Fuzzy N-Bipolar Soft Sets for Multi-Criteria Decision-Making: Theory and Application

1  Introduction

Handling uncertainty and imprecision in data has been a fundamental challenge in various scientific and engineering disciplines. Several mathematical frameworks have been proposed to address these challenges, each offering unique perspectives. Among them, Zadeh’s fuzzy set theory [1] stands out as a pioneering model for representing vagueness. Fuzzy sets allow for degrees of membership rather than binary classification, enabling a nuanced representation of imprecise information. This framework has found applications in control systems, decision-making (DM), and image processing, among others. On the other hand, Pawlak’s rough set theory [2] addresses uncertainty arising from indiscernibility or incomplete information. By employing equivalence classes, rough sets approximate imprecise concepts using lower and upper approximations. These models have been instrumental in areas such as feature selection, data mining, and knowledge discovery. Other advancements, such as intuitionistic fuzzy sets [3] and vague sets [4], have further enriched the landscape, each tailored to address specific types of uncertainties.

S-set theory, introduced by Molodtsov [5], provides a parameterized framework for addressing uncertainty in a versatile and straightforward manner. Unlike fuzzy sets and rough sets, which operate within fixed mathematical structures, S-sets pair objects with parameters to model problems flexibly. This approach has been extended in various ways to enhance its applicability. Fuzzy soft (FS) sets, proposed by Maji et al. [6], integrate the concept of fuzzy membership functions into the S-set framework, allowing for the representation of partial truths associated with each parameter. Other extensions include interval-valued FS sets [7], and Einstein q-rung orthopair FS sets [8], which expand the utility of S-sets in diverse applications such as DM [9] and medical diagnosis [10].

The initial work on operations of S-set theory by Maji et al. [11] laid a strong foundation, enabling applications across various domains. Ali et al. [12] further defined several new operations on S-sets. Shabir et al. extended this theory to introduce bipolar soft (BS) sets [13], emphasizing scenarios where attributes can simultaneously express positive and negative aspects. Naz et al. [14] expanded the concept by developing fuzzy BS (FBS) sets, a model incorporating fuzziness and bipolarity to address algebraic structures and real-world problems. These advancements underscored the flexibility and applicability of S-set theory in handling uncertainties. Recent developments have integrated S-set theory with DM strategies to tackle multi-criteria problems. For example, Musa et al. [15] introduced the concept of bipolar hypersoft sets. In a subsequent study [16], they utilized this model in DM applications, demonstrating its practical relevance and effectiveness. Asaad et al. [17] extended this framework to incorporate fuzzy evaluations. Rahman et al. [18] proposed a synergistic multi-criteria decision-making (MCDM) strategy using parameterized single-valued neutrosophic S-sets for selecting sustainable educational institution sites. Similarly, Ihsan et al. [19] applied Pythagorean FS expert sets to MCDM contexts. In supply chain management, Saeed et al. [20] employed possibility single-valued neutrosophic soft settings for efficient DM. Senapati [21] introduced an Aczel-Alsina aggregation-based outranking method for MCDM using single-valued neutrosophic numbers. These contributions demonstrate the potential of S-set theory and its extensions in solving complex DM and uncertainty-related problems. Alcantud et al. [22] provided a systematic literature review of S-set theory, further highlighting its extensive development and applications. Zulqarnain et al. [23] defined interval-valued Pythagorean FS sets and Wang et al. [24] proposed a novel concept of generalized Pythagorean FS sets in decision systems.

Building on these foundations, Fatimah et al. [25] introduced N-soft (N-S) set to enable multi-level graded evaluations. This framework is particularly suitable for problems requiring discrete grading systems, such as academic grading, performance evaluation, and ranking systems. The N-S set framework has been significantly extended to address diverse DM challenges. Akram et al. [26] introduced fuzzy N-soft set (FN-S) set, incorporating fuzziness to accommodate partial memberships under uncertainty. Additionally, Korkmaz et al. [27] proposed a novel approach to FN-S sets, applying them to the identification and sanctioning of cyber harassment on social media platforms, demonstrating their applicability in digital DM contexts. Rehman et al. [28] introduced picture FN-S set, extending fuzziness by including neutral and abstain membership values. Musa et al. [29] proposed N-hypersoft set, while Musa [30] introduced N-bipolar hypersoft set, and Musa et al. [31] defined N-bipolar hypersoft topology. Rehman et al. [32] suggested complex intuitionistic FN-S set, and Akram et al. [33] introduced complex q-rung orthopair FN-S set, enhancing DM with advanced fuzzy systems. Farooq et al. [34] developed complex bipolar FN-S set emphasizing bipolarity in collaborative and group DM.

Recently, Shabir et al. [35] proposed N-bipolar soft (N-BS) set. Furthermore, Riaz et al. [36,37] introduced M-parametrized N-S set, which provide a versatile approach to MCDM by incorporating an additional layer of parameterization. Building on this, Musa et al. [38] defined bipolar M-parametrized N-S set, enhancing the framework to address problems involving both positive and negative evaluations in DM contexts. In addition, Kamaci et al. [39] presented m-polar N-S set, demonstrating its applicability in MCDM. Khan et al. [40] presented a synergistic method for evaluating educational institutions using similarity measures of possibility Pythagorean fuzzy hypersoft sets, contributing to the broader application of advanced fuzzy systems in MCDM. Alballa et al. [41] proposed a solid waste management approach utilizing fuzzy parameterized possibility single-valued neutrosophic hypersoft expert settings, highlighting the use of fuzzy and neutrosophic sets in practical MCDM applications. Gul [42] extended the VIKOR approach for MCDM (VIKOR stands for “VIsekriterijumsko KOmpromisno Rangiranje”), incorporating a bipolar fuzzy preference δ-covering-based bipolar fuzzy rough set model, enhancing the DM process in complex environments with bipolar attributes. Wang et al. [43] introduced group DM methods based on probabilistic hesitant N-S sets, further refining the probabilistic approach in S-set-based DM methodologies. Musa et al. [44] proposed N-BS expert sets, expanding the DM paradigm in robust MCDM applications. These contributions further advance the field of MCDM, integrating new methodologies and frameworks that enhance the handling of uncertainty in real-world scenarios.

The FN-BS set framework provides a significant advancement over existing uncertainty theories, such as fuzzy sets, S-sets, and their various extensions. Traditional uncertainty models, while useful, often exhibit key limitations in handling the complexities of real-world DM scenarios. Specifically, many of these models struggle to simultaneously account for both positive and negative aspects (bipolarity) and to deal with evaluations involving more than two states (multinary evaluations). These shortcomings are particularly problematic in fields like MCDM, where decision-makers are frequently confronted with conflicting, uncertain, and imprecise data.

The FN-BS set model addresses these issues by integrating three critical components: fuzziness, bipolarity, and multi-graded levels of assessment. Fuzziness allows for modeling uncertain, imprecise information; bipolarity enables the capture of both positive and negative factors that influence decisions; and multinary evaluations provide the flexibility to consider more than just binary outcomes, thus reflecting the complexities of real-world decision contexts. For instance, in a healthcare MCDM scenario, FN-BS sets facilitate the simultaneous modeling of positive attributes such as treatment effectiveness and negative attributes like adverse effects, each with varying degrees of certainty and importance. This capability is particularly crucial in healthcare, where both the benefits and risks must be carefully weighed, and the evaluation of treatments or interventions involves more than just a yes/no decision.

Moreover, FN-BS sets allow for the inclusion of expert input, which can be essential for navigating complex DM processes where quantitative data may be scarce or difficult to interpret. The model’s ability to handle varying degrees of expert judgment further enhances its applicability in uncertain environments. By offering a more nuanced, comprehensive framework for DM, FN-BS sets provide decision-makers with a clearer understanding of the trade-offs involved in their choices, improving the reliability and robustness of their decisions.

These innovative features make the FN-BS framework highly relevant to modern research on uncertainty theories and their applications in DM. Unlike traditional models, which may be limited to simpler DM contexts, FN-BS sets are particularly suited for complex, uncertain, and multi-dimensional problems. Their ability to simultaneously incorporate multiple sources of uncertainty, such as both positive and negative factors and varying levels of expert input, provides a more holistic and realistic approach to DM under uncertainty. This novel approach represents a significant step forward in the development of DM models, demonstrating the potential of FN-BS sets to transform MCDM in areas such as healthcare, finance, and beyond.

1.1 Objectives

The primary objectives of this research are as follows:

•   To develop a robust and generalized framework for MCDM under uncertainty by integrating fuzziness, bipolarity, and multi-level graded evaluations.

•   To address the limitations of existing uncertainty theories by proposing a novel model, FN-BS sets, which combines the strengths of fuzzy logic, bipolar evaluations, and parameterized frameworks.

•   To establish a solid theoretical foundation for FN-BS sets by defining their operations, properties, and algebraic structures.

•   To demonstrate the practical utility of FN-BS sets through a real-world case study, showcasing their effectiveness in handling complex decision scenarios with both positive and negative attributes.

•   To provide comparative analysis and sensitivity evaluation of the FN-BS framework against existing approaches, highlighting its advantages and applicability in diverse MCDM contexts.

1.2 Motivation

DM in uncertain and complex environments has become a critical area of research due to its wide-ranging applications. Traditional DM models often fall short in addressing uncertainty, imprecision, and the intricate interplay of attributes, especially in scenarios where attributes exhibit opposing (bipolar) effects or require nuanced evaluations beyond binary classifications. The significance of this research lies in addressing these limitations by developing a more generalized and robust framework for MCDM.

FN-BS sets are important because they bridge these gaps by integrating fuzzy logic, bipolar evaluations, and non-binary assessments, offering a powerful tool for representing real-world problems more accurately. For instance, FN-BS sets enable simultaneous modeling of positive factors and negative factors, while incorporating graded levels of evaluation to reflect the complexities of outcomes. Such an approach is crucial for advancing MCDM methodologies and ensuring more reliable and informed decisions in uncertain environments.

1.3 Main Contributions

This paper introduces FN-BS sets as a novel and comprehensive framework designed to enhance MCDM processes in uncertain and complex environments. The key contributions of this work are as follows:

•   Addressing the Limitations of Existing Models: FN-BS sets extend traditional S-sets and other uncertainty theories by incorporating fuzziness, bipolarity, and multi-level graded evaluations, which are critical for capturing the nuanced nature of real-world decision problems.

•   Theoretical Development: This paper formally defines FN-BS sets, introduces their set-theoretic operations, and explores their algebraic properties, providing a solid theoretical foundation for further research.

•   Novel Applications: FN-BS sets are applied to a practical case study, demonstrating their effectiveness in modeling complex scenarios with both positive and negative attributes. Two algorithms are proposed for evaluating and selecting optimal choices in DM scenarios, showcasing the practical utility of the FN-BS framework.

•   Comparative Analysis: The FN-BS model is compared against existing approaches, highlighting its advantages in handling binary and non-binary evaluations, bipolar settings, and membership degrees.

These contributions collectively establish the FN-BS framework as a significant advancement in the field of uncertainty theories and MCDM, addressing critical gaps and enabling more robust DM processes.

1.4 Structure of the Paper

The structure of this paper is organized as follows:

•   Section 2: Reviews the foundational concepts of fuzzy sets, S-sets, and their extensions, providing the theoretical background necessary for understanding FN-BS sets.

•   Section 3: Introduces the novel framework of FN-BS sets, starting with their formal definition and foundational concepts. This section also explores set-theoretic operations and algebraic properties related to FN-BS sets.

•   Section 4: Demonstrates the application of FN-BS sets in MCDM, presenting two algorithms for evaluating and selecting optimal choices in decision scenarios. A case study of selecting the best vaccination program is used to illustrate these algorithms.

•   Section 5: Compares and discusses the outcomes of the two proposed algorithms, highlighting their strengths and the consistency of results across different approaches.

•   Section 6: Presents a sensitivity analysis of the FN-BS model against existing approaches, emphasizing its advantages in handling binary and non-binary evaluations, bipolar settings, and membership degrees.

•   Section 7: Provides the conclusions, summarizing the findings of the paper, discussing limitations of the FN-BS model, and suggesting directions for future work, including possible extensions of the FN-BS framework and its integration with emerging technologies.

Through this structure, the paper systematically presents the theoretical development, application, and comparative evaluation of FN-BS sets, providing a comprehensive perspective on their usefulness for MCDM.

2  Preliminaries

This section reviews the foundational concepts of fuzzy sets, S-sets, and their extensions, which underpin the study. Here, S denotes a universal set of objects, A represents a set of attributes (parameters), and G={0,1,…,N−1} a set of ordered grades, where N∈{2,3,…}.

Definition 1. [1] A fuzzy set μ over S is defined by a membership function μ:S→[0,1], which assigns to each element s∈S a membership degree in the interval [0,1]. The value μs represents the degree to which s belongs to S. The collection of all fuzzy sets on S is denoted by ℱ(S).

Definition 2. [5] An S-set is represented as a pair (Γ,A), where Γ:A→P(S), and P(S) denotes the power set of S, i.e., the set of all crisp subsets of S.

Definition 3. [6] A pair (Λ,A) is called an FS set, where Λ:A→ℱ(S).

Definition 4. [25] An N-S set is represented as a triple (Θ,A,N), where Θ:A→P(S×G) satisfies the condition that for each a∈A, there exists a unique pair (s,ga)∈S×G such that (s,ga)∈Θ(a) or equivalently, Θ(a)(s)=ga, where s∈S and ga∈G. Here, P(S×G) denotes the set of all crisp subsets of S×G.

Definition 5. [26] An FN-S set is expressed as a triple (Ω,A,N), where Ω:A→ℱ(S×G), and for each a∈A, ⟨(s,ga),Ωga⟩∈Ω(a) or equivalently, Ω(a)(s)=⟨ga,Ωga⟩. Here, s∈S, ga∈G, Ωga∈[0,1], and ℱ(S×G) represents the collection of all fuzzy subsets of S×G.

Definition 6. [11] For a set of attributes A={a1,a2,…,an}, the NOT set of A, denoted ¬A, is defined as ¬A={¬a1,¬a2,…,¬an}, where ¬ai indicates the negation of ai for i=1,2,…,n.

Definition 7. [13] A BS set is a triple (ψ,ζ,A), where ψ:A→P(S) and ζ:¬A→P(S) such that, for all a∈A, ψ(a)∩ζ(¬a)=∅, where ψ(a),ζ(¬a)∈S.

Definition 8. [14] An FBS set is represented by a triple (γ,λ,A), where γ:A→ℱ(S) and λ:¬A→ℱ(S) such that, for all a∈A, the condition 0≤γ(a)(s)+λ(¬a)(s)≤1 holds, where s∈S and γ(a)(s),λ(¬a)(s)∈[0,1].

Definition 9. [35] An N-BS set is a quadruple (f,h,A,N), where f:A→P(S×G) and h:¬A→P(S×G) satisfy the following conditions: For each a∈A, there exists a unique pair (s,ga)∈S×G such that (s,ga)∈f(a), or equivalently, f(a)(s)=ga and for each ¬a∈¬A, there exists a unique pair (s,g¬a)∈S×G such that (s,g¬a)∈h(¬a), or equivalently, h(¬a)(s)=g¬a, subject to the condition ga+g¬a≤N−1, where s∈S and ga,g¬a∈G.

3  Fuzzy N-Bipolar Soft Sets

This section introduces and systematically analyzes the novel framework of FN-BS sets. It is structured into three subsections: foundational concepts, set-theoretic operations, and algebraic properties. The aim is to establish a robust mathematical basis for this model.

3.1 Definition and Fundamental Concepts of Fuzzy N-Bipolar Soft Sets

In this subsection, the novel hybrid model of FN-BS sets is formally defined. Key foundational concepts are introduced, including the representation of FN-BS sets and their components, providing a comprehensive framework for their application in DM scenarios.

Definition 10. An FN-BS set on a universe of objects S, a set of attributes A⊆E, and a set of ordered grades G is defined as a quadruple (μ,η,A,N), where μ:A→ℱ(S×G) and η:¬A→ℱ(S×G), such that for each a∈A and s∈S, ⟨(s,ga),μga⟩∈μ(a), or equivalently, μ(a)(s)=⟨ga,μga⟩ and for each ¬a∈¬A and s∈S, ⟨(s,g¬a),ηg¬a⟩∈η(¬a), or equivalently, η(¬a)(s)=⟨g¬a,ηg¬a⟩, with the conditions:

ga+g¬a≤N−1,(1)

0≤μga+ηg¬a≤1,(2)

where ga,g¬a∈G, and μga,ηg¬a∈[0,1].

It is assumed that both S and A are finite unless otherwise specified. In such cases, the FN-BS set can be represented in a tabular format. Here, ⟨gij+,μij⟩ and ⟨gij−,ηij⟩ represent ⟨(si,gij+),μij⟩∈μ(aj) and ⟨(si,gij−),ηij⟩∈η(¬aj), respectively. This tabular representation is shown in Table 1.

Remark 1. The following points require attention in relation to Definition 10:

1.   The condition ga+g¬a≤N−1 introduces a restriction to prevent contradictions or inconsistent evaluations within the model. In a bipolar evaluation system, an object cannot be both fully present (for a given attribute a) and fully absent (for its negation ¬a) to the same extreme degree. For example, if an attribute's grade is very high, the negation of the attribute must have a relatively low grade to maintain this balance.

2.   The condition 0≤μga+ηg¬a≤1 ensures that the total membership degree for an object with respect to an attribute a and its negation ¬a does not exceed 1. Essentially, it says that for each attribute a, if an object is highly present with respect to the attribute a (i.e., a high membership degree for ga), it should be less present with respect to the negation of that attribute (i.e., a low membership degree for g¬a), and vice versa.

3.   The relationship ga∩g¬a=∅ may not hold in general. This is because, in certain cases, an object might have evaluations for both the attribute a and its negation ¬a that are not mutually exclusive. For example, an object might have a certain degree of presence or relevance for both the attribute a and its negation ¬a. In such cases, the evaluations for both the attribute and its negation coexist and overlap, meaning the intersection of ga and g¬a is not empty.

4.   The grades and their corresponding values are considered proportional and may vary. This variation in the grades is due to the context and sensitivity of the evaluation process, where the assignment of grades and their corresponding values can adapt to different scales or evaluation requirements.

To better understand the essential aspects of our new model, consider the following example.

Example 1. Consider a medical institution evaluating patients S={s1,s2,s3,s4} for a specific treatment based on the set of attributes A={a1=good health,a2=high mobility,a3=strong immune system} and their negations ¬A={¬a1=poor health,¬a2=low mobility,¬a3=weak immune system}. From Definition 9, the 4-BS set can be obtained from Table 2, where

•   One square "□" represents least favorable.

•   One triangle "△" represents slightly favorable.

•   Two triangles "△△" represent moderately favorable.

•   Three triangles "△△△" represent most favorable.

This graded evaluation by symbols can easily be identified with numbers as in G={0,1,2,3} where

•   0 corresponds to ◻.

•   1 corresponds to △.

•   2 corresponds to △△.

•   3 corresponds to △△△.

The tabular representation of the 4-BS set (μ′,η′,A,4) is shown in Table 3.

This information is sufficient when derived from precise data; however, when the data is vague or uncertain, the FN-BS set becomes essential to provide insights into how these grades are assigned to patients. The selection panel assigns membership values based on the evaluation grade of the patients as follows:

For positive membership values (μga):

0≤μga<0.2 when ga=0,0.2≤μga<0.5 when ga=1,0.5≤μga<0.8 when ga=2,0.8≤μga≤1 when ga=3.(3)

For negative membership values (ηg¬a):

0≤ηg¬a<0.2 when g¬a=0,0.2≤ηg¬a<0.5 when g¬a=1,0.5≤ηg¬a<0.8 when g¬a=2,0.8≤ηg¬a≤1 when g¬a=3.(4)

Then, the F4-BS set (μ,η,A,4) for the same patients and attributes is presented in Table 4. To illustrate, for s1, the grade ga1=3 reflects excellent health, with the membership value μga1=0.8 quantifying its strong alignment. In contrast, for ¬a1, the grade g¬a1=0 indicates minimal concern for poor health, with ηg¬a1=0.1 confirming this minimal association.

Remark 2. The grade 0∈G, as defined in Definition 10, does not signify a lack of information or incompleteness. Instead, it denotes the minimum grade within the hierarchy of ordered grades.

Remark 3. Any FN-BS set can naturally be viewed as an F(N + 1)-BS set or, more generally, as an FM-BS set where M > N.

The rationale for this concept lies in the fact that, in some situations, the highest grades may appear in (μ,A,N) but not in (η,¬A,N), and vice versa. In certain cases, the highest grades might be found in both sets. Based on this insight, the following definitions are provided.

Definition 11. An FN-BS set (μ,η,A,N) is called positively efficient if, for some a∈A and s∈S, μ(a)(s)=⟨N−1,μN−1⟩, and for some ¬a∈¬A and s∈S, η(¬a)(s)=⟨0,η0⟩, where μN−1,η0∈[0,1].

Definition 12. An FN-BS set (μ,η,A,N) is termed negatively efficient if, for some a∈A and s∈S, μ(a)(s)=⟨0,μ0⟩, and for some ¬a∈¬A and s∈S, η(¬a)(s)=⟨N−1,ηN−1⟩, where μ0,ηN−1∈[0,1].

Definition 13. An FN-BS set (μ,η,A,N) is said to be totally efficient if it is both positively efficient and negatively efficient.

Definition 14. If (μ,η,A,N) is not a totally efficient FN-BS set, then its minimized FN-BS set, denoted by (μm,ηm,A,M), is defined as follows: for every a∈A, ¬a∈¬A, and s∈S, M=max{ga,g¬a}+1, gam=ga, g¬am=g¬a, μgamm=μga, and ηg¬amm=ηg¬a, where ⟨ga,μga⟩∈μ(a)(s), ⟨gam,μgamm⟩∈μm(a)(s), ⟨g¬a,ηg¬a⟩∈η(¬a)(s), and ⟨g¬am,ηg¬amm⟩∈ηm(¬a)(s).

Remark 4. Any minimized FN-BS set (μm,ηm,A,M) of a non-totally efficient FN-BS set (μ,η,A,N) is always totally efficient.

Remark 5. For each value of 0<𝒯<N, there exists an associated FBS set for every FN-BS set.

Definition 15. Given a threshold 0<𝒯<N and an FN-BS set (μ,η,A,N), the corresponding FBS set is denoted as (μ𝒯,η𝒯,A) and is defined by the following expressions, for all a∈A and s∈S:

μ𝒯(a)(s)={μga,if ⟨ga,μga⟩∈μ(a)(s) such that ga≥T0,otherwise(5)

and for all ¬a∈¬A and s∈S:

η𝒯(¬a)(s)={ηg¬a,if ⟨g¬a,ηg¬a⟩∈η(¬a)(s) such that g¬a≥N−𝒯0,otherwise(6)

In particular, we define (μ𝒯=1,η𝒯=1,A) as the bottom FBS set corresponding to (μ,η,A,N), while (μ𝒯=N−1,η𝒯=N−1,A) is known as the top FBS set associated with (μ,η,A,N).

Definition 16. An FN-BS set (μO,ηO,A,N) is called a relative null FN-BS set if, for every a∈A and s∈S, μO(a)(s)=⟨0,0⟩, and for every ¬a∈¬A and s∈S, ηO(¬a)(s)=⟨N−1,1⟩.

Definition 17. An FN-BS set (μW,ηW,A,N) is referred to as a relative whole FN-BS set if, for all a∈A and s∈S, μW(a)(s)=⟨N−1,1⟩, and for all ¬a∈¬A and s∈S, ηW(¬a)(s)=⟨0,0⟩.

3.2 Set-Theoretic Operations on Fuzzy N-Bipolar Soft Sets

This subsection investigates various set-theoretic operations within the framework of FN-BS sets, such as complement, subset, union, intersection, and others. The properties of these operations are analyzed, with examples to illustrate their behavior and potential applications.

In the context of FN-BS sets, there are four types of complementary operations, starting with the most significant one as follows:

Definition 18. The FN-BS complement of (μ,η,A,N), denoted by (μ,η,A,N)c~, is defined as (μ,η,A,N)c~=(μc~,ηc~,A,N), such that, for all a∈A and s∈S, gac~=g¬a and μgac~c~=ηg¬a, and for all ¬a∈¬A and s∈S, g¬ac~=ga and ηg¬ac~c~=μga.

Definition 19. The FN-BS weak complement of (μ,η,A,N) is any FN-BS set (μ,η,A,N)ω~=(μω~,ηω~,A,N), where, for all a∈A and s∈S, gaω~∩ga=∅ and μgaω~ω~∩μga=∅, and for all ¬a∈¬A and s∈S, g¬aω~∩g¬a=∅ and ηg¬aω~ω~∩ηg¬a=∅.

Definition 20. The FN-BS top weak complement of (μ,η,A,N) is (μ,η,A,N)t~=(μt~,ηt~,A,N) where

μt~(a)(s)={⟨N−1,μga⟩,if ga<N−1⟨0,μga⟩,if ga=N−1(7)

and

ηt~(¬a)(s)={⟨0,ηg¬a⟩,ifg¬a>0⟨0,ηg¬a⟩,ifg¬a=0andga<N−1⟨N−1,ηg¬a⟩,otherwise(8)

Here, μga,ηg¬a∈[0,1].

Definition 21. The FN-BS bottom weak complement of (μ,η,A,N) is (μ,η,A,N)b~=(μb~,ηb~,A,N) where

μb~(a)(s)={⟨0,μga⟩,if ga>0⟨0,μga⟩,if ga=0and g¬a<N−1⟨N−1,μga⟩,otherwise(9)

and

μb~(¬a)(s)={⟨N−1,ηg¬a⟩,if g¬a<N−1⟨0,ηg¬a⟩,if g¬a=N−1(10)

Here, μga,ηg¬a∈[0,1].

Example 2. Consider the F4-BS set (μ,η,A,4) as in Example 1. The F4-BS complement, weak complement, top weak complement, and bottom weak complement of this set are presented in tabular form in Tables 5–8.

Definition 22. An FN-BS set (μ1,η1,A1,N) is considered a subset of (μ2,η2,A2,N), denoted as (μ1,η1,A1,N)⊑~(μ2,η2,A2,N), if the following conditions are satisfied:

1.   A1⊆A2.

2.   For every a∈A1 and s∈S, ga1≤ga2 and μga11≤μga22, where ⟨ga1,μga11⟩∈μ1(a)(s) and ⟨ga2,μga22⟩∈μ2(a)(s).

3.   For every ¬a∈¬A1 and s∈S, g¬a2≤g¬a1 and ηg¬a22≤ηg¬a21, where ⟨g¬a1,ηg¬a11⟩∈η1(¬a)(s) and ⟨g¬a2,ηg¬a22⟩∈η2(¬a)(s).

Definition 23. Two FN-BS sets (μ1,η1,A1,N) and (μ2,η2,A2,N) over S are said to be FN-BS equal if the following conditions hold:

1.   A1=A2.

2.   For every a∈A1=A2 and s∈S, ga1=ga2 and μga11=μga22, where ⟨ga1,μga11⟩∈μ1(a)(s) and ⟨ga2,μga22⟩∈μ2(a)(s).

3.   For every ¬a∈¬A1=¬A2 and s∈S, g¬a1=g¬a2 and ηg¬a11=ηg¬a22, where ⟨g¬a1,ηg¬a11⟩∈η1(¬a)(s) and ⟨g¬a2,ηg¬a22⟩∈η2(¬a)(s).

Definition 24. The FN-BS extended union of (μ1,η1,A1,N1) and (μ2,η2,A2,N2) is denoted and defined as (μ1,η1,A1,N1) ⊔~ε (μ2,η2,A2,N2)= (μ,η,A1∪A2,max(N1,N2)), where for all a∈A1∪A2 and s∈S:

μ(a)(s)={μ1(a)(s),if a∈A1\A2μ2(a)(s),if a∈A2\A1⟨ga,μga⟩,such that ga=max{ga1,ga2} and μga=max{μga11,μga22}where ⟨ga1,μga11⟩∈μ1(a)(s) and ⟨ga2,μga22⟩∈μ2(a)(s).(11)

and for all ¬a∈¬A1∪¬A2 and s∈S:

η(¬a)(s)={η1(¬a)(s),if ¬a∈¬A1\¬A2η2(¬a)(s),if ¬a∈¬A2\¬A1⟨g¬a,ηg¬a⟩,such that g¬a=min{g¬a1,g¬a2} and ηg¬a=min{ηg¬a11,ηg¬a22}where ⟨g¬a1,ηg¬a11⟩∈η1(¬a)(s)and⟨g¬a2,ηg¬a22⟩∈η2(¬a)(s).(12)

Definition 25. The FN-BS extended intersection of (μ1,η1,A1,N1) and (μ2,η2,A2,N2) is denoted and defined as (μ1,η1,A1,N1) ⊓~ε (μ2,η2,A2,N2)= (μ,η,A1∪A2,max(N1,N2)), where for all a∈A1∪A2 and s∈S:

μ(a)(s)={μ1(a)(s),if a∈A1\A2μ2(a)(s),if a∈A2\A1⟨ga,μga⟩,such that ga=min{ga1,ga2} and μga=min{μga11,μga22}where ⟨ga1,μga11⟩∈μ1(a)(s) and ⟨ga2,μga22⟩∈μ2(a)(s).(13)

and for all ¬a∈¬A1∪¬A2 and s∈S:

η(¬a)(s)={η1(¬a)(s),if ¬a∈¬A1\¬A2η2(¬a)(s),if ¬a∈¬A2\¬A1⟨g¬a,ηg¬a⟩,such that g¬a=max{g¬a1,g¬a2} and ηg¬a=max{ηg¬a11,ηg¬a22}where ⟨g¬a1,ηg¬a11⟩∈η1(¬a)(s) and ⟨g¬a2,ηg¬a22⟩∈η2(¬a)(s).(14)

Definition 26. The FN-BS restricted union of (μ1,η1,A1,N1) and (μ2,η2,A2,N2) is denoted and defined as (μ1,η1,A1,N1) ⊔~ℜ (μ2,η2,A2,N2)= (μ,η,A1∩A2,max(N1,N2)), where for all a∈A1∩A2≠∅ and s∈S:

μ(a)(s)=⟨ga,μga⟩(15)

where ga=max{ga1,ga2} and μga=max{μga11,μga22}, with ⟨ga1,μga11⟩∈μ1(a)(s) and ⟨ga2,μga22⟩∈μ2(a)(s), and for all ¬a∈¬A1∩¬A2≠∅ and s∈S:

η(¬a)(s)=⟨g¬a,ηg¬a⟩(16)

where g¬a=min{g¬a1,g¬a2} and ηg¬a=min{ηg¬a11,ηg¬a22}, with ⟨g¬a1,ηg¬a11⟩∈η1(¬a)(s) and ⟨g¬a2,ηg¬a22⟩∈η2(¬a)(s).

Definition 27. The FN-BS restricted intersection of (μ1,η1,A1,N1) and (μ2,η2,A2,N2) is denoted and defined as (μ1,η1,A1,N1) ⊓~ℜ (μ2,η2,A2,N2)= (μ,η,A1∩A2,max(N1,N2)), where for all a∈A1∩A2≠∅ and s∈S:

μ(a)(s)=⟨ga,μga⟩(17)

where ga=min{ga1,ga2} and μga=min{μga11,μga22}, with ⟨ga1,μga11⟩∈μ1(a)(s) and ⟨ga2,μga22⟩∈μ2(a)(s), and for all ¬a∈¬A1∩¬A2≠∅ and s∈S:

η(¬a)(s)=⟨g¬a,ηg¬a⟩(18)

where g¬a=min{g¬a1,g¬a2} and ηg¬a=min{ηg¬a11,ηg¬a22}, with ⟨g¬a1,ηg¬a11⟩∈η1(¬a)(s) and ⟨g¬a2,ηg¬a22⟩∈η2(¬a)(s).

Example 3. Consider F4-BS set (μ,η,A,4) in Example 1 and the F3-BS set (μ1,η1,A1,3) described in tabular form by Table 9. The FN-BS extended union (intersection) and the FN-BS restricted union (intersection) are given in Tables 10–13.

3.3 Algebraic Properties of Operations on Fuzzy N-Bipolar Soft Sets

The algebraic properties associated with operations on FN-BS sets are explored in this subsection. These include distributive, associative, and commutative properties, among others. The relationships between these properties are highlighted, and their implications for the structure of FN-BS sets are discussed.

Proposition 1. Let (μ1,η1,A,N), (μ2,η2,A,N), and (μ3,η3,A,N) be three FN-BS sets. Then

1.   (μ1,η1,A,N) ⊑~ (μW,ηW,A,N).

2.   (μO,ηO,A,N) ⊑~ (μ1,η1,A,N).

3.   If (μ1,η1,A,N) ⊑~ (μ2,η2,A,N) and (μ2,η2,A,N) ⊑~ (μ3,η3,A,N), then (μ1,η1,A,N) ⊑~ (μ3,η3,A,N).

Proof. Straightforward. ◼

Proposition 2. Let (μ1,η1,A1,N) and (μ2,η2,A2,N) be two FN-BS sets. Then

1.   (μ1,η1,A1,N) ⊔~ε (μ2,η2,A2,N) is the smallest FN-BS set which contains both (μ1,η1,A1,N) and (μ2,η2,A2,N).

2.   (μ1,η1,A1,N) ⊓~ℜ (μ2,η2,A2,N) is the largest FN-BS set which is contained in both (μ1,η1,A1,N) and (μ2,η2,A2,N).

Proof. Straightforward. ◼

Proposition 3. Let (μ1,η1,A,N) and (μ2,η2,A,N) be two FN-BS sets. Then

1.   (μO,ηO,A,N)c~ = (μW,ηW,A,N).

2.   (μW,ηW,A,N)c~ = (μO,ηO,A,N).

3.   ((μ1,η1,A,N)c~)c~ = (μ1,η1,A,N).

4.   If (μ1,η1,A,N) ⊑~ (μ2,η2,A,N), then (μ2,η2,A,N)c~ ⊑~ (μ1,η1,A,N)c~.

5.   (μO,ηO,A,N) ⊑~ (μ1,η1,A,N) ⊓~ℜ (μ1,η1,A,N)c~ ⊑~ (μ1,η1,A,N) ⊔~ℜ (μ1,η1,A,N)c~ ⊑~ (μW,ηW,A,N).

6.   If (μ1,η1,A,N) ⊑~ (μ2,η2,A,N), then (μ1,η1,A,N) ⊓~ℜ (μ2,η2,A,N)= (μ1,η1,A,N).

7.   If (μ1,η1,A,N) ⊑~ (μ2,η2,A,N), then (μ1,η1,A,N) ⊔~ℜ (μ2,η2,A,N)= (μ2,η2,A,N).

Proof. Straightforward. ◼

Proposition 4. Let (μ1,η1,A1,N) and (μ2,η2,A2,N) be two FN-BS sets. Then

1.   ((μ1,η1,A1,N) ⊔~ε (μ2,η2,A2,N))c~ = (μ1,η1,A1,N)c~ ⊓~ε (μ2,η2,A2,N)c~.

2.   ((μ1,η1,A1,N) ⊓~ε (μ2,η2,A2,N))c~ = (μ1,η1,A1,N)c~ ⊔~ε (μ2,η2,A2,N)c~.

3.   ((μ1,η1,A1,N) ⊔~ℜ (μ2,η2,A2,N))c~ = (μ1,η1,A1,N)c~ ⊓~ℜ (μ2,η2,A2,N)c~.

4.   ((μ1,η1,A1,N) ⊓~ℜ (μ2,η2,A2,N))c~ = (μ1,η1,A1,N)c~ ⊔~ℜ (μ2,η2,A2,N)c~.

Proof. (1) Let (μ1,η1,A1,N) ⊔~ε (μ2,η2,A2,N) = (μ3,η3,A1∪A2,N). Then, ((μ1,η1,A1,N) ⊔~ε (μ2,η2,A2,N))c~ = (μ3,η3,A1∪A2,N)c~ = (μ3c~,η3c~,A1∪A2,N). By definition, for all a∈A1∪A2 and s∈S:

μ3(a)(s)={μ1(a)(s),if a∈A1\A2μ2(a)(s),if a∈A2\A1⟨ga3,μga33⟩,such that ga3=max{ga1,ga2} and μga33=max{μga11,μga22}where ⟨ga1,μga11⟩∈μ1(a)(s) and ⟨ga2,μga22⟩∈μ2(a)(s).(19)