Cubic Pythagorean fuzzy sets and their application to multi-attribute decision making with unknown weight information

Abstract

Pythagorean fuzzy sets (PFSs) and interval-valued Pythagorean fuzzy sets (IVPFSs) play a vital role in decision-making processes. In this paper based on PFS and IVPFS we introduce the concept of Cubic Pythagorean fuzzy set in which membership degree is an IVPFS and non-membership degree is a PFS. We define some basic operation of Cubic Pythagorean fuzzy numbers (CPFNs). We define score and accuracy functions to compare CPFNs. We also define distance between CPFNs. Based on the defined operations we develop Cubic Pythagorean fuzzy weighted averaging (CPFWA) operator and Cubic Pythagorean fuzzy weighted geometric (CPFWG) operator. We discuss some properties of the developed operators such as idempotancy, boundedness and monotonicity. Moreover, we give a multi-attribute decision making, to show the validity and effectiveness of the developed approach. Finally, we compare our approach with the existing methods.

Keywords

Pythagorean fuzzy sets interval-valued Pythagorean fuzzy sets Cubic Pythagorean fuzzy sets Cubic Pythagorean fuzzy weighted averaging (CPFWA) operator Cubic Pythagorean fuzzy weighted geometric (CPFWG) operator decision making

1 Introduction

The Intuitionistic Fuzzy Set denoted as (IFS) was originated by Atanassov [1, 2], which generalizes Zadeh fuzzy set [25]. The multi-attribute decision making (MADM) problems is a branch of theory of decision. It has been discovered in true living decision situations [3 , 24]. In [19] Wei using Intuitionistic Fuzzy weighted averaging (IFWA) operator and method of maximizing deviation with partial information of attribute weights, scrutinized MADM problem for which the attribute values are prescribed by IFN. Yager [21 –23] initiated the idea of Pythagorean fuzzy set (PFS), categorized by a membership and non-membership degree, also imposed additional criteria that the squared addition of its membership and non-membership degree is smaller than or equal to 1. The example for illustration was constructed by Yager [22]. In the real world PFS has more capability for modeling vagueness. After that the idea of TOPSIS to MADM problem with PFS information, was projected by Zhang and Xu [26]. Their relationship was discussed by Peng and Yang [12], they also suggested dominance and subordination ranking multiple-attribute group decision-making (MAGDM) method. Due to insufficient, it may be tacky for decision makers to reasonably measure the view by using crisp numbers, however they can be categorized by sub-interval of [0, 1]. Due to this the notion of interval valued Pythagorean fuzzy sets (IVPFSs) is given by Peng and Yang [13], composed of the sub-intervals of [0, 1] for membership and non-membership degrees. In [11] the notion of interval-valued Pythagorean fuzzy weighted averaging (IVPFWA) operator and interval-valued Pythagorean fuzzy weighted geometric (IVPFWG) operators respectively are given by Liang et al. along with a novel decision making method for the treatment of MAGDM problem. They presented the idea of interval-valued Pythagorean fuzzy weighted arithmetic averaging (IVPFWAA) operator. In [14] Peng and Yang presented the idea of Choquet integral operators under Pythagorean fuzzy environment. For handling MADM problems, Khan et al. [9], designed the interval valued Pythagorean fuzzy Choquet integral geometric (IVPFCIG) operator. As a generalization of IFS, in [5, 6] Jun gave the idea of cubic set and cubic IFS as an extension of cubic set defined as its membership degree in an interval valued IFS and non-membership degree as an IFS. Since IFSs and IVIFS full fill the conditions stated as, the sum of its membership degrees is smaller than 1 respectively. However, Decision makers (DMs) deal with the situation stated as the sum of membership’s degrees is greater than 1. Therefore motivated by the concept of CIFS, in this paper we initiated the concept of cubic Pythagorean fuzzy sets where the membership degree is IVPFS and non-membership degree is PFS. Therefore, based on the IVPFS and PFS we introduce the idea of Cubic Pythagorean fuzzy set (CPFS) as a generalization of Cubic IFS in the current paper. In CPFS the membership degree is an IVPFS and non-membership degree is PFS. We define some basic operations, score function and accuracy function between CPFNs. We also define distance between two CPFNs. Based on the proposed operations we also introduced aggregation operators of two types, such as Cubic Pythagorean fuzzy weighted averaging (CPFWA) operator and Cubic Pythagorean fuzzy weighted geometric operators denoted as (CPFWG). Then we illustrated the suggested operator for MADM problem by quoting an example. Remainder of the article is structured as follow.

We gave basic definitions and operations in Section 2, in Section 3 we introduce the concept of CPFS and define some basic operations. We also develop score and accuracy function between CPFNs. In Section 4 we develop CPFWA and CPFWG operators and discuss some of its properties such as idempotancy, boundedness, monotonicity. In Section 5 we develop MADM approach with partially known weight information. Under Section 6 we provide a numerical sample for the sake of validity and implementation of the projected method. Similarly, a comparison analysis of proposed and existing methods is mention. Finally, Section 7 presents the conclusion.

2 Some basic definitions and operations

As a generalization of intuitionistic fuzzy set [1], Yagar [22] gave the idea of Pythagorean fuzzy set. Pythagorean fuzzy set can be defined by:

Definition 1. [22] Taking X as a fixed set, then a PFS P in X can be defined as: $P = {(x, μ_{P} (x), υ_{P} (x) | x \in X}$ (1) where μ_P (x) and υ_P (x) are the mappings from X to [0, 1], represent the degrees of membership and nonmembership of element x ∈ X to set X with the condition 0 ⩽ μ_P (x) ⩽1, 0 ⩽ υ_P (x) ⩽1, and $0 ⩽ μ_{P}^{2} (x) + υ_{P}^{2} (x) ⩽ 1,$ for all x ∈ X. The Pythagorean fuzzy index of element x ∈ X to set P is given as $π_{P} (x) = \sqrt{1 - μ_{P}^{2} (x) - υ_{P}^{2} (x)}$ which gives the degree of indeterminacy of x to P. π_P (x) satisfies 0 ⩽ π_P (x) ⩽1 for every x ∈ X.

Since we know that in real world, due to insufficiency information about the real world decision making problem, there are difficulties for the DMs to exactly figure out their opinion by a crisp number but can range over the sub-interval of [0,1]. For the expression of this idea, as a generalization of interval-valued intuitionistic fuzzy set (IFVIFS) [3] and PFS [22] Peng and Yang [13] and Zhang [27] introduced interval valued Pythagorean fuzzy set (IVPFS). The IVPFS is defined as follows.

Definition 2. [13] Taking X a fixed set. Then an interval-valued Pythagorean fuzzy set on X is denoted and defined as: $\tilde{P} = {〈 x, μ_{\tilde{P}} (x), υ_{\tilde{P}} (x) 〉 | x \in X},$ (2) where $μ_{\tilde{P}} (x) = [μ_{\tilde{P}}^{L} (x), μ_{\tilde{P}}^{U} (x)] \subset [0, 1],$ (3) and $υ_{\tilde{P}} (x) = [υ_{\tilde{P}}^{L} (x), υ_{\tilde{P}}^{U} (x)] \subset [0, 1],$ (4) treated as intervals, with $μ_{\tilde{P}}^{L} (x) = \inf μ_{\tilde{P}} (x)$ and $μ_{\tilde{P}}^{U} (x) = sup μ_{\tilde{P}} (x)$ likewise $υ_{\tilde{P}}^{L} (x) = inf υ_{\tilde{P}} (x)$ and $υ_{\tilde{P}}^{U} (x) = sup υ_{\tilde{P}}^{U} (x)$ for all x ∈ X. with condition given as

$0 ⩽ (μ_{\tilde{P}}^{U} (x))^{2} + (υ_{\tilde{P}}^{U} (x))^{2} ⩽ 1 .$ (5)

If $π_{\tilde{P}} (x) = [π_{\tilde{P}}^{L} (x), π_{\tilde{P}}^{U} (x)],$ (6) for all x ∈ X, then it is said to be the interval-valued Pythagorean fuzzy index of x to $\tilde{P}$ , where $π_{\tilde{P}}^{L} (x) = \sqrt{1 - (μ_{\tilde{P}}^{U} (x))^{2} - (υ_{\tilde{P}}^{U} (x))^{2}}$ (7) and $π_{\tilde{P}}^{U} (x) = \sqrt{1 - (μ_{\tilde{P}}^{L} (x))^{2} - (υ_{\tilde{P}}^{L} (x))^{2}}$ (8)

There are two special cases of interval-valued Pythagorean fuzzy set.

(i) Interval-valued Pythagorean fuzzy set reduces to Pythagorean fuzzy set if

$μ_{\tilde{P}}^{L} (x) = μ_{\tilde{P}}^{U} (x)$ and $υ_{\tilde{P}}^{L} (x) = υ_{\tilde{P}}^{U} (x)$ ,

(ii) Interval-valued Pythagorean fuzzy set reduces to an interval-valued intuitionistic fuzzy set if $μ_{\tilde{P}}^{U} (x) + υ_{\tilde{P}}^{U} (x) ⩽ 1,$

Definition 3. [13] Let $\tilde{P} = 〈 [μ_{\tilde{P}}^{L}, μ_{\tilde{P}}^{U}], [υ_{\tilde{P}}^{L}, υ_{\tilde{P}}^{U}] 〉,$ ${\tilde{P}}_{1} = 〈 [μ_{{\tilde{P}}_{1}}^{L}, μ_{{\tilde{P}}_{1}}^{U}], [υ_{{\tilde{P}}_{1}}^{L}, υ_{{\tilde{P}}_{1}}^{U}] 〉,$ ${\tilde{P}}_{2} = 〈 [μ_{{\tilde{P}}_{2}}^{L},$ $μ_{{\tilde{P}}_{2}}^{U}], [υ_{{\tilde{P}}_{2}}^{L}, υ_{{\tilde{P}}_{2}}^{U}] 〉,$ be the three interval-valued Pythagorean fuzzy numbers and λ > 0, then the following operational laws holds: $λ \tilde{P} = ([\sqrt{1 - (1 - (μ_{\tilde{P}}^{L})^{2})^{λ}}, \sqrt{1 - (1 - (υ_{\tilde{P}}^{L})^{2})^{λ}}], [(μ_{\tilde{P}}^{U})^{λ}, (υ_{\tilde{P}}^{U})^{λ}])$ (9) ${\tilde{P}}^{λ} = ([(μ_{\tilde{P}}^{L})^{λ}, (υ_{\tilde{P}}^{L})^{λ}], [\sqrt{1 - (1 - (μ_{\tilde{P}}^{U})^{2})^{λ}}, \sqrt{1 - (1 - (υ_{\tilde{P}}^{U})^{2})^{λ}}])$ (10) ${\tilde{P}}_{1} \otimes {\tilde{P}}_{2} = (\begin{matrix} [μ_{{\tilde{P}}_{1}}^{L} μ_{{\tilde{P}}_{2}}^{L}, υ_{{\tilde{P}}_{1}}^{L} υ_{{\tilde{P}}_{2}}^{L}], \\ [\sqrt{(μ_{{\tilde{P}}_{1}}^{U})^{2} + (μ_{{\tilde{P}}_{2}}^{U})^{2} - (μ_{{\tilde{P}}_{1}}^{U})^{2} (μ_{{\tilde{P}}_{2}}^{U})^{2}}, \sqrt{(υ_{{\tilde{P}}_{1}}^{U})^{2} + (υ_{{\tilde{P}}_{2}}^{U})^{2} - (υ_{{\tilde{P}}_{1}}^{U})^{2} (υ_{{\tilde{P}}_{2}}^{U})^{2}}] \end{matrix})$ (11) ${\tilde{P}}_{1} \oplus {\tilde{P}}_{2} = (\begin{matrix} [\sqrt{(μ_{{\tilde{P}}_{1}}^{L})^{2} + (μ_{{\tilde{P}}_{2}}^{L})^{2} - (μ_{{\tilde{P}}_{1}}^{L})^{2} (μ_{{\tilde{P}}_{2}}^{L})^{2}}, \sqrt{(υ_{{\tilde{P}}_{1}}^{L})^{2} + (υ_{{\tilde{P}}_{2}}^{L})^{2} - (υ_{{\tilde{P}}_{1}}^{L})^{2} (υ_{{\tilde{P}}_{2}}^{L})^{2}}], \\ [μ_{{\tilde{P}}_{1}}^{U} μ_{{\tilde{P}}_{2}}^{U}, υ_{{\tilde{P}}_{1}}^{L} υ_{{\tilde{P}}_{2}}^{L}] \end{matrix})$ (12)

For interval-valued Pythagorean fuzzy number Peng and Yang [24], introduced the score function and accuracy function as follows:

Definition 4. [13] Let ${\tilde{P}}_{i} = 〈 [μ_{{\tilde{P}}_{i}}^{L}, υ_{{\tilde{P}}_{i}}^{L}], [μ_{{\tilde{P}}_{i}}^{U}, υ_{{\tilde{P}}_{i}}^{U}] 〉$ (i = 1, 2) be two IVPFNs, then the score and accuracy functions of the IVPFNs, can be define in Equations (13) and (14) respectively. $S ({\tilde{P}}_{1}) = \frac{1}{2} [(μ_{{\tilde{P}}_{1}}^{L})^{2} + (μ_{{\tilde{P}}_{1}}^{U})^{2} - (υ_{{\tilde{P}}_{1}}^{L})^{2} - (υ_{{\tilde{P}}_{1}}^{U})^{2}],$ (13) and $H ({\tilde{P}}_{1}) = \frac{1}{2} [(μ_{{\tilde{P}}_{1}}^{L})^{2} + (μ_{{\tilde{P}}_{1}}^{U})^{2} + (υ_{{\tilde{P}}_{1}}^{L})^{2} + (υ_{{\tilde{P}}_{1}}^{U})^{2}],$ (14)

Zang [27] present an interval-valued Pythagorean fuzzy distance measure for IVPFNs:

Definition 5. [27] Let ${\tilde{P}}_{i} = 〈 [μ_{{\tilde{P}}_{i}}^{L}, υ_{{\tilde{P}}_{i}}^{L}], [μ_{{\tilde{P}}_{i}}^{U}, υ_{{\tilde{P}}_{i}}^{U}] 〉$ (i = 1, 2) be two IVPFNs, then the distance between ${\tilde{P}}_{1}$ and ${\tilde{P}}_{2}$ is defined as follows: $d ({\tilde{P}}_{1}, {\tilde{P}}_{2}) = \frac{1}{4} [\begin{matrix} | (μ_{{\tilde{P}}_{1}}^{L})^{2} - (μ_{{\tilde{P}}_{2}}^{L})^{2} | + | (υ_{{\tilde{P}}_{1}}^{L})^{2} - (υ_{{\tilde{P}}_{2}}^{L})^{2} | + | (μ_{{\tilde{P}}_{1}}^{U})^{2} - (μ_{{\tilde{P}}_{2}}^{U})^{2} | + \\ | (υ_{{\tilde{P}}_{1}}^{U})^{2} - (υ_{{\tilde{P}}_{2}}^{U})^{2} | + | (π_{{\tilde{P}}_{1}}^{L})^{2} - (π_{{\tilde{P}}_{2}}^{L})^{2} | + | (π_{{\tilde{P}}_{1}}^{U})^{2} - (π_{{\tilde{P}}_{2}}^{U})^{2} | \end{matrix}]$ (15) where $[π_{{\tilde{P}}_{1}}^{L}, π_{{\tilde{P}}_{1}}^{U}] = [\sqrt{1 - (μ_{{\tilde{P}}_{1}}^{U})^{2} - (υ_{{\tilde{P}}_{1}}^{U})^{2}}, \sqrt{1 - (μ_{{\tilde{P}}_{1}}^{L})^{2} - (υ_{{\tilde{P}}_{1}}^{L})^{2}}]$ (16) and $[π_{{\tilde{P}}_{2}}^{L}, π_{{\tilde{P}}_{2}}^{U}] = [\sqrt{1 - (μ_{{\tilde{P}}_{2}}^{U})^{2} - (υ_{{\tilde{P}}_{2}}^{U})^{2}}, \sqrt{1 - (μ_{{\tilde{P}}_{2}}^{L})^{2} - (υ_{{\tilde{P}}_{2}}^{L})^{2}}]$ (17)

3 Cubic Pythagorean fuzzy sets

In this section we introduce the concept of Cubic Pythagorean fuzzy set which is another extension of Cubic set [5] and Cubic intuitionistic fuzzy set [6] introduced by Jun.

Definition 6. Let X be a nonempty set. By a Cubic Pythagorean fuzzy set (abbreviated CPFS) in X we mean a structure $C_{P} = {(x, \tilde{P} (x), P (x)) | x \in X},$ (18) in which $\tilde{P}$ is an interval-valued Pythagorean fuzzy set in X and P is an Pythagorean fuzzy set in X. We denote a Cubic Pythagorean fuzzy number (abbreviated CPFN) by $\tilde{c} = ({\tilde{p}}_{\tilde{c}}, p_{\tilde{c}}) = (〈 [μ_{\tilde{c}}^{L}, υ_{\tilde{c}}^{L}], [μ_{\tilde{c}}^{U}, υ_{\tilde{c}}^{U}] 〉; (μ_{\tilde{c}}, υ_{\tilde{c}})) .$ Let $π_{\tilde{c}} (x) = 〈 [π_{\tilde{c}}^{L} (x), π_{\tilde{c}}^{U} (x)]; φ_{\tilde{c}} (x) 〉 .$ Then $π_{\tilde{c}} (x)$ is said to be Cubic Pythagorean fuzzy index of element x ∈ X to set $\tilde{c}$ , where

$φ_{\tilde{c}} (x) = \sqrt{1 - μ_{\tilde{c}}^{2} (x) - υ_{\tilde{c}}^{2} (x)}$ (19) $π_{\tilde{c}}^{L} (x) = \sqrt{1 - (μ_{\tilde{c}}^{U} (x))^{2} - (υ_{\tilde{c}}^{U} (x))^{2}}$ (20) $π_{\tilde{c}}^{U} (x) = \sqrt{1 - (μ_{\tilde{c}}^{L} (x))^{2} - (υ_{\tilde{c}}^{L} (x))^{2}} .$ (21)

Example 1. For X = {1, 2, 3, 4}, the pair $C_{P} = ((x, \tilde{P} (x), P (x))$ with the tabular representation in the following Table 1, is a Cubic Pythagorean fuzzy set in X.

Table 1

Cubic Pythagorean fuzzy set

X	$\tilde{p} = 〈 μ_{\tilde{p}}, υ_{\tilde{p}} 〉$	p = 〈μ_p, υ_p〉
1	〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 6] 〉	(0 . 8, 0 . 5)
2	〈 [0 . 3, 0 . 4] , [0 . 8, 0 . 9] 〉	(0 . 7, 0 . 4)
3	〈 [0 . 7, 0 . 8] , [0 . 4, 0 . 5] 〉	(0 . 8, 0 . 3)
4	〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉	(0 . 6, 0 . 7)

In the following, we define some operation on CPFNs such as union, intersection, complement, sum, multiplication and scaler multiplication.

Definition 7. Let $\tilde{c} = (\tilde{p}, p_{\tilde{c}}),$ ${\tilde{c}}_{1} = ({\tilde{p}}_{{\tilde{c}}_{1}}, p_{{\tilde{c}}_{1}}),$ ${\tilde{c}}_{2} = ({\tilde{p}}_{{\tilde{c}}_{2}}, p_{{\tilde{c}}_{2}}),$ are three CPFNs, and δ > 0 . Then the following operational laws holds: ${\tilde{c}}_{1} \cup {\tilde{c}}_{2} = (\begin{matrix} 〈 [max {μ_{{\tilde{c}}_{1}}^{L}, μ_{{\tilde{c}}_{2}}^{L}}, max {υ_{{\tilde{c}}_{1}}^{L}, υ_{{\tilde{c}}_{2}}^{L}}], [min {μ_{{\tilde{c}}_{1}}^{U}, μ_{{\tilde{c}}_{2}}^{U}}, min {υ_{{\tilde{c}}_{1}}^{U}, υ_{{\tilde{c}}_{2}}^{U}}] 〉; \\ (max {μ_{{\tilde{c}}_{1}}, μ_{{\tilde{c}}_{2}}}, min {υ_{{\tilde{c}}_{1}}, υ_{{\tilde{c}}_{2}}}) \end{matrix})$ (22) ${\tilde{c}}_{1} \cap {\tilde{c}}_{2} = (\begin{matrix} 〈 [min {μ_{{\tilde{c}}_{1}}^{L}, μ_{{\tilde{c}}_{2}}^{L}}, min {υ_{{\tilde{c}}_{1}}^{L}, υ_{{\tilde{c}}_{2}}^{L}}], [max {μ_{{\tilde{c}}_{1}}^{U}, μ_{{\tilde{c}}_{2}}^{U}}, max {υ_{{\tilde{c}}_{1}}^{U}, υ_{{\tilde{c}}_{2}}^{U}}] 〉; \\ (min {μ_{{\tilde{c}}_{1}}, μ_{{\tilde{c}}_{2}}}, max {υ_{{\tilde{c}}_{1}}, υ_{{\tilde{c}}_{2}}}) \end{matrix})$ (23) $δ \tilde{c} = (\begin{matrix} 〈 [\sqrt{1 - {(1 - (μ_{\tilde{c}}^{L})^{2})}^{δ}}, \sqrt{1 - (1 - (μ_{\tilde{c}}^{U})^{2})^{δ}}], [(υ_{\tilde{c}}^{L})^{δ}, (υ_{\tilde{c}}^{U})^{δ}] 〉; \\ (\sqrt{1 - (1 - μ_{\tilde{c}}^{2})^{δ}}, υ_{\tilde{c}}^{δ}) \end{matrix})$ (24) ${\tilde{c}}^{δ} = (\begin{matrix} 〈 [(μ_{\tilde{c}}^{L})^{δ}, (μ_{\tilde{c}}^{U})^{δ}], [\sqrt{1 - (1 - (υ_{\tilde{c}}^{L})^{2})^{δ}}, \sqrt{1 - (1 - (υ_{\tilde{c}}^{U})^{2})^{δ}}] 〉; \\ (μ_{\tilde{c}}^{δ}, \sqrt{1 - (1 - υ_{\tilde{c}}^{2})^{δ}}) \end{matrix})$ (25) ${\tilde{c}}_{1} \oplus {\tilde{c}}_{2} = (\begin{matrix} 〈 [\begin{matrix} \sqrt{(μ_{{\tilde{c}}_{1}}^{L})^{2} + (μ_{{\tilde{c}}_{2}}^{L})^{2} - (μ_{{\tilde{c}}_{1}}^{L})^{2} (μ_{{\tilde{c}}_{2}}^{L})^{2}}, \\ \sqrt{(μ_{{\tilde{c}}_{1}}^{U})^{2} + (μ_{{\tilde{c}}_{2}}^{U})^{2} - (μ_{{\tilde{c}}_{1}}^{U})^{2} (μ_{{\tilde{c}}_{2}}^{U})^{2}} \end{matrix}], [υ_{{\tilde{c}}_{1}}^{L} υ_{{\tilde{c}}_{2}}^{L}, υ_{{\tilde{c}}_{1}}^{U} υ_{{\tilde{c}}_{2}}^{U}] 〉; \\ (\sqrt{μ_{c_{1}}^{2} + μ_{c_{2}}^{2} - μ_{c_{1}}^{2} μ_{c_{2}}^{2}}, υ_{c_{1}} υ_{c_{2}}) \end{matrix})$ (26) ${\tilde{c}}_{1} \otimes {\tilde{c}}_{2} = (\begin{matrix} 〈 [μ_{{\tilde{c}}_{1}}^{L} μ_{{\tilde{c}}_{2}}^{L}, μ_{{\tilde{c}}_{1}}^{U} μ_{{\tilde{c}}_{2}}^{U}], [\begin{matrix} \sqrt{(υ_{{\tilde{c}}_{1}}^{L})^{2} + (υ_{{\tilde{c}}_{2}}^{L})^{2} - (υ_{{\tilde{c}}_{1}}^{L})^{2} (υ_{{\tilde{c}}_{2}}^{L})^{2}}, \\ \sqrt{(υ_{{\tilde{c}}_{1}}^{U})^{2} + (υ_{{\tilde{c}}_{2}}^{U})^{2} - (υ_{{\tilde{c}}_{1}}^{U})^{2} (υ_{{\tilde{c}}_{2}}^{U})^{2}} \end{matrix}] 〉; \\ (μ_{{\tilde{c}}_{1}} μ_{{\tilde{c}}_{2}}, \sqrt{υ_{{\tilde{c}}_{1}}^{2} + υ_{{\tilde{c}}_{2}}^{2} - υ_{{\tilde{c}}_{1}}^{2} υ_{{\tilde{c}}_{2}}^{2}}) \end{matrix})$ (27)

Theorem 1. Let ${\tilde{c}}_{1} = ({\tilde{p}}_{{\tilde{c}}_{1}}, p_{{\tilde{c}}_{1}}),$ ${\tilde{c}}_{2} = ({\tilde{p}}_{{\tilde{c}}_{2}}, p_{{\tilde{c}}_{2}}),$ are three CPFNs, then

$({\tilde{c}}_{1} \cup {\tilde{c}}_{2}) \oplus ({\tilde{c}}_{1} \cap {\tilde{c}}_{2}) = {\tilde{c}}_{1} \oplus {\tilde{c}}_{2}$

Proof. Easy to prove.

Theorem 2. Let ${\tilde{c}}_{1} = ({\tilde{p}}_{{\tilde{c}}_{1}}, p_{{\tilde{c}}_{1}}),$ ${\tilde{c}}_{2} = ({\tilde{p}}_{{\tilde{c}}_{2}}, p_{{\tilde{c}}_{2}}),$ ${\tilde{c}}_{3} = ({\tilde{p}}_{{\tilde{c}}_{3}}, p_{{\tilde{c}}_{3}})$ are three CPFNs, then

$(1) ({\tilde{c}}_{1} \cup {\tilde{c}}_{2}) \cap {\tilde{c}}_{3} = ({\tilde{c}}_{1} \cap {\tilde{c}}_{3}) \cup ({\tilde{c}}_{2} \cap {\tilde{c}}_{3})$

$(2) ({\tilde{c}}_{1} \cap {\tilde{c}}_{2}) \cup {\tilde{c}}_{3} = ({\tilde{c}}_{1} \cup {\tilde{c}}_{3}) \cap ({\tilde{c}}_{2} \cup {\tilde{c}}_{3})$ $(3) ({\tilde{c}}_{1} \cup {\tilde{c}}_{2}) \oplus {\tilde{c}}_{3} = ({\tilde{c}}_{1} \oplus {\tilde{c}}_{3}) \cup ({\tilde{c}}_{2} \oplus {\tilde{c}}_{3})$ $(4) ({\tilde{c}}_{1} \cap {\tilde{c}}_{2}) \oplus {\tilde{c}}_{3} = ({\tilde{c}}_{1} \oplus {\tilde{c}}_{3}) \cap ({\tilde{c}}_{2} \oplus {\tilde{c}}_{3})$ $(5) ({\tilde{c}}_{1} \cup {\tilde{c}}_{2}) \otimes {\tilde{c}}_{3} = ({\tilde{c}}_{1} \oplus {\tilde{c}}_{3}) \cup ({\tilde{c}}_{2} \otimes {\tilde{c}}_{3})$ $(6) ({\tilde{c}}_{1} \cap {\tilde{c}}_{2}) \otimes {\tilde{c}}_{3} = ({\tilde{c}}_{1} \oplus {\tilde{c}}_{3}) \cap ({\tilde{c}}_{2} \otimes {\tilde{c}}_{3})$

Proof. Easy to prove.

To compare CPFNs in the following we define Score and accuracy function between CPFNs.

Definition 8. Let $\tilde{c} = ({\tilde{p}}_{\tilde{c}}, {\tilde{p}}_{\tilde{c}})$ be a cubic Pythagorean fuzzy number. The score for CPFN is defined as follows: $S (\tilde{c}) = \frac{1}{2} [\frac{1}{2} [(μ_{\tilde{c}}^{L})^{2} + (μ_{\tilde{c}}^{U})^{2} - (υ_{\tilde{c}}^{L})^{2} - (υ_{\tilde{c}}^{U})^{2}] + ((μ_{\tilde{c}})^{2} - (υ_{\tilde{c}})^{2})]$ (28) where $S (\tilde{c}) \in [- 1, 1] .$

Definition 9. Let $\tilde{c} = ({\tilde{p}}_{\tilde{c}}, p_{\tilde{c}})$ be a cubic Pythagorean fuzzy number. The accuracy function of CPFN is defined as follows: $α (\tilde{c}) = \frac{1}{2} [\frac{1}{2} [(μ_{\tilde{c}}^{L})^{2} + (μ_{\tilde{c}}^{U})^{2} + (υ_{\tilde{c}}^{L})^{2} + (υ_{\tilde{c}}^{U})^{2}] + ((μ_{\tilde{c}})^{2} + (υ_{\tilde{c}})^{2})]$ (29) where $α (\tilde{c}) \in [0, 1] .$

Definition 10. Let ${\tilde{c}}_{1} = ({\tilde{p}}_{{\tilde{c}}_{1}}, p_{{\tilde{c}}_{1}}),$ and ${\tilde{c}}_{2} = ({\tilde{p}}_{{\tilde{c}}_{2}}, p_{{\tilde{c}}_{2}})$ be two CPFNs. Then the distance between two CPFNs is defined as: $d ({\tilde{c}}_{1}, {\tilde{c}}_{2}) = \frac{1}{6} [\begin{matrix} | (μ_{{\tilde{c}}_{1}}^{L})^{2} - (μ_{{\tilde{c}}_{2}}^{L})^{2} | + | (μ_{{\tilde{c}}_{1}}^{U})^{2} - (μ_{{\tilde{c}}_{2}}^{U})^{2} | + | (υ_{{\tilde{c}}_{1}}^{L})^{2} - (υ_{{\tilde{c}}_{2}}^{L})^{2} | \\ + | (υ_{{\tilde{c}}_{1}}^{U})^{2} - (υ_{{\tilde{c}}_{2}}^{U})^{2} | + | (π_{{\tilde{c}}_{1}}^{L})^{2} - (π_{{\tilde{c}}_{2}}^{L})^{2} | + | (π_{{\tilde{c}}_{1}}^{U})^{2} - (π_{{\tilde{c}}_{2}}^{U})^{2} | + \\ | μ_{{\tilde{c}}_{1}}^{2} - μ_{{\tilde{c}}_{2}}^{2} | + | υ_{{\tilde{c}}_{1}}^{2} - υ_{{\tilde{c}}_{2}}^{2} | + | π_{{\tilde{c}}_{1}}^{2} - π_{{\tilde{c}}_{2}}^{2} | \end{matrix}]$ (30)

Proposition 3. Let ${\tilde{c}}_{1} = ({\tilde{p}}_{{\tilde{c}}_{1}}, p_{{\tilde{c}}_{1}}),$ ${\tilde{c}}_{2} = ({\tilde{p}}_{{\tilde{c}}_{2}}, p_{{\tilde{c}}_{2}})$ and ${\tilde{c}}_{3} = ({\tilde{p}}_{{\tilde{c}}_{3}}, p_{{\tilde{c}}_{3}})$ be two CPFNs. Then,

$0 ⩽ d ({\tilde{c}}_{1}, {\tilde{c}}_{2}) ⩽ 1 .$

$d ({\tilde{c}}_{1}, {\tilde{c}}_{2}) = 0$ if and only if ${\tilde{c}}_{1} = {\tilde{c}}_{2} .$

$d ({\tilde{c}}_{1}, {\tilde{c}}_{2}) = d ({\tilde{c}}_{2}, {\tilde{c}}_{1}) .$

If ${\tilde{c}}_{1} ⩽ {\tilde{c}}_{2} ⩽ {\tilde{c}}_{3}$ , then $d ({\tilde{c}}_{1}, {\tilde{c}}_{2}) ⩽ d ({\tilde{c}}_{1}, {\tilde{c}}_{3})$ and $d ({\tilde{c}}_{2}, {\tilde{c}}_{3}) ⩽ d ({\tilde{c}}_{1}, {\tilde{c}}_{3}) .$

Proof. Easy to prove.

4 Cubic Pythagorean fuzzy aggregation operators

Under this heading we described the Cubic Pythagorean fuzzy aggregation operators and some properties of these aggregation operators like boundedness, idempotency, and monotonicity, which are the consequences of series of a number system.

Definition 11. Let ${\tilde{c}}_{i} = ({\tilde{p}}_{{\tilde{c}}_{i}}, p_{{\tilde{c}}_{i}}) (i = 1, 2, \dots, n)$ be the family CPFN, s with w = (w₁, w₂, …, w_n) ^T the weight vectors of ${\tilde{c}}_{i} (i = 1, 2, \dots, n)$ satisfying $\sum_{i = 1}^{n} w_{i} = 1 .$ Then cubic Pythagorean fuzzy weighted average (CPFWA) operator is a mapping $CPFWA : {\tilde{c}}^{n} \to \tilde{c}$ , where

$\begin{matrix} CPFWA ({\tilde{c}}_{1} {\tilde{c}}_{2} \dots, {\tilde{c}}_{n}) \\ = w_{1} {\tilde{c}}_{1} \oplus w_{2} {\tilde{c}}_{2} \oplus \dots \oplus w_{n} {\tilde{c}}_{n} \\ = (\begin{matrix} 〈 [\sum_{i = 1}^{n} w_{i} μ_{{\tilde{c}}_{i}}^{L}, \sum_{i = 1}^{n} w_{i} μ_{{\tilde{c}}_{i}}^{U}], [\sum_{i = 1}^{n} w_{i} υ_{{\tilde{c}}_{i}}^{L}, \sum_{i = 1}^{n} w_{i} υ_{{\tilde{c}}_{i}}^{U}] 〉; \\ (\sum_{i = 1}^{n} w_{i} μ_{{\tilde{c}}_{i}}, \sum_{i = 1}^{n} w_{i} υ_{{\tilde{c}}_{i}}) \end{matrix}) \end{matrix}$ (31)

Theorem 3. Let ${\tilde{c}}_{i} = ({\tilde{p}}_{{\tilde{c}}_{i}}, p_{{\tilde{c}}_{i}})$ for (i = 1, 2, …, n) be a collection of CPFNs with w = (w₁, w₂, …, w_n) ^T be the weight vectors of ${\tilde{c}}_{i} (i = 1, 2, \dots, n)$ satisfying $\sum_{i = 1}^{n} w_{i} = 1 .$ Then the aggregation result by using CPFWA operator is also CPFN and

$\begin{matrix} CPFWA ({\tilde{c}}_{1} {\tilde{c}}_{2} \dots, {\tilde{c}}_{n}) \\ = (\begin{matrix} 〈 \begin{matrix} \sqrt{1 - Π_{i = 1}^{n} (1 - (μ_{{\tilde{c}}_{i}}^{L})^{2})^{w_{i}}}, \sqrt{1 - Π_{i = 1}^{n} (1 - (μ_{{\tilde{c}}_{i}}^{U})^{2})^{w_{i}}}] \\ [Π_{i = 1}^{n} (υ_{{\tilde{c}}_{i}}^{L})^{w_{i}}, Π_{i = 1}^{n} (υ_{{\tilde{c}}_{i}}^{U})^{w_{i}}] \end{matrix} 〉; \\ (\sqrt{1 - Π_{i = 1}^{n} (1 - (μ_{{\tilde{c}}_{i}})^{2})^{w_{i}}}, Π_{i = 1}^{n} (υ_{{\tilde{c}}_{i}})^{w_{i}}) \end{matrix}) \end{matrix}$ (32)

Proof. By mathematical induction we can easily prove that Equation (42) holds for all n.

Definition 12. Let ${\tilde{c}}_{i} = ({\tilde{p}}_{{\tilde{c}}_{i}}, p_{{\tilde{c}}_{i}})$ for (i = 1, 2, …, n) be a collection of CPFNs with w = (w₁, w₂, …, w_n) ^T be the weight vectors of ${\tilde{c}}_{i} (i = 1, 2, \dots, n)$ satisfying $\sum_{i = 1}^{n} w_{i} = 1$ then cubic Pythagorean fuzzy weighted Geometric (CPFWG) operator is a mapping $CPFWG : {\tilde{c}}^{n} \to \tilde{c}$ , where $\begin{matrix} CPFWG ({\tilde{c}}_{1}, {\tilde{c}}_{2} \dots, {\tilde{c}}_{n}) \\ = ({\tilde{c}}_{1})^{w_{1}} \otimes ({\tilde{c}}_{2})^{w_{2}} \otimes ({\tilde{c}}_{2})^{w_{2}} \dots \otimes ({\tilde{c}}_{n})^{w_{n}} \\ = (\begin{matrix} 〈 \begin{matrix} [Π_{i = 1}^{n} (μ_{{\tilde{c}}_{i}}^{L})^{w_{i}}, Π_{i = 1}^{n} (μ_{{\tilde{c}}_{i}}^{U})^{w_{i}}], \\ [Π_{i = 1}^{n} (υ_{{\tilde{c}}_{i}}^{L})^{w_{i}}, Π_{i = 1}^{n} (υ_{{\tilde{c}}_{i}}^{U})^{w_{i}}] \end{matrix} 〉; \\ (Π_{i = 1}^{n} (μ_{{\tilde{c}}_{i}})^{w_{i}}, Π_{i = 1}^{n} (υ_{{\tilde{c}}_{i}})^{w_{i}}) \end{matrix}) \end{matrix}$ (33)

Theorem 4. Let ${\tilde{c}}_{i} = ({\tilde{p}}_{{\tilde{c}}_{i}}, p_{{\tilde{c}}_{i}}) (i = 1, 2, \dots, n)$ be a collection of CPFNs and w = (w₁, w₂, …, w_n) ^T be the weight vectors of ${\tilde{c}}_{i} (i = 1, 2, \dots, n)$ with $\sum_{i = 1}^{n} w_{i} = 1 .$ Then the aggregation result by using CPFWA operator is also CPFN and $\begin{matrix} CPFWG ({\tilde{c}}_{1}, {\tilde{c}}_{2} \dots, {\tilde{c}}_{n}) \\ = (\begin{matrix} 〈 [Π_{i = 1}^{n} (μ_{{\tilde{c}}_{i}}^{L})^{w_{i}}, Π_{i = 1}^{n} (μ_{{\tilde{c}}_{i}}^{U})^{w_{i}}], [\begin{matrix} \sqrt{1 - Π_{i = 1}^{n} (1 - (υ_{{\tilde{c}}_{i}}^{L})^{2})^{w_{i}}}, \\ \sqrt{1 - Π_{i = 1}^{n} (1 - (υ_{{\tilde{c}}_{i}}^{U})^{2})^{w_{i}}} \end{matrix}] 〉; \\ (Π_{i = 1}^{n} (μ_{{\tilde{c}}_{i}})^{w_{i}}, \sqrt{1 - Π_{i = 1}^{n} (1 - (υ_{{\tilde{c}}_{i}})^{2})^{w_{i}}}) \end{matrix}) \end{matrix}$ (34)

Proof. Proof of the Theorem is same as Theorem 4.

Lemma 1. For all ${\tilde{c}}_{i}, w_{i} \in [0, 1]$ the following holds: $\sum_{i = 1}^{n} ({\tilde{c}}_{i})^{w_{i}} ⩽ \sum_{i = 1}^{n} w_{i} {\tilde{c}}_{i}$

Equality holds only if ${\tilde{c}}_{1} = {\tilde{c}}_{2} = . . . = {\tilde{c}}_{n} .$

The following theorem shows the relations between CPFWA operator and CPFWG operator.

Theorem 5. Let ${\tilde{c}}_{i} = ({\tilde{p}}_{{\tilde{c}}_{i}}, p_{{\tilde{c}}_{i}})$ for (i = 1, 2, …, n) be a collection of CPFNs then $CPFWG ({\tilde{c}}_{1}, {\tilde{c}}_{2}, \dots, {\tilde{c}}_{n}) ⩽ CPFWA ({\tilde{c}}_{1}, {\tilde{c}}_{2}, \dots, {\tilde{c}}_{n})$ where w = (w₁, w₂, …, w_n) ^T be the weight vectors of ${\tilde{c}}_{i} (i = 1, 2, \dots, n)$ with $\sum_{i = 1}^{n} w_{i} = 1$ .

Proof. Using Lemma 1. We can prove the result.

In the following we discuss some desirable properties of CPFWA operator and CPFWG operator.

Theorem 6. Let ${\tilde{c}}_{i} = ({\tilde{p}}_{{\tilde{c}}_{i}}, p_{{\tilde{c}}_{i}}) (i = 1, 2, 3, \dots, n)$ be a collection of Cubic Pythagorean fuzzy numbers and w = (w₁, w₂, …, w_n) ^T is the weighted vector of ${\tilde{c}}_{i} (i = 1, 2, 3, \dots, n)$ with w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$ Then we have

(1) (Idempotency): If all ${\tilde{c}}_{i} (i = 1, 2, 3, \dots, n)$ are equal, i.e., ${\tilde{c}}_{i} = \tilde{c},$ for all i. Then $CPFWA ({\tilde{c}}_{1}, {\tilde{c}}_{2}, \dots, {\tilde{c}}_{n}) = \tilde{c} .$ (35) and $CPFWG ({\tilde{c}}_{1}, {\tilde{c}}_{2}, \dots, {\tilde{c}}_{n}) = \tilde{c}$ (36)

(2) (Monotonicity): Let ${\tilde{c}}_{i} = (μ_{{\tilde{c}}_{i}}, υ_{{\tilde{c}}_{i}})$ (i = 1, 2, 3, …, n) and ${\tilde{c}}_{i}^{*} = (μ_{{\tilde{c}}_{i}^{*}}, υ_{{\tilde{c}}_{i}^{*}})$ (i = 1, 2, 3, …, n) be two collections of Cubic Pythagorean fuzzy numbers, if $μ_{{\tilde{c}}_{i}} ⩽ μ_{{\tilde{c}}_{i}^{*}}$ and $υ_{{\tilde{c}}_{i}} ⩾ υ_{{\tilde{c}}_{i}^{*}}$ for all i. Then for every w we have $CPFWA ({\tilde{c}}_{1}, {\tilde{c}}_{2}, \dots, {\tilde{c}}_{n}) ⩽ CPFWA ({\tilde{c}}_{1}^{*}, {\tilde{c}}_{2}^{*}, \dots, {\tilde{c}}_{n}^{*})$ (37) and $CPFWG ({\tilde{c}}_{1}, {\tilde{c}}_{2}, \dots, {\tilde{c}}_{n}) ⩽ CPFWG ({\tilde{c}}_{1}^{*}, {\tilde{c}}_{2}^{*}, \dots, {\tilde{c}}_{n}^{*})$ (38)

(3) (Boundedness): If ${\tilde{c}}^{+} = (\begin{matrix} 〈 [{max}_{i}^{{} μ_{{\tilde{c}}_{i}}^{L}}, {max}_{i}^{{} μ_{{\tilde{c}}_{i}}^{U}}], [{min}_{i}^{{} υ_{{\tilde{c}}_{i}}^{L}}, {min}_{i}^{{} υ_{{\tilde{c}}_{i}}^{U}}] 〉; \\ ({max}_{i}^{{} μ_{{\tilde{c}}_{i}}}, {min}_{i}^{{} υ_{{\tilde{c}}_{i}}}) \end{matrix})$ and ${\tilde{c}}^{-} = (\begin{matrix} 〈 [{min}_{i}^{{} μ_{{\tilde{c}}_{i}}^{L}}, {min}_{i}^{{} μ_{{\tilde{c}}_{i}}^{U}}], [{max}_{i}^{{} υ_{{\tilde{c}}_{i}}^{L}}, {max}_{i}^{{} υ_{{\tilde{c}}_{i}}^{U}}] 〉; \\ ({min}_{i}^{{} μ_{{\tilde{c}}_{i}}}, {max}_{i}^{{} υ_{{\tilde{c}}_{i}}}) \end{matrix})$ then ${\tilde{c}}^{-} ⩽ CPFWA ({\tilde{c}}_{1}, {\tilde{c}}_{2}, \dots, {\tilde{c}}_{n}) ⩽ {\tilde{c}}^{+}$ (39) and ${\tilde{c}}^{-} ⩽ CPFWG ({\tilde{c}}_{1}, {\tilde{c}}_{2}, \dots, {\tilde{c}}_{n}) ⩽ {\tilde{c}}^{+}$ (40)

Proof. Proof of the Theorem is easy and we omit here.

5 Multi-attribute decision making based on Cubic Pythagorean fuzzy information

A matrix can be used to represent multi-attribute decision-making problem whose elements represent the information about alternatives versus attributes. A Cubic Pythagorean fuzzy decision matrix of CPFNs can be constructed which gives the information of the alternative X_i satisfy the attributes A_i We construct a Cubic Pythagorean fuzzy decision matrix, whose elements are CPFNs, which are given not only the information that the alternative X_i satisfying the attribute A_i. Let X = {X₁, X₂, …, X_m} be the set of m alternatives each of which having attributes set A = {A₁, A₂, …, A_n} of n members. For the evaluation of the efficiency of alternative X_i under the influence of attribute A_j, the decision maker must have to provide a range of information over the sub-interval of [0,1] and a fuzzy value for both the membership and non-membership. We can express as a CPFNs the efficiency of X_i using attribute A_i such as ${\tilde{c}}_{ij} = ({\tilde{p}}_{{\tilde{c}}_{ij}}, p_{{\tilde{c}}_{ij}})$ satisfying $0 ⩽ (μ_{{\tilde{c}}_{ij}}^{U} (x))^{2} + (υ_{{\tilde{c}}_{ij}}^{U} (x))^{2} ⩽ 1 .$ and $0 ⩽ μ_{P_{ij}}^{2} (x) + υ_{P_{ij}}^{2} (x) ⩽ 1$ for all i = 1, 2, …, m, j = 1, 2, …, n Cubic Pythagorean fuzzy decision mentioned as $C = [\begin{matrix} {\tilde{c}}_{11} & {\tilde{c}}_{12} & \dots & {\tilde{c}}_{1 n} \\ {\tilde{c}}_{21} & {\tilde{c}}_{22} & \dots & {\tilde{c}}_{2 n} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {\tilde{c}}_{m 1} & {\tilde{c}}_{m 2} & {\tilde{c}}_{mn} \end{matrix}]$ with the weight vector w = (w₁, w₂, …, w_n) ^T of ${\tilde{c}}_{i} (i = 1, 2, \dots, n)$ given by the DMs, satisfying $\sum_{i = 1}^{n} w_{i} = 1,$ 0 ⩽ w_j ⩽ 1 where w_j is the degree of importance of each attribute. Generally, DMs are interested for the evaluation of degree of preference of each attribute. Because of complexity in decision making problem and human thinking variations the information of weight may not be completed. Let us take the case that the DMs may provide the attribute weight information may as follow for i ≠ j.

A weak ranking: {w_i ⩾ w_j};

A strict ranking: {w_i - w_j ⩾ λ_i (>0)};

A ranking with multiples: {w_i ⩾ λ_iw_j}, 0 ⩽ λ_i ⩽ 1;

An interval form: {λ_i ⩽ w_i ⩽ λ_i + δ_i}, 0 ⩽ λ_i ⩽ λ_i + δ_i ⩽ 1;

A ranking of differences: {w_i - w_j ⩾ w_k - w_l}, for j ≠ k ≠ l.

5.1 Determining the optimal weight of attributes by maximizing deviation method

Due to the central role of attribute weight for the evaluation MADM, Wang [22], proposed a maximizing deviation method to calculate the attribute weight for finding the solution of MADM with numerical information. Which states that an attribute with greater value should be given higher weight and vice versa. Hence, by using the method of maximizing deviation we construct an optimization model for the determination of optimal attribute weight in Cubic Pythagorean fuzzy environment. The deviation between X_j and all other alternatives for the attribute A_i, is given as:

$D_{ij} (w) = \sum_{k = 1}^{m} w_{i} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj}),$ for all j = 1, 2, …, n and i = 1, 2, …, m where $d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj})$ denotes the Cubic Pythagorean fuzzy distance between the CPFNs ${\tilde{c}}_{ij}$ and ${\tilde{c}}_{kj}$ . The deviation value of all alternatives to other alternatives for the attribute A_j ∈ A, is defined by $\begin{matrix} D_{j} (w) = \sum_{k = 1}^{m} D_{ij} (w) \\ = \sum_{k = 1}^{m} \sum_{i = 1}^{m} w_{i} (\frac{1}{6} [\begin{matrix} | (μ_{{\tilde{c}}_{ij}}^{L})^{2} - (μ_{{\tilde{c}}_{kj}}^{L})^{2} | + | (μ_{{\tilde{c}}_{ij}}^{U})^{2} - (μ_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | (υ_{{\tilde{c}}_{ij}}^{L})^{2} - (υ_{{\tilde{c}}_{kj}}^{L})^{2} | + | (υ_{{\tilde{c}}_{ij}}^{U})^{2} - (υ_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | (π_{{\tilde{c}}_{ij}}^{L})^{2} - (π_{{\tilde{c}}_{kj}}^{L})^{2} | + | (π_{{\tilde{c}}_{ij}}^{U})^{2} - (π_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | μ_{{\tilde{c}}_{ij}}^{2} - μ_{{\tilde{c}}_{kj}}^{2} | + | υ_{{\tilde{c}}_{ij}}^{2} - υ_{{\tilde{c}}_{kj}}^{2} | + | π_{{\tilde{c}}_{ij}}^{2} - π_{{\tilde{c}}_{kj}}^{2} | \end{matrix}]) \end{matrix}$

We can create a non-linear programming model using above analysis to select weight vector w for the maximization of all deviation values of all attributes, as follows: $\begin{matrix} (M 1) max D (w) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} w_{i} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj}) \\ s . t \sum_{i = 1}^{n} w_{i} = 1 w_{j} ⩾ 0 j = 1, 2, \dots, n \end{matrix}$

For the solution of the model mention above we take L (w, ξ) =0 where $L (w, ξ) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} w_{i} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj}) + \frac{ξ}{2} (\sum_{j = 1}^{n} w_{j} - 1)$ which is a form of Lagrangian function for constrained optimization problem (M1), with Lagrange multiplier n. The partial derivatives of L are $\frac{\partial L}{\partial w_{j}} = \sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj}) + ξ w_{i} = 0$ (41) $\frac{\partial L}{\partial ξ} = \frac{1}{2} (\sum_{j = 1}^{n} w_{j} - 1) = 0$ (42) $w_{j} = \frac{- \sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj})}{ξ}$ (43) put (53) in (51) we get $ξ = - \sqrt[2]{\sum_{j = 1}^{n} {(\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj}))}^{2}}$ (44) clearly ξ < 0 where $\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj})$ and $\sqrt{\sum_{j = 1}^{n} {(\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj})}^{2}}$ represent the sum of deviations of all the alternatives with respect to the A_j attribute and the sum of deviations of all the alternatives with respect to all the attributes respectively. Combining Equations (53) and (54), we can get

$w_{j} = \frac{\sum_{j = 1}^{n} (\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj})}{\sqrt{\sum_{j = 1}^{n} (\sum_{i = 1}^{m} \sum_{k = 1}^{m} d ({\tilde{c}}_{ij}, {\tilde{c}}_{kj})^{2}}}$

By normalizing w_j (j = 1, 2, …, n), i.e. $\sum_{i = 1}^{n} w_{i} = 1$ , we get

$w_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} (w_{i} (\frac{1}{6} [\begin{matrix} | (μ_{{\tilde{c}}_{ij}}^{L})^{2} - (μ_{{\tilde{c}}_{kj}}^{L})^{2} | + | (μ_{{\tilde{c}}_{ij}}^{U})^{2} - (μ_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | (υ_{{\tilde{c}}_{ij}}^{L})^{2} - (υ_{{\tilde{c}}_{kj}}^{L})^{2} | + | (υ_{{\tilde{c}}_{ij}}^{U})^{2} - (υ_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | (π_{{\tilde{c}}_{ij}}^{L})^{2} - (π_{{\tilde{c}}_{kj}}^{L})^{2} | + | (π_{{\tilde{c}}_{ij}}^{U})^{2} - (π_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | μ_{{\tilde{c}}_{ij}}^{2} - μ_{{\tilde{c}}_{kj}}^{2} | + | (υ_{{\tilde{c}}_{ij}})^{2} - υ_{{\tilde{c}}_{kj}}^{2} | \\ + | π_{{\tilde{c}}_{ij}}^{2} - π_{{\tilde{c}}_{kj}}^{2} | \end{matrix}])}{\sum_{j = 1}^{n} [\sum_{i = 1}^{m} \sum_{k = 1}^{m} (w_{i} (\frac{1}{6} [\begin{matrix} | (μ_{{\tilde{c}}_{ij}}^{L})^{2} - (μ_{{\tilde{c}}_{kj}}^{L})^{2} | + | (μ_{{\tilde{c}}_{ij}}^{U})^{2} - (μ_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | (υ_{{\tilde{c}}_{ij}}^{L})^{2} - (υ_{{\tilde{c}}_{kj}}^{L})^{2} | + | (υ_{{\tilde{c}}_{ij}}^{U})^{2} - (υ_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | (π_{{\tilde{c}}_{ij}}^{L})^{2} - (π_{{\tilde{c}}_{kj}}^{L})^{2} | + | (π_{{\tilde{c}}_{ij}}^{U})^{2} - (π_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | μ_{{\tilde{c}}_{ij}}^{2} - μ_{{\tilde{c}}_{kj}}^{2} | + | υ_{{\tilde{c}}_{ij}}^{2} - υ_{{\tilde{c}}_{kj}}^{2} | + | π_{{\tilde{c}}_{ij}}^{2} - π_{{\tilde{c}}_{kj}}^{2} | \end{matrix}])]}$ (45)

For partially known weight information, we construct the following constrained optimization model:

$(M 2) {\begin{matrix} MaxD (w) \\ = \sum_{j = 1}^{n} [\sum_{i = 1}^{m} \sum_{k = 1}^{m} (\begin{matrix} w_{i} (\frac{1}{6} [| (μ_{\tilde{cij}}^{L})^{2} - (μ_{{\tilde{c}}_{kj}}^{L})^{2} | + | (μ_{{\tilde{c}}_{ij}}^{U})^{2} - (μ_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | (υ_{{\tilde{c}}_{ij}}^{L})^{2} - (υ_{{\tilde{c}}_{kj}}^{L})^{2} | + | (υ_{{\tilde{c}}_{ij}}^{U})^{2} - (υ_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | (π_{{\tilde{c}}_{ij}}^{L})^{2} - (π_{{\tilde{c}}_{kj}}^{L})^{2} | + | (π_{{\tilde{c}}_{ij}}^{U})^{2} - (π_{{\tilde{c}}_{kj}}^{U})^{2} | \\ + | μ_{{\tilde{c}}_{ij}}^{2} - μ_{{\tilde{c}}_{kj}}^{2} | + | υ_{{\tilde{c}}_{ij}}^{2} - υ_{{\tilde{c}}_{kj}}^{2} | + | π_{{\tilde{c}}_{ij}}^{2} - π_{{\tilde{c}}_{kj}}^{2} |])] \end{matrix})] \\ s . t w \in \nabla, w_{j} ⩾ 0, j = 1, 2, . . ., n, \sum_{i = 1}^{n} w_{i} = 1 \end{matrix}}$ (46) where ∇ is the set containing constraint of the weight w_j, which should be satisfied. Using MALTLAB we can simulate (M2) for the optimal solution of attribute weight vector w = (w₁, w₂, …, w_n).

5.2 An approach to multi-attribute decision making using Cubic Pythagorean fuzzy information

Using the presented model, we solve MADM problem, with incompletely known or completely unknown information about attributes weight having CPFN form. The algorithm has the following steps:

Step 1. Construct CPF decision matrices as $C = [{\tilde{c}}_{ij}]_{m \times n}$ where ${\tilde{c}}_{ij} = ({\tilde{p}}_{{\tilde{c}}_{ij}}, p_{{\tilde{c}}_{ij}}), i = 1, 2, \dots, n$ and j = 1, 2, …, m .

If there are cost and benefit attributes. Then convert Cubic Pythagorean decision-matrix into the normalized Cubic Pythagorean fuzzy decision-matrix. D_N = (λ_ij) _m×n such that $λ_{ij} = {\begin{matrix} {\tilde{c}}_{ij} if the attribute is benefit \\ {\tilde{c}}_{ij}^{c} if the attribute is cost \end{matrix}}$

Where ${\tilde{c}}_{ij}^{c} = (〈 [υ_{{\tilde{c}}_{ij}}^{L}, υ_{{\tilde{c}}_{ij}}^{U}], [μ_{{\tilde{c}}_{ij}}^{L}, μ_{{\tilde{c}}_{ij}}^{U}] 〉, (υ_{{\tilde{c}}_{ij}}, μ_{{\tilde{c}}_{ij}}))$ for i = 1, 2, …, n ; j = 1, 2, …, m .

Step 2. Use the proposed aggregation operators to collect all the decision matrices.

Step 3. For completely unknown attribute weights information, achieve the attribute weights using Equation (45); if the information is partially known, then use (M2) for attribute weights.

Step 4. Use the proposed aggregation operators to obtain the CPFN as a preference values ${\tilde{c}}_{i} (i = 1, 2, \dots, n)$ of X_i.

Step 5. Calculate the score function $S ({\tilde{c}}_{i}) (i = 1, 2, \dots, n)$ and the accuracy degree, of all ${\tilde{c}}_{i}$ (i = 1, 2, …, n) . , using expression (28) and (29) respectively.

Step 6. Select the best one by making the Ranking of alternatives X_i (i = 1, 2, …, n) .

6 An illustrative example

In this section, we are going to present an illustrative example of the new approach in a decision-making problem. We analyze a company that operates in Europe and North America that wants to invest some money in a new market. They consider five possible alternatives:

X₁: Invest in the Asian market.

X₂: Invest in the South American market.

X₃: Invest in the African market.

X₄: Invest in all three markets.

To evaluate these alternatives, the investor has brought together a group of three alternatives. After analyzing the information, this group considers that the key factor is the economic situation of the world economy for the next period. They consider five main possible states of nature that could happen in the future:

A₁: Very bad economic situation.

A₂: Bad economic situation.

A₃: Regular economic situation.

A₄: Good economic situation.

A₅: Very good economic situation.

The experts of the government evaluate and offer their own opinions regarding the results obtained with each alternative. The weights of experts of is given as w = (0.35, 0.4, 0.25) ^T. As the environment is very uncertain, the group of experts needs to assess the available information by using PFNs. The expected results given in the form of CPFNs depending on the characteristic A_i and the alternative X_j are shown in Tables 2 –4.

Table 2
Cubic Pythagorean fuzzy decision matrix by expert e₁

X ₁ X ₂ X ₃ X ₄

A ₁ (〈 [0 . 3, 0 . 4] , [0 . 7, 0 . 8] 〉 ; (0 . 5, 0 . 8)) (〈 [0 . 6, 0 . 70] , [0 . 5, 0 . 7] 〉 ; (0 . 6, 0 . 7)) (〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 7, 0 . 5)) (〈 [0 . 4, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 9, 0 . 4))

A ₂ (〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 8, 0 . 6)) (〈 [0 . 4, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 3, 0 . 9)) (〈 [0 . 7, 0 . 8] , [0 . 4, 0 . 5] 〉 ; (0 . 9, 0 . 3)) (〈 [0 . 5, 0 . 6] , [0 . 7, 0 . 8] 〉 ; (0 . 7, 0 . 5))

A ₃ (〈 [0 . 4, 0 . 5] , [0 . 7, 0 . 8] 〉 ; (0 . 3, 0 . 9)) (〈 [0 . 5, 0 . 6] , [0 . 7, 0 . 8] 〉 ; (0 . 8, 0 . 4)) (〈 [0 . 6, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 6, 0 . 7)) (〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 7] 〉 ; (0 . 8, 0 . 6))

A ₄ (〈 [0 . 7, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 7, 0 . 6)) (〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 5, 0 . 8)) (〈 [0 . 3, 0 . 4] , [0 . 8, 0 . 9] 〉 ; (0 . 8, 0 . 5)) (〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 6, 0 . 7))

A ₅ (〈 [0 . 6, 0 . 7] , [0 . 4, 0 . 5] 〉 ; (0 . 9, 0 . 4)) (〈 [0 . 3, 0 . 4] , [0 . 8, 0 . 9] 〉 ; (0 . 7, 0 . 6)) (([0 . 5, 0 . 6] , [0 . 7, 0 . 8]) ; (0 . 9, 0 . 4)) (〈 [0 . 3, 0 . 4] , [0 . 8, 0 . 9] 〉 ; (0 . 5, 0 . 8))

	X ₁	X ₂	X ₃	X ₄
A ₁	(〈 [0 . 3, 0 . 4] , [0 . 7, 0 . 8] 〉 ; (0 . 5, 0 . 8))	(〈 [0 . 6, 0 . 70] , [0 . 5, 0 . 7] 〉 ; (0 . 6, 0 . 7))	(〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 7, 0 . 5))	(〈 [0 . 4, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 9, 0 . 4))
A ₂	(〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 8, 0 . 6))	(〈 [0 . 4, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 3, 0 . 9))	(〈 [0 . 7, 0 . 8] , [0 . 4, 0 . 5] 〉 ; (0 . 9, 0 . 3))	(〈 [0 . 5, 0 . 6] , [0 . 7, 0 . 8] 〉 ; (0 . 7, 0 . 5))
A ₃	(〈 [0 . 4, 0 . 5] , [0 . 7, 0 . 8] 〉 ; (0 . 3, 0 . 9))	(〈 [0 . 5, 0 . 6] , [0 . 7, 0 . 8] 〉 ; (0 . 8, 0 . 4))	(〈 [0 . 6, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 6, 0 . 7))	(〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 7] 〉 ; (0 . 8, 0 . 6))
A ₄	(〈 [0 . 7, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 7, 0 . 6))	(〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 5, 0 . 8))	(〈 [0 . 3, 0 . 4] , [0 . 8, 0 . 9] 〉 ; (0 . 8, 0 . 5))	(〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 6, 0 . 7))
A ₅	(〈 [0 . 6, 0 . 7] , [0 . 4, 0 . 5] 〉 ; (0 . 9, 0 . 4))	(〈 [0 . 3, 0 . 4] , [0 . 8, 0 . 9] 〉 ; (0 . 7, 0 . 6))	(([0 . 5, 0 . 6] , [0 . 7, 0 . 8]) ; (0 . 9, 0 . 4))	(〈 [0 . 3, 0 . 4] , [0 . 8, 0 . 9] 〉 ; (0 . 5, 0 . 8))

Table 3

Cubic Pythagorean fuzzy decision matrix by expert e₂

	X ₁	X ₂	X ₃	X ₄
A ₁	(〈 [0 . 5, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 4, 0 . 9))	(〈 [0 . 4, 0 . 5] , [0 . 7, 0 . 8] 〉 ; (0 . 7, 0 . 6))	(〈 [0 . 3, 0 . 4] , [0 . 8, 0 . 9] 〉 ; (0 . 3, 0 . 8))	(〈 [0 . 2, 0 . 3] , [0 . 8, 0 . 9] 〉 ; (0 . 7, 0 . 5))
A ₂	(〈 [0 . 7, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 5, 0 . 8))	(〈 [0 . 5, 0 . 6] , [0 . 6, 0 . 7] 〉 ; (0 . 6, 0 . 5))	(〈 [0 . 4, 0 . 5] , [0 . 7, 0 . 8] 〉 ; (0 . 4, 0 . 7))	(〈 [0 . 4, 0 . 5] , [0 . 6, 0 . 8] 〉 ; (0 . 8, 0 . 6))
A ₃	(〈 [0 . 6, 0 . 7] , [0 . 6, 0 . 8] 〉 ; (0 . 6, 0 . 7))	(〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 6] 〉 ; (0 . 5, 0 . 7))	(〈 [0 . 5, 0 . 6] , [0 . 6, 0 . 7] 〉 ; (0 . 5, 0 . 6))	(〈 [0 . 6, 0 . 8] , [0 . 4, 0 . 6] 〉 ; (0 . 9, 0 . 4))
A ₄	(〈 [0 . 4, 0 . 5] , [0 . 4, 0 . 7] 〉 ; (0 . 7, 0 . 6))	(〈 [0 . 7, 0 . 8] , [0 . 4, 0 . 5] 〉 ; (0 . 4, 0 . 8))	(〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 6] 〉 ; (0 . 6, 0 . 5))	(〈 [0 . 8, 0 . 9] , [0 . 2, 0 . 4] 〉 ; (0 . 5, 0 . 8))
A ₅	(〈 [0 . 3, 0 . 4] , [0 . 8, 0 . 9] 〉 ; (0 . 8, 0 . 5))	(〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 4, 0 . 9))	(〈 [0 . 7, 0 . 8] , [0 . 4, 0 . 5] 〉 ; (0 . 7, 0 . 4))	(〈 [0 . 5, 0 . 7] , [0 . 6, 0 . 7] 〉 ; (0 . 6, 0 . 7))

Table 4

Cubic Pythagorean fuzzy decision matrix e₃

	X ₁	X ₂	X ₃	X ₄
A ₁	(〈 [0 . 7, 0 . 8] , [0 . 4, 0 . 6] 〉 ; (0 . 9, 0 . 4))	(〈 [0 . 8, 0 . 9] , [0 . 3, 0 . 4] 〉 ; (0 . 8, 0 . 3))	(〈 [0 . 5, 0 . 8] , [0 . 5, 0 . 6] 〉 ; (0 . 8, 0 . 5))	(〈 [0 . 3, 0 . 4] , [0 . 7, 0 . 8] 〉 ; (0 . 4, 0 . 9))
A ₂	(〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 7] 〉 ; (0 . 8, 0 . 5))	(〈 [0 . 7, 0 . 8] , [0 . 4, 0 . 5] 〉 ; (0 . 9, 0 . 4))	(〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 6] 〉 ; (0 . 7, 0 . 6))	(〈 [0 . 4, 0 . 6] , [0 . 6, 0 . 7] 〉 ; (0 . 5, 0 . 8))
A ₃	(〈 [0 . 5, 0 . 6] , [0 . 6, 0 . 7] 〉 ; (0 . 7, 0 . 6))	(〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 6] 〉 ; (0 . 6, 0 . 4))	(〈 [0 . 4, 0 . 6] , [0 . 6, 0 . 8] 〉 ; (0 . 9, 0 . 3))	(〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 6] 〉 ; (0 . 6, 0 . 5))
A ₄	(〈 [0 . 3, 0 . 4] , [0 . 7, 0 . 8] 〉 ; (0 . 6, 0 . 7))	(〈 [0 . 5, 0 . 6] , [0 . 6, 0 . 7] 〉 ; (0 . 5, 0 . 7))	(〈 [0 . 7, 0 . 8] , [0 . 4, 0 . 5] 〉 ; (0 . 6, 0 . 7))	(〈 [0 . 5, 0 . 8] , [0 . 4, 0 . 5] 〉 ; (0 . 7, 0 . 6))
A ₅	(〈 [0 . 4, 0 . 5] , [0 . 5, 0 . 7] 〉 ; (0 . 5, 0 . 8))	(〈 [0 . 5, 0 . 8] , [0 . 4, 0 . 6] 〉 ; (0 . 4, 0 . 8))	(〈 [0 . 4, 0 . 5] , [0 . 6, 0 . 7] 〉 ; (0 . 5, 0 . 4))	(〈 [0 . 6, 0 . 7] , [0 . 5, 0 . 7] 〉 ; (0 . 8, 0 . 4))

Suppose the information about the attribute weights is partly known and the known weight information is given as follows:

$\begin{matrix} Δ = {0.2 ⩽ w_{1}, 0.1 ⩽ w_{2} ⩽ w_{1}, 00.3 ⩽ w_{3} ⩽ 00.4, \\ 0.3 ⩾ w_{4}, 0.20 ⩽ w_{5} ⩽ 00.3, \\ w_{4} - w_{2} ⩽ w_{3} - w_{5}, \sum_{i = 1}^{5} w_{i} = 1} \end{matrix}$

For Cubic Pythagorean fuzzy weighted averaging (CPFWA) operator

Step 2. By using CPFWA operator we get the collective decision matrix (see Table 5)

Table 5

Cubic Pythagorean fuzzy decision matrix by using CPFWA operator

	X ₁	X ₂	X ₃	X ₄
A ₁	$\begin{matrix} (〈 [0 . 5208, 0 . 7181], [0 . 5320, 0 . 6636] 〉, \\ (0 . 6658, 0 . 7052)) \end{matrix}$	$\begin{matrix} (〈 [0 . 6181, 0 . 7315], [0 . 5035, 0 . 6420] 〉; \\ (0 . 7028, 0 . 5325)) \end{matrix}$	$\begin{matrix} (〈 [0 . 6109, 0 . 7720], [0 . 5046, 0 . 6123] 〉; \\ (0 . 6409, 0 . 6034)) \end{matrix}$	$\begin{matrix} (〈 [0 . 3098, 0 . 5960], [0 . 6564, 0 . 7583] 〉; \\ (0 . 7688, 0 . 5356)) \end{matrix}$
A ₂	$\begin{matrix} (〈 [0 . 7226, 0 . 8699], [0 . 3409, 0 . 4601] 〉; \\ (0 . 7191, 0 . 6432)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5398, 0 . 7395], [0 . 5086, 0 . 6097] 〉; \\ (0 . 6824, 0 . 5809)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5839, 0 . 6879], [0 . 5291, 0 . 6316] 〉; \\ (0 . 7478, 0 . 5007)) \end{matrix}$	$\begin{matrix} (〈 [0 . 4389, 0 . 5640], [0 . 6333, 0 . 7737] 〉; \\ (0 . 7151, 0 . 6049)) \end{matrix}$
A ₃	$\begin{matrix} (〈 [0 . 5173, 0 . 6182], [0 . 6333, 0 . 7737] 〉; \\ (0 . 5762, 0 . 7355)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5687, 0 . 6692], [0 . 5625, 0 . 6636] 〉; \\ (0 . 6652, 0 . 5004)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5197, 0 . 6905], [0 . 5629, 0 . 6857] 〉; \\ (0 . 7047, 0 . 5325)) \end{matrix}$	$\begin{matrix} (〈 [0 . 6000, 0 . 7459], [0 . 4573, 0 . 6333] 〉; \\ (0 . 8235, 0 . 4874)) \end{matrix}$
A ₄	$\begin{matrix} (〈 [0 . 5295, 0 . 6350], [0 . 4974, 0 . 6857] 〉; \\ (0 . 6784, 0 . 6236)) \end{matrix}$	$\begin{matrix} (〈 [0 . 7091, 0 . 8171], [0 . 4003, 0 . 5030] 〉; \\ (0 . 4639, 0 . 7737)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5622, 0 . 6658], [0 . 5574, 0 . 6607] 〉; \\ (0 . 6905, 0 . 5439)) \end{matrix}$	$\begin{matrix} (〈 [0 . 7533, 0 . 8815], [0 . 2741, 0 . 4229] 〉; \\ (0 . 5964, 0 . 7105)) \end{matrix}$
A ₅	$\begin{matrix} (〈 [0 . 4598, 0 . 5606], [0 . 5581, 0 . 6880] 〉; \\ (0 . 8088, 0 . 5201)) \end{matrix}$	$\begin{matrix} (〈 [0 . 6338, 0 . 7905], [0 . 4544, 0 . 5880] 〉; \\ (0 . 5428, 0 . 7583)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5820, 0 . 6863], [0 . 5385, 0 . 6411] 〉; \\ (0 . 7762, 0 . 4000)) \end{matrix}$	$\begin{matrix} (〈 [0 . 4782, 0 . 6266], [0 . 6340, 0 . 7644] 〉; \\ (0 . 6435, 0 . 6377)) \end{matrix}$

Step 3. Utilizing model (M2) to construct single objective model as: $(M 2) = {\begin{matrix} max : 1.4003 w_{1} + 1.7651 w_{2} + 1.3722 w_{3} \\ + 2.1846 w_{4} + 1.9084 w_{5} \\ s . t . w \in Δ, w_{i} ⩾ 0, i = 1, 2, 3, 4, 5 \end{matrix}$ the optimum solution for this problem is achieved by using MATLAB which is w = (0.2, 0.1, 0.3, 0.2, 0.2) ^T

Step 4. Using CPFWA operator to obtain the overall preference values of the alternatives X_j (j = 1, 2, 3, 4) using average, we get

$\begin{matrix} {\tilde{c}}_{1} = (〈 [0.5394, 0.6756], [0.5341, 0.6792] 〉; (0.6901, 0.6496)) \\ {\tilde{c}}_{2} = (〈 [0.6226, 0.7501], [0.4876, 0.6036] 〉; (0.6236, 0.6098)) \\ {\tilde{c}}_{3} = (〈 [0.5672, 0.7039], [0.5414, 0.6511] 〉; (0.7122, 0.5146)) \\ {\tilde{c}}_{4} = (〈 [0.5712, 0.7327], [0.4894, 0.6415] 〉; (0.7376, 0.5775)) \end{matrix}$

Step 5. Calculate the score $S ({\tilde{c}}_{j}) (j = 1, 2, 3, 4)$ of the overall CPFNs ${\tilde{c}}_{j} (j = 1, 2, 3, 4) :$ $\begin{matrix} S ({\tilde{c}}_{1}) = & - 0.0583, S ({\tilde{c}}_{2}) = & - 0.0669, \\ S ({\tilde{c}}_{3}) = & 0.0451, S ({\tilde{c}}_{4}) = & 0.0096 \end{matrix}$

Step 6. Rank all the alternatives X_j (j = 1, 2, 3, 4) according to the score $S ({\tilde{c}}_{j}) (j = 1, 2, 3, 4)$ of the overall cubic Pythagorean fuzzy preference values we have., $S ({\tilde{c}}_{3}) > S ({\tilde{c}}_{4}) > S ({\tilde{c}}_{2}) > S ({\tilde{c}}_{1})$ which shows that X₃ > X₄ > X₂ > X₁, that is the most desirable alternative is X₃.

For Cubic Pythagorean fuzzy weighted geometric (CPFWG) operator

Step 2. By using CPFWG operator we get the collective decision matrix (see Table 6).

Table 6

Pythagorean fuzzy decision matrix by using CPFWG operator

	X ₁	X ₂	X ₃	X ₄
A ₁	$\begin{matrix} (〈 [0 . 4548, 0 . 6277], [0 . 5708, 0 . 6905] 〉; \\ (0 . 5297, 0 . 8096)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5482, 0 . 6515], [0 . 5704, 0 . 7053] 〉; \\ (0 . 6857, 0 . 5954)) \end{matrix}$	$\begin{matrix} (〈 [0 . 4805, 0 . 6318], [0 . 6338, 0 . 7530] 〉; \\ (0 . 5157, 0 . 6639)) \end{matrix}$	$\begin{matrix} (〈 [0 . 2821, 0 . 4544], [0 . 7016, 0 . 8118] 〉; \\ (0 . 6646, 0 . 6681)) \end{matrix}$
A ₂	$\begin{matrix} (〈 [0 . 7058, 0 . 8452], [0 . 3646, 0 . 5084] 〉; \\ (0 . 6629, 0 . 6863)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5030, 0 . 7130], [0 . 5252, 0 . 6260] 〉; \\ (0 . 5210, 0 . 7231)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5385, 0 . 6411], [0 . 5755, 0 . 6801] 〉; \\ (0 . 6111, 0 . 5822)) \end{matrix}$	$\begin{matrix} (〈 [0 . 4325, 0 . 5578], [0 . 6394, 0 . 7793] 〉; \\ (0 . 6788, 0 . 6435)) \end{matrix}$
A ₃	$\begin{matrix} (〈 [0 . 4974, 0 . 5987], [0 . 6394, 0 . 7793] 〉; \\ (0 . 4892, 0 . 7861)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5629, 0 . 6632], [0 . 5871, 0 . 6905] 〉; \\ (0 . 6169, 0 . 5586)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5040, 0 . 6636], [0 . 5687, 0 . 7028] 〉; \\ (0 . 6173, 0 . 5954)) \end{matrix}$	$\begin{matrix} (〈 [0 . 6000, 0 . 7384], [0 . 4639, 0 . 6394] 〉; \\ (0 . 7804, 0 . 5075)) \end{matrix}$
A ₄	$\begin{matrix} (〈 [0 . 4528, 0 . 5574], [0 . 5361, 0 . 7028] 〉; \\ (0 . 6735, 0 . 6287)) \end{matrix}$	$\begin{matrix} (〈 [0 . 6743, 0 . 7758], [0 . 4392, 0 . 5398] 〉; \\ (0 . 4573, 0 . 7793)) \end{matrix}$	$\begin{matrix} (〈 [0 . 4892, 0 . 5950], [0 . 6350, 0 . 7515] 〉; \\ (0 . 6636, 0 . 5647)) \end{matrix}$	$\begin{matrix} (〈 [0 . 7113, 0 . 8739], [0 . 2979, 0 . 4283] 〉; \\ (0 . 5797, 0 . 7283)) \end{matrix}$
A ₅	$\begin{matrix} (〈 [0 . 4109, 0 . 5145], [0 . 6467, 0 . 7789] 〉; \\ (0 . 7412, 0 . 5920)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5046, 0 . 6579], [0 . 5960, 0 . 7304] 〉; \\ (0 . 4865, 0 . 8118)) \end{matrix}$	$\begin{matrix} (〈 [0 . 5410, 0 . 6432], [0 . 5839, 0 . 6879] 〉; \\ (0 . 7027, 0 . 4000)) \end{matrix}$	$\begin{matrix} (〈 [0 . 4376, 0 . 5755], [0 . 6750, 0 . 7994] 〉; \\ (0 . 6049, 0 . 6990)) \end{matrix}$

Step 3. Utilizing model (M2) to construct single objective model as: $M 2 = {\begin{matrix} \begin{matrix} max : 1.4070 w_{1} + 1.8401 w_{2} + 1.4698 w_{3} \\ + 2.3159 w_{4} + 1.5668 w_{5} \end{matrix} \\ s . t . w \in Δ, w_{i} ⩾ 0, i = 1, 2, 3, 4, 5 \end{matrix}$ the optimum solution for this problem is achieved by using MATLAB which is w = (0.2, 0.1, 0.3, 0.2, 0.2) ^T.

Step 4. Using CPFWG operator to obtain the overall preference values of the alternatives X_j (j = 1, 2, 3, 4) using average, we get $\begin{matrix} {\tilde{c}}_{1} = (〈 [0.4779, 0.5983], [0.5895, 0.7305] 〉; (0.5935, 0.7271)) \\ {\tilde{c}}_{2} = (〈 [0.5616, 0.6858], [0.5551, 0.6730] 〉; (0.5565, 0.6897)) \\ {\tilde{c}}_{3} = (〈 [0.5067, 0.6367], [0.6003, 0.7192] 〉; (0.6194, 0.5745)) \\ {\tilde{c}}_{4} = (〈 [0.4850, 0.6411], [0.5748, 0.7132] 〉; (0.6674, 0.6488)) \end{matrix}$

Step 5. Calculate the score $S ({\tilde{c}}_{j}) (j = 1, 2, 3, 4)$ of the overall CPFNs ${\tilde{c}}_{j} (j = 1, 2, 3, 4) :$ $\begin{matrix} S ({\tilde{c}}_{1}) = & - 0.1671, S ({\tilde{c}}_{2}) = & - 0.1579, \\ S ({\tilde{c}}_{3}) = & - 0.0496, S ({\tilde{c}}_{4}) = & - 0.0763 \end{matrix}$

Step 6. Rank all the alternatives X_j (j = 1, 2, 3, 4) according to the score $S ({\tilde{c}}_{j}) (j = 1, 2, 3, 4)$ of the overall cubic Pythagorean fuzzy preference values we have $S ({\tilde{c}}_{3}) > S ({\tilde{c}}_{4}) > S ({\tilde{c}}_{2}) > S ({\tilde{c}}_{1})$ which shows that X₃ > X₄ > X₂ > X₁, that is the most desirable alternative is X₃.

6.1 Comparison Analysis

In order to verify the validity and effectiveness of the proposed approach, a comparative study is conducted using the methods of PFNs Yager et al. [24] and IVPFNs Peng and Yang [15] which are special cases of CPFNs, to the same illustrative example.

6.1.1 Comparison with Pythagorean fuzzy numbers

PFNs can be considered as a special case of PHFNs when we take only non-membership degree. For comparison, the PHNs can be transformed to PFNs by removing membership degrees.

6.1.1.1 For Pythagorean fuzzy weighted averaging operator [23]

After transformation, the Pythagorean fuzzy information can be shown in Table 7.

Table 7
Pythagorean fuzzy averaging decision matrix

X ₁ X ₂ X ₃ X ₄

A ₁ (0.6658, 0.7052) (0.7028, 0.5325) (0.6409, 0.6034) (0.7688, 0.5356)

A ₂ (0.7191, 0.6432) (0.6824, 0.5809) (0.7478, 0.5007) (0.7151, 0.6049)

A ₃ (0.5762, 0.7355) (0.6652, 0.5004) (0.7047, 0.5325) (0.8235, 0.4874)

A ₄ (0.6784, 0.6236) (0.4639, 0.7737) (0.6905, 0.5439) (0.5964, 0.7105)

A ₅ (0.8088, 0.5201) (0.5428, 0.7583) (0.7762, 0.4000) (0.6435, 0.6377)

	X ₁	X ₂	X ₃	X ₄
A ₁	(0.6658, 0.7052)	(0.7028, 0.5325)	(0.6409, 0.6034)	(0.7688, 0.5356)
A ₂	(0.7191, 0.6432)	(0.6824, 0.5809)	(0.7478, 0.5007)	(0.7151, 0.6049)
A ₃	(0.5762, 0.7355)	(0.6652, 0.5004)	(0.7047, 0.5325)	(0.8235, 0.4874)
A ₄	(0.6784, 0.6236)	(0.4639, 0.7737)	(0.6905, 0.5439)	(0.5964, 0.7105)
A ₅	(0.8088, 0.5201)	(0.5428, 0.7583)	(0.7762, 0.4000)	(0.6435, 0.6377)

Step 2. By using Pythagorean fuzzy Averaging Operator [23]: $PFWA = \sqrt{1 - Π_{i = 1}^{n} (1 - (μ_{c_{i}})^{2})^{w_{i}}}, Π_{i = 1}^{n} (υ_{c_{i}})^{w_{i}})$ and consider weight w = (0.2, 0.1, 0.3, 0.2, 0.2) ^T we get the overall Pythagorean fuzzy preference values as: $\begin{matrix} p_{1} = (0.6901, 0.6496), p_{2} = (0.6236, 0.6098), \\ p_{3} = (0.7122, 0.5146), p_{4} = (0.7376, 0.5775) \end{matrix}$

Step 3. Calculate the score S (p_i) (i = 1, 2, 3, 4) of overall PFNs p_i (i = 1, 2, 3, 4) $\begin{matrix} S (p_{1}) = & 0.0543, S (p_{2}) = & 0.0170, \\ S (p_{3}) = & 0.2424, S (p_{4}) = & 0.2106 \end{matrix}$

Step 4. Rank all the alternatives X_j (j = 1, 2, 3, 4) according to the score $S (p_{j}) = μ_{j}^{2} - υ_{j}^{2} (j = 1, 2, 3, 4)$ of the overall Pythagorean fuzzy preference values we have., S (p₃) > S (p₄) > S (p₂) > S (p₁) which shows that X₃ > X₄ > X₂ > X₁, that is the most desirable alternative is X₃, which is the same as our approach but in our approach in membership degree there is an interval valued Pythagorean fuzzy set and in non-membership degree there is a Pythagorean fuzzy set. While the approach developed in [23], has only single element in both memberships degrees. Thus the approach developed in this paper is more reliable than developed in [23].

For Pythagorean fuzzy weighted geometric operator [23]

After transformation, the Pythagorean fuzzy information can be shown in Table 8

Table 8

Pythagorean fuzzy geometric decision matrix

	X ₁	X ₂	X ₃	X ₄
A ₁	(0 . 5297, 0 . 8096)	(0 . 6857, 0 . 5954)	(0 . 5157, 0 . 6639)	(0 . 6646, 0 . 6681)
A ₂	(0 . 6629, 0 . 6863)	(0 . 5210, 0 . 7231)	(0 . 6111, 0 . 5822)	(0 . 6788, 0 . 6435)
A ₃	(0 . 4892, 0 . 7861)	(0 . 6169, 0 . 5586)	(0 . 6173, 0 . 5954)	(0 . 7804, 0 . 5075)
A ₄	(0 . 6735, 0 . 6287)	(0 . 4573, 0 . 7793)	(0 . 6636, 0 . 5647)	(0 . 5797, 0 . 7283)
A ₅	(0 . 7412, 0 . 5920)	(0 . 4865, 0 . 8118)	(0 . 7027, 0 . 4000)	(0 . 6049, 0 . 6990)

Step 2. By using Pythagorean fuzzy Averaging Operator [23]: $PFWG = (Π_{i = 1}^{n} (μ_{c_{i}})^{w_{i}}, \sqrt{1 - Π_{i = 1}^{n} (1 - (υ_{i})^{2})^{w_{i}}})$ and consider weight w = (0.2, 0.1, 0.3, 0.2, 0.2) ^T we get the overall Pythagorean fuzzy preference values as: $\begin{matrix} p_{1} = (0.5935, 0.7271), p_{2} = (0.5565, 0.6897), \\ p_{3} = (0.6194, 0.5745), p_{4} = (0.6674, 0.6488) \end{matrix}$

Step 3. Calculate the score S (p_i) (i = 1, 2, 3, 4) of overall PFNs p_i (i = 1, 2, 3, 4) $\begin{matrix} S (p_{1}) = & - 0.1764, S (p_{2}) = & - 0.1660, \\ S (p_{3}) = & 0.0536, S (p_{4}) = & 0.0245 \end{matrix}$

6.1.2 Comparison with Interval-valued Pythagorean fuzzy numbers

IVPFNs can be considered as a special case of PHFNs when we take only non-membership degree. For comparison, the CPFNs can be transformed to IVPFNs by removing non-membership degrees.

For Interval valued Pythagorean fuzzy weighted averaging operator [15]:

After transformation, the interval valued Pythagorean fuzzy information can be shown in Table 9.

Table 9
Interval valued averaging Pythagorean fuzzy decision matrix

X ₁ X ₂ X ₃ X ₄

A ₁ $\begin{matrix} 〈 [0 . 5208, 0 . 7181], \\ [0 . 5320, 0 . 6636] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 6181, 0 . 7315], \\ [0 . 5035, 0 . 6420] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 6109, 0 . 7720], \\ [0 . 5046, 0 . 6123] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 3098, 0 . 5960], \\ [0 . 6564, 0 . 7583] 〉 \end{matrix}$

A ₂ $\begin{matrix} 〈 [0 . 7226, 0 . 8699], \\ [0 . 3409, 0 . 4601] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 5398, 0 . 7395], \\ [0 . 5086, 0 . 6097] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 5839, 0 . 6879], \\ [0 . 5291, 0 . 6316] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 4389, 0 . 5640], \\ [0 . 6333, 0 . 7737] 〉 \end{matrix}$

A ₃ $\begin{matrix} 〈 [0 . 5173, 0 . 6182], \\ [0 . 6333, 0 . 7737] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 5687, 0 . 6692], \\ [0 . 5625, 0 . 6636] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 5197, 0 . 6905], \\ [0 . 5629, 0 . 6857] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 6000, 0 . 7459], \\ [0 . 4573, 0 . 6333] 〉 \end{matrix}$

A ₄ $\begin{matrix} 〈 [0 . 5295, 0 . 6350], \\ [0 . 4974, 0 . 6857] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 7091, 0 . 8171], \\ [0 . 4003, 0 . 5030] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 5622, 0 . 6658], \\ [0 . 5574, 0 . 6607] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 7533, 0 . 8815], \\ [0 . 2741, 0 . 4229] 〉 \end{matrix}$

A ₅ $\begin{matrix} 〈 [0 . 4598, 0 . 5606], \\ [0 . 5581, 0 . 6880] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 6338, 0 . 7905], \\ [0 . 4544, 0 . 5880] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 5820, 0 . 6863], \\ [0 . 5385, 0 . 6411] 〉 \end{matrix}$ $\begin{matrix} 〈 [0 . 4782, 0 . 6266], \\ [0 . 6340, 0 . 7644] 〉 \end{matrix}$

	X ₁	X ₂	X ₃	X ₄
A ₁	$\begin{matrix} 〈 [0 . 5208, 0 . 7181], \\ [0 . 5320, 0 . 6636] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 6181, 0 . 7315], \\ [0 . 5035, 0 . 6420] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 6109, 0 . 7720], \\ [0 . 5046, 0 . 6123] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 3098, 0 . 5960], \\ [0 . 6564, 0 . 7583] 〉 \end{matrix}$
A ₂	$\begin{matrix} 〈 [0 . 7226, 0 . 8699], \\ [0 . 3409, 0 . 4601] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5398, 0 . 7395], \\ [0 . 5086, 0 . 6097] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5839, 0 . 6879], \\ [0 . 5291, 0 . 6316] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 4389, 0 . 5640], \\ [0 . 6333, 0 . 7737] 〉 \end{matrix}$
A ₃	$\begin{matrix} 〈 [0 . 5173, 0 . 6182], \\ [0 . 6333, 0 . 7737] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5687, 0 . 6692], \\ [0 . 5625, 0 . 6636] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5197, 0 . 6905], \\ [0 . 5629, 0 . 6857] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 6000, 0 . 7459], \\ [0 . 4573, 0 . 6333] 〉 \end{matrix}$
A ₄	$\begin{matrix} 〈 [0 . 5295, 0 . 6350], \\ [0 . 4974, 0 . 6857] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 7091, 0 . 8171], \\ [0 . 4003, 0 . 5030] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5622, 0 . 6658], \\ [0 . 5574, 0 . 6607] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 7533, 0 . 8815], \\ [0 . 2741, 0 . 4229] 〉 \end{matrix}$
A ₅	$\begin{matrix} 〈 [0 . 4598, 0 . 5606], \\ [0 . 5581, 0 . 6880] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 6338, 0 . 7905], \\ [0 . 4544, 0 . 5880] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5820, 0 . 6863], \\ [0 . 5385, 0 . 6411] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 4782, 0 . 6266], \\ [0 . 6340, 0 . 7644] 〉 \end{matrix}$

Step 2. By using Interval valued Pythagorean fuzzy Averaging Operator [15]: $IVPFWA = 〈 \begin{matrix} , \\ [Π_{i = 1}^{n} (υ_{c_{i}}^{L})^{w_{i}}, Π_{i = 1}^{n} (υ_{c_{i}}^{U})^{w_{i}}] \end{matrix} 〉$ and consider weight w = (0.2, 0.1, 0.3, 0.2, 0.2) ^T we get the overall interval valued Pythagorean fuzzy preference values as: $\begin{matrix} {\tilde{p}}_{1} = 〈 [0.5394, 0.6756], [0.5341, 0.6792] 〉, \\ {\tilde{p}}_{2} = 〈 [0.6226, 0.7501], [0.4876, 0.6036] 〉 \\ {\tilde{p}}_{3} = 〈 [0.5672, 0.7039], [0.5414, 0.6511] 〉, \\ {\tilde{p}}_{4} = 〈 [0.5712, 0.7327], [0.4894, 0.6415] 〉 \end{matrix}$

Step 3. Calculate the score $S ({\tilde{p}}_{i}) (i = 1, 2, 3, 4)$ of overall IVPFNs ${\tilde{p}}_{i} (i = 1, 2, 3, 4)$ by using Definition [4]. $\begin{matrix} S ({\tilde{p}}_{1}) = & - 0.1708, S ({\tilde{p}}_{2}) = & - 0.1508, \\ S ({\tilde{p}}_{3}) = & - 0.1523, S ({\tilde{p}}_{4}) = & - 0.1913 \end{matrix}$

Step 4. Rank all the alternatives X_j (j = 1, 2, 3, 4) according to the score $S ({\tilde{p}}_{j}) (j = 1, 2, 3, 4)$ of the overall interval valued Pythagorean fuzzy preference values we have., $S ({\tilde{p}}_{2}) > S ({\tilde{p}}_{3}) > S ({\tilde{p}}_{1}) > S ({\tilde{p}}_{4})$ which shows that X₂ > X₃ > X₁ > X₄, that is the most desirable alternative is X₂, which is different from our approach as in our approach, the membership degree is an interval valued Pythagorean fuzzy set and in non-membership degree there is a Pythagorean fuzzy set. While the approach developed in [15], has only intervals in both memberships degrees. Thus the approach developed in this paper is more reliable than developed in [15].

For Interval valued Pythagorean fuzzy weighted geometric operator [15]:

After transformation, the interval valued Pythagorean fuzzy information can be shown in Table 10.

Table 10

Interval valued geometric Pythagorean fuzzy decision matrix

	X ₁	X ₂	X ₃	X ₄
A ₁	$\begin{matrix} 〈 [0 . 4548, 0 . 6277], \\ [0 . 5708, 0 . 6905] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5482, 0 . 6515], \\ [0 . 5704, 0 . 7053] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 4805, 0 . 6318], \\ [0 . 6338, 0 . 7530] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 2821, 0 . 4544], \\ [0 . 7016, 0 . 8118] 〉 \end{matrix}$
A ₂	$\begin{matrix} 〈 [0 . 7058, 0 . 8452], \\ [0 . 3646, 0 . 5084] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5030, 0 . 7130], \\ [0 . 5252, 0 . 6260] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5385, 0 . 6411], \\ [0 . 5755, 0 . 6801] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 4325, 0 . 5578], \\ [0 . 6394, 0 . 7793] 〉 \end{matrix}$
A ₃	$\begin{matrix} 〈 [0 . 4974, 0 . 5987], \\ [0 . 6394, 0 . 7793] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5629, 0 . 6632], \\ [0 . 5871, 0 . 6905] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5040, 0 . 6636], \\ [0 . 5687, 0 . 7028] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 6000, 0 . 7384], \\ [0 . 4639, 0 . 6394] 〉 \end{matrix}$
A ₄	$\begin{matrix} 〈 [0 . 4528, 0 . 5574], \\ [0 . 5361, 0 . 7028] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 6743, 0 . 7758], \\ [0 . 4392, 0 . 5398] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 4892, 0 . 5950], \\ [0 . 6350, 0 . 7515] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 7113, 0 . 8739], \\ [0 . 2979, 0 . 4283] 〉 \end{matrix}$
A ₅	$\begin{matrix} 〈 [0 . 4109, 0 . 5145], \\ [0 . 6467, 0 . 7789] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5046, 0 . 6579], \\ [0 . 5960, 0 . 7304] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 5410, 0 . 6432], \\ [0 . 5839, 0 . 6879] 〉 \end{matrix}$	$\begin{matrix} 〈 [0 . 4376, 0 . 5755], \\ [0 . 6750, 0 . 7994] 〉 \end{matrix}$

Table 11

Comparison analysis with existing methods

Method	Ranking of alternatives
PFWA operator [23]	X₃ > X₄ > X₂ > X₁
PFWG operator [23]	X₃ > X₄ > X₂ > X₁
IVPFWA operator [15]	X₂ > X₃ > X₁ > X₄
IVPFWG operator [15]	X₃ > X₄ > X₂ > X₁
CPFWA operator	X₃ > X₄ > X₂ > X₁
CPFWG operator	X₃ > X₄ > X₂ > X₁

Step 2. By using Interval valued Pythagorean fuzzy Averaging Operator [15]: $IVPFWG = 〈 \begin{matrix} , \\ [Π_{i = 1}^{n} (μ_{c_{i}}^{L})^{w_{i}}, Π_{i = 1}^{n} (μ_{c_{i}}^{U})^{w_{i}}], \\ [\sqrt{1 - Π_{i = 1}^{n} (1 - (υ_{i}^{L})^{2})^{w_{i}}}, \\ \sqrt{1 - Π_{i = 1}^{n} (1 - (υ_{i}^{U})^{2})^{w_{i}}}] \end{matrix} 〉$ and consider weight w = (0.2, 0.1, 0.3, 0.2, 0.2) ^T we get the overall interval valued Pythagorean fuzzy preference values as: $\begin{matrix} {\tilde{p}}_{1} = 〈 [0.4779, 0.5983], [0.5895, 0.7305] 〉, \\ {\tilde{p}}_{2} = 〈 [0.5616, 0.6858], [0.5551, 0.6730] 〉, \\ {\tilde{p}}_{3} = 〈 [0.5067, 0.6367], [0.6003, 0.7192] 〉, \\ {\tilde{p}}_{4} = 〈 [0.4850, 0.6411], [0.5748, 0.7132] 〉 \end{matrix}$

Step 3. Calculate the score $S ({\tilde{p}}_{i}) (i = 1, 2, 3, 4)$ of overall IVPFNs ${\tilde{p}}_{i} (i = 1, 2, 3, 4)$ by using Definition [4]. $\begin{matrix} S ({\tilde{p}}_{1}) = & - 0.1579, S ({\tilde{p}}_{2}) = & - 0.1498, \\ S ({\tilde{p}}_{3}) = & - 0.1528, S ({\tilde{p}}_{4}) = & - 0.1770 \end{matrix}$

Step 4. Rank all the alternatives X_j (j = 1, 2, 3, 4) according to the score $S ({\tilde{p}}_{j}) (j = 1, 2, 3, 4)$ of the overall interval valued Pythagorean fuzzy preference values we have., $S ({\tilde{p}}_{2}) > S ({\tilde{p}}_{3}) > S ({\tilde{p}}_{1}) > S ({\tilde{p}}_{4})$ which shows that X₃ > X₄ > X₂ > X₁, that is the most desirable alternative is X₃, which is different from our approach as in our approach, the membership degree is an interval valued Pythagorean fuzzy set and in non-membership degree there is a Pythagorean fuzzy set. While the approach developed in [15], has only intervals in both memberships degrees. Thus the approach developed in this paper is more reliable than developed in [15]

7 Conclusion

Yager in [23], introduced the concept of Pythagorean fuzzy set as a generalization of intuitionistic fuzzy set [1]. In [15], Peng and Yang initiated the concept of interval valued Pythagorean fuzzy set. Based on the PFS and IVPFS in this paper we introduced the concept of Cubic Pythagorean fuzzy set, in which the membership degree is an IVPFS and non-membership degree is a PFS. We defined some basic operations and discussed some properties of the proposed operation. To compare two cubic Pythagorean fuzzy numbers (CPFNs) we introduced score function and accuracy function. We also defined distance between two CPFNs. Based on the developed operation we proposed Cubic Pythagorean fuzzy weighted averaging (CPFWA) operator and Cubic Pythagorean fuzzy weighted geometric (CPFWG) operator and discussed some of its properties. Moreover, we developed a multi-attribute decision making based on the proposed aggregation operators. Also we presented a numerical example to show the validity and effectiveness of the proposed approach. Finally, we compared the proposed approach to existing methods.

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