Some Studies of Covering based Rough Bipolar Fuzzy Sets, Fuzzy Bipolar Soft Sets and Bipolar Fuzzy Soft Sets and their Applications

Abstract

The escalating need to address decisions, coupled with their mounting complexity and inherent uncertainty, requires robust models of intelligent decision-making. These models predominantly manifest as modeling activities involving assessments, significantly contributing to understanding the intricate interactions within specific decision-making (𝔻𝕄) application domains. Consequently, the structure of these models has unavoidably grown in complexity. However, the often limited availability of data in various cases necessitates tools to enhance the treatment of existing uncertainty in these problems. Multi-criteria decision making (𝕄ℂ𝔻𝕄), a well-established branch of decision theory, endeavors to achieve multiple, often conflicting, objectives in decision problems typically characterized by high uncertainty. As a response to the pervasive uncertainty in real-world applications, fuzzy 𝕄ℂ𝔻𝕄 has emerged as a paradigm for handling uncertainty in𝕄ℂ𝔻𝕄problems. This area of research has evolved into a substantial field with a well-established scientific community, producing numerous extensions, approaches, and tools applied across a diverse range of real-world applications. The foundation of fuzzy set (𝔽𝕊) theory [68], laid by Lotfi Zadeh in 1965, serves as a means to represent uncertainty and ambiguity in real-world systems. An extension of 𝔽𝕊 theory remains a crucial method for representing the non-probabilistic nature of uncertain, incomplete, imprecise, or ambiguous information. Contemporary information processing trends highlight the importance of bipolar information representation, both in terms of knowledge representation and processing. In 1994, Zhang introduced the concept of YinYang bipolar fuzzy sets (𝔹𝔽𝕊s) [70], providing a bipolar rational method for handling two-sided information. In 𝔹𝔽𝕊s, each member is associated with two components, one representing membership values and the other representing counter-attributes. A 𝔹𝔽𝕊s, extending the ideas of 𝔽𝕊 to include membership levels within the range of [−1, 1]. A 𝔹𝔽𝕊 assigns a membership degree of 0 to an element when it is unrelated to the property in question, a membership degree in the range (0, 1] indicates some degree of satisfaction with the property, and a membership degree within the range [−1, 0] suggests partial satisfaction of the implicit counter property. For instance, consider the evaluation of the dark shade levels of cosmetic items. While an 𝔽𝕊 can describe the dark shade by assigning a degree (ranging from 0 to 1), e.g. 0.7, to a cosmetic item, it may not accurately represent the light shade in the item, which could be less than 0.3 or even 0. In such cases, 𝔹𝔽𝕊s offer a superior approach, as they can represent the degree of dark shade and light shade separately. For example, a 𝔹𝔽𝕊 ̃𝜗 = (̃𝜗(𝑃 ), ̃𝜗(𝑁)), may assign the values (0.7, −0.1) to a cosmetic item, indicating a dark shade degree of 0.7 and a lightness degree of −0.1. The ability of 𝔹𝔽𝕊s to accommodate bipolar information makes them valuable in a wide array of applications. Human actions often rely on bipolar judgments, motivating research in this do main, as discussed by Lee [32], who defined fundamental operations on 𝔹𝔽𝕊s and compared 𝔹𝔽𝕊s to other 𝔽𝕊 extensions. Another challenge in data analysis involves identifying objects possessing a particular property or attribute within a universe of discourse. Molodtsov’s soft set (𝕊𝕊) theory [46], introduced in 1999, addresses this issue. 𝕊𝕊s, defined through a mapping, differ from 𝔽𝕊s and involve set-valued mappings associating each attribute with a set of objects from a universe of discourse. Both 𝔽𝕊s and 𝕊𝕊s address problems of uncertainty and imprecision, bridging the gap between classical mathematical methodologies and the vague data of the real world. Bipolar soft sets (𝔹𝕊𝕊s) were later introduced by iii Shabir and Naz [54]. In 1982, Pawlak introduced rough sets (ℝ𝕊) [50], evolving from foundational research into the logical properties of intelligent programs. ℝ𝕊 theory became a technique for database mining and information exploration in relational databases, offering an alternative to the inherent mathematical uncertainty in fuzzy theory. However, ℝ𝕊 analysis faces limitations with discrete data when handling abstract or continuous data. To address this, fuzzy rough sets (𝔽ℝ𝕊) and rough 𝔽𝕊s (ℝ𝔽𝕊) were introduced, enabling the treatment of real-valued data. 𝔽ℝ𝕊 theory extends ℝ𝕊 theory to handle numerical constant attributes and finds applications in various practical scenarios. The Pawlak model, utilizing equivalence relations for indiscernibility, may not always be feasible in real-world situations due to certain constraints. Generalized models for ℝ𝕊 have been proposed, including binary relation-based ℝ𝕊, neighborhood ℝ𝕊, covering-based rough sets (ℂℝ𝕊), ℝ𝔽𝕊, and 𝔽ℝ𝕊. Among these extensions, ℂℝ𝕊 has gained considerable attention in ℝ𝕊 theory. In 1983, Zakowski [69] presented a covering-based generalization of the Pawlak approximation operator, leading to the development of covering-based fuzzy rough sets (ℂ𝔽ℝ𝕊). Further discussions on fuzzy neighborhoods based on ℂ𝔽ℝ𝕊 have sparked interest in unified versions of ℝ𝕊 and 𝔽𝕊 theories. The definition of fuzzy covering 𝔽ℂ is given in [21, 35] as. Let ℵ be an arbitrary universal set, and̃𝐹𝑃 ℵ be the fuzzy power set of ℵ. We call ̂𝕜 = { 𝕜1, 𝕜2, ...., 𝕜𝑟 } with 𝕜𝑗 ∈̃𝐹𝑃 ℵ(𝑗 = 1, 2,…, 𝑟), a 𝔽ℂ of ℵ, if ( ⋃𝑟 𝑗=1 𝕜𝑗 ) (𝑡) = 1 for each 𝑡 ∈ ℵ. In fact, there exist some limits of this definition in the practical applications. For example, Suppose ℵ = { 𝑡1, 𝑡2, 𝑡3, 𝑡4 } be the set of four candidates apply to a vacant post in X- company. The company hire an expert which evaluates each candidate on the basis of four attributes, 𝑎1 = personality, 𝑎2 = experience, 𝑎3 = grades, 𝑎4 = communication skill. For 𝑎𝑖(𝑖 = 1, 2, 3, 4), we have ̂𝕜 = { 𝕜1, 𝕜2, 𝕜3, 𝕜4 } , which can be listed as follows. 𝕜1 = 0.6 𝑡1 + 0.46 𝑡2 + 0.4 𝑡3 + 0.1 𝑡4 , 𝕜2 = 0.3 𝑡1 + 0.5 𝑡2 + 0.67 𝑡3 + 0.9 𝑡4 , 𝕜3 = 0.85 𝑡1 + 0.6 𝑡2 + 0.65 𝑡3 + 0.34 𝑡4 ‚ 𝕜4 = 0.7 𝑡1 + 0.82 𝑡2 + 0.43 𝑡3 + 0.26 𝑡4 . If the company chooses only one candidate of ℵ for a vacant post, then there naturally exists a problem, which is the critical value given by the expert. This example is a typical evaluation question. It is easy to find that critical “1” is a meaningless existence for this example. In other words, the expert could not choose a suitable candidate for a vacant post in the company by using fuzzy evaluation methods. To overcome these limits, Ma [40] generalized the 𝔽ℂ to fuzzy 𝛽-covering by replacing 1 with a parameter 𝛽 ∈ (0, 1]. Ma [41] described two types of 𝔽ℂ rough set models in 2016 that seems to draw a bridge between ℂℝ𝕊 theory and 𝔽ℝ𝕊 theory. This work extended the models and its representation of the matrix to an L-fuzzy covering rough set, too. Yang and Hu [64, 65] and D’eer et al. [15] also studied 𝔽ℂ based ℝ𝕊 and proposed three kinds of 𝔽ℂ based ℝ𝕊 models based on Ma’s models [41]. They put forward fuzzy neighborhoods and fuzzy minimal descriptions and obtained several properties. Tozlu [47] suggested a topological approach to soft covering rough space. Zhang and Zhan [72] introduced the idea of Fuzzy soft 𝛽−ℂ𝔽ℝ𝕊 theory in 2019. It has become very important in some fields, because of its ability to combine the above ingredients. However, a wide variety of human 𝔻𝕄 is based on bipolar judgmental thinking. Naz and Shabir [47] contributed toward the algebraic structure of fuzzy bipolar soft sets. Malik and Shabir [45] introduced the idea of rough fuzzy 𝔹𝕊𝕊s in 2019. Yager and Rybalov [63] introduced uninorms as aggregation operators that generalize t-norms and t-conorms, finding applications in various fields such as expert systems and fuzzy logic. These opiv erators, with a neutral element 𝑒 in the unit interval [0, 1], play a crucial role in both theoretical investigations and practical applications. Idempotent uninorms on [0, 1], presented by Martin, Mayor, and Torrens, enhance the theorem of Czogala and Drewniak [14] on idempotent, associative, and increasing operations with a neutral element. Associative, monotonic, and idempotent uninorms are a special combination of minimum and maximum operations, forming locally internal structures. Two important classes of uninorms are characterized by the unit interval [0, 1] based on minimum and maximum operations. The structures of t-norms and t-conorms on bounded lattices have been extensively studied, and methods for constructing uninorms on such lattices are discussed by Karaçal and Çaylı et al. [12, 28, 29]. In the last chapter, we focus on uninorms on pseudo-ordered sets, particularly the subclass of trellises, also known as tournament lattices, non-associative lattices, or weekly associative lattices. Trellises generalize lattices, and psosets generalize posets, replacing the partial order relation with a more general reflexive and antisymmetric relation. However, trellises lack associativity and increasingness in their meet and join operations due to the non-transitive nature of the pseudo-order relation. Skala [55, 56] and other authors have made significant contributions to the study of trellises. Gladstein [23] demonstrated that trellises of finite length are complete if and only if every cycle has a least upper bound and a greatest lower bound, a description later extended to include pseudo chains and joins of cycles for psosets [49, 51, 52]. This paper aims to introduce and study uninorms on pseudo-ordered sets, specifically the subclass of trellises, to explore their mathematical properties and contribute to the understanding of non-transitive relations. v Chapter-wise Study There are nine chapters in this thesis which are briefly described below The first Chapter serves an introductory purpose. we present several fundamental definitions and results that will be needed throughout this thesis. In Chapter 2, we extend ℂ𝔽ℝ𝕊 model in [15] to three types of models, (i) covering based bipolar fuzzy rough set ℂ𝔹𝔽ℝ𝕊 model, (ii) monotone ℂ𝔹𝔽ℝ𝕊 model and (iii) (𝔹𝔽ℂ𝔹𝔽ℝ𝕊) model, which enrich the rough set models. Meanwhile, the related properties of these models are investigated. By integrating the 𝔹𝔽ℂ𝔹𝔽ℝ𝕊 model with the conventional (𝔻𝕄) methods, namely the 𝕎𝔸𝕊ℙ𝔸𝕊 method, we introduce a fresh approach to address 𝕄ℂ𝔻𝕄 problems. We put this extension to the test by applying it to find out the exact diagnosis in agriculture. The effectiveness of our proposed method is validated through a comparison with existing methods. In Chapter 3, we propose a hybrid model for 𝕄ℂ𝔻𝕄 based on bipolar fuzzy soft 𝛽−covering based bipolar fuzzy rough sets (𝔹𝔽𝕊𝛽ℂ𝔹𝔽ℝ𝕊s) using 𝕍𝕀𝕂𝕆ℝ technique. It consists of a suitable redesign of the 𝕍𝕀𝕂𝕆ℝ approach so that it can use information with bipolar configurations. This method focuses on selecting and ranking from a set of feasible alternatives and determines a compromise solution for a problem with conflicting criteria to help the decision-maker in reaching a final course of action. It determines the compromise ranking list based on the particular measure of closeness to the ideal solution. For illustration, the proposed technique is applied to a 𝔻𝕄 problems, namely, the selection of sites for renewable energy projects ( solar power plants). A comparison of this method with another aggregation operator method and with the existing𝔻𝕄algorithm Fuzzy 𝕍𝕀𝕂𝕆ℝ method [48], Bipolar fuzzy 𝕋𝕆ℙ𝕊𝕀𝕊 method [2] and Bipolar fuzzy 𝔼𝕃𝔼ℂ𝕋ℝ𝔼 method [2] are also presented. The theory presented in Chapter 4, a novel hybrid 𝕄ℂ𝔻𝕄 model, leveraging fuzzy bipolar soft 𝛽- covering based fuzzy bipolar soft rough sets (𝔽𝔹𝕊-𝛽-ℂ𝔽𝔹𝕊ℝ𝕊s) in conjunction with the 𝕋𝕆ℙ𝕊𝕀𝕊 and 𝔼𝕃𝔼ℂ𝕋ℝ𝔼 techniques. The approach involves an adept adaptation of 𝕋𝕆ℙ𝕊𝕀𝕊 and 𝔼𝕃𝔼ℂ𝕋ℝ𝔼 methodologies to accommodate bipolar information configurations. In the context of medical diagnosis, a process aimed at identifying the specific disease or disorder underlying an individualís symptoms or clinical indicators, the inherent unpredictability and vulnerability of diagnostic data present substantial challenges. The principal aim of this research is to provide a framework for evaluating the health condition of a patient and the factors that impact it in the context of a fuzzy bipolar soft 𝛽- covering-based rough set environment. The theoretical foundation of 𝔽𝔹𝕊−𝛽 − ℂ𝔽𝔹𝕊ℝ𝕊s proves invaluable in constructing information-driven frameworks for 𝕄ℂ𝔻𝕄 methods. Additionally, the paper provides a comparative analysis of 𝕋𝕆ℙ𝕊𝕀𝕊 and 𝔼𝕃𝔼ℂ𝕋ℝ𝔼−1, showcasing their practicality, feasibility, and sustainability in the diagnostic process. Both 𝕄ℂ𝔻𝕄 methods yield a single disease as the final diagnosis. Chapter 5, develops a new methodology, the theory of Bipolar soft covering-based rough sets (𝔹𝕊ℂ𝔹- ℝ𝕊s), which will be used to propose a new technique to solve 𝔻𝕄 problems. The idea introduced in this study has never been discussed earlier. Furthermore, this concept has been explored using a detailed study of the structural properties. By combining the 𝔹𝕊ℂ𝔹-ℝ𝕊s model with two traditional 𝔻𝕄 methods (the ℙℝ𝕆𝕄𝔼𝕋ℍ𝔼-II method and the 𝕋𝕆ℙ𝕊𝕀𝕊 method), we introduce a novel method vi for addressing multi-criteria group decision making (𝕄ℂ𝔾𝔻𝕄) problems. We give an application in 𝕄ℂ𝔾𝔻𝕄to showthat the proposed technique can be successfully applied to some real-world problems including uncertainty, namely, the selection of a site for a renewable energy project (Earth Dam). The effectiveness of the proposed method is validated by comparing it with existing methods. The techniques exhibit the practicability, feasibility, and sustainability of Site selection. Both 𝕄ℂ𝔾𝔻𝕄 methods give one Site as a conclusion. Chapter 6, introduces a novel methodology, termed the Theory of Bipolar Soft Covering-Based Rough Sets (𝔹𝕊ℂ𝔹-ℝ𝕊s), applied to two distinct universes, to propose an innovative 𝔻𝕄 technique. This concept has not been previously explored, and its structural properties are thoroughly examined. By integrating the 𝔹𝕊ℂ𝔹-ℝ𝕊s model with two conventional𝔻𝕄methods, namely the𝕎𝔸𝕊ℙ𝔸𝕊 method and the 𝔼𝕃𝔼ℂ𝕋ℝ𝔼−1 method, we introduce a fresh approach to address 𝕄ℂ𝔾𝔻𝕄 problems. The proposed method, rooted in two different universes, offers two distinct 𝔻𝕄 pathways. We put this extension to the test by applying it to the complex task of selecting an appropriate energy project and identifying the best site for this project. The effectiveness of our proposed method is validated through a comparison with existing methods. The results highlight the practicality, feasibility, and sustainability of employing our approach to choose the right energy project and its optimal location. Both 𝕄ℂ𝔾𝔻𝕄 methods converge on the selection of a wind power plant and a specific site as their final recommendations. In Chapter 7, we introduce the notion of an uninorm on bounded pseudo-ordered sets (psosets, for short) and bounded trellises. We propose some new constructions to obtain idempotent uninorms on bounded psosets and trellises with a neutral element, where some necessary and sufficient conditions on the construction subintervals are required. We also provide some corresponding examples to understand these new classes of idempotent uninorms. Finally, we provide a characterization of ∨- preserving and ∧-preserving idempotent uninorms on bounded trellises in terms of a decreasing unary operator.