Q-Multi Cubic Pythagorean Fuzzy Sets and Their Correlation Coefficients for Multi-Criteria Group Decision Making

Abstract

Q-multi cubic Pythagorean fuzzy sets (Q-mCPFSs) are influential, effective and symmetrical for representing uncertain and imprecise information in decision making processes. Q-mCPFSs extend the concept of Q-multi fuzzy sets by introducing the notion of cubic Pythagorean membership functions, which provide a more flexible and accurate representation of uncertainty. First, we will introduce the concepts of Q-mPFSs and Q-mIVPFSs. With the combination of Q-mPFSs and Q-mIVPFSs, we will present the concept of Q-mCPFSs. Then, we propose two correlation coefficients for Q-mCPFSs. Furthermore, multi-criteria GDM methods using Q-mCPFSs will be discussed, highlighting their advantages in handling uncertain and imprecise information. Finally, we will provide an illustrative example, to demonstrate the effectiveness of Q-mCPFSs in decision making processes.The main contributions of the Q-mCPFS information expression, correlation coefficients and GDM methods in the Q-mCPFS setting of both uncertainty and certainty are thus highlighted in this study. These contributions provide valuable insights into the application of Q-mCPFSs in decision making processes, allowing decision makers to make more informed and effective choices. Additionally, the illustrative example serves as a practical demonstration of how these methods can be applied in real-world scenarios, further emphasizing their effectiveness and relevance.

1. Introduction

Most of our real-life problems require dealing with uncertainty and imprecision. By allowing the representation of ambiguous and uncertain information, fuzzy sets offer a mathematical framework to deal with such issues. We utilize Pythagorean fuzzy sets to better and more accurately characterize uncertain information. This extension of fuzzy sets incorporates the concept of Pythagorean [1,2] membership degrees, which allows for a more precise representation of uncertainty. Q-Pythagorean sets [3] allow for the simultaneous consideration of multiple membership degrees and Pythagorean membership degrees, resulting in a more comprehensive and nuanced characterization of uncertainty. The concept of Q-multi fuzzy sets [4,5] extends the traditional fuzzy set theory by incorporating multiple membership degrees for each element, enabling a more flexible representation of uncertainty. Combining the concepts of Q-multi fuzzy sets and Pythagorean fuzzy sets provides a powerful framework for representing uncertain information. Pythagorean m-polar fuzzy sets [6,7,8] contribute to handling complex and ambiguous information, making them suitable for real-world problems where uncertainty is inherent. Similarly, IVPFs [9,10,11,12,13,14,15,16] and CIFs [17,18,19,20,21] provide additional tools for modeling uncertainty in different contexts, such as decision making and pattern recognition. The concept of CPFs [22] further expands the applicability of fuzzy sets by allowing for the representation of both uncertainty and inconsistency in a unified framework. CPFs combine the advantages of PFSs and IVPFs, making them a versatile tool for addressing a wide range of real-world problems. This comprehensive approach enables decision makers to effectively handle complex and uncertain information while considering conflicting opinions or preferences. This study presents some definitions for Q-multi Pythagorean sets, Q-multi interval-valued Pythagorean fuzzy sets and Q-multi cubic Pythagorean fuzzy sets and some of their relations that played an important role in helping us hypothesize correlation coefficients of Q-mCPSs. The correlation coefficients for most extensions of fuzzy sets [23,24,25,26,27,28,29,30,31] also contributed to this assumption. Q-mCPSs can describe the opinions that several experts or decision makers propose on GDM problems, allowing for a more comprehensive and collaborative approach to decision making. Additionally, Q-mCPSs can also capture the dynamic nature of opinions and adapt to changes in the decision making process, making them highly adaptable in real-world scenarios.

Furthermore, Q-mCPSs provide a structured framework for aggregating and prioritizing the opinions of multiple experts or decision makers, ensuring a more robust and reliable decision making process. This not only enhances the quality of the final decision but also promotes transparency and accountability in the decision making process.

2. Preliminaries

In this section, we will review previous concepts, regarding Q-mFSs, Q-PFSs and CFSs.

Definition 1

([4]). Let G be a fixed set, $I=[0,1]$ be the unit interval, $z^{+}$ be a positive integer and Q be a nonempty set. Then, the set of ordered Sequences (denoted by Q-mFS) is called a Q-multi fuzzy set in G and Q. It takes the following form:

$$m{F}_Q=\left\{\left((g,q),{\psi }_1(g,q),{\psi }_2(g,q),\dots ,{\psi }_{z^{+}}(g,q)\right):g\in G,q\in Q\right\},$$

where ${\psi }_n:G\times Q\to I^{z^{+}},n=1,2,\dots ,z^{+}$;

$${\psi }_1(g,q)+{\psi }_2(g,q)+\dots +{\psi }_{z^{+}}(g,q)⩽1.$$

The function $\left({\psi }_1(g,q),{\psi }_2(g,q),\dots ,{\psi }_{z^{+}}(g,q)\right)$ is called the membership function.

We denote Q-mF(G) for the set of all Q-multi fuzzy sets of dimension $z^{+}$ in G and Q.

Definition 2

([3]). Let G be a fixed set and Q be a nonempty set. We call $N_q$ a Q-Pythagorean fuzzy set (Q-PFS) if it has the following form:

$$\begin{array}{cc}\hfill N_q=& \left\{(g,q),{\psi }_{N_q}(g,q),{\tau }_{N_q}(g,q):g\in G,q\in Q\right\},\hfill \\ & N_q:G\times Q⟶I,{\psi }_{N_q},{\tau }_{N_q}\in I,\hfill \end{array}$$

where ${\psi }_{N_q},{\tau }_{N_q}$ are the degree of membership and non-membership of the pair $(g,q)$, respectively.

The following inequality should hold:

$$0⩽{\left({\psi }_{N_q}\right)}^2+{\left({\tau }_{N_q}\right)}^2⩽1,\forall g\in G,q\in Q.$$

The degree of indeterminacy of $(g,q)$ is given as

$${\xi }_{N_q}(g,q)=\sqrt{1-{\left({\psi }_{N_q}(g,q)\right)}^2-{\left({\tau }_{N_q}(g,q)\right)}^2},\forall g\in G,q\in Q.$$

Definition 3

([17]). Let G a fixed set; then, the ordered pair of the interval-valued fuzzy set and the fuzzy set called a cubic fuzzy set are defined as

$$c=\left\{g,{\tilde{\psi }}_c\left(g\right),{\tau }_c\left(g\right):g\in G\right\},$$

where ${\tilde{\psi }}_c\left(g\right)=\left[b_c\left(g\right),c_c\left(g\right)\right]\subseteq [0,1]$ and ${\tau }_c\left(g\right)\in [0,1]$.

The pair $\left({\tilde{\psi }}_c\left(g\right),{\tau }_c\left(g\right)\right)$ is called a cubic-fuzzy-numbers pair.

3. Q-Multi Interval-Valued Pythagorean Fuzzy Sets

This section presents the concept of Q-mIVPFSs as an extension of Q-mPFSs.

Definition 4.

Let G be a fixed set; then, a Q-multi interval-valued Pythagorean fuzzy set (Q-mIVPFS) under G can be defined as follows:

$$\begin{array}{cc}\hfill {\tilde{P}}_Q=& \left\{\left((g,q),\left({\tilde{\psi }}_1(g,q),{\tilde{\psi }}_2(g,q),\dots ,{\tilde{\psi }}_{z^{+}}(g,q)\right),\left({\tilde{\tau }}_1(g,q),{\tilde{\tau }}_2(g,q),\dots \right.\right.\right.\hfill \\ & \left.\left.\left.,{\tilde{\tau }}_{z^{+}}(g,q)\right)\right);g\in G,q\in Q\right\},{\tilde{\psi }}_n,{\tilde{\tau }}_n\in I,\forall n=1,2,\dots ,z^{+},\hfill \end{array}$$

where ${\tilde{\psi }}_n(g,q)=\left[b_n(g,q),c_n(g,q)\right]\subseteq [0,1],{\tilde{\tau }}_n(g,q)=\left[d_n(g,q),e_n(g,q)\right]\subseteq [0,1]$ are the degree of membership and nonmembership intervals of the Q-mIVPFS, respectively. This satisfies the following conditions:

(i) 

${\displaystyle \sum _{n=1}^{z^{+}}}{\tilde{\psi }}_n(g,q)\subseteq [0,1]$ and ${\displaystyle \sum _{n=1}^{z^{+}}}{\tilde{\tau }}_n(g,q)\subseteq [0,1]$;

(ii) 

$0\le {\displaystyle \sum _{n=1}^{z^{+}}}\left({\left(c_n(g,q)\right)}^2+{\left(e_n(g,q)\right)}^2\right)\le 1$.

The degree of indeterminacy of $(g,q)$ is given as

$${\xi }_{{\tilde{P}}_Q}(g,q)=\left[f_n(g,q),h_n(g,q)\right],\forall g\in G,q\in Q,$$

where

$$f_n(g,q)=\sqrt{1-{\left(c_n(g,q)\right)}^2-{\left(e_n(g,q)\right)}^2},$$

and

$$h_n(g,q)=\sqrt{1-{\left(b_n(g,q)\right)}^2-{\left(d_n(g,q)\right)}^2}.$$

The pair $\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)$ is called a Q-multi interval-valued Pythagorean fuzzy number (Q-mIVPFN).

Remark 1.

If $b_n(g,q)=c_n(g,q)$ and $d_n(g,q)=e_n(g,q)$, then we obtain the Q-multi Pythagorean fuzzy set.

Example 1.

Let $G=\left\{g_1,g_2,g_3\right\}$ and $Q=\{q,d,s\}$ with $z^{+}=2$; then, ${\tilde{P}}_Q:G\times Q⟶I^2$.

$$\begin{array}{cc}\hfill {\tilde{P}}_Q=& \left\{\left(\left(g_1,q\right),\left([0.31,0.51],[0.41,0.46]\right),\left([0.11,0.15],[0.22,0.52]\right)\right),\right.\hfill \\ & \left.\left(\left(g_3,s\right),\left([0.15,0.22],[0.37,0.68]\right),\left([0.24,43],[0.17,0.39]\right)\right)\right\},\hfill \end{array}$$

is a Q-multi interval-valued Pythagorean fuzzy set in G and Q.

Definition 5.

Let ${\tilde{P}}_Q$ and ${\tilde{T}}_Q$ be two Q-mIVPNs, such that

$${\tilde{P}}_Q=\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right),{\tilde{T}}_Q=\left(\left[k_n,l_n\right],\left[r_n,s_n\right]\right),\forall n=1,2,\dots ,z^{+}.$$

Then, the next relations can be found as:

(1) 

${\tilde{P}}_Q={\tilde{T}}_Q$ if and only if $b_n=k_n,c_n=l_n,d_n=r_n$, and  $e_n=s_n$;

(2) 

${\tilde{P}}_Q⩽{\tilde{T}}_Q$ if and only if $b_n⩽k_n,c_n⩽l_n,d_n⩾r_n$, and  $e_n⩾s_n$;

(3) 

${\tilde{P}}_Q\cup {\tilde{T}}_Q=\left(\left[\max ,\left\{b_n,k_n\right\},\max \left\{c_n,l_n\right\}\right],\left[\min \left\{d_n,r_n\right\},\min \left\{e_n,s_n\right\}\right]\right)$;

(4) 

${\tilde{P}}_Q\cap {\tilde{T}}_Q=\left(\left[\min \left\{b_{n,}{k}_n\right\},\min \left\{c_n,l_n\right\}\right],\left[\max \left\{d_n,r_n\right\},\max \left\{e_n,s_n\right\}\right]\right)$;

(5) 

${\tilde{P}}_Q^c=\left(\left[d_n,e_n\right],\left[b_n,c_n\right]\right)$.

Definition 6.

A Q-mIVPFN $\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)$ is denoted by:

(i) 

$\tilde{\mathsf{\Phi }}$ if $\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)=([0,0],[1,1])$,  $\forall n$ (null Q-mIVPFs);

(ii) 

$\tilde{\breve{x}}$ if $\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)=([1,1],[0,0]),\forall n$ (absolute Q-mIPFs).

Proposition 1.

If ${\tilde{P}}_Q,{\tilde{T}}_Q$ and ${\tilde{S}}_Q$ are a Q-mIVPFSs underling G, then:

(1) 

$\tilde{\mathsf{\Phi }}\cup {\tilde{P}}_Q={\tilde{P}}_Q$ and $\tilde{\mathsf{\Phi }}\cap {\tilde{P}}_Q=\tilde{\mathsf{\Phi }}$;

(2) 

$\tilde{\breve{x}}\cup {\tilde{P}}_Q={\tilde{\breve{x}}}^c$ and $\tilde{\breve{x}}\cap {\tilde{P}}_Q={\tilde{P}}_Q$;

(3) 

${\tilde{P}}_Q\cap {\tilde{P}}_Q={\tilde{P}}_Q$ and ${\tilde{P}}_Q\cup {\tilde{P}}_Q={\tilde{P}}_Q$;

(4) 

${\tilde{P}}_Q\cup {\tilde{T}}_Q={\tilde{T}}_Q\cup {\tilde{P}}_Q$ and ${\tilde{P}}_Q\cap {\tilde{T}}_Q={\tilde{T}}_Q\cap {\tilde{P}}_Q$;

(5) 

${\tilde{P}}_Q\cup \left({\tilde{T}}_Q\cup {\tilde{S}}_Q\right)=\left({\tilde{P}}_Q\cup {\tilde{T}}_Q\right)\cup {\tilde{S}}_Q$ and ${\tilde{P}}_Q\cap \left({\tilde{T}}_Q\cap {\tilde{S}}_Q\right)=\left({\tilde{P}}_Q\cap {\tilde{T}}_Q\right)\cap {\tilde{S}}_Q$;

(6) 

${\tilde{P}}_Q\cup \left({\tilde{T}}_Q\cap {\tilde{S}}_Q\right)=\left({\tilde{P}}_Q\cup {\tilde{T}}_Q\right)\cap \left({\tilde{P}}_Q\cup {\tilde{S}}_Q\right)$ and ${\tilde{P}}_Q\cap \left({\tilde{T}}_Q\cup {\tilde{S}}_Q\right)=\left({\tilde{P}}_Q\cap {\tilde{T}}_Q\right)\cup \left({\tilde{P}}_Q\cap {\tilde{S}}_Q\right)$.

Proof. 

The above is clear from Definition 5, but we can only demonstrate (6): for

$${\tilde{P}}_Q=\left\{(g,q),\left({\tilde{\psi }}_n,{\tilde{\tau }}_n\right)\right\},{\tilde{T}}_Q=\left\{(g,q),\left({\tilde{\alpha }}_n,{\tilde{\beta }}_n\right)\right\}\mathrm{and}{\tilde{S}}_Q=\left\{(g,q),\left({\tilde{\gamma }}_n,{\tilde{\eta }}_n\right)\right\},$$

where $\left({\tilde{\psi }}_n,{\tilde{\tau }}_n\right)=\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right),\left({\tilde{\alpha }}_n,{\tilde{\beta }}_n\right)=\left(\left[k_n,l_n\right],\left[r_n,s_n\right]\right)$ and $\left({\tilde{\gamma }}_n,{\tilde{\eta }}_n\right)=\left(\left[t_n,u_n\right],\left[v_n,x_n\right]\right)$, assuming

$$\max \left({\tilde{\psi }}_n,{\tilde{\alpha }}_n\right)={\tilde{\psi }}_n,\max \left({\tilde{\psi }}_n,{\tilde{\gamma }}_n\right)={\tilde{\psi }}_n,\max \left({\tilde{\alpha }}_n,{\tilde{\gamma }}_n\right)={\tilde{\alpha }}_n,$$

we establish that

$${\tilde{P}}_Q\cup \left({\tilde{T}}_Q\cap {\tilde{S}}_Q\right)\stackrel{?}{=}\left({\tilde{P}}_Q\cup {\tilde{T}}_Q\right)\cap \left({\tilde{P}}_Q\cup {\tilde{S}}_Q\right).$$

For the left side,

$$\begin{array}{cc}\hfill {\tilde{T}}_Q\cap {\tilde{S}}_Q& =\left\{(g,q),\left(\left[\min \left\{k_n,t_n\right\},\min \left\{l_n,u_n\right\}\right],\left[\max \left\{r_n,v_n\right\},\max \left\{s_n,x_n\right\}\right]\right)\right\}\hfill \\ \hfill & =\left\{(g,q),\left(\left[t_n,u_n\right],\left[v_n,x_n\right]\right)\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{P}}_Q\cup \left({\tilde{T}}_Q\cap {\tilde{S}}_Q\right)& =\left\{(g,q),\left(\left[\max \left\{b_n,t_n\right\},\max \left\{c_n,u_n\right\}\right],\left[\min \left\{d_n,v_n\right\},\min \left\{e_n,x_n\right\}\right]\right)\right\}\hfill \\ \hfill & =\left\{(g,q),\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)\right\}.\hfill \end{array}$$

(1)

For the right side,

$$\begin{array}{cc}\hfill {\tilde{P}}_Q\cup {\tilde{T}}_Q& =\left\{(g,q),\left(\left[\max \left\{b_n,k_n\right\},\max \left\{c_n,l_n\right\}\right],\left[\min \left\{d_n,r_n\right\},\min \left\{e_n,s_n\right\}\right]\right)\right\}\hfill \\ \hfill & =\left\{(g,q),\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\tilde{P}}_Q\cup {\tilde{S}}_Q& =\left\{(g,q),\left(\left[\max \left\{b_n,t_n\right\},\max \left\{c_n,u_n\right\}\right],\left[\min \left\{d_n,v_n\right\},\min \left\{e_n,x_n\right\}\right]\right)\right\}\hfill \\ \hfill & =\left\{(g,q),\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left({\tilde{P}}_Q\cup {\tilde{T}}_Q\right)\cap \left({\tilde{P}}_Q\cup {\tilde{S}}_Q\right)& =\left\{(g,q),\left(\left[\min \left\{b_n,b_n\right\},\min \left\{c_n,c_n\right\}\right],\left[\max \left\{d_n,d_n\right\},\max \left\{e_n,e_n\right\}\right]\right)\right\}\hfill \\ \hfill & =\left\{(g,q),\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)\right\}.\hfill \end{array}$$

(2)

From (1) and (2), we confirm ${\tilde{P}}_Q\cup \left({\tilde{T}}_Q\cap {\tilde{S}}_Q\right)=\left({\tilde{P}}_Q\cup {\tilde{T}}_Q\right)\cap \left({\tilde{P}}_Q\cup {\tilde{S}}_Q\right)$. The proofs of other equations are similar.    □

Corollary 1.

$\tilde{\mathsf{\Phi }}\cup \tilde{\breve{x}}=\tilde{\breve{x}}$ and $\tilde{\mathsf{\Phi }}\cap \tilde{\breve{x}}=\tilde{\mathsf{\Phi }}$.

Proposition 2.

If ${\tilde{P}}_Q$ and ${\tilde{T}}_Q$ are a Q-mIVPSs underling G, then we have:

(1) 

${\tilde{P}}_Q\cap {\tilde{T}}_Q\subseteq {\tilde{P}}_Q\subseteq {\tilde{P}}_Q\cup {\tilde{T}}_Q$;

(2) 

${\tilde{P}}_Q\cap {\tilde{T}}_Q\subseteq {\tilde{T}}_Q\subseteq {\tilde{P}}_Q\cup {\tilde{T}}_Q$.

Proof. 

The results are easily deduced from the properties of max and min that we see for (1):

$$\left[\min \left\{b_n,k_n\right\},\min \left\{c_n,l_n\right\}\right]⩽\left[b_n,c_n\right]⩽\left[\max \left\{b_n,k_n\right\},\max \left\{c_n,l_n\right\}\right]$$

and

$$\left[\max \left\{d_n,r_n\right\},\max \left\{e_n,s_n\right\}\right]⩾\left[d_n,e_n\right]⩾\left[\min \left\{d_n,r_n\right\},\min \left\{e_n,s_n\right\}\right],$$

and similarly for (2).    □

Proposition 3.

If ${\tilde{P}}_Q$ and ${\tilde{T}}_Q$ are a Q-mIVPFSs underling G, then the De’ Morgan laws hold:

(1) 

${\left({\tilde{P}}_Q\cup {\tilde{T}}_Q\right)}^c={\tilde{P}}_Q^c\cap {\tilde{T}}_Q^c$;

(2) 

${\left({\tilde{P}}_Q\cap {\tilde{T}}_Q\right)}^c={\tilde{P}}_Q^c\cup {\tilde{T}}_Q^c$.

Proof. 

We verify from (1) and (2) by similar steps. For $g\in G,q\in Q$ and $\forall n=1,2,\dots ,z^{+}$,

$${\left({\tilde{P}}_Q\cup {\tilde{T}}_Q\right)}^c\stackrel{?}{=}{\tilde{P}}_Q^c\cap {\tilde{T}}_Q^c.$$

For the left side:

$$\begin{array}{cc}\hfill {\left({\tilde{P}}_Q\cup {\tilde{T}}_Q\right)}^c& ={\left\{(g,q),\left(\left[\max \left\{b_n,k_n\right\},\max \left\{c_n,l_n\right\}\right],\left[\min \left\{d_n,r_n\right\},\min \left\{e_n,s_n\right\}\right]\right)\right\}}^c\hfill \\ \hfill & ={\left\{(g,q),\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)\right\}}^c=\left\{(g,q),\left(\left[d_n,e_n\right],\left[b_n,c_n\right]\right)\right\}.\hfill \end{array}$$

(3)

For the right side:

$$\begin{array}{cc}\hfill {\tilde{P}}_Q^c\cap {\tilde{T}}_Q^c& ={\left\{(g,q),\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)\right\}}^c\cap {\left\{(g,q),\left(\left[k_n,l_n\right],\left[r_n,s_n\right]\right)\right\}}^c\hfill \\ \hfill & =\left\{(g,q),\left(\left[d_n,e_n\right],\left[b_n,c_n\right]\right)\right\}\cap \left\{(g,q),\left(\left[r_n,s_n\right],\left[k_n,l_n\right]\right)\right\}\hfill \\ \hfill & =\left\{(g,q),\left(\left[\min \left\{d_n,r_n\right\},\min \left\{e_n,s_n\right\}\right],\left[\max \left\{b_n,k_n\right\},\max \left\{c_n,l_n\right\}\right]\right)\right\}\hfill \\ \hfill & =\left\{(g,q),\left(\left[d_n,e_n\right],\left[b_n,c_n\right]\right)\right\}.\hfill \end{array}$$

(4)

From (3) and (4), we confirm ${\left({\tilde{P}}_Q\cup {\tilde{T}}_Q\right)}^c={\tilde{P}}_Q^c\cap {\tilde{T}}_Q^c$.    □

Remark 2.

For ${\tilde{P}}_Q$, a Q-mIVPFSs underling G, we observe that

$${\tilde{P}}_Q\cup {\tilde{P}}_Q^c\ne \tilde{\breve{x}}\mathit{and}{\tilde{P}}_Q\cap {\tilde{P}}_Q^c\ne \tilde{\mathsf{\Phi }}.$$

Proposition 4.

(1) 

${\left({\tilde{P}}_Q^c\right)}^c={\tilde{P}}_Q$;

(2) 

${\left(\tilde{\breve{x}}\right)}^c=\tilde{\mathsf{\Phi }}$ and $\tilde{\breve{x}}={\left(\tilde{\mathsf{\Phi }}\right)}^c$.

Proof. 

Straightforward.    □

Proposition 5.

If ${\tilde{P}}_Q$ is a Q-mIVPFS underling G, then $\tilde{\mathsf{\Phi }}\subseteq {\tilde{P}}_Q\subseteq \tilde{\breve{x}}$.

Proof. 

Straightforward.    □

4. Q-Multi Cubic Pythagorean Fuzzy Sets

This section presents the concept of Q-mCPFSs and defines its relations, based on the hybrid concepts of Q-mPFSs and Q-mIVPFSs.

Definition 7.

Let G be a fixed set; then, a Q-multi cubic Pythagorean (Q-mCPFS) fuzzy set can be defined as follows:

$$\begin{array}{c}{P}_c=\left\{\left((g,q),\left({\mathsf{\Psi }}_1(g,q),{\mathsf{\Psi }}_2(g,q),\dots ,{\mathsf{\Psi }}_{z^{+}}(g,q)\right),\left({\mathcal{T}}_1(g,q),{\mathcal{T}}_2(g,q),\dots \right.\right.\right.\\ \left.\left.\left.,{\mathcal{T}}_{z^{+}}(g,q)\right)\right),g\in G,q\in Q\right\},\forall n=1,2,\dots ,z^{+},\end{array}$$

where

$${\mathsf{\Psi }}_n(g,q)=\left(B_n(g,q);{\psi }_n(g,q)\right)=\left\{\left[b_n(g,q),c_n(g,q)\right];{\psi }_n(g,q)\right),$$

and

$${\mathcal{T}}_n(g,q)=\left(D_n(g,q);{\tau }_n(g,q)\right)=\left(\left[d_n(g,q),e_n(g,q)\right];{\tau }_n(g,q)\right)$$

are the degree membership and nonmembership of $(g,q)$, respectively. The following conditions must hold:

(1) 

$0\le {\displaystyle \sum _{n=1}^{z^{+}}}\left({\left(c_n(g,q)\right)}^2+{\left(e_n(g,q)\right)}^2\right)\le 1$;

(2) 

$0⩽{\left({\psi }_n(g,q)\right)}^2+{\left({\tau }_n(g,q)\right)}^2⩽1$.

The degree of indeterminacy of $(g,q)$ is given as

$$\begin{array}{cc}\hfill {\xi }_{P_c}& =\left({\xi }_{{\tilde{P}}_Q},{\xi }_{P_Q}\right)\hfill \\ & =\left(\left[f_n(g,q),h_n(g,q)\right],{\xi }_{P_Q}(g,q)\right),\forall g\in G,q\in Q,\hfill \end{array}$$

such that

$$\begin{array}{cc}& f_n(g,q)=\sqrt{1-{\left(c_n(g,q)\right)}^2-{\left(e_n(g,q)\right)}^2},\hfill \\ & h_n(g,q)=\sqrt{1-{\left(b_n(g,q)\right)}^2-{\left(d_n(g,q)\right)}^2}\hfill \end{array}$$

and

$${\xi }_{p_Q}(g,q)=\sqrt{1-{\left({\psi }_n(g,q)\right)}^2-{\left({\tau }_n(g,q)\right)}^2}.$$

The pair $\left(\left(B_n,{\psi }_n\right),\left(D_n,{\tau }_n\right)\right)$ is called a Q-mCPFN, while the pair $\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)$ represents the Q-mIVFN, and $\left({\psi }_n,{\tau }_n\right)$ represents a Q-mPFN.

Example 2.

Consider the G and Q in Example 1 with $z^{+}=2$; thus, we have

$$\begin{array}{cc}\hfill P_c=& \left\{\left(\left(g_2,q\right),(([0.31,0.51],0.42),([0.41,0.46],0.31)),\right.\right.\hfill \\ \hfill & \left.\left(\left([0.11,0.15],0.55\right),([0.22,0.52],0.33)\right)\right)\}.\hfill \end{array}$$

Definition 8.

Let $P_c$ and $T_c$ be two Q-mCPFN, such that

$$P_c=\left(\left(\left[b_n,c_n\right],{\psi }_n\right),\left(\left[d_n,e_n\right],{\tau }_n\right)\right),T_c=\left(\left(\left[k_n,l_n\right],{\alpha }_n\right),\left(\left[r_n,s_n\right],{\beta }_n\right)\right).$$

Then, the next relations can be found as:

(1) 

(Equality) $P_c=T_c\iff \left[b_n,c_n\right]=\left[k_n,l_n\right],\left[d_n,e_n\right]=\left[r_n,s_n\right],{\psi }_n={\alpha }_n\mathit{and}{\tau }_n={\beta }_n$;

(2) 

(p-order) $P_c{\subseteq }_p{T}_c$ if $\left[b_n,c_n\right]\subseteq \left[k_n,l_n\right],\left[d_n,e_n\right]\supseteq \left[r_n,s_n\right]$,  ${\psi }_n⩽{\alpha }_n$ and ${\tau }_n⩾{\beta }_n$;

(3) 

(R-order) $P_c{\subseteq }_R{T}_c$ if $\left[b_n,c_n\right]\subseteq \left[k_n,l_n\right],\left[d_n,e_n\right]\supseteq \left[r_n,s_n\right]$,  ${\psi }_n⩾{\alpha }_n$ and ${\tau }_n⩽{\beta }_n$;

(4) 

 

$$\begin{array}{cc}\hfill P_c{\cup }_p{T}_c=& \left(\left[\max \left\{b_n,k_n\right\},\max \left\{c_n,l_n\right\}\right],\left[\min \left\{d_n,r_n\right\},\min \left\{e_n,s_n\right\}\right]\right.;\hfill \\ & \left.\left(\max \left\{{\psi }_n,{\alpha }_n\right\},\min \left\{{\tau }_n,{\beta }_n\right\}\right)\right);\hfill \end{array}$$

(5) 

$$\begin{array}{cc}\hfill P_c{\cap }_P{T}_c=& \left(\left[\min \left\{b_n,k_n\right\},\min \left\{c_n,l_n\right\}\right],\left[\max \left\{d_n,r_n\right\},\max \left\{e_n,s_n\right\}\right]\right.;\hfill \\ & \left.\left(\min \left\{{\psi }_n,{\alpha }_n\right\},\max \left\{{\tau }_n,{\beta }_n\right\}\right)\right);\hfill \end{array}$$

(6) 

$P_c^c=\left(\left(\left[d_n,e_n\right],{\tau }_n\right),\left(\left[b_n,c_n\right],{\psi }_n\right)\right)$;

(7) 

$$\begin{array}{cc}\hfill P_c{\cup }_R{T}_c=& \left(\left[\max \left\{b_n,k_n\right\},\max \left\{c_n,l_n\right\}\right],\left[\min \left\{d_n,r_n\right\},\min \left\{e_n,s_n\right\}\right]\right.;\hfill \\ & \left.\left(\min \left\{{\psi }_n,{\alpha }_n\right\},\min \left\{{\tau }_n,{\beta }_n\right\}\right)\right);\hfill \end{array}$$

(8) 

$$\begin{array}{cc}\hfill P_c{\cap }_R{T}_c=& \left(\left[\min \left\{b_n,k_n\right\},\min \left\{c_n,l_n\right\}\right],\left[\max \left\{d_n,r_n\right\},\max \left\{e_n,s_n\right\}\right]\right.;\hfill \\ & \left.\left(\max \left\{{\psi }_n,{\alpha }_n\right\},\min \left\{{\tau }_n,{\beta }_n\right\}\right)\right).\hfill \end{array}$$

Theorem 1.

Let $P_c,T_c$ and $S_c$ be a Q-mCPFSs underling G; then,

(1) 

If $P_c{\subseteq }_p{T}_c$ and $T_c{\subseteq }_p{S}_c$, then $P_c{\subseteq }_p{S}_c$;

(2) 

If $P_c{\subseteq }_p{T}_c$, then $T_c^c{\subseteq }_p{P}_c^c$;

(3) 

If $P_c{\subseteq }_p{T}_c$ and $P_c\subseteq S_c$, then $P_c{\subseteq }_p{T}_c{\cap }_p{S}_c$;

(4) 

If $P_c{\subseteq }_p{T}_c$ and $S_c\subseteq T_c$, then $P_c{\cup }_p{S}_c{\subseteq }_p{T}_c$,

and the same relations for the R-order.

Proof. 

Straightforward.    □

Definition 9.

A Q-mCPFS defined as:

(1) 

External Q-mCPFS if ${\psi }_n(g,q)\notin \left[b_n(g,q),c_n(g,q)\right]$ and ${\tau }_n(g,q)\notin \left[d_n(g,q),e_n(g,q)\right],\forall g\in G,q\in Q$;

(2) 

Internal Q-mCPFS if ${\psi }_n(g,q)\in \left[b_n(g,q),c_n(g,q)\right]$ and ${\tau }_n(g,q)\in \left[d_n(g,q),e_n(g,q)\right],\forall g\in G,q\in Q$.

Theorem 2.

Let $P_c$ be a Q-mCPFS underling G. If $P_c$ is an (external or internal) Q-mCPFS, then $P_c^c$ is also an (external or internal) Q-mCPFS.

Proof. 

As $P_c=\left(\left(\left[b_n,c_n\right],{\psi }_n\right)\right.$, $\left.\left[\left[d_n,e_n\right],{\tau }_n\right)\right)$ is an external, then

$$P_c^c=\left(\left(\left[d_n,e_n\right],{\tau }_n\right),\left(\left[b_n,c_n\right],{\psi }_n\right)\right)$$

is also an external, and likewise for internal Q-mCPFS.    □

Remark 3.

For $P_c=\left(\left(\left[b_n,c_n\right],{\psi }_n\right),\left(\left[d_n,e_n\right],{\tau }_n\right)\right)$, we find:

(1) 

If $\left({\psi }_n,{\tau }_n\right)=(0,1)$, then R-order Q-mCPFS becomes Q-mIVPFS;

(2) 

If $\left(\left[b_n,c_n\right],\left[d_n,e_n\right]\right)=([1,1],[0,1])$, then p-order Q-mCPFS becomes Q-mPFS.

5. Correlation Coefficients of Q-mCPSs

In this section, we introduce the correlation coefficients of Q-mCPFSs and their properties. In addition, we discuss the application of Q-mCPFSs in various fields, and highlight their importance in decision making processes in the next section, through an illustrative example.

Definition 10.

Let $P_c=\left\{\left(\left(B_n;{\psi }_n\right),\left(D_n;{\tau }_n\right)\right)\right\},T_c=\left\{\left(\left(K_n;{\alpha }_n\right),\left(R_n;{\beta }_n\right)\right)\right\}$ be two Q-mCPSs, where $B_n=\left[b_n,c_n\right],D_n=\left[d_n,e_n\right],K_n=\left[k_n,l_n\right]$ and $R_n=\left[r_n,s_n\right]$. Then, the correlation coefficients ${\sigma }_1$ and ${\sigma }_2$ between $P_c$ and $T_c$ are given as follows:

$$\begin{array}{cc}\hfill {\sigma }_1(P_c,T_c)& =\frac{{\displaystyle \sum _{n=1}^{z^{+}}}{\displaystyle \sum _{q=1}^j}{\displaystyle \sum _{g=1}^m}\left\{\begin{array}{c}{b}_n(g,q)k_n(g,q)+c_n(g,q)l_n(g,q)+d_n(g,q)r_n(g,q)\hfill \\ +e_n(g,q)s_n(g,q)+2\left({\psi }_n{\alpha }_n+{\tau }_n{\beta }_n\right)\hfill \end{array}\right\}}{\sqrt{{\displaystyle \sum _{n=1}^{z^{+}}}{\displaystyle \sum _{q=1}^j}{\displaystyle \sum _{g=1}^m}\left\{\begin{array}{c}{b}_n^2(g,q)+c_n^2(g,q)\hfill \\ +d_n^2(g,q)+e_n^2(g,q)\hfill \\ +2\left({\psi }_n^2(g,q)+{\tau }_n^2(g,q)\right)\hfill \end{array}\right\}}·\sqrt{{\displaystyle \sum _{n=1}^{z^{+}}}{\displaystyle \sum _{q=1}^j}{\displaystyle \sum _{g=1}^m}\left\{\begin{array}{c}{k}_n^2(g,q)+l_n^2(g,q)\hfill \\ +r_n^2(g,q)+s_n^2(g,q)\hfill \\ +2\left({\alpha }_n^2(g,q)+{\beta }_n^2(g,q)\right)\hfill \end{array}\right\}}}\hfill \\ \hfill & =\frac{\mathrm{cov}\left(P_c,T_c\right)}{\sqrt{V\left(P_c,P_c\right)}\sqrt{V\left(T_c,T_c\right)}};\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\sigma }_2(P_c,T_c)& =\frac{{\displaystyle \sum _{n=1}^{z^{+}}}{\displaystyle \sum _{q=1}^j}{\displaystyle \sum _{g=1}^m}\left\{\begin{array}{c}{b}_n(g,q)k_n(g,q)+c_n(g,q)l_n(g,q)+d_n(g,q)r_n(g,q)\hfill \\ +e_n(g,q)s_n(g,q)+2\left({\psi }_n{\alpha }_n+{\tau }_n{\beta }_n\right)\hfill \end{array}\right\}}{\max \left\{{\displaystyle \sum _{n=1}^{z^{+}}}{\displaystyle \sum _{q=1}^j}{\displaystyle \sum _{g=1}^m}\left\{\begin{array}{c}{b}_n^2(g,q)+c_n^2(g,q)\hfill \\ +d_n^2(g,q)+e_n^2(g,q)\hfill \\ +2\left({\psi }_n^2(g,q)+{\tau }_n^2(g,q)\right)\hfill \end{array}\right\},{\displaystyle \sum _{n=1}^{z^{+}}}{\displaystyle \sum _{q=1}^j}{\displaystyle \sum _{g=1}^m}\left\{\begin{array}{c}{k}_n^2(g,q)+l_n^2(g,q)\hfill \\ +r_n^2(g,q)+s_n^2(g,q)\hfill \\ +2\left({\alpha }_2^2(g,q)+{\beta }_n^2(g,q)\right)\hfill \end{array}\right\}\right\}}\hfill \\ \hfill & =\frac{\mathrm{cov}\left(P_c,T_c\right)}{\max \left\{V\left(P_c,P_c\right),V\left(T_c,T_c\right)\right\}}.\hfill \end{array}$$

(6)

Theorem 3.

If ${\sigma }_r$ for $r=1,2$, then the following properties of correlation coefficients ${\sigma }_r$ satisfy for all r as:

(1) 

$0⩽{\sigma }_r\left(P_c,T_c\right)⩽1$;

(2) 

${\sigma }_r\left(P_c,T_c\right)={\sigma }_r\left(T_c,P_c\right)$;

(3) 

${\sigma }_r\left(P_c,T_c\right)=1;$  if  $P_c=T_c$.

Proof. 

It is clear that for (2) and (3),

we verify from (1). The inequality ${\sigma }_r\left(P_c,T_c\right)⩾0$ is straightforward, so we will deduce that ${\sigma }_r\left(P_c,T_c\right)\le 1$. By using the Cauchy–Schwarz inequality for $\mathrm{cov}\left(P_c,T_c\right)$, we obtain

$$\begin{array}{cc}\hfill {\left(\mathrm{cov}\left(P_c,T_c\right)\right)}^2& ⩽\sum _{n=1}^{z^{+}}\sum _{q=1}^j\sum _{g=1}^m\left\{\begin{array}{c}{b}_n^2(g,q)+c_n^2(g,q)+d_n^2(g,q)+e_n^2(g,q)\hfill \\ +2\left({\psi }_n^2(g,q)+{\tau }_n^2(g,q)\right)\hfill \end{array}\right\}\hfill \\ & \times \sum _{n=1}^{z^{+}}\sum _{q=1}^j\sum _{g=1}^m\left\{\begin{array}{c}{k}_n^2(g,q)+l_n^2(g,q)+r_n^2(g,q)+s_n^2(g,q)\hfill \\ +2\left({\alpha }_n^2(g,q)+{\beta }_n^2(g,q)\right)\hfill \end{array}\right\}\hfill \\ & =V\left(P_c,P_c\right),V\left(T_c,T_c\right).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill {\left(\mathrm{cov}\left(P_c,T_c\right)\right)}^2& \le V\left(P_c\right)·V\left(T_c\right)\hfill \\ \hfill \Rightarrow \mathrm{cov}\left(P_c,T_c\right)& \le \sqrt{V\left(P_c\right)}·\sqrt{V\left(T_c\right)};\hfill \end{array}$$

hence, ${\sigma }_1\left(P_c,T_c\right)⩽1$.

And, similarly, for ${\sigma }_2\left(P_c,T_c\right)$, we find

$$\mathrm{cov}\left(P_c,T_c\right)⩽\sqrt{{\left(\max \left\{V\left(P_c\right),V\left(T_c\right)\right\}\right)}^2}=\max \left\{V\left(P_c\right),V\left(T_c\right)\right\};$$

hence, ${\sigma }_2\left(P_c,T_c\right)⩽1$.    □

6. Proposed Method

This section presents the development of multi-criteria group decision making (GDM) methods within the framework of Q-mCPFS, utilizing correlation coefficients. We consider a GDM problem with the following characteristics:

(1)

Alternatives $\left\{P_{c_1},P_{c_2},\dots ,P_{c_i}\right\}$;

(2)

Criteria $\left\{{\delta }_{11},{\delta }_{12},\dots ,{\delta }_{21},{\delta }_{22},\dots {\delta }_{mj}\right\}$.

Assume an expert has been assigned to evaluate each $P_{c_i}$ under ${\delta }_{mj}$ and given by Q-mCPNs. The values are denoted by

$${\mu }_{rgq}^n=\left(\left(B_{rgq}^n;{\psi }_{rgq}^n\right),\left(D_{rgq}^n;{\tau }_{rgq}^n\right)\right),$$

with

$$B_{rgq}^n=\left[b_{rgq}^n,c_{rgq}^n\right],D_{rgq}^n=\left[d_{rgq}^n,e_{rgq}^n\right]\subseteq [0,1],{\psi }_{rgq}^n,{\tau }_{rgq}^n\in [0,1],$$

and

$$0⩽{\left(c_{rgq}^n\right)}^2+{\left(e_{rgq}^n\right)}^2⩽1,0\le {\left({\psi }_{rgq}^n\right)}^2+{\left({\tau }_{rgq}^n\right)}^2⩽1$$

for $r=1,2,\dots ,i;g=1,2,\dots ,m;q=1,2,\dots ,j,n=1,2,\dots ,z^{+}$.

The decision algorithm of the multicriteria GDM methods is presented by the following steps:

Step 1:

Collect the information as a decision matrix of Q-mCPSs (corresponding to each alternative-criteria pair),

$$P=\begin{array}{cc}& \begin{array}{cccccc} {\delta }_{11}& \hspace{0.17em}{\delta }_{12}& \hspace{0.17em}\cdots & {\delta }_{21} & \cdots & \hspace{0.17em}{\delta }_{mj} \end{array}\\ \begin{array}{c}{P}_{c_1}\\ P_{c_2}\\ ⋮\\ P_{c_i}\end{array}& \left(\begin{array}{cccccc}{\mu }_{111}& {\mu }_{112}& \cdots & {\mu }_{121}& \cdots & {\mu }_{1mj}\\ {\mu }_{211}& {\mu }_{212}& \cdots & {\mu }_{221}& \cdots & {\mu }_{2mj}\\ ⋮& ⋮& \ddots & ⋮& & ⋮\\ {\mu }_{i11}& {\mu }_{i12}& \cdots & {\mu }_{i21}& \cdots & {\mu }_{imj}\end{array}\right)\end{array};$$

Step 2:

Take the perfect set, which was suggested by experts, as an ideal alternative $P_c^{*}$;

Step 3:

Use the proposed correlation coefficients ${\sigma }_1$ and ${\sigma }_2$, as given in Equations (5) and (6), to calculate the measurement degree between $P_{c_i}$ and $P_c^{*}$;

Step 4:

Determine the best alternative, based on the results of ${\sigma }_1$ and ${\sigma }_2$, which provide a measure of the performance or effectiveness of each alternative. A value closer to 1 indicates a higher level of suitability, making it the preferable choice;

Step 5:

End.

Example 3.

Paying attention to education is a must for cultures that aspire to join the ranks of nations, as it is the most significant pillar of industrialized countries that strive for leadership and greatness. The Saudi Arabia Vision 2030 AD includes the ambitious goal of making education in the Kingdom a pioneering model by raising the level of quality of education and improving outcomes, in line with the country’s ambitious goal of reaching the ranks of the developed countries, clearly demonstrating the Kingdom’s great interest in education and in making it a fundamental focus, pillar and starting point for construction and development.

Quality standards are based on fundamental principles and foundations, and they differ, based on the fields in which they are applied and the evaluation systems that oversee them. However, they all share numerous specifications and standards that address the quality of the finished product throughout the production process. Quality education follows this framework exactly, focusing on the requirements for graduates, their educational achievement throughout the process and their capacity to overcome any challenges that may stand in their way.

The most important factors influencing the quality of education in schools are the curriculum and educational efficiency. There are numerous other factors as well: evaluation, monitoring, efficiency in education technology and resources, the educational setting and inspiration and support.

In order to execute the first step of the proposed method on page 9, we will gather data for Q-mCPSs. This study, which is hypothetical, will take the following characteristics into account:

1. 

Curriculum (represented by $g_1$). The curriculum needs to be current and compliant with current international standards for educational excellence.

2. 

Educational competence (represented by $g_2$) is the ability of teachers to instruct pupils in ways that are appropriate to the demands and changes of the modern world.

3. 

Public and private schools will stand in for $q_1$ and $q_2$, respectively.

4. 

The Education Departments of Riyadh and Sharqia will stand in for $P_{c_1}$ and $P_{c_2}$, respectively, as alternatives.

The quality of education in the regions of Riyadh and Sharqiya $\left(P_{c_1}\right)$ and $\left(P_{c_2}\right)$, respectively, will be assessed using the Q-mCPNs displayed in the Table 1.

Table 1. The decision matrix of Q-mCPSs.

	${\mathit{P}}_{{\mathit{c}}_1}$	${\mathit{P}}_{{\mathit{c}}_2}$
$\left(g_1,q_1\right)$	$\begin{array}{cc}& \left(\left(\left([0.60,0.70],[0.25,0.30]\right),(0.60,0.35)\right),\right.\hfill \\ & \left(\left([0.40,0.50],[0.25,0.30]\right),(0.55,0.30)\right),\hfill \\ & \left.\left(\left([0.30,0.40],[0.40,0.45]\right),(0.50,0.25)\right)\right)\hfill \end{array}$	$\begin{array}{cc}& \left(\left(\left([0.50,0.60],[0.30,0.35]\right),(0.50,0.40)\right),\right.\hfill \\ & \left(\left([0.30,0.40],[0.30,0.35]\right),(0.45,0.40)\right),\hfill \\ & \left.\left(\left([0.20,0.30],[0.45,0.50]\right),(0.40,0.35)\right)\right)\hfill \end{array}$
$\left(g_1,q_2\right)$	$\begin{array}{cc}& \left(\left(\left([0.50,0.60],[0.30,0.35]\right),(0.60,0.30)\right),\right.\hfill \\ & \left(\left([0.30,0.40],[0.35,0.40]\right),(0.60,0.25)\right),\hfill \\ & \left.\left(\left([0.25,0.35],[0.40,0.45]\right),(0.50,0.20)\right)\right)\hfill \end{array}$	$\begin{array}{cc}& \left(\left(\left([0.40,0.50],[0.35,0.40]\right),(0.50,0.40)\right),\right.\hfill \\ & \left(\left([0.20,0.30],[0.40,0.45]\right),(0.50,0.35)\right),\hfill \\ & \left.\left(\left([0.15,0.25],[0.45,0.50]\right),(0.40,0.30)\right)\right)\hfill \end{array}$
$\left(g_2,q_1\right)$	$\begin{array}{cc}& \left(\left(\left([0.75,0.85],[0.20,0.25]\right),(0.70,0.30)\right),\right.\hfill \\ & \left(\left([0.60,0.70],[0.30,0.35]\right),(0.60,0.35)\right),\hfill \\ & \left.\left(\left([0.50,0.60],[0.35,0.40]\right),(0.60,0.40)\right)\right)\hfill \end{array}$	$\begin{array}{cc}& \left(\left(\left([0.60,0.70],[0.25,0.30]\right),(0.60,0.40)\right),\right.\hfill \\ & \left(\left([0.50,0.60],[0.35,0.40]\right),(0.50,0.45)\right),\hfill \\ & \left.\left(\left([0.40,0.50],[0.40,0.45]\right),(0.50,0.50)\right)\right)\hfill \end{array}$
$\left(g_2,q_2\right)$	$\begin{array}{cc}& \left(\left(\left([0.60,0.70],[0.20,0.25]\right),(0.75,0.25)\right),\right.\hfill \\ & \left(\left([0.50,0.60],[0.30,0.35]\right),(0.70,0.30)\right),\hfill \\ & \left.\left(\left([0.45,0.55],[0.40,0.45]\right),(0.60,0.35)\right)\right)\hfill \end{array}$	$\begin{array}{cc}& \left(\left(\left([0.50,0.60],[0.30,0.35]\right),(0.60,0.30)\right),\right.\hfill \\ & \left(\left([0.45,0.55],[0.40,0.45]\right),(0.60,0.35)\right),\hfill \\ & \left.\left(\left([0.35,0.45],[0.45,0.50]\right),(0.50,0.40)\right)\right)\hfill \end{array}$

We shall discuss the Q-mCPN of $P_{c_1}$ under $(g_1,q_1)$. The following are the percentages of academic achievement across all school levels for students, as well as the percentage of satisfied and dissatisfied educational supervisors:

$$\left(\left([0.60,0.70],[0.25,0.30]\right),(0.60,0.35)\right).$$

This means that when evaluating the effect of the curriculum’s quality $\left(g_1\right)$ and ongoing development on primary school students’ achievement levels, we discover that in certain government schools $\left(q_1\right)$, this level increased at a rate ranging from $60\%$ to $70\%$ and, in other schools, it decreased at a rate ranging from $25\%$ to $30\%$. The first educational supervisor is $35\%$ dissatisfied and has a $60\%$ satisfaction rating with the degree of achievement:

$$\left(\left([0.40,0.50],[0.25,0.30]\right),(0.55,0.30)\right).$$

In reference to middle school student achievement, we find that it increased at a pace of between $40\%$ and $50\%$ in certain schools and fell at a rate ranging from $25\%$ to $30\%$ in other schools. The second supervisor of education had a satisfaction rate of $55\%$ and a dissatisfaction rate of $30\%$, regarding the achievement score:

$$\left(\left([0.30,0.40],[0.40,0.45]\right),(0.50,0.25)\right).$$

Regarding secondary school pupils’ accomplishment levels, we find that they went up by a rate of between $30\%$ and $40\%$ in certain schools and down by a rate ranging from $40\%$ to $45\%$ in other schools. The third supervisor of education had a satisfaction rate of $50\%$ and a dissatisfaction rate of $25\%$, regarding the achievement score.

We will proceed in a similar manner with the remaining data for the Riyadh region $\left(P_{c_1}\right)$ and interpret the data for the Sharqia province $\left(P_{c_2}\right)$, keeping in mind that there are three different experts, each of whom has a focus on a particular stage for each $(g,q)$ in the rows.

Proposing the ideal alternative, $P_c^{*}$, which represents the necessary student development for all levels in all schools in any chosen region, is the second step that educational supervisors will take, to ensure that students achieve at the ideal educational achievement level. This is how it will appear:

$$\begin{array}{cc}\hfill P_c^{*}=& \left\{\right(\left(g_1,q_1\right),\left(\left([0.70,0.80],[0.15,0.20]\right),(0.70,0.25)\right),\hfill \\ \hfill & \left(\left([0.50,0.60],[0.20,0.25]\right),(0.65,0.20)\right),\left(\left([0.40,0.50],[0.30,0.35]\right),(0.60,0.15)\right)),\hfill \\ \hfill & (\left(g_1,q_2\right),\left(\left([0.70,0.80],[0.20,0.25]\right),(0.75,0.25)\right),\hfill \\ \hfill & \left(\left([0.45,0.55],[0.25,0.30]\right),(0.70,0.15)\right),\left(\left([0.35,0.45],[0.35,0.40]\right),(0.60,0.10)\right)),\hfill \\ \hfill & (\left(g_2,q_1\right),\left(\left([0.80,0.90],[0.10,0.15]\right),(0.80,0.20)\right),\hfill \\ \hfill & \left(\left([0.75,0.85],[0.20,0.25]\right),(0.70,0.25)\right),\left(\left([0.60,0.70],[0.25,0.30]\right),(0.70,0.30)\right)),\hfill \\ \hfill & (\left(g_2,q_2\right),\left(\left([0.70,0.80],[0.15,0.20]\right),(0.85,0.15)\right),\hfill \\ \hfill & \left(\left([0.65,0.75],[0.25,0.30]\right),(0.80,0.20)\right),\left(\left([0.55,0.65],[0.30,0.35]\right),(0.75,0.20)\right)\left)\right\}.\hfill \end{array}$$

Supervisors will compute the correlation coefficients provided in Equations (5) and (6) in the third step. The findings will be as follows:

$${\sigma }_1\left(P_{c_1},P_c^{*}\right)=\frac{22.94}{23.4321}=0.978999,{\sigma }_1\left(P_{c_2},P_c^{*}\right)=\frac{20.315}{21.8116}=0.931385.$$

In order to carry out the fourth step, we can deduce that $P_{c_1}>P_{c_2}$ for ${\sigma }_1$. This indicates that Riyadh students’ achievement level is nearly identical to 1 and slightly superior to that of Sharqia students. This indicates a high degree of success in achieving the intended outcomes. Therefore, $P_{c_1}$ represents the best alternative. To make the results more accurate, we will find ${\sigma }_2$, and we will obtain the following results:

$${\sigma }_2\left(P_{c_1},P_c^{*}\right)=\frac{22.94}{\max \{21.385,25.675\}},{\sigma }_2\left(P_{c_2},P_c^{*}\right)=\frac{22.94}{\max \{18.53,25.675\}},$$

$${\sigma }_2\left(P_{c_1},P_c^{*}\right)=0.893476={\sigma }_2\left(P_{c_2},P_c^{*}\right).$$

As a result, we can conclude that the educational achievement of students in Riyadh and the Sharqia Region is identical.

The ability to collect more precise and thorough data is what sets Q-mCPFS apart from earlier research and demonstrates the depth of the study’s significance. For instance, we cannot use a larger amount of data to achieve accuracy while utilizing correlation coefficients under the cubic intuitionistic fuzzy set (CIFS); therefore, this needs to be considered. Suppose that $P_c$ and $T_c$ have the following correlation coefficients, ${\delta }_1$ and ${\delta }_2$, respectively:

$$\begin{array}{cc}\hfill {\delta }_1(P_c,T_c)& =\frac{{\displaystyle \sum _{t=1}^m}\left\{\begin{array}{c}b\left(g_t\right)k\left(g_t\right)+c\left(g_t\right)l\left(g_t\right)+d\left(g_t\right)r\left(g_t\right)\hfill \\ +e\left(g_t\right)s\left(g_t\right)+2\left(\psi \alpha +\tau \beta \right)\hfill \end{array}\right\}}{\sqrt{{\displaystyle \sum _{t=1}^m}\left\{\begin{array}{c}{b}^2\left(g_t\right)+c^2\left(g_t\right)\hfill \\ +d^2\left(g_t\right)+e^2\left(g_t\right)\hfill \\ +2\left({\psi }^2\left(g_t\right)+{\tau }^2\left(g_t\right)\right)\hfill \end{array}\right\}}·\sqrt{{\displaystyle \sum _{t=1}^m}\left\{\begin{array}{c}{k}^2\left(g_t\right)+l^2\left(g_t\right)\hfill \\ +r^2\left(g_t\right)+s^2\left(g_t\right)\hfill \\ +2\left({\alpha }^2\left(g_t\right)+{\beta }^2\left(g_t\right)\right)\hfill \end{array}\right\}}}\hfill \\ \hfill {\delta }_2(P_c,T_c)& =\frac{{\displaystyle \sum _{t=1}^m}\left\{\begin{array}{c}b\left(g_t\right)k\left(g_t\right)+c\left(g_t\right)l\left(g_t\right)+d\left(g_t\right)r\left(g_t\right)\hfill \\ +e\left(g_t\right)s\left(g_t\right)+2\left(\psi \alpha +\tau \beta \right)\hfill \end{array}\right\}}{\max \left\{{\displaystyle \sum _{t=1}^m}\left\{\begin{array}{c}{b}^2\left(g_t\right)+c^2\left(g_t\right)\hfill \\ +d^2\left(g_t\right)+e^2\left(g_t\right)\hfill \\ +2\left({\psi }^2\left(g_t\right)+{\tau }^2\left(g_t\right)\right)\hfill \end{array}\right\},{\displaystyle \sum _{t=1}^m}\left\{\begin{array}{c}{k}^2\left(g_t\right)+l^2\left(g_t\right)\hfill \\ +r^2\left(g_t\right)+s^2\left(g_t\right)\hfill \\ +2\left({\alpha }^2\left(g_t\right)+{\beta }^2\left(g_t\right)\right)\hfill \end{array}\right\}\right\}}.\hfill \end{array}$$

We observe that

$$P_c=\left\{\left(\right(B;\psi ),(D;\tau \left)\right)\right\},T_c=\left\{\left(\right(K;\alpha ),(R;\beta \left)\right)\right\},$$

where $B=[b,c],D=[d,e],K=[k,l]$ and $R=[r,s]$.

If we utilize the data from Example 3, we can see the following differences in the degree of inaccuracy of the data:

• The data for all educational stages will be chosen at random because we are unable to get information at each stage independently, as the CIFS is not multiple.
• The Q set, which is ignored in this case, is essential for classifying schools, as it helps us resolve any ambiguities and conduct more data analysis.
• In comparison to IFS decision making, the decision making process for PFS outcomes is superior.

Decision makers will greatly and widely benefit from Q-mCPFSs and their correlation coefficients, in order to make accurate decisions. A Q-mCPFS provides decision makers with a comprehensive and efficient tool to analyze data and extract meaningful insights. By utilizing its correlation coefficients, decision makers can gain a deeper understanding of the relationships between variables, enabling them to make informed and accurate decisions. It is evident from the following notes that decision makers will benefit greatly from this study:

Improved decision making: The article introduces a novel approach, namely Q-mCPFSs and their correlation coefficients, which can be applied in multi-criteria group decision making processes. The managerial implication here would be that decision makers can utilize this methodology to enhance the quality and accuracy of their decision making processes in complex and uncertain situations.

Enhanced analysis of fuzzy information: The use of Q-mCPFSs and their correlation coefficients provides a framework for handling fuzzy information and evaluating alternatives. This can help managers in various domains, such as finance, marketing or operations, to better analyze and interpret uncertain data and make informed decisions.

Facilitation of group decision making: The article focuses on multi-criteria group decision making, implying that it offers a method for managing group dynamics and incorporating multiple perspectives when making decisions. The managerial implication here is that organizations can leverage this approach, to promote collaboration, consensus-building and more effective decision making within teams or committees.

7. Conclusions

Q-mCPFSs build upon the concepts of Q-mPFSs and introduce a wider range of contention through Q-PFSs values. This allows for a more comprehensive interpretation of data using the Q-mIVPFSs format. To facilitate this analysis, we utilized the Q-mCPFSs, to derive correlation coefficients and to inform our decision making process. This study shows that decision makers can choose the option that is most similar to the ideal by using the proposed method and the correlation coefficients of Q-mCPSs. It is important to keep in mind that Q-mCPFSs can be used in other applications by utilizing aggregation operators, which is a promising new area of study.