Urban Transportation Strategy Selection for Multi-Criteria Group Decision-Making Using Pythagorean Fuzzy N-Bipolar Soft Expert Sets

1  Introduction

1.1 Research Background

Designing successful urban mobility system solutions is a problem of multifaceted complexity driven by competing priorities such as cost, environmental conservation, optimal traffic flow, and social justice. Decision-makers must balance inputs from a broad spectrum of stakeholders with varying concerns and preferences, typically based on personal intuition or untested data. These complexities are compounded further by the dynamic context of cities, where policy must be continuously adapted to respond to shifting demands. As a result, conventional decision-making (DM) models—typically using strict or dichotomous logic—fall short to capture reality-based contexts’ complexities, and hence more and more reliance is being placed on computational models capable of coping with uncertainty and vagueness.

A foundational mathematical approach for modeling such uncertainty and vagueness is fuzzy set (FS) theory, introduced by Zadeh [1], which allows partial membership and captures the imprecision of human reasoning. Intuitionistic FSs (IFSs), developed by Atanassov [2], extend FS by introducing independent membership degrees (MDs) and non-membership degrees (NMDs). Yager’s Pythagorean FSs (PFSs) [3] further generalize IFSs by allowing the squared sum of MDs and NMDs to be at most one, offering enhanced flexibility. PFSs have gained wide recognition in multi-criteria group decision-making (MCGDM) due to their expressive power, with applications in distance measures [4–6], structural modeling [7,8], and decision support systems [9].

1.2 Related Work

Soft set (SS) theory, introduced by Molodtsov [10], provides a parameterized framework for uncertainty modeling, independent of probabilistic or fuzzy principles. By associating elements with parameter sets, SSs effectively handle vague and incomplete data. The framework was further developed by Maji et al. [11] and Ali et al. [12], who formalized core operations. Extensions such as fuzzy SSs [13], intuitionistic fuzzy SSs [14], and Pythagorean fuzzy SSs [15] have broadened its applicability. These hybrid models have been successfully applied in medical diagnosis [16], multi-attribute aggregation [17], structural analysis [18], clustering [19], and biological networks [20], highlighting their suitability for complex MCGDM environments [21].

Bipolarity reflects the dual nature of human reasoning, where evaluations consider both positive and negative dimensions. Shabir and Naz [22] introduced bipolar soft sets (BSSs), extending SSs to accommodate contradictory information. Subsequent refinements by Karaaslan and Karataş [23] and Mahmood and Jaballah [24] improved operational definitions and adaptability. Fuzzy extensions [25,26] have enhanced modeling precision. In particular, the works by Zeb et al. [27,28] stand out by developing advanced aggregation operators for fuzzy and Pythagorean fuzzy BSSs, significantly advancing practical DM methodologies. Applications in decision models [29] and error-reduction frameworks [30] confirm the practical value of these theoretical advances.

Many existing models rely on binary or real-valued evaluations within [0,1], which often fail to reflect symbolic or multilevel assessments used in real-world scenarios (e.g., star ratings or badges). To address this, Fatimah et al. [31] introduced N-soft sets (NSSs), which support multinary or symbolic evaluations linked to parameters. NSSs enable a more nuanced expression of uncertainty and preferences, applicable to areas like three-way decision theory [32]. Further developments include aggregation methods [33,34], m-polar NSSs [35], and M-parameterized topology-based models [36]. Bipolar extensions—such as bipolar M-parameterized NSSs [37], N-bipolar soft sets (NBSSs) [38] and fuzzy NBSSs [39]—allow simultaneous modeling of positive and negative information.

In situations that require the active involvement of multiple experts—each contributing unique insights shaped by their individual backgrounds, expertise, and value systems—modeling and aggregating conflicting or diverse opinions becomes a critical component of the DM process. Soft expert sets (SESs), introduced by Alkhazaleh and Salleh [40], provide a foundational structure by embedding expert evaluations directly into SS theory. This integration supports group DM under uncertainty, where collective judgments must be synthesized despite partial knowledge, imprecise preferences, and subjective bias.

To better capture the nuances of expert-based assessments, several fuzzy generalizations have been proposed, including fuzzy SESs (FSESs) [41], intuitionistic fuzzy SESs (IFSESs) [42], and Pythagorean fuzzy SESs (PFSESs) [43], each offering richer semantics for expressing uncertainty through MDs and NMDs. These models have been further extended by incorporating bipolar reasoning, enabling the representation of both favorable and unfavorable expert views within a unified framework. Notable developments in this regard include bipolar SESs (BSESs) [44], fuzzy bipolar SESs (FBSESs) [45], and soft expert models built upon multinary evaluations, such as (fuzzy) N-soft expert sets ((F)NSESs) [46] and their Pythagorean counterparts, Pythagorean fuzzy NSESs (PFNSESs) [47].

Recent advances have combined the expressiveness of N-soft structures and expert and bipolar assessment to generate models like N-bipolar soft expert sets (NBSESs) [48], fuzzy NBSESs (FNBSESs) [49], and intuitionistic fuzzy NBSESs (IFNBSESs) [50]. These enhanced frameworks are designed to handle at the same time symbolic or linguistic scales, bipolar assessments, various expert opinions, and various forms of uncertainty. Collectively, these models constitute a comprehensive and integrated MCGDM methodology that can more effectively represent real-world complexities than can classical methodologies. As they currently exist, SESs and their modern extensions represent a powerful class of tools for implementing consensus-based decision methods under uncertainty and multi-dimensionality conditions.

From a broader methodological standpoint, Kumar and Pamucar [51] presented a large-scale bibliometric analysis of MCGDM developments over the past two decades, highlighting the rise of hybrid fuzzy methods and identifying application gaps across various sectors. The review confirms that while fuzzy MCGDM methods have diversified in terms of structure and application, challenges remain in modeling heterogeneous expert knowledge, symbolic preferences, and bipolar trade-offs.

In parallel, domain-specific applications of fuzzy and MCGDM methods for sustainable DM have gained momentum. For instance, Nayeb-Pashaei et al. [52] applied the fuzzy logarithm methodology of additive weights using triangular fuzzy numbers to identify and rank sustainability criteria in urban transport systems. Their model effectively captures uncertainty in sustainability dimensions such as health, safety, and emissions. However, it does not accommodate expert-level disagreement or contradictory evaluations, which are often critical in sustainability planning.

These gaps motivate the development of Pythagorean fuzzy N-bipolar soft expert set (PFNBSES), which uniquely integrates Pythagorean fuzzy logic with NBSES, allowing for more nuanced, expert-driven, and context-sensitive sustainability DM frameworks.

1.3 Research Gap and Motivation

The motivation for developing the PFNBSES model is that DM environments in the real world are growing more and more complex, particularly those involving multiple experts, uncertainty, bipolarity, and qualitative diversity. Classical SS paradigms, as essential for handling uncertainty, cannot handle at the same time bipolarity, multinary evaluation, and expert-based discrimination. BSSs handle dual-valued information but are usually limited to binary evaluation. NSSs are supportive of multinary symbolic evaluation but are not necessarily supportive of expert consensus or bipolar scales. SESs are supportive of multiple expert opinions but are usually developed from lower-order fuzzy or intuitionistic structures.

This layered design enables PFNBSES to reflect the intrinsic dualities and subjectivities of modern DM problems while maintaining computational rigor and interpretability. Its development is driven by the practical need for a scalable, expressive, and expert-driven framework applicable to domains such as urban planning, medical diagnostics, and cybersecurity.

1.4 Novelties and Advantages of the Proposed Model

The PFNBSES model offers a unified and flexible framework that integrates Pythagorean fuzzy logic, multinary evaluation scales, bipolar reasoning, and multi-expert aggregation. These features collectively enable more nuanced, balanced, and reliable DM compared to existing SS-based models.

A detailed comparative analysis of PFNBSES’s strengths and advantages is provided in Section 5.2, where its superior handling of uncertainty, expert diversity, and bipolarity is rigorously demonstrated.

This succinct overview highlights PFNBSES’s significant contribution to advancing MCGDM methodologies.

1.5 Objectives and Key Contributions

This study aims to develop a comprehensive DM framework for MCGDM environments characterized by uncertainty, bipolar evaluations, and expert diversity. The PFNBSES model unifies and extends several SS-based methodologies to improve flexibility, precision, and applicability. It addresses methodological gaps in SSs, BSSs, and soft expert frameworks by incorporating Pythagorean fuzzy logic, multinary evaluation, and expert consensus within a single model.

The main contributions of this paper are as follows:

•   Formal definition of the PFNBSES model with foundational operations including union, intersection, complement, agreement, and disagreement, supported by algebraic properties.

•   Development of a complete algorithmic framework for MCGDM using PFNBSES, enabling systematic processing of expert evaluations involving both positive and negative attributes.

•   Application of the proposed model to a real-world case study in urban transportation strategy planning, demonstrating its effectiveness in handling conflicting criteria and supporting transparent DM.

•   Comparative analysis benchmarking PFNBSES against existing SS-based models, highlighting its superior capability in addressing bipolarity, multinary evaluations, and expert involvement.

•   Critical evaluation of the computational and practical challenges associated with PFNBSES to inform future research and implementation.

These contributions collectively position PFNBSES as a significant advancement in soft computing-based DM, especially in contexts requiring nuanced judgment, expert collaboration, and reconciliation of conflicting objectives.

The remainder of the paper is organized as follows. Section 2 presents the foundational background and literature review. Section 3 introduces the proposed PFNBSES model, detailing its formal structure and fundamental operations. Section 4 outlines the complete DM framework based on PFNBSES, including a step-by-step algorithm and a case study in urban transportation planning. Section 5 offers a critical assessment of PFNBSES, covering its advantages, comparative performance, and computational considerations. Section 6 concludes the paper with key findings and directions for future research.

To summarize, the PFNBSES model addresses a critical need for more expressive and integrative DM tools capable of handling symbolic, bipolar, and expert-based evaluations simultaneously. While the model improves over existing frameworks in flexibility and structure, it may introduce computational complexity and practical challenges in large-scale settings. Nonetheless, its potential impact spans multiple domains—such as urban planning, healthcare, and risk analysis—where DM under conflicting, uncertain, and multidimensional criteria is essential.

2  Basic Definitions and Notation

This section presents the key notations and basic concepts used throughout the study. Let ¥ be the universal set of alternatives (or objects), and ℋ the set of attributes (or parameters). The ordered evaluation scale is given by ℧={0,1,…,N−1}, where N∈{2,3,…}. Let ℰ be the set of experts, and 𝒪={0=disagree,1=agree} the set of opinions. Furthermore, we define Λ=ℋ×ℰ×𝒪, with λ⊂Λ.

Definition 1. [3] Let γ⊕,γ⊖:¥→[0,1] be the membership and non-membership functions of ν∈¥. Then, ψ={⟨ν,γ⊕(ν),γ⊖(ν)⟩:ν∈¥} is called a PFS if 0≤(γ⊕(ν))2+(γ⊖(ν))2≤1.

Definition 2. [9] Let Ω=(ρ⊕,ρ⊖) be a Pythagorean fuzzy number (PFN). Then:

i.   Its score value is S(Ω)=(ρ⊕)2−(ρ⊖)2, with S(Ω)∈[−1,1].

ii.   Its accuracy value is A(Ω)=(ρ⊕)2+(ρ⊖)2, with A(Ω)∈[0,1].

Definition 3. [9] Let Ω1=(ρ1⊕,ρ1⊖) and Ω2=(ρ2⊕,ρ2⊖) be two PFNs, with score values S(Ω1), S(Ω2) and accuracy values A(Ω1), A(Ω2). Then:

i.   If S(Ω1)>S(Ω2), then Ω1≻Ω2.

ii.   If S(Ω1)=S(Ω2), then:

•   if A(Ω1)>A(Ω2), then Ω1≻Ω2;

•   if A(Ω1)=A(Ω2), then Ω1=Ω2.

Definition 4. [10] An SS is defined as a pair (ω,ℋ), where ω:ℋ→2¥, and 2¥ denotes the family of all crisp subsets of ¥. The pair can be expressed as

(ω,ℋ)={(ℏ,ω(ℏ)):ℏ∈ℋ, ω(ℏ)⊆¥}.

Definition 5. [40] An SES is defined as a pair (ω,λ), where ω:λ→2¥ and λ⊂Λ=ℋ×ℰ×𝒪. The SES can be written as

(ω,λ)={(t,ω(t)):t∈λ, ω(t)⊆¥},

where t=(ℏ,ς,o), with ℏ∈ℋ, ς∈ℰ, and o∈𝒪.

Definition 6. [40] The NOT set associated with a set λ, denoted by ¬λ, is expressed as ¬λ={¬t∣t∈λ}, where ¬t=(¬ℏ,ς,o) signifies the negation (opposite) of an element t=(ℏ,ς,o) with ℏ∈ℋ, ς∈ℰ, and o∈𝒪.

Definition 7. A quadruple (μ,π,λ,N) is called:

i.     an NBSES [48], if μ:λ→2¥×℧ and π:¬λ→2¥×℧ such that for each t∈λ, there is a unique pair (ν,♭t)∈¥×℧ with (ν,♭t)∈μ(t), and for each ¬t∈¬λ, a unique pair (ν,♭¬t)∈¥×℧ with (ν,♭¬t)∈π(¬t), satisfying ♭t+♭¬t≤N−1, where ν∈¥ and ♭t,♭¬t∈℧. Here, 2¥×℧ denotes the collection of all crisp subsets of ¥×℧.

ii.     an FNBSES [49], if μ:λ→F¥×℧ and π:¬λ→F¥×℧ such that for each t∈λ, there exists a unique pair (ν,♭t)∈¥×℧ with ⟨(ν,♭t),μ(ν,♭t)⟩∈μ(t), and for each ¬t∈¬λ, a unique pair (ν,♭¬t)∈¥×℧ with ⟨(ν,♭¬t),π(ν,♭¬t)⟩∈π(¬t), satisfying 0≤μ(ν,♭t)+π(ν,♭¬t)≤1, where ν∈¥, ♭t,♭¬t∈℧, and μ(ν,♭t),π(ν,♭¬t)∈[0,1]. Here, F¥×℧ denotes the set of all FSs defined over ¥×℧.

iii.     an IFNBSES [50], if μ:λ→I¥×℧ and π:¬λ→I¥×℧ such that for each t∈λ, there is a unique pair (ν,♭t)∈¥×℧ with ⟨(ν,♭t),μ⊕(ν,♭t),μ⊖(ν,♭t)⟩∈μ(t), and for each ¬t∈¬λ, a unique pair (ν,♭¬t)∈¥×℧ with ⟨(ν,♭¬t),π⊕(ν,♭¬t),π⊖(ν,♭¬t)⟩∈π(¬t), subject to 0≤μ⊕(ν,♭t)+π⊕(ν,♭¬t)≤1 and 0≤μ⊖(ν,♭t)+π⊖(ν,♭¬t)≤1, where ν∈¥, ♭t,♭¬t∈℧, and all μ⊕(ν,♭t),μ⊖(ν,♭t),π⊕(ν,♭¬t),π⊖(ν,♭¬t)∈[0,1]. Clearly, μ⊕(ν,♭t) and π⊕(ν,♭¬t) are MDs, while μ⊖(ν,♭t) and π⊖(ν,♭¬t) are NMDs. Here, I¥×℧ denotes the set of all IFSs on ¥×℧.

3  Pythagorean Fuzzy N-Bipolar Soft Expert Sets

In this section, we introduce the PFNBSES model and define its fundamental operations, including the null set, the whole set, complement, subset, equality, agree, disagree, union, and intersection. Each of these operations is presented together with its associated algebraic properties.

Definition 8. A quadruple (ξ¯,ξ_,λ,N) is called a PFNBSES if ξ¯:λ→P¥×℧ and ξ_:¬λ→P¥×℧, such that for every t∈λ, there exists a unique pair (ν,♭t)∈¥×℧ with ⟨(ν,♭t),ξ¯⊕(ν,♭t),ξ¯⊖(ν,♭t)⟩∈ξ¯(t), and for each ¬t∈¬λ, there exists a unique pair (ν,♭¬t)∈¥×℧ such that ⟨(ν,♭¬t),ξ_⊕(ν,♭¬t),ξ_⊖(ν,♭¬t)⟩∈ξ_(¬t), subject to the following conditions:

0≤(ξ¯⊕(ν,♭t))2+(ξ_⊕(ν,♭¬t))2≤1,(1)

0≤(ξ¯⊖(ν,♭t))2+(ξ_⊖(ν,♭¬t))2≤1,(2)

where ν∈¥, ♭t,♭¬t∈℧, and ξ¯⊕(ν,♭t),ξ¯⊖(ν,♭t),ξ_⊕(ν,♭¬t),ξ_⊖(ν,♭¬t)∈[0,1]. Here, ξ¯⊕(ν,♭t) and ξ_⊕(ν,♭¬t) denote the MDs, while ξ¯⊖(ν,♭t) and ξ_⊖(ν,♭¬t) represent the NMDs.

Henceforth, we consider the sets ¥={ν1,ν2,…,νn}, ℋ={ℏ1,ℏ2,…,ℏm}, and ℰ={ς1,ς2,…,ςt} to be finite, unless stated otherwise. Under this assumption, the PFNBSES (ξ¯,ξ_,λ,N) can be efficiently represented in a tabular format. In particular, we adopt the notations ξ¯(ti)(νj)=⟨♭ijti,ξ¯ij⊕,ξ¯ij⊖⟩ and ξ_(¬ti)(νj)=⟨♭ij¬ti,ξ_ij⊕,ξ_ij⊖⟩ as abbreviations for the elements ⟨(νj,♭ijti),ξ¯⊕(νj,♭ijti),ξ¯⊖(νj,♭ijti)⟩∈ξ¯(ti) and ⟨(νj,♭ij¬ti),ξ_⊕(νj,♭ij¬ti),ξ_⊖(νj,♭ij¬ti)⟩∈ξ_(¬ti), respectively. The associated tabular format is shown in Table 1.

To demonstrate the structure and functionality of our proposed model, we present the following illustrative example.

Example 1. Suppose a government agency seeks to identify the most secure cloud storage provider among five candidates: ¥={ν1,ν2,ν3,ν4,ν5}. A set of critical criteria is considered:

ℋ={ℏ1=data encryption quality, ℏ2=compliance with regulations, ℏ3=system availability},

along with their corresponding negative counterparts:

¬ℋ={¬ℏ1=encryption vulnerabilities, ¬ℏ2=non-compliance issues, ¬ℏ3=frequent downtimes}.

Initially, an internal IT team conducts a detailed assessment of all five providers based on the above parameters and prepares a technical report. This report is then independently reviewed by a panel of cybersecurity experts, denoted by ℰ={ς1,ς2,ς3}. Each expert uses their domain knowledge to express their evaluations symbolically over both the positive and negative criteria.

Each evaluation is expressed using a graded symbolic scale to capture the intensity of satisfaction or risk associated with each parameter. The symbols used in Table 2 are interpreted as follows:

•   “∘” denotes extremely poor performance (very high risk or total lack of compliance).

•   “∗” denotes below average performance (not meeting expectations or partially risky).

•   “∗∗” denotes moderate performance (acceptable but with noticeable shortcomings).

•   “∗∗∗” denotes good performance (satisfies most expectations with minor issues).

•   “∗∗∗∗” denotes excellent performance (fully meets or exceeds expectations).

Each symbolic grade is associated with a corresponding numerical value in the set ℧={0,1,2,3,4}, as follows:

•   0 corresponds to ∘ (extremely poor).

•   1 corresponds to ∗ (below average).

•   2 corresponds to ∗∗ (moderate).

•   3 corresponds to ∗∗∗ (good).

•   4 corresponds to ∗∗∗∗ (excellent).

According to Definition 7 (ii), the tabular representation of the 5BSES (μ,π,λ,5) is presented in Table 3.

To achieve a thorough assessment, the experts express their evaluations of the systems in the form of a PF5BSES. Following the predefined grading scheme, they assign MDs and NMDs in accordance with the principles of PFSs, as illustrated in Table 4.

Therefore, the PF5BSES is displayed in Table 5.

We now introduce a set of fundamental operations defined on PFNBSESs. These operations include the null and whole sets, subset relation, equality, agree and disagree notions, complement as well as union and intersection.

Definition 9. A PFNBSES (ξ¯N,ξ_N,λ,N) is called a relative null set if, for all t∈λ and ν∈¥, we have ξ¯N(t)(ν)=⟨0,0.0,1.0⟩, and for all ¬t∈¬λ and ν∈¥, we have ξ_N(¬t)(ν)=⟨N−1,1.0,0.0⟩.

Definition 10. A PFNBSES (ξ¯W,ξ_W,λ,N) is called a relative whole set if, for all t∈λ and ν∈¥, we have ξ¯W(t)(ν)=⟨N−1,1.0,0.0⟩, and for all ¬t∈¬λ and ν∈¥, we have ξ_W(¬t)(ν)=⟨0,0.0,1.0⟩.

Definition 11. A PFNBSES (ξ¯1,ξ_1,λ1,N) is said to be a subset of (ξ¯2,ξ_2,λ2,N), denoted by (ξ¯1,ξ_1,λ1,N) ⊆^ (ξ¯2,ξ_2,λ2,N), if the following conditions are satisfied:

1.   λ1⊆λ2.

2.   For every t∈λ1 and ν∈¥: ♭1t≤♭2t, ξ¯1⊕(ν,♭1t)≤ξ¯2⊕(ν,♭2t), and ξ¯2⊖(ν,♭2t)≤ξ¯1⊖(ν,♭1t), where ⟨(ν,♭1t),ξ¯1⊕(ν,♭1t),ξ¯1⊖(ν,♭1t)⟩∈ξ¯1(t) and ⟨(ν,♭2t),ξ¯2⊕(ν,♭2t),ξ¯2⊖(ν,♭2t)⟩∈ξ¯2(t).

3.   For every ¬t∈¬λ1 and ν∈¥: ♭2¬t≤♭1¬t, ξ_2⊕(ν,♭2¬t)≤ξ_1⊕(ν,♭1¬t), and ξ_1⊖(ν,♭1¬t)≤ξ_2⊖(ν,♭2¬t), where ⟨(ν,♭1¬t),ξ_1⊕(ν,♭1¬t),ξ_1⊖(ν,♭1¬t)⟩∈ξ_1(¬t) and ⟨(ν,♭2¬t),ξ_2⊕(ν,♭2¬t),ξ_2⊖(ν,♭2¬t)⟩∈ξ_2(¬t).

Definition 12. Two PFNBSESs (ξ¯1,ξ_1,λ1,N) and (ξ¯2,ξ_2,λ2,N) are said to be equal if both (ξ¯1,ξ_1,λ1,N) ⊆^ (ξ¯2,ξ_2,λ2,N) and (ξ¯2,ξ_2,λ2,N) ⊆^ (ξ¯1,ξ_1,λ1,N) are satisfied.

Definition 13. Let (ξ¯,ξ_,λ,N) be a PFNBSES. The positive agree, denoted as (ξ¯,ξ_,λ,N)+1, is a subset of (ξ¯,ξ_,λ,N) and is defined as:

(ξ¯,ξ_,λ,N)+1={ξ¯+1(t)∣t∈ℋ×ℰ×{1}}.(3)

Definition 14. Let (ξ¯,ξ_,λ,N) be a PFNBSES. The positive disagree, denoted as (ξ¯,ξ_,λ,N)+0, is a subset of (ξ¯,ξ_,λ,N) and is formulated as:

(ξ¯,ξ_,λ,N)+0={ξ¯+0(t)∣t∈ℋ×ℰ×{0}}.(4)

Definition 15. Consider a PFNBSES (ξ¯,ξ_,λ,N). The negative agree, represented as (ξ¯,ξ_,λ,N)−1, is a subset of (ξ¯,ξ_,λ,N) given by:

(ξ¯,ξ_,λ,N)−1={ξ_−1(¬t)∣¬t∈¬ℋ×ℰ×{1}}.(5)

Definition 16. Given a PFNBSES (ξ¯,ξ_,λ,N), the negative disagree, symbolized by (ξ¯,ξ_,λ,N)−0, is a subset of (ξ¯,ξ_,λ,N) and is defined as:

(ξ¯,ξ_,λ,N)−0={ξ_−0(¬t)∣¬t∈¬ℋ×ℰ×{0}}.(6)

Definition 17. The complement of (ξ¯,ξ_,λ,N), denoted by (ξ¯,ξ_,λ,N)c^, is defined as (ξ¯,ξ_,λ,N)c^=(ξ¯c^,ξ_c^,λ,N), where for each t∈λ and ν∈¥, it holds that ξ¯c^(t)=ξ_(¬t), meaning ♭tc^=♭¬t, ξ¯⊕c^(ν,♭t)=ξ_⊕(ν,♭¬t), and ξ¯⊖c^(ν,♭t)=ξ_⊖(ν,♭¬t). Likewise, for every ¬t∈¬λ and ν∈¥, we have ξ_c^(¬t)=ξ¯(t), i.e., ♭¬tc^=♭t, ξ_⊕c^(ν,♭¬t)=ξ¯⊕(ν,♭t), and ξ_⊖c^(ν,♭¬t)=ξ¯⊖(ν,♭t).

Definition 18. The extended union of (ξ¯1,ξ_1,λ1,N1) and (ξ¯2,ξ_2,λ2,N2) is denoted and defined as (ξ¯1,ξ_1,λ1,N1) ∪^e^ (ξ¯2,ξ_2,λ2,N2) = (ξ¯,ξ_,λ1∪λ2,max(N1,N2)), where for all t∈λ1∪λ2:

ξ¯(t)={ξ¯1(t),if t∈λ1∖λ2,ξ¯2(t),if t∈λ2∖λ1,⟨(ν,max{♭1t,♭2t}),max{ξ¯1⊕(ν,♭1t),ξ¯2⊕(ν,♭2t)},min{ξ¯1⊖(ν,♭1t),ξ¯2⊖(ν,♭2t)}⟩,if t∈λ1∩λ2,(7)

where ⟨(ν,♭1t),ξ¯1⊕(ν,♭1t),ξ¯1⊖(ν,♭1t)⟩∈ξ¯1(t) and ⟨(ν,♭2t),ξ¯2⊕(ν,♭2t),ξ¯2⊖(ν,♭2t)⟩∈ξ¯2(t).

Similarly, for all ¬t∈¬λ1∪¬λ2:

ξ_(¬t)={ξ_1(¬t),if ¬t∈¬λ1∖¬λ2,ξ_2(¬t),if ¬t∈¬λ2∖¬λ1,⟨(ν,min{♭1¬t,♭2¬t}),min{ξ_1⊕(ν,♭1¬t),ξ_2⊕(ν,♭2¬t)},max{ξ_1⊖(ν,♭1¬t),ξ_2⊖(ν,♭2¬t)}⟩,if ¬t∈¬λ1∩¬λ2,(8)

where ⟨(ν,♭1¬t),ξ_1⊕(ν,♭1¬t),ξ_1⊖(ν,♭1¬t)⟩∈ξ_1(¬t) and ⟨(ν,♭2¬t),ξ_2⊕(ν,♭2¬t),ξ_2⊖(ν,♭2¬t)⟩∈ξ_2(¬t).

Definition 19. The extended intersection of (ξ¯1,ξ_1,λ1,N1) and (ξ¯2,ξ_2,λ2,N2) is denoted and defined as (ξ¯1,ξ_1,λ1,N1) ∩^e^ (ξ¯2,ξ_2,λ2,N2) = (ξ¯,ξ_,λ1∪λ2,max(N1,N2)), where for all t∈λ1∪λ2:

ξ¯(t)={ξ¯1(t),if t∈λ1∖λ2,ξ¯2(t),if t∈λ2∖λ1,⟨(ν,min{♭1t,♭2t}),min{ξ¯1⊕(ν,♭1t),ξ¯2⊕(ν,♭2t)},max{ξ¯1⊖(ν,♭1t),ξ¯2⊖(ν,♭2t)}⟩,if t∈λ1∩λ2,(9)

where ⟨(ν,♭1t),ξ¯1⊕(ν,♭1t),ξ¯1⊖(ν,♭1t)⟩∈ξ¯1(t) and ⟨(ν,♭2t),ξ¯2⊕(ν,♭2t),ξ¯2⊖(ν,♭2t)⟩∈ξ¯2(t).

Similarly, for all ¬t∈¬λ1∪¬λ2:

ξ_(¬t)={ξ_1(¬t),if ¬t∈¬λ1∖¬λ2,ξ_2(¬t),if ¬t∈¬λ2∖¬λ1,⟨(ν,max{♭1¬t,♭2¬t}),max{ξ_1⊕(ν,♭1¬t),ξ_2⊕(ν,♭2¬t)},min{ξ_1⊖(ν,♭1¬t),ξ_2⊖(ν,♭2¬t)}⟩,if ¬t∈¬λ1∩¬λ2,(10)

where ⟨(ν,♭1¬t),ξ_1⊕(ν,♭1¬t),ξ_1⊖(ν,♭1¬t)⟩∈ξ_1(¬t) and ⟨(ν,♭2¬t),ξ_2⊕(ν,♭2¬t),ξ_2⊖(ν,♭2¬t)⟩∈ξ_2(¬t).

Definition 20. The restricted union of (ξ¯1,ξ_1,λ1,N1) and (ξ¯2,ξ_2,λ2,N2) is denoted and defined as (ξ¯1,ξ_1,λ1,N1) ∪^r^ (ξ¯2,ξ_2,λ2,N2) = (ξ¯,ξ_,λ1∩λ2,max(N1,N2)), where for all t∈λ1∩λ2≠∅:

ξ¯(t)=⟨(ν,max{♭1t,♭2t}),max{ξ¯1⊕(ν,♭1t),ξ¯2⊕(ν,♭2t)},min{ξ¯1⊖(ν,♭1t),ξ¯2⊖(ν,♭2t)}⟩,(11)

where ⟨(ν,♭1t),ξ¯1⊕(ν,♭1t),ξ¯1⊖(ν,♭1t)⟩∈ξ¯1(t) and ⟨(ν,♭2t),ξ¯2⊕(ν,♭2t),ξ¯2⊖(ν,♭2t)⟩∈ξ¯2(t).

Similarly, for all ¬t∈¬λ1∩¬λ2≠∅:

ξ_(¬t)=⟨(ν,min{♭1¬t,♭2¬t}),min{ξ_1⊕(ν,♭1¬t),ξ_2⊕(ν,♭2¬t)},max{ξ_1⊖(ν,♭1¬t),ξ_2⊖(ν,♭2¬t)}⟩,(12)

where ⟨(ν,♭1¬t),ξ_1⊕(ν,♭1¬t),ξ_1⊖(ν,♭1¬t)⟩∈ξ_1(¬t) and ⟨(ν,♭2¬t),ξ_2⊕(ν,♭2¬t),ξ_2⊖(ν,♭2¬t)⟩∈ξ_2(¬t).

Definition 21. The restricted intersection of (ξ¯1,ξ_1,λ1,N1) and (ξ¯2,ξ_2,λ2,N2) is denoted and defined as (ξ¯1,ξ_1,λ1,N1) ∩^r^ (ξ¯2,ξ_2,λ2,N2) = (ξ¯,ξ_,λ1∩λ2,max(N1,N2)), where for all t∈λ1∩λ2≠∅:

ξ¯(t)=⟨(ν,min{♭1t,♭2t}),min{ξ¯1⊕(ν,♭1t),ξ¯2⊕(ν,♭2t)},max{ξ¯1⊖(ν,♭1t),ξ¯2⊖(ν,♭2t)}⟩,(13)

where ⟨(ν,♭1t),ξ¯1⊕(ν,♭1t),ξ¯1⊖(ν,♭1t)⟩∈ξ¯1(t) and ⟨(ν,♭2t),ξ¯2⊕(ν,♭2t),ξ¯2⊖(ν,♭2t)⟩∈ξ¯2(t).

Similarly, for all ¬t∈¬λ1∩¬λ2≠∅:

ξ_(¬t)=⟨(ν,max{♭1¬t,♭2¬t}),max{ξ_1⊕(ν,♭1¬t),ξ_2⊕(ν,♭2¬t)},min{ξ_1⊖(ν,♭1¬t),ξ_2⊖(ν,♭2¬t)}⟩,(14)

where ⟨(ν,♭1¬t),ξ_1⊕(ν,♭1¬t),ξ_1⊖(ν,♭1¬t)⟩∈ξ_1(¬t) and ⟨(ν,♭2¬t),ξ_2⊕(ν,♭2¬t),ξ_2⊖(ν,♭2¬t)⟩∈ξ_2(¬t).

Next, we present several key propositions that describe essential algebraic properties of the PFNBSES model, including complementarity, commutativity, associativity, distributivity, absorption, and De Morgan’s laws, which together establish the foundational structure of the framework.

Proposition 1. Let (ξ¯,ξ_,λ,N) be a PFNBSES, and let (ξ¯N,ξ_N,λ,N) and (ξ¯W,ξ_W,λ,N) represent the relative null and relative whole, respectively. Then,

1.   ((ξ¯,ξ_,λ,N)c^)c^=(ξ¯,ξ_,λ,N).

2.   (ξ¯N,ξ_N,λ,N)c^=(ξ¯W,ξ_W,λ,N).

3.   (ξ¯W,ξ_W,λ,N)c^=(ξ¯N,ξ_N,λ,N).

Proof. Straightforward. □

Proposition 2. Let (ξ¯1,ξ_1,λ1,N1), (ξ¯2,ξ_2,λ2,N2), and (ξ¯3,ξ_3,λ3,N3) be PFN1BSES, PFN2BSES, and PFN3BSES, respectively. Let ⊛∈{∪^e^,∩^e^,∪^r^,∩^r^}, then

1.   (ξ¯1,ξ_1,λ1,N1)⊛(ξ¯2,ξ_2,λ2,N2)=(ξ¯2,ξ_2,λ2,N2)⊛(ξ¯1,ξ_1,λ1,N1).

2.   (ξ¯1,ξ_1,λ1,N1)⊛((ξ¯2,ξ_2,λ2,N2)⊛(ξ¯3,ξ_3,λ3,N3))=((ξ¯1,ξ_1,λ1,N1)⊛(ξ¯2,ξ_2,λ2,N2))⊛(ξ¯3,ξ_3,λ3,N3).

Proof. Straightforward. □

Proposition 3. Let (ξ¯1,ξ_1,λ,N) and (ξ¯2,ξ_2,λ,N) be two PFNBSESs. Then,

1.   (ξ¯1,ξ_1,λ,N) ∪^e^ (ξ¯2,ξ_2,λ,N)=(ξ¯1,ξ_1,λ,N) ∪^r^ (ξ¯2,ξ_2,λ,N).

2.   (ξ¯1,ξ_1,λ,N) ∩^e^ (ξ¯2,ξ_2,λ,N)=(ξ¯1,ξ_1,λ,N) ∩^r^ (ξ¯2,ξ_2,λ,N).

Proof. Straightforward. □

Proposition 4. Let (ξ¯1,ξ_1,λ1,N) and (ξ¯2,ξ_2,λ2,N) be two PFNBSESs. Then,

1.   ((ξ¯1,ξ_1,λ1,N) ∪^e^ (ξ¯2,ξ_2,λ2,N))c^=(ξ¯1,ξ_1,λ1,N)c^ ∩^e^ (ξ¯2,ξ_2,λ2,N)c^.

2.   ((ξ¯1,ξ_1,λ1,N) ∩^e^ (ξ¯2,ξ_2,λ2,N))c^=(ξ¯1,ξ_1,λ1,N)c^ ∪^e^ (ξ¯2,ξ_2,λ2,N)c^.