Multi-Q Cubic Bipolar Fuzzy Soft Sets and Cosine Similarity Methods for Multi-Criteria Decision Making

Abstract

This study introduces a novel mathematical tool for representing imprecise and ambiguous data: the multi-q cubic bipolar fuzzy soft set. Building upon established bipolar fuzzy sets and soft sets, this paper fist defines the concept of multi-q cubic bipolar fuzzy sets and their fundamental properties. Mathematical operations such as complement, union, and intersection are then developed for these sets. The core contribution lies in the introduction of multi-q cubic bipolar fuzzy soft sets. This new tool allows for a more nuanced representation of imprecise data compared to existing approaches. Key operations for manipulating these sets, including complement, restriction, and expansion, are defined. The applicability of multi-q cubic bipolar fuzzy soft sets extends to various domains, including multi-criteria decision making and problem solving. Illustrative examples demonstrate the practical utility of this innovative concept.

1. Introduction

In our everyday lives, we often come into decision-making circumstances that are unclear and uncertain. These scenarios arise because of gaps in our knowledge, incomplete information, inconsistent data, asymmetry, and shortages of information [1,2]. Zadeh’s approach has long proven highly beneficial in decision analysis, effectively managing complex and ambiguous data. Fuzzy sets, employing fuzzy logic and neural networks for flaw detection, find applications in various computer science domains, including traffic control systems, weather forecasting, and commercial appliances such as washing machines and air conditioners. Extensive research has led to the emergence of several fuzzy set extensions, including interval-valued set theory, axiomatic fuzzy set theory, bipolar fuzzy set theory, neuromorphic set theory, and soft set theory. These extensions have been specifically developed for particular data types and have a wide range of uses in many industries [3,4,5,6].

For the development of mathematical frameworks, bipolarity is essential in data analysis. It encompasses both the favorable and unfavorable features of a topic, demonstrating the ambivalent nature of human decision-making. Illustrations encompass happiness and sadness, pleasantness and unpleasantness, and advantages and disadvantages. Ensuring a harmonious social environment requires effectively managing these conflicting components. Bipolar-valued fuzzy sets, an expansion of bipolar fuzzy sets, were first described by Chang (1998) and are specifically advantageous for handling information that encompasses both a characteristic and its opposite [3,4,5].

The literature explores the issue of bipolar ambiguity and uncertainty by describing several strategies for bipolar fuzzy decision-making. These strategies provide clear and precise bipolar fuzzy descriptions of alternatives using a variety of degrees. Nevertheless, the fundamental idea of bipolar fuzzy sets frequently falls short in providing accurate information regarding classifications or scores, mostly because of inadequate data and an inability to thoroughly elucidate uncertainty and ambiguity, particularly in delicate decision-making scenarios. Interval-valued bipolar fuzzy sets (IVBFSs) do not provide accurate assessments for expert judgments based on alternative qualities, as indicated by previous studies [7,8,9].

Pattern recognition and set analysis make use of the cubic bipolar fuzzy set correlation coefficient, a mathematical technique for measuring the similarity or difference between items represented by fuzzy sets. When dealing with ambiguous or uncertain data, this technique expands upon standard correlation coefficients. Important points are:

• Correlation Coefficients: The degree of association between two variables is indicated by statistical measurements that range from $-1$ (complete negative correlation) to 1 (perfect positive correlation).
• Cubic Bipolar Fuzzy Sets: These enhance basic fuzzy sets by incorporating cubic membership functions, allowing membership values from $-1$ (no membership) to 1 (full membership) and thus addressing data uncertainty and ambiguity.
• Applications in Pattern Recognition and Cluster Analysis: These involve using correlation coefficients to determine item similarity within fuzzy sets, facilitating pattern identification and cluster analysis, which are essential for analyzing and categorizing data with inherent uncertainty.

Figure 1 illustrates Hierarchy for Multi-q Cubic Bipolar Fuzzy Soft Sets and Cosine Similarity Methods for Multi-Criteria Decision Making in this paper.

Figure 1. Hierarchy for Multi-q Cubic Bipolar Fuzzy Soft Sets and Cosine Similarity Methods for Multi-Criteria Decision Making.

2. Related Work

The paper referenced [10] introduced three new cosine similarity metrics for cubic bipolar fuzzy sets (CBFSs). It also developed a TOPSIS methodology that uses cosine similarity measures for multi-criteria group decision-making (MCGDM) scenarios. Additionally, the paper presented a specific case study on selecting a sustainable plastic recycling method to demonstrate the effectiveness of the proposed MCGDM procedure.

The paper proposes new measures for cosine similarity and distance in cubic Fermatean fuzzy sets (CFFSs). It introduces a novel method for creating alternative similarity measures for CFFSs based on the proposed measures. The cosine distance metric for CFFSs is derived from the relationship between similarity and distance metrics, as discussed in the paper by [11] titled “Improved CFFSs”.

The authors of [12] proposed an improved cosine similarity metric for interval neutrosophic numbers that takes into account the degrees of truth, indeterminacy, and falsity. In addition, they developed a method for selecting selections by considering many criteria using a modified cosine similarity measure. They utilized the technique of ordered weighted averaging (OWA) to merge and summarize the neutrosophic information associated with each option.

The researchers in [13] presented a decision-making model that incorporates modified Jaccard and Dice similarity metrics in the framework of dual hesitant fuzzy sets. Their approach involves using updated Jaccard and Dice similarity metrics in a dual hesitant fuzzy sets (DHFSs) framework as parts of a multiple-criteria decision-making (MCDM) model. The approach entails the assessment of alternatives by employing dual hesitant fuzzy elements (DHFEs). The authors further introduced a methodology for determining the weights of criteria by employing an objective function and linear programming (LP) technique, as opposed to relying on decision makers to assign them directly.

This technique use Dice and Jaccard weighted similarity measures to evaluate the comparison between each option and the ideal, and then ranks the alternatives to identify the most optimum one.

A recent study [14] Proposed a new hybrid model, called the multi-fuzzy N-soft set, to tackle decision-making difficulties related to imprecise and graded data. This technique combines the benefits of multi-fuzzy set theory with N-soft sets, allowing for the incorporation of ratings or grades inside a multi-fuzzy framework. The article presents the theoretical underpinning of multi-fuzzy N-soft sets and introduces a flexible decision-making mechanism particularly tailored for this novel environment. The technique utilizes a conversion mechanism to transform data into a careful N-soft environment, enabling the use of scores in decision-making.

Table 1 shows Summary of Related Work the most prominent objectives, Contribution and Potential Drawbacks addressed by the studies mentioned above.

Table 1. Summary of Related Work.

Paper Ref	Methodology	Contribution	Objectives	Potential Drawbacks
1	1. Propose three new cosine similarity measures for cubic bipolar fuzzy sets (CBFSs). 2. Develop a TOPSIS approach based on the proposed cosine similarity measures.3. Provide an illustrative example on the selection of a sustainable plastic recycling process to demonstrate the efficiency of the suggested MCGDM technique.	Proposed three new cosine similarity measures for CBFSs. Developed a TOPSIS approach based on the proposed cosine similarity measures for MCGDM problems. Provided an illustrative example to demonstrate the effectiveness of the proposed MCGDM technique.	Proposed three new cosine similarity measures for cubic bipolar fuzzy sets (CBFSs) and developed a TOPSIS approach based on the proposed similarity measures for multi-criteria group decision-making (MCGDM) problems.	Computational complexity of the proposed methods is not discussed. The proposed methods are not compared with existing methods to assess their performance. The illustrative example is specific to the domain of plastic recycling and may not be generalizable to other MCGDM problems.
2	1. Develop novel measures for cosine similarity and distance applicable to CFFSs.2. Introduce a method for constructing alternative similarity measures based on the proposed measures.3. Establish a cosine distance metric for CFFSs using the relationship between similarity and distance measures.	Proposed new cosine similarity and distance measures for CFFSs. Developed a method for constructing alternative similarity measures. Established a cosine distance metric for CFFSs.	Proposed new measures for cosine similarity and distance specifically for cubic Fermatean fuzzy sets (CFFSs).	The study lacks comparison with existing similarity and distance measures for CFFSs. The computational complexity of the proposed methods is not discussed. The applicability of the methods to real-world decision-making problems requires further investigation with practical examples.
3	1. Proposes an improved cosine similarity measure for interval neutrosophic numbers (INNs) that considers truth, indeterminacy, and falsity memberships.2. Develops a multi-criteria decision-making (MCDM) method based on the improved cosine similarity measure.3. Utilizes ordered weighted averaging (OWA) to aggregate the neutrosophic information associated with each decision alternative.	Introduced an improved cosine similarity measure for interval neutrosophic numbers (INNs). Proposed a multi-criteria decision-making (MCDM) method based on the improved similarity measure. Utilized ordered weighted averaging (OWA) for neutrosophic information aggregation.	Introduced an improved cosine similarity measure between interval neutrosophic numbers, considering truth, indeterminacy, and falsity memberships. Established a multi-criteria decision-making method based on the improved cosine similarity measure. Adopted ordered weighted averaging (OWA) to aggregate the neutrosophic information related to each alternative.	The study does not explicitly compare the proposed similarity measure with existing ones. The computational complexity of the MCDM method with OWA aggregation is not discussed. The effectiveness of the method in real-world decision-making scenarios would benefit from demonstration with practical examples.
4	1. Integrates modified Jaccard and Dice similarity measures into an MCDM model based on DHFSs.2. Evaluates decision alternatives represented by dual hesitant fuzzy elements (DHFEs).3. Employs an objective function and linear programming (LP) to determine criteria weights objectively.4. Applies weighted Dice and Jaccard similarity measures to compare each alternative to an ideal solution and rank the alternatives.	Proposed new cosine similarity and distance measures for CFFSs. Developed a method for constructing alternative similarity measures. Established a cosine distance metric for CFFSs.	Proposed a multiple-criteria decision-making model using modified Jaccard and Dice similarity measures under the framework of dual hesitant fuzzy sets.	The applicability of the methods to real-world decision-making problems requires further investigation with practical examples.
Our study	1. Defines the concept of MQCBFSS, building upon bipolar fuzzy sets and soft sets.2. Established fundamental properties of the multi-q cubic bipolar fuzzy set.3. Develops mathematical operations for MQCBFSS, including complement, union, and intersection.	Develops an MCDM model using modified Jaccard and Dice similarity measures under the framework of dual hesitant fuzzy sets (DHFSs). Employs an objective function and linear programming (LP) to determine criteria weights objectively. Enables ranking and selection of the best alternative based on the calculated similarity measures.	Introduces the concept of multi-q cubic bipolar fuzzy soft sets. The fundamental properties of the multi-q cubic bipolar fuzzy set are outlined. Mathematical operations such as complement, union, and intersection are developed for these sets.	No specific future direction that can be implemented to rise up and improve decision-making.

3. Preliminaries

This section presents the fundamental concepts and mathematical frameworks necessary for analyzing multi-q cubic bipolar fuzzy soft sets. This portion will cover the definitions, features, and many expansions of fuzzy sets, cubic fuzzy sets, and bipolar fuzzy sets.

3.1. Fuzzy Set [2]

Definition 1. 

A fuzzy set $\mathcal{B}$ over a set Y is defined as follows:

$$\mathcal{B}=\{(\mathit{y},{\mathit{M}}_{\mathit{B}}\left(\mathit{y}\right))\mid \mathit{y}\in \mathit{Y}\}$$

where the membership function ${\mathit{M}}_{\mathit{B}}\left(\mathit{y}\right):\mathit{Y}\to [\mathbf{0},\mathbf{1}]$ denotes the degree to which the element $\mathit{y}$ belongs to $\mathcal{B}$, indicating how much $\mathit{y}$ is considered a member of $\mathcal{B}$.

3.2. Cubic Fuzzy Set [2]

Definition 2. 

The concept of cubic fuzzy set $\mathcal{B}$ over the set $\mathit{Y}$ can be described as

$$\mathcal{B}=\left\{\right(\mathit{y},\mathcal{B}\left(\mathit{y}\right),\mu \left(\mathit{y}\right))\mid \mathit{y}\in \mathit{Y}\}$$

where the interval-valued fuzzy set $\mathcal{B}\left(\mathit{y}\right)=[{\mathcal{B}}^{-}\left(\mathit{y}\right),{\mathcal{B}}^{+}\left(\mathit{y}\right)]\subseteq [\mathbf{0},\mathbf{1}]$, and the fuzzy set $\mu \left(\mathit{y}\right)\in [\mathbf{0},\mathbf{1}]$. This combination is represented as $\mathcal{B}=(\mathcal{B},\mu )$, which means that $\mathcal{B}$ consists of both interval-valued and regular fuzzy sets.

3.3. Bipolar Fuzzy Set [15]

Definition 3. 

The concept of bipolar fuzzy set $\mathcal{B}$ over the universe $\mathit{Y}$ is defined as

$$\mathcal{B}=\{(\mathit{y},{\Delta }_{\mathit{B}}^{+}\left(\mathit{y}\right),{\Delta }_{\mathit{B}}^{-}\left(\mathit{y}\right))\mid \mathit{y}\in \mathit{Y}\}$$

where the positive membership function is ${\Delta }_{\mathit{B}}^{+}\left(\mathit{y}\right):\mathit{Y}\to [\mathbf{0},\mathbf{1}]$, and the negative membership function is ${\Delta }_{\mathit{B}}^{-}\left(\mathit{y}\right):\mathit{Y}\to [-\mathbf{1},\mathbf{0}]$. For each element $\mathit{y}\in \mathit{Y}$, the condition $-\mathbf{1}\le {\Delta }_{\mathit{B}}^{+}\left(\mathit{y}\right)+{\Delta }_{\mathit{B}}^{-}\left(\mathit{y}\right)\le \mathbf{1}$ must be satisfied, ensuring that the sum of positive and negative memberships lies within the specified range.

3.4. Cubic Bipolar Fuzzy Set [16]

Definition 4. 

The concept of cubic bipolar fuzzy set $\mathcal{B}$ over the universe $\mathit{Y}$ can be defined as

$$\mathcal{B}=\left\{\right(\mathit{y},\mathit{M}\left(\mathit{y}\right),\mu \left(\mathit{y}\right))\mid \mathit{y}\in \mathit{Y}\}$$

where $\mathit{M}\left(\mathit{y}\right)$ is an interval-valued bipolar set, and $\mu \left(\mathit{y}\right)$ is a bipolar fuzzy set on $\mathit{Y}$. Therefore, a cubic bipolar fuzzy set can be expressed as

$$\mathcal{B}=\{(\mathit{y},[{\Delta }_{\mathit{mB}}^{+}\left(\mathit{y}\right),{\Delta }_{\mathit{uB}}^{+}\left(\mathit{y}\right)],[{\Delta }_{\mathit{mB}}^{-}\left(\mathit{y}\right),{\Delta }_{\mathit{uB}}^{-}\left(\mathit{y}\right)],\{{\Delta }_{\mathit{B}}^{+}\left(\mathit{y}\right),{\Delta }_{\mathit{B}}^{-}\left(\mathit{y}\right)\})\mid \mathit{y}\in \mathit{Y}\}$$

The intervals $[{\Delta }_{\mathit{mB}}^{+}\left(\mathit{y}\right),{\Delta }_{\mathit{uB}}^{+}\left(\mathit{y}\right)]\subseteq [\mathbf{0},\mathbf{1}]$ and $[{\Delta }_{\mathit{mB}}^{-}\left(\mathit{y}\right),{\Delta }_{\mathit{uB}}^{-}\left(\mathit{y}\right)]\subseteq [-\mathbf{1},\mathbf{0}]$ denote the degrees of positive and negative membership, respectively. The values ${\Delta }_{\mathit{B}}^{+}\left(\mathit{y}\right)\in [\mathbf{0},\mathbf{1}]$ and ${\Delta }_{\mathit{B}}^{-}\left(\mathit{y}\right)\in [-\mathbf{1},\mathbf{0}]$ indicate the membership levels of the element $\mathit{y}$, specifying how strongly $\mathit{y}$ is considered a member of the set in both positive and negative contexts.

Thus, a cubic bipolar fuzzy number is represented as

$$([{\Delta }_{\mathit{mB}}^{+}\left(\mathit{y}\right),{\Delta }_{\mathit{uB}}^{+}\left(\mathit{y}\right)],[{\Delta }_{\mathit{mB}}^{-}\left(\mathit{y}\right),{\Delta }_{\mathit{uB}}^{-}\left(\mathit{y}\right)],\{{\Delta }_{\mathit{B}}^{+}\left(\mathit{y}\right),{\Delta }_{\mathit{B}}^{-}\left(\mathit{y}\right)\})\mid \mathit{y}\in \mathit{Y}$$

3.5. Multi-Q Fuzzy Set [17]

Definition 5. 

For a positive integer $\mathit{k}$, the concept of multi-q fuzzy set ${\tilde{\mathit{G}}}_{\mathit{Q}}$ in a set $\mathit{V}$ and a non-empty set $\mathit{Q}$ is described as a collection of ordered pairs:

$${\tilde{\mathit{G}}}_{\mathit{Q}}=\{(\mathit{v},\mathit{q}),{\Delta }_{\mathit{i}}(\mathit{v},\mathit{q})\mid \mathit{v}\in \mathit{V},\mathit{q}\in \mathit{Q}\}$$

where ${\Delta }_{\mathit{i}}:\mathit{V}\times \mathit{Q}\to {\mathit{I}}_{\mathit{k}}$ for $\mathit{i}=\mathbf{1},\mathbf{2},\dots ,\mathit{k}$. The membership function of the multi-q fuzzy set ${\tilde{\mathit{G}}}_{\mathit{Q}}$ is given by the tuple $({\Delta }_{\mathbf{1}}(\mathit{v},\mathit{q}),{\Delta }_{\mathbf{2}}(\mathit{v},\mathit{q}),\dots ,{\Delta }_{\mathit{k}}(\mathit{v},\mathit{q}))$, which satisfies the condition $({\Delta }_{\mathbf{1}}(\mathit{v},\mathit{q})+{\Delta }_{\mathbf{2}}(\mathit{v},\mathit{q})+\cdots +{\Delta }_{\mathit{k}}(\mathit{v},\mathit{q}))\le \mathbf{1}$. The notation ${\Delta }_{\mathit{Q}}^{\mathit{k}}\mathit{G}\left(\mathit{V}\right)$ represents the set of all multi-q fuzzy sets of dimension $\mathit{k}$ in $\mathit{V}$ and $\mathit{Q}$, encompassing all such sets within these parameters.

3.6. Bipolar Multi-Q Fuzzy Set [18]

Definition 6. 

For the sets $\mathit{H}$ and $\mathit{Q}$, which are non-empty, the concept of bipolar multi-q fuzzy set Δ in $\mathit{H}$ is described as

$$\Delta =\{(\mathit{h},\mathit{q}),{\Delta }_{\mathit{i}}^{+}(\mathit{h},\mathit{q}),{\Delta }_{\mathit{i}}^{-}(\mathit{h},\mathit{q})\mid \mathit{h}\in \mathit{H},\mathit{q}\in \mathit{Q}\}$$

where ${\Delta }_{\mathit{i}}^{+}:\mathit{H}\times \mathit{Q}\to [\mathbf{0},\mathbf{1}]$ and ${\Delta }_{\mathit{i}}^{-}:\mathit{H}\times \mathit{Q}\to [-\mathbf{1},\mathbf{0}]$ are functions. The value ${\Delta }_{\mathit{i}}^{+}(\mathit{h},\mathit{q})$ represents the extent to which an element $(\mathit{h},\mathit{q})$ fulfills the associated property in the bipolar Q fuzzy set Δ. Conversely, ${\Delta }_{\mathit{i}}^{-}(\mathit{h},\mathit{q})$ measures the degree of non-fulfillment of the same element, providing a comprehensive assessment of the element’s membership in positive and negative terms.

4. Multi-$\mathit{Q}$ Cubic Bipolar Fuzzy Soft Set

We introduce the concept of a multi-q cubic bipolar fuzzy set (MQCBFSS) as a novel approach to describe intricate data that encompasses both positive and negative attributes. It takes into account elements within an original universe combined with supplementary factors. The MQCBFSS assigns two sets of membership degrees: one interval-valued bipolar fuzzy set that captures both positive and negative characteristics simultaneously and a separate bipolar fuzzy set that represents positive and negative memberships separately. This enables a more comprehensive and refined method of evaluating the extent to which an element aligns with a specific category or set of criteria.

Definition 7. 

The concept of multi-$\mathit{Q}$ cubic bipolar fuzzy set $\mathcal{B}$ over the initial universe $\mathit{Y}$ and $\mathit{Q}$ can be defined as

$$\mathcal{B}=\{⟨(\mathit{y},\mathit{q}),\mathit{D}(\mathit{y},\mathit{q}),\mathit{E}(\mathit{y},\mathit{q})\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\},$$

where $\mathit{D}$ is an interval-valued bipolar fuzzy set (IVBF), and $\mathit{E}$ is a bipolar fuzzy set (BF). This can also be written as

$$\begin{array}{c}\hfill \mathcal{B}=\{⟨(\mathit{y},\mathit{q}),[{\Delta }_{{\mathit{l}}_{\mathit{B}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{u}}_{\mathit{B}}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{{\mathit{l}}_{\mathit{B}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{u}}_{\mathit{B}}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q})\}\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\},\end{array}$$

where the intervals $[{\Delta }_{{\mathit{l}}_{\mathit{B}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{u}}_{\mathit{B}}}^{+}(\mathit{y},\mathit{q})]\in \mathit{I}[\mathbf{0},\mathbf{1}]$ and $[{\Delta }_{{\mathit{l}}_{\mathit{B}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{u}}_{\mathit{B}}}^{-}(\mathit{y},\mathit{q})]\in {\mathit{I}}^{*}[-\mathbf{1},\mathbf{0}]$ represent the interval-valued positive and negative membership degrees, respectively. The values ${\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q})\in [\mathbf{0},\mathbf{1}]$ and ${\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q})\in [-\mathbf{1},\mathbf{0}]$ indicate the positive and negative membership levels for an element $\mathit{y}\in \mathit{Y}$ and $\mathit{q}\in \mathit{Q}$.

Definition 8. 

Definition 2 introduces the concept of a multi-$\mathit{Q}$ cubic bipolar fuzzy number, which is a single element within a multi-$\mathit{Q}$ cubic bipolar fuzzy set (MQCBFSS) as defined in Definition 2. This number represents the intricate membership details of a particular element $\mathit{y}$ that is part of a factor $\mathit{q}$ within the MQCBFSS. It serves as a data point in the MQCBFSS framework.

The concept of multi-$\mathit{Q}$ cubic bipolar fuzzy numbers can be represented as

$$⟨(\mathit{y},\mathit{q}),[{\Delta }_{{\mathit{l}}_{\mathit{B}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{u}}_{\mathit{B}}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{{\mathit{l}}_{\mathit{B}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{u}}_{\mathit{B}}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q})\}⟩.$$

Example 1. 

Consider $\mathit{k}=⟨\mathbf{N},\mathit{M}⟩$ as a multi-$\mathit{Q}$ cubic bipolar fuzzy set on $\mathit{v}$:

$$\mathit{k}=\{⟨(\mathit{y},\mathit{q}),[\mathbf{0}.\mathbf{23},\mathbf{0}.\mathbf{61}],[-\mathbf{0}.\mathbf{63},-\mathbf{0}.\mathbf{45}],\{\mathbf{0}.\mathbf{71},-\mathbf{0}.\mathbf{68}\}⟩\}.$$

Example 1 illustrates a particular component (represented as N) in a multi-q cubic bipolar fuzzy set (MQCBFSS). This element, associated with factor M, possesses both favorable and unfavorable characteristics. The interval [0.23, 0.61] represents the extent of its positive correlation with M, whereas [−0.63, −0.45] represents the extent of its negative correlation. Furthermore, distinct positive (0.71) and negative (−0.68) membership degrees offer additional insights into its correlation with factor M.

Definition 9. 

Consider the initial universe $\mathcal{B}$; let

$$\begin{array}{cc}\hfill \mathcal{B}& =\left\{⟨(\mathit{y},\mathit{q}),[{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q})\}⟩\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\right\},\hfill \\ \hfill \mathit{E}& =\left\{⟨(\mathit{y},\mathit{q}),[{\Delta }_{\mathit{l}\mathit{E}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathit{E}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}\mathit{E}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathit{E}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{\mathit{E}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{E}}^{-}(\mathit{y},\mathit{q})\}⟩\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\right\},\hfill \end{array}$$

be two multi-Q cubic bipolar fuzzy sets on $\mathit{y}\in \mathit{Y}$, $q\in Q$, and $\lambda >\mathbf{0}$.

Then, the operations on multi-Q cubic bipolar fuzzy sets under P-order are given below:

(1) 

$$\begin{array}{cc}\hfill \mathcal{B}{\cup }_{\mathit{P}}\mathit{E}& =\left\{⟨(\mathit{y},\mathit{q}),\right.\hfill \\ \hfill & [max\{{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}\mathit{E}}^{+}(\mathit{y},\mathit{q})\},max\{{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathit{E}}^{+}(\mathit{y},\mathit{q})\}]\hfill \\ \hfill & [min\{{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}\mathit{E}}^{-}(\mathit{y},\mathit{q})\},min\{{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathit{E}}^{-}(\mathit{y},\mathit{q})\}]\hfill \\ \hfill & \left.\{max\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{E}}^{+}(\mathit{y},\mathit{q})\},min\{{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{E}}^{-}(\mathit{y},\mathit{q})\}\}⟩\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\right\}.\hfill \end{array}$$

(2) 

$$\begin{array}{cc}\hfill \mathcal{B}{\cap }_{\mathit{P}}\mathit{E}& =\left\{⟨(\mathit{y},\mathit{q}),\right.\hfill \\ \hfill & [min\{{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}\mathit{E}}^{+}(\mathit{y},\mathit{q})\},min\{{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathit{E}}^{+}(\mathit{y},\mathit{q})\}]\hfill \\ \hfill & [max\{{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}\mathit{E}}^{-}(\mathit{y},\mathit{q})\},max\{{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathit{E}}^{-}(\mathit{y},\mathit{q})\}]\hfill \\ \hfill & \left.\{min\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{E}}^{+}(\mathit{y},\mathit{q})\},max\{{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{E}}^{-}(\mathit{y},\mathit{q})\}\}⟩\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\right\}.\hfill \end{array}$$

(3) 

$$\begin{array}{cc}\hfill \mathcal{B}{\oplus }_{\mathit{P}}\mathit{E}& =\left\{⟨(\mathit{y},\mathit{q}),\right.\hfill \\ \hfill & [{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q})+{\Delta }_{\mathit{l}\mathit{E}}^{+}(\mathit{y},\mathit{q})\hfill \\ \hfill & -{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}){\Delta }_{\mathit{l}\mathit{E}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q})+{\Delta }_{\mathit{u}\mathit{E}}^{+}(\mathit{y},\mathit{q})\hfill \\ \hfill & -{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q}){\Delta }_{\mathit{u}\mathit{E}}^{+}(\mathit{y},\mathit{q})],[-{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}){\Delta }_{\mathit{l}\mathit{E}}^{-}(\mathit{y},\mathit{q}),\hfill \\ \hfill & -\left({\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q}){\Delta }_{\mathit{u}\mathit{E}}^{-}(\mathit{y},\mathit{q})\right)],\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q})+{\Delta }_{\mathit{E}}^{+}(\mathit{y},\mathit{q})\hfill \\ \hfill & -{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}){\Delta }_{\mathit{E}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q}){\Delta }_{\mathit{E}}^{-}(\mathit{y},\mathit{q})\}\hfill \\ \hfill & \left.⟩\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\right\}.\hfill \end{array}$$

(4) 

$$\begin{array}{cc}\hfill \mathcal{B}{\otimes }_{\mathit{P}}\mathit{E}& =\left\{⟨(\mathit{y},\mathit{q}),\right.\hfill \\ \hfill & [{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}){\Delta }_{\mathit{l}\mathit{E}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q}){\Delta }_{\mathit{u}\mathit{E}}^{+}(\mathit{y},\mathit{q})]\hfill \\ \hfill & [-(\mathbf{1}-(\mathbf{1}-(-{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q})){\Delta }_{\mathit{l}\mathit{E}}^{-}(\mathit{y},\mathit{q}))-{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q}){\Delta }_{\mathit{u}\mathit{E}}^{-}(\mathit{y},\mathit{q}))]\hfill \\ \hfill & [-{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q}){\Delta }_{\mathit{u}\mathit{E}}^{-}(\mathit{y},\mathit{q})-{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q}){\Delta }_{\mathit{u}\mathit{E}}^{-}(\mathit{y},\mathit{q}))]\hfill \\ \hfill & \left.\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}){\Delta }_{\mathit{E}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q}){\Delta }_{\mathit{E}}^{-}(\mathit{y},\mathit{q})\}⟩\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\right\}.\hfill \end{array}$$

(5) 

$$\begin{array}{cc}\hfill {\mathcal{B}}^{\lambda }& =\left\{⟨(\mathit{y},\mathit{q}),\right.\hfill \\ \hfill & [{\left({\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q})\right)}^{\lambda },{\left({\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q})\right)}^{\lambda }]\hfill \\ \hfill & [-(\mathbf{1}-(\mathbf{1}-{(-{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}))}^{\lambda })),-(\mathbf{1}-(\mathbf{1}-{(-{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q}))}^{\lambda }))]\hfill \\ \hfill & \left.\{{\Delta }_{\mathit{B}}^{+}{(\mathit{y},\mathit{q})}^{\lambda },-(\mathbf{1}-(\mathbf{1}-{(-{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q}))}^{\lambda }))\}⟩\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\right\}.\hfill \end{array}$$

(6) 

$$\begin{array}{cc}\hfill \lambda \mathcal{B}& =\left\{⟨(\mathit{y},\mathit{q}),\right.\hfill \\ \hfill & [\mathbf{1}-{(\mathbf{1}-{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}))}^{\lambda },\mathbf{1}-{(\mathbf{1}-{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q}))}^{\lambda }]\hfill \\ \hfill & [-{\left({\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q})\right)}^{\lambda },-{\left({\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q})\right)}^{\lambda }]\hfill \\ \hfill & \left.\{{\Delta }_{\mathit{B}}^{+}{(\mathit{y},\mathit{q})}^{\lambda },-(\mathbf{1}-{(\mathbf{1}-{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q}))}^{\lambda })\}⟩\mid \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\right\}.\hfill \end{array}$$

(7) 

$$\begin{array}{cc}\hfill \mathcal{B}{\subseteq }_{\mathit{P}}\mathit{E}& =[{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q})]\le [{\Delta }_{\mathit{l}\mathit{E}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathit{E}}^{+}(\mathit{y},\mathit{q})]\hfill \\ \hfill & \mathit{and}[{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q})]\ge [{\Delta }_{\mathit{l}\mathit{E}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathit{E}}^{-}(\mathit{y},\mathit{q})]\hfill \\ \hfill & \mathit{and}{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q})\le {\Delta }_{\mathit{E}}^{+}(\mathit{y},\mathit{q})\hfill \\ \hfill & \mathit{and}{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q})\ge {\Delta }_{\mathit{E}}^{-}(\mathit{y},\mathit{q});\forall \mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}.\hfill \end{array}$$

Definition 10. 

Let

$$\begin{array}{cc}\hfill \mathcal{B}=\{& ⟨(\mathit{y},\mathit{q}),[{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q})\}⟩:\mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\}\hfill \end{array}$$

be a multi-Q cubic bipolar fuzzy set.

The complement of $\mathcal{B}$ can be defined as:

$$\begin{array}{cc}\hfill {\mathcal{B}}^{\mathit{c}}=\{& ⟨(\mathit{y},\mathit{q}),[\mathbf{1}-{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),\mathbf{1}-{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q})],\hfill \\ \hfill & [-\mathbf{1}+(-{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q})),-\mathbf{1}+(-{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}))],\hfill \\ \hfill & \{\mathbf{1}-{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),-\mathbf{1}+(-{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q}))\}⟩:\mathit{y}\in \mathit{Y},\mathit{q}\in \mathit{Q}\}.\hfill \end{array}$$

Theorem 1. 

Let

$$\begin{array}{cc}\hfill \mathcal{B}& =\left([{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q})\}\right),\hfill \\ \hfill {\mathit{B}}_{\mathbf{1}}& =\left([{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})\}\right),\hfill \\ \hfill {\mathit{B}}_{\mathbf{2}}& =\left([{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\}\right),\hfill \end{array}$$

be three multi-Q cubic bipolar fuzzy sets, and let $\lambda >\mathbf{0}$; then, the following are valid under P-order:

(1) 

${\left({\mathcal{B}}^{\mathit{c}}\right)}^{\lambda }={\left(\lambda \mathcal{B}\right)}^{\mathit{c}}.$

(2) 

$\lambda \left({\mathcal{B}}^{\mathit{c}}\right)={\left({\mathcal{B}}^{\lambda }\right)}^{\mathit{c}}.$

(3) 

$\lambda \left({\mathit{B}}_{\mathbf{1}}{\cup }_{\mathit{P}}{\mathit{B}}_{\mathbf{2}}\right)=\lambda {\mathit{B}}_{\mathbf{1}}{\cup }_{\mathit{P}}\lambda {\mathit{B}}_{\mathbf{2}}.$

(4) 

${\left({\mathit{B}}_{\mathbf{1}}{\cup }_{\mathit{P}}{\mathit{B}}_{\mathbf{2}}\right)}^{\lambda }={\mathit{B}}_{\mathbf{1}}^{\lambda }{\cup }_{\mathit{P}}{\mathit{B}}_{\mathbf{2}}^{\lambda }.$

(5) 

$\lambda \left({\mathit{B}}_{\mathbf{1}}{\cap }_{\mathit{P}}{\mathit{B}}_{\mathbf{2}}\right)=\lambda {\mathit{B}}_{\mathbf{1}}{\cap }_{\mathit{P}}\lambda {\mathit{B}}_{\mathbf{2}}.$

(6) 

${\left({\mathit{B}}_{\mathbf{1}}{\cap }_{\mathit{P}}{\mathit{B}}_{\mathbf{2}}\right)}^{\lambda }={\mathit{B}}_{\mathbf{1}}^{\lambda }{\cap }_{\mathit{P}}{\mathit{B}}_{\mathbf{2}}^{\lambda }.$

Proof. 

We shall prove (1), (3), (5), while (2), (4), and (6) can be proved analogously:

(1)

$$\begin{array}{cc}\hfill {\left({\mathcal{B}}^{\mathit{c}}\right)}^{\lambda }& =\left(\right[\mathbf{1}-{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),\mathbf{1}-{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q})],\hfill \\ \hfill & [-\mathbf{1}+(-{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q})),-\mathbf{1}+(-{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}))],\hfill \\ \hfill & \{\mathbf{1}-{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),-\mathbf{1}+(-{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q}))\}{)}^{\lambda }\hfill \\ \hfill & =\left(\right[{(\mathbf{1}-{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q}))}^{\lambda },{(\mathbf{1}-{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}))}^{\lambda }],\hfill \\ \hfill & \left[-\left(\mathbf{1}-{\left(\mathbf{1}-(-{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q}))\right)}^{\lambda }\right),\right.\hfill \\ \hfill & \left.-\left(\mathbf{1}-{\left(\mathbf{1}-(-{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}))\right)}^{\lambda }\right)\right],\hfill \\ \hfill & \{{(\mathbf{1}-{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}))}^{\lambda },-\mathbf{1}+{(-{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q}))}^{\lambda }\})\hfill \\ \hfill & ={\left({\mathcal{B}}^{\mathit{c}}\right)}^{\lambda }\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \lambda \left({\mathit{B}}_{\mathbf{1}}{\cup }_{\mathit{P}}{\mathit{B}}_{\mathbf{2}}\right)& =\left(\right[\mathbf{1}-{\left(\mathbf{1}-max\{{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\}\right)}^{\lambda },\hfill \\ \hfill & \mathbf{1}-{\left(\mathbf{1}-max\{{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\}\right)}^{\lambda }],\hfill \\ \hfill & [-min{\{{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\}}^{\lambda },\hfill \\ \hfill & -min{\{{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\}}^{\lambda }],\hfill \\ \hfill & \{\mathbf{1}-{\left(\mathbf{1}-max\{{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\}\right)}^{\lambda },\hfill \\ \hfill & -min{\{{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\}}^{\lambda }\left\}\right)\hfill \\ \hfill & =\left(\right[\mathbf{1}-{\left(\mathbf{1}-max\{{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\}\right)}^{\lambda },\hfill \\ \hfill & \mathbf{1}-{\left(\mathbf{1}-max\{{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\}\right)}^{\lambda }],\hfill \\ \hfill & [-min{\{{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\}}^{\lambda },\hfill \\ \hfill & -min{\{{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\}}^{\lambda }],\hfill \\ \hfill & \{\mathbf{1}-{\left(\mathbf{1}-max\{{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\}\right)}^{\lambda },\hfill \\ \hfill & -min{\{{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\}}^{\lambda }\left\}\right)\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \lambda ({\mathit{B}}_{\mathbf{1}}{\cap }_{\mathit{P}}{\mathit{B}}_{\mathbf{2}})& =([\mathbf{1}-{\left(\mathbf{1}-min\{{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\}\right)}^{\lambda },\hfill \\ \hfill & \mathbf{1}-{\left(\mathbf{1}-min\{{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\}\right)}^{\lambda }],\hfill \\ \hfill & [-{(max\{{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\})}^{\lambda },\hfill \\ \hfill & -{(max\{{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\})}^{\lambda }],\hfill \\ \hfill & \{\mathbf{1}-{\left(\mathbf{1}-min\{{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\}\right)}^{\lambda },\hfill \\ \hfill & -{(max\{{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\})}^{\lambda }\})\hfill \\ \hfill & =([min\{\mathbf{1}-{\left(\mathbf{1}-{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})\right)}^{\lambda },\hfill \\ \hfill & \mathbf{1}-{\left(\mathbf{1}-{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\right)}^{\lambda }\},\hfill \\ \hfill & min\{\mathbf{1}-{\left(\mathbf{1}-{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})\right)}^{\lambda },\hfill \\ \hfill & \mathbf{1}-{\left(\mathbf{1}-{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\right)}^{\lambda }\left\}\right],\hfill \\ \hfill & [max\{-{({\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}))}^{\lambda },-{({\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}))}^{\lambda }\},\hfill \\ \hfill & max\{-{({\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}))}^{\lambda },-{({\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}))}^{\lambda }\}],\hfill \\ \hfill & \{min\{\mathbf{1}-{\left(\mathbf{1}-{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})\right)}^{\lambda },\hfill \\ \hfill & \mathbf{1}-{\left(\mathbf{1}-{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\right)}^{\lambda }\},\hfill \\ \hfill & max\{-{({\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}))}^{\lambda },-{({\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}))}^{\lambda }\})\hfill \end{array}$$

□

Theorem 2. 

Let

$$\begin{array}{cc}\hfill \mathcal{B}& =\left([{\Delta }_{\mathit{l}\mathcal{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}\mathcal{B}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}\mathcal{B}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{\mathit{B}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{B}}^{-}(\mathit{y},\mathit{q})\}\right),\hfill \\ \hfill {\mathit{B}}_{\mathbf{1}}& =\left([{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})\}\right),\hfill \\ \hfill {\mathit{B}}_{\mathbf{2}}& =\left([{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})\}\right),\hfill \end{array}$$

be three multi-q cubic bipolar fuzzy sets, and let $\lambda >\mathbf{0}$; then, the following are valid under $\mathit{R}$-order:

(1) 

${\left({\mathcal{B}}^{\mathit{c}}\right)}^{\lambda }={\left(\lambda \mathcal{B}\right)}^{\mathit{c}}.$

(2) 

$\lambda \left({\mathcal{B}}^{\mathit{c}}\right)={\left({\mathcal{B}}^{\lambda }\right)}^{\mathit{c}}.$

(3) 

$\lambda \left({\mathit{B}}_{\mathbf{1}}{\cup }_{\mathit{R}}{\mathit{B}}_{\mathbf{2}}\right)=\lambda {\mathit{B}}_{\mathbf{1}}{\cup }_{\mathit{R}}\lambda {\mathit{B}}_{\mathbf{2}}.$

(4) 

${\left({\mathit{B}}_{\mathbf{1}}{\cup }_{\mathit{R}}{\mathit{B}}_{\mathbf{2}}\right)}^{\lambda }={\mathit{B}}_{\mathbf{1}}^{\lambda }{\cup }_{\mathit{R}}{\mathit{B}}_{\mathbf{2}}^{\lambda }.$

(5) 

$\lambda \left({\mathit{B}}_{\mathbf{1}}{\cap }_{\mathit{R}}{\mathit{B}}_{\mathbf{2}}\right)=\lambda {\mathit{B}}_{\mathbf{1}}{\cap }_{\mathit{R}}\lambda {\mathit{B}}_{\mathbf{2}}.$

(6) 

${\left({\mathit{B}}_{\mathbf{1}}{\cap }_{\mathit{R}}{\mathit{B}}_{\mathbf{2}}\right)}^{\lambda }={\mathit{B}}_{\mathbf{1}}^{\lambda }{\cap }_{\mathit{R}}{\mathit{B}}_{\mathbf{2}}^{\lambda }.$

Proof. 

Straightforward. □

Definition 11. 

Let ${\mathit{B}}_{\mathit{i}}=([{\Delta }_{\mathit{l}{\mathit{B}}_{\mathit{i}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathit{i}}}^{+}(\mathit{y},\mathit{q})],[{\Delta }_{\mathit{l}{\mathit{B}}_{\mathit{i}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathit{i}}}^{-}(\mathit{y},\mathit{q})],\{{\Delta }_{{\mathit{B}}_{\mathit{i}}}^{+}(\mathit{y},\mathit{q}),$${\Delta }_{{\mathit{B}}_{\mathit{i}}}^{-}(\mathit{y},\mathit{q})\left\}\right),$$\mathrm{for}(i=1,2)$ be two multi-q cubic bipolar fuzzy sets. Then

(1) 

${\mathit{B}}_{\mathbf{1}}={\mathit{B}}_{\mathbf{2}}$ if

$$\begin{array}{cc}\hfill & [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})]=[{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})]\hfill \\ \hfill & [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})]=[{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})]\hfill \\ \hfill & {\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})={\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\hfill \\ \hfill & {\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})={\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}).\hfill \end{array}$$

(2) 

${\mathit{B}}_{\mathbf{1}}{\subseteq }_{\mathit{P}}{\mathit{B}}_{\mathbf{2}}$ if

$$\begin{array}{cc}\hfill & [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})]\le [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})]\hfill \\ \hfill & [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})]\ge [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})]\hfill \\ \hfill & {\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})\le {\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\hfill \\ \hfill & {\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})\ge {\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}).\hfill \end{array}$$

(3) 

${\mathit{B}}_{\mathbf{1}}{\subseteq }_{\mathit{R}}{\mathit{B}}_{\mathbf{2}}$ if

$$\begin{array}{cc}\hfill & [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})]\le [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})]\hfill \\ \hfill & [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})]\ge [{\Delta }_{\mathit{l}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}),{\Delta }_{\mathit{u}{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q})]\hfill \\ \hfill & {\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{+}(\mathit{y},\mathit{q})\ge {\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{+}(\mathit{y},\mathit{q})\hfill \\ \hfill & {\Delta }_{{\mathit{B}}_{\mathbf{1}}}^{-}(\mathit{y},\mathit{q})\le {\Delta }_{{\mathit{B}}_{\mathbf{2}}}^{-}(\mathit{y},\mathit{q}).\hfill \end{array}$$

Definition 12. 

For any multi-Q cubic bipolar fuzzy set, the score function under $\mathit{P}$-order can be defined as

$$\begin{array}{cc}\hfill {\mathit{S}}_{\mathit{P}}\left(\mathcal{B}\right)& ={\displaystyle \frac{1}{6}}\left([{\Delta }_{l\mathcal{B}}^{+}(y,q)+{\Delta }_{u\mathcal{B}}^{+}(y,q)]+[2+{\Delta }_{l\mathcal{B}}^{-}(y,q)+{\Delta }_{u\mathcal{B}}^{-}(y,q)]-{\Delta }_B^{+}(y,q)+1+{\Delta }_B^{+}(y,q)\right).\hfill \end{array}$$

Likewise, the score function under $\mathit{R}$-order is given by

$$\begin{array}{cc}\hfill {\mathit{S}}_{\mathit{R}}\left(\mathcal{B}\right)& ={\displaystyle \frac{1}{6}}\left([{\Delta }_{l\mathcal{B}}^{+}(y,q)+{\Delta }_{u\mathcal{B}}^{+}(y,q)]+[2+{\Delta }_{l\mathcal{B}}^{-}(y,q)+{\Delta }_{u\mathcal{B}}^{-}(y,q)]+{\Delta }_B^{+}(y,q)+1+{\Delta }_B^{-}(y,q)\right).\hfill \end{array}$$

where ${\mathit{S}}_{\mathit{P},\mathit{R}}\left(\mathcal{B}\right)\in [\mathbf{0},\mathbf{1}]$.

Definition 13. 

For any multi-Q cubic bipolar fuzzy set, the accuracy function can be defined as

$$\begin{array}{cc}\hfill \mathcal{B}\left(\mathcal{B}\right)& ={\displaystyle \frac{1}{6}}\left([{\Delta }_{l\mathcal{B}}^{+}(y,q)+{\Delta }_{u\mathcal{B}}^{+}(y,q)]-[{\Delta }_{l\mathcal{B}}^{-}(y,q)+{\Delta }_{u\mathcal{B}}^{-}(y,q)]+{\Delta }_B^{+}(y,q)-{\Delta }_B^{-}(y,q)\right).\hfill \end{array}$$

where $\mathcal{B}\left(\mathcal{B}\right)\in [\mathbf{0},\mathbf{1}]$. The score and accuracy functions are used to compare two multi-q cubic bipolar fuzzy sets.

(i) 

If ${\mathit{S}}_{\mathit{P},\mathit{R}}\left({\mathit{B}}_{\mathbf{1}}\right)<{\mathit{S}}_{\mathit{P},\mathit{R}}\left({\mathit{B}}_{\mathbf{2}}\right)$, then ${\mathit{B}}_{\mathbf{1}}$ is considered smaller than ${\mathit{B}}_{\mathbf{2}}$, denoted by ${\mathit{B}}_{\mathbf{1}}<{\mathit{B}}_{\mathbf{2}}$.

(ii) 

If ${\mathit{S}}_{\mathit{P},\mathit{R}}\left({\mathit{B}}_{\mathbf{1}}\right)>{\mathit{S}}_{\mathit{P},\mathit{R}}\left({\mathit{B}}_{\mathbf{2}}\right)$, then ${\mathit{B}}_{\mathbf{1}}$ is considered greater than ${\mathit{B}}_{\mathbf{2}}$, denoted by ${\mathit{B}}_{\mathbf{1}}>{\mathit{B}}_{\mathbf{2}}$.

(iii) 

If ${\mathit{S}}_{\mathit{P},\mathit{R}}\left({\mathit{B}}_{\mathbf{1}}\right)={\mathit{S}}_{\mathit{P},\mathit{R}}\left({\mathit{B}}_{\mathbf{2}}\right)$, then if $\mathcal{B}\left({\mathit{B}}_{\mathbf{1}}\right)>\mathcal{B}\left({\mathit{B}}_{\mathbf{2}}\right)$, ${\mathit{B}}_{\mathbf{1}}$ is considered greater than ${\mathit{B}}_{\mathbf{2}}$. If $\mathcal{B}\left({\mathit{B}}_{\mathbf{1}}\right)=\mathcal{B}\left({\mathit{B}}_{\mathbf{2}}\right)$, then ${\mathit{B}}_{\mathbf{1}}\sim {\mathit{B}}_{\mathbf{2}}$.

5. Multi-$\mathit{Q}$ Cubic Bipolar Fuzzy Soft Set

Definition 14. 

Let $\mathit{V}$ be an initial universe, $\mathit{P}$ be a set of parameters, and $\mathcal{B}\subset \mathit{P}$. The pair $(\mathit{G},\mathcal{B})$ is defined as a soft set if $\mathit{G}:\mathcal{B}\to \mathit{P}\left(\mathit{V}\right)$, where $\mathit{P}\left(\mathit{V}\right)$ represents the power set of $\mathit{V}$.

Definition 15. 

Consider $\mathit{V}$ as a universe, $\mathit{P}$ as a set of parameters, and $\mathcal{B}\subset \mathit{P}$. Define ${\tilde{\mathit{G}}}_{\mathit{R}}:\mathcal{B}\to \mathit{mRCBFS}\left({\tilde{\mathit{G}}}_{\mathit{R}}^{\mathit{V}}\right)$, where $\mathit{mRCBFS}\left({\tilde{\mathit{G}}}_{\mathit{R}}^{\mathit{V}}\right)$ denotes the set of all multi-R cubic bipolar fuzzy subsets within $\mathit{V}$. The pair $({\tilde{\mathit{G}}}_{\mathit{R}},\mathcal{B})$ is then referred to as a multi-R cubic bipolar fuzzy soft set over the universe $\mathit{V}$, and it is described by

$$({\tilde{\mathit{G}}}_{\mathit{R}},\mathcal{B})=\{⟨\mathit{f},[{\Gamma }_{{\mathit{l}}_{\mathit{B}}\mathit{f}}^{+}(\mathit{y},\mathit{r}),{\Gamma }_{{\mathit{u}}_{\mathit{B}}\mathit{f}}^{+}(\mathit{y},\mathit{r})],[{\Gamma }_{{\mathit{l}}_{\mathit{B}}\mathit{f}}^{-}(\mathit{y},\mathit{r}),{\Gamma }_{{\mathit{u}}_{\mathit{B}}\mathit{f}}^{-}(\mathit{y},\mathit{r})],\{{\Gamma }_{\mathit{f}\mathcal{B}}^{+},{\Gamma }_{\mathit{f}\mathcal{B}}^{-}\}⟩\mid \mathit{y}\in \mathit{V},\mathit{r}\in \mathit{R},\mathit{f}\in \mathcal{B}\}.$$

Definition 16. 

Let $\mathit{V}$ be a universe, $\mathit{P}$ be a set of parameters, and $\mathit{Y}$ be a set of experts. Define $\mathit{S}=\{0=\mathit{disagree},1=\mathit{agree}\}$ as a set of two-valued opinions, and let $\mathit{W}=\mathit{P}\times \mathit{Y}\times \mathit{S}$. If $\mathcal{B}\subseteq \mathit{W}$, then the pair $(\mathit{G},\mathcal{B})$ is referred to as a soft expert set over $\mathit{V}$, where $\mathit{G}$ is a mapping such that $\mathit{G}:\mathcal{B}\to \mathit{P}\left(\mathit{V}\right)$, with $\mathit{P}\left(\mathit{V}\right)$ representing the power set of $\mathit{V}$.

Example 2. 

Suppose a company intends to purchase three types of products from two different brands and seeks the opinions of two specialists regarding these products. Let $\mathit{V}=\{{\mathit{v}}_{\mathbf{1}},{\mathit{v}}_{\mathbf{2}},{\mathit{v}}_{\mathbf{3}}\}$ represent the set of products, $\mathit{R}=\{{\mathit{r}}_{\mathbf{1}},{\mathit{r}}_{\mathbf{2}}\}$ denote the set of brands, and $\mathit{P}=\{{\mathit{p}}_{\mathbf{1}}=\mathit{ease}\mathit{of}\mathit{use},{\mathit{p}}_{\mathbf{2}}=\mathit{quality},{\mathit{p}}_{\mathbf{3}}=\mathit{design}\}$ be the set of decision parameters. The multi-$\mathit{R}$ cubic bipolar fuzzy soft set $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ can be defined as follows:

$$\begin{array}{cc}\hfill ({\mathit{G}}_{\mathit{R}},\mathcal{B})=\{& ⟨{\mathit{p}}_{\mathbf{1}},\left({\displaystyle \frac{{\mathit{v}}_{\mathbf{1}}\mathit{r}}{[\mathbf{0}.\mathbf{16},\mathbf{0}.\mathbf{28}],[-\mathbf{0}.\mathbf{69},-\mathbf{0}.\mathbf{53}],\{\mathbf{0}.\mathbf{36},-\mathbf{0}.\mathbf{61}\}}}\right),\left({\displaystyle \frac{{\mathit{v}}_{\mathbf{1}}\mathbf{s}}{[\mathbf{0}.\mathbf{13},\mathbf{0}.\mathbf{22}],[-\mathbf{0}.\mathbf{54},-\mathbf{0}.\mathbf{43}],\{\mathbf{0}.\mathbf{34},-\mathbf{0}.\mathbf{41}\}}}\right),\hfill \\ & \left({\displaystyle \frac{{\mathit{v}}_{\mathbf{3}}\mathit{r}}{[\mathbf{0}.\mathbf{32},\mathbf{0}.\mathbf{46}],[-\mathbf{0}.\mathbf{62},-\mathbf{0}.\mathbf{42}],\{\mathbf{0}.\mathbf{58},-\mathbf{0}.\mathbf{33}\}}}\right),\left({\displaystyle \frac{{\mathit{v}}_{\mathbf{3}}\mathbf{s}}{[\mathbf{0}.\mathbf{66},\mathbf{0}.\mathbf{46}],[-\mathbf{0}.\mathbf{71},-\mathbf{0}.\mathbf{63}],\{\mathbf{0}.\mathbf{42},-\mathbf{0}.\mathbf{56}\}}}\right)⟩\hfill \\ & ⟨{\mathit{p}}_{\mathbf{2}},\left({\displaystyle \frac{{\mathit{v}}_{\mathbf{2}}\mathit{r}}{[\mathbf{0}.\mathbf{66},\mathbf{0}.\mathbf{78}],[-\mathbf{0}.\mathbf{36},-\mathbf{0}.\mathbf{22}],\{\mathbf{0}.\mathbf{51},-\mathbf{0}.\mathbf{44}\}}}\right),\left({\displaystyle \frac{{\mathit{v}}_{\mathbf{2}}\mathbf{s}}{[\mathbf{0}.\mathbf{72},\mathbf{0}.\mathbf{81}],[-\mathbf{0}.\mathbf{42},-\mathbf{0}.\mathbf{33}],\{\mathbf{0}.\mathbf{63},-\mathbf{0}.\mathbf{74}\}}}\right),\hfill \\ & \left({\displaystyle \frac{{\mathit{v}}_{\mathbf{3}}\mathit{r}}{[\mathbf{0}.\mathbf{62},\mathbf{0}.\mathbf{71}],[-\mathbf{0}.\mathbf{59},-\mathbf{0}.\mathbf{42}],\{\mathbf{0}.\mathbf{26},-\mathbf{0}.\mathbf{31}\}}}\right),\left({\displaystyle \frac{{\mathit{v}}_{\mathbf{3}}\mathbf{s}}{[\mathbf{0}.\mathbf{56},\mathbf{0}.\mathbf{67}],[-\mathbf{0}.\mathbf{46},-\mathbf{0}.\mathbf{39}],\{\mathbf{0}.\mathbf{23},-\mathbf{0}.\mathbf{47}\}}}\right)⟩\hfill \\ & ⟨{\mathit{p}}_{\mathbf{3}},\left({\displaystyle \frac{{\mathit{v}}_{\mathbf{1}}\mathit{r}}{[\mathbf{0}.\mathbf{37},\mathbf{0}.\mathbf{55}],[-\mathbf{0}.\mathbf{69},-\mathbf{0}.\mathbf{86}],\{\mathbf{0}.\mathbf{45},-\mathbf{0}.\mathbf{53}\}}}\right),\left({\displaystyle \frac{{\mathit{v}}_{\mathbf{1}}\mathbf{s}}{[\mathbf{0}.\mathbf{42},\mathbf{0}.\mathbf{63}],[-\mathbf{0}.\mathbf{35},-\mathbf{0}.\mathbf{28}],\{\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{7}\}}}\right),\hfill \\ & \left({\displaystyle \frac{{\mathit{v}}_{\mathbf{2}}\mathit{r}}{[\mathbf{0}.\mathbf{64},\mathbf{0}.\mathbf{82}],[-\mathbf{0}.\mathbf{57},-\mathbf{0}.\mathbf{44}],\{\mathbf{0}.\mathbf{8},-\mathbf{0}.\mathbf{9}\}}}\right),\left({\displaystyle \frac{{\mathit{v}}_{\mathbf{2}}\mathbf{s}}{[\mathbf{0}.\mathbf{53},\mathbf{0}.\mathbf{74}],[-\mathbf{0}.\mathbf{61},-\mathbf{0}.\mathbf{32}],\{\mathbf{0}.\mathbf{56},-\mathbf{0}.\mathbf{73}\}}}\right)⟩\hfill \end{array}$$

Definition 17. 

Consider two soft sets $(\mathit{G},\mathcal{B})$ and $(\mathit{H},\mathit{C})$ over the universe $\mathit{V}$. The soft set $(\mathit{G},\mathcal{B})$ is regarded as a soft subset of $(\mathit{H},\mathit{C})$ if the following conditions hold (denoted as $\mathit{T}$-order):

1 

$\mathcal{B}\subseteq \mathit{C}$.

2 

For each $\mathit{b}\in \mathcal{B}$, $\mathit{G}\left(\mathit{b}\right)\subseteq \mathit{H}\left(\mathit{b}\right)$, ${\overline{\Gamma }}_{\mathit{G}\left(\mathit{b}\right)}\left(y\right)\le {\overline{\Gamma }}_{\mathit{H}\left(\mathit{b}\right)}\left(y\right)$, and ${\lambda }_{\mathit{G}\left(\mathit{b}\right)}\left(y\right)\ge {\lambda }_{\mathit{H}\left(\mathit{b}\right)}\left(y\right)$ for all $\mathit{y}\in \mathit{V}$.

In a similar manner, $(\mathit{G},\mathcal{B})$ is termed a soft superset of $(\mathit{H},\mathit{C})$ if $(\mathit{H},\mathit{C})$ is a soft subset of $(\mathit{G},\mathcal{B})$. This relation is denoted by $(\mathit{G},\mathcal{B})\begin{array}{c}\supset \\ \sim \end{array}(\mathit{H},\mathit{C})$.

Definition 18. 

Given two soft sets $(\mathit{G},\mathcal{B})$ and $(\mathit{H},\mathit{C})$ over the universe $\mathit{V}$, $(\mathit{G},\mathcal{B})$ is considered a soft subset of $(\mathit{H},\mathit{C})$ if the following conditions are met (denoted as $\mathit{S}$-order):

(1) 

$\mathcal{B}\subseteq \mathit{C}$.

(2) 

For each $\mathit{b}\in \mathcal{B}$, $\mathit{G}\left(\mathit{b}\right){\subset }_S\mathit{H}\left(\mathit{b}\right)$, ${\overline{\Gamma }}_{\mathit{G}\left(\mathit{b}\right)}\left(y\right)\le {\overline{\Gamma }}_{\mathit{H}\left(\mathit{b}\right)}\left(y\right)$, and ${\lambda }_{\mathit{G}\left(\mathit{b}\right)}\left(y\right)\le {\lambda }_{\mathit{H}\left(\mathit{b}\right)}\left(y\right)$ for all $\mathit{y}\in \mathit{V}$.

Definition 19. 

Let $({\tilde{\mathit{G}}}_R,\mathcal{B})$ and $({\tilde{\mathit{H}}}_R,\mathit{C})$ be two multi-$\mathit{R}$ cubic bipolar fuzzy soft sets. The set $({\tilde{\mathit{G}}}_R,\mathcal{B})$ is considered a multi-$\mathit{R}$ cubic bipolar fuzzy soft subset of $({\tilde{\mathit{H}}}_R,\mathit{C})$ if the following conditions are met:

(1) 

$\mathcal{B}\subseteq \mathit{C}$.

(2) 

For every $\mathit{f}\in \mathcal{B}$, $\mathit{G}\left(\mathit{f}\right)$ is a multi-$\mathit{R}$ cubic bipolar fuzzy subset of $\mathit{H}\left(\mathit{f}\right)$.

This relationship is denoted by $({\tilde{\mathit{G}}}_R,\mathcal{B})\subset ({\tilde{\mathit{H}}}_R,\mathit{C})$.

Definition 20. 

A multi-$\mathit{R}$ cubic bipolar fuzzy soft set $({\tilde{\mathit{G}}}_R,\mathcal{B})$ is termed a null multi-$\mathit{R}$ cubic bipolar fuzzy soft set, represented by the empty set φ, if for every $\mathit{f}\in \mathcal{B}$ and $\mathit{r}\in \mathit{R}$, ${\tilde{\mathit{G}}}_R\left(\mathit{f}\right)=\phi$.

Definition 21. 

A multi-$\mathit{R}$ cubic bipolar fuzzy soft set $({\mathit{G}}_R,\mathcal{B})$ is called an absolute multi-$\mathit{R}$ cubic bipolar fuzzy soft set if ${\mathit{G}}_R\left(\mathit{f}\right)=C{\mathit{BF}}^V$ for every $\mathit{f}\in \mathcal{B}$.

Definition 22. 

Given two multi-$\mathit{R}$ cubic bipolar fuzzy soft sets $({\tilde{\mathit{G}}}_R,\mathcal{B})$ and $({\tilde{\mathit{H}}}_R,\mathit{C})$ over the universe $\mathit{V}$, then:

-

The $\mathit{T}$-union of $({\tilde{\mathit{G}}}_R,\mathcal{B})$ and $({\tilde{\mathit{H}}}_R,\mathit{C})$ is defined as the soft set $({\tilde{\mathit{J}}}_R,\mathit{D})$, where $\mathit{D}=\mathcal{B}\cup \mathit{C}$ and for each $\mathit{d}\in \mathit{D}$:

$${\tilde{\mathit{J}}}_R\left(\mathit{d}\right)=\left\{\begin{array}{cc}{\tilde{\mathit{G}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathcal{B}\setminus \mathit{C}\hfill \\ {\tilde{\mathit{H}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathit{C}\setminus \mathcal{B}\hfill \\ {\tilde{\mathit{G}}}_R\left(\mathit{d}\right){\cup }_T{\tilde{\mathit{H}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}\right.$$

-

The $\mathit{S}$-union of $({\tilde{\mathit{G}}}_R,\mathcal{B})$ and $({\tilde{\mathit{H}}}_R,\mathit{C})$ is defined to be the soft set $({\tilde{\mathit{J}}}_R,\mathit{D})$, where $\mathit{D}=\mathcal{B}\cup \mathit{C}$ and $\mathit{d}\in \mathit{D}$:

$${\tilde{\mathit{J}}}_R\left(\mathit{d}\right)=\left\{\begin{array}{cc}{\tilde{\mathit{G}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathcal{B}\setminus \mathit{C}\hfill \\ {\tilde{\mathit{H}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathit{C}\setminus \mathcal{B}\hfill \\ {\tilde{\mathit{G}}}_R\left(\mathit{d}\right){\cup }_S{\tilde{\mathit{H}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}\right.$$

for all $\mathit{d}\in \mathit{D}$; this is denoted by

• $({\tilde{\mathit{J}}}_R,\mathit{D})=({\tilde{\mathit{G}}}_R,\mathcal{B}){⋓}_T({\tilde{\mathit{H}}}_R,\mathit{C})$.
• $({\tilde{\mathit{J}}}_R,\mathit{D})=({\tilde{\mathit{G}}}_R,\mathcal{B}){⋓}_S({\tilde{\mathit{H}}}_R,\mathit{C})$.

Definition 23. 

If $({\tilde{\mathit{G}}}_R,\mathcal{B})$ and $({\tilde{\mathit{H}}}_R,\mathit{C})$ are multi-$\mathit{R}$ cubic bipolar fuzzy soft sets over $\mathit{V}$, then:

-

The $\mathit{T}$ intersection of $({\tilde{\mathit{G}}}_R,\mathcal{B})$ and $({\tilde{\mathit{H}}}_R,\mathit{C})$ is defined by the soft set $({\tilde{\mathit{J}}}_R,\mathit{D})$, where $\mathit{D}=\mathcal{B}\cap \mathit{C}$: $\mathit{d}\in \mathit{D}$:

$${\tilde{\mathit{J}}}_R\left(\mathit{d}\right)=\left\{\begin{array}{cc}{\tilde{\mathit{G}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathcal{B}\setminus \mathit{C}\hfill \\ {\tilde{\mathit{H}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathit{C}\setminus \mathcal{B}\hfill \\ {\tilde{\mathit{G}}}_R\left(\mathit{d}\right){\cap }_T{\tilde{\mathit{H}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}\right.$$

-

The $\mathit{S}$ intersection of $({\tilde{\mathit{G}}}_R,\mathcal{B})$ and $({\tilde{\mathit{H}}}_R,\mathit{C})$ is defined by the soft set $({\tilde{\mathit{J}}}_R,\mathit{D})$, where $\mathit{D}=\mathcal{B}\cap \mathit{C}$, $\mathit{d}\in \mathit{D}$:

$${\tilde{\mathit{J}}}_R\left(\mathit{d}\right)=\left\{\begin{array}{cc}{\tilde{\mathit{G}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathcal{B}\setminus \mathit{C}\hfill \\ {\tilde{\mathit{H}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathit{C}\setminus \mathcal{B}\hfill \\ {\tilde{\mathit{G}}}_R\left(\mathit{d}\right){\cap }_S{\tilde{\mathit{H}}}_R\left(\mathit{d}\right)\hfill & \mathit{if}\mathit{d}\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}\right.$$

for all $\mathit{d}\in \mathit{D}$; this is denoted by

-

$({\tilde{\mathit{J}}}_R,\mathit{D})=({\tilde{\mathit{G}}}_R,\mathcal{B}){⋒}_T({\tilde{\mathit{H}}}_R,\mathit{C})$.

-

$({\tilde{\mathit{J}}}_R,\mathit{D})=({\tilde{\mathit{G}}}_R,\mathcal{B}){⋒}_S({\tilde{\mathit{H}}}_R,\mathit{C})$.

Definition 24. 

The complement of a multi-$\mathit{R}$ cubic bipolar fuzzy soft set $({\mathit{G}}_R,\mathcal{B})$, denoted by ${({\mathit{G}}_R,\mathcal{B})}^c$, is defined as ${({\mathit{G}}_R,\mathcal{B})}^c=({\mathit{G}}_R^c,\mathcal{B})$, where ${\mathit{G}}_R:\mathcal{B}\to \mathit{MRCBFS}\left(\mathit{V}\right)$ is a mapping given by

$${\mathit{G}}_R^c\left(\mathit{f}\right)={\left({\mathit{G}}_R\left(\mathit{f}\right)\right)}^c\forall \mathit{f}\in \mathcal{B}$$

Remark 1. 

If $({\mathit{G}}_R,\mathcal{B})$ is a multi-$\mathit{R}$ cubic bipolar fuzzy soft set, then

$${\left({({\mathit{G}}_R,\mathcal{B})}^c\right)}^c=({\mathit{G}}_R,\mathcal{B})$$

Theorem 3. 

Let $({\mathit{G}}_R,\mathcal{B})$ be a multi-$\mathit{R}$ cubic bipolar fuzzy soft set over a common universe $\mathit{V}$; then ($\mathit{T}$-order and $\mathit{S}$-order):

(*1) 

$({\mathit{G}}_R,\mathcal{B}){\cup }_T({\mathit{G}}_R,\mathcal{B})=({\mathit{G}}_R,\mathcal{B})$.

(2) 

$({\mathit{G}}_R,\mathcal{B}){\cup }_S({\mathit{G}}_R,\mathcal{B})=({\mathit{G}}_R,\mathcal{B})$.

(*3) 

$({\mathit{G}}_R,\mathcal{B}){\cap }_T({\mathit{G}}_R,\mathcal{B})=({\mathit{G}}_R,\mathcal{B})$.

(4) 

$({\mathit{G}}_R,\mathcal{B}){\cap }_S({\mathit{G}}_R,\mathcal{B})=({\mathit{G}}_R,\mathcal{B})$.

(5) 

$({\mathit{G}}_R,\mathcal{B}){\cup }_T\phi =({\mathit{G}}_R,\mathcal{B})$.

(6) 

$({\mathit{G}}_R,\mathcal{B}){\cup }_S\phi =({\mathit{G}}_R,\mathcal{B})$.

(7) 

$({\mathit{G}}_R,\mathcal{B}){\cap }_T\phi =\phi$.

(8) 

$({\mathit{G}}_R,\mathcal{B}){\cap }_S\phi =\phi$.

Proof. 

(1) 

$({\mathit{G}}_R,\mathcal{B}){\cup }_T({\mathit{G}}_R,\mathcal{B})=({\mathit{G}}_R,\mathcal{B})$.

The union of two multi-$\mathit{R}$ cubic bipolar fuzzy soft sets $({\mathit{J}}_R,\mathit{D})$ is given by:

$$({\mathit{J}}_R,\mathit{D})=({\mathit{G}}_R,\mathcal{B}){\cup }_T({\mathit{G}}_R,\mathcal{B})\mathrm{where}\mathit{D}=\mathcal{B}{\cup }_T\mathcal{B}.$$

(1)

For each $b\in \mathit{D}$:

$$\begin{array}{cc}\hfill {\mathit{J}}_R\left(b\right)& ={\mathit{G}}_R\left(b\right)\mathrm{if}b\in \mathcal{B}\setminus \mathcal{B},\hfill \\ \hfill {\mathit{J}}_R\left(b\right)& ={\mathit{G}}_R\left(b\right)\mathrm{if}b\in \mathcal{B}\setminus \mathcal{B},\hfill \\ \hfill {\mathit{J}}_R\left(b\right)& ={\mathit{G}}_R\left(b\right){\cup }_T{\mathit{G}}_R\left(b\right)\mathrm{if}b\in \mathcal{B}\cap \mathcal{B}.\hfill \end{array}$$

This leads to:

$$\begin{array}{cc}\hfill \mathbf{Case}\mathbf{1}:& \mathrm{If}b\in \mathcal{B}\setminus \mathcal{B},\hfill \\ \hfill {\mathit{J}}_R\left(b\right)& ={\mathit{G}}_R\left(b\right),\hfill \\ \hfill \mathbf{Case}\mathbf{2}:& \mathrm{If}b\in \mathcal{B}\setminus \mathcal{B},\hfill \\ \hfill {\mathit{J}}_R\left(b\right)& ={\mathit{G}}_R\left(b\right),\hfill \\ \hfill \mathbf{Case}\mathbf{3}:& \mathrm{If}b\in \mathcal{B}\cap \mathcal{B},\hfill \\ \hfill {\mathit{J}}_R\left(b\right)& ={\mathit{G}}_R\left(b\right){\cup }_T{\mathit{G}}_R\left(b\right),\hfill \\ \hfill & ={\mathit{G}}_R\left(b\right).\hfill \end{array}$$

Therefore, ${\mathit{J}}_R\left(b\right)={\mathit{G}}_R\left(b\right)$ for all $b\in \mathcal{B}$. Hence, $({\mathit{J}}_R,\mathit{D})=({\mathit{G}}_R,\mathcal{B})$ from (1), implying $({\mathit{G}}_R,\mathcal{B}){\cup }_T({\mathit{G}}_R,\mathcal{B})=({\mathit{G}}_R,\mathcal{B})$.

(2)

$({\mathit{G}}_R,\mathcal{B}){\cap }_T({\mathit{G}}_R,\mathcal{B})=({\mathit{G}}_R,\mathcal{B})$.

The intersection of two multi-$\mathit{R}$ cubic bipolar fuzzy soft sets $({\mathit{J}}_R,\mathit{D})$ is given by:

$$({\mathit{J}}_R,\mathit{D})=({\mathit{G}}_R,\mathcal{B}){\cap }_T({\mathit{G}}_R,\mathcal{B}),$$

(2)

where $\mathit{D}=\mathcal{B}\cap \mathcal{B}$.

For each $b\in \mathit{D}$:

$$\begin{array}{cc}\hfill {\mathit{J}}_R\left(b\right)& ={\mathit{G}}_R\left(b\right){\cap }_T{\mathit{G}}_R\left(b\right)\mathrm{if}b\in \mathit{D}=\mathcal{B}\cap \mathcal{B}.\hfill \end{array}$$

Let $\mathcal{B}\in \mathit{D}=\mathcal{B}\cap \mathcal{B}$:

$$\begin{array}{cc}\hfill {\mathit{J}}_R\left(b\right)& ={\mathit{G}}_R\left(b\right){\cap }_T{\mathit{G}}_R\left(b\right)\mathrm{if}b\in \mathcal{B},\hfill \\ \hfill & ={\mathit{G}}_R\left(b\right)\mathrm{if}b\in \mathcal{B},\hfill \\ \hfill & ={\mathit{G}}_R\left(b\right){\cap }_T{\mathit{G}}_R\left(b\right)\mathrm{if}b\in \mathcal{B}.\hfill \end{array}$$

Therefore, ${\mathit{J}}_R\left(b\right)={\mathit{G}}_R\left(b\right)$, leading to $({\mathit{J}}_R,\mathit{D})=({\mathit{G}}_R,\mathcal{B})$ from (2). Hence, $({\mathit{G}}_R,\mathcal{B}){\cap }_T({\mathit{G}}_R,\mathcal{B})=({\mathit{G}}_R,\mathcal{B})$.

□

Definition 25. 

The intersection of two multi-$\mathit{R}$ cubic bipolar fuzzy soft sets $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $(H_R,C)$ produces a new multi-R cubic bipolar fuzzy soft set $({\mathit{J}}_{\mathit{R}},\mathit{D})$, where $\mathit{D}=\mathcal{B}\cap \mathit{C}\ne \mathit{\phi }$. The mapping $\mathit{J}:\mathit{D}\to \mathit{MRCBFS}$ is given by $\mathit{J}\left(\mathit{f}\right)={\mathit{G}}_{\mathit{R}}\left(\mathit{f}\right)\cap {\mathit{H}}_{\mathit{R}}\left(\mathit{f}\right)$ for all $\mathit{f}\in \mathit{D}$. This relationship is represented as $({\mathit{J}}_{\mathit{R}},\mathit{D})=({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C})$.

Example 3. 

Consider the set of bikes $\mathit{V}=\{{\mathit{b}}_{\mathbf{1}},{\mathit{b}}_{\mathbf{2}},{\mathit{b}}_{\mathbf{3}}\}$ and the set of parameters $\mathit{P}=\{{\mathit{p}}_{\mathbf{1}}=\mathit{Light},{\mathit{p}}_{\mathbf{2}}=\mathit{Beautiful},{\mathit{p}}_{\mathbf{3}}=\mathit{Good}\mathit{mileage},{\mathit{p}}_{\mathbf{4}}=\mathit{Modern}\mathit{Technology}\}$. Let $\mathcal{B}=\{{\mathit{p}}_{\mathbf{1}},{\mathit{p}}_{\mathbf{2}}\}\subseteq \mathit{P}$ and $\mathit{C}=\{{\mathit{p}}_{\mathbf{1}},{\mathit{p}}_{\mathbf{2}},{\mathit{p}}_{\mathbf{3}}\}\subseteq \mathit{P}$. Let $\mathit{R}=\{\mathit{r},\mathbf{s}\}$:

$$\begin{array}{cc}\hfill ({\mathit{G}}_{\mathit{R}},\mathcal{B})& =\{⟨{\mathit{p}}_{\mathbf{1}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathit{r}}{[\mathbf{0}.\mathbf{1},\mathbf{0}.\mathbf{5}][-\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{4}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{7}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{4}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{2}]\{\mathbf{0}.\mathbf{8},-\mathbf{0}.\mathbf{9}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathit{r}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{3}][-\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{5}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{7}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathbf{s}}{[-\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{4}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{6}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathit{r}}{[\mathbf{0}.\mathbf{6},\mathbf{0}.\mathbf{9}][-\mathbf{0}.\mathbf{7},-\mathbf{0}.\mathbf{3}]\{\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{8}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{8}][-\mathbf{0}.\mathbf{4},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{7}\}}}⟩\hfill \\ \hfill & ⟨{\mathit{p}}_{\mathbf{2}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathit{r}}{[\mathbf{0}.\mathbf{4},\mathbf{0}.\mathbf{6}][-\mathbf{0}.\mathbf{8},-\mathbf{0}.\mathbf{2}]\{\mathbf{0}.\mathbf{1},-\mathbf{0}.\mathbf{5}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{7}][-\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{4}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{5}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathit{r}}{[\mathbf{0}.\mathbf{6},\mathbf{0}.\mathbf{8}][-\mathbf{0}.\mathbf{7},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{4}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathbf{s}}{[\mathbf{0}.\mathbf{4},\mathbf{0}.\mathbf{5}][-\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{7}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathit{r}}{[\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{8}][-\mathbf{0}.\mathbf{9},-\mathbf{0}.\mathbf{2}]\{\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{8}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathbf{s}}{[\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{8}][-\mathbf{0}.\mathbf{9},-\mathbf{0}.\mathbf{5}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{5}\}}}⟩\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill ({\mathit{H}}_{\mathit{R}},\mathcal{B})& =\{⟨{\mathit{p}}_{\mathbf{1}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathit{r}}{[\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{5}][-\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{7}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathbf{s}}{[\mathbf{0}.\mathbf{5},\mathbf{0}.\mathbf{9}][-\mathbf{0}.\mathbf{8},-\mathbf{0}.\mathbf{2}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{7}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathit{r}}{[\mathbf{0}.\mathbf{8},\mathbf{0}.\mathbf{9}][-\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{3}]\{\mathbf{0}.\mathbf{1},-\mathbf{0}.\mathbf{2}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{5}][-\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{3}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{9}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathit{r}}{[\mathbf{0}.\mathbf{7},\mathbf{0}.\mathbf{8}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{3}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{6}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathbf{s}}{[\mathbf{0}.\mathbf{4},\mathbf{0}.\mathbf{9}][-\mathbf{0}.\mathbf{8},-\mathbf{0}.\mathbf{4}]\{\mathbf{0}.\mathbf{1},-\mathbf{0}.\mathbf{8}\}}}⟩\hfill \\ \hfill & ⟨{\mathit{p}}_{\mathbf{2}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathit{r}}{[\mathbf{0}.\mathbf{5},\mathbf{0}.\mathbf{9}][-\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{7}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathbf{s}}{[\mathbf{0}.\mathbf{5},\mathbf{0}.\mathbf{6}][-\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{6}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathit{r}}{[\mathbf{0}.\mathbf{4},\mathbf{0}.\mathbf{2}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{2}]\{\mathbf{0}.\mathbf{1},-\mathbf{0}.\mathbf{5}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{6}][-\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{3}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{6}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathit{r}}{[\mathbf{0}.\mathbf{6},\mathbf{0}.\mathbf{8}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{3}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{6}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathbf{s}}{[\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{4}][-\mathbf{0}.\mathbf{8},-\mathbf{0}.\mathbf{4}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{7}\}}}⟩\hfill \\ \hfill & ⟨{\mathit{p}}_{\mathbf{3}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathit{r}}{[\mathbf{0}.\mathbf{5},\mathbf{0}.\mathbf{9}][-\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{7}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathbf{s}}{[\mathbf{0}.\mathbf{5},\mathbf{0}.\mathbf{7}][-\mathbf{0}.\mathbf{7},-\mathbf{0}.\mathbf{5}]\{\mathbf{0}.\mathbf{8},-\mathbf{0}.\mathbf{9}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathit{r}}{[\mathbf{0}.\mathbf{6},\mathbf{0}.\mathbf{7}][-\mathbf{0}.\mathbf{9},-\mathbf{0}.\mathbf{4}]\{\mathbf{0}.\mathbf{4},-\mathbf{0}.\mathbf{5}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathbf{s}}{[\mathbf{0}.\mathbf{8},\mathbf{0}.\mathbf{9}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{7},-\mathbf{0}.\mathbf{8}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathit{r}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{3}][-\mathbf{0}.\mathbf{7},-\mathbf{0}.\mathbf{5}]\{\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{8}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{6}][-\mathbf{0}.\mathbf{4},-\mathbf{0}.\mathbf{3}]\{\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{7}\}}}⟩\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill ({\mathit{J}}_{\mathit{R}},\mathit{D})& =({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}),\mathit{D}=\mathcal{B}\cap \mathit{C}=\{{\mathit{p}}_{\mathbf{1}},{\mathit{p}}_{\mathbf{2}}\}\hfill \\ \hfill & =\{⟨{\mathit{p}}_{\mathbf{1}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathit{r}}{[\mathbf{0}.\mathbf{1},\mathbf{0}.\mathbf{5}][-\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{7}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{4}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{2}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{7}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathit{r}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{3}][-\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{3}]\{\mathbf{0}.\mathbf{1},-\mathbf{0}.\mathbf{2}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{5}][-\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{7}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathit{r}}{[\mathbf{0}.\mathbf{6},\mathbf{0}.\mathbf{8}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{3}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{6}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{8}][-\mathbf{0}.\mathbf{4},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{1},-\mathbf{0}.\mathbf{7}\}}}⟩\hfill \\ \hfill & ⟨{\mathit{p}}_{\mathbf{2}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathit{r}}{[\mathbf{0}.\mathbf{4},\mathbf{0}.\mathbf{6}][-\mathbf{0}.\mathbf{8},-\mathbf{0}.\mathbf{2}]\{\mathbf{0}.\mathbf{1},-\mathbf{0}.\mathbf{5}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{1}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{7}][-\mathbf{0}.\mathbf{6},-\mathbf{0}.\mathbf{4}]\{\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{5}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathit{r}}{[\mathbf{0}.\mathbf{6},\mathbf{0}.\mathbf{6}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{1},-\mathbf{0}.\mathbf{4}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{2}}\mathbf{s}}{[\mathbf{0}.\mathbf{2},\mathbf{0}.\mathbf{5}][-\mathbf{0}.\mathbf{3},-\mathbf{0}.\mathbf{1}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{4}\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathit{r}}{[\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{8}][-\mathbf{0}.\mathbf{5},-\mathbf{0}.\mathbf{2}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{6}\}}},{\displaystyle \frac{{\mathit{b}}_{\mathbf{3}}\mathbf{s}}{[\mathbf{0}.\mathbf{3},\mathbf{0}.\mathbf{4}][-\mathbf{0}.\mathbf{8},-\mathbf{0}.\mathbf{4}]\{\mathbf{0}.\mathbf{2},-\mathbf{0}.\mathbf{5}\}}}⟩\}\hfill \end{array}$$

Lemma 1. 

(1) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup \left(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C})\right)=({\mathit{G}}_{\mathit{R}},\mathcal{B}).$

(2) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap \left(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C})\right)=({\mathit{G}}_{\mathit{R}},\mathcal{B}).$

Proof. 

(1)

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup \left(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C})\right)=({\mathit{G}}_{\mathit{R}},\mathcal{B})$.

Let multi-$\mathit{R}$ cubic bipolar fuzzy soft set $({\mathit{J}}_{\mathit{R}},\mathit{D})$ be an intersection of two multi-$\mathit{R}$ cubic bipolar fuzzy soft sets $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$, where $\mathit{D}=\mathcal{B}\cap \mathit{C}$:

$$({\mathit{J}}_{\mathit{R}},\mathit{D})=({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}),$$

(3)

where $\mathit{D}=\mathcal{B}\cap \mathit{C}$. Define by ${\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)$ if $\mathit{b}\in \mathit{D}=\mathcal{B}\cap \mathit{C}$.

Let multi-$\mathit{R}$ cubic bipolar fuzzy soft set $({\mathit{K}}_{\mathit{R}},\mathit{M})$ be the union of two multi-$\mathit{R}$ cubic bipolar fuzzy soft sets $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{J}}_{\mathit{R}},\mathit{D})$, which is:

$$\begin{array}{cc}\hfill ({\mathit{K}}_{\mathit{R}},\mathit{M})& =({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{J}}_{\mathit{R}},\mathit{D}),\hfill \end{array}$$

where $\mathit{M}=\mathcal{B}\cup \mathit{D}$. Define by:

$$\begin{array}{cc}\hfill {\mathit{K}}_{\mathit{R}}\left(\mathit{b}\right)& ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\mathrm{if}b\in \mathcal{B}\setminus \mathit{D},\hfill \\ \hfill & ={\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)\mathrm{if}b\in \mathit{C}\setminus \mathcal{B},\hfill \\ \hfill & ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cup {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)\mathrm{if}b\in \mathcal{B}\cap \mathit{D}.\hfill \end{array}$$

There are three cases:

Case 1: If $\mathit{b}\in \mathcal{B}\setminus \mathit{D}$

$$\begin{array}{cc}\hfill {\mathit{K}}_{\mathit{R}}\left(\mathit{b}\right)& ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\mathrm{if}b\in \mathcal{B}\setminus \mathit{D},\hfill \\ \hfill & ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\mathrm{if}b\in \mathcal{B}.\hfill \end{array}$$

${\mathit{K}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\to ({\mathit{K}}_{\mathit{R}},\mathit{M})=({\mathit{G}}_{\mathit{R}},\mathcal{B})$ from (3).

Case 2: If $\mathit{b}\in \mathit{D}\setminus \mathcal{B}=\mathcal{B}\cap \mathit{C}-\mathcal{B}$

$$\begin{array}{cc}\hfill {\mathit{K}}_{\mathit{R}}\left(\mathit{b}\right)& ={\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)\mathrm{if}b\in \mathit{D}\setminus \mathcal{B},\hfill \\ \hfill & =\mathit{\phi }\mathrm{if}b\in \mathit{\phi }.\hfill \end{array}$$

${\mathit{K}}_{\mathit{R}}\left(\mathit{b}\right)=\mathit{\phi }$ if $\mathit{b}\in \mathit{\phi }\to ({\mathit{K}}_{\mathit{R}},\mathit{M})=\mathit{\phi }$ from (3).

Case 3: If $\mathit{b}\in \mathcal{B}\cap \mathit{D}$

$$\begin{array}{cc}\hfill {\mathit{K}}_{\mathit{R}}\left(\mathit{b}\right)& ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cup {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)\mathrm{if}b\in \mathcal{B}\cap \mathit{D},\hfill \\ \hfill & ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cup ({\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right))\mathrm{from}\left(3\right),\hfill \\ \hfill & ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\sin \mathrm{ce}({\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right))\subset {\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right),\hfill \\ \hfill & ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right).\hfill \end{array}$$

${\mathit{K}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)$.

$({\mathit{K}}_{\mathit{R}},\mathit{M})=({\mathit{G}}_{\mathit{R}},\mathcal{B})$ from (3).

This is satisfied in all three cases, hence $({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup \left(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C})\right)=({\mathit{G}}_{\mathit{R}},\mathcal{B})$.

(2)

Same as above.

□

Theorem 4. 

The commutative property of multi-R cubic bipolar fuzzy soft sets $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$:

(1) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C})=({\mathit{H}}_{\mathit{R}},\mathit{C})\cap ({\mathit{G}}_{\mathit{R}},\mathcal{B})$.

(2) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C})=({\mathit{H}}_{\mathit{R}},\mathit{C})\cup ({\mathit{G}}_{\mathit{R}},\mathcal{B})$.

Proof. 

Show that $({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C})=({\mathit{H}}_{\mathit{R}},\mathit{C})\cap ({\mathit{G}}_{\mathit{R}},\mathcal{B})$.

A multi-R cubic bipolar fuzzy soft set $({\mathit{J}}_{\mathit{R}},\mathit{D})$ is an intersection of two multi-R cubic bipolar fuzzy soft sets, $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$, where $\mathit{D}=\mathcal{B}\cap C$.

4.1 

${\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)$ if $b\in \mathit{D}=\mathcal{B}\cap C$.

A multi-R cubic bipolar fuzzy soft set $({\mathit{K}}_{\mathit{R}},\mathit{E})$ is an intersection of two multi-R cubic bipolar fuzzy soft sets, $({\mathit{H}}_{\mathit{R}},\mathit{C})$ and $({\mathit{G}}_{\mathit{R}},\mathcal{B})$, where $\mathit{E}=C\cap \mathcal{B}$.

4.2 

${\mathit{K}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)$ if $b\in \mathit{E}=C\cap \mathcal{B}$.

To show that $({\mathit{J}}_{\mathit{R}},\mathit{D})=({\mathit{K}}_{\mathit{R}},\mathit{E})$:

$$\begin{array}{cccc}\hfill {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)& =G_R\left(b\right)\cap H_R\left(b\right)\hfill & \hfill & \mathrm{for}\mathrm{all}b\in \mathit{D}=\mathcal{B}\cap C.\hfill \\ \hfill & =H_R\left(b\right)\cap G_R\left(b\right)\hfill & \hfill & \sin \mathrm{ce}{G}_R\left(b\right)\cap H_R\left(b\right)=H_R\left(b\right)\cap G_R\left(b\right).\hfill \\ \hfill & =H_R\left(b\right)\cap G_R\left(b\right)\hfill & \hfill & \forall b\in \mathit{D}=\mathcal{B}\cap C=C\cap \mathcal{B}=\mathit{E}.\hfill \\ \hfill & =K_R\left(b\right)\hfill & \hfill & \forall b\in C\cap \mathcal{B}=\mathit{E}.\hfill \\ \hfill & =K_R\left(b\right).\hfill \end{array}$$

Thus, $({\mathit{J}}_{\mathit{R}},\mathit{D})=({\mathit{K}}_{\mathit{R}},\mathit{E})$ from 4.1 and 4.2 $\to ({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C})=({\mathit{H}}_{\mathit{R}},\mathit{C})\cap ({\mathit{G}}_{\mathit{R}},\mathcal{B})$ from 4.1 and 4.2.

A multi-R cubic bipolar fuzzy soft set $({\mathit{J}}_{\mathit{R}},\mathit{D})$ is the union of two multi-R cubic bipolar fuzzy soft sets, $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$, over a common universe ∪:

4.3 

$({\mathit{J}}_{\mathit{R}},\mathit{D})=({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C})$ where $\mathit{D}=\mathcal{B}\cup C$ defined by:

4.4 

${\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)$ if $b\in \mathcal{B}\setminus C$.

4.5 

${\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)$ if $b\in C\setminus \mathcal{B}$.

4.6 

${\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cup {\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)$ if $b\in \mathcal{B}\cap C$.

There are three cases:

• Case 1: If $b\in \mathcal{B}\setminus C$:

  4.7 

  ${\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)$ if $b\in \mathcal{B}\setminus C$ from 4.4.

• Case 2: If $b\in C\setminus \mathcal{B}$:

  4.8 

  ${\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)$ if $b\in C\setminus \mathcal{B}$ from 4.5.

• Case 3: If $b\in \mathcal{B}\cap C$:

  $$\begin{array}{cccc}\hfill {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)& =G_R\left(b\right)\cup H_R\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\cap C=C\cap \mathcal{B}\mathrm{from}4.6.\hfill \\ \hfill & =H_R\left(b\right)\cup G_R\left(b\right)\hfill & \hfill & \mathrm{if}b\in C\cap \mathcal{B}.\hfill \end{array}$$

Combining 4.4, 4.5, and 4.6, we get:

$$\begin{array}{cccc}\hfill {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)& =G_R\left(b\right)\cup H_R\left(b\right)\hfill & \hfill & \mathrm{if}b\in C\setminus \mathcal{B}.\hfill \\ \hfill & =G_R\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\cap C.\hfill \\ \hfill & =H_R\left(b\right)\cup G_R\left(b\right)\hfill & \hfill & \mathrm{if}b\in C\cap \mathcal{B}.\hfill \end{array}$$

Thus, $({\mathit{J}}_{\mathit{R}},\mathit{D})$ becomes:

$$\begin{array}{cccc}\hfill {\mathit{J}}_{\mathit{R}}(\mathit{b},\mathit{D})& =({\mathit{H}}_{\mathit{R}},\mathit{C})\cup ({\mathit{G}}_{\mathit{R}},\mathcal{B})\hfill & \hfill & \mathrm{where}\mathit{D}=C\cap \mathcal{B}.\hfill \\ \hfill & =\mathrm{R}.\mathrm{H}.\mathrm{S}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C})=({\mathit{H}}_{\mathit{R}},\mathit{C})\cup ({\mathit{G}}_{\mathit{R}},\mathcal{B}).\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\square \hfill \end{array}$$

Theorem 5. 

The associative law of multi-R cubic bipolar fuzzy soft set $({\mathit{G}}_{\mathit{R}},\mathcal{B})$, $({\mathit{H}}_{\mathit{R}},\mathit{C})$, and $({\mathit{J}}_{\mathit{R}},\mathit{D})$:

(1) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap (({\mathit{H}}_{\mathit{R}},\mathit{C})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D}))=(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cap ({\mathit{J}}_{\mathit{R}},\mathit{D})$.

(2) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup (({\mathit{H}}_{\mathit{R}},\mathit{C})\cup ({\mathit{J}}_{\mathit{R}},\mathit{D}))=(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cup ({\mathit{J}}_{\mathit{R}},\mathit{D})$.

Proof. 

(1)

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap (({\mathit{H}}_{\mathit{R}},\mathit{C})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D}))=(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cap ({\mathit{J}}_{\mathit{R}},\mathit{D})$.

A multi-R cubic bipolar fuzzy soft set $({\mathit{L}}_{\mathit{R}},\mathit{E})$ is an intersection of two multi-R cubic bipolar fuzzy soft sets, $({\mathit{H}}_{\mathit{R}},\mathit{C})$ and $({\mathit{J}}_{\mathit{R}},\mathit{D})$, which is

$$({\mathit{H}}_{\mathit{R}},\mathit{C})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D})=({\mathit{L}}_{\mathit{R}},\mathit{E})$$

where $\mathit{E}=C\cap D$ defined by

$${\mathit{L}}_{\mathit{R}}\left(\mathit{b}\right)=H_R\left(b\right)\cap J_R\left(b\right)\mathrm{if}b\in \mathit{E}=C\cap D.$$

A multi-R cubic bipolar fuzzy soft set $({\mathit{M}}_{\mathit{R}},\mathit{X})$ is an intersection of two multi-R cubic bipolar fuzzy soft sets, $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{L}}_{\mathit{R}},\mathit{E})$, which is

$$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{L}}_{\mathit{R}},\mathit{E})=({\mathit{M}}_{\mathit{R}},\mathit{X})$$

where $\mathit{X}=\mathcal{B}\cap E$ defined by

$${\mathit{M}}_{\mathit{R}}\left(\mathit{b}\right)=G_R\left(b\right)\cap L_R\left(b\right)\mathrm{if}b\in \mathit{X}=\mathcal{B}\cap E.$$

$$\begin{array}{cccc}\hfill {\mathit{M}}_{\mathit{R}}\left(\mathit{b}\right)& ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{L}}_{\mathit{R}}\left(\mathit{b}\right)\hfill & \hfill & \mathrm{for}\mathrm{all}\mathit{b}\in \mathit{X}=\mathcal{B}\cap \mathit{E}\mathrm{by}4.13,\hfill \\ \hfill & ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap ({\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right))\hfill & \hfill & \mathrm{by}4.11,\hfill \\ \hfill & =({\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right))\cap {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)\hfill & \hfill & \mathrm{for}\mathrm{all}\mathit{b}\in \mathit{X}=\mathcal{B}\cap \mathit{E}=\mathcal{B}\cap (\mathit{C}\cap \mathit{D}),\hfill \\ \hfill & =({\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right))\cap {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)\hfill & \hfill & \mathrm{for}\mathrm{all}\mathit{b}\in \mathcal{B}\cap (\mathit{C}\cap \mathit{D}).\hfill \end{array}$$

$$\begin{array}{cccc}\hfill ({\mathit{M}}_{\mathit{R}},\mathit{X})& =(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cap ({\mathit{J}}_{\mathit{R}},\mathit{D})\hfill & \hfill & \mathrm{by}4.12,\hfill \\ \hfill & ({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{L}}_{\mathit{R}},\mathit{E})=(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cap ({\mathit{J}}_{\mathit{R}},\mathit{D})\hfill & \hfill & \mathrm{by}4.12,\hfill \\ \hfill & ({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap (({\mathit{H}}_{\mathit{R}},\mathit{C})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D}))=(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cap ({\mathit{J}}_{\mathit{R}},\mathit{D}).\hfill \end{array}$$

(2)

Same as above.

Hence,

$$\begin{array}{c}\hfill ({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup (({\mathit{H}}_{\mathit{R}},\mathit{C})\cup ({\mathit{J}}_{\mathit{R}},\mathit{D}))=(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cup ({\mathit{J}}_{\mathit{R}},\mathit{D}).\end{array}$$

□

Theorem 6. 

The distributive law of multi-R cubic bipolar fuzzy soft set $({\mathit{G}}_{\mathit{R}},\mathcal{B})$, $({\mathit{H}}_{\mathit{R}},\mathit{C})$, and $({\mathit{J}}_{\mathit{R}},\mathit{D})$:

(1) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap (({\mathit{H}}_{\mathit{R}},\mathit{C})\cup ({\mathit{J}}_{\mathit{R}},\mathit{D}))=(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cup (({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D}))$.

(2) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup (({\mathit{H}}_{\mathit{R}},\mathit{C})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D}))=(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cap (({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{J}}_{\mathit{R}},\mathit{D}))$.

Proof. 

(1)

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap (({\mathit{H}}_{\mathit{R}},\mathit{C})\cup ({\mathit{J}}_{\mathit{R}},\mathit{D}))=(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cup (({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D}))$.

A multi-R cubic bipolar fuzzy soft set $({\mathit{L}}_{\mathit{R}},\mathit{E})$ is the union of two multi-R cubic bipolar fuzzy soft sets, $({\mathit{H}}_{\mathit{R}},\mathit{C})$ and $({\mathit{J}}_{\mathit{R}},\mathit{D})$, over a common universe U.

$$({\mathit{H}}_{\mathit{R}},\mathit{C})\cup ({\mathit{J}}_{\mathit{R}},\mathit{D})=({\mathit{L}}_{\mathit{R}},\mathit{E}),$$

where $\mathit{E}=C\cup D$, defined by

$$\begin{array}{cccc}\hfill {\mathit{L}}_{\mathit{R}}\left(\mathit{b}\right)& ={\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)\hfill & \hfill & \mathrm{if}b\in C\setminus D,\hfill \\ \hfill & ={\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)\hfill & \hfill & \mathrm{if}b\in D\setminus C,\hfill \\ \hfill & ={\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)\cup {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right)\hfill & \hfill & \mathrm{if}b\in C\cap D.\hfill \end{array}$$

A multi-R cubic bipolar fuzzy soft set $({\mathit{M}}_{\mathit{R}},\mathit{v})$ is an intersection of two multi-R cubic bipolar fuzzy soft sets: $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{L}}_{\mathit{R}},\mathit{E})$.

$$({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{L}}_{\mathit{R}},\mathit{E})=({\mathit{M}}_{\mathit{R}},\mathit{v}),$$

where $\mathit{v}=\mathcal{B}\cap E$, defined by

$${\mathit{M}}_{\mathit{R}}\left(\mathit{b}\right)={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{L}}_{\mathit{R}}\left(\mathit{b}\right),\mathrm{for}\mathrm{all}b\in \mathit{v}=\mathcal{B}\cap E.$$

$$\begin{array}{cccc}\hfill {\mathit{M}}_{\mathit{R}}\left(\mathit{b}\right)& ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{L}}_{\mathit{R}}\left(\mathit{b}\right)\hfill & \hfill & \mathrm{for}\mathrm{all}b\in \mathit{v}=\mathcal{B}\cap E\mathrm{by}4.16,\hfill \\ \hfill & ={\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap ({\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right)\cup {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right))\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\cap (C\cap D),\hfill \\ \hfill & =({\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{H}}_{\mathit{R}}\left(\mathit{b}\right))\cup ({\mathit{G}}_{\mathit{R}}\left(\mathit{b}\right)\cap {\mathit{J}}_{\mathit{R}}\left(\mathit{b}\right))\hfill & \hfill & \mathrm{for}\mathrm{all}b\in \mathcal{B}.\hfill \end{array}$$

$$\begin{array}{cccc}\hfill ({\mathit{M}}_{\mathit{R}},\mathit{v})& =(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cup (({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D}))\hfill & \hfill & \mathrm{by}4.15,\hfill \\ \hfill ({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{L}}_{\mathit{R}},\mathit{E})& =(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cup (({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D}))\hfill & \hfill & \mathrm{by}4.15.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill ({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap (({\mathit{H}}_{\mathit{R}},\mathit{C})\cup ({\mathit{J}}_{\mathit{R}},\mathit{D}))& =(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C}))\cup (({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{J}}_{\mathit{R}},\mathit{D})).\hfill \end{array}$$

(2)

Same as above.

□

Lemma 2. 

Let $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$ be two multi-R cubic bipolar fuzzy soft sets. Then:

(1) 

If $({\mathit{G}}_{\mathit{R}},\mathcal{B})\subset ({\mathit{H}}_{\mathit{R}},\mathit{C})$, then $({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C})=({\mathit{G}}_{\mathit{R}},\mathcal{B})$.

(2) 

If $({\mathit{G}}_{\mathit{R}},\mathcal{B})\subset ({\mathit{H}}_{\mathit{R}},\mathit{C})$, then $({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C})=({\mathit{H}}_{\mathit{R}},\mathit{C})$.

5.1. De Morgan Law of Multi-R Cubic Bipolar Fuzzy Soft Sets

The De Morgan laws for multi-R cubic bipolar fuzzy soft sets $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$ are:

(1)

${\left(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C})\right)}^c={({\mathit{G}}_{\mathit{R}},\mathcal{B})}^c\cap {({\mathit{H}}_{\mathit{R}},\mathit{C})}^c$.

(2)

${\left(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cap ({\mathit{H}}_{\mathit{R}},\mathit{C})\right)}^c={({\mathit{G}}_{\mathit{R}},\mathcal{B})}^c\cup {({\mathit{H}}_{\mathit{R}},\mathit{C})}^c$.

Proof. 

Let $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$ be two multi-R cubic bipolar fuzzy soft sets over a common universe $\mathit{U}$.

The union of $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$ is a multi-R cubic bipolar fuzzy soft set $({\mathit{J}}_{\mathit{R}},\mathit{D})$, where $\mathit{D}=\mathcal{B}\cup \mathit{C}$, and $({\mathit{J}}_{\mathit{R}},\mathit{D})$ is defined by

$$\begin{array}{cccc}\hfill {\mathit{J}}_{\mathit{R}}\left(b\right)& ={\mathit{G}}_{\mathit{R}}\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\setminus \mathit{C},\hfill \end{array}$$

(4)

$$\begin{array}{cccc}\hfill & ={\mathit{H}}_{\mathit{R}}\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathit{C}\setminus \mathcal{B},\hfill \end{array}$$

(5)

$$\begin{array}{cccc}\hfill & ={\mathit{G}}_{\mathit{R}}\left(b\right)\cup {\mathit{H}}_{\mathit{R}}\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}$$

(6)

The intersection of $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$ is a multi-R cubic bipolar fuzzy soft set $({\mathit{k}}_{\mathit{R}},\mathit{E})$, where $\mathit{E}=\mathcal{B}\cap \mathit{C}$, and $({\mathit{k}}_{\mathit{R}},\mathit{E})$ is defined by:

$$\begin{array}{cccc}\hfill {\mathit{k}}_{\mathit{R}}\left(b\right)& ={\mathit{G}}_{\mathit{R}}\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\setminus \mathit{C},\hfill \end{array}$$

(7)

$$\begin{array}{cccc}\hfill & ={\mathit{H}}_{\mathit{R}}\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathit{C}\setminus \mathcal{B},\hfill \end{array}$$

(8)

$$\begin{array}{cccc}\hfill & ={\mathit{G}}_{\mathit{R}}\left(b\right)\cap {\mathit{H}}_{\mathit{R}}\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}$$

(9)

To show that ${\left(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C})\right)}^c={({\mathit{G}}_{\mathit{R}},\mathcal{B})}^c\cap {({\mathit{H}}_{\mathit{R}},\mathit{C})}^c$, consider the following cases:

(1)

If $b\in \mathcal{B}\setminus \mathit{C}$, then $b\in \mathit{D}$ and

$$\begin{array}{cccc}\hfill {\mathit{J}}_{\mathit{R}}\left(b\right)& ={\mathit{G}}_{\mathit{R}}\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\setminus \mathit{C}.\hfill \end{array}$$

Taking the complement, we get

$$\begin{array}{cccc}\hfill {\left({\mathit{J}}_{\mathit{R}}\left(b\right)\right)}^c& ={\left({\mathit{G}}_{\mathit{R}}\left(b\right)\right)}^c\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\setminus \mathit{C}.\hfill \end{array}$$

(10)

(2)

If $b\in \mathit{C}\setminus \mathcal{B}$, then $b\in \mathit{D}$ and

$$\begin{array}{cccc}\hfill {\mathit{J}}_{\mathit{R}}\left(b\right)& ={\mathit{H}}_{\mathit{R}}\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathit{C}\setminus \mathcal{B}.\hfill \end{array}$$

Taking the complement, we get

$$\begin{array}{cccc}\hfill {\left({\mathit{J}}_{\mathit{R}}\left(b\right)\right)}^c& ={\left({\mathit{H}}_{\mathit{R}}\left(b\right)\right)}^c\hfill & \hfill & \mathrm{if}b\in \mathit{C}\setminus \mathcal{B}.\hfill \end{array}$$

(11)

(3)

If $b\in \mathcal{B}\cap \mathit{C}$, then $b\in \mathit{D}$ and

$$\begin{array}{cccc}\hfill {\mathit{J}}_{\mathit{R}}\left(b\right)& ={\mathit{G}}_{\mathit{R}}\left(b\right)\cup {\mathit{H}}_{\mathit{R}}\left(b\right)\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}$$

Taking the complement, we get

$$\begin{array}{cccc}\hfill {\left({\mathit{J}}_{\mathit{R}}\left(b\right)\right)}^c& ={({\mathit{G}}_{\mathit{R}}\left(b\right)\cup {\mathit{H}}_{\mathit{R}}\left(b\right))}^c\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}$$

(12)

Define ${\mathit{G}}_{\mathit{R}}\left(b\right)$ and ${\mathit{H}}_{\mathit{R}}\left(b\right)$ as:

$$\begin{array}{cc}\hfill {\mathit{G}}_{\mathit{R}}\left(b\right)& =\{⟨\mathit{x},\mathit{q}⟩,[{\Gamma }_{\mathit{l}\mathcal{B}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathcal{B}}^{+}(\mathit{x},\mathit{q})],[{\Gamma }_{\mathit{l}\mathcal{B}}^{-}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathcal{B}}^{-}(\mathit{x},\mathit{q})],\{{\Gamma }_{\mathit{B}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{B}}^{-}(\mathit{x},\mathit{q})\}\}\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill {\mathit{H}}_{\mathit{R}}\left(b\right)& =\{⟨\mathit{x},\mathit{q}⟩,[{\Gamma }_{\mathit{l}\mathit{C}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathit{C}}^{+}(\mathit{x},\mathit{q})],[{\Gamma }_{\mathit{l}\mathit{C}}^{-}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathit{C}}^{-}(\mathit{x},\mathit{q})],\{{\Gamma }_{\mathit{C}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{C}}^{-}(\mathit{x},\mathit{q})\}\}\hfill \end{array}$$

(14)

Combining Equations (12)–(14), we get:

$$\begin{array}{cc}\hfill {\left({\mathit{J}}_{\mathit{R}}\left(b\right)\right)}^c& =\left(⟨\mathit{x},\mathit{q}⟩,[{\Gamma }_{\mathit{l}\mathcal{B}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathcal{B}}^{+}(\mathit{x},\mathit{q})],[{\Gamma }_{\mathit{l}\mathcal{B}}^{-}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathcal{B}}^{-}(\mathit{x},\mathit{q})],\{{\Gamma }_{\mathit{B}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{B}}^{-}(\mathit{x},\mathit{q})\}\right.\hfill \\ \hfill & \cup {\left.\{⟨\mathit{x},\mathit{q}⟩,[{\Gamma }_{\mathit{l}\mathit{C}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathit{C}}^{+}(\mathit{x},\mathit{q})],[{\Gamma }_{\mathit{l}\mathit{C}}^{-}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathit{C}}^{-}(\mathit{x},\mathit{q})],\{{\Gamma }_{\mathit{C}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{C}}^{-}(\mathit{x},\mathit{q})\}\}\right)}^c\hfill \\ \hfill & =\left(⟨\mathit{x},\mathit{q}⟩,[max({\Gamma }_{\mathit{l}\mathcal{B}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{l}\mathit{C}}^{+}(\mathit{x},\mathit{q})),max({\Gamma }_{\mathit{u}\mathcal{B}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathit{C}}^{+}(\mathit{x},\mathit{q}))],\right.\hfill \\ \hfill & [min({\Gamma }_{\mathit{l}\mathcal{B}}^{-}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{l}\mathit{C}}^{-}(\mathit{x},\mathit{q})),min({\Gamma }_{\mathit{u}\mathcal{B}}^{-}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathit{C}}^{-}(\mathit{x},\mathit{q}))],\hfill \\ \hfill & {\left.\{max({\Gamma }_{\mathit{B}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{C}}^{+}(\mathit{x},\mathit{q})),min({\Gamma }_{\mathit{B}}^{-}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{C}}^{-}(\mathit{x},\mathit{q}))\}\right)}^c\hfill \\ \hfill & =\left(⟨\mathit{x},\mathit{q}⟩,\left(1-max({\Gamma }_{\mathit{l}\mathcal{B}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{l}\mathit{C}}^{+}(\mathit{x},\mathit{q})),1-(-min({\Gamma }_{\mathit{u}\mathcal{B}}^{-}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathit{C}}^{-}(\mathit{x},\mathit{q})))\right)\right)\hfill \\ \hfill & =\left(⟨\mathit{x},\mathit{q}⟩,\left(max(1-{\Gamma }_{\mathit{l}\mathcal{B}}^{+}(\mathit{x},\mathit{q}),1-{\Gamma }_{\mathit{l}\mathit{C}}^{+}(\mathit{x},\mathit{q})),min(-1-(-{\Gamma }_{\mathit{u}\mathcal{B}}^{-}(\mathit{x},\mathit{q}),\right.\right.\hfill \\ \hfill & \left.\left.-1-(-{\Gamma }_{\mathit{u}\mathit{C}}^{-}(\mathit{x},\mathit{q})))\right)\right)\hfill \\ \hfill & =\left(⟨\mathit{x},\mathit{q}⟩,1-{\Gamma }_{\mathit{l}\mathcal{B}}^{+}(\mathit{x},\mathit{q}),-1-(-{\Gamma }_{\mathit{u}\mathcal{B}}^{-}(\mathit{x},\mathit{q}))\right)\hfill \\ \hfill & \cap \left(⟨\mathit{x},\mathit{q}⟩,1-{\Gamma }_{\mathit{l}\mathit{C}}^{+}(\mathit{x},\mathit{q}),-1-(-{\Gamma }_{\mathit{u}\mathit{C}}^{-}(\mathit{x},\mathit{q}))\right)\hfill \\ \hfill & ={\left(⟨\mathit{x},\mathit{q}⟩,{\Gamma }_{\mathit{l}\mathcal{B}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathcal{B}}^{-}(\mathit{x},\mathit{q})\right)}^c\cap {\left(⟨\mathit{x},\mathit{q}⟩,{\Gamma }_{\mathit{l}\mathit{C}}^{+}(\mathit{x},\mathit{q}),{\Gamma }_{\mathit{u}\mathit{C}}^{-}(\mathit{x},\mathit{q})\right)}^c\hfill \\ \hfill & ={\left({\mathit{G}}_{\mathit{R}}\left(b\right)\right)}^c\cap {\left({\mathit{H}}_{\mathit{R}}\left(b\right)\right)}^c\mathrm{if}b\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}$$

From (10), (11), and (12), we get:

$$\begin{array}{cccc}\hfill {\left({\mathit{J}}_{\mathit{R}}\left(b\right)\right)}^c& ={\left({\mathit{G}}_{\mathit{R}}\left(b\right)\right)}^c\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\setminus \mathit{C},\hfill \\ \hfill & ={\left({\mathit{H}}_{\mathit{R}}\left(b\right)\right)}^c\hfill & \hfill & \mathrm{if}b\in \mathit{C}\setminus \mathcal{B},\hfill \\ \hfill & ={\left({\mathit{G}}_{\mathit{R}}\left(b\right)\right)}^c\cap {\left({\mathit{H}}_{\mathit{R}}\left(b\right)\right)}^c\hfill & \hfill & \mathrm{if}b\in \mathcal{B}\cap \mathit{C}.\hfill \end{array}$$

Thus, ${\left({\mathit{J}}_{\mathit{R}}\left(b\right)\right)}^c={\left({\mathit{G}}_{\mathit{R}}\left(b\right)\right)}^c\cap {\left({\mathit{H}}_{\mathit{R}}\left(b\right)\right)}^c$ and ${\left(({\mathit{G}}_{\mathit{R}},\mathcal{B})\cup ({\mathit{H}}_{\mathit{R}},\mathit{C})\right)}^c={({\mathit{G}}_{\mathit{R}},\mathcal{B})}^c\cap {({\mathit{H}}_{\mathit{R}},\mathit{C})}^c$.

The proof for the second De Morgan law follows similarly. □

5.2. Or and Add Operation on Multi-R Cubic Bipolar Fuzzy Soft Set

Definition 26. 

Let $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ and $({\mathit{H}}_{\mathit{R}},\mathit{C})$ be two multi-R cubic bipolar fuzzy soft sets:

(1) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\wedge ({\mathit{H}}_{\mathit{R}},\mathit{C})$ is a multi-R cubic bipolar fuzzy soft set defined by $({\mathit{G}}_{\mathit{R}},\mathcal{B})\wedge ({\mathit{H}}_{\mathit{R}},\mathit{C})=(\tilde{\mathit{J}},\mathcal{B}\times \mathit{C})$, where ${\mathit{J}}_{\mathit{R}}(b,c)=({\mathit{G}}_{\mathit{R}}\left(b\right)\cap {\mathit{H}}_{\mathit{R}}\left(c\right))$∀$(b,c)\in \mathit{d}=\mathcal{B}\times \mathit{C}$.

(2) 

$({\mathit{G}}_{\mathit{R}},\mathcal{B})\vee ({\mathit{H}}_{\mathit{R}},\mathit{C})$ is a multi-R cubic bipolar fuzzy soft set defined by $({\mathit{G}}_{\mathit{R}},\mathcal{B})\vee ({\mathit{H}}_{\mathit{R}},\mathit{C})=(\mathit{J},\mathcal{B}\times \mathit{C})$, where ${\mathit{J}}_{\mathit{R}}(b,c)=({\mathit{G}}_{\mathit{R}}\left(b\right)\cup {\mathit{H}}_{\mathit{R}}\left(c\right))$∀$(b,c)\in \mathit{d}=\mathcal{B}\times \mathit{C}$.

Example 4. 

Let $\mathit{U}=\{{\mathit{f}}_{\mathbf{1}},{\mathit{f}}_{\mathbf{2}}\}$ be a set of two men under consideration, $\mathit{E}=\{{\mathit{b}}_{\mathbf{1}}=\mathit{Educated},{\mathit{b}}_{\mathbf{2}}=\mathit{Government}\mathit{employed},{\mathit{b}}_{\mathbf{3}}=\mathit{Businessman},{\mathit{b}}_{\mathbf{4}}=\mathit{Smart},{\mathit{b}}_{\mathbf{5}}=\mathit{Weak}\}$ be the set of parameters, and $\mathcal{B}=\{{\mathit{b}}_{\mathbf{1}},{\mathit{b}}_{\mathbf{2}}\}\subseteq \mathit{E}$, $\mathit{C}=\{{\mathit{b}}_{\mathbf{4}},{\mathit{b}}_{\mathbf{5}}\}\subseteq \mathit{E}$, $\mathit{R}=\{\mathit{p},\mathit{q}\}$; then:

$$\begin{array}{cc}\hfill ({\mathit{G}}_{\mathit{R}},\mathcal{B})=\{& ⟨{\mathit{b}}_{\mathbf{1}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.1,0.7][-0.6,-0.4]\{0.3,-0.5\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.3,0.5][-0.5,-0.3]\{0.2,-0.3\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.4,0.9][-0.5,-0.2]\{0.1,-0.4\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.4,0.6][-0.3,-0.2]\{0.8,-0.4\}}}⟩\hfill \\ \hfill & ⟨{\mathit{b}}_{\mathbf{2}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.4,0.8][-0.4,-0.1]\{0.3,-0.9\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.6,0.7][-0.8,-0.5]\{0.3,-0.6\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.2,0.5][-0.7,-0.4]\{0.7,-0.9\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.2,0.6][-0.6,-0.4]\{0.3,-0.9\}}}⟩\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill ({\mathit{H}}_{\mathit{R}},\mathit{C})=\{& ⟨{\mathit{b}}_{\mathbf{4}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.3,0.6][-0.4,-0.3]\{0.1,-0.6\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.6,0.8][-0.3,-0.2]\{0.4,0.5\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.2,0.7][-0.3,-0.1]\{0.8,-0.9\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.5,0.8][-0.4,-0.3]\{0.5,-0.8\}}}⟩\hfill \\ \hfill & ⟨{\mathit{b}}_{\mathbf{5}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.6,0.9][-0.8,-0.1]\{0.2,-0.5\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.4,0.5][-0.6,-0.5]\{0.3,-0.9\}}}\hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.5,0.6][-0.3,-0.1]\{0.2,-0.8\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.2,0.9][-0.8,-0.4]\{0.4,-0.5\}}}⟩\},\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill ({\mathit{G}}_{\mathit{R}},\mathcal{B})\wedge ({\mathit{H}}_{\mathit{R}},\mathit{C})& =(\mathit{J},\mathcal{B}\times \mathit{C}),\mathit{D}=\mathcal{B}\times \mathit{C}\hfill \\ \hfill & =\{{\mathit{b}}_{\mathbf{1}},{\mathit{b}}_{\mathbf{2}}\}\times \{{\mathit{b}}_{\mathbf{4}},{\mathit{b}}_{\mathbf{5}}\}\hfill \\ \hfill & =\{({\mathit{b}}_{\mathbf{1}},{\mathit{b}}_{\mathbf{4}}),({\mathit{b}}_{\mathbf{1}},{\mathit{b}}_{\mathbf{5}}),({\mathit{b}}_{\mathbf{2}},{\mathit{b}}_{\mathbf{4}}),({\mathit{b}}_{\mathbf{2}},{\mathit{b}}_{\mathbf{5}})\},\hfill \\ \hfill {\mathit{J}}_{\mathit{R}}(b,c)& ={\mathit{G}}_{\mathit{R}}\left(b\right)\cap {\mathit{H}}_{\mathit{R}}\left(b\right),(b,c)\in \mathit{D}.\hfill \end{array}$$

•$({\mathit{J}}_{\mathit{R}},\mathit{D})=({\mathit{G}}_{\mathit{R}},\mathcal{B})\wedge ({\mathit{H}}_{\mathit{R}},\mathit{C})=$

$$\begin{array}{cc}\hfill {\mathit{J}}_{\mathit{R}},({\mathit{b}}_{\mathbf{1}},{\mathit{b}}_{\mathbf{4}})=& \{⟨{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.1,0.6][-0.4,-0.3]\{0.1,-0.5\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.3,0.5][-0.3,-0.2]\{0.2,-0.3\}}}\cap \hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.2,0.7][-0.3,-0.1]\{0.1,-0.4\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.4,0.6][-0.3,-0.2]\{0.5,-0.4\}}}⟩\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill •{\mathit{J}}_{\mathit{R}},({\mathit{b}}_{\mathbf{1}},{\mathit{b}}_{\mathbf{5}})=& \{⟨{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.1,0.7][-0.6,-0.1]\{0.2,-0.5\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.3,0.5][-0.5,-0.3]\{0.2,-0.3\}}}\cap \hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.4,0.6][-0.3,-0.1]\{0.1,-0.4\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.2,0.6][-0.3,-0.2]\{0.3,-0.5\}}}⟩\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill •{\mathit{J}}_{\mathit{R}},({\mathit{b}}_{\mathbf{2}},{\mathit{b}}_{\mathbf{4}})=& \{⟨{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.3,0.6][-0.4,-0.1]\{0.1,-0.1\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.6,0.7][-0.3,-0.2]\{0.3,-0.6\}}}\cap \hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.2,0.5][-0.3,-0.1]\{0.7,-0.9\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.2,0.6][-0.4,-0.3]\{0.3,-0.8\}}}⟩\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill •{\mathit{J}}_{\mathit{R}},({\mathit{b}}_{\mathbf{2}},{\mathit{b}}_{\mathbf{5}})=& \{⟨{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.4,0.8][-0.4,-0.1]\{0.2,-0.5\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.4,0.5][-0.4,-0.3]\{0.3,-0.6\}}}\cap \hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.2,0.5][-0.7,-0.1]\{0.2,-0.8\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.2,0.6][-0.6,-0.4]\{0.3,-0.3\}}}⟩\},\hfill \end{array}$$

Example 5. 

Consider that either $({\mathit{G}}_{\mathit{R}},\mathcal{B})$ or $({\mathit{H}}_{\mathit{R}},\mathit{C})$ is

$$\begin{array}{cc}\hfill •{\mathit{J}}_{\mathit{R}},({\mathit{b}}_{\mathbf{1}},{\mathit{b}}_{\mathbf{4}})=& \{⟨{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.3,0.7][-0.6,-0.4]\{0.3,-0.6\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.6,0.8][-0.5,-0.3]\{0.9,-0.5\}}}\cap \hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.4,0.9][-0.5,-0.2]\{0.8,-0.9\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.5,0.8][-0.4,-0.3]\{0.8,-0.8\}}}⟩\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill •{\mathit{J}}_{\mathit{R}},({\mathit{b}}_{\mathbf{1}},{\mathit{b}}_{\mathbf{5}})=& \{⟨{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.6,0.9][-0.8,-0.4]\{0.3,-0.5\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.4,0.5][-0.6,-0.3]\{0.3,-0.9\}}}\cap \hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.5,0.9][-0.5,-0.2]\{0.2,-0.5\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.4,0.9][-0.8,-0.4]\{0.8,-0.5\}}}⟩\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill •{\mathit{J}}_{\mathit{R}},({\mathit{b}}_{\mathbf{2}},{\mathit{b}}_{\mathbf{4}})=& \{⟨{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.4,0.8][-0.4,-0.8]\{0.3,-0.9\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.6,0.8][-0.8,-0.5]\{0.4,-0.6\}}}\cap \hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.2,0.7][-0.7,-0.4]\{0.8,-0.9\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.5,0.8][-0.6,-0.4]\{0.5,-0.9\}}}⟩\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill •{\mathit{J}}_{\mathit{R}},({\mathit{b}}_{\mathbf{2}},{\mathit{b}}_{\mathbf{5}})=& \{⟨{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{p}}{[0.6,0.9][-0.8,-0.1]\{0.3,-0.9\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{1}}\mathit{q}}{[0.6,0.7][-0.8,-0.5]\{0.3,-0.9\}}}\cap \hfill \\ \hfill & {\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{p}}{[0.5,0.6][-0.7,-0.4]\{0.7,-0.9\}}},{\displaystyle \frac{{\mathit{f}}_{\mathbf{2}}\mathit{q}}{[0.2,0.9][-0.8,-0.4]\{0.3,-0.6\}}}⟩\}.\hfill \end{array}$$

6. Proposed Method

This section examines the advancement of multi-criteria group decision-making strategies utilizing cosine similarity within the context of multi-Q cubic bipolar fuzzy soft sets (MQCBFSs) and provides theoretical arguments for our approach.

Multi-criteria group decision-making refers to a situation in which many decision-makers collaborate to choose the optimal option from a range of possibilities, taking into account multiple criteria. Cosine similarity is a mathematical method employed to quantify the similarity between two vectors. In our specific scenario, decision-makers use it to compare various alternatives or their preferences. Multi-q cubic bipolar fuzzy soft sets (MQCBFSs) are a sophisticated mathematical frameworks used to express uncertain and imprecise information in a decision-making context.

Here are some theoretical justifications for our proposed method:

• Managing Ambiguity and Indeterminacy MQCBFSs are highly suitable for effectively managing the inherent uncertainty and ambiguity in decision-making difficulties encountered in real-world scenarios. Additionally, the cosine similarity can be modified to gauge the similarity between fuzzy sets, offering a reliable method to assess different alternatives when faced with uncertainty [19]
• Integrating Diverse Viewpoints Multi-criteria decision-making frequently involves several stakeholders with varying preferences, and MQCBFSs can efficiently capture these distinct perspectives. Cosine similarity can then be used to combine these perspectives into a consensus [20].
• Enhancing Decision Precision Our strategy enhances the accuracy and dependability of decision-making outputs in comparison to conventional methods by leveraging the capabilities of MQCBFSs and cosine similarity [20].

Definition 27 

([10]). (Cosine Similarity) Let $\mathit{R}=\{{\mathit{e}}_{\mathbf{1}},{\mathit{e}}_{\mathbf{2}},\dots \}$ be a finite universe of discourse. Consider two CBFSs on $\mathit{R}$, defined as follows:

$$\mathit{U}=⟨{\mathit{e}}_{\mathit{i}},\left[{\mathit{p}}_{\mathit{U}}^{+}\left({\mathit{e}}_{\mathit{i}}\right),{\mathit{p}}_{\mathit{U}}^{-}\left({\mathit{e}}_{\mathit{i}}\right)\right],\left[{\mathit{p}}_{\overline{\mathit{U}}}^{+}\left({\mathit{e}}_{\mathit{i}}\right),{\mathit{p}}_{\overline{\mathit{U}}}^{-}\left({\mathit{e}}_{\mathit{i}}\right)\right],\left({\mathit{p}}_{\mathit{U}}^{+}\left({\mathit{e}}_{\mathit{i}}\right),{\mathit{p}}_{\overline{\mathit{U}}}\left({\mathit{e}}_{\mathit{i}}\right)\right)⟩:{\mathit{e}}_{\mathit{i}}\in \mathit{R}$$

and

$$\mathit{B}=⟨{\mathit{e}}_{\mathit{i}},\left[{\mathit{p}}_{\mathit{B}}^{+}\left({\mathit{e}}_{\mathit{i}}\right),{\mathit{p}}_{\mathit{B}}^{-}\left({\mathit{e}}_{\mathit{i}}\right)\right],\left[{\mathit{p}}_{\overline{\mathit{B}}}^{+}\left({\mathit{e}}_{\mathit{i}}\right),{\mathit{p}}_{\overline{\mathit{B}}}^{-}\left({\mathit{e}}_{\mathit{i}}\right)\right],\left({\mathit{p}}_{\mathit{B}}^{+}\left({\mathit{e}}_{\mathit{i}}\right),{\mathit{p}}_{\overline{\mathit{B}}}^{-}\left({\mathit{e}}_{\mathit{i}}\right)\right)⟩:{\mathit{e}}_{\mathit{i}}\in \mathit{R}$$

The cosine similarity (SM), which is based on the cosine of the angle between two vectors, is then given by

$${\mathcal{S}}_{\mathbf{1}}(\mathcal{U},\mathcal{S})={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left[{\displaystyle \frac{{\rho }_{{\mathit{e}}_{\mathit{u}}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right){\rho }_{\mathcal{S}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)+{\rho }_{{\mathit{e}}_{{\mathit{u}}_{\mathbf{t}}}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right){\rho }_{{\mathit{u}}_{\mathcal{S}}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)+{\rho }_{{\mathit{e}}_{\mathit{u}}}^{-}\left({\overline{\mathit{e}}}_{\mathit{i}}\right){\rho }_{\mathcal{S}}^{-}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)+{\rho }_{{\mathit{u}}_{\mathbf{1}}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right){\rho }_{\mathcal{S}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)+{\rho }_{\mathit{u}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right){\rho }_{\mathcal{S}}^{-}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)+{\rho }_{\mathit{u}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right){\rho }_{\mathcal{S}}^{-}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)}{\sqrt{{\left({\rho }_{{\mathit{e}}_{\mathit{u}}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2+{\left({\rho }_{{\mathit{u}}_{{\mathit{e}}_{\mathit{i}}}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2+{\left({\rho }_{{\mathit{u}}_{\mathcal{S}}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2+{\left({\rho }_{{\mathit{u}}_{\mathbf{1}}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2+{\left({\rho }_{\mathit{u}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2}\times \sqrt{{\left({\rho }_{\mathcal{S}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2+{\left({\rho }_{\mathcal{S}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2+{\left({\rho }_{\mathcal{S}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2+{\left({\rho }_{\mathcal{S}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2+{\left({\rho }_{\mathcal{S}}^{+}\left({\overline{\mathit{e}}}_{\mathit{i}}\right)\right)}^2}}}\right]$$

• Step 1: Collect information for MQCBFSs and determine variables $({\mathit{c}}_{\mathbf{1}},{\mathit{c}}_{\mathbf{2}},\dots ,{\mathit{c}}_{\mathit{m}})$ and the calibration ${\mathit{Q}}_{\mathit{i}}\forall i=1,2,3,\dots$
• Step 2: Write MCQBFSs.
• Step 3: Take the ideal group suggested by experts as an ideal alternative.
• Step 4: Use the cosine similarity proposed in Equation (1).
• Step 5: Determine the yield for each of the four products.
• Step 6: Choose the highest value.
• Step 7: The end.

Example 6. 

Emphasizing agriculture is crucial for attaining the Kingdom’s Vision 2030 and its developmental goals. Agriculture plays a vital role in guaranteeing long-lasting and nourishing food provision, bolstering food safety, and fostering economic development. The Kingdom enjoys the advantages of a varied natural environment, characterized by vast expanses of fertile land. The careful selection of suitable soil is a crucial element in promoting the progress of the agricultural industry and guaranteeing the long-term preservation of the environment.

Quality standards are established based on fundamental principles and vary depending on the specific areas of application and the evaluation techniques employed. Although they have distinct variations, they share similar parameters and criteria that ensure that the quality of the end product remains consistent throughout the production process. Sustainable agriculture adheres to these criteria, fulfilling essential criteria and tackling sector-specific obstacles.

Soil and climate are major factors influencing agricultural product quality, but other factors such as water availability, farming techniques, and processing methods also play significant roles.

• Given

A company plans to purchase three types of products from two different brands and takes into account the opinions of two specialists. Let ${\mathit{u}}_{\mathbf{1}}$ and ${\mathit{u}}_{\mathbf{2}}$ represent the set of brands. Let ${\mathit{q}}_{\mathbf{1}}$ and ${\mathit{q}}_{\mathbf{2}}$ denote the set of products. The decision parameters are given by ${\mathit{e}}_{\mathbf{1}}$ (water drainage), ${\mathit{e}}_{\mathbf{2}}$ (aeration), and ${\mathit{e}}_{\mathbf{3}}$ (humidity).

• Step-by-Step Derivation

1-  

Parameters and Products:

• U1 and U2 represent two different brands.
• Q1 and Q2 represent two different types of products.
• e1, e2, and e3 represent the evaluation criteria—water drainage, aeration, and humidity— respectively.

	$e_1$	$e_2$	$e_3$
$U_1{Q}_1$	$[0.40,0.50]$$[-0.35,-0.15]$$\{0.15,-0.13\}$	$[0.40,1.00]$$[-0.70,-0.64]$$\{0.61,-0.34\}$	$[0.35,0.45]$$[-0.6,-0.34]$$\{0.50,-0.20\}$
$U_1{Q}_2$	$[0.60,1.00]$$[-0.55,-0.45]$$\{0.50,-0.40\}$	$[0.51,0.67]$$[-0.60,-0.44]$$\{0.70,-0.30\}$	$[0.2,1.00]$$[-0.67,-0.55]$$\{0.60,-0.20\}$

	$e_1$	$e_2$	$e_3$
$U_2{Q}_1$	$[0.60,0.72]$$[-0.55,-0.32]$$\{0.60,-0.5\}$	$[0.33,0.55]$$[-0.45,-0.40]$$\{0.35,-0.30\}$	$[0.50,0.64]$$[-0.65,-0.53]$$\{0.60,-0.45\}$
$U_2{Q}_2$	$[0.90,1.00]$$[-0.67,-0.43]$$\{0.5,-0.2\}$	$[1.00,0.66]$$[-0.32,-0.14]$$\{0.60,-0.40\}$	$[0.78,0.80]$$[-0.40,-0.43]$$\{0.30,-0.20\}$

2-  

MQCBFSs: The data are represented using MQCBFSs, which capture the uncertainty and imprecision in the decision-making process.

$$\begin{array}{cc}\hfill \mathbf{s}({\mathit{U}}_{\mathbf{1}},{\mathit{U}}_{\mathbf{2}})=& \left({\displaystyle \frac{{\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{1}}}{\left\{<{\mathit{e}}_{\mathbf{1}}[0.40,0.50][-0.35,-0.15]\{0.15,-0.13\}>\right\}}}{\displaystyle \frac{{\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{2}}}{\left\{<[0.60,1.00][-0.55,-0.45]\{0.50,-0.40\}>\right\}}}\right.,\hfill \\ \hfill & \left.{\displaystyle \frac{{\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{1}}}{\left\{<[0.60,0.72][-0.55,-0.32]\{0.60,-0.5\}>\right\}}},{\displaystyle \frac{{\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{2}}}{\left\{<[0.90,1.00][-0.67,-0.43]\{0.50,-0.20\}>\right\}}}\right)\hfill \\ \hfill & =\left({\displaystyle \frac{{\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{1}}}{⟨{\mathit{e}}_{\mathbf{2}}[0.40,1.00][-0.70,-0.64]\{0.61,-0.34\}⟩}},{\displaystyle \frac{{\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{2}}}{⟨[0.51,0.67][-0.60,-0.44]\{0.70,-0.30\}⟩}}\right)\hfill \\ \hfill & =\left({\displaystyle \frac{{\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{1}}}{⟨[0.33,0.55][-0.45,-0.40]\{0.35,-0.30\}⟩}},{\displaystyle \frac{{\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{2}}}{⟨[1.00,0.66][-0.32,-0.14]\{0.60,-0.40\}⟩}}\right)\hfill \\ \hfill & =\left({\displaystyle \frac{{\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{1}}}{⟨{\mathit{e}}_{\mathbf{3}}[0.35,0.45][-0.6,-0.34]\{0.50,-0.20\}⟩}},{\displaystyle \frac{{\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{2}}}{⟨[0.2,1.00][-0.67,-0.55]\{0.60,-0.20\}⟩}}\right)\hfill \\ \hfill & =\left({\displaystyle \frac{{\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{1}}}{⟨[0.50,0.64][-0.65,-0.53]\{0.60,-0.45\}⟩}},{\displaystyle \frac{{\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{2}}}{⟨[0.78,0.80][-0.40,-0.43]\{0.30,-0.20\}⟩}}\right)\hfill \end{array}$$

3-  

Take the Perfect Set:

	$e_1$	$e_2$	$e_3$
$UQ$	[0.31, 0.50] [−0.60, −0.30] {0.43, −0.52}	[0.54, 0.73] [−0.37, −0.15] {0.43, −0.42}	[0.44, 0.67] [−0.47, −0.33] {0.57, −0.46}

4-  

Using Cosine Similarity:

• Cosine similarity is used to measure the similarity between different product–brand combinations based on the evaluation criteria.
• The formula for calculating cosine similarity between two vectors is provided.
• Detailed calculations for each product–brand–criteria combination are shown.

$$\begin{array}{cc}\hfill & {\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.4\times 0.31]+[0.5\times 0.5]+[-0.35\times -0.6]+[-0.15\times -0.3]+[0.15\times 0.43]+[-0.13\times -0.52]}{\sqrt{{\left(0.4\right)}^2+{\left(0.5\right)}^2+{(-0.35)}^2+{(-0.15)}^2+{\left(0.15\right)}^2+{(-0.13)}^2}\times \sqrt{{\left(0.31\right)}^2+{\left(0.5\right)}^2+{(-0.6)}^2+{(-0.3)}^2+{\left(0.43\right)}^2+{(-0.52)}^2}}}\right\}=0.9047\hfill \\ \hfill & {\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{1}}{\mathit{e}}_{\mathbf{2}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.4\times 0.54]+[1\times 0.73]+[-0.7\times -0.37]+[-0.64\times -0.15]+[0.61\times 0.43]+[-0.34\times -0.42]}{\sqrt{{\left(0.4\right)}^2+{\left(1\right)}^2+{(-0.7)}^2+{(-0.64)}^2+{\left(0.61\right)}^2+{(-0.34)}^2}\times \sqrt{{\left(0.54\right)}^2+{\left(0.73\right)}^2+{(-0.37)}^2+{(-0.15)}^2+{\left(0.43\right)}^2+{(-0.42)}^2}}}\right\}=0.9221\hfill \\ \hfill & {\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{1}}{\mathit{e}}_{\mathbf{3}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.35\times 0.44]+[0.45\times 0.67]+[-0.6\times -0.47]+[-0.34\times -0.33]+[0.5\times 0.57]+[-0.2\times -0.36]}{\sqrt{{\left(0.35\right)}^2+{\left(0.45\right)}^2+{(-0.6)}^2+{(-0.34)}^2+{\left(0.5\right)}^2+{(-0.2)}^2}\times \sqrt{{\left(0.44\right)}^2+{\left(0.67\right)}^2+{(-0.47)}^2+{(-0.33)}^2+{\left(0.57\right)}^2+{(-0.46)}^2}}}\right\}=0.9407\hfill \\ \hfill & {\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{2}}{\mathit{e}}_{\mathbf{1}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.6\times 0.31]+[1\times 0.5]+[-0.55\times -0.6]+[-0.45\times -0.3]+[0.5\times 0.43]+[-0.4\times -0.52]}{\sqrt{{\left(0.6\right)}^2+{\left(1\right)}^2+{(-0.55)}^2+{(-0.45)}^2+{\left(0.5\right)}^2+{(-0.4)}^2}\times \sqrt{{\left(0.31\right)}^2+{\left(0.5\right)}^2+{(-0.6)}^2+{(-0.3)}^2+{\left(0.43\right)}^2+{(-0.52)}^2}}}\right\}=0.9341\hfill \\ \hfill & {\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.51\times 0.54]+[0.67\times 0.73]+[-0.6\times -0.37]+[-0.44\times -0.15]+[0.7\times 0.43]+[-0.3\times -0.42]}{\sqrt{{\left(0.51\right)}^2+{\left(0.67\right)}^2+{(-0.6)}^2+{(-0.44)}^2+{\left(0.7\right)}^2+{(-0.3)}^2}\times \sqrt{{\left(0.54\right)}^2+{\left(0.73\right)}^2+{(-0.37)}^2+{(-0.15)}^2+{\left(0.43\right)}^2+{(-0.42)}^2}}}\right\}=0.9408\hfill \\ \hfill & {\mathit{U}}_{\mathbf{1}}{\mathit{Q}}_{\mathbf{2}}{\mathit{e}}_{\mathbf{3}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.2\times 0.44]+[0.1\times 0.67]+[-0.67\times -0.47]+[-0.55\times -0.33]+[0.6\times 0.57]+[-0.2\times -0.36]}{\sqrt{{\left(0.2\right)}^2+{\left(0.1\right)}^2+{(-0.67)}^2+{(-0.55)}^2+{\left(0.6\right)}^2+{(-0.2)}^2}\times \sqrt{{\left(0.44\right)}^2+{\left(0.67\right)}^2+{(-0.47)}^2+{(-0.33)}^2+{\left(0.57\right)}^2+{(-0.36)}^2}}}\right\}=0.8299\hfill \\ \hfill & {\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{1}}{\mathit{e}}_{\mathbf{1}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.6\times 0.31]+[0.72\times 0.5]+[-0.55\times -0.6]+[-0.32\times -0.3]+[0.6\times 0.43]+[-0.5\times -0.52]}{\sqrt{{\left(0.6\right)}^2+{\left(0.72\right)}^2+{(-0.55)}^2+{(-0.32)}^2+{\left(0.6\right)}^2+{(-0.5)}^2}\times \sqrt{{\left(0.31\right)}^2+{\left(0.5\right)}^2+{(-0.6)}^2+{(-0.3)}^2+{\left(0.43\right)}^2+{(-0.52)}^2}}}\right\}=0.9694\hfill \\ \hfill & {\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{1}}{\mathit{e}}_{\mathbf{2}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.33\times 0.54]+[0.55\times 0.73]+[-0.45\times -0.37]+[-0.4\times -0.15]+[0.35\times 0.43]+[-0.3\times -0.42]}{\sqrt{{\left(0.33\right)}^2+{\left(0.55\right)}^2+{(-0.45)}^2+{(-0.4)}^2+{\left(0.35\right)}^2+{(-0.3)}^2}\times \sqrt{{\left(0.54\right)}^2+{\left(0.73\right)}^2+{(-0.37)}^2+{(-0.15)}^2+{\left(0.43\right)}^2+{(-0.42)}^2}}}\right\}=0.940\hfill \\ \hfill & {\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{1}}{\mathit{e}}_{\mathbf{3}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.5\times 0.44]+[0.64\times 0.67]+[-0.65\times -0.47]+[-0.53\times -0.33]+[0.6\times 0.57]+[-0.45\times -0.36]}{\sqrt{{\left(0.5\right)}^2+{\left(0.64\right)}^2+{(-0.65)}^2+{(-0.53)}^2+{\left(0.6\right)}^2+{(-0.45)}^2}\times \sqrt{{\left(0.44\right)}^2+{\left(0.67\right)}^2+{(-0.47)}^2+{(-0.33)}^2+{\left(0.57\right)}^2+{(-0.46)}^2}}}\right\}=0.9577\hfill \\ \hfill & {\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{2}}{\mathit{e}}_{\mathbf{1}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.9\times 0.31]+[1\times 0.5]+[-0.67\times -0.6]+[-0.43\times -0.3]+[0.5\times 0.43]+[-0.2\times -0.52]}{\sqrt{{\left(0.9\right)}^2+{\left(1\right)}^2+{(-0.67)}^2+{(-0.43)}^2+{\left(0.5\right)}^2+{(-0.2)}^2}\times \sqrt{{\left(0.31\right)}^2+{\left(0.5\right)}^2+{(-0.6)}^2+{(-0.3)}^2+{\left(0.43\right)}^2+{(-0.52)}^2}}}\right\}=0.8798\hfill \\ \hfill & {\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{2}}{\mathit{e}}_{\mathbf{2}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[1\times 0.54]+[0.66\times 0.73]+[-0.32\times -0.37]+[-0.14\times -0.15]+[0.6\times 0.43]+[-0.4\times -0.42]}{\sqrt{{\left(1\right)}^2+{\left(0.66\right)}^2+{(-0.32)}^2+{(-0.14)}^2+{\left(0.6\right)}^2+{(-0.4)}^2}\times \sqrt{{\left(0.54\right)}^2+{\left(0.73\right)}^2+{(-0.37)}^2+{(-0.15)}^2+{\left(0.43\right)}^2+{(-0.42)}^2}}}\right\}=0.9497\hfill \\ \hfill & {\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{2}}{\mathit{e}}_{\mathbf{3}}={\displaystyle \frac{1}{2}}\sum _{i=1}^2\left\{{\displaystyle \frac{[0.78\times 0.44]+[0.8\times 0.67]+[-0.4\times -0.47]+[-0.43\times -0.33]+[0.3\times 0.57]+[-0.2\times -0.36]}{\sqrt{{\left(0.78\right)}^2+{\left(0.8\right)}^2+{(-0.4)}^2+{(-0.43)}^2+{\left(0.3\right)}^2+{(-0.2)}^2}\times \sqrt{{\left(0.44\right)}^2+{\left(0.67\right)}^2+{(-0.47)}^2+{(-0.33)}^2+{\left(0.57\right)}^2+{(-0.46)}^2}}}\right\}=0.8945\hfill \end{array}$$

5-  

Results:

• The calculated cosine similarity values for each product–brand–criteria combination are presented in Table 2.
  Table 2. Cosine similarity values for each product–brand–criteria combination.

  	${\mathit{U}}_1{\mathit{Q}}_1$	${\mathit{U}}_1{\mathit{Q}}_2$	${\mathit{U}}_2{\mathit{Q}}_1$	${\mathit{U}}_2{\mathit{Q}}_2$
  $e_1$	0.9047	0.9341	0.9694	0.8798
  $e_2$	0.9221	0.9408	0.940	0.9497
  $e_3$	0.9407	0.8299	0.9577	0.8945
  	0.9225	0.9016	0.9557	0.9080

• The product with the highest average cosine similarity across all criteria is selected as the best option.
• In this case, U2Q1 has the highest average cosine similarity, indicating it is the preferred choice.

Based on our empirical findings for the four products, we note that the product $({\mathit{U}}_{\mathbf{2}}{\mathit{Q}}_{\mathbf{1}}=\mathbf{0}.\mathbf{9557})$ exhibits the highest proximity to the value 1, therefore making it the optimal choice. Therefore, we can conclude that the third product outperforms all other evaluated products, rendering it the superior option. The MQCBFS can collect more accurate and extensive data than previous methods. The relative importance of distinct factors can greatly influence the ultimate decision in multi-criteria decision-making (MCDM) problems. Weighting methods offer a means to allocate varying degrees of significance to each criterion. Sensitivity analysis is an essential process for assessing the resilience of a decision model. We methodically altered input parameters to observe their effect on the outcome. In the realm of multi-criteria decision making, sensitivity analysis is a valuable tool for evaluating the impact of modifications to criteria weights, attribute values, or other model parameters on the ultimate ranking of alternatives. The enumeration is concluded.

7. Conclusions

This study examined the use of cosine similarity in the context of multi-q cubic bipolar fuzzy soft sets (MQCBFSs). Cosine similarity is highly relevant in a range of real-world innovation processes, especially in content-based filtering (CBF) systems. Design writers stress the importance of accuracy and reliability in electrical applications that use CBF.

We overcame the constraints of current methods by developing an innovative approach for computing cosine similarity specifically for MQCBFSs. This strategy exhibits promise for addressing current global concerns. In addition, we conducted a comparison between the efficacy of our suggested approaches and the currently available cosine similarity algorithms. The results presented indisputable proof of the enhanced dependability of our novel techniques.

In addition to utilizing cosine similarity, this study presents potential opportunities for future investigation. We have introduced the innovative concept of MQCBFSs and examined its essential integration and intersection operations. The given examples aptly demonstrate the advantages of these techniques for intricate data processing. Furthermore, we created and implemented a novel algorithm to address a decision-making issue, showcasing the practical effectiveness of MQCBFSs in real-life situations.

The potential of multi-queue circular buffer file systems in various fields is considerable. The versatility of MQCBFSs allows for their application in diverse sectors that extend beyond the focus of this work. Here are some intriguing prospects for future investigation:

• Sentiment Analysis: By integrating a sentiment vocabulary that encompasses a range of positive and negative sentiments, MQCBFSs are capable of analyzing intricate emotions conveyed in textual data.
• Pattern Recognition: We can use MQCBFSs to detect patterns in data with inherent bipolar attributes, such as medical diagnoses or financial market trends.
• Recommendation Systems: Enhancing MQCBFSs to incorporate user preferences with different levels of reluctance can result in more tailored and efficient recommendation systems. These examples are only a small sample, and the possible uses of MQCBFSs are extensive.

Subsequent investigations can delve deeper into these potentialities and enhance the framework for wider use across many fields of study.

In summary, this study introduces a new framework that utilizes MQCBFSs and emphasizes its potential for a wide range of applications. Subsequent studies can investigate additional MQCBFS expansions and their integration with other similarity measures to tackle even more complex decision-making issues.