On Multi-Granulation Rough Bipolar Sets and Their Applications in Decision Making

Abstract

Conventional tools for formal reasoning, computation, and modelling are characterized by precision and determinism. However, numerous real-world problems across various domains involve uncertainties that cannot be effectively addressed using traditional mathematical tools. Probability theory has traditionally served as a valuable mechanism for addressing uncertainty, with the prerequisite that a system must exhibit stochastic stability. Achieving stochastic stability typically demands a significant number of trials, consuming considerable time. In today’s fast-paced life style, where time is a scarce resource, alternative approaches are needed to address the challenges associated with uncertainty. Recognizing this need, experts in various fields of science and technology are exploring non-traditional methods to extract valuable information, dedicating step-by-step efforts to appeal to specialists and professionals. In this context, Zadeh [72] undertook a commendable initiative by introducing the theory of fuzzy sets. Fuzzy sets not only address uncertainty but also possess the capability to express human linguistic terms in a mathematical frame work. This development marked as a significant leap forward and contributed to rapid advancement. Subsequently, various generalizations of fuzzy sets, including intuitionistic fuzzy sets, bipolar fuzzy sets, neutrosophic fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and q-rung fuzzy sets have emerged. Each of these hypotheses is acknowledged and proves valuable in addressing uncertainty. However, each of these theories also presents specific challenges, as Moldsove [39] has illustrated. In 1994, Zhang [74] presented the bipolar fuzzy set which is a direct extension of fuzzy set that uses membership degrees as the specification degrees for a constraint and its counter-constraint. Bipolar fuzzy information has been used to express the property of an object as well as its counter property. Moldsove [39] introduced the soft set theory as a novel mathematical mechanism for addressing uncertainty. This innovative concept serves as a key notion, providing an alternative approach free from the challenges associated with established methods. The theory offers a set of parameters, ensuring flexibility, and encompasses operations that prove useful in han- 5 dling diverse situations. Several authors have proposed applications and expansions of soft set theory. Essentially, the primary objective of soft set theory is to furnish a tool equipped with an ample number of parameters to address the inherent uncertainty in data, while concurrently possessing the capability to represent data effectively. The rapid advancement of soft set theory with the introduction of new operations [7, 33] had led to its swift development and practical applications. This progression enhances our foresight and understanding of the theory’s applications. Shabir and Naz [52] put forth the idea of bipolar soft set and some algebraic operations on it. The idea of fuzzy bipolar soft set was put up by Naz and Shabir [43], who also looked into their algebraic structure. Another theory that addresses uncertainty in a non-traditional manner is known as rough set theory, introduced by Pawlak [44]. This numerical approach is designed to handle vague or uncertain information. The subset derived from the lower approximation consists of objects that are certain to be part of an interest subset, while the upper approximation comprises objects that may potentially be the part of interest subset. Each subset defined through upper and lower approximations is referred to as a rough set. Practical applications in various areas, such as information disclosure, data analysis, machine learning, approximate classification, and conflict analysis, have been successfully addressed using rough set theory. Following this, a key focus on research within rough set theory typically resolves around the Pawlak rough set theory. By leveraging soft binary relations, Pawlak’s rough set theory can be regarded as a distinct instance of soft rough sets. Since its inception, it has captivated the interest of researchers and scholars. The original rough set theory continues to have the potential to offer solutions to various problems even today. An important application of rough set theory lies in its ability to condense data without sacrificing valuable information. The advantageous applications of rough set models have demonstrated their efficiency in addressing numerous issues [8, 17, 29, 34–36, 46, 51]. Through the lens of granular computing, as introduced by Zadeh [73], prevailing rough set models typically represent a general concept using the upper and lower approximations within 6 a single granulation. This implies that the concept is elucidated by knowledge derived from a single relation such as an equivalence relation [35–37], tolerance relation [45], soft binary relation [50, 51], and reflexive relation [60], applied to the universe. This category of rough set models, which utilizes a single granulation, is conveniently referred to as single granulation rough sets. The Pawlak RST and several of its variations are constructed upon a single granular structure formed by a binary relation. Consequently, these models are termed as single granulation rough sets. Qian et al. [47] highlight that in various scenarios, there is a need to characterize a target idea concurrently from distinct independent situations. This suggest the necessity for multigranulation space. Consequently, Qian et al. [47–49] introduced the concept of multigranulation rough sets, wherein the approximations of a target set are defined using multi equivalence relations. In addition to the theoretical motivation discussed earlier, there ia also a practical motivation for real-world applications. When considering the applications of rough sets, the multigranulation rough set theory proves highly advantageous in various practical scenarios. This includes applications in multi-source data analysis, knowledge discovery from high-dimensional data, and distributive information systems. For instance, the development of multigranulation rough sets is particularly desirable in the following two cases: (1) When employing rough set theory for data mining and knowledge discovery from multisource data, a crucial challenge lies in determining how to handle knowledge representation and rough approximation within the framework of a multi-source information system. To efficiently unearth knowledge online, there is no need to aggregate and treat all information systems from various sources as a unified information system for data analysis. A more practical approach involves directly analyzing these multi-source information systems. In this scenario, the classical single granulation rough set theory has limitations, notably in terms of extended computational times, hindering efficient knowledge discovery from multi-source information systems. (2) Examining data with high dimensions presents a challenge for knowledge discovery due 7 to the abundance of attributes. There are two issues: (i) granulation data using all attributes results in significantly longer intensions of information granules and much smaller extensions, thereby affecting the generalization ability of a rule-based classifier; and (ii) the presence of numerous attributes leads to algorithmic inefficiency in rough set theory. These drawbacks are critical, rendering existing rough set models unsuitable for effective use in rough set-based data analysis for high-dimensional data. The application of this theory has yielded fruitful results in various domains, including decision making [3, 9, 10, 13], medial diagnosis [18, 26, 40, 42, 59, 71, 75], and other related fields [28, 31, 32, 54, 57, 65, 76]. A soft binary relation extends the concept of regular binary relations within the context of soft sets. RST deals with individual binary relations using rough approximations. Soft binary relations offer a perspective to handle various binary relations by employing rough approximations. Pawlaks’s RST, when viewed through the lens of soft binary relations, can be interpreted as soft rough sets. This thesis explores the application of rough approximations within the context of multi-soft binary relations. The study involves approximation of a bipolar fuzzy set, a fuzzy bipolar soft set, and a bipolar Pythagorean fuzzy set, resulting in two sets each of bipolar fuzzy sets, fuzzy bipolar soft sets, and bipolar Pythagorean fuzzy sets. These sets, termed lower and upper approximations, are determined based on aftersets and foresets, respectively.