Some studies of Rough Pythagorean Fuzzy sets and Rough q-Rung Orthopair Fuzzy sets based on Soft Binary Relations, and their Applications

Abstract

Fuzzy Set (FzS), Rough Set (RfS), and Soft Set (SfS) methodologies are invaluable tools for examining uncertainty and incompleteness within information systems. These frameworks offer distinct advantages and find wide-ranging applications across artificial intelligence, realworld problem-solving, and computer science domains. In the realms of both natural and social sciences, a plethora of Fuzzy concepts abound. Zadeh [70] introduced Fuzzy Set (FzS) theory in 1965, providing a mathematical framework to represent fuzziness. By adeptly portraying fuzziness using formal mathematical language, Zadeh ushered in a groundbreaking approach to processing Fuzzy and uncertain information. The structured and well-formulated mathematical methods inherent in FzS theory have empowered humanity to grapple with data and information residing within uncertain boundaries. Notably, FzS theory aligns seamlessly with the cognitive processes of human perception and reasoning, thereby enriching our understanding of complex phenomena. Molodstove [40] introduced the concept of Soft Set (SfS), distinguished by its innovative parameterization technique, which distinguishes it from traditional methods. This novel approach has fostered a broad spectrum of applications across various disciplines, including Riemann Integration, operational research, game theory, probability theory, and measure theory. Building upon Molodstove’s foundational work, Maji et al. [42, 43] further enhanced the theoretical underpinnings of SfSs by introducing various operations, thus enriching the methodological toolkit available for SfS analysis. Subsequent advancements by Ali et al. [3] refined and extended the operations proposed by Maji et al. [43], demonstrating the validity of De-Morgan’s laws within the context of SfS theory. Maji et al. [41] embarked on a pioneering effort to amalgamate the structural framework of SfSs with that of Fuzzy Sets (FzSs), giving rise to the innovative concept of Fuzzy Soft Set (FzSfS). This fusion of methodologies opened new avenues for tackling decision- 2 making problems, as demonstrated by Feng et al. [23]. Further explorations into SfS theory, as documented in [6, 44], have elucidated additional operations, thereby refining our understanding and expanding the applicability of SfS methodologies. Ali and Shabir [5] contributed to the field by defining logic connectives tailored specifically for SfSs and FzSfSs, enhancing the logical coherence of these methodologies. Additionally, in their subsequent work [4], Ali and Shabir presented improvements to the operations of FzSfS as defined in [41], further refining the methodological toolkit available for FzSfS analysis. Pawlak [46] introduced Rough Set (RfS) theory as a mathematical framework for addressing vagueness and uncertainty. Central to RfS theory is the fundamental assumption that each object in a universe possesses some associated knowledge. For instance, in the context of citizens of a country like Pakistan, the information system might be structured around National Identity Cards, allowing for the classification of objects as indiscernible based on their similarity. The indiscernibility relation forms the mathematical foundation of RfS theory, generating equivalence classes that represent concepts within the knowledge base. Every idea is characterized by a couple of ideas known as the lower estimation and the upper guess, working with extracting significant experiences from unsure information. RfS theory has garnered significant interest across various disciplines, including artificial intelligence, cognitive sciences, and machine learning. Intelligent systems, pattern recognition, and decision analysis are just a few of the many fields in which it can be used. This expansiveness features its flexibility and highlights its pertinence in different fields. RfS theory broadens traditional set theory and offers unmistakable benefits. Dissimilar to statistical methods, it does not need extra information, for example, probability measures, or MmDgs as in FzSs. The combination of RfSs with other mathematical structures, such as FzSs, has further broadened its utility, enabling the description of attribute sets with ease. Various extensions and refinements of RfS theory have been proposed in the literature, including covering-based 3 RfSs, variable precision RfSs, and RfSs based on Binary Relations [66]. Dubois and Prade [20] integrated RfSs and FzSs based on Pawlak approximation space, enhancing the analytical capabilities of both methodologies. Additionally, RfS models have found applications in diverse problem domains, as evidenced by their utilization in various research endeavors [12, 13, 14, 15, 18, 20, 30, 50, 67, 73]. Further advancements in RfS theory include the exploration of roughness in groups and subgroups [16], the development of generalized RfS based on Binary Relations [72], and the investigation of roughness in semigroups (SmG) [34] and order SmGs based on pseudo-order [53]. Besides, mathematicians have examined the idea of rough Ideal (Rf Id) of rings [18]. In addition, efforts have been made to integrate RfSs, SfSs, and FzSs to define soft approximation spaces, [24]. In decision-making scenarios, diverse experts often yield varying evaluation results, necessitating a comprehensive framework beyond the conventional membership degree (MmDg) offered by Fuzzy Sets (FzSs). In light of this need, Atanassov [7] presented the idea of the Intuitionistic Fuzzy Set (ItFzS), where ItFzS incorporates both the MmDg as well as the nonmembership degree (NnMmDg). Each element in an ItFzS U is characterized by its MmDg (UY (¨)) and NnMmDg (UN(¨)). The relationship UY (¨) + UN(¨) ≤ 1 characterizes the level of certainty and non-trust in the element’s classification inside the universal set ξ. The novelty of ItFzS theory has spurred extensive research endeavors aimed at exploring its theoretical foundations and practical applications. Feng et al. [22] gave the concept of Intuitionistic Fuzzy Soft Set (ItFzSfS), expanding upon ItFzSfS theory by considering various operations on it. However, the constraint UY (¨)+UN(¨) ≤ 1 confines the feasible selection of UY (¨) and UN(¨) to form a triangular region in the first quadrant, as illustrated in Figure 1. This limitation poses challenges in scenarios where UY (¨) + UN(¨) > 1, within the permissible range UY (¨), UN(¨) ∈ [0, 1]. To address this issue, Yager [61] proposed the concept of Pythagorean Fuzzy Set (PyFzS), where the MmDg UY (¨) and NnMmDg UN(¨) adhere to the relation 4 U2 Y (¨)+U2N (¨) ≤ 1. This relation delineates a unit quarter circle in the first quadrant, as depicted in Figure 2, providing a broader spectrum for expressing the MmDg and the NnMmDg of a Pythagorean Fuzzy Set. By making use of PyFzS, we get an expanded set of membership and NnMmDgs, effectively capturing the uncertainty associated with decision-making processes. Compared to ItFzSs, PyFzSs encompass a larger set of points, as illustrated in Figure 3, thereby offering a more comprehensive representation of uncertain information and decision-making scenarios. While Figure 1 ItFzSs and PyFzS offer valuable frameworks for handling incomplete information, certain limitations persist. For instance, consider a scenario where a decision maker provides UY (¨) = 0.7 and UN(¨) = 0.9. In this case, U2 Y (¨) + U2N (¨) = 1.30 > 1, highlighting a discrepancy that ItFzSs and PyFzSs cannot accommodate. To this end, Yager [62] introduced the novel idea of a q-Rung Orthopair Fuzzy Set (qROFzS), which extends the capabilities of both ItFzSs and PyFzSs. In qROFzSs, the MmDg UY (¨) and NnMmDg UN(¨) adhere to the relation Uq Y (¨) + Uq N(¨) ≤ 1, where q ≥ 1. For instance, the pair (0.5, 0.4) represents an intuitionistic MmDg since 0.5 + 0.4 ≤ 1. Conversely, if the NnMmDg is 0.6, then 0.5+0.6 ≥ 1, indicating a Pythagorean MmDg which is not an intuitionistic MmDg. However, scenarios such as (0.5, 0.9) cannot be adequately described by either 5 Figure 2 Figure 3 ItFzSs or PyFzSs. In this case, with 0.5 + 0.9 ≥ 1 and 0.52 + 0.92 ≥ 1, (0.5, 0.9) represents a q-Rung Orthopair MmDg (q ≥ 3), making qROFzS suitable for resolving decision-making dilemmas. Notably, for q = 1, qROFzS reduces to an ItFzS, while for q = 2, it aligns with a PyFzS, illustrating the generalization of both models. qROFzSs offer a broader spectrum of Fuzzy information expression, providing increased flexibility and suitability for navigating uncertain environments. In [69], Yager and Alajlan proposed the fundamental features of qROFzS, which play a crucial role in knowledge representation. Ali [8], leveraging the concept of orbits, offered an alternative interpretation of qROFzS. Meanwhile, Peng et al. [51] scrutinized the exponential and aggregate operators of qROFzSs, 6 endorsing various scoring functions and applying them to the selection of education management systems. Shaheen et al. [55] devised an alternative approach supporting qROFzS for deriving score functions. Hussain et al. [28] established covering-based rough qROFzS and presented a methodology for resolving decision-making problems. Additionally, Bilal et al. [10] formulated a generalized rough qROFzS based on the Binary Relation of the dual universe, utilizing this framework to elucidate and determine certain fundamental concepts. The qROFz aggregation operators for averaging information were subsequently introduced by Liu and Wang [35]. This foundational work has spurred further research into qROFz theory, with contributions from various scholars [8, 19, 37, 38, 39, 59, 68]. This ongoing exploration holds the promise of enhancing our capacity to manage uncertainty and incomplete information in decision-making processes. In summary, this dissertation aims to advance the study of extended Fuzzy sets by introducing novel frameworks, algorithms, and applications. The theoretical contributions made in this work provide a strong foundation for addressing real-world problems in decision-making, pattern recognition, and more. Motivation The exploration of Pythagorean Fuzzy Sets (PyFzSs) has been extensive, with numerous researchers delving into its theory and applications across various fields. Peng et al. [47] laid down the foundations by defining Pythagorean Fuzzy Soft Sets (PyFzSfS), elucidating basic operations, and showcasing applications. Zhang et al. [71] proposed an application of TOPSIS over PyFzS, expanding its utility in decision-making contexts. Additionally, Hussain et al. [28] introduced Pythagorean Fuzzy soft rough sets (PyFzSfRfSs), further enriching the theoretical framework. Olgun et al. [45] contributed to the advancement of PyFzS theory by defining Pythagorean Fuzzy Topological Spaces (PyFzTpSs) and exploring Pythagorean Fuzzy 7 continuity between two PyFzTpSs. Mean operators for PyFzSs have also been developed by several authors, as documented in [58, 60]. Recent research has seen a surge in the study of q-Rung Orthopair Fuzzy Sets (qROFzSs). Peng et al. [52] introduced an algorithm for emergency decision-making using qROFzSs, demonstrating their effectiveness in critical scenarios. Hussain et al. [29] innovatively combined qROFzS with Soft Sets, presenting aggregation operators that enhance their applicability. Furthermore, in [36, 51], researchers defined various mean operators tailored for qROFzS, such as qROFz Bonferroni mean operators and exponential operators, contributing to the versatility of the model. Chapter-wise Study Chapter one serves as an introduction to foundational concepts essential for the subsequent chapters. In the second chapter, inspired by the pioneering work of Yager [61], we extend the framework presented by Kanwal and Shabir [32] to the realm of Pythagorean Fuzzy Sets (PyFzSs). We delve into lower and upper approximations of PyFzSs using soft Binary Relations concerning FrS and AfSs, establishing their properties. Additionally, we introduce two types of Pythagorean Fuzzy Topological Spaces (PyFzTpSs) derived from soft Binary Relation (SfBnR) and explore similarity relations between PyFzSs. The notion of roughness and accuracy for Pythagorean MmDgs with respect to FrS and AfSs is also introduced. Furthermore, an algorithm for decision-making using PyFzSs is provided, accompanied by an illustrative example demonstrating its application in real-world scenarios. The third chapter focuses on the lower approximation (LoAp) and the upper approximation (UpAp) of a q-Rung Orthopair Fuzzy Set (qROFzS) utilizing Crisp Binary Relations (CrBnR) with respect to FrS and AfSs, along with their associated properties. We delve into two types of q-Rung Orthopair Topological Spaces (qROFzTpSs) induced by CrBnRs and explore simi- 8 larity relations between qROFzSs based on CrBnRs. Additionally, we introduce the concept of roughness and accuracy for q-Rung Orthopair MmDgs with respect to FrS and AfSs. An algorithm for decision-making using qROFzSs is presented, along with a practical example illustrating its efficacy in decision-making contexts. In the fourth chapter, we extend the approach outlined by Kanwal and Shabir [31] and Bilal et al. [9, 10] to the realm of qROFzS utilizing SfBnRs on dual universes, following Yager’s pioneering idea. The chapter explores the properties of lower and upper approximations of qROFzSs based on SfBnRs concerning FrS and AfSs. We discuss various types of q-Rung Orthopair Fuzzy Topologies (qROFzTps) constructed using soft Reflexive Relations (SfRfR) and introduce similarity relations between qROFzSs based on SfBnRs. Moreover, we present a graphical solution for decision-making problems using qROFzSs, accompanied by an illustrative example to demonstrate its application. The fifth chapter delves into the approximation of PyFzSs in terms of SfRfRs. We discuss approximations of PyFzSs by FrS and AfSs, resulting in upper and lower PyFzSfSs. Furthermore, we explore the approximation of PyFz subSmGs (SbSmGs), Pythaorean Fuzzy Left Ideals (PyFzLf Ids), Pythagorean Fuzzy Right Ideals (PyFzRiIds), Pythagorean Fuzzy Interior Ideals (PyFzItIds), and Pythagorean Fuzzy Bi-Ideals (PyFzBiIds) of SmGs, accompanied by illustrative examples. In the sixth chapter, we extend our discussion to the approximation of qROFzSs in terms of SfRfRs. Similar to the previous chapter, we explore approximations of qROFzSs sets by FrS and AfSs, yielding upper and lower qROFzSfSs. Additionally, we delve into the approximation of qROFzSbSmGs, qROFz left (right) Ideals, qROFzItIds, and qROFzBiIds of SmGs, complemented by relevant examples. 9