Medical Diagnosis under Effective Bipolar-Valued Multi-Fuzzy Soft Settings

Abstract

The Molodtsov-initiated soft set theory plays an important role as a powerful mathematical tool for handling uncertainty. As an extension of the soft set, the fuzzy soft set can be seen to be more generic and flexible than utilizing the soft set only that fails to represent problem parameters fuzziness. Through this progress, the fuzzy soft set theory cannot deal with decision-making problems involving multi-attribute sets, bipolarity, or some effective considered parameters. Therefore, the goal of this article is to adapt effectiveness and bipolarity concepts with the multi-fuzzy soft set of order n. One can see that this approach generates a novel, extended, effective decision-making environment that is more applicable than any previously introduced one. In addition, types, concepts, and operations of effective bipolar-valued multi-fuzzy soft sets of dimension n are provided, each with an example. Furthermore, properties like absorption, associative, distributive, commutative, and De Morgan’s laws of those new sets are investigated. Moreover, a decision-making methodology under effective bipolar-valued multi-fuzzy soft settings is established. This technique facilitates reaching the final decision that this student is qualified to take a certain education level, or this patient is suffering from a certain disease, etc. In addition, a case study represented in a medical diagnosis example is discussed in detail to make the proposed algorithm clearer. Applying matrix techniques in this example as well as using MATLAB^®, not only makes it easier and faster in doing calculations, but also gives more accurate, optimal, and effective decisions. Finally, the sensitivity analysis, as well as a comparison with the existing methods, are conducted in detail and are summarized in a chart to show the difference between them and the current one.

1. Introduction

In real life, ambiguity and uncertainty are the most typical contributing factors to complexity when generating judgments. Uncertain data are basic and common in many vital fields, including environmental research, corporate management, engineering, economics, medical science, sociology, and numerous others. This uncertainty is produced by missing data updates, inadequate information, data randomness, measurement device barriers, and so on.

Because of the huge quantity of uncertain data that is being gathered and accumulated, as well as the significance of these applications, research on effective methods for illustrating uncertain data and handling uncertainties has sparked plenty of interest in recent years, but it continues to be difficult. As a result, we always have numerous complex challenges in these and other fields. We cannot solve the challenges that arise from uncertainties in those cases using normal mathematical methods.

Decision-making is a process employed at the managerial level of any company to identify and pick alternatives based on individual preferences. Every decision context can be defined as a collection of data, replacement possibilities, choices, and values that are readily accessible at the time of the decision. Because the work and time necessary to obtain statistics or locate alternatives limit both knowledge and its replacements, any conclusions reached must be made within such a constrained framework.

Decision-making has become one of the most important aspects of life and work in recent years, owing to its tight relationship to success and effectiveness. Successful people achieve their life and work goals through effective and efficient decision-making. Individual perspectives, values, attitudes, and concepts, as well as ideas, are commonly utilized to guide decision-making. While a person can arrive at decisions based on a range of principles, they should exercise extreme caution in selecting one that is productive and contributes to significant performance. However, these theories exist in order to help people become better decision-makers in the world. The decision-making issue in an uncertain environment has attracted attention in recent years.

There are a lot of research studies, as well as applications in the literature, about many special mathematical tools, like probability theory, fuzzy sets [1], intuitionistic fuzzy set theory [2], soft set theory [3], and other mathematical methods, which are helpful ways for modeling uncertain data and making successful and useful decisions. Nevertheless, each of them faces particular difficulties while handling uncertainty. The probability theory is an old and useful strategy for tackling uncertainty, but it is only capable of being applied to circumstances involving random processes, or processes in which the occurrence of events is purely determined by chances.

In 1965, Zadeh [1] introduced a very important extension of the well-known crisp set to represent and overcome appearing uncertainty, which is the fuzzy concept approach. In fuzzy set theory, one can measure the degree of membership of an element by the element’s membership (indicator, or characteristic) function from the domain X to the interval $[0,1]$.

A well-known crisp set on the initial universe X can be measured by the element’s membership (indicator, or characteristic) function from X to $\{0,1\}$. Despite the fact that the fuzzy set has been recognized as a viable mathematical method of handling uncertainty, it has the following drawback: this particular number tells us almost nothing concerning how precise it is. The specific number (membership percent) involves reasoning both in favor of as well as against an object belonging, with no detail about exactly how much more of each there actually is. To overcome this limitation, an extended or generalized new concept of the term “fuzzy set” was provided by Atanassov [2] and was cleared by an example, which is the “intuitionistic fuzzy set”. The new concept represented in the soft set idea was first formulated, in 1999, by Molodtsov [3]. There has been a claim that made them introduce the soft set theory. The claim was that the inadequate parametrization tools of the previously introduced theories might be one of the important reasons for the issues and difficulties.

The novel-introduced softness concept or the soft set concept is a new practical mathematical tool, which is free from the above difficulties. Then, it was used to facilitate dealing with uncertainties for a long time. After that, in 2002, Maji et al. ([4,5]) examined and analyzed the Molodtsov-proposed soft set theory. They looked into many soft set-related ideas, developed an in-depth theoretical outline of the discipline, and then implemented it in a decision-making situation.

The bipolarity in information when dealing with decision-making issues, on the other hand, seemed to be an essential factor to take into account. That is because it is a very helpful component when building the mathematical structure for the majority of cases in problems with decision-making. According to bipolarity theory, bipolar sensibility beliefs cover a range of different decision-making processes. Love versus hate, advantages versus drawbacks, sweet versus salty, and finally, starvation versus satisfaction, are a few examples of different analyses of decision approaches.

By way of example, if we have an effective drug, this cannot prevent it from resulting in probable serious side effects. On the other hand, if it is an ineffective drug, it can have no dangerous side effects. As an innovative generalization or extension of the fuzzy set concept, Lee [6] developed the bipolar-valued fuzzy idea in 2000. In this scenario, the membership range that defines the element’s belonging was expanded from the interval $[0,1]$ to the interval $[-1,1]$.

Later, Maji et al. [7] created, in 2001, the novel concept of the fuzzy soft set theory by integrating the principles of fuzzy sets into the softness concept. Additionally, Roy and Maji [8] generated a decision-making method based on a fuzzy soft set to choose the ideal (best) item to purchase out of a variety of options. Furthermore, Yang et al. [9] provided the fuzzy soft set matrix formulation depending on the fuzzy soft set notion.

Moreover, a$\breve{g}$man et al. [10] examined fuzzy soft matrixes and various algebraic operations, and performed a theoretical investigation in fuzzy soft contexts. In their studies of fuzzy soft matrixes, Basu et al. [11] and Kumar and Kaur [12] developed novel concepts and operations. For more novel information about the fuzzy soft extension and its properties, one can refer to [13,14,15,16,17,18,19,20] to obtain many more theorems, results, and examples.

After that, Maji et al. [21] constructed, in 2004, the new notion of the intuitionistic fuzzy soft sets being an original extension of the soft sets. A few more operations were also offered on intuitionistic fuzzy soft sets and several their properties were recognized. A straightforward example was also provided to illustrate how to use this mathematical tool. The intuitionistic fuzzy soft matrixes idea was then developed by Chetia et al. [22] in order to readily express intuitionistic fuzzy soft sets and facilitate operations on them. They also detailed their higher functional operations in order to conduct theoretical research in the intuitionistic fuzzy soft set environment and produce some conclusions.

Moreover, Abdullah et al. [23] presented, in 2014, the idea of the bipolar fuzzy soft set and provided its fundamental properties. Further, the fundamental principles of the bipolar fuzzy soft sets were discovered. In addition, they overcame problems that occurred in decision-making by using the bipolar fuzzy soft set. In fact, the bipolar fuzzy soft sets and intuitionistic fuzzy soft sets are different from each other, contrary to their appearances. Following that, Sebastian and Ramakrishnan [24] suggested the idea of the multi-fuzzy set concept by employing the multi-characteristic function, which is an ordered sequence of the aforementioned characteristic functions.

Actually, the multi-fuzzy sets can solve some particular issues which are exceedingly challenging for other fuzzy set extensions to describe. For example, the three-dimensional characteristic function, whose components are the characteristic functions representing the three primary known colors; red, green, and blue, can describe the color of pixels in a two-dimensional image in a way that the characteristic function of the regular fuzzy set cannot. As a result, any image may be generally represented as a set of arranged pixels with a multi-characteristic function. Moreover, Yang et al. [25] introduced the multi-fuzzy soft sets and proposed many applications using decision-making techniques based on those new sets.

Santhi and Shyamala [26] also described the bipolar-valued multi-fuzzy set and provided some observations on the bipolar-valued multi-fuzzy subgroups of a group. Furthermore, Yang et al. [27] offered various decision-making applications based on their concept of the bipolar-valued multi-fuzzy soft set. In addition, Sakr et al. ([28,29]) have introduced the bipolar-valued vague soft sets, the bipolar-valued multi-vague soft sets and their applications. The vague set is a generalization for a fuzzy set, in which any membership value is an interval subset from $[0,1]$, not only a specific single membership value lying within the range of 0 to 1 as known in fuzzy sets. Furthermore, for additional knowledge on those topics and many more interesting related topics, refer to [30,31,32,33].

In addition, using well-known techniques, Chakraborty et al. [34] have constructed the sense of de-bipolarization for a triangular bipolar neutrosophic number, such that any bipolar neutrosophic fuzzy number of any type can be smoothly turned into a real number quickly. Using bipolar neutrosophic perception to create an issue is a more accurate, reliable and trusted way than others. They have also considered a multi-criteria decision-making problem (MCDM) for several users in the bipolar neutrosophic area.

Moreover, Haque et al. [35] have investigated a novel scheme to detect the best cloud service provider using logarithmic operational law in the generalized spherical fuzzy environment. Furthermore, for more information about other decision-making techniques, one can refer to [36,37].

Moreover, Xiao [38] introduced the complex evidential distance (CED), which is a strict distance metric with the properties of nonnegativity, nondegeneracy, symmetry and triangle inequality that satisfies the axioms of a distance. For a long time, evidence theory has been an effective methodology for modeling and processing uncertainty that has been widely applied in various fields. In evidence theory, several distance measures have been presented, which play an important role in representing the degree of difference between pieces of evidence. The complex evidential distance (CED) has been considered to be a generalization of the traditional evidential distance.

Furthermore, Alkhazaleh [39] recently noticed that in the fuzzy soft set theory, the final decision in the given decision-making problems depends only on the usual parameters without considering the effect of any other external parameters. To overcome this limitation, he proposed a new concept to represent those external parameters which is the effective parameter set. In addition, he defined the effective fuzzy soft set concept built on the effective set definition. He also established the operations of the effective fuzzy soft sets and studied some of their properties. Moreover, he gave an application of the effective fuzzy soft set in decision-making problems. Finally, he introduced an application of this novel theory to medical diagnosis and exhibited the technique with a hypothetical case study.

Work Motivation

As stated above, we have the fuzzy set, the soft set, the bipolarity, the effective set, and the multi-attribute concepts. If we have all those concepts in one decision-making problem as its circumstances are described, then any one, two, or even more combined sets of those stated above will fail to handle this issue. Then, there is a need to have a new extension that collects all those concepts in one combined set to deal easily with this type of problem.

In other words, the idea of this research comes from a need to generalize the multi-fuzzy soft set, the bipolar-valued fuzzy soft set, and the multipolar fuzzy soft set. This is necessary when we have multi attributes and bipolar attributes together with fuzzy soft information and effective needed parameters. The effective bipolar-valued multi-fuzzy soft set of dimension n is the best one to satisfy this need by combining all needed circumstances in one novel generalized definition. Therefore, in this paper, we define the effective bipolar-valued multi-fuzzy soft sets of dimension n along with their types, properties, operations, and real-life medical applications. The rest of this paper is organized as follows:

Section 2 is nominated to state the needed preliminary definitions and concepts. In Section 3, the effective bipolar-valued multi-fuzzy soft set, its types and some novel associated concepts are inferred. Moreover, their operations, like union and intersection, are presented in Section 4. After that, Section 5 concludes some related properties, for example, commutative properties, absorption properties, associative properties, De Morgan’s laws, and distributive laws.

Moreover, the purpose of Section 6 is to derive a decision-making algorithm based on the effective bipolar-valued multi-fuzzy soft sets. This helps us to conduct the decision that this patient is suffering from this disease or this student is qualified or accepted to take or study at this education level, … The technique steps are introduced using matrixes to make it easier to do computations.

Furthermore, the MATLAB^® program is used to do the addition and multiplication operations of matrixes, to obtain effective sets, or to make any calculations quickly, accurately, and easily. In addition, we give the sensitivity analysis as well as the comparative analysis at the end of Section 6. This detailed comparison between the existing methodologies and the current one is presented to highlight the distinctions between them. The results of this comparison are also summarized in a chart. Finally, Section 7 is set up for concluding remarks and some predicted future works. The structure of the paper content is given by a graphical tree diagram shown below in Figure 1.

2. Preliminaries

The fundamental preliminary definitions, required in the subsequent results, are discussed in this section. These definitions are about the fuzzy set, bipolar-valued fuzzy set, multi-fuzzy set, bipolar-valued multi-fuzzy set, soft set, effective set, and effective fuzzy soft set. One can refer to [1,3,6,24,26,39] to find more detailed results and examples about those above concepts.

Definition 1

((Fuzzy set) [1]). Assume that Ξ is an initial universe. Then, we can define the fuzzy class (set) ϝ over Ξ as a set characterized by a characteristic function ${\eta }_F:\mathsf{\Xi }\to [0,1]$. We can call ${\eta }_F$ the indicator function, or the membership function of the fuzzy set ϝ. In addition, ${\eta }_F\left(u\right)$ is called the degree of membership, or the membership grade value of $\xi \in \mathsf{\Xi }$ in ϝ. We can represent the fuzzy set ϝ over an initial universe Ξ by one of the following two formulas:

$$F=\{({\eta }_F\left(\xi \right)/\xi ):\xi \in \mathsf{\Xi },{\eta }_F\left(\xi \right)\in [0,1]\},or$$

$$F=\{(\xi ,{\eta }_F\left(\xi \right)):\xi \in \mathsf{\Xi },{\eta }_F\left(\xi \right)\in [0,1]\}.$$

Definition 2

((Bipolar-valued fuzzy set) [6]). For the positive characteristic function ${\eta }_{\mathcal{A}}^{+}:\mathsf{\Xi }\to [0,1]$ and the negative characteristic function ${\eta }_{\mathcal{A}}^{-}:\mathsf{\Xi }\to [-1,0]$, the formula

$$\mathcal{A}=\{(\xi ,{\eta }_{\mathcal{A}}^{+}\left(\xi \right),{\eta }_{\mathcal{A}}^{-}\left(\xi \right)):\xi \in \mathsf{\Xi }\}$$

represents the bipolar-valued fuzzy set $\mathcal{A}$ on Ξ. ${\eta }_{\mathcal{A}}^{+}:\mathsf{\Xi }\to [0,1]$ can describe the satisfaction degree of ξ to the property corresponding to $\mathcal{A}$ and ${\eta }_{\mathcal{A}}^{-}:\mathsf{\Xi }\to [-1,0]$ can describe the satisfaction degree of ξ to the counter-property of $\mathcal{A}$.

Definition 3

((Multi-fuzzy set) [24]). A multi-fuzzy set $\mathcal{N}$ of dimension n over Ξ is characterized by a set of ordered sequences in the following structure:

$$\mathcal{N}=\{(\xi ,{\eta }_{1\mathcal{N}}\left(\xi \right),{\eta }_{2\mathcal{N}}\left(\xi \right),\dots ,{\eta }_{n\mathcal{N}}\left(\xi \right)):\xi \in \mathsf{\Xi }\},$$

taking into account that, for $i=1,2,\cdots ,n$, ${\eta }_{i\mathcal{N}}:\mathsf{\Xi }\to [0,1]$ represent the characteristic or the membership functions. We can call the function ${\eta }_{\mathcal{N}}=({\eta }_{1\mathcal{N}},{\eta }_{2\mathcal{N}},\dots ,{\eta }_{n\mathcal{N}})$, the fuzzy multi-membership function of a multi-fuzzy set $\mathcal{N}$ of dimension n.

Definition 4

((Bipolar-valued multi-fuzzy set) [26]). The following formula represents the bipolar-valued multi-fuzzy set $\mathcal{B}$ of dimension n over an initial universe Ξ :

$$\mathcal{B}=\{(\xi ,{\eta }_{1\mathcal{B}}^{+}\left(\xi \right),{\eta }_{2\mathcal{B}}^{+}\left(\xi \right),\dots ,{\eta }_{n\mathcal{B}}^{+}\left(\xi \right),{\eta }_{1\mathcal{B}}^{-}\left(\xi \right),{\eta }_{2\mathcal{B}}^{-}\left(\xi \right),\dots ,{\eta }_{n\mathcal{B}}^{-}\left(\xi \right)):\xi \in \mathsf{\Xi }\},$$

taking into account that, for $i=1,2,\cdots ,n$, ${\eta }_{i\mathcal{B}}^{+}:\mathsf{\Xi }\to [0,1]$ represent the positive characteristic (or membership) functions indicating the satisfaction degrees of ξ to some properties corresponding to $\mathcal{B}$ and ${\eta }_{i\mathcal{B}}^{-}:\mathsf{\Xi }\to [-1,0]$ represent the negative characteristic (or membership) functions indicating the satisfaction degrees of ξ to some implicit counter-properties of $\mathcal{B}$.

Definition 5

((Soft set) [3]). Let Ξ be an initial universe, Υ be a set of parameters (or attributes), and $\mathsf{\Lambda }\subseteq \mathsf{{\rm Y}}$. The power set of Ξ is obtained from $P\left(\mathsf{\Xi }\right)=2^{\mathsf{\Xi }}$. A pair $(\mathsf{\Gamma },\mathsf{\Lambda })$ or ${\mathsf{\Gamma }}_{\mathsf{\Lambda }}$ is called a soft set over Ξ, taking into account that Γ is a mapping represented by $\mathsf{\Gamma }:\mathsf{\Lambda }\to P\left(\mathsf{\Xi }\right)$. In addition, we can formulate ${\mathsf{\Gamma }}_{\mathsf{\Lambda }}$ as a set of ordered pairs ${\mathsf{\Gamma }}_{\mathsf{\Lambda }}=\{(\lambda ,{\mathsf{\Gamma }}_{\mathsf{\Lambda }}\left(\lambda \right)):\lambda \in \mathsf{\Lambda },{\mathsf{\Gamma }}_{\mathsf{\Lambda }}\left(\lambda \right)\in P\left(\mathsf{\Xi }\right)\}$. Λ is said to be the support of ${\mathsf{\Gamma }}_{\mathsf{\Lambda }}$, as well as ${\mathsf{\Gamma }}_{\mathsf{\Lambda }}\left(\lambda \right)\ne \varphi$, for any $\lambda \in \mathsf{\Lambda }$ and ${\mathsf{\Gamma }}_{\mathsf{\Lambda }}\left(\lambda \right)=\varphi$ for any $\lambda \notin \mathsf{\Lambda }$. That is to say that a soft set $(\mathsf{\Gamma },\mathsf{\Lambda })$ over Ξ can be considered to be a parameterized family of subsets of Ξ.

Definition 6

((Effective set) [39]). An effective set is defined as a fuzzy set ℸ over the universal set Δ in which ℸ is given by the mapping $\daleth :\mathsf{\Delta }\to [0,1]$. We can say that Δ is the set of all effective attributes or parameters that can affect the value of membership of every element. It has a positive effect on the membership values of the elements after applying it to them. Note that, in some cases, some membership values remain as is, even after implementation. One can define ℸ by the following formula: $\daleth =\{(\delta ,{\varrho }_{\daleth }\left(\delta \right)),\delta \in \mathsf{\Delta }\}$.

Definition 7

((Effective fuzzy soft set) [39]). For a given initial universal set Ξ, we can indicate the set of all fuzzy subsets of Ξ by $\mathfrak{F}\left(\mathsf{\Xi }\right)$. Suppose that ${\upsilon }_i\in \mathsf{{\rm Y}}$ are the usual parameters, Δ is the effective parameter set and ℸ is the effective set over Δ. Then, we call the pair $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ an effective fuzzy soft set over Ξ, taking into account that the mapping $\mathsf{\Psi }:\mathsf{\Xi }\to \mathfrak{F}\left(\mathsf{\Xi }\right)$ is given by the following formula: $\mathsf{\Psi }{\left({\delta }_i\right)}_{\daleth }=\{({\xi }_j,{\eta }_{\mathsf{\Psi }}{\left({\xi }_j\right)}_{\daleth }),{\xi }_j\in \mathsf{\Xi },{\delta }_i\in \mathsf{\Delta }\}$, where, for all ${\delta }_k\in \mathsf{\Delta }$, we have:

$$\begin{array}{cc}\hfill {\eta }_{\mathsf{\Psi }}{({\xi }_j,{\upsilon }_i)}_{\daleth }& =\left\{\begin{array}{c}{\eta }_{\mathsf{\Psi }}({\xi }_j,{\upsilon }_i)+\frac{(1-{\eta }_{\mathsf{\Psi }}({\xi }_j,{\upsilon }_i)){\sum }_k{\varrho }_{{\daleth }_{{\xi }_j}}\left({\delta }_k\right)}{\left|\mathsf{\Delta }\right|},if{\eta }_{\mathsf{\Psi }}({\xi }_j,{\upsilon }_i)\in (0,1),\hfill \\ \\ {\eta }_{\mathsf{\Psi }}\left({\xi }_j\right),otherwise.\hfill \end{array}\right.\hfill \end{array}$$

(1)

where $\left|\mathsf{\Delta }\right|$ is the number of elements in the given effective parameter set Δ ${\eta }_{\mathsf{\Psi }}({\xi }_j,{\upsilon }_i)$ is the membership degree value of the item ${\xi }_j$ for the parameter ${\upsilon }_i$ and ${\sum }_k{\varrho }_{{\daleth }_{{\xi }_j}}\left({\delta }_k\right)$ is the summation of all effective parameters values of ${\xi }_j$.

Example 1.

If we have an initial universal set $\mathsf{\Xi }=\{{\xi }_1,{\xi }_2,{\xi }_3\}$ and the set of parameters $\mathsf{{\rm Y}}=\{{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\}$. Let the fuzzy soft set for the parameter ${\upsilon }_1$ be

$$(\mathsf{\Psi },\mathsf{{\rm Y}})\left({\upsilon }_1\right)=\{({\xi }_1,0.3),({\xi }_2,0.7),({\xi }_3,0.5)\}.$$

Then, to compute the effective membership value for the first item ${\xi }_1$ with a membership value $0.3$ for the first parameter ${\upsilon }_1$ and the following given effective set ℸ for ${\xi }_1$:

$$\daleth \left({\xi }_1\right)=\{({\delta }_1,0.8),({\delta }_2,1),({\delta }_3,0),({\delta }_4,0.2)\},$$

where ${\delta }_1$, ${\delta }_2$, ${\delta }_3$ and ${\delta }_4$ are the given effective parameters, we do the following computation according to Formula (1) from Definition 7:

$${\eta }_{\mathsf{\Psi }}{({\xi }_1,{\upsilon }_1)}_{\daleth }=0.3+\frac{(1-0.3)[0.8+1+0+0.2]}{4}=0.3+\frac{0.7\times 2}{4}=0.3+0.35=0.65.$$

Similarly, we can calculate the effective membership values for the remaining membership values of ${\xi }_1$ for the last two parameters ${\upsilon }_2$ and ${\upsilon }_3$ and also for the other two items ${\xi }_2$ and ${\xi }_3$. The reader can refer to [39] page 3 to find the full illustrative example to understand this definition well.

Remark 1.

For simplicity, instead of writing the full complex Formula (1) from Definition 7, one can write the effective membership value ${\eta }_{\mathsf{\Psi }}{({\xi }_j,{\upsilon }_i)}_{\daleth }$ corresponding to the membership value ${\eta }_{\mathsf{\Psi }}({\xi }_j,{\upsilon }_i)$ of a specific item ${\xi }_j$ for a certain parameter ${\upsilon }_i$ as ${\eta }_{\daleth }$, when we know that Ψ is the only fuzzy soft set we talking about. In case we have two fuzzy soft sets ${\mathsf{\Psi }}_1$ and ${\mathsf{\Psi }}_2$ or more, we must write the full formulas ${\eta }_{{\mathsf{\Psi }}_1}({\xi }_j,{\upsilon }_i)$ and ${\eta }_{{\mathsf{\Psi }}_2}({\xi }_j,{\upsilon }_i)$, respectively, to distinguish between them.

3. Effective Bipolar-Valued Multi-Fuzzy Soft Sets

The major goal of the current section is to formulate the definition of the effective bipolar-valued fuzzy soft set and the effective multi-fuzzy soft set. Furthermore, the definition of the effective bipolar-valued multi-fuzzy soft set of dimension n is derived and reflected by an illustrative example. In addition, their kinds and some associated concepts are conducted.

Definition 8

(Effective bipolar-valued fuzzy soft set). For a given initial universal set Ξ, we can indicate the set of all bipolar-valued fuzzy subsets of Ξ by $\mathfrak{BF}\left(\mathsf{\Xi }\right)$. Suppose that Υ is the parameter set, Δ is the effective parameter set, and ℸ is the effective set over Δ. Then, we call the pair $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ an effective bipolar-valued fuzzy soft set over Ξ, taking into account that the mapping $\mathsf{\Psi }:\mathsf{\Xi }\to \mathfrak{BF}\left(\mathsf{\Xi }\right)$ is given by the following formula:

$$\mathsf{\Psi }{\left({\delta }_i\right)}_{\daleth }=\{({\xi }_j,{\eta }_{\mathsf{\Psi }}^{+}{\left({\xi }_j\right)}_{\daleth },{\eta }_{\mathsf{\Psi }}^{-}{\left({\xi }_j\right)}_{\daleth }),{\xi }_j\in \mathsf{\Psi },{\delta }_i\in \mathsf{\Delta }\},$$

where, for all ${\delta }_k\in \mathsf{\Delta }$, we have the positive and negative effective membership values ${\eta }_{\daleth }^{+}$ and ${\eta }_{\daleth }^{-}$ corresponding to the positive and negative membership values ${\eta }^{+}\in (0,1)$ and ${\eta }^{-}\in (-1,0)$ of the item ${\xi }_j$ for the parameter ${\upsilon }_l$ given, respectively, by the following two formulas:

$$\begin{array}{cc}\hfill {\eta }_{\daleth }^{+}& ={\eta }^{+}+\frac{(1-{\eta }^{+}){\sum }_k{\varrho }_{\daleth }\left({\delta }_k\right)}{\left|\mathsf{\Delta }\right|},\hfill \end{array}$$

(2)

and

$$\begin{array}{cc}\hfill {\eta }_{\daleth }^{-}& ={\eta }^{-}+\frac{(-1-{\eta }^{+}){\sum }_k{\varrho }_{\daleth }\left({\delta }_k\right)}{\left|\mathsf{\Delta }\right|},\hfill \end{array}$$

(3)

where $\left|\mathsf{\Delta }\right|$ is the number of elements in the given effective parameter set Δ.

In case that ${\eta }^{+}=0or1$, then ${\eta }_{\daleth }^{+}={\eta }^{+}$. Similarly, if ${\eta }^{-}=0or-1$, then ${\eta }_{\daleth }^{-}={\eta }^{-}$.

Remark 2.

Formulas (2) and (3), stated in Definition 8, can be combined into one formula as follows:

$$\begin{array}{cc}\hfill {\eta }_{\daleth }& =\left\{\begin{array}{c}\eta +\frac{(1-\eta ){\sum }_k{\varrho }_{{\daleth }_{{\xi }_j}}\left({\delta }_k\right)}{\left|\mathsf{\Delta }\right|},if\eta \in (0,1),\hfill \\ \\ \eta +\frac{(-1-\eta ){\sum }_k{\varrho }_{{\daleth }_{{\xi }_j}}\left({\delta }_k\right)}{\left|\mathsf{\Delta }\right|},if\eta \in (-1,0),\hfill \\ \\ \eta ,otherwise.\hfill \end{array}\right.\hfill \end{array}$$

(4)

regarding that

$$\begin{array}{cc}\hfill {\eta }_{\daleth }& =\left\{\begin{array}{c}{\eta }_{\daleth }^{+},if\eta \in [0,1],\hfill \\ \\ {\eta }_{\daleth }^{-},if\eta \in [-1,0],\hfill \end{array}\right.\hfill \end{array}$$

where $\left|\mathsf{\Delta }\right|$ is the number of elements in the given effective parameter set Δ.

Definition 9

(Effective multi-fuzzy soft set of order n). For a given initial universal set Ξ, we can indicate the set of all multi-fuzzy subsets of order n on Ξ by $\mathfrak{MF}\left(\mathsf{\Xi }\right)$. Suppose that Υ is the parameter set, Δ is the effective parameter set, and ℸ is the effective set over Δ. Then, we call the pair $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ an effective multi-fuzzy soft set of order n over Ξ, taking into account that the mapping $\mathsf{\Psi }:\mathsf{\Xi }\to \mathfrak{MF}\left(\mathsf{\Xi }\right)$ is given by the following formula:

$$\mathsf{\Psi }{\left({\delta }_i\right)}_{\daleth }=\{({\xi }_j,{\eta }_{1\mathsf{\Psi }}{\left({\xi }_j\right)}_{\daleth },{\eta }_{2\mathsf{\Psi }}{\left({\xi }_j\right)}_{\daleth },\cdots ,{\eta }_{n\mathsf{\Psi }}{\left({\xi }_j\right)}_{\daleth }),{\xi }_j\in \mathsf{\Xi },{\delta }_i\in \mathsf{\Delta }\},$$

where, for all ${\delta }_k\in \mathsf{\Delta }$, we have:

$$\begin{array}{cc}\hfill {\eta }_{r\daleth }& =\left\{\begin{array}{c}{\eta }_r+\frac{(1-{\eta }_r){\sum }_k{\varrho }_{{\daleth }_{{\xi }_j}}\left({\delta }_k\right)}{\left|\mathsf{\Delta }\right|},if{\eta }_r\in (0,1),\hfill \\ \\ {\eta }_r,otherwise.\hfill \end{array}\right.\hfill \end{array}$$

(5)

regarding that $r:1,2,\cdots ,n$ and $\left|\mathsf{\Delta }\right|$ is the number of elements in the given effective parameter set Δ.

Definition 10

(Effective bipolar-valued multi-fuzzy soft set of order n). For a given initial universal set Ξ, we can indicate the set of all bipolar-valued multi-fuzzy subsets of order n on Ξ by $\mathfrak{BMF}\left(\mathsf{\Xi }\right)$. Suppose that Υ is the parameter set, Δ is the effective parameter set, and ℸ is the effective set over Δ. Then, we call the pair $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ an effective bipolar-valued multi-fuzzy soft set of order n over Ξ, taking into account that the mapping $\mathsf{\Psi }:\mathsf{\Xi }\to \mathfrak{BMF}\left(\mathsf{\Xi }\right)$ is given by the following formula:

$$\begin{array}{cc}\hfill \mathsf{\Psi }{\left({\delta }_i\right)}_{\daleth }& =\{({\xi }_j,{\eta }_{1\mathsf{\Psi }}^{+}{\left({\xi }_j\right)}_{\daleth },{\eta }_{2\mathsf{\Psi }}^{+}{\left({\xi }_j\right)}_{\daleth },\cdots ,{\eta }_{n\mathsf{\Psi }}^{+}{\left({\xi }_j\right)}_{\daleth },{\eta }_{1\mathsf{\Psi }}^{-}{\left({\xi }_j\right)}_{\daleth },{\eta }_{2\mathsf{\Psi }}^{-}{\left({\xi }_j\right)}_{\daleth },\cdots ,{\eta }_{n\mathsf{\Psi }}^{-}{\left({\xi }_j\right)}_{\daleth }),\hfill \\ & {\xi }_j\in \mathsf{\Xi },{\delta }_i\in \mathsf{\Delta }\},\hfill \end{array}$$

where, for all ${\delta }_k\in \mathsf{\Delta }$, we have:

$$\begin{array}{cc}\hfill {\eta }_{r\daleth }& =\left\{\begin{array}{c}{\eta }_r+\frac{(1-{\eta }_r){\sum }_k{\varrho }_{{\daleth }_{{\xi }_j}}\left({\delta }_k\right)}{\left|\mathsf{\Delta }\right|},if{\eta }_r\in (0,1),\hfill \\ \\ {\eta }_r+\frac{(-1-{\eta }_r){\sum }_k{\varrho }_{{\daleth }_{{\xi }_j}}\left({\delta }_k\right)}{\left|\mathsf{\Delta }\right|},if{\eta }_r\in (-1,0),\hfill \\ \\ {\eta }_r,otherwise,\hfill \end{array}\right.\hfill \end{array}$$

(6)

where

$$\begin{array}{cc}\hfill {\eta }_{r\daleth }& =\left\{\begin{array}{c}{\eta }_{r\daleth }^{+},if{\eta }_r\in [0,1],\hfill \\ \\ {\eta }_{r\daleth }^{-},if{\eta }_r\in [-1,0],\hfill \end{array}\right.\hfill \end{array}$$

taking into account that $r:1,2,\cdots ,n$ and $\left|\mathsf{\Delta }\right|$ are the number of elements in the given effective parameter set Δ.

Example 2.

Consider a universal set of houses $\mathsf{\Xi }=\{{\xi }_1,{\xi }_2,{\xi }_3\}$ which are considered to be bought. Let the two major sets of parameters (attributes) that describe their features be $\mathsf{{\rm Y}}=\{{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\}$, ${\upsilon }_i(i=1,2,3)$ and its opposite set $-\mathsf{{\rm Y}}=\{-{\upsilon }_1,-{\upsilon }_2,-{\upsilon }_3\}$, $-{\upsilon }_i(i=1,2,3)$ stand for the features and the opposite-features, respectively. These features can be classified into the following three main types of parameters: Location affairs, financial affairs, and design affairs, respectively. Location affairs and their opposite features are as follows: (“near the main road” and “far from the main road”), (“close to the city center” and “far from the city center”), (“in a green surrounding” and “in an industrial surrounding”). Financial affairs and their opposite features are as follows: (“expensive” and “cheap”), (“cash payment” and “payment facilities”), (“long-term installment” and “short-term installment”). Design affairs and their opposite features are as follows: (“large house” and “small house”), (“within stacked apartments” and “within unstacked apartments”), (“luxurious design” and “poor design”). In addition, suppose that $\mathsf{\Delta }=\{{\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4\}$ is a set of effective attributes, where ${\delta }_1=$ the house has not been licensed yet, ${\delta }_2=$ there have been people living in this house before, ${\delta }_3=$ there is no an elevator and ${\delta }_4=$ there is a broker fee. Let the effective set over Δ for ${\xi }_i,i=1,2,3$, be as follows according to experts’ evaluation:

$$\daleth \left({\xi }_1\right)=\{({\delta }_1,0.7),({\delta }_2,0.2),({\delta }_3,0.5),({\delta }_4,0.4)\},$$

$$\daleth \left({\xi }_2\right)=\{({\delta }_1,0.5),({\delta }_2,0.1),({\delta }_3,0),({\delta }_4,0.8)\},$$

$$\daleth \left({\xi }_3\right)=\{({\delta }_1,1),({\delta }_2,0.6),({\delta }_3,0.3),({\delta }_4,0.9)\}.$$

Furthermore, the attractiveness of the given houses according to the purchaser’s preferences can be described by a bipolar-valued multi-fuzzy soft set $(\mathsf{\Psi },\mathsf{{\rm Y}})$ of order 3 over a universal set Ξ as follows:

$$\begin{array}{cc}\hfill (\mathsf{\Psi },\mathsf{{\rm Y}})& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.4,0.8,0.6,-0.3,-0.9,-0.2),({\xi }_2,0.1,0.5,0.3,-0.7,-0.6,-0.4),\hfill \\ & ({\xi }_3,0.9,0.4,0.5,-0.7,-0.2,-0.3)\left\}\right),({\upsilon }_2,\{({\xi }_1,0.6,0.3,1,-0.7,-0.5,-0.3),\hfill \\ & ({\xi }_2,0.5,1,0.9,-0.2,-1,-0.4),({\xi }_3,0.2,0.4,0.6,-0.3,-0.8,-0.1)\left\}\right),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,1,0.6,0.3,-0.2,-0.5,-0.4),({\xi }_2,0.7,0.9,0.1,-0.9,-1,-0.3),\hfill \\ & ({\xi }_3,0.1,0.7,0.2,-0.2,-0.5,-0.8)\left\}\right)\}.\hfill \end{array}$$

Then, using Formula (6) from Definition 10, we have the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ of order 3 over a universal set Ξ that describes the attractiveness of the of above houses, effectively, as the following:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.67,0.89,0.78,-0.61,-0.94,-0.56),\hfill \\ & ({\xi }_2,0.41,0.67,0.54,-0.8,-0.74,-0.61),\hfill \\ & ({\xi }_3,0.97,0.82,0.85,-0.91,-0.76,-0.79)\left\}\right),\hfill \\ & ({\upsilon }_2,\{({\xi }_1,0.78,0.61,1,-0.83,-0.72,-0.61),\hfill \\ & ({\xi }_2,0.67,1,0.93,-0.48,-1,-0.61),\hfill \\ & ({\xi }_3,0.76,0.82,0.88,-0.79,-0.94,-0.73)\left\}\right),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,1,0.78,0.61,-0.56,-0.72,-0.67),\hfill \\ & ({\xi }_2,0.8,0.93,0.41,-0.93,-1,-0.54),\hfill \\ & ({\xi }_3,0.73,0.91,0.76,-0.76,-0.85,-0.94)\left\}\right)\}.\hfill \end{array}$$

(7)

$({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ description can help the purchaser decide which house is the best choice for him/her. This decision-making technique comes from extracting the matrix corresponding to every positive pole and every negative pole of the effective bipolar-valued multi-fuzzy soft set that contains the membership values of the given items. After that, by doing some matrix operations like multiplication and addition, one can easily obtain the final decision from the final resulting matrix.

Remark 3.

The above effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ of order 3 (7) from Example 2 can be represented in a matrix form to be easy to deal with. It can be divided into two matrixes; one represents the positive poles, say $A^{+}$ and the other represents the negative poles, say $A^{-}$, as follows:

$$A^{+}=\begin{array}{cc}& \begin{array}{ccccccccc}{\upsilon }_1^{\prime }\hfill & {\upsilon }_2^{\prime }& {\upsilon }_3^{\prime }& {\upsilon }_1^{″}& {\upsilon }_2^{″}& {\upsilon }_3^{″}& {\upsilon }_1^{‴}& {\upsilon }_2^{‴}& {\upsilon }_3^{‴}\end{array}\\ \begin{array}{c}{\xi }_1\\ {\xi }_2\\ {\xi }_3\end{array}& \left(\begin{array}{ccccccccc}0.67& 0.78& 1& 0.89& 0.61& 0.78& 0.78& 1& 0.61\\ 0.41& 0.67& 0.8& 0.67& 1& 0.93& 0.54& 0.93& 0.41\\ 0.97& 0.76& 0.73& 0.82& 0.82& 0.91& 0.85& 0.88& 0.76\end{array}\right)\end{array},$$

$$A^{-}=\begin{array}{cc}& \begin{array}{ccccccccc}{\upsilon }_1^{\prime }\hfill & {\upsilon }_2^{\prime }& {\upsilon }_3^{\prime }& {\upsilon }_1^{″}& {\upsilon }_2^{″}& {\upsilon }_3^{″}& {\upsilon }_1^{‴}& {\upsilon }_2^{‴}& {\upsilon }_3^{‴}\end{array}\\ \begin{array}{c}{\xi }_1\\ {\xi }_2\\ {\xi }_3\end{array}& \left(\begin{array}{ccccccccc}\mathrm{-0.61}& \mathrm{-0.83}& \mathrm{-0.56}& \mathrm{-0.94}& \mathrm{-0.72}& \mathrm{-0.72}& \mathrm{-0.56}& \mathrm{-0.61}& \mathrm{-0.67}\\ \mathrm{-0.8}& \mathrm{-0.48}& \mathrm{-0.93}& \mathrm{-0.74}& \mathrm{-1}\hfill & \mathrm{-1}\hfill & \mathrm{-0.61}& \mathrm{-0.61}& \mathrm{-0.54}\\ \mathrm{-0.91}& \mathrm{-0.79}& \mathrm{-0.76}& \mathrm{-0.76}& \mathrm{-0.94}& \mathrm{-0.85}& \mathrm{-0.79}& \mathrm{-0.73}& \mathrm{-0.94}\end{array}\right)\end{array}.$$

We can call $(A^{+},A^{-})$ the effective bipolar-valued multi-fuzzy soft matrix corresponding to the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$.

Definition 11

(Complete effective bipolar-valued multi-fuzzy soft set). Assume that Ξ is an initial universe. Suppose that Υ is a parameter set. Then, any effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ of dimension n on an initial universe Ξ, constructed by an effective set ℸ, is called absolute (or complete), stand for $({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})$, if for all $\upsilon \in \mathsf{{\rm Y}}$, we have ${\mathsf{\Psi }}_{\mathsf{{\rm Y}}}{\left(\upsilon \right)}_{\daleth }=\mathfrak{BMF}\left(\mathsf{\Xi }\right)$. That is to say that, for $i=1,2,\dots ,n$, we have ${\eta }_{i{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth }=1$ and ${\eta }_{i{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }=-1$, for all $\upsilon \in \mathsf{{\rm Y}}$ and for all $\xi \in \mathsf{\Xi }$. i.e.,

$$({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})=\{(\upsilon ,\left\{(\xi ,1,\stackrel{n-times}{\dots },1,-1,\stackrel{n-times}{\dots },-1)\right\}):\upsilon \in \mathsf{{\rm Y}},\xi \in \mathsf{\Xi }\}.$$

Definition 12

(Null effective bipolar-valued multi-fuzzy soft set). Given that, Ξ is an initial universe. Assume that Υ is a parameter set. Then, any effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ of dimension n on an initial universe Ξ, constructed by an effective set ℸ, is called empty (or null), stand for $({\varphi }_{\daleth },\mathsf{{\rm Y}})$, if for all $\upsilon \in \mathsf{{\rm Y}}$, we have ${\mathsf{\Psi }}_{\mathsf{{\rm Y}}}{\left(\upsilon \right)}_{\daleth }=\varphi$. That is to say that, for $i=1,2,\dots ,n$, we have ${\eta }_{i{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth }=0$ and ${\eta }_{i{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }=0$, for all $\upsilon \in \mathsf{{\rm Y}}$ and for all $\xi \in \mathsf{\Xi }$. i.e.,

$$({\varphi }_{\daleth },\mathsf{{\rm Y}})=\{(\upsilon ,\left\{(\xi ,0,\stackrel{2n-times}{\dots },0)\right\}):\upsilon \in \mathsf{{\rm Y}},\xi \in \mathsf{\Xi }\}.$$

4. Operations on Effective Bipolar-Valued Multi-Fuzzy Soft Sets

The basic objective of this section is to propose the operations on effective bipolar-valued multi-fuzzy soft sets. Operations like the union, the intersection, the complement, the subset, and many more are established. Furthermore, an example for each operation is given to illustrate how this operation can be.

Definition 13

(Union of two effective bipolar-valued multi-fuzzy soft sets). Assume that Ξ is an initial universe. Suppose that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets. Let ${\daleth }_1$ and ${\daleth }_2$ be two effective parameter sets over Δ. Then, the operation of the union of two effective bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_{1{\daleth }_1},{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_{2{\daleth }_2},{\mathsf{{\rm Y}}}_2)$ of dimension n on an initial universe Ξ is defined as a new effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{{\daleth }^U}^U,{\mathsf{{\rm Y}}}^U)$ of dimension n, where ${\daleth }^U:\mathsf{\Delta }\to [0,1]$ is a mapping characterized by ${\daleth }^U={\daleth }_1\tilde{\cup }{\daleth }_2$, as well as, ${(\mathsf{\Psi },\mathsf{{\rm Y}})}^U=({\mathsf{\Psi }}^U,{\mathsf{{\rm Y}}}^U)=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, taking into account that ${\mathsf{{\rm Y}}}^U={\mathsf{{\rm Y}}}_1\cup {\mathsf{{\rm Y}}}_2$.

The two formulas that compute ${\daleth }^U={\daleth }_1\tilde{\cup }{\daleth }_2$ and $({\mathsf{\Psi }}^U,{\mathsf{{\rm Y}}}^U)=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, respectively, can be determined, for each $\xi \in \mathsf{\Xi }$, as follows:

$$\begin{array}{cc}\hfill {\varrho }_{{\daleth }_{\xi }^U}\left(\delta \right)& =\left\{\begin{array}{c}{\varrho }_{{\daleth }_{1\xi }}\left(\delta \right),ife\in {\daleth }_1-{\daleth }_2,\hfill \\ {\varrho }_{{\daleth }_{2\xi }}\left(\delta \right),ife\in {\daleth }_2-{\daleth }_1,\hfill \\ max\{{\varrho }_{{\daleth }_{1\xi }}\left(\delta \right),{\varrho }_{{\daleth }_{2\xi }}\left(\delta \right)\},ife\in {\daleth }_1\cap {\daleth }_2,\hfill \end{array}\right.\hfill \end{array}$$

(8)

for each $\delta \in \mathsf{\Delta }$ and

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}^U,{\mathsf{{\rm Y}}}^U)& =\left\{\begin{array}{c}=\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ {\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}),\xi \in \mathsf{\Xi }\},if\upsilon \in {\mathsf{{\rm Y}}}_1-{\mathsf{{\rm Y}}}_2,\hfill \\ =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ {\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}),\xi \in \mathsf{\Xi }\},if\upsilon \in {\mathsf{{\rm Y}}}_2-{\mathsf{{\rm Y}}}_1,\hfill \\ =\left\{\right(\upsilon ,\left\{\right(\xi ,max{\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },max{\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },\dots ,\hfill \\ {\max \{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },min{\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth },\hfill \\ min{\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth },\dots ,min{\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth }\left)\right\}),\xi \in \mathsf{\Xi }\},\hfill \\ if\upsilon \in {\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2,\hfill \end{array}\right.\hfill \end{array}$$

(9)

for each $\upsilon \in {\mathsf{{\rm Y}}}^U$.

Example 3.

Under assumptions of Example 2, one can define two effective sets ${\daleth }_1$ and ${\daleth }_2$ over $\mathsf{\Delta }=\{{\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4\}$, for $h_1$ and $h_2$, as follows:

$${\daleth }_1\left({\xi }_1\right)=\{({\delta }_1,0.35),({\delta }_2,0),({\delta }_3,0.91),({\delta }_4,0.46)\},$$

$${\daleth }_1\left({\xi }_2\right)=\{({\delta }_1,0.75),({\delta }_2,0.52),({\delta }_3,1),({\delta }_4,0.29)\},$$

$${\daleth }_2\left({\xi }_1\right)=\{({\delta }_1,0.62),({\delta }_2,0.13),({\delta }_3,0.22),({\delta }_4,0.38)\},$$

$${\daleth }_2\left({\xi }_2\right)=\{({\delta }_1,0.57),({\delta }_2,0.88),({\delta }_3,0),({\delta }_4,1)\},$$

respectively, associated with the following two bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, each of order 3, over a universal set Ξ:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.27,0,0.11,-0.8,-0.55,-0.4),({\xi }_2,0.62,0.2,0.47,-0.19,-1,-0.72),\hfill \\ & ({\upsilon }_2,\{({\xi }_1,0.76,1,0.5,-0.67,-0.45,-0.33),({\xi }_2,0.15,1,0.29,0,-0.7,-0.44),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,0,0.36,0.3,-0.24,-0.85,-1),({\xi }_2,0.97,0.19,0.1,-0.69,0,-0.1)\})\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.34,0,0.1,-0.51,-0.15,-0.94),({\xi }_2,0.23,0.76,0.73,-0.09,-0.35,0),\hfill \\ & ({\upsilon }_2,\{({\xi }_1,0.9,0.74,0.6,-0.34,-0.54,-0.3),({\xi }_2,0.5,0.65,0.32,-0.2,-0.9,-0.4),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,0.3,0.6,0.53,-0.28,-0.75,0),({\xi }_2,0.77,0.11,0.51,-0.43,-1,-0.66)\})\}.\hfill \end{array}$$

Then, compute the union of the two effective sets, namely ${\daleth }^U={\daleth }_1\tilde{\cup }{\daleth }_2$, applying Formula (8) from Definition 13, as follows:

$${\daleth }^U\left({\xi }_1\right)=\{({\delta }_1,0.62),({\delta }_2,0.13),({\delta }_3,0.91),({\delta }_4,0.46)\},$$

$${\daleth }^U\left({\xi }_2\right)=\{({\delta }_1,0.75),({\delta }_2,0.88),({\delta }_3,1),({\delta }_4,1)\}.$$

In addition, compute the union of the two bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$ of order 3, namely ${(\mathsf{\Psi },\mathsf{{\rm Y}})}^U=({\mathsf{\Psi }}^U,{\mathsf{{\rm Y}}}^U)=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, where ${\mathsf{{\rm Y}}}^U={\mathsf{{\rm Y}}}_1\cup {\mathsf{{\rm Y}}}_2$, applying Formula (9) from Definition 13, as follows:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}^U,{\mathsf{{\rm Y}}}^U)& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.34,0,0.11,-0.8,-0.55,-0.94),({\xi }_2,0.62,0.76,0.73,-0.19,-1,0),\hfill \\ & ({\upsilon }_2,\{({\xi }_1,0.9,1,0.6,-0.67,-0.54,-0.33),({\xi }_2,0.5,1,0.32,-0.2,-0.9,-0.44),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,0.3,0.6,0.53,-0.28,-0.85,-1),({\xi }_2,0.97,0.19,0.51,-0.69,-1,-0.66)\})\}.\hfill \end{array}$$

Finally, computing effective union of bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_{{\daleth }^U}^U,{\mathsf{{\rm Y}}}^U)$ of order 3, using Formula (6) from Definition 10, results the following:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{{\daleth }^U}^U,{\mathsf{{\rm Y}}}^U)& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.6898,0,0.5817,-0.906,-0.7885,-0.9718),\hfill \\ & ({\xi }_2,0.96485,0.9778,0.975025,-0.925075,-1,0),\hfill \\ & ({\upsilon }_2,\{({\xi }_1,0.953,1,0.812,-0.8449,-0.7838,-0.6851),\hfill \\ & ({\xi }_2,0.95375,1,0.9371,-0.926,-0.99075,-0.9482),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,0.671,0.812,0.7791,-0.6616,-0.9295,-1),\hfill \\ & ({\xi }_2,0.997225,0.925075,0.954675,-0.971325,-1,-0.96855)\left\}\right)\}.\hfill \end{array}$$

Definition 14

(Restricted union of two effective bipolar-valued multi-fuzzy soft sets). Assume that Ξ is an initial universe. Suppose that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets. Let ${\daleth }_1$ and ${\daleth }_2$ be two effective parameter sets over Δ. Then, the restricted union of two effective bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_{1{\daleth }_1},{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_{2{\daleth }_2},{\mathsf{{\rm Y}}}_2)$ of dimension n on an initial universe Ξ is defined as a new effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{{\daleth }^{U_R}}^{U_R},{\mathsf{{\rm Y}}}^{U_R})$ of dimension n, where ${\daleth }^{U_R}:\mathsf{\Delta }\to [0,1]$ is a mapping characterized by ${\daleth }^{U_R}={\daleth }_1\widetilde{{\cup }_R}{\daleth }_2$, as well as, ${(\mathsf{\Psi },\mathsf{{\rm Y}})}^{U_R}=({\mathsf{\Psi }}^{U_R},{\mathsf{{\rm Y}}}^{U_R})=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\widetilde{{\cup }_R}({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, taking into account that ${\mathsf{{\rm Y}}}^{U_R}={\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2\ne \varphi$ and ${\daleth }_1\cap {\daleth }_2\ne \varphi$. The two formulas that compute ${\daleth }^{U_R}={\daleth }_1\widetilde{{\cup }_R}{\daleth }_2$ and $({\mathsf{\Psi }}^{U_R},{\mathsf{{\rm Y}}}^{U_R})=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\widetilde{{\cup }_R}({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, respectively, can be determined, for each $\xi \in \mathsf{\Xi }$, as ${\varrho }_{{\daleth }_{\xi }^{U_R}}\left(\delta \right)=max\{{\varrho }_{{\daleth }_{1\xi }}\left(\delta \right),{\varrho }_{{\daleth }_{2\xi }}\left(\delta \right)\}$, for each $\delta \in \mathsf{\Delta }$ and

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}^{U_R},{\mathsf{{\rm Y}}}^{U_R})& =\left\{\right(\upsilon ,\left\{\right(\xi ,max{\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },max{\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },\dots ,\hfill \\ & {\max \{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },min{\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth },\hfill \\ & {\min \{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth },\dots ,min{\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth }\left)\right\}),\xi \in \mathsf{\Xi }\},\hfill \end{array}$$

for each $\upsilon \in {\mathsf{{\rm Y}}}^{U_R}$.

Definition 15

(Intersection of two effective bipolar-valued multi-fuzzy soft sets). Assume that Ξ is an initial universe. Suppose that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets. Let ${\daleth }_1$ and ${\daleth }_2$ be two effective parameter sets over Δ. Then, the operation of the intersection of two effective bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_{1{\daleth }_1},{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_{2{\daleth }_2},{\mathsf{{\rm Y}}}_2)$ of dimension n on an initial universe Ξ is defined as a new effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{{\daleth }^I}^I,{\mathsf{{\rm Y}}}^I)$ of dimension n, where ${\daleth }^I:\mathsf{\Delta }\to [0,1]$ is a mapping characterized by ${\daleth }^I={\daleth }_1\tilde{\cap }{\daleth }_2$, as well as, ${(\mathsf{\Psi },\mathsf{{\rm Y}})}^I=({\mathsf{\Psi }}^I,{\mathsf{{\rm Y}}}^I)=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\tilde{\cap }({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, taking into account that ${\mathsf{{\rm Y}}}^I={\mathsf{{\rm Y}}}_1\cup {\mathsf{{\rm Y}}}_2$.

The two formulas that compute ${\daleth }^I={\daleth }_1\tilde{\cap }{\daleth }_2$ and $({\mathsf{\Psi }}^I,{\mathsf{{\rm Y}}}^I)=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\tilde{\cap }({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, respectively, can be determined, for each $\xi \in \mathsf{\Xi }$, as follows:

$$\begin{array}{cc}\hfill {\varrho }_{{\daleth }_{\xi }^I}\left(\delta \right)& =\left\{\begin{array}{c}{\varrho }_{{\daleth }_{1\xi }}\left(\delta \right),ife\in {\daleth }_1-{\daleth }_2,\hfill \\ {\varrho }_{{\daleth }_{2\xi }}\left(\delta \right),ife\in {\daleth }_2-{\daleth }_1,\hfill \\ min\{{\varrho }_{{\daleth }_{1\xi }}\left(\delta \right),{\varrho }_{{\daleth }_{2\xi }}\left(\delta \right)\},ife\in {\daleth }_1\cap {\daleth }_2,\hfill \end{array}\right.\hfill \end{array}$$

(10)

for each $\delta \in \mathsf{\Delta }$ and

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}^I,{\mathsf{{\rm Y}}}^I)& =\left\{\begin{array}{c}=\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ {\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}),\xi \in \mathsf{\Xi }\},if\upsilon \in {\mathsf{{\rm Y}}}_1-{\mathsf{{\rm Y}}}_2,\hfill \\ =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ {\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}),\xi \in \mathsf{\Xi }\},if\upsilon \in {\mathsf{{\rm Y}}}_2-{\mathsf{{\rm Y}}}_1,\hfill \\ =\left\{\right(\upsilon ,\left\{\right(\xi ,min{\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },min{\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },\dots ,\hfill \\ min{\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },max{\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth },\hfill \\ max{\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth },\dots ,max{\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth }\left)\right\}),\xi \in \mathsf{\Xi }\},\hfill \\ if\upsilon \in {\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2,\hfill \end{array}\right.\hfill \end{array}$$

(11)

for each $\upsilon \in {\mathsf{{\rm Y}}}^I$.

Example 4.

Compute the intersection of the two effective sets stated in Example 3, say ${\daleth }^I={\daleth }_1\tilde{\cap }{\daleth }_2$, applying Formula (10) from Definition 15, as the following:

$${\daleth }^I\left({\xi }_1\right)=\{({\delta }_1,0.35),({\delta }_2,0),({\delta }_3,0.22),({\delta }_4,0.38)\},$$

$${\daleth }^I\left({\xi }_2\right)=\{({\delta }_1,0.57),({\delta }_2,0.52),({\delta }_3,0),({\delta }_4,0.29)\}.$$

Also, compute the intersection of the two bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$ of order 3 stated in Example 3, say ${(\mathsf{\Psi },\mathsf{{\rm Y}})}^I=({\mathsf{\Psi }}^I,{\mathsf{{\rm Y}}}^I)=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\tilde{\cap }({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, where ${\mathsf{{\rm Y}}}^I={\mathsf{{\rm Y}}}_1\cup {\mathsf{{\rm Y}}}_2$, applying Formula (11) from Definition 15, as follows:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}^I,{\mathsf{{\rm Y}}}^I)& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.27,0,0.1,-0.51,-0.15,-0.4),({\xi }_2,0.23,0.2,0.47,-0.09,-0.35,0),\hfill \\ & ({\upsilon }_2,\{({\xi }_1,0.76,0.74,0.5,-0.34,-0.45,-0.3),({\xi }_2,0.15,0.65,0.29,0,-0.7,-0.4),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,0,0.36,0.3,-0.24,-0.75,0),({\xi }_2,0.77,0.11,0.1,-0.43,0,-0.1)\})\},\hfill \end{array}$$

Then, computing effective intersection of bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_{{\daleth }^I}^I,{\mathsf{{\rm Y}}}^I)$ of order 3, using Formula (6) from Definition 10; the results are as follows:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{{\daleth }^I}^I,{\mathsf{{\rm Y}}}^I)& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.443375,0,0.31375,-0.626375,-0.351875,-0.5425),\hfill \\ & ({\xi }_2,0.49565,0.476,0.65285,-0.40395,-0.57425,0),\hfill \\ & ({\upsilon }_2,\{({\xi }_1,0.817,0.80175,0.61875,-0.49675,-0.580625,-0.46625),\hfill \\ & ({\xi }_2,0.44325,0.77075,0.53495,0,-0.8035,-0.607),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,0,0.512,0.46625,-0.4205,-0.809375,0),\hfill \\ & ({\xi }_2,0.84935,0.41705,0.4105,-0.62665,0,-0.4105)\left\}\right)\}.\hfill \end{array}$$

Definition 16

(Restricted intersection of two effective bipolar-valued multi-fuzzy soft sets). Assume that Ξ is an initial universe. Suppose that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets. Let ${\daleth }_1$ and ${\daleth }_2$ be two effective parameter sets over Δ. Then, the restricted intersection of two effective bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_{1{\daleth }_1},{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_{2{\daleth }_2},{\mathsf{{\rm Y}}}_2)$ of dimension n on an initial universe Ξ is defined as a new effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{{\daleth }^{I_R}}^{I_R},{\mathsf{{\rm Y}}}^{I_R})$ of dimension n, where ${\daleth }^{I_R}:\mathsf{\Delta }\to [0,1]$ is a mapping characterized by ${\daleth }^{I_R}={\daleth }_1\widetilde{{\cap }_R}{\daleth }_2$, as well as, ${(\mathsf{\Psi },\mathsf{{\rm Y}})}^{I_R}=({\mathsf{\Psi }}^{I_R},{\mathsf{{\rm Y}}}^{I_R})=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\widetilde{{\cap }_R}({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, taking into account that ${\mathsf{{\rm Y}}}^{I_R}={\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2\ne \varphi$ and ${\daleth }_1\cap {\daleth }_2\ne \varphi$. The two formulas that compute ${\daleth }^{I_R}={\daleth }_1\widetilde{{\cap }_R}{\daleth }_2$ and $({\mathsf{\Psi }}^{I_R},{\mathsf{{\rm Y}}}^{I_R})=({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\widetilde{{\cap }_R}({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$, respectively, can be determined, for each $\xi \in \mathsf{\Xi }$, as ${\varrho }_{{\daleth }_{\xi }^{I_R}}\left(\delta \right)=min\{{\varrho }_{{\daleth }_{1\xi }}\left(\delta \right),{\varrho }_{{\daleth }_{2\xi }}\left(\delta \right)\}$, for each $\delta \in \mathsf{\Delta }$ and

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}^{I_R},{\mathsf{{\rm Y}}}^{I_R})& =\left\{\right(\upsilon ,\left\{\right(\xi ,min{\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },min{\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },\dots ,\hfill \\ & min{\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right),{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)\}}_{\daleth },max{\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth },\hfill \\ & max{\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth },\dots ,max{\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right),{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)\}}_{\daleth }\left)\right\}),\xi \in \mathsf{\Xi }\},\hfill \end{array}$$

for each $\upsilon \in {\mathsf{{\rm Y}}}^{I_R}$.

Definition 17

(Subset of effective bipolar-valued multi-fuzzy soft set). Suppose that Ξ is an initial universe. Assume that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets. Given that ${\daleth }_1$ and ${\daleth }_2$ are two effective parameter sets over Δ. Let $({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$ are two effective bipolar-valued multi-fuzzy soft sets of dimension n on a universal set Ξ. Then, $({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)$ is called an effective bipolar-valued multi-fuzzy soft subset of $({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$ if the following are satisfied:

1. 

${\daleth }_1\subseteq {\daleth }_2$,

2. 

${\mathsf{{\rm Y}}}_1\subseteq {\mathsf{{\rm Y}}}_2$ and

3. 

${\mathsf{\Psi }}_1\left(\upsilon \right)\subseteq {\mathsf{\Psi }}_2\left(\upsilon \right)$, for all $\upsilon \in {\mathsf{{\rm Y}}}_1$.

(1)

means that, for $\xi \in \mathsf{\Xi }$ and each $\delta \in \mathsf{\Delta }$, we have ${\varrho }_{{\daleth }_{1\xi }}\left(\delta \right)\le {\varrho }_{{\daleth }_{2\xi }}\left(\delta \right)$.

(2)

means ordinary inclusion (usual subset).

(3)

means that, for $i=1,2,\dots ,n$, ${\eta }_{i{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right)\le {\eta }_{i{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)$ and ${\eta }_{i{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right)\ge {\eta }_{i{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)$,

i.e., ${\eta }_{i{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}\left(\xi \right)\le {\eta }_{i{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}\left(\xi \right)$ and ${\eta }_{i{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}\left(\xi \right)\ge {\eta }_{i{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}\left(\xi \right)$, for each $\upsilon \in {\mathsf{{\rm Y}}}_1$ and for each $\xi \in \mathsf{\Xi }$

One can write $({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\tilde{\subseteq }({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$. In this case, $({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$ is called an effective bipolar-valued multi-fuzzy soft superset of $({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)$, denoted by $({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)\tilde{\supseteq }({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)$.

Definition 18

(Equality of two effective bipolar-valued multi-fuzzy soft sets). Given that Ξ is an initial universe, suppose that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets. Assume that ${\daleth }_1$ and ${\daleth }_2$ are two effective parameter sets over Δ. Then, two effective bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$ of dimension n on an initial universe Ξ are called effective bipolar-valued multi-fuzzy soft equal if they are effective bipolar-valued multi-fuzzy soft subsets of each other as stated in Definition 17, i.e., $({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)\tilde{\subseteq }({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)$ and $({\mathsf{\Psi }}_2,{\mathsf{{\rm Y}}}_2)\tilde{\subseteq }({\mathsf{\Psi }}_1,{\mathsf{{\rm Y}}}_1)$.

Definition 19

(Complement of effective bipolar-valued multi-fuzzy soft set). For a parameter set Υ and an effective parameter set Δ the operation of the complement of an effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ of dimension n on an initial universe Ξ is defined by ${({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})}^c=({\mathsf{\Psi }}_{{\daleth }^c}^c,\mathsf{{\rm Y}})$. We have ${\daleth }^c:\mathsf{\Delta }\to [0,1]$ is characterized by ${\varrho }_{{\daleth }_{\xi }^c}\left(\delta \right)=1-{\varrho }_{{\daleth }_{\xi }}\left(\delta \right)$, for each $\xi \in \mathsf{\Xi }$ and for each $\delta \in \mathsf{\Delta }$. In addition, we have ${\mathsf{\Psi }}^c:\mathsf{{\rm Y}}\to \mathfrak{BMF}\left(\mathsf{\Xi }\right)$ is described, for $i=1,2,\cdots ,n$, as follows: ${\eta }_{i{\mathsf{\Psi }}^c\left(\upsilon \right)}^{+}\left(\xi \right)=1-{\eta }_{i\mathsf{\Psi }\left(\upsilon \right)}^{+}\left(\xi \right)$ and ${\eta }_{i{\mathsf{\Psi }}^c\left(\upsilon \right)}^{-}\left(\xi \right)=1-{\eta }_{i\mathsf{\Psi }\left(\upsilon \right)}^{-}\left(\xi \right)$, for each $\xi \in \mathsf{\Xi }$ and for each $\upsilon \in \mathsf{{\rm Y}}$. That is to say that we have the following:

$$\begin{array}{cc}\hfill {({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})}^c& =\left\{\right(\upsilon ,\left\{\right(\xi ,1-{\eta }_{1{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{+}{\left(\xi \right)}_{{\daleth }^c},1-{\eta }_{2{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{+}{\left(\xi \right)}_{{\daleth }^c},\dots ,1-{\eta }_{n{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{+}{\left(\xi \right)}_{{\daleth }^c},\hfill \\ & 1-{\eta }_{1{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{-}{\left(\xi \right)}_{{\daleth }^c},1-{\eta }_{2{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{-}{\left(\xi \right)}_{{\daleth }^c},\dots ,1-{\eta }_{n{\mathsf{\Psi }}_{\mathsf{{\rm Y}}}\left(\upsilon \right)}^{-}{\left(\xi \right)}_{{\daleth }^c}\left)\right\}):\upsilon \in \mathsf{{\rm Y}},\xi \in \mathsf{\Xi }\}.\hfill \end{array}$$

(12)

Example 5.

The complement of $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ in Example 2 can be calculated as follows:

$$\begin{array}{cc}\hfill {({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})}^c=({\mathsf{\Psi }}_{{\daleth }^c}^c,\mathsf{{\rm Y}})& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.82,0.64,0.73,-0.865,-0.595,-0.91),\hfill \\ & ({\xi }_2,0.965,0.825,0.895,-0.755,-0.79,-0.86),\hfill \\ & ({\xi }_3,0.37,0.72,0.65,-0.51,-0.86,-0.79)\left\}\right),\hfill \\ & ({\upsilon }_2,\{({\xi }_1,0.73,0.865,0,-0.685,-0.775,-0.82),\hfill \\ & ({\xi }_2,0.825,0,0.685,-0.93,0,-0.86),\hfill \\ & ({\xi }_3,0.86,0.72,0.58,-0.79,-0.44,-0.93)\left\}\right),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,0,0.73,0.865,-0.91,-0.775,-0.82),\hfill \\ & ({\xi }_2,0.755,0.685,0.965,-0.685,0,-0.895),\hfill \\ & ({\xi }_3,0.93,0.51,0.86,-0.86,-0.65,-0.44)\left\}\right)\}.\hfill \end{array}$$

Remark 4.

The above definitions can be extended from the case of just two sets to the case of a family of sets. One can easily infer the formulas that describe those definitions and can give an example for each one.

5. Properties of Effective Bipolar-Valued Multi-Fuzzy Soft Sets

In this section, we give many significant properties for effective bipolar-valued multi-fuzzy soft sets of dimension n like associative, commutative, distributive, absorption, and De Morgan’s properties. Using Definitions 11–13, 15–17 and 19 of Section 4 makes the following theorems hold. By applying formulas and operations stated in those definitions, one can easily prove these theorems directly.

Theorem 1.

Given that Ξ is an initial universe, assume that Υ is a parameter set. Suppose that $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ is an effective bipolar-valued multi-fuzzy soft set of dimension n on an initial universe Ξ, constructed by an effective set ℸ. Let $({\varphi }_{\daleth },\mathsf{{\rm Y}})$ and $({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})$ be, respectively, the null and the absolute effective bipolar-valued multi-fuzzy soft set of dimension n on a common initial universe Ξ. Then, we have the following satisfied for them:

1. 

$({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})\tilde{\cup }({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})=({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})\tilde{\cap }({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})=({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$.

2. 

$({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})\tilde{\cap }({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})=({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})\tilde{\cup }({\varphi }_{\daleth },\mathsf{{\rm Y}})=({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$.

3. 

$({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})\tilde{\cup }({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})=({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})\tilde{\cup }({\varphi }_{\daleth },\mathsf{{\rm Y}})=({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})$.

4. 

$({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})\tilde{\cap }({\varphi }_{\daleth },\mathsf{{\rm Y}})=({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})\tilde{\cap }({\varphi }_{\daleth },\mathsf{{\rm Y}})=({\varphi }_{\daleth },\mathsf{{\rm Y}})$.

Proof. 

We prove $\left(4\right)$. Similarly, $\left(1\right)$, $\left(2\right)$ and $\left(3\right)$ can be proved by using the same technique. For $\left(4\right)$, we prove that $({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})\tilde{\cap }({\varphi }_{\daleth },\mathsf{{\rm Y}})=({\varphi }_{\daleth },\mathsf{{\rm Y}})$ and by following the same method $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})\tilde{\cap }({\varphi }_{\daleth },\mathsf{{\rm Y}})=({\varphi }_{\daleth },\mathsf{{\rm Y}})$ can be proved. From Definitions 11 and 12, $({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})=\{(\upsilon ,\left\{(\xi ,1,\stackrel{n-times}{\dots },1,-1,\stackrel{n-times}{\dots },-1)\right\}):\upsilon \in \mathsf{{\rm Y}},\xi \in \mathsf{\Xi }\}$ and $({\varphi }_{\daleth },\mathsf{{\rm Y}})=\{(\upsilon ,\left\{(\xi ,0,\stackrel{2n-times}{\dots },0)\right\}):\upsilon \in \mathsf{{\rm Y}},\xi \in \mathsf{\Xi }\}$, respectively. Assume, for $\mathsf{{\rm Y}}=\mathsf{{\rm Y}}\cup \mathsf{{\rm Y}}=\mathsf{{\rm Y}}$, that

$$\begin{array}{cc}\hfill & ({\mathcal{C}}_{\daleth },\mathsf{{\rm Y}})\tilde{\cap }({\varphi }_{\daleth },\mathsf{{\rm Y}})\hfill \\ & =({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})\hfill \\ & =\{(\upsilon ,\left\{(\xi ,{\eta }_{1\mathsf{\Psi }\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2\mathsf{\Psi }\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n\mathsf{\Psi }\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{1\mathsf{\Psi }\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2\mathsf{\Psi }\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n\mathsf{\Psi }\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth })\right\}):\hfill \\ & \upsilon \in \mathsf{{\rm Y}},\xi \in \mathsf{\Xi }\}\hfill \\ & =\{(\upsilon ,\left\{(\xi ,min{\{1,0\}}_{\daleth },\stackrel{n-times}{\dots },min{\{1,0\}}_{\daleth },max{\{-1,0\}}_{\daleth },\stackrel{n-times}{\dots },max{\{-1,0\}}_{\daleth })\right\}):\hfill \\ & \upsilon \in \mathsf{{\rm Y}},\xi \in \mathsf{\Xi }\}\hfill \\ & =\{(\upsilon ,\left\{(\xi ,(0,\stackrel{n-times}{\dots },0),(0,\stackrel{n-times}{\dots },0))\right\}):\upsilon \in \mathsf{{\rm Y}},\xi \in \mathsf{\Xi }\}\hfill \\ & =\{(\upsilon ,\left\{(\xi ,0,\stackrel{2n-times}{\dots },0)\right\}):\upsilon \in \mathsf{{\rm Y}},\xi \in \mathsf{\Xi }\}=({\varphi }_{\daleth },\mathsf{{\rm Y}}).\hfill \end{array}$$

Then, this is true for $\upsilon \in \mathsf{{\rm Y}}\cap \mathsf{{\rm Y}}=\mathsf{{\rm Y}}$, which is the third case in Definition 15. But, we have no parameters for the first and second cases since $\upsilon \in \mathsf{{\rm Y}}-\mathsf{{\rm Y}}=\varphi$. □

Theorem 2.

Let Ξ be an initial universe. Suppose that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets. For a common effective set ℸ, let $({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)$ be two effective bipolar-valued multi-fuzzy soft sets of dimension n on a universal set Ξ. Then, we have the following absorption properties are true:

1. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1){\tilde{\cap }}_R({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)=({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$.

2. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1){\tilde{\cap }}_R\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)=({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$.

Proof. 

To prove $\left(1\right)$, assume that

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)& =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ & {\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_2,\xi \in \mathsf{\Xi }\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)& =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ & {\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_1,\xi \in \mathsf{\Xi }\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)& =({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1){\tilde{\cap }}_R({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2),{\mathsf{{\rm Y}}}_3={\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2,\hfill \\ & =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ & {\eta }_{1{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_3,\xi \in \mathsf{\Xi }\},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{4\daleth },{\mathsf{{\rm Y}}}_4)& =({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3),{\mathsf{{\rm Y}}}_4={\mathsf{{\rm Y}}}_1\cup {\mathsf{{\rm Y}}}_3,\hfill \\ & =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_4\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_4\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_4\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ & {\eta }_{1{\mathsf{\Psi }}_4\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_4\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_4\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_4,\xi \in \mathsf{\Xi }\}.\hfill \end{array}$$

We must prove that $\left(1\right)$ is true for all following three cases, according to Definition 13:

(i) 

If $\upsilon \in {\mathsf{{\rm Y}}}_1-{\mathsf{{\rm Y}}}_2$, therefore, from Definition 16, we have:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)& =({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1){\tilde{\cap }}_R({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\hfill \\ & =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ & {\eta }_{1{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_1-{\mathsf{{\rm Y}}}_2,\xi \in \mathsf{\Xi }\}=\varphi .\hfill \end{array}$$

Then, by using $\left(3\right)$ from Theorem 1, we have:

$$\begin{array}{c}\hfill ({\mathsf{\Psi }}_{4\daleth },{\mathsf{{\rm Y}}}_4)=({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)=({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }\varphi =({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1).\end{array}$$

(ii) 

If $\upsilon \in {\mathsf{{\rm Y}}}_2-{\mathsf{{\rm Y}}}_1$, then we obtain from Definition 16 that:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)& =({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1){\tilde{\cap }}_R({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\hfill \\ & =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ & {\eta }_{1{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_2-{\mathsf{{\rm Y}}}_1,\xi \in \mathsf{\Xi }\}=\varphi .\hfill \end{array}$$

Then, by using $\left(3\right)$ from Theorem 1, we have:

$$\begin{array}{c}\hfill ({\mathsf{\Psi }}_{4\daleth },{\mathsf{{\rm Y}}}_4)=({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)=({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }\varphi =({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1).\end{array}$$

(iii) 

If $\upsilon \in {\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2$, then we obtain from Definition 16 that:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)& =({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1){\tilde{\cap }}_R({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\hfill \\ & =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_3\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ & {\eta }_{1{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_3\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2,\xi \in \mathsf{\Xi }\}\hfill \\ & =\left\{\right(\upsilon ,\left\{\right(\xi ,min\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth }\},min\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth }\},\dots ,\hfill \\ & min\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth }\},max\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\},\hfill \\ & max\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\},\dots ,max\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\}\left)\right\}):\hfill \\ & \upsilon \in {\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2,\xi \in \mathsf{\Xi }\}.\hfill \end{array}$$

Since,

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{4\daleth },{\mathsf{{\rm Y}}}_4)& =({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)\hfill \\ & =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_4\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_4\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_4\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ & {\eta }_{1{\mathsf{\Psi }}_4\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_4\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_4\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2,\xi \in \mathsf{\Xi }\}.\hfill \end{array}$$

Therefore, we have from Definition 13 that:

$$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{4\daleth },{\mathsf{{\rm Y}}}_4)& =\left\{\right(\upsilon ,\left\{\right(\xi ,max\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },min\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth }\}\},\hfill \\ & max\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },min\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth }\}\},\dots ,\hfill \\ & max\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },min\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth }\}\},\hfill \\ & min\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },max\{{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{1{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\}\},\hfill \\ & min\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },max\{{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\}\},\dots ,\hfill \\ & min\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },max\{{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{n{\mathsf{\Psi }}_2\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\}\}\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2,\xi \in \mathsf{\Xi }\}\hfill \\ & =\left\{\right(\upsilon ,\left\{\right(\xi ,{\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{+}{\left(\xi \right)}_{\daleth },\hfill \\ & {\eta }_{1{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },{\eta }_{2{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth },\dots ,{\eta }_{n{\mathsf{\Psi }}_1\left(\upsilon \right)}^{-}{\left(\xi \right)}_{\daleth }\left)\right\}):\upsilon \in {\mathsf{{\rm Y}}}_1\cap {\mathsf{{\rm Y}}}_2,\xi \in \mathsf{\Xi }\}\hfill \\ & =({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1).\hfill \end{array}$$

To prove $\left(2\right)$, one can follow the same steps as $\left(1\right)$. □

Corollary 1.

Given that, Ξ is an initial universe. Let ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ be two parameter sets. For two effective bipolar-valued multi-fuzzy soft sets of dimension n on a common initial universe $({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)$, generated by a common effective set ℸ, we obtain that:

$$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1){\tilde{\cap }}_R({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)=({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1){\tilde{\cap }}_R\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)=({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1).$$

Proof. 

This corollary can be proved directly as the above Theorem 2. □

Theorem 3.

Let Ξ be an initial universe. Assume that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets. Suppose that we have a common effective set ℸ, associated with two effective bipolar-valued multi-fuzzy soft sets of dimension n, namely $({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)$. Then, we obtain that the abelian (commutative) property hold as below:

1. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cap }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)=({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\tilde{\cap }({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$.

2. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)=({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\tilde{\cup }({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$.

Proof. 

Applying the same technique stated in Theorem 2, one can easily prove this result using Definitions 13 and 15. □

Proposition 1.

Suppose that Ξ is an initial universe. Given that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets, assume that we have a common effective set ℸ, associated with two effective bipolar-valued multi-fuzzy soft sets of dimension n, namely $({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)$. If $({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\subseteq }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)$, then

1. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1){\tilde{\cap }}_R({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)=({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$.

2. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)=({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)$.

Proof. 

This proposition can be proved directly like Theorem 2, applying Definitions 14 and 16. □

Theorem 4.

Let Ξ be an initial universe and ${\mathsf{{\rm Y}}}_1$, ${\mathsf{{\rm Y}}}_2$ and ${\mathsf{{\rm Y}}}_3$ be three parameter sets. For a common effective set ℸ, suppose that $({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$, $({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)$ and $({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)$ are effective bipolar-valued multi-fuzzy soft sets of dimension n on a common initial universe Ξ. Then, we have the associative and distributive laws, respectively, satisfied as the following:

1. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cap }\left(({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\tilde{\cap }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)\right)=\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cap }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)\tilde{\cap }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)$.

2. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }\left(({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\tilde{\cup }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)\right)=\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)\tilde{\cup }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)$.

3. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cap }\left(({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\tilde{\cup }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)\right)=\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cap }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)\tilde{\cup }\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cap }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)\right)$.

4. 

$({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }\left(({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\tilde{\cap }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)\right)=\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)\tilde{\cap }\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{3\daleth },{\mathsf{{\rm Y}}}_3)\right)$.

Proof. 

Using Definitions 13 and 15 and applying the same technique stated in Theorem 2, we can prove this theorem. □

Theorem 5.

Assume that Ξ is an initial universe. Suppose that ${\mathsf{{\rm Y}}}_1$ and ${\mathsf{{\rm Y}}}_2$ are two parameter sets. For a common effective set ℸ, we have the following De Morgan’s laws hold for any two effective bipolar-valued multi-fuzzy soft sets $({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)$ and $({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)$ of dimension n on a common initial universe Ξ:

1. 

${\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cup }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)}^c={({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)}^c\tilde{\cap }{({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)}^c$.

2. 

${\left(({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)\tilde{\cap }({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)\right)}^c={({\mathsf{\Psi }}_{1\daleth },{\mathsf{{\rm Y}}}_1)}^c\tilde{\cup }{({\mathsf{\Psi }}_{2\daleth },{\mathsf{{\rm Y}}}_2)}^c$.

Proof. 

One can easily prove this theorem with the help of Theorem 2’s technique by using Definitions 13, 15 and 19. □

6. Medical Diagnosis

The aim of this section is to focus on a real-life issue of diagnosis. An algorithm for medical diagnosis, or educational evaluation, …, using the effective bipolar-valued multi-fuzzy soft set of dimension n is introduced. One can apply this technique using matrixes operations and properties to diagnose the case. This diagnosis includes determining what student, or what patient, …, respectively, is succeeding in taking which education level, or is suffering from which disease, …

Furthermore, a case study example of medical diagnosis is discussed in detail. The steps of the initiated method are framed under matrix operations to facilitate doing computations. Moreover, the addition and the multiplication of matrixes, as well as calculations of effective memberships, are made with the help of the MATLAB^® program to make them faster, more accurate, and easy to do.

6.1. Methodology and Algorithm

Suppose that there is a set of n students or patients, …$\mathsf{\Pi }=\{{\pi }_1,{\pi }_2,\dots ,{\pi }_n\}$, say. In addition, assume that we have two sets of m exams or symptoms, … and their opposites ${\mathsf{{\rm Y}}}_{=}\{{\upsilon }_1,{\upsilon }_2,\dots ,{\upsilon }_m\}$ and $-{\mathsf{{\rm Y}}}_{=}\{-{\upsilon }_1,-{\upsilon }_2,\dots ,-{\upsilon }_m\}$, respectively. Furthermore, let those sets be related to a set of k levels or diseases, …$\mathsf{\Xi }=\{{\xi }_1,{\xi }_2,\dots ,{\xi }_k\}$. Moreover, consider $\mathsf{\Delta }=\{{\delta }_1,{\delta }_2,\dots ,{\delta }_r\}$ is a set of r effective parameters or attributes proposed according to the problem.

The effective set ℸ can be constructed according to the students’ says or the patients’ says. In addition, the bipolar-valued multi-fuzzy soft set $(\mathsf{\Psi },\mathsf{{\rm Y}})$ can be obtained by asking every student or patient many questions and subjecting him/her to some tests or analyses by experts. Moreover, the multi-fuzzy soft set $(\mathsf{\Gamma },\mathsf{{\rm Y}})$ that indicates an approximate description of the given levels or diseases and their exams or symptoms, respectively, can be built from expert documentation.

Then, under these given assumptions, we can start the algorithm’s steps to determine what student, or what patient, …, respectively, is succeeding in taking which education level, or is suffering from which disease, … The first step is to compute the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ from the given bipolar-valued multi-fuzzy soft set $(\mathsf{\Psi },\mathsf{{\rm Y}})$ and the given effective set ℸ for the given students or patients by using Formula (6) from Definition 10. After that, the second step is to extract the matrix corresponding to every positive pole of the effective bipolar-valued multi-fuzzy soft set that contains the membership values of the given items, say $A_i$. Again, similarly, the third step is to extract the matrix corresponding to every negative pole of the effective bipolar-valued multi-fuzzy soft set that contains the membership values of the given items, say $B_i$.

Also, similarly, the fourth step is to extract the matrix corresponding to every pole of the given multi-fuzzy soft set, say $C_j$, where $j=2i$. Then, the fifth step is to multiply every $A_i$ and $B_i$ matrix by its $C_j$ corresponding matrix (or multiply it by the transpose of its $C_j$ corresponding matrix, if necessary according to the problem conditions). Finally, the sixth step is to add all resulting matrixes, say $D_j$, to obtain the final diagnosis matrix, say D, in which the diagnosis for each student or patient is regarded as the maximum value in their row. For simplicity, Figure 2 briefly represents the proposed algorithm’s steps as a simple flowchart.

6.2. Case Study

This section is devoted to discussing a medical example, in which we follow the above algorithm steps to determine the best diagnosis for every patient who has some known symptoms by a specific degree. In every step that needs calculations, we use the MATLAB^® program to perform any computations like calculating effective values, the matrix addition operation, and the matrix multiplication operation.

Example 6.

Consider a universal set of patients $\mathsf{\Pi }=\{{\pi }_1,{\pi }_2,{\pi }_3\}$ who are predicted to be possibly suffering from one of four proposed diseases according to their symptoms and circumstances.

Let the two major sets of parameters (attributes) that describe the symptoms be $\mathsf{{\rm Y}}=\{{\upsilon }_1,{\upsilon }_2,{\upsilon }_3\}$, ${\upsilon }_i(i=1,2,3)$ and its opposite set $-\mathsf{{\rm Y}}=\{-{\upsilon }_1,-{\upsilon }_2,-{\upsilon }_3\}$, $-{\upsilon }_i(i=1,2,3)$ stand for the symptoms and the opposite-symptoms, respectively. These symptoms can be classified, according to their association with different human systems, into the following three main types of parameters:

1. 

Respiratory symptoms, digestive symptoms, and neurological symptoms, respectively. Respiratory symptoms and their opposite symptoms are as follows:

(“difficult and slow breath” and “easy and fast breath”), (“runny nose” and “stuffy nose”), (“difficult swallow” and “easy swallow”).

2. 

Digestive symptoms and their opposite symptoms are as follows:

(“diarrhea” and “constipation”), (“nausea” and “appetite”), (“abdominal pain” and “abdominal relax”).

3. 

Neurological symptoms and their opposite symptoms are as follows:

(“headache” and “head relax”), (“increased sweating” and “decreased sweating”), (“fatigue and pain” and “ability and well-being”).

In addition, let the universal set $\mathsf{\Xi }=\{{\xi }_1,{\xi }_2,{\xi }_3,{\xi }_4\}$ be a set of possible proposed diseases, where ${\xi }_1=$ Malaria, ${\xi }_2=$ Dengue fever, ${\xi }_3=$ Corona virus $(COVID-19)$. and ${\xi }_4=$ respiratory syncytial virus $\left(RSV\right)$.

Furthermore, suppose that $\mathsf{\Delta }=\{{\delta }_1,{\delta }_2,{\delta }_3,{\delta }_4\}$ is a set of effective attributes, where ${\delta }_1=$ the patient has closely contacted with anyone who was suffering from COVID-19, ${\delta }_2=$ the patient has used to sleep without a mosquito net, or any other cover ${\delta }_3=$ the patient works in a hospital, or a medical center and ${\delta }_4=$ the patient has closely contacted with anyone who was suffering from $RSV$. After talking to the patients, we can construct the effective set ℸ over Δ for ${\pi }_i,i=1,2,3$, as follows according to the patient’s words and an expert medical evaluation:

$$\daleth \left({\pi }_1\right)=\{({\delta }_1,0.7),({\delta }_2,0.2),({\delta }_3,0.5),({\delta }_4,0.4)\},$$

$$\daleth \left({\pi }_2\right)=\{({\delta }_1,0.5),({\delta }_2,0.1),({\delta }_3,0),({\delta }_4,0.8)\},$$

$$\daleth \left({\pi }_3\right)=\{({\delta }_1,1),({\delta }_2,0.6),({\delta }_3,0.3),({\delta }_4,0.9)\}.$$

Moreover, after asking every patient many questions as well as subjecting him/her to some medical tests by a medical committee, we have the following bipolar-valued multi-fuzzy soft set $(\mathsf{\Psi },\mathsf{{\rm Y}})$ of order 3:

$$\begin{array}{cc}\hfill (\mathsf{\Psi },\mathsf{{\rm Y}})& =\left\{\right({\upsilon }_1,\{({\pi }_1,0.4,0.8,0.6,-0.3,-0.9,-0.2),({\pi }_2,0.1,0.5,0.3,-0.7,-0.6,-0.4),\hfill \\ & ({\pi }_3,0.9,0.4,0.5,-0.7,-0.2,-0.3)\left\}\right),({\upsilon }_2,\{({\pi }_1,0.6,0.3,1,-0.7,-0.5,-0.3),\hfill \\ & ({\pi }_2,0.5,1,0.9,-0.2,-1,-0.4),({\pi }_3,0.2,0.4,0.6,-0.3,-0.8,-0.1)\left\}\right),\hfill \\ & ({\upsilon }_3,\{({\pi }_1,1,0.6,0.3,-0.2,-0.5,-0.4),({\pi }_2,0.7,0.9,0.1,-0.9,-1,-0.3),\hfill \\ & ({\pi }_3,0.1,0.7,0.2,-0.2,-0.5,-0.8)\left\}\right)\}.\hfill \end{array}$$

Furthermore, from expert medical documentation, we have a multi-fuzzy soft set $(\mathsf{\Gamma },\mathsf{{\rm Y}})$ of order 6 indicating an approximate description of the four diseases and their symptoms.

$$\begin{array}{cc}\hfill (\mathsf{\Gamma },\mathsf{{\rm Y}})& =\left\{\right({\upsilon }_1,\{({\xi }_1,0.1,0.2,0.3,0.8,0.3,0.2),({\xi }_2,0.2,0.3,0.1,0.1,0.2,0.1),\hfill \\ & ({\xi }_3,0.4,0.5,0.7,0.4,0.5,0.1),({\xi }_4,1,0.8,0.6,0,0.2,0.1)\left\}\right),\hfill \\ & ({\upsilon }_2,\{({\xi }_1,0.7,0.8,0.9,0.1,0.2,0.1),({\xi }_2,0.1,0.9,0.9,0.1,0.1,0.1),\hfill \\ & ({\xi }_3,0.9,0.6,0.7,0.8,0.1,0.1)\left\}\right),({\xi }_4,0.6,0.5,0.2,0.1,0.1,0.2)\left\}\right),\hfill \\ & ({\upsilon }_3,\{({\xi }_1,0.9,0.1,1,0.2,0.1,0.1),({\xi }_2,0.8,0.3,0.7,0.1,0.3,0.2),\hfill \\ & ({\xi }_3,0.7,0.9,0.8,0.2,0.1,0.2),({\xi }_4,0.7,0.1,0.6,0.1,0,0.2)\left\}\right)\}.\hfill \end{array}$$

What is the best medical diagnosis for all patients?

Solution.

• $Step\left(1\right)$: Compute the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ of order 3, that describes the above patients’ cases, by using Formula (6) from Definition 10, as follows:

  $$\begin{array}{cc}\hfill ({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})& =\left\{\right({\upsilon }_1,\{({\pi }_1,0.67,0.89,0.78,-0.61,-0.94,-0.56),\hfill \\ & ({\pi }_2,0.41,0.67,0.54,-0.8,-0.74,-0.61),\hfill \\ & ({\pi }_3,0.97,0.82,0.85,-0.91,-0.76,-0.79)\left\}\right),\hfill \\ & ({\upsilon }_2,\{({\pi }_1,0.78,0.61,1,-0.83,-0.72,-0.61),\hfill \\ & ({\pi }_2,0.67,1,0.93,-0.48,-1,-0.61),\hfill \\ & ({\pi }_3,0.76,0.82,0.88,-0.79,-0.94,-0.73)\left\}\right),\hfill \\ & ({\upsilon }_3,\{({\pi }_1,1,0.78,0.61,-0.56,-0.72,-0.67),\hfill \\ & ({\pi }_2,0.8,0.93,0.41,-0.93,-1,-0.54),\hfill \\ & ({\pi }_3,0.73,0.91,0.76,-0.76,-0.85,-0.94)\left\}\right)\}.\hfill \end{array}$$
• $Step\left(2\right)$: Extract the matrixes $A_1$, $A_2$ and $A_3$ representing the patient-symptom $(+ve)$ relations from the membership values of the first, second and third positive poles of the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ of order 3, respectively, as the following:

  $$A_1=\begin{array}{cc}& \begin{array}{ccc}{\upsilon }_1^{\prime }& {\upsilon }_2^{\prime }& {\upsilon }_3^{\prime }\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{ccc}0.67& 0.78& 1\\ 0.41& 0.67& 0.8\\ 0.97& 0.76& 0.73\end{array}\right)\end{array},$$

  $$A_2=\begin{array}{cc}& \begin{array}{ccc}{\upsilon }_1^{″}& {\upsilon }_2^{″}& {\upsilon }_3^{″}\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{ccc}0.89& 0.61& 0.78\\ 0.67& 1& 0.93\\ 0.82& 0.82& 0.91\end{array}\right)\end{array},$$

  $$A_3=\begin{array}{cc}& \begin{array}{ccc}{\upsilon }_1^{‴}& {\upsilon }_1^{‴}& {\upsilon }_1^{‴}\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{ccc}0.78& 1& 0.61\\ 0.54& 0.93& 0.41\\ 0.85& 0.88& 0.76\end{array}\right)\end{array}.$$
• $Step\left(3\right)$: Similarly, extract the matrixes $B_1$, $B_2$ and $B_3$ representing the patient-symptom $(-ve)$ relations from the membership values of the first, second and third negative poles of the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ of order 3, respectively, as the following:

  $$B_1=\begin{array}{cc}& \begin{array}{ccc}-{\upsilon }_1^{\prime }& -{\upsilon }_2^{\prime }& -{\upsilon }_3^{\prime }\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{ccc}\mathrm{-0.61}& \mathrm{-0.83}& \mathrm{-0.56}\\ \mathrm{-0.8}& \mathrm{-0.48}& \mathrm{-0.93}\\ \mathrm{-0.91}& \mathrm{-0.79}& \mathrm{-0.76}\end{array}\right)\end{array},$$

  $$B_2=\begin{array}{cc}& \begin{array}{ccc}-{\upsilon }_1^{″}& -{\upsilon }_2^{″}& -{\upsilon }_3^{″}\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{ccc}\mathrm{-0.94}& \mathrm{-0.72}& \mathrm{-0.72}\\ \mathrm{-0.74}& \mathrm{-1}\hfill & \mathrm{-1}\hfill \\ \mathrm{-0.76}& \mathrm{-0.94}& \mathrm{-0.85}\end{array}\right)\end{array},$$

  $$B_3=\begin{array}{cc}& \begin{array}{ccc}-{\upsilon }_1^{‴}& -{\upsilon }_2^{‴}& -{\upsilon }_3^{‴}\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{ccc}\mathrm{-0.56}& \mathrm{-0.61}& \mathrm{-0.67}\\ \mathrm{-0.61}& \mathrm{-0.61}& \mathrm{-0.54}\\ \mathrm{-0.79}& \mathrm{-0.73}& \mathrm{-0.94}\end{array}\right)\end{array}.$$
• $Step\left(4\right)$: In addition, extract the matrixes $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, and $C_6$, representing the symptom-disease relations from the membership values of the six poles of the multi-fuzzy soft set $(\mathsf{\Gamma },\mathsf{{\rm Y}})$ of order 6, respectively, as the following:

  $$C_1=\begin{array}{cc}& \begin{array}{ccc}{\upsilon }_1^{\prime }& {\upsilon }_2^{\prime }& {\upsilon }_3^{\prime }\end{array}\\ \begin{array}{c}{\xi }_1\\ {\xi }_2\\ {\xi }_3\\ {\xi }_4\end{array}& \left(\begin{array}{ccc}0.1& 0.7& 0.9\\ 0.2& 0.1& 0.8\\ 0.4& 0.9& 0.7\\ 1& 0.6& 0.7\end{array}\right)\end{array},$$

  $$C_2=\begin{array}{cc}& \begin{array}{ccc}{\upsilon }_1^{″}& {\upsilon }_2^{″}& {\upsilon }_2^{″}\end{array}\\ \begin{array}{c}{\xi }_1\\ {\xi }_2\\ {\xi }_3\\ {\xi }_4\end{array}& \left(\begin{array}{ccc}0.2& 0.8& 0.1\\ 0.3& 0.9& 0.3\\ 0.5& 0.6& 0.9\\ 0.8& 0.5& 0.1\end{array}\right)\end{array},$$

  $$C_3=\begin{array}{cc}& \begin{array}{ccc}{\upsilon }_1^{‴}& {\upsilon }_2^{‴}& {\upsilon }_3^{‴}\end{array}\\ \begin{array}{c}{\xi }_1\\ {\xi }_2\\ {\xi }_3\\ {\xi }_4\end{array}& \left(\begin{array}{ccc}0.3& 0.9& 1\\ 0.1& 0.9& 0.7\\ 0.7& 0.7& 0.8\\ 0.6& 0.2& 0.6\end{array}\right)\end{array},$$

  $$C_4=\begin{array}{cc}& \begin{array}{ccc}-{\upsilon }_1^{\prime }& -{\upsilon }_2^{\prime }& -{\upsilon }_3^{\prime }\end{array}\\ \begin{array}{c}{\xi }_1\\ {\xi }_2\\ {\xi }_3\\ {\xi }_4\end{array}& \left(\begin{array}{ccc}0.8& 0.1& 0.2\\ 0.1& 0.1& 0.1\\ 0.4& 0.8& 0.2\\ 0& 0.1& 0.1\end{array}\right)\end{array},$$

  $$C_5=\begin{array}{cc}& \begin{array}{ccc}-{\upsilon }_1^{″}& -{\upsilon }_2^{″}& -{\upsilon }_3^{″}\end{array}\\ \begin{array}{c}{\xi }_1\\ {\xi }_2\\ {\xi }_3\\ {\xi }_4\end{array}& \left(\begin{array}{ccc}0.3& 0.2& 0.1\\ 0.2& 0.1& 0.3\\ 0.5& 0.1& 0.1\\ 0.2& 0.1& 0\end{array}\right)\end{array},$$

  $$C_6=\begin{array}{cc}& \begin{array}{ccc}-{\upsilon }_1^{‴}& -{\upsilon }_3^{‴}& -{\upsilon }_3^{‴}\end{array}\\ \begin{array}{c}{\xi }_1\\ {\xi }_2\\ {\xi }_3\\ {\xi }_4\end{array}& \left(\begin{array}{ccc}0.2& 0.1& 0.1\\ 0.1& 0.1& 0.2\\ 0.1& 0.1& 0.2\\ 0.1& 0.2& 0.2\end{array}\right)\end{array}.$$
• $Step\left(5\right)$: To obtain the patient-disease matrixes (patient-diagnosis matrixes) $D_1$, $D_2$, $D_3$, $D_4$, $D_5$ and $D_6$, we take the transpose for $C_1$, $C_2$, $C_3$, $C_4$, $C_5$ and $C_6$, then find the products $D_1=A_1\times C_1^T$, $D_2=A_2\times C_2^T$, $D_3=A_3\times C_3^T$, $D_4=B_1\times C_4^T$, $D_5=B_2\times C_5^T$ and $D_6=B_3\times C_6^T$, respectively, as follows:

  $$D_1=A_1\times C_1^T=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}1.513& 1.012& 1.67& 1.838\\ 1.23& 0.789& 1.327& 1.327\\ 1.286& 0.854& 1.583& 1.937\end{array}\right)\end{array},$$

  $$D_2=A_2\times C_2^T=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}0.744& 1.05& 1.513& 1.095\\ 0.997& 1.29& 1.502& 1.099\\ 0.911& 1.257& 1.721& 1.157\end{array}\right)\end{array},$$

  $$D_3=A_3\times C_3^T=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}1.744& 1.405& 1.734& 1.034\\ 1.409& 1.178& 1.357& 0.756\\ 1.807& 1.409& 1.819& 1.142\end{array}\right)\end{array},$$

  $$D_4=B_1\times C_4^T=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}\mathrm{-0.683}& \mathrm{-0.2}& \mathrm{-1.02}& \mathrm{-0.139}\\ \mathrm{-0.874}& \mathrm{-0.221}& \mathrm{-0.89}& \mathrm{-0.141}\\ \mathrm{-0.959}& \mathrm{-0.246}& \mathrm{-1.148}& \mathrm{-0.155}\end{array}\right)\end{array},$$

  $$D_5=B_2\times C_5^T=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}\mathrm{-0.498}& \mathrm{-0.476}& \mathrm{-0.614}& \mathrm{-0.26}\\ \mathrm{-0.522}& \mathrm{-0.548}& \mathrm{-0.57}& \mathrm{-0.248}\\ \mathrm{-0.501}& \mathrm{-0.501}& \mathrm{-0.559}& \mathrm{-0.246}\end{array}\right)\end{array},$$

  $$D_6=B_3\times C_6^T=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}\mathrm{-0.24}& \mathrm{-0.251}& \mathrm{-0.251}& \mathrm{-0.312}\\ \mathrm{-0.237}& \mathrm{-0.23}& \mathrm{-0.23}& \mathrm{-0.291}\\ \mathrm{-0.325}& \mathrm{-0.34}& \mathrm{-0.34}& \mathrm{-0.413}\end{array}\right)\end{array},$$
• $Step\left(6\right)$: Finally, to obtain the final diagnosis matrix D, we calculate the summation of $D_i,i=1,2,\cdots ,6$ as the following:

  $$\begin{gathered} D=D_1+D_2+D_3+D_4+D_5+D_6= \\ \begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}2.58& 2.54& 3.032& \mathbf{3.256}\\ 2.003& 2.258& 2.496& \mathbf{2.547}\\ 2.219& 2.433& 3.076& \mathbf{3.422}\end{array}\right)\end{array}. \end{gathered}$$

  It is clear from the above final diagnosis matrix D that the maximum value in each row is the fourth one. That is, the values $3.256$, $2.547$ and $3.422$, respectively, are the maximum values for the patients ${\pi }_1$, ${\pi }_2$ and ${\pi }_3$ corresponding to the disease ${\xi }_4$.
  Consequently, we conclude that the patients ${\pi }_1$, ${\pi }_2$ and ${\pi }_3$ are suffering from the disease ${\xi }_4$, which is $RSV$. Then, the best medical diagnosis for all those patients is $RSV$.
  If more than one patient is suffering from the same disease, as occurred in the current example, one can determine which patient is in the most need of treatment. According to the above final diagnosis matrix D, the order of alternatives (patients) is as follows: ${\pi }_3>{\pi }_1>{\pi }_2$.
  This shows that the third patient must be the first one to be treated, followed by the first patient and finally, the second patient. Normally, we give the necessary treatment to every needing patient, but in case of a lack of treatments or medical devices (like ventilators needed to treat $RSV$), we follow that priority.

6.3. Sensitivity Analysis

In the above Example 6, the computation of the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ depends on the given effective set ℸ, which arises from the patients’ words and the expert medical evaluation. This means that if the experts evaluate parameters satisfying by different values, then the effective set ℸ values will be different.

Consequently, this leads to different values for parameters considered in the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$. Then, the final decisions and the ranking will also be different. On the one hand, to avoid this problem, we can consider the evaluations of more than one expert who take the patient’s words and then calculate the arithmetic mean to be more accurate.

On the other hand, if we notice that one of the nominated experts usually gives us inordinate evaluations like 0 or 1, we can cancel their opinion and not refer to them again in any future evaluations. For example, if all values of the effective set are zeros, then all values of the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ remain the same. Therefore, the final diagnosis matrix D becomes as the following:

$$D=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}2.32& 2.17& 2.48& \mathbf{2.57}\\ 1.59& 2.02& \mathbf{2.3}& 1.79\\ 0.63& 1.11& 1.62& \mathbf{1.84}\end{array}\right)\end{array}.$$

Consequently, the patients ${\pi }_1$ and ${\pi }_3$ are suffering from the disease ${\xi }_4$ ($RSV$) and the patient ${\pi }_2$ is suffering from the disease ${\xi }_3$ (COVID-19). In addition, the alternatives’ order according to treatment need is ${\pi }_1>{\pi }_2>{\pi }_3$. Furthermore, if all values of the effective set are ones, then the values of the effective bipolar-valued multi-fuzzy soft set $({\mathsf{\Psi }}_{\daleth },\mathsf{{\rm Y}})$ are 1 for positive poles and $-1$ for negative poles. Therefore, the final diagnosis matrix D becomes the following:

$$D=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}2.9& 3& 3.7& \mathbf{4.1}\\ 2.9& 3& 3.7& \mathbf{4.1}\\ 2.9& 3& 3.7& \mathbf{4.1}\end{array}\right)\end{array}.$$

Hence, the diagnosis remains the same, but the ranking order of the three patients (alternatives) becomes ${\pi }_1={\pi }_2={\pi }_3$.

Similarly, the evaluation of how much each patient suffers from each symptom is represented in the two major sets of parameters that describe the symptoms. This means that if the medical devices that measure some symptoms have any problem, then the values of the membership will vary. This also can affect the final decision because we will have a different initial bipolar-valued multi-fuzzy soft set $(\mathsf{\Psi },\mathsf{{\rm Y}})$. To overcome this issue, we must be sure that all medical devices work properly before starting the decision-making process.

Finally, proposing the parameters that can serve as effective parameters or as the parameters representing the symptoms can also affect the decision. This arises from the fact that these proposed parameters may not represent a true measure for the proposed diseases. In addition, the connection between every symptom and every disease is represented in a multi-fuzzy soft set $(\mathsf{\Gamma },\mathsf{{\rm Y}})$. Then, if the values of membership in $(\mathsf{\Gamma },\mathsf{{\rm Y}})$ vary according to different expert medical documentation, it will affect the final decision again. Therefore, the process of choosing suitable experts for the medical problem is very important first of all.

6.4. Comparison

A comparative analysis is conducted to compare decision-making under the effective bipolar-valued multi-fuzzy soft set of dimension n environment with previous existing different settings or models. We solve the same Example 6 under those previous existing different settings. The results of this comparative analysis are outlined as the following:

• If we make the final decision under the multi-fuzzy soft set, offered by Yang et al. [25] using algorithm steps, then the results are given as follows. The final diagnosis matrix D is provided as the following:

  $$D=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}3.2& 2.8& \mathbf{3.88}& 3.05\\ 2.98& 2.86& \mathbf{3.56}& 2.3\\ 1.68& 1.7& \mathbf{2.61}& 2.22\end{array}\right)\end{array}.$$

  Therefore, from the final obtained diagnosis matrix D, the maximum value for all patients is the third one in each row, which is $3.88$, $3.56$, and $2.61$, for patients ${\pi }_1$, ${\pi }_2$, and ${\pi }_3$, respectively. That is to say that all patients are diagnosed with ${\xi }_3$, which is COVID-19. In addition, the patients’ order as alternatives according to the need of treatment is as follows: ${\pi }_1>{\pi }_2>{\pi }_3$. Finally, we notice that those three patients are diagnosed with $RSV$ under our proposed model. On the other hand, they are diagnosed with COVID-19 under this model of Yang et al. [25]. This may occur because these two diseases have many similar symptoms. Therefore, the effectiveness of our model is to distinguish between those similar diseases.
• When one makes the final decision under the bipolar-valued fuzzy soft set, presented by Abdullah et al. [23] using method steps, then we have the results as the following. The final diagnosis matrix D is given by

  $$D=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}1.01& 0.82& 0.68& \mathbf{1.37}\\ 0.23& 0.45& 0.36& \mathbf{0.78}\\ \mathrm{-0.31}& 0.16& 0.05& \mathbf{1.04}\end{array}\right)\end{array}.$$

  Then, from the above final diagnosis matrix D, one can find that the maximum value for all patients ${\pi }_1$, ${\pi }_2$, and ${\pi }_3$ is the fourth one in each row, which is, respectively, as follows: $1.37$, $0.78$, and $1.04$. That is to say that all patients are diagnosed with ${\xi }_4$, which is $RSV$. In addition, the alternatives’ order according to treatment need is as the following: ${\pi }_1>{\pi }_3>{\pi }_2$.
• If the final decision is made under the bipolar-valued multi-fuzzy soft set, introduced by Yang et al. [27] using process steps, then the results are obtained as below. We obtain the final diagnosis matrix D as follows:

  $$D=\begin{array}{cc}& \begin{array}{cccc}{\xi }_1& {\xi }_2& {\xi }_3& {\xi }_4\end{array}\\ \begin{array}{c}{\pi }_1\\ {\pi }_2\\ {\pi }_3\end{array}& \left(\begin{array}{cccc}2.32& 2.17& 2.48& \mathbf{2.57}\\ 1.59& 2.02& \mathbf{2.3}& 1.79\\ 0.63& 1.11& 1.62& \mathbf{1.84}\end{array}\right)\end{array}.$$

  Hence, from this final diagnosis matrix D, we have the maximum value for the first patient ${\pi }_1$ is $2.57$, occurred by the fourth disease ${\xi }_4$. Then, the patient ${\pi }_1$ is suffering from the disease ${\xi }_4$, which is $RSV$. In addition, the maximum value in the second patient’s row ${\pi }_2$ is $2.3$, obtained by the fourth disease ${\xi }_4$. Therefore, the patient ${\pi }_2$ is suffering from the disease ${\xi }_3$, which is COVID-19. Moreover, the maximum value for the third patient ${\pi }_3$ is $1.84$, scored by the fourth disease ${\xi }_4$. Then, the patient ${\pi }_3$ is also suffering from $RSV$. Furthermore, the order of alternatives (patients) according to their need of treatment is as follows: ${\pi }_1>{\pi }_2>{\pi }_3$.

Finally, we can summarize the final medical decisions and the ranking order of the three patients in the following comparative table, namely

Table 1. Decisions and ranking of alternatives using different models on Example 6.

Models	${\mathit{\pi }}_1$	${\mathit{\pi }}_2$	${\mathit{\pi }}_3$	Ranking Order
Yang et al. [25]	3.88 $\to {\xi }_3$	3.56 $\to {\xi }_3$	2.61 $\to {\xi }_3$	${\pi }_1>{\pi }_2>{\pi }_3$
Abdullah et al. [23]	1.37 $\to {\xi }_4$	0.78 $\to {\xi }_4$	1.04 $\to {\xi }_4$	${\pi }_1>{\pi }_3>{\pi }_2$
Yang et al. [27]	2.57 $\to {\xi }_4$	2.3 $\to {\xi }_3$	1.84 $\to {\xi }_4$	${\pi }_1>{\pi }_2>{\pi }_3$
Proposed model	3.256 $\to {\xi }_4$	2.547 $\to {\xi }_4$	3.422 $\to {\xi }_4$	${\pi }_3>{\pi }_1>{\pi }_2$

, as well as one can find a chart that shows different models’ comparative results below in Figure 3:

7. Concluding Notes and Future Researches

The aim of this article is to derive a new hybrid extension of the ordinary or the crisp set, which is the effective bipolar-valued multi-fuzzy soft set of dimension n. The types and the related novel important concepts and operations, have been discussed. Furthermore, De Morgan’s laws and distributive laws, as well as associative properties, absorption properties, and commutative properties, have been conducted. Moreover, a decision-making approach has been provided based on the effective bipolar-valued multi-fuzzy soft sets of dimension n.

In addition, a real example of medical diagnosis has been illustrated to show how to use the proposed technique. To reach the best diagnosis easily, we have formulated the technique steps using matrixes instead of set extensions to be easier to deal with. Furthermore, to reach more accurate and faster results, we have used MATLAB^® to add and multiply matrixes through the paper, as well as to compute the effective values or any other calculations. This method facilitates obtaining the optimal decision that this patient is suffering from this disease or this student is accepted to take this education level, … In addition, the sensitivity analysis on parameters has been conducted. Finally, a comparison with other existing algorithms in terms of application to better demonstrate the advantages of the proposed algorithm has been established.

One of the advantages of the suggested model is that it is a generalization of many previous models like multi-fuzzy soft, bipolar-valued fuzzy soft, and multipolar fuzzy soft set. That is to say that either the multi-fuzzy soft set, the bipolar-valued fuzzy soft set, or the multipolar fuzzy soft set is a special case of the effective bipolar-valued multi-fuzzy soft set. This implies that using any one of them in decision-making applications may face limitations when the problem contains more complicated circumstances like bipolar attributes and/or multi attributes. Therefore, combining bipolarity and multi-set with fuzzy soft and effectiveness concepts increases the decision’s accuracy and uniqueness.

In certain cases, the suggested approach, like any other technique or framework, may have inherent restrictions, limitations, or drawbacks. In particular, one of those limitations happens if there are a significant number of attributes (parameters) or/and items (patients or students), resulting in a huge number of computations when using the method being proposed. In order to overcome this restriction, several mathematical programs, such as MATLAB^® or Wolfram Mathematica^®, which are capable of processing massive amounts of data quickly and efficiently, can be used.

Furthermore, another limitation is that the effective bipolar-valued multi-fuzzy soft model works effectively when combining bipolarity with multi-fuzzy soft data, but it cannot be effective when combining bipolarity with multi-vague soft data noticed in a variety of situations in the real world. That is to say that the effective bipolar-valued multi-fuzzy soft set definition alone is unable to communicate the vagueness, which is a generalization of the fuzziness.

Therefore, as a future idea, authors can define the effective bipolar-valued multi-vague soft set and use it in applications to overcome this limitation. In addition, applying either the effective bipolar-valued multi-fuzzy soft set or the effective bipolar-valued multi-vague soft set to real data and comparing the two results also may be an interesting future work because it will clarify the effectiveness of the two methods in reality. Moreover, in future research, authors can extend the ideas to picture effective bipolar-valued multi-fuzzy soft sets of dimension n, spherical effective bipolar-valued multi-fuzzy soft sets of dimension n, and Pythagorean effective bipolar-valued multi-fuzzy soft sets of dimension n.