Extended topsis method based on Pythagorean cubic fuzzy multi-criteria decision making with incomplete weight information

Abstract

The purpose of this article is to propose a new technique to Pythagorean cubic fuzzy (PCF) multiple criteria decision making (MCDM) using the Topsis method with incomplete weight information. Firstly, the maximum deviation model is proposed to find the criteria for determining the optimal weights. Based on the developed method, a MCDM approach is proposed using PCF information. Furthermore, a numerical example is presented to show the feasibility and effectiveness of the proposed information. Finally, a systematic evaluation analysis is given to compare the present work with the existing work.

Keywords

Pythagorean cubic fuzzy set maximize deviation model unknown weight information multi criteria decision-making

1 Introduction

Multi criteria group decision making (MCGDM) is an important tool for selecting the most significant option from all of the possible alternatives in the estimation and selection process. It has been widely applied to a variety of real-world situations. Zhang and Guo [35] developed a VIKOR based method uncertain preference ordinals and incomplete weight information to solve group decision making (GDM) problems. In [36] Zhang et al. developed a novel computational model based on the use of extended linguistic hierarchies, which not only can be used to operate with multigranular linguistic distribution assessments but also can provide interpretable linguistic results to decision makers. Zhang et al. [29] presented a comprehensive review regarding the different approaches to consensus reaching processes (CRP) and presented a series of CRPs as the comparison objects. Because of increasingly complex manufacturing processes and requirements, it is often infeasible for decision makers (DMs) or experts to consider every relevant factor during the evaluation and selection process. Zadeh [28] proposed fuzzy sets to solve a real-life decision making problem characterizing them with membership function. After their introduction, fuzzy sets many authors have extended the concept and applied to MCGDM problems. For instance Yu et al. [27] developed a new method to deal with multi-criteria group decision making (MCGDM) problems with unbalanced hesitant fuzzy linguistic term sets (HFLTSs) by considering the psychological behavior of decision makers. In [33] Zhang et al., proposed an approach to deriving a priority weight vector from an incomplete HFPR using the logarithmic least squares method. Zhang et al. [30] developed a consensus-based group decision-making framework for failure mode and effect analysis with the aim of classifying failure modes into several ordinal risk classes in which we assumed that failure mode and effect analysis participants provide their preferences in a linguistic way using possibilistic hesitant fuzzy linguistic information. Fuzzy set were then generalized to intuitionistic fuzzy set (IFS) [1] which are described by membership degree and non-membership degrees with the condition that the sum of the membership degree and non-membership degrees is less than or equal to one. Xu [22] developed intuitionistic fuzzy aggregation operators. Zhang and Guo [34] proposed a group decision making with intuitionistic multiplicative preference relations. IFS is further extended to Pythagorean fuzzy set (PFS) by Yager [23 –26] which can be described by membership degree and non-membership degrees with the condition that the square sum of the membership degree and non-membership degrees is less than or equal to one. Yager [23] demonstrated that PFS are able to model types of imprecision in decision making problems that IFS cannot. PFS has been applied to MCGDM problems. Khan et al. [10] developed a MCDM approach based on Pythagorean fuzzy Einstein prioritized aggregation operators. Khan [11] developed Pythagorean fuzzy Einstein Choquet integral averaging operator and Pythagorean fuzzy Einstein Choquet integral geometric operator to deal with multi-criteria group decision making problems. PFS has been further generalized to interval valued Pythagorean fuzzy set (IVPFS) [31], relaxing the conditions on membership degree and non-membership degrees such that the sum of their squares is less than or equal to one. Researchers have developed numerous extensions to these techniques and models to address uncertainty in MCDM problems.

TOPSIS is often a crucial part of the decision-making processes. TOPSIS collects and compares a groups information by finding weight for each criteria and then using a distance measure formula for the ideal solution to a decision making problem. TOPSIS method relies on the assumption that the function of the criteria is monotonic. If the parameters to the TOPSIS method are not in the form required by MCDM problems than normalization is necessary. An advantage of the TOPSIS technique is that it allows unnecessary parameters to be replaced by parameters that are required for the decision-making problems in ways that other models are unsuitable for.

TOPSIS method has been applied to a variety of decision making problems. For example, it has been used to determine the best locations to plant crops [22], for identifying fluids that best performed heat transfer [5], and in the supplier selection process for an investment firm [3]. Khan et al. [7, 8], used the Choquet integral TOPSIS technique with IVPFNs to solve MCGDM problems and IVPF GRA method for MCDM with incomplete weight information. Other important works presenting generalizations of TOPSIS with FS theory for MCDM problems include [2 , 32]. Wu and Xu [38] developed hesitant fuzzy linguistic weighted average operator and hesitant fuzzy linguistic ordered weighted average operator to address multiple attribute GDM with hesitant fuzzy linguistic information. In [39] Wu et al. developed a MCGDM approach based on VIKOR and TOPSIS method under hesitant fuzzy linguistic information. In [13] Khan et al., generalized the concept of PFS with hesitant fuzzy set and introduced the concept of Pythagorean hesitant fuzzy sets (PHFSs). The PHFS is characterized by a MD and a NMD are sets of some possible values between 0 and 1 respectively. The authors developed some elementary operational laws for PHFSs and based on these operational laws they proposed some aggregation operators to aggregate the PHFNs. Moreover Khan et al. [14 –16] developed some more aggregation operators for dealing with Pythagorean hesitant fuzzy MADM problems. Furthermore Khan et al. [17, 18] proposed TOPSIS method and VIKOR method under PHFS environments to deal with MADM problem. Jun in 2012 generalized the concept of IFS and initiated the notion of cubic set. Khan et al. [9] generalized the concept PFS and IVPFS, and introduced the concept of Pythagorean cubic fuzzy set (PCFS), which is a better tool to deal with imprecision and uncertainty as it contain both the PFS and IVPFS. To derive the PCFS practically, suppose there is a person who want to invest some money. Before investing, he wants to analyze the estimation of the interest on his investment and found that it would be 50–60% tending towards the profit side, while 45% towards the loss side. However, after the completion of the certain months, they found that his return to be 35–40% agreeing to his earlier profit estimate, while towards the loss estimates it is 70% disagreeing. Such information is represented as (〈[0.50, 0.60] , 0.45〉, 〈[0.35, 0.40] , 0.70〉). Following the methods proposed in [21], we present a new extension to TOPSIS for solving MCDM problems.

Section 2 reviews the important properties of PFS and PCFS. Section 3 introduces a novel PCFS approach to extend TOPSIS method for handling MCGDM problems and Section 4 demonstrates a practical use case. Section 5 provides a comparative analysis of the proposed technique with other well-known decision-making methods to demonstrate the effectiveness of the approach.

2 Preliminaries

In this section, important properties and definitions PFS and PCFS for later use in proposed technique.

Definition 1. [1] Let X be a universe of discourse, an intuitionistic fuzzy set (IFS) I can be defined on X as follows: $I = {x, 〈 μ_{I} (x), ν_{I} (x) 〉 | x \in X},$ (1)

Where μ_I (x) and ν_I (x) are mappings from X → [0, 1] satisfies the condition 0 ⩽ μ_I (x) + ν_I (x) ⩽1 for all x ∈ X.

Definition 2. [6] Let X be a universe of discourse, an cubic set C can be defined on X as follows: $C = {x, 〈 \tilde{μ} (x), ν (x) 〉 | x \in X}$ (2)

Where $\tilde{μ} (x)$ is an interval-valued fuzzy set in X and ν (x) is a fuzzy set in X.

Definition 3. [23] Let X be a universe of discourse, a Pythagorean fuzzy set (PFS) P can be defined on X as follows: $P = {x, 〈 μ_{P} (x), ν_{P} (x) 〉 | x \in X},$ (3)

Where μ_P (x) and ν_P (x) are mappings from X:→ [0, 1], such that $0 ⩽ μ_{P}^{2} (x) + ν_{P}^{2} (x) ⩽ 1$ for all x ∈ X.

Definition 4. [9] Let X be a universe of discourse, a Pythagorean fuzzy set (PFS) P can be defined on X as follows: $P_{c} = {〈 x, (μ_{c} (x), ν_{c} (x)) 〉 | x \in X},$ (4)

Where μ_c (x) = 〈A (x) , λ (x) 〉, $ν_{c} (x) = 〈 \tilde{A} (x), μ$ (x) 〉 and $0 ⩽ (μ_{c} (x))^{2} + (ν_{c} (x))^{2} ⩽ [\tilde{1}, 1] .$ Or 0 $⩽ (\sup (A (x)))^{2} + (\sup (\tilde{A} (x)))^{2} ⩽ 1$ , 0⩽ λ² (x) + μ² (x) ⩽1. The degree of indeterminacy of the PCFS can be defined as: $π_{P_{c}} = 〈 \begin{matrix} \sqrt{1 - (\sup (A (x)))^{2} - (\sup (\tilde{A} (x)))^{2}}, \\ \sqrt{1 - λ^{2} (x) - μ^{2} (x)} \end{matrix} 〉$

For simplicity, we call (μ_{c
₁}, ν_{c
₂}) (PCFN) denoted by $p_{c} = (〈 A, λ 〉, 〈 \tilde{A}, μ 〉)$ .

Definition 5. [9] Let $p_{c_{1}} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ , $p_{c_{2}} = (〈 A_{2}, λ_{2} 〉, 〈 {\tilde{A}}_{2}, μ_{2} 〉)$ and $p_{c} = (〈 A, λ 〉, 〈 \tilde{A},$ μ〉) is three PCFNs and δ > 0, where A₁ = [a₁, b₁], ${\tilde{A}}_{1} = [{\tilde{a}}_{1}, {\tilde{b}}_{1}]$ , A₂ = [a₂, b₂], ${\tilde{A}}_{2} = [{\tilde{a}}_{2}, {\tilde{b}}_{2}]$ , A = [a, b], $\tilde{A} = [\tilde{a}, \tilde{b}]$ the operation laws are:

p_c
₁ ⊕ p_c
₂ $= (\begin{matrix} 〈 [\sqrt{a_{1}^{2} + a_{2}^{2} - a_{1}^{2} a_{2}^{2}}, \sqrt{b_{1}^{2} + b_{2}^{2} - b_{1}^{2} b_{2}^{2}}], \\ \sqrt{λ_{1}^{2} + λ_{2}^{2} - λ_{1}^{2} λ_{2}^{2}} 〉, 〈 [{\tilde{a}}_{1} {\tilde{a}}_{2}, {\tilde{b}}_{1} {\tilde{b}}_{2}], μ_{1} μ_{2} 〉 \end{matrix}),$

p_c
₁ ⊗ p_c
₂ $= (\begin{matrix} 〈 [a_{1} a_{2}, b_{1} b_{2}], λ_{1} λ_{2} 〉, 〈 [\sqrt{{\tilde{a}}_{1}^{2} + {\tilde{a}}_{2}^{2} - {\tilde{a}}_{1}^{2} {\tilde{a}}_{2}^{2}}, \\ \sqrt{{\tilde{b}}_{1}^{2} + {\tilde{b}}_{2}^{2} - {\tilde{b}}_{1}^{2} {\tilde{b}}_{2}^{2}}], \sqrt{μ_{1}^{2} + μ_{2}^{2} - μ_{1}^{2} μ_{2}^{2}} 〉 \end{matrix}),$

δp _c
₁ $= (\begin{matrix} 〈 [\sqrt{1 - (1 - a_{1}^{2})^{δ}}, \sqrt{1 - (1 - b_{1}^{2})^{δ}}], \\ \sqrt{1 - (1 - λ_{1}^{2})^{δ}} 〉, 〈 [({\tilde{a}}_{1} {\tilde{a}}_{2})^{δ}, ({\tilde{b}}_{1} {\tilde{b}}_{2})^{δ}], (μ_{1} μ_{2})^{δ} 〉 \end{matrix}),$

$p_{c_{1}}^{δ}$ $= (\begin{matrix} 〈 [(a_{1} a_{2})^{δ}, (b_{1} b_{2})^{δ}], (λ_{1} λ_{2})^{δ} 〉, 〈 [\sqrt{1 - (1 - {\tilde{a}}_{1}^{2})^{δ}}, \\ \sqrt{1 - (1 - {\tilde{b}}_{1}^{2})^{δ}}], \sqrt{1 - (1 - μ_{1}^{2})^{δ}}) 〉 \end{matrix}) .$

$p_{c_{1}}^{c} = 〈 {\tilde{A}}_{1}, μ_{1} 〉, 〈 A_{1}, λ_{1} 〉$ .

Definition 6. [9] Let $p_{c_{1}} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ , $p_{c_{2}} = (〈 A_{2}, λ_{2} 〉, 〈 {\tilde{A}}_{2}, μ_{2} 〉)$ and $p_{c} = (〈 A, λ 〉, 〈 \tilde{A},$ μ〉) are 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

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

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

Definition 7. [9] Let $p_{c} = (〈 A, λ 〉, 〈 \tilde{A}, μ 〉)$ be a PCFN, where A = [a, b], $\tilde{A} = [\tilde{a}, \tilde{b}]$ . We can introduce the score function of p_c as: $S (p_{c}) = {(\frac{a + b - λ}{3})}^{2} - {(\frac{\tilde{a} + \tilde{b} - μ}{3})}^{2} .$ (5)

Where S (p_c) ∈ [-1, 1].

Definition 8. [9] 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}.

i) If S (p_{c
₁})< S (p_{c
₂}) ⇒ p_{c
₁} < p_{c
₂}.

ii) If S (p_{c
₁})> S (p_{c
₂}) ⇒ p_{c
₁} > p_{c
₂}.

iii) If S (p_{c
₁}) = S (p_{c
₂})⇒ p_{c
₁} ∼ p_{c
₂}.

Definition 9. [9] Let $p_{c} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ be a PCFN. Then we define the accuracy degree of p_c, denoted by α (p_c), where A₁ = [a₁, b₁], ${\tilde{A}}_{1} = [{\tilde{a}}_{1}, {\tilde{b}}_{1}]$ , can be defined as: $α (p_{c}) = {(\frac{a_{1} + b_{1} - λ_{1}}{3})}^{2} + {(\frac{{\tilde{a}}_{1} + {\tilde{b}}_{1} - μ_{1}}{3})}^{2} .$ (6)

Where α (p_c) ∈ [0, 1].

Definition 10. [9] Let p_{c
₁} and p_{c
₂} be any two PCFNs on a set X = {x₁, x₂, . . . , x_n}. The distance measure between p_{c
₁} and p_{c
₂} can be defined in the following way: $d (p_{c_{1}}, p_{c_{2}}) = \frac{1}{6} [\begin{matrix} | a_{1}^{2} - a_{2}^{2} | + | b_{1}^{2} - b_{2}^{2} | + | {\tilde{a}}_{1}^{2} - {\tilde{a}}_{2}^{2} | + \\ | {\tilde{b}}_{1}^{2} - {\tilde{b}}_{2}^{2} | + | λ_{1}^{2} - λ_{2}^{2} | + | μ_{1}^{2} - μ_{2}^{2} | \end{matrix}]$ (7)

Definition 11. [9] Let p_{c
_i} = (μ_{c
_ij}, ν_{c
_ij}) be a collection of all PCFNs and w = (w_1, w_2,_…, w_n)^T be the weight vector of ${\hat{c}}_{i}$ with w_i ⩾ 0 such that w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ . Then (PCFWA) operator is a mapping PCFWA : PCFNⁿ → PCFN $\begin{matrix} PCFWA (p_{c_{1}}, p_{c_{2}}, . . ., p_{c_{n}}) \\ = w_{1} p_{c_{1}} \oplus w_{2} p_{c_{2}} \oplus, . . ., \oplus w_{n} p_{c_{n}} \\ = (\begin{matrix} 〈 \begin{matrix} [\sqrt{1 - \prod_{i = 1}^{n} (1 - a_{i}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - b_{i}^{2})^{w_{i}}}]; \\ \sqrt{1 - \prod_{i = 1}^{n} (1 - λ_{i}^{2})^{w_{i}}} \end{matrix} 〉, \\ 〈 [\prod_{i = 1}^{n} {\tilde{a}}_{i}^{w_{i}}, \prod_{i = 1}^{n} {\tilde{b}}_{i}^{w_{i}}]; \prod_{i = 1}^{n} μ_{i}^{w_{i}} 〉 \end{matrix}) \end{matrix}$ (8) and the PCFWA operator is said to be a Pythagorean cubic fuzzy weighted averaging operator.

Definition 12. [9] Let p_{c
_i} = (μ_{c
_ij}, ν_{c
_ij}) be a collection 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 Pythagorean cubic fuzzy (PCFWG) operator is a mapping PCFWG : PCFNⁿ → PCFN can be defined as $\begin{matrix} PCFWG (p_{c_{1}}, p_{c_{2}}, . . ., p_{c_{n}}) = p_{c_{1}}^{w_{1}} \otimes p_{c_{2}}^{w_{2}} \otimes, . . ., \otimes p_{c_{n}}^{w_{n}} \\ = (\begin{matrix} 〈 [\prod_{i = 1}^{n} a_{i}^{w_{i}}, \prod_{i = 1}^{n} b_{i}^{w_{i}}]; \prod_{i = 1}^{n} λ_{i}^{w_{i}} 〉, \\ 〈 \begin{matrix} [\sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{a}}_{i}^{2})^{w_{i}}}, \sqrt{1 - \prod_{i = 1}^{n} (1 - {\tilde{b}}_{i}^{2})^{w_{i}}}]; \\ \sqrt{1 - \prod_{i = 1}^{n} (1 - μ_{i}^{2})^{w_{i}}} \end{matrix} 〉 \end{matrix}) \end{matrix}$ (9) and the PCFWG operator is said to be a Pythagorean cubic fuzzy weighted geometric operator.

3 Multi-criteria group decision-making based on Pythagorean cubic fuzzy environment

In this section we present a multi-criteria group decision making approach based on Pythagorean cubic fuzzy Topsis method with unknown weight information.

3.1 Description of the problem

The MCGDM problems are represented as a decision making process which give ranking information for the attributes with respect to the criteria. We propose a Pythagorean cubic fuzzy decision making process, which specifies not only the information about the alternatives X_i that satisfy the criteria A_j, but also the extent to which X_i does not fulfill the criteria A_j. Suppose that we have a MCDM function with a set of m alternatives X ={ X₁, X₂, . . . , X_m }. Let X be a collection of alternatives and A = {A₁, A₂, . . . , A_n} be the collection of criteria. To compute the efficiency of the i-th alternative X_i in the j-th criteria A_j, the decision-maker must use knowledge about alternative X_is fulfillment of criteria A_j, but about its non-fullfillment of A_j. This alternative A_j and criteria X_i can also be represented by μ_{c
_ij} and v_{c
_ij} which indicate the membership degree and non-membership degree. The efficiency of the alternative X based on the criteria A_j is denoted by a PCFN p_{c
_ij} =〈 μ_{c
_ij}, v_{c
_ij} 〉 with the condition that for all p_{1c_ij} ∈ μ_{c
_ij}, ∃ $p_{1_{c_{ij}}}^{'} \in v_{c_{ij}}$ such that $0 ⩽ {(p_{c_{ij}})}^{2} + {(p_{c_{ij}}^{'})}^{2} ⩽ 1$ , and for all p_{1c_ij} ∈ μ_{c
_ij}, ∃ $p_{1_{c_{ij}}}^{'} \in v_{c_{ij}}$ such that $0 ⩽ {(p_{c_{ij}})}^{2} + {(p_{c_{ij}}^{'})}^{2} ⩽ 1$ . The PCF decision-matrix C: $C = [\begin{matrix} p_{c_{11}} & p_{c_{12}} & \dots & p_{c_{1 n}} \\ p_{c_{21}} & p_{c_{22}} & \dots & p_{c_{2 n}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ p_{c_{m 1}} & p_{c_{m 2}} & \dots & p_{c_{mn}} \end{matrix}]$

Considering that the attributes have different importance degrees, the weight vector of all the attributes, given by the DMs, is defined by $w = {(w_{1}, w_{2}, \dots, w_{n})}^{T}, 1 ⩾ w_{j} ⩾ 0, j = 1, 2, \dots, n, \sum_{j = 1}^{n} w_{j} = 1$ and w_j is the importance degree of each attribute. Due to the complexity and uncertainty of practical decision making problems and the inherent subjective nature of human thinking, the information about attribute weights is usually incomplete. For convenience, let Δ be a set of the known weight information [12, 19], where Δ can be constructed by the following forms, 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.

3.2 Determining the optimal weight of criteria by maximizing deviation method

The Optimal weight information plays an important an role in MCDM. Motivated by the idea propsed by Wu and Chen [37] in this section we present a maximizing deviation methodology to define the criteria weights for explaining MCDM problems with numerical information, the criteria having a greater deviation value compared to the alternatives must be allocated a greater weight, while the criteria having a small distance value compared to the alternatives would be assigned a lesser weight. Hence, we make an optimization model constructed using the maximizing deviation approach to define the optimal weight of criteria. For the criteria, A_j ∈ A, the distance of the alternatives X_i can be defined as:

$D_{ij} (w) = \sum_{k = 1}^{m} w_{i} d (p_{c_{ij}}, p_{c_{kj}}),$ i=1,2, ... ,m and j = 1,2, ... ,n Where

$d (p_{c_{ij}}, p_{c_{kj}}) = \frac{1}{6} [\begin{matrix} | a_{ij}^{2} - a_{kj}^{2} | + | b_{ij}^{2} - b_{kj}^{2} | + | {\tilde{a}}_{ij}^{2} - {\tilde{a}}_{kj}^{2} | + \\ | {\tilde{b}}_{ij}^{2} - {\tilde{b}}_{kj}^{2} | + | λ_{ij}^{2} - λ_{kj}^{2} | + | μ_{ij}^{2} - μ_{kj}^{2} | \end{matrix}]$ denotes the Pythagorean cubic fuzzy distance between the PCFNs p_{c
_ij} and p_{c
_kj} While $(p_{c_{ij}}, p_{c_{kj}}) = (〈 A_{ij}, λ_{ij} 〉, 〈 {\tilde{A}}_{kj}, μ_{kj} 〉)$ be a PCFNs, where A_ij = [a_ij, b_ij], ${\tilde{A}}_{kj} = [{\tilde{a}}_{kj}, {\tilde{b}}_{kj}]$ .

Definition 13. Let $D_{j} (w) = \sum_{i = 1}^{m} D_{ij} (w)$

$\sum_{i = 1}^{m} \sum_{k = 1}^{m} w_{j} (\begin{matrix} \frac{1}{6} [| a_{ij}^{2} - a_{kj}^{2} | + | b_{ij}^{2} - b_{kj}^{2} | + | {\tilde{a}}_{ij}^{2} - {\tilde{a}}_{kj}^{2} | + \\ | {\tilde{b}}_{ij}^{2} - {\tilde{b}}_{kj}^{2} | + | λ_{ij}^{2} - λ_{kj}^{2} | + | μ_{ij}^{2} - μ_{kj}^{2} |] \end{matrix})$

j=1,2...,n. Then D_j (w) denotes the distance of the alternatives for the criteria A_j ∈ A. On the basis of the proposed model to define a non-linear model to choose the weight vector w which maximizes the deviation: $M (1) (\begin{matrix} maxD (w) = \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} w_{j} d (p_{c_{ij}}, p_{c_{kj}}) \\ s . t w_{j} ⩾ 0, j = 1, 2, . . ., n, \sum_{j = 1}^{n} w_{j} = 1 \end{matrix}$

To explain this model, we have $\begin{matrix} L (w, ζ) = & \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} w_{j} d (p_{c_{ij}}, p_{c_{kj}}) \\ + \frac{ζ}{2} (\sum_{j = 1}^{n} w_{j}^{2} - 1) = 0 \end{matrix}$

which indicates the Lagrange function of the constrained optimization problem (M-1), where ζ is a real number, denoting the Lagrange multiplier variable. The partial derivatives of L are computed as: $\frac{\partial L}{\partial w_{j}} = \sum_{i = 1}^{m} \sum_{k = 1}^{m} d (p_{c_{ij}}, p_{c_{kj}}) + ζ \sum_{j = 1}^{n} w_{j} = 0$ (10)

$\frac{\partial L}{\partial ζ} = \frac{1}{2} (\sum_{j = 1}^{n} w_{j}^{2} - 1) = 0$ (11)

This implies that $w_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (p_{c_{ij}}, p_{c_{kj}})}{ζ}, j = 1, 2, . . ., n$ (12)

Putting (12) in (11) $ζ = - \sqrt{\sum_{j = 1}^{n} (\sum_{i = 1}^{m} \sum_{k = 1}^{m} d {(p_{c_{i j}}, p_{c_{i j}})}^{2})}$ (13)

Clearly ζ < 0, $\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (p_{c_{ij}}, p_{c_{kj}})$ which is the sum of the distance measures of the alternatives with respect to the jth criteria and $\sqrt{\sum_{j = 1}^{n} (\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (p_{c_{ij}}, p_{c_{kj}}))^{2}}$ means the sum of deviations of all the alternatives with respect to all the criteria. Combining Equations (12) and (13), we have, $w_{j} = \frac{\sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{k = 1}^{m} {\hat{w}}_{j} d (p_{c_{ij}}, p_{c_{kj}})}{\sqrt{\sum_{j = 1}^{n} (\sum_{i = 1}^{m} \sum_{k = 1}^{m} d (p_{c_{ij}}, p_{c_{kj}}))^{2}}}$ (14)

By normalizing w_j (j = 1,2,..., n), we make their sum into a unit, and get $w_{j} = \frac{\sum_{i = 1}^{m} \sum_{k = 1}^{m} (\frac{1}{6} [\begin{matrix} | a_{ij}^{2} - a_{kj}^{2} | + | b_{ij}^{2} - b_{kj}^{2} | + | {\tilde{a}}_{ij}^{2} - {\tilde{a}}_{kj}^{2} | + \\ | {\tilde{b}}_{ij}^{2} - {\tilde{b}}_{kj}^{2} | + | λ_{ij}^{2} - λ_{kj}^{2} | + | μ_{ij}^{2} - μ_{kj}^{2} | \end{matrix}])}{\sum_{j = 1}^{n} (\sum_{j = 1}^{m} \sum_{k = 1}^{m} (\frac{1}{6} [\begin{matrix} | a_{ij}^{2} - a_{kj}^{2} | + | b_{ij}^{2} - b_{kj}^{2} | + | {\tilde{a}}_{ij}^{2} - {\tilde{a}}_{kj}^{2} | + \\ | {\tilde{b}}_{ij}^{2} - {\tilde{b}}_{kj}^{2} | + | λ_{ij}^{2} - λ_{kj}^{2} | + | μ_{ij}^{2} - μ_{kj}^{2} | \end{matrix}]))}$ (15)

However, there are actual situations that the information about the weight vector is not completely unknown but partially known. For these cases, based on the set of the known weight information Δ, we construct the following constrained optimization model: $M (2) (\begin{matrix} maxD (w) = \sum_{j = 1}^{n} (\sum_{j = 1}^{m} \sum_{k = 1}^{m} (\frac{1}{6} [\begin{matrix} | a_{ij}^{2} - a_{kj}^{2} | + | b_{ij}^{2} - b_{kj}^{2} | + | {\tilde{a}}_{ij}^{2} - {\tilde{a}}_{kj}^{2} | \\ + | {\tilde{b}}_{ij}^{2} - {\tilde{b}}_{kj}^{2} | + | λ_{ij}^{2} - λ_{kj}^{2} | + | μ_{ij}^{2} - μ_{kj}^{2} | \end{matrix}])) \\ s . t w_{j} ⩾ 0, j = 1, 2, . . ., n, \sum_{j = 1}^{n} w_{j} = 1 \end{matrix}$ (16)

Where Δ is also a set of constraint conditions that the weight value w_j. should satisfy according to the requirements in real situations. The model (M-2) is a linear programming model that can be executed by utilizing the LINGO 11.0 software. By solving this model, we get the optimal solution w = (w₁,w₂, ... ,w_n)^T, which can be used as the weight vector of criteria.

3.3 Algorithm

In the process of Pythagorean cubic fuzzy aggregation operators [9], it produces the loss of too much information due to the complexity of the aggregation process of Pythagorean cubic fuzzy aggregation operators, which implies a deficiency of precision in the final results. Therefore, in order to overcome this disadvantage, we have extended the TOPSIS method to take Pythagorean cubic fuzzy information into account and used the distance measures of PCFNs to obtain the final ranking of the alternatives. TOPSIS is a kind of method to solve MCGDM problems, which aims at choosing the alternative with the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS), and is widely used for deal with the ranking problems in real situations. Under the PCF notation, the Pythagorean cubic fuzzy positive ideal solution (PCF-PIS), is represented by p+, and the Pythagorean cubic fuzzy negative ideal solution (PCF-NIS) p^-, can be written with:

Let $p_{c_{1}} = (〈 A_{1}, λ_{1} 〉, 〈 {\tilde{A}}_{1}, μ_{1} 〉)$ are PCFNs, where A₁ = [a₁, b₁], ${\tilde{A}}_{1} = [{\tilde{a}}_{1}, {\tilde{b}}_{1}]$ .

$\begin{matrix} p^{+} = & {X_{j}, 〈 \max_{i} {A_{1}, {\tilde{A}}_{1}}, \min_{j} {λ_{1}, μ_{1}} 〉 | \\ j = 1, 2, . . ., n} \end{matrix}$ (17)

$\begin{matrix} p^{-} = & {X_{j}, 〈 \min_{i} {A_{1}, {\tilde{A}}_{1}}, \max_{j} {λ_{1}, μ_{1}} 〉 | \\ j = 1, 2, . . ., n}^{T} \end{matrix}$ (18)

The separation of the alternatives can be derived by using equation (7). The separation measures d⁺ and d^-, of each alternative for PCF-PIS p⁺ and the PCF-NIS p^-, separately,

$\begin{matrix} d^{+} = & \sum_{j = 1}^{n} w_{j} d (p_{c_{ij}}, p_{c_{j}^{+}}) \\ = & \frac{1}{6} \sum_{j = 1}^{n} w_{j} [\begin{matrix} | a_{ij}^{2} - a_{j}^{+^{2}} | + | b_{ij}^{2} - b_{j}^{+^{2}} | + | {\tilde{a}}_{ij}^{2} - {\tilde{a}}_{j}^{+^{2}} | + \\ | {\tilde{b}}_{ij}^{2} - {\tilde{b}}_{j}^{+^{2}} | + | λ_{ij}^{2} - λ_{j}^{+^{2}} | + | μ_{ij}^{2} - μ_{j}^{+^{2}} | \end{matrix}] \\ i = 1, 2, . . ., n \end{matrix}$ (19)

$\begin{matrix} d^{-} = \sum_{j = 1}^{n} w_{j} d (p_{c_{ij}}, p_{c_{j}^{-}}) \\ = \frac{1}{6} \sum_{j = 1}^{n} w_{j} [\begin{matrix} | a_{ij}^{2} - a_{j}^{-^{2}} | + | b_{ij}^{2} - b_{j}^{-^{2}} | + | {\tilde{a}}_{ij}^{2} - {\tilde{a}}_{j}^{-^{2}} | \\ + | {\tilde{b}}_{ij}^{2} - {\tilde{b}}_{j}^{-^{2}} | + | λ_{ij}^{2} - λ_{j}^{-^{2}} | + | μ_{ij}^{2} - μ_{j}^{-^{2}} | \end{matrix}] \end{matrix}$ (20)

The relative closeness coefficient for the X_i to the PCF (PIS p⁺): $C_{i} = \frac{d_{i}^{-}}{d_{i}^{+} + d_{i}^{-}}$ (21) where 0 ⩽ C_i ⩽ 1 ; i = 1, 2, . . . , m. Clearly alternative X_i is nearer to the PCF-PIS p⁺ and further from the PCF-NIS p^- as C_i approaches 1. Therefore, according to the closeness coefficient C_i, we can determine the ranking orders of all alternatives and select the best one from a set of feasible alternatives.

Based on the above models, we shall develop a practical approach for solving MCDM problems, in which the information about attribute weights is incompletely known or completely unknown, and the attribute values take the form of PCFNs.

Step 1. In the MCDM problem, we construct the decision matrices C = (p_{c
_ij}) _m×n where all the argument p_{c
_ij} = 〈μ_{c
_ij}, ν_{c
_ij}〉, (i = 1, 2, . . . , m, j = 1, 2, . . . , n) are PCFNs, given by the decision makers, for the alternative X_i ∈ X and criteria A_j ∈ A.

Step 2. Aggregate all the Pythagorean cubic fuzzy decision matrices by using the PCFWG operator.

Step 3. If the knowlege about the criteria weights are totally unknown, they can be obtained using the M(1) model; If the knowledge about the criteria weights is incompletely known, then we use model M(2) to calculate the criteria weights.

Step 4. Use equation (17) and equation (18) to get the PCF PIS p⁺ and the PCF NIS p^-.

Step 5. Use equation (19) and equation (20) to compute the separation measures $d_{i}^{+}$ and $d_{i}^{-}$ , separately.

Step 6. Utilize the equation (21) to compute the relative closeness coefficient C_i.

Step 7. Rank the obtained alternatives X_i and then choose the best one.

4 Illustrative example

In this section we shall present a numerical example to show potential evaluation of emerging technology commercialization with Pythagorean cubic fuzzy information in order to illustrate the method proposed in this paper. There is a panel with three possible emerging technology enterprises X_i (i = 1, 2, 3, 4, 5) to select. The experts selects three attribute to evaluate the three possible emerging technology enterprises: (1) A₁ is the technical advancement; (2) A₂ is the potential market and market risk; (3) A₃ is the industrialization infrastructure (4) A₃ is the human resources and financial conditions. The five possible emerging technology enterprises F(i = 1, 2, 3) are to be evaluated using the Pythagorean cubic fuzzy information by three decision makers whose weighted vector is (0.35, 0.4, 0.25)^T and the Pythagorean cubic fuzzy decision matrices are presented in Table 1 –3.

Table 1
PCFDM C₁ by DMs d₁

A ₁ A ₂ A ₃ A ₄

X ₁ (〈 [0.7, 0.8] ; 0.7 〉 , 〈 [0.4, 0.5] ; 0.5 〉) (〈 [0.6, 0.7] ; 0.6 〉 , 〈 [0.5, 0.6] ; 0.5 〉) (〈 [0.6, 0.8] ; 0.7 〉 , 〈 [0.5, 0.6] ; 0.5 〉) (〈 [0.6, 0.7] ; 0.8 〉 , 〈 [0.5, 0.7] ; 0.4 〉)

X ₂ (〈 [0.5, 0.8] ; 0.2 〉 , 〈 [0.5, 0.6] ; 0.9 〉) (〈 [0.7, 0.8] ; 0.4 〉 , 〈 [0.4, 0.5] ; 0.8 〉) (〈 [0.6, 0.7] ; 0.8 〉 , 〈 [0.5, 0.6] ; 0.4 〉) (〈 [0.6, 0.7] ; 0.9 〉 , 〈 [0.5, 0.6] ; 0.2 〉)

X ₃ (〈 [0.7, 0.8] ; 0.5 〉 , 〈 [0.4, 0.5] ; 0.7 〉) (〈 [0.6, 0.7] ; 0.8 〉 , 〈 [0.5, 0.6] ; 0.5 〉) (〈 [0.7, 0.8] ; 0.8 〉 , 〈 [0.4, 0.6] ; 0.4 〉) (〈 [0.6, 0.7] ; 0.3 〉 , 〈 [0.4, 0.6] ; 0.8 〉)

X ₄ (〈 [0.7, 0.8] ; 0.7 〉 , 〈 [0.4, 0.5] ; 0.6 〉) (〈 [0.7, 0.8] ; 0.6 〉 , 〈 [0.4, 0.5] ; 0.7 〉) (〈 [0.7, 0.8] ; 0.7 〉 , 〈 [0.4, 0.5] ; 0.4 〉) (〈 [0.7, 0.8] ; 0.9 〉 , 〈 [0.4, 0.5] ; 0.2 〉)

	A ₁	A ₂	A ₃	A ₄
X ₁	(〈 [0.7, 0.8] ; 0.7 〉 , 〈 [0.4, 0.5] ; 0.5 〉)	(〈 [0.6, 0.7] ; 0.6 〉 , 〈 [0.5, 0.6] ; 0.5 〉)	(〈 [0.6, 0.8] ; 0.7 〉 , 〈 [0.5, 0.6] ; 0.5 〉)	(〈 [0.6, 0.7] ; 0.8 〉 , 〈 [0.5, 0.7] ; 0.4 〉)
X ₂	(〈 [0.5, 0.8] ; 0.2 〉 , 〈 [0.5, 0.6] ; 0.9 〉)	(〈 [0.7, 0.8] ; 0.4 〉 , 〈 [0.4, 0.5] ; 0.8 〉)	(〈 [0.6, 0.7] ; 0.8 〉 , 〈 [0.5, 0.6] ; 0.4 〉)	(〈 [0.6, 0.7] ; 0.9 〉 , 〈 [0.5, 0.6] ; 0.2 〉)
X ₃	(〈 [0.7, 0.8] ; 0.5 〉 , 〈 [0.4, 0.5] ; 0.7 〉)	(〈 [0.6, 0.7] ; 0.8 〉 , 〈 [0.5, 0.6] ; 0.5 〉)	(〈 [0.7, 0.8] ; 0.8 〉 , 〈 [0.4, 0.6] ; 0.4 〉)	(〈 [0.6, 0.7] ; 0.3 〉 , 〈 [0.4, 0.6] ; 0.8 〉)
X ₄	(〈 [0.7, 0.8] ; 0.7 〉 , 〈 [0.4, 0.5] ; 0.6 〉)	(〈 [0.7, 0.8] ; 0.6 〉 , 〈 [0.4, 0.5] ; 0.7 〉)	(〈 [0.7, 0.8] ; 0.7 〉 , 〈 [0.4, 0.5] ; 0.4 〉)	(〈 [0.7, 0.8] ; 0.9 〉 , 〈 [0.4, 0.5] ; 0.2 〉)

Table 2

PCFDM C₂ by d₂

	A ₁	A ₂	A ₃	A ₄
X ₁	(〈 [0.4, 0.5] ; 0.5 〉 , 〈 [0.7, 0.8] ; 0.7 〉)	(〈 [0.4, 0.6] ; 0.6 〉 , 〈 [0.7, 0.8] ; 0.5 〉)	(〈 [0.8, 0.9] ; 0.6 〉 , 〈 [0.3, 0.4] ; 0.7 〉)	(〈 [0.7, 0.8] ; 0.7 〉 , 〈 [0.4, 0.5] ; 0.4 〉)
X ₂	(〈 [0.4, 0.5] ; 0.8 〉 , 〈 [0.7, 0.8] ; 0.5 〉)	(〈 [0.3, 0.4] ; 0.8 〉 , 〈 [0.7, 0.8] ; 0.5 〉)	(〈 [0.5, 0.7] ; 0.4 〉 , 〈 [0.6, 0.7] ; 0.7 〉)	(〈 [0.7, 0.8] ; 0.7 〉 , 〈 [0.3, 0.6] ; 0.6 〉)
X ₃	(〈 [0.7, 0.8] ; 0.4 〉 , 〈 [0.4, 0.5] ; 0.7 〉)	(〈 [0.4, 0.5] ; 0.7 〉 , 〈 [0.7, 0.8] ; 0.5 〉)	(〈 [0.5, 0.8] ; 0.7 〉 , 〈 [0.6, 0.7] ; 0.4 〉)	(〈 [0.5, 0.6] ; 0.5 〉 , 〈 [0.6, 0.7] ; 0.6 〉)
X ₄	(〈 [0.7, 0.8] ; 0.7 〉 , 〈 [0.4, 0.5] ; 0.6 〉)	(〈 [0.6, 0.8] ; 0.5 〉 , 〈 [0.5, 0.6] ; 0.6 〉)	(〈 [0.6, 0.8] ; 0.8 〉 , 〈 [0.5, 0.7] ; 0.4 〉)	(〈 [0.4, 0.5] ; 0.8 〉 , 〈 [0.7, 0.8] ; 0.3 〉)

Table 3

PCFDM C₃ by d₃

	A ₁	A ₂	A ₃	A ₄
X ₁	(〈 [0.6, 0.7] ; 0.7 〉 , 〈 [0.5, 0.6] ; 0.5 〉)	(〈 [0.7, 0.8] ; 0.6 〉 , 〈 [0.4, 0.5] ; 0.7 〉)	(〈 [0.4, 0.5] ; 0.4 〉 , 〈 [0.7, 0.6] ; 0.7 〉)	(〈 [0.5, 0.6] ; 0.8 〉 , 〈 [0.6, 0.7] ; 0.4 〉)
X ₂	(〈 [0.6, 0.8] ; 0.3 〉 , 〈 [0.4, 0.5] ; 0.9 〉)	(〈 [0.4, 0.5] ; 0.3 〉 , 〈 [0.7, 0.8] ; 0.8 〉)	(〈 [0.6, 0.7] ; 0.5 〉 , 〈 [0.5, 0.6] ; 0.6 〉)	(〈 [0.6, 0.7] ; 0.5 〉 , 〈 [0.5, 0.6] ; 0.7 〉)
X ₃	(〈 [0.7, 0.8] ; 0.4 〉 , 〈 [0.4, 0.5] ; 0.7 〉)	(〈 [0.6, 0.7] ; 0.4 〉 , 〈 [0.5, 0.6] ; 0.7 〉)	(〈 [0.7, 0.8] ; 0.8 〉 , 〈 [0.4, 0.5] ; 0.4 〉)	(〈 [0.6, 0.7] ; 0.3 〉 , 〈 [0.4, 0.6] ; 0.8 〉)
X ₄	(〈 [0.6, 0.7] ; 0.7 〉 , 〈 [0.5, 0.6] ; 0.6 〉)	(〈 [0.6, 0.8] ; 0.6 〉 , 〈 [0.4, 0.5] ; 0.7 〉)	(〈 [0.7, 0.8] ; 0.8 〉 , 〈 [0.4, 0.5] ; 0.3 〉)	(〈 [0.5, 0.6] ; 0.9 〉 , 〈 [0.5, 0.7] ; 0.3 〉)

Case 1. Assume that the information about the attribute weights is completely unknown; we get the most desirable alternative(s) according to the following steps:

Step 1. The decision-makers provide judgement in Table 1 –3.

Step 2. Aggregate all the Pythagorean cubic fuzzy decision matrices by using the PCFWG operator.

Step 3: By using equation (16) with the matrix from Table 4 we obtain the weight vector: $w = (0.2990, 0.2288, 0.2102, 0.2620)^{T}$

Table 4

Collective PCFDM by using PCFWG operator

	A ₁	A ₂	A ₃	A ₄
X ₁	$(\begin{matrix} 〈 [0.5343, 0.6369]; 0.6119 〉, \\ 〈 [0.5788, 0.6832]; 0.5977 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5343, 0.6850]; 0.6000 〉, \\ 〈 [0.6538, 0.6696]; 0.5761 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5961, 0.7283]; 0.5564 〉, \\ 〈 [0.5275, 0.5352]; 0.6538 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6042, 0.7050]; 0.7584 〉, \\ 〈 [0.5017, 0.6363]; 0.4000 〉 \end{matrix})$
X ₂	$(\begin{matrix} 〈 [0.4830, 0.6629]; 0.3933 〉, \\ 〈 [0.5788, 0.7011]; 0.8191 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.4217, 0.5266]; 0.4842 〉, \\ 〈 [0.5032, 0.7425]; 0.7191 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5578, 0.7000]; 0.5266 〉, \\ 〈 [0.5734, 0.6446]; 0.6049 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6382, 0.7050]; 0.6587 〉, \\ 〈 [0.4639, 0.6000]; 0.5699 〉 \end{matrix})$
X ₃	$(\begin{matrix} 〈 [0.7000, 0.8000]; 0.4277 〉, \\ 〈 [0.4000, 0.5000]; 0.7000 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5102, 0.6119]; 0.6160 〉, \\ 〈 [0.5977, 0.7011]; 0.5761 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6119, 0.8000]; 0.7584 〉, \\ 〈 [0.4966, 0.6222]; 0.4000 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5578, 0.6581]; 0.3680 〉, \\ 〈 [0.4966, 0.6446]; 0.7395 〉 \end{matrix})$
X ₄	$(\begin{matrix} 〈 [0.7686, 0.8000]; 0.7000 〉, \\ 〈 [0.4337, 0.5337]; 0.6000 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6284, 0.8000]; 0.5578 〉, \\ 〈 [0.4441, 0.5442]; 0.6645 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.6581, 0.8000]; 0.7686 〉, \\ 〈 [0.4441, 0.5977]; 0.3736 〉 \end{matrix})$	$(\begin{matrix} 〈 [0.5059, 0.6081]; 0.8586 〉, \\ 〈 [0.5788, 0.7085]; 0.2744 〉 \end{matrix})$

Step 4: Using equation (17) and equation (18) gives the PIS p⁺ and the NIS p^-: $p^{+} = {\begin{matrix} (\begin{matrix} 〈 [0.7686, 0.8000]; 0.7000 〉, \\ 〈 [0.4337, 0.5337]; 0.6000 〉 \end{matrix}), (\begin{matrix} 〈 [0.6284, 0.8000]; 0.5578 〉, \\ 〈 [0.4441, 0.5442]; 0.6645 〉 \end{matrix}), \\ (\begin{matrix} 〈 [0.6581, 0.8000]; 0.7686 〉, \\ 〈 [0.4441, 0.5977]; 0.3736 〉 \end{matrix}), (\begin{matrix} 〈 [0.6382, 0.7050]; 0.6587 〉, \\ 〈 [0.4639, 0.6000]; 0.5699 〉 \end{matrix}) \end{matrix}}$

$p^{-} = {\begin{matrix} (\begin{matrix} 〈 [0.4830, 0.6629]; 0.3933 〉, \\ 〈 [0.5788, 0.7011]; 0.8191 〉 \end{matrix}), (\begin{matrix} 〈 [0.4217, 0.5266]; 0.4842 〉, \\ 〈 [0.5032, 0.7425]; 0.7191 〉 \end{matrix}), \\ (\begin{matrix} 〈 [0.5578, 0.7000]; 0.5266 〉, \\ 〈 [0.5734, 0.6446]; 0.6049 〉 \end{matrix}), (\begin{matrix} 〈 [0.5059, 0.6081]; 0.8586 〉, \\ 〈 [0.5788, 0.7085]; 0.2744 〉 \end{matrix}) \end{matrix}}$

Step 5: Utilize equation (19) and equation (20) to compute the $d_{i}^{+}$ and $d_{i}^{-}$ :

$\begin{matrix} d_{1}^{+} = 0.1363, d_{2}^{+} = 0.1525, d_{3}^{+} = 0.1051, d_{4}^{+} = 0.0477 \\ d_{1}^{-} = 0.1091, d_{2}^{-} = 0.0515, d_{3}^{-} = 0.1683, d_{4}^{-} = 0.1525 \end{matrix}$

Step 6: Use equation (21) to compute the C_i:

$C_{1} = 0.4445; C_{2} = 0.2524; C_{3} = 0.6155; C_{4} = 0.7617$

Step 7: Rank the alternatives X_i (i = 1, 2, 3, 4), w.r.t C_i $X_{4} > X_{3} > X_{1} > X_{2} .$

Thus the most desirable option is X₄.

Case 2. If the weights are partially-known and the known weights can represented as:

$Ω = {\begin{matrix} 0.18 ⩽ w_{1} ⩽ 0.24, 0.2 ⩽ w_{2} ⩽ 0.3, \\ 0.3 ⩽ w_{3} ⩽ 0.35, 0.2 ⩽ w_{4} ⩽ 0.28, \sum_{j = 1}^{n} w_{i} = 1 \end{matrix}}$

Step 1: Apply the M(2) model: ${\begin{matrix} \max (D) = 2.0092 w_{1} + 1.5373 w_{2} \\ + 1.4122 w_{3} + 1.7603 w_{4} \\ s . t . w \in Ω, w_{j} ⩾ 0, j = 1, 2, 3, 4 \end{matrix}}$

By solving this model, we acquire the optimal weight vector w = (0.24, 0.20, 0.30, 0.26) ^T.

Step 2: Use equation (17) and equation (18) to obtain CPF-PIS p⁺ and CPF-NIS p^-, respectively from Case 1:

Step 3: Apply equation (19) and equation (20) to compute $d_{i}^{+}$ and $d_{i}^{-}$ :

$\begin{matrix} d_{1}^{+} = 0.1383, d_{2}^{+} = 0.1472, d_{3}^{+} = 0.0971, d_{4}^{+} = 0.0474 \\ d_{1}^{-} = 0.1044, d_{2}^{-} = 0.0507, d_{3}^{-} = 0.1664, d_{4}^{-} = 0.1472 \end{matrix}$

Step 4: Using equation (21) to compute C_i:

$C_{1} = 0.4301; C_{2} = 0.2561; C_{3} = 0.6314; C_{4} = 0.7564$

Step 5: Ranking the alternatives X_i (i = 1, 2, 3, 4), to the C_i (i = 1, 2, 3, 4): X₄ > X₃ > X₁ > X₂.

Therefore the best option is X₄.

5 Comparison analysis and discussions

The proposed method is compared with existing approaches and demonstrated to be more general while providing the same results as classic techniques. To accomplish this we convert the more general PCFN to IVPFN. To accomplish this non-membership values are removed, resulting in Pythagorean fuzzy numbers (PFNs). Examples of this follow in the remaining subsections.

5.1 Comparison analysis with Interval Valued Pythagorean fuzzy approach

PCFNs can be converted to IVPFNs by removing the non-membership degree. The interval valued Pythagorean fuzzy information is presented in Table 5.

Table 5
IVPFD matrix

A ₁ A ₂ A ₃ A ₄

X ₁ ([0.5343, 0.6369] , [0.5788, 0.6832]) ([0.5343, 0.6850] , [0.6538, 0.6696]) ([0.5961, 0.7283] , [0.5275, 0.5352]) ([0.6042, 0.7050] , [0.5017, 0.6363])

X ₂ ([0.4830, 0.6629] , [0.5788, 0.7011]) ([0.4217, 0.5266] , [0.5032, 0.7425]) ([0.5578, 0.7000] , [0.5734, 0.6446]) ([0.6382, 0.7050] , [0.4639, 0.6000])

X ₃ ([0.7000, 0.8000] , [0.4000, 0.5000]) ([0.5102, 0.6119] , [0.5977, 0.7011]) ([0.6119, 0.8000] , [0.4966, 0.6222]) ([0.5578, 0.6581] , [0.4966, 0.6446])

X ₄ ([0.7686, 0.8000] , [0.4337, 0.5337]) ([0.6284, 0.8000] , [0.4441, 0.5442]) ([0.6581, 0.8000] , [0.4441, 0.5977]) ([0.5059, 0.6081] , [0.5788, 0.7085])

	A ₁	A ₂	A ₃	A ₄
X ₁	([0.5343, 0.6369] , [0.5788, 0.6832])	([0.5343, 0.6850] , [0.6538, 0.6696])	([0.5961, 0.7283] , [0.5275, 0.5352])	([0.6042, 0.7050] , [0.5017, 0.6363])
X ₂	([0.4830, 0.6629] , [0.5788, 0.7011])	([0.4217, 0.5266] , [0.5032, 0.7425])	([0.5578, 0.7000] , [0.5734, 0.6446])	([0.6382, 0.7050] , [0.4639, 0.6000])
X ₃	([0.7000, 0.8000] , [0.4000, 0.5000])	([0.5102, 0.6119] , [0.5977, 0.7011])	([0.6119, 0.8000] , [0.4966, 0.6222])	([0.5578, 0.6581] , [0.4966, 0.6446])
X ₄	([0.7686, 0.8000] , [0.4337, 0.5337])	([0.6284, 0.8000] , [0.4441, 0.5442])	([0.6581, 0.8000] , [0.4441, 0.5977])	([0.5059, 0.6081] , [0.5788, 0.7085])

By utlizing IVPF TOPSIS method, IVPF (PIS) p⁺ and IVPF (NIS) p^- are:

$p^{+} = {\begin{matrix} (\begin{matrix} [0.7686, 0.8000], \\ [0.4337, 0.5337] \end{matrix}), (\begin{matrix} [0.6284, 0.8000], \\ [0.4441, 0.5442] \end{matrix}), \\ (\begin{matrix} [0.6581, 0.8000], \\ [0.4441, 0.5977] \end{matrix}), (\begin{matrix} [0.6382, 0.7050], \\ [0.4639, 0.6000] \end{matrix}) \end{matrix}}$ $p^{-} = {\begin{matrix} (\begin{matrix} [0.4830, 0.6629], \\ [0.5788, 0.7011] \end{matrix}), (\begin{matrix} [0.4217, 0.5266], \\ [0.5032, 0.7425] \end{matrix}), \\ (\begin{matrix} [0.5578, 0.7000], \\ [0.5734, 0.6446] \end{matrix}), (\begin{matrix} [0.5059, 0.6081], \\ [0.5788, 0.7085] \end{matrix}) \end{matrix}}$

The separation measures $d_{i}^{+}$ and $d_{i}^{-}$ IVPF PIS p⁺ and the IVPF NIS p^-, respectively are:

$\begin{matrix} d_{1}^{+} = 0.2249, d_{2}^{+} = 0.2793, d_{3}^{+} = 0.2168, d_{4}^{+} = 0.1756 \\ d_{1}^{-} = 0.2089, d_{2}^{-} = 0.1964, d_{3}^{-} = 0.2668, d_{4}^{-} = 0.2853 \end{matrix}$

The C_i (relative closeness) of X_i to the IVPFSs (PIS p⁺) are:

$\begin{matrix} C_{1} = 0.4815, C_{2} = 0.4128, C_{3} = 0.5516, C_{4} = 0.6190 \end{matrix}$

Ranking the options X_i (i = 1, 2, 3, 4), X₄ > X₃ > X₁ > X₂. The best option is X₄.

In this paper, TOPSIS method was extended to the Pathagorean cubic fuzzy sets. It has also been shown that this method incoporates more general information than previous techniques. In cases where the information required to make a decision is charactorized by many inconsistant and/or unknown variables, this method is able to find the best decision, handling certain ambiguity that other methods cannot, thus allowing decision makers to make more informed decisions.

5.2 Comparison analysis with Pythagorean fuzzy approach

PFNs are special forms of PCFNs where decision-makers only evaluate membership and non-membership functions. Table 6 shows the membership and non membership of a PCFN that has been converted to a PFN by removing the interval component of the PCFN.

Table 6
PFD matrix

A ₁ A ₂ A ₃ A ₄

X ₁ (0.6119, 0.5977) (0.6000, 0.5761) (0.5564, 0.6538) (0.7584, 0.4000)

X ₂ (0.3933, 0.8191) (0.4842, 0.7191) (0.5266, 0.6049) (0.6587, 0.5699)

X ₃ (0.4277, 0.7000) (0.6160, 0.5761) (0.7584, 0.4000) (0.3680, 0.7395)

X ₄ (0.7000, 0.6000) (0.5578, 0.6645) (0.7686, 0.3736) (0.8586, 0.2744)

	A ₁	A ₂	A ₃	A ₄
X ₁	(0.6119, 0.5977)	(0.6000, 0.5761)	(0.5564, 0.6538)	(0.7584, 0.4000)
X ₂	(0.3933, 0.8191)	(0.4842, 0.7191)	(0.5266, 0.6049)	(0.6587, 0.5699)
X ₃	(0.4277, 0.7000)	(0.6160, 0.5761)	(0.7584, 0.4000)	(0.3680, 0.7395)
X ₄	(0.7000, 0.6000)	(0.5578, 0.6645)	(0.7686, 0.3736)	(0.8586, 0.2744)

Based on the values in Table 6, PF TOPSIS, is used to calculate The PF (PIS p⁺) and PF (NIS p^-) as: $p^{+} = {\begin{matrix} (0.7000, 0.6000), (0.6160, 0.5761), \\ (0.7686, 0.3736), (0.8586, 0.2744) \end{matrix}}$ $p^{-} = {\begin{matrix} (0.3933, 0.8191), (0.4842, 0.7191), \\ (0.5266, 0.6049), (0.3680, 0.7395) \end{matrix}}$

The separation measures $d_{i}^{+}$ , $d_{i}^{-}$ and relative closeness, are:

$\begin{matrix} d_{1}^{+} = 0.1608, d_{2}^{+} = 0.2949, d_{3}^{+} = 0.2363, d_{4}^{+} = 0.0219 \\ d_{1}^{-} = 0.2548, d_{2}^{-} = 0.0776, d_{3}^{-} = 0.1698, d_{4}^{-} = 0.3463 \\ C_{1} = 0.6130, C_{2} = 0.2083, C_{3} = 0.4181, C_{4} = 0.9405 \end{matrix}$

Rank the alternatives X_i (i = 1, 2, 3, 4) $X_{4} > X_{1} > X_{3} > X_{2} .$

Thus the best option is X₄.

Using this method, the order of the decision list is different from the previous method in this paper. Specifically, alternatitives X₁ and X₃ have swapped places. This is because because PFN does not contain as much information as it only incorporates membership and nonmembership which can result in loss of data, resulting in a different outcome.

Despite the differences in rank, in this case the best option was the same in all studied cases and the order is different in 1 out of 3 approaches.

From the above analysis, we have the following advantages:

1: PCFNs can more precisely express the uncertainty in the MCDM than the IVPFNs. In other words, PCFNs is the extension of the IVPFNs. So, we can know that the PCFNs has more extensive application prospect than the IVPFNs.

2: The proposed method incorporates the preferences and intuition of the decision maker, reducing risk in MCDM problems.

3: The approach suggested in this paper is a new extension of an existing technique which can deal with a greater variaty of MCDM problems than previous TOPSIS approaches.

6 Conclusion

As many practical MCGDM problems take place in a complex environment and usually adhere to imprecise data and uncertainty. The PCFS is a very effective tool for dealing with the fuzziness of the expert’s judgments over alternative with respect to the criteria. In this paper, we have first developed a method called the maximizing deviation method to determine the optimal relative weights of criteria based on Pythagorean cubic fuzzy setting. An important advantage of the proposed method is its ability to relieve the influence of subjectivity of the experts and at the same time remain the original decision information sufficiently. Then we have proposed an extended approach on the basis of TOPSIS to solve MCGDM problems with Pythagorean cubic fuzzy information. The approach is based on the relative closeness of each alternative to determine the ranking order of all alternatives, which avoids producing the loss of too much information in the process of information aggregation. Finally, the effectiveness and applicability of the proposed method has been illustrated with an example. Apparently, our approach is straightforward and has less loss of information, and can be applied easily to other managerial decision making problems under Pythagorean hesitant fuzzy environment.

In future, we will introduce the concept TODIM methods under Pythagorean cubic fuzzy environment. Further we will define Pythagorean cubic fuzzy linguistic sets and will propose a MCGDM based TOPSIS and TODIM methods under Pythagorean cubic fuzzy linguistic environment. We can also extended the developed concept to consensus issue [29, 30].

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