₂, …, w_n) ^T be the weight vector of p_{c
_i} (i = 1, 2, …, n) with w_i ⩾ 0 (i = 1, 2, 3 … , n) where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then Pythagorean cubic fuzzy weighted geometric (PCFWG) operator is a mapping PCFWG : PCFNⁿ → PCFN $\begin{matrix} PCFWG (p_{c_{1}}, p_{c_{2}}, . . ., p_{c_{n}}) \\ = p_{c_{1}}^{w_{1}} \otimes p_{c_{2}}^{w_{2}} \otimes, . . ., \otimes p_{c_{n}}^{w_{n}} \end{matrix}$ (10)

where PCFWG is called Pythagorean cubic fuzzy weighted geometric (PCFWG) operator.

Theorem 2. Let $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ (i = 1, 2, …, n) be a collection of all PCFNs, and w = (w₁, w₂, …, w_n) ^T be the weight vector of p_{c
_i} (i = 1, 2, …, n) with w_i ⩾ 0 where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the aggregation result using PCFWA/PCFWG operator is also a PCFN and $\begin{matrix} PCFWA (p_{c_{1}}, p_{c_{2}}, . . ., p_{c_{n}}) = \\ (\begin{matrix} 〈 \begin{matrix} [\sqrt{1 - \prod_{i = 1}^{n} (1 - a_{i}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - b_{i}^{2})^{w_{i}}}]; \\ \sqrt{1 - \prod_{i = 1}^{n} (1 - λ_{i}^{2})^{w_{i}}} \end{matrix} 〉, \\ 〈 [\prod_{i = 1}^{n} {\tilde{a}}_{i}^{w_{i}}, \prod_{i = 1}^{n} {\tilde{b}}_{i}^{w_{i}}]; \prod_{i = 1}^{n} μ_{i}^{w_{i}} 〉 \end{matrix}) \end{matrix}$ (11)

and $\begin{matrix} PCFWG (p_{c_{1}}, p_{c_{2}}, . . ., p_{c_{n}}) = \\ (\begin{matrix} 〈 [\prod_{i = 1}^{n} a_{i}^{w_{i}}, \prod_{i = 1}^{n} b_{i}^{w_{i}}]; \prod_{i = 1}^{n} λ_{i}^{w_{i}} 〉, \\ 〈 \begin{matrix} [\sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{a}}_{i}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{b}}_{i}^{2})^{w_{i}}}]; \\ \sqrt{1 - \prod_{i = 1}^{n} (1 - μ_{i}^{2})^{w_{i}}} \end{matrix} 〉 \end{matrix}) \end{matrix}$ (12)

Proof. Straightforward similar proof.

Theorem 3. Let $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ (i = 1, 2, …, n) be a group of all PCFNs, and w = (w₁, w₂, …, w_n) ^T be the weight vector of p_{c
_i} with where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then we have

(Idempotency): If all p_{c
_i} are equal, i.e., p_{c
_i} = p_c, for all I. Then

$PCFWA / PCFWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) = p_{c} .$ (13)

(Boundary):

$p_{c_{min}} ⩽ PCFWA / PCFWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) ⩽ p_{c_{max}},$ (14)

for all w. where $\begin{matrix} p_{c_{min}} = (min_{j} (μ_{i}), max_{j} (ν_{i})), \\ p_{c_{max}} = (max_{j} (μ_{i}), min_{j} (ν_{i})) . \end{matrix}$

(Monotonicity): $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ and $p_{c_{i}}^{*} = (〈 A_{i}^{*}; λ_{i}^{*} 〉, 〈 {\tilde{A}}_{i}^{*}; μ_{i}^{*} 〉)$ (i = 1, 2, 3, …, n) be the groups of pythagorean cubic fuzzy values,if $A_{i} ⩽ A_{i}^{*}$ and λ_i ⩾ λ^*_i, ${\tilde{A}}_{i} ⩽ {\tilde{A}}_{i}^{*}$ and $μ_{i} ⩾ μ_{i}^{*}$ for all i. Then $\begin{matrix} PCFWA / PCFWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) \\ ⩽ PCFWA / PCFWG (p_{c}^{*_{1}}, p_{c}^{*_{2}}, \dots, p_{c}^{*_{n}}) \end{matrix}$ (15)

for every w. Similarly for PCFWG operator, we have.

Proof. Proof of the Theorem follows from Theorm [2,3, 2,3] of [29], and Theorem [4] of [3].

Lemma 1. Let p_{c
_i} > 0, w_i > 0 (i = 1, 2, … ; , n) and $\sum_{i = 1}^{n} w_{i} = 1$ . Then $\prod_{i = 1}^{n} (p_{c_{i}})^{w_{i}} ⩽ \sum_{i = 1}^{n} w_{i} p_{c_{i}} .$ (16)

Where the equality holds if and only if p_{c
₁} = p_{c
₂} = p_{c
₃} = ⋯ = p_{c
_n} .

Theorem 4. Let $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ (i = 1, 2, …, n) be a collection of all PCFNs, then $PCFWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) ⩽ PCFWA (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) .$

Where w = (w₁, w₂, …, w_n) ^T is the weighted vector of p_{c
_i} (i = 1, 2, 3, …, n) ∃ w_i ∈ [0, 1] (i = 1, 2, 3, …, n) and $\sum_{i = 1}^{n} w_{i} = 1$ .

Proof. Based on Lemma 1, we can easily prove the Theorem.

Definition 22. Let $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ (i = 1, 2, …, n) be a group of all PCFNs, while w = (w₁, w₂, …, w_n) ^T be the weight vector of p_{c
_i} (i = 1, 2, …, n) with w_i ⩾ 0 (i = 1, 2, …, n) where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then (PCFOWA) operator is a mapping PCFOWA : PCFNⁿ → PCFN $\begin{matrix} PCFOWA (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) \\ = w_{1} p_{c_{1}} \oplus w_{2} p_{c_{2}} \oplus, \dots, \oplus w_{n} p_{c_{n}} \end{matrix}$ (17)

and the PCFOWA operator is said to be a Pythagorean cubic fuzzy ordered weighted averaging operator.

Definition 23. Let $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ (i = 1, 2, …, n) be a group of all PCFNs, while w = (w₁, w₂, …, w_n) ^T be the weight vector of p_{c
_i} (i = 1, 2, …, n) with w_i ⩾ 0 (i = 1, 2, …, n) where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then the Pythagorean cubic fuzzy (PCFOWG) operator is a mapping PCFOWG : PCFNⁿ → PCFN can be defined as $\begin{matrix} PCFOWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) \\ = p_{c_{1}}^{w_{1}} \otimes p_{c_{2}}^{w_{2}} \otimes, \dots, \otimes p_{c_{n}}^{w_{n}} \end{matrix}$ (18)

and the PCFOWG operator is said to be a Pythagorean cubic fuzzy ordered weighted geometric operator.

Theorem 5. Let $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ (i = 1, 2, 3, …, n) be a collection of all PCFNs, and w = (w₁, w₂, …, w_n) ^T be the weight vector of p_{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 $\begin{matrix} PCFOWA (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) = \\ (\begin{matrix} 〈 \begin{matrix} [\sqrt{1 - \prod_{i = 1}^{n} (1 - a_{σ (i)}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - b_{σ (i)}^{2})^{w_{i}}}]; \\ \sqrt{1 - \prod_{i = 1}^{n} (1 - λ_{σ (i)}^{2})^{w_{i}}} \end{matrix} 〉, \\ 〈 [\prod_{i = 1}^{n} {\tilde{a}}_{σ (i)}^{w_{i}}, \prod_{i = 1}^{n} {\tilde{b}}_{σ (i)}^{w_{i}}]; \prod_{i = 1}^{n} μ_{σ (i)}^{w_{i}} 〉 \end{matrix}) \end{matrix}$ (19)

And the aggregation result using PCFOWG operator is also a PCFN and $\begin{matrix} PCFOWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) = \\ (\begin{matrix} 〈 [\prod_{i = 1}^{n} a_{σ (i)}^{w_{i}}, \prod_{i = 1}^{n} b_{σ (i)}^{w_{i}}], \prod_{i = 1}^{n} λ_{σ (i)}^{w_{i}} 〉, \\ 〈 \begin{matrix} [\sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{a}}_{σ (i)}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{b}}_{σ (i)}^{2})^{w_{i}}}], \\ \sqrt{1 - \prod_{i = 1}^{n} (1 - μ_{σ (i)}^{2})^{w_{i}}} \end{matrix} 〉 \end{matrix}) \end{matrix}$ (20)

Proof. Similar proof.

Theorem 6. Let $p_{c_{i}} = (〈 A_{i}; λ_{i} 〉, 〈 {\tilde{A}}_{i}; μ_{i} 〉)$ (i = 1, 2, 3, …, n) be a group of all PCFNs, and w = (w₁, w₂, …, w_n) ^T be the weight vector of p_{c
_i} with w_i ⩾ 0 where w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then we have

(Idempotency): If all p_{c
_i} are equal, i.e., p_{c
_i} = p_c, for all i. Then $PCFOWA / PCFOWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) = p_{c} .$ (21)

(Boundary): $p_{c_{min}} ⩽ PCFOWA / PCFOWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) ⩽ p_{c_{max}},$ (22)

for all w, where $\begin{matrix} p_{c_{min}} = (min_{j} (μ_{i}), max_{j} (ν_{i})), \\ p_{c_{max}} = (max_{j} (μ_{i}), min_{j} (ν_{i})) . \end{matrix}$

(Monotonicity): p_{c
_i} = 〈μ_i, ν_i〉, and $p_{c}^{*_{i}} = p_{c_{i}} = 〈 μ^{*_{i}}, ν^{*_{i}} 〉$ , (i = 1, 2, 3, … . , n) be the groups of pythagorean cubic fuzzy values, if μ_i ⩽ μ^*_i and ν_i ⩾ ν^*_i for all i. Then $\begin{matrix} \begin{matrix} PCFOWA / PCFOWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) ⩽ \\ PCFOWA / PCFOWG (p_{c}^{*_{1}}, p_{c}^{*_{2}}, \dots, p_{c}^{*_{n}}) \end{matrix} \end{matrix}$

for every w.

Proof. Straightforwad.

Lemma 2. Let p_{c
_i} > 0, w_i > 0 (i = 1, 2, 3, …, n) and $\sum_{i = 1}^{n} w_{i} = 1$ . Then $\prod_{i = 1}^{n} (p_{c_{i}})^{w_{i}} ⩽ \sum_{i = 1}^{n} w_{i} p_{c_{i}} .$ (23)

Where the equality holds if and only if p_{c
₁} = p_{c
₂} = p_{c
₃} = ⋯ = p_{c
_n}.

Theorem 7. Let p_{c
_i} = 〈μ_{c
_ij}, ν_{c
_ij}〉, (i = 1, 2, 3, …, n) be a collection of all PCFNs, then $\begin{matrix} PCFOWG (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) \\ ⩽ PCFOWA (p_{c_{1}}, p_{c_{2}}, \dots, p_{c_{n}}) . \end{matrix}$ (24)

Where w = (w₁, w₂, …, w_n) ^T is the weighted vector of p_{c
_i} (i = 1, 2, 3, …, n) ∃ w_i ∈ [0, 1] (i = 1, 2, 3 … , n) and $\sum_{i = 1}^{n} w_{i} = 1$ .

Proof. Based on Lemma 2, we can easily prove the Theorem.

5 Application to multi-criteria decision-making under on Pythagorean cubic fuzzy aggregation operators

In this section, we use the Pythagorean cubic fuzzy aggregation operators to multi-criteria decision-making process. Assume, we have an n options X ={ x_1,x₂, …, x_n } and m criteria A ={ A_1,A₂, …, A_m } to be evaluated having weight vector w = (w₁, w₂, …, w_m) ^T ∃ w_j ∈ [0, 1], and $\sum_{j = 1}^{m} w_{j} = 1$ . To assess the performance of the alternative x_i under the criteria A_j, the decision makers are required to convey not only the information that the alternative x_i satisfy the criteria A_j, but also the alternative x_i does not satisfy the criteria A_j. These two part information can be stated by 〈A_ij ; λ_ij〉 and $〈 {\tilde{A}}_{ij}; μ_{ij} 〉$ which represent the degrees that the alternative x_i satisfy the criterion A_j and does not satisfy the criterion A_j, then the performance of the alternative x_i under the criteria A_j can be stated by an PCFN $p_{c_{ij}} = (〈 A_{ij}; λ_{ij} 〉, 〈 {\tilde{A}}_{ij}; μ_{ij} 〉)$ with the condition that for all A_{c
_ij} ∈ p_{c
_ij}, ∃ ${\tilde{A}}_{c_{ij}} \in p_{c_{ij}}^{'}$ such that $0 ⩽ {(A_{c_{ij}})}^{2} + {({\tilde{A}}_{c_{ij}})}^{2} ⩽ 1$ , i = 1, 2, …, n, j = 1, 2, …, m and k = 1, 2, …, t. To get the ranking of the alternatives, the following steps are stated:

Step 1. First, First we make the Pythagorean cubic fuzzy decision matrix C = (p_{c
_ij}) _m×n where $p_{c_{ij}} = (〈 A_{ij}; λ_{ij} 〉, 〈 {\tilde{A}}_{ij}; μ_{ij} 〉)$ (i = 1, 2, … , n ; j = 1, 2, …, m). If the criteria have two kinds, such as cost and benefit criteria. Then the Pythagorean cubic decision matrix can be changed into the normalized Pythagorean fuzzy decision-matrix is $D_{N} = {[(〈 A_{ij}; λ_{ij} 〉, 〈 {\tilde{A}}_{ij}; μ_{ij} 〉)]}_{m \times n},$

where $β_{c_{ij}} = {\begin{matrix} p_{c_{ij}} if the attribute is of benefit type \\ p_{c_{ij}}^{c} if the attribute is of cost type \end{matrix}$ where $p_{c_{ij}}^{c} = (〈 {\tilde{A}}_{ij}; μ_{ij} 〉, 〈 A_{ij}; λ_{ij} 〉)$ . If all the criteria are of same type than there is not needed to normalized the decision matrix.

Step 2. Use the proposed aggregation operators to obtain the PCFN p_{c
_i} (i = 1, 2, …, n) for the alternatives X_i. That is the developed operators to stem the collective overall preference values p_{c
_i} (i = 1, 2, …, n) of the alternative x_i, where w = (w₁, w₂, …, w_n) ^T is the weighting vector of the criteria.

Step 3. Use Equation (8) we compute the scores S (p_{c
_i}) (i = 1, 2, …, n) and the deviation degree $\bar{α} (p_{c_{i}})$ of all the overall values p_{c
_i}.

Step 4. Rank all the alternatives and select the best one(s).

Table 1
Pythagorean cubic fuzzy decision matrix C by decision maker D₁

A₁ A₂ A₃ A₄

X₁ $(\begin{matrix} 〈 [0.6, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.6, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.3, 0.4]; 0.6 〉, \\ 〈 [0.8, 0.9]; 0.7 〉 \end{matrix})$

X₂ $(\begin{matrix} 〈 [0.5, 0.7]; 0.5 〉, \\ 〈 [0.4, 0.5]; 0.6 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.3, 0.4]; 0.8 〉, \\ 〈 [0.8, 0.9]; 0.6 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.5, 0.7]; 0.8 〉, \\ 〈 [0.4, 0.6]; 0.6 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.6, 0.7]; 0.8 〉, \\ 〈 [0.5, 0.6]; 0.5 〉 \end{matrix})$

X₃ $(\begin{matrix} 〈 [0.5, 0.6]; 0.8 〉, \\ 〈 [0.4, 0.7]; 0.6 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.6, 0.7]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.6, 0.8]; 0.2 〉, \\ 〈 [0.5, 0.6]; 0.9 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.6, 0.8]; 0.7 〉, \\ 〈 [0.5, 0.6]; 0.6 〉 \end{matrix})$

X₄ $(\begin{matrix} 〈 [0.6, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.6, 0.7]; 0.9 〉, \\ 〈 [0.5, 0.6]; 0.4 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.5, 0.7]; 0.7 〉, \\ 〈 [0.4, 0.6]; 0.6 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.4, 0.5]; 0.8 〉, \\ 〈 [0.7, 0.8]; 0.6 〉 \end{matrix})$

X₅ $(\begin{matrix} 〈 [0.8, 0.9]; 0.3 〉, \\ 〈 [0.3, 0.4]; 0.9 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.7, 0.8]; 0.9 〉, \\ 〈 [0.5, 0.6]; 0.3 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.6, 0.8]; 0.5 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$ $(\begin{matrix} 〈 [0.6, 0.7]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$

	A₁	A₂	A₃	A₄
X₁	$(\begin{matrix} 〈 [0.6, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.3, 0.4]; 0.6 〉, \\ 〈 [0.8, 0.9]; 0.7 〉 \end{matrix})$
X₂	$(\begin{matrix} 〈 [0.5, 0.7]; 0.5 〉, \\ 〈 [0.4, 0.5]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.3, 0.4]; 0.8 〉, \\ 〈 [0.8, 0.9]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.7]; 0.8 〉, \\ 〈 [0.4, 0.6]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.8 〉, \\ 〈 [0.5, 0.6]; 0.5 〉 \end{matrix})$
X₃	$(\begin{matrix} 〈 [0.5, 0.6]; 0.8 〉, \\ 〈 [0.4, 0.7]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.2 〉, \\ 〈 [0.5, 0.6]; 0.9 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.7 〉, \\ 〈 [0.5, 0.6]; 0.6 〉 \end{matrix})$
X₄	$(\begin{matrix} 〈 [0.6, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.9 〉, \\ 〈 [0.5, 0.6]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.7]; 0.7 〉, \\ 〈 [0.4, 0.6]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.8 〉, \\ 〈 [0.7, 0.8]; 0.6 〉 \end{matrix})$
X₅	$(\begin{matrix} 〈 [0.8, 0.9]; 0.3 〉, \\ 〈 [0.3, 0.4]; 0.9 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.9 〉, \\ 〈 [0.5, 0.6]; 0.3 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.5 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$

Table 2

Pythagorean cubic fuzzy decision matrix C by decision maker D₂

	A₁	A₂	A₃	A₄
X₁	$(\begin{matrix} 〈 [0.4, 0.6]; 0.5 〉, \\ 〈 [0.7, 0.8]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.3, 0.6]; 0.6 〉, \\ 〈 [0.7, 0.8]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.8, 0.9]; 0.5 〉, \\ 〈 [0.3, 0.4]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.4, 0.5]; 0.5 〉 \end{matrix})$
X₂	$(\begin{matrix} 〈 [0.5, 0.6]; 0.8 〉, \\ 〈 [0.7, 0.8]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.8 〉, \\ 〈 [0.7, 0.9]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.4 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.3, 0.6]; 0.5 〉 \end{matrix})$
X₃	$(\begin{matrix} 〈 [0.7, 0.8]; 0.4 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.7 〉, \\ 〈 [0.7, 0.8]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.8]; 0.7 〉, \\ 〈 [0.6, 0.7]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.6]; 0.4 〉, \\ 〈 [0.6, 0.7]; 0.6 〉 \end{matrix})$
X₄	$(\begin{matrix} 〈 [0.7, 0.8]; 0.7 〉, \\ 〈 [0.4, 0.5]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.5 〉, \\ 〈 [0.4, 0.5]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.7 〉, \\ 〈 [0.5, 0.7]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.7 〉, \\ 〈 [0.7, 0.8]; 0.5 〉 \end{matrix})$
X₅	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.4, 0.5]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.7 〉, \\ 〈 [0.5, 0.6]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.5]; 0.7 〉, \\ 〈 [0.5, 0.7]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.5 〉, \\ 〈 [0.4, 0.5]; 0.6 〉 \end{matrix})$

Table 3

Pythagorean cubic fuzzy decision matrix C by decision maker D₃

	A₁	A₂	A₃	A₄
X₁	$(\begin{matrix} 〈 [0.6, 0.7]; 0.7 〉, \\ 〈 [0.5, 0.6]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.5 〉, \\ 〈 [0.7, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.6]; 0.8 〉, \\ 〈 [0.6, 0.7]; 0.5 〉 \end{matrix})$
X₂	$(\begin{matrix} 〈 [0.6, 0.8]; 0.2 〉, \\ 〈 [0.4, 0.5]; 0.9 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.4 〉, \\ 〈 [0.7, 0.8]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.8 〉, \\ 〈 [0.5, 0.6]; 0.3 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.8 〉, \\ 〈 [0.5, 0.6]; 0.4 〉 \end{matrix})$
X₃	$(\begin{matrix} 〈 [0.7, 0.8]; 0.4 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.4 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.8 〉, \\ 〈 [0.5, 0.6]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.7 〉, \\ 〈 [0.4, 0.6]; 0.4 〉 \end{matrix})$
X₄	$(\begin{matrix} 〈 [0.6, 0.7]; 0.7 〉, \\ 〈 [0.5, 0.6]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.5 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.7 〉, \\ 〈 [0.4, 0.5]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.9 〉, \\ 〈 [0.4, 0.5]; 0.3 〉 \end{matrix})$
X₅	$(\begin{matrix} 〈 [0.5, 0.6]; 0.6 〉, \\ 〈 [0.6, 0.7]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.5 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.4 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.5 〉, \\ 〈 [0.4, 0.6]; 0.7 〉 \end{matrix})$

Table 4

Ordered Pythagorean cubic fuzzy decision matrix C by decision maker D₁

	A₁	A₂	A₃	A₄
X₁	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.3, 0.4]; 0.6 〉, \\ 〈 [0.8, 0.9]; 0.7 〉 \end{matrix})$
X₂	$(\begin{matrix} 〈 [0.5, 0.7]; 0.5 〉, \\ 〈 [0.4, 0.5]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.6]; 0.8 〉, \\ 〈 [0.4, 0.6]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.8 〉, \\ 〈 [0.5, 0.6]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.3, 0.4]; 0.8 〉, \\ 〈 [0.8, 0.9]; 0.6 〉 \end{matrix})$
X₃	$(\begin{matrix} 〈 [0.6, 0.8]; 0.2 〉, \\ 〈 [0.5, 0.6]; 0.9 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.7 〉, \\ 〈 [0.5, 0.6]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.6]; 0.8 〉, \\ 〈 [0.4, 0.7]; 0.6 〉 \end{matrix})$
X₄	$(\begin{matrix} 〈 [0.6, 0.8]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.7]; 0.7 〉, \\ 〈 [0.4, 0.6]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.9 〉, \\ 〈 [0.5, 0.6]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.8 〉, \\ 〈 [0.7, 0.8]; 0.6 〉 \end{matrix})$
X₅	$(\begin{matrix} 〈 [0.8, 0.9]; 0.3 〉, \\ 〈 [0.3, 0.4]; 0.9 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.5 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.6 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.9 〉, \\ 〈 [0.5, 0.6]; 0.3 〉 \end{matrix})$

Table 5

Ordered Pythagorean cubic fuzzy decision matrix C by decision maker D₂

	A₁	A₂	A₃	A₄
X₁	$(\begin{matrix} 〈 [0.8, 0.9]; 0.5 〉, \\ 〈 [0.3, 0.4]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.4, 0.5]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.6]; 0.5 〉, \\ 〈 [0.7, 0.8]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.3, 0.6]; 0.6 〉, \\ 〈 [0.7, 0.8]; 0.7 〉 \end{matrix})$
X₂	$(\begin{matrix} 〈 [0.7, 0.8]; 0.4 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.3, 0.6]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.6]; 0.8 〉, \\ 〈 [0.7, 0.8]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.8 〉, \\ 〈 [0.7, 0.9]; 0.5 〉 \end{matrix})$
X₃	$(\begin{matrix} 〈 [0.7, 0.8]; 0.4 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.6]; 0.4 〉, \\ 〈 [0.6, 0.7]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.8]; 0.7 〉, \\ 〈 [0.6, 0.7]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.7 〉, \\ 〈 [0.7, 0.8]; 0.5 〉 \end{matrix})$
X₄	$(\begin{matrix} 〈 [0.7, 0.8]; 0.5 〉, \\ 〈 [0.4, 0.5]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.7 〉, \\ 〈 [0.4, 0.5]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.7 〉, \\ 〈 [0.5, 0.7]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.7 〉, \\ 〈 [0.7, 0.8]; 0.5 〉 \end{matrix})$
X₅	$(\begin{matrix} 〈 [0.7, 0.8]; 0.5 〉, \\ 〈 [0.4, 0.5]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.4, 0.5]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.7 〉, \\ 〈 [0.5, 0.6]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.5]; 0.7 〉, \\ 〈 [0.5, 0.7]; 0.6 〉 \end{matrix})$

Table 6

Ordered Pythagorean cubic fuzzy decision matrix C by decision maker D₃

	A₁	A₂	A₃	A₄
X₁	$(\begin{matrix} 〈 [0.7, 0.8]; 0.6 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.7 〉, \\ 〈 [0.5, 0.6]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.5 〉, \\ 〈 [0.7, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.6]; 0.8 〉, \\ 〈 [0.6, 0.7]; 0.5 〉 \end{matrix})$
X₂	$(\begin{matrix} 〈 [0.6, 0.8]; 0.2 〉, \\ 〈 [0.4, 0.5]; 0.9 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.8 〉, \\ 〈 [0.5, 0.6]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4, 0.5]; 0.4 〉, \\ 〈 [0.7, 0.8]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.8 〉, \\ 〈 [0.5, 0.6]; 0.3 〉 \end{matrix})$
X₃	$(\begin{matrix} 〈 [0.7, 0.8]; 0.4 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.4 〉, \\ 〈 [0.5, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.7]; 0.7 〉, \\ 〈 [0.4, 0.6]; 0.4 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.8 〉, \\ 〈 [0.5, 0.6]; 0.4 〉 \end{matrix})$
X₄	$(\begin{matrix} 〈 [0.7, 0.8]; 0.5 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.7 〉, \\ 〈 [0.4, 0.5]; 0.5 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.7 〉, \\ 〈 [0.5, 0.6]; 0.6 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.8]; 0.9 〉, \\ 〈 [0.4, 0.5]; 0.3 〉 \end{matrix})$
X₅	$(\begin{matrix} 〈 [0.7, 0.8]; 0.4 〉, \\ 〈 [0.4, 0.5]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.7, 0.8]; 0.5 〉, \\ 〈 [0.4, 0.6]; 0.7 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6, 0.7]; 0.5 〉, \\ 〈 [0.5, 0.6]; 0.8 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5, 0.6]; 0.6 〉, \\ 〈 [0.6, 0.7]; 0.5 〉 \end{matrix})$

Table 7

Comparative study and ranking of the alternatives

Overall Score Values of the Alternatives
Methods	A₁	A₂	A₃	A₄	A₅	Ranking
PCFWA	0.0186	−0.0092	0.0165	0.0063	0.0568	X₅ > X₁ > X₃ > X₄ > X₂
PCFWG	−0.0027	−0.0163	0.0353	0.0028	0.0656	X₅ > X₃ > X₄ > X₁ > X₂
PCFOWA	0.0160	−0.0164	0.0183	0.0050	0.0510	X₅ > X₃ > X₁ > X₄ > X₂
PCFOWG	−0.0063	−0.0210	0..0297	−0.0011	0.0587	X₅ > X₃ > X₄ > X₁ > X₂

6 Numerical example

Suppose there is an investment company, which wants to invest a sum of money in the best choice (alternative). There is a board with five possible choices (alternatives) to invest the money: {X₁, X₂, X₃, X₄, X₅}. X₁ is a car company, X₂ is a food company, X₃ is a computer company, X₄ is an arms company and X₅ is a TV company.

Four criteria are taken into account, including the risk analysis (A₁), the growth analysis (A₂), the social-political impact analysis (A₃) and the environmental impact analysis (A₄). w = (0.21, 0.26, 0.23, 0.3) ^T is the weight vector of the criteria A_j (j = 1, 2, …, n). A committee of three decision-makers assesses the five possible alternatives X_i (i = 1, 2, 3, 4, 5) under above four criteria A_j (j = 1, 2, 3, 4). Then, to evaluate the investment company {X₁, X₂, X₃, X₄, X₅}, the ranking is required. These experts provide the decision matrices in the form of Pythagorean cubic fuzzy values are represented by the following matrices:

For Pythagorean Cubic Fuzzy Weighted averaging aggregation Operator.

Step 1. The decision-makers gives his decision in Tables 1–3.

Step 2. Utilize the PCFWA aggregation operator in Equation (11), we get the collective PCFN for the alternatives X_i. $\begin{matrix} \begin{matrix} C_{1} = (〈 [0.5857, 0.7059]; 0.6284 〉, 〈 [0.5375, 0.6238]; 0.6393 〉) \\ C_{2} = (〈 [0.5534, 0.6822]; 0.7014 〉, 〈 [0.5066, 0.6429]; 0.5429 〉) \\ C_{3} = (〈 [0.5727, 0.7253]; 0.6282 〉, 〈 [0.5106, 0.6155]; 0.5780 〉) \\ C_{4} = (〈 [0.6189, 0.7326]; 0.7528 〉, 〈 [0.4736, 0.5891]; 0.5133 〉) \\ C_{5} = (〈 [0.6660, 0.7657]; 0.6142 〉, 〈 [0.4533, 0.5719]; 0.6289 〉) \end{matrix} \end{matrix}$

Step 3. Use Equation (8) we compute the scores S (p_{c
_i}) of p_{c
_i} (i = 1, 2, 3, 4, 5) which is. $\begin{matrix} S (C_{1}) = 0.0186; S (C_{2}) = - 0.0092; \\ S (C_{3}) = 0.0165; S (C_{4}) = 0.0063; S (C_{5}) = 0.0568 \end{matrix}$

Step 4. Place the scores in the form of descending order and pick that alternative, which is the highest. Which is C₅ > C₁ > C₃ > C₄ > C₂. Hence X₅ > X₁ > X₃ > X₄ > X₂. Thus X₅ is the best option.

For Pythagorean Cubic Fuzzy Weighted Geometric aggregation Operator

Step 1. The decision-makers provides his judgement in Tables 1–3.

Step 2. Utilize the PCFWG aggregation operator in Equation (12), we collect overall preference values which as, $\begin{matrix} \begin{matrix} C_{1} = (〈 [0.5207, 0.6623]; 0.6064 〉, 〈 [0.5929, 0.6816]; 0.6772 〉) \\ C_{2} = (〈 [0.5170, 0.6414]; 0.6019 〉, 〈 [0.5825, 0.7066]; 0.6138 〉) \\ C_{3} = (〈 [0.5748, 0.7041]; 0.5347 〉, 〈 [0.4987, 0.6315]; 0.6440 〉) \\ C_{4} = (〈 [0.5876, 0.7105]; 0.6992 〉, 〈 [0.5083, 0.6244]; 0.5556 〉) \\ C_{5} = (〈 [0.6506, 0.7391]; 0.5420 〉, 〈 [0.4689, 0.5876]; 0.6983 〉), \end{matrix} \end{matrix}$

Step 3. By utlizing Equation (8) we compute the scores $S ({\hat{c}}_{i})$ of the values p_{c
_i} (i = 1, 2, 3, 4, 5) Given as. $\begin{matrix} S (C_{1}) = - 0.0027; S (C_{2}) = - 0.0163; \\ S (C_{3}) = 0.0353; S (C_{4}) = 0.0028; S (C_{5}) = 0.0656 \end{matrix}$

Step 4. Organize the scores of the alternatives in descending order, choice the highest score function. Since C₅ > C₃ > C₄ > C₁ > C₂. Hence X₅ > X₃ > X₄ > X₁ > X₂. Thus X₅ is the best option again for the investment company.

For Pythagorean Cubic Fuzzy Ordered Geometric averaging Aggregation Operator

First we calculate the score and then ordered the Tables 4–6.

Step 2. Utilize the PCFOGA aggregation operator in Equation (19), we obtain the PCFN for the alternatives X_i $\begin{matrix} \begin{matrix} C_{1} = (〈 [0.5765, 0.7155]; 0.6306 〉, 〈 [0.5453, 0.6312]; 0.6349 〉) \\ C_{2} = (〈 [0.5469, 0.6689]; 0.7183 〉, 〈 [0.5124, 0.6485]; 0.5323 〉) \\ C_{3} = (〈 [0.5919, 0.7188]; 0.6424 〉, 〈 [0.4867, 0.6194]; 0.5753 〉) \\ C_{4} = (〈 [0.6032, 0.7414]; 0.7534 〉, 〈 [0.4734, 0.5899]; 0.5116 〉) \\ C_{5} = (〈 [0.6633, 0.7608]; 0.6279 〉, 〈 [0.4498, 0.5785]; 0.6096 〉) \end{matrix} \end{matrix}$

Step 3. Use Equation (6) we compute the scores S (p_{c
_i}) of p_{c
_i} (i = 1, 2, 3, 4, 5) which is. S (C₁) = 0.0160 ; S (C₂) = -0.0164 ; S (C₃) = 0 . 0183 ; S (C₄) =0 . 0050 ; S (C₅) =0 . 0510.

Step 4. Place the scores in the form of descending order and pick that alternative, which is the uppermost. Which is C₅ > C₃ > C₁ > C₄ > C₂. Hence X₅ > X₃ > X₁ > X₄ > X₂. Thus X₅ is the best option.

For Pythagorean Cubic Fuzzy Ordered Geometric Weighted Aggregation Operator

Step 1. The decision-makers provides his judgement in Tables 1–3.

Step 2. Utilize the PCFOGW aggregation operator in Equation (24), we collect overall preference values which as, $\begin{matrix} \begin{matrix} C_{1} = (〈 [0.5124, 0.6574]; 0.6083 〉, 〈 [0.5984, 0.6873]; 0.6757 〉) \\ C_{2} = (〈 [0.5098, 0.6265]; 0.6108 〉, 〈 [0.5769, 0.7156]; 0.6105 〉) \\ C_{3} = (〈 [0.5732, 0.6965]; 0.5468 〉, 〈 [0.5078, 0.6365]; 0.6394 〉) \\ C_{4} = (〈 [0.5751, 0.7097]; 0.7050 〉, 〈 [0.5081, 0.6250]; 0.5521 〉) \\ C_{5} = (〈 [0.6461, 0.7298]; 0.5517 〉, 〈 [0.4768, 0.5959]; 0.6844 〉) \end{matrix} \end{matrix}$

Step 3. By utlizing Equation (8) we compute the scores $S ({\hat{c}}_{i})$ of the values p_{c
_i} (i = 1, 2, 3, 4, 5) Given as. S (C₁) = -0.0063 ; S (C₂) = -0.0210 ; S (C₃) =0 . 0297 ; S (C₄) = -0 . 0011 ; S (C₅) =0 . 0587.

Step 4. Form the scores of the alternatives in descending order, choice the highest score function. Since C₅ > C₃ > C₄ > C₁ > C₂. Hence X₅ > X₃ > X₄ > X₁ > X₂. Thus X₅ is the best option again for the investment company.

It can be easily seen from Table 7 that the ranking of the four possible alternatives obtained by the proposed aggregation operator is relatively similar to each other. The best alternative obtained by these operators is the same, namely, X₅. These methods are appropriate to solve the situations that the input, opinions are cooperating, which could consider the interaction between the experts and criteria, which is more reasonable to handle such type of problems.

7 Conclusion

In this article, we have presented the notion of the Pythagorean cubic fuzzy set which is the generalization of interval valued Pythagorean fuzzy set. Some Pythagorean cubic fuzzy operational laws have been developed. We have defined score and accuracy degree for the comparison of Pythagorean cubic fuzzy numbers. We also defined Pythagorean cubic fuzzy distance between Pythagorean cubic fuzzy numbers. To aggregate the Pythagorean cubic fuzzy information, we proposed (PCFWA) operator, (PCFWG), (PCFOWA) and (PCFOWG) operator under Pythagorean cubic fuzzy environment, we also discussed some of its properties like Idempotency, Boundary, Monotonicity and showed a relation between these developed operators. Moreover, we proposed a multi criteria decision-making (MCDM) approach to show the strength and effectiveness of the developed operators. Moreover, we have applied the developed aggregation operators to explain the decision-making problems. 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. Finally, we have provided some comparison with the existence operators to show the validity, practicality, and effectiveness of the novel methodology.

In the future, we will combine other methods with PCFSs, such as the Einstein product and introduce the notion of Pythagorean cubic fuzzy Einstein weighted averaging (PCFEWA), Pythagorean cubic fuzzy Einstein ordered weighted averaging (PCFEOWA), Pythagorean cubic fuzzy Einstein weighted geometric (PCFEWG), Pythagorean cubic fuzzy Einstein ordered weighted geometric (PCFEOWG) and some more generalized operators in multi criteria decision-making process.

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