Uncertainty is an essential factor in any decision-making process. A q-rung orthopair fuzzy set (q-ROFS) is more practical and robust than a fuzzy set (FS), intuitionistic fuzzy set (IFS), and Pythagorean fuzzy set (PFS) to describe uncertainty in numerous decision-making issues. The most attractive aspect of q-ROFSs is that they provide a wider space for membership and non-membership degrees and provide decision-makers more liberty in expressing their legitimate opinions. This study introduces the conception of q-rung orthopair fuzzy relation (q-ROFR), which will help to remove certain limitations associated with intuitionistic fuzzy relation (IFR) and Pythagorean fuzzy relation (PFR). Some basic operations in this regard are given for q-ROFRs. The set of all q-ROFRs leads to several algebraic structures for these operations (semi-group, semi-ring, hemi-ring, and bounded distributive lattice). Additionally, an application of the designed approach is proposed for predicting scores in cricket. Moreover, the comparative analysis of the proposed method with some existing techniques is also presented by giving several examples to authenticate the feasibility and superiority of the proposed approach.

Aggregation operators (AOs) are employed in various fuzzy environments to accommodate uncertain information. Aczel-Alsina (AA) t-norm and t-conorm inherently flexible endows Aczel-Alsina AOs with greater adaptability and robustness in the aggregation process than operators rooted in other t-norm and t-conorm families. Further, (p, q)-rung linear Diophantine fuzzy set ((p, q)-RLDFS) is one of the advanced versions of the fuzzy set (FS), which has been revealed to be very successful at combating uncertain information and has grown in prominence in decision analysis. The article aims to establish AA operations laws for (p, q)-RLDFSs. Furthermore, we expose the (p, q)-rung linear Diophantine fuzzy Aczel-Alsina weighted average ((p, q)-RLDFAAWA), (p, q)-rung linear Diophantine fuzzy Aczel-Alsina weighted geometric ((p, q)-RLDFAAWG) operators and systematically analyze their unique features and results with concrete examples. Additionally, based on the devised AOs, a multi-criteria decision making (MCDM) approach is designed to choose optimal agricultural technological systems under the (p, q)-RLDFS framework. The proposed model is subjected to a comprehensive comparison against various existing studies to demonstrate its superiority in terms of reliability and accuracy. Moreover, the influence of key parameters, designated as Lambda, p, and q, on the ranking outcomes is thoroughly examined to assess the applicability and robustness of the results derived from the developed method.

Graph structure (GS) is an advancement of the graph concept which effectively represents intricate situations with various connections, frequently used in computer science and mathematics to illustrate relationships among objects and extensively researched in fuzzy sets (FS), intuitionistic fuzzy set (IFS), pythagorean fuzzy set (PFS) and q-rung orthopair fuzzy set (q-ROFS). Meanwhile, a linear Diophantine fuzzy set (LDFS) is a remarkable extension of the existing notions of a FS, IFS, PFS and q-ROFS by comporting reference parameters that removed all the limitations related to membership degree (MD) and non-membership degree (NMD). According to the best of our knowledge, there is a lack of elegantly proposed GS extension for LDFSs in the current literature. As a result, this research focuses on introducing first linear Diophantine fuzzy graph structure (LDFGS) concept which extends the existing notions of GS in various contexts of FSs. Several key concepts in LDFGSs are presented, such as ˘ρi-edge, ˘ρi-path, strength of ˘ρi-path, ˘ρi-strength of connectedness, ˘ρi-degree of a vertex, vertex degree, total ρ˘i-degree of a vertex, and total vertex degree in an LDFGS. In addition, we introduce the ˘ρi-size, size, and order of an LDFGS. Moreover, this article presents the ideas of the maximal product of two LDFGSs, strong LDFGS, degree and ˘ρi-degree of the maximal product, ˘ρi-regular and regular LDFGSs, along with examples for clarification. Certain significant results related to the proposed concepts also demonstrated with explanatory examples such as the maximal product of two strong LDFGSs is also a strong LDFGS, the maximal product of two connected LDFGSs is also a connected LDFGS but the maximal product of two regular LDFGS may not be a regular LDGS. Moreover, many interesting and alternative formulas for calculating ˘ρi-degrees of an LDFGS in various situations are proved with examples. LDFGSs are highly beneficial for solving numerous combinatorial problems involving multiple relations, and they surpass existing concepts of GSs within the FS context due to their flexibility in selecting MD and NMD alongside their reference parameters.

Aczel-Alsina t-norm and t-conorm are intrinsically flexible and endow Aczel-Alsina aggregation operators with greater versatility and robustness in the aggregation process than operators rooted in other t-norms and t-conorm families. Moreover, the linear Diophantine fuzzy set (LD-FS) is one of the resilient extensions of the fuzzy sets (FSs), intuitionistic fuzzy sets (IFSs), Pythagorean fuzzy sets (PyFSs), and q-rung orthopair fuzzy sets (q-ROFSs), which has acquired prominence in decision analysis due to its exceptional efficacy in resolving ambiguous data. Keeping in view the advantages of both LD-FSs and Aczel-Alsina aggregation operators, this article aims to establish Aczel-Alsina operation rules for LD-FSs, such as Aczel-Alsina sum, Aczel-Alsina product, Aczel-Alsina scalar multiplication, and Aczel-Alsina exponentiation. Based on these operation rules, we expose the linear Diophantine fuzzy Aczel-Alsina weighted average (LDFAAWA) operator, and linear Diophantine fuzzy Aczel-Alsina weighted geometric (LDFAAWG) operator and scrutinize their distinctive characteristics and results. Additionally, based on these aggregation operators (AOs), a multi-criteria decision-making (MCDM) approach is designed and tested with a practical case study related to forecasting weather under an LD-FS setting. The developed model undergoes a comparative analysis with several prevailing approaches to demonstrate the superiority and accuracy of the proposed model. Besides, the influence of the parameter Λ on the ranking order is successfully highlighted.

The primary purpose of this paper is to introduce the concept of the fuzzy module of fractions and investigate a relationship between fuzzy modules of fractions and rough sets (RSs). In this respect, first, the concept of the soft module of fractions is introduced. Then, the idea of fuzzy approximations of the FS of a module of fractions is defined by using a soft module of fractions and obtaining a new hybrid model called multi-granulation soft rough FSs (MGSR-FSs) in a module of fractions. Moreover, the idea of a fuzzy upper rough module of fractions is introduced. However, it is important to note that the fuzzy lower approximation of a fuzzy module of fractions is not a fuzzy lower rough module of fractions. Several examples are provided to elaborate on the proposed notions.

The notion of linear Diophantine fuzzy sets (LD-FSs) is a novel mechanism to combat uncertainties in decision analysis. Due to reference parameters associated with membership grade (MG) and non-membership grade (NMG), LD-FS is more efficient and reliable than ideas of the fuzzy set (FS), intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PyFS), and q-rung orthopair fuzzy set (q-ROFS). The main goal of this article is to present an innovative roughness strategy for LD-FSs using a fuzzy relation (FR) over dual universes, known as a linear Diophantine fuzzy rough set (LD-FRS). The lower and upper approximations of an LD-FS are formulated using fuzzy relation (FR) over dual universes, and several axiomatic systems are investigated. The suggested model of LD-FRS is more flexible to address fuzziness and roughness. Meanwhile, a relationship is made between LD-FRSs and linear Diophantine fuzzy topologies (LDF-topologies). It is shown that the collection of all lower approximations based on a reflexive FR leads to an LDF-topology. Moreover, several similarity relations among LD-FSs are also examined based on their lower and upper approximations. An application of multi-criteria group decision-making (MCGDM) is demonstrated by a supplier selection problem. Finally, a detailed comparative analysis with certain existing methods is given to verify the feasibility and superiority of the suggested model.

Rough sets (RSs) and fuzzy sets (FSs) are designed to tackle the uncertainty in the data. By taking into account the control or reference parameters, the linear Diophantine fuzzy set (LD-FS) is a novel approach to decision making (DM), broadens the previously dominant theories of the intuitionistic fuzzy set (IFS), Pythagorean fuzzy set (PyFS), and q-rung orthopair fuzzy set (q-ROFS), and allows for a more flexible representation of uncertain data. A promising avenue for RS theory is to investigate RSs within the context of LD-FS, where LD-FSs are approximated by an intuitionistic fuzzy relation (IFR). The major goal of this article is to create a novel method of roughness for LD-FSs employing an IFR over dual universes. The notions of lower and upper approximations of an LD-FS are established by using an IFR, and some axiomatic systems are carefully investigated in detail. Moreover, a link between LD-FRSs and linear Diophantine fuzzy topology (LDF-topology) has been established. Eventually, based on lower and upper approximations of an LD-FS, several similarity relations are investigated. Meanwhile, we apply the recommended model of LD-FRSs over dual universes for solving the DM problem. Furthermore, a real-life case study is given to demonstrate the practicality and feasibility of our designed approach. Finally, we conduct a detailed comparative analysis with certain existing methods to explore the effectiveness and superiority of the established technique.

Preference analysis is a significant component in decision-making (DM) when selecting an optimal alternative. By comparing any two alternatives pairwise, preference relations (PRs) effectively depict the preference degrees of decision-makers (DMrs). The rough set theory (RST) has been effectively applied to cope with preference analysis by swapping the equivalence relation (Er) with the dominance relation (DR). In this study, we propose new transfer functions to construct alternatives’ upward/downward fuzzy preference degree (FPD) for evaluating upward and downward fuzzy PRs (FPRs). Based on these newly proposed transfer functions, we present a novel method for fuzzifying RSs called the upward α- fuzzified preference rough sets (α↑-FPRSs). The basic properties of the proposed α↑-FPRSs are thoroughly studied. Moreover, several uncertainty measures related to α↑-FPRSs are presented. Meanwhile, we offered the notion of upward fuzzy β-covering (UFβC) and upward fuzzy β-neighborhood (UFβ-nghd), upward β-neighborhood (Uβ-nghd), and several related properties are explored. Based on UFβ-nghd and Uβ-nghd, we construct two new models of UFβC rough sets (UFβ-CRSs) along with their properties. We formulate a novel technique of multi-attribute DM (MADM). To legitimise the practicality of our proposed model, we provide a real-life example of selecting an appropriate medication to treat a specific disease. Finally, we look into the efficacy of the launched scheme through a comparison study.

Theories of the rough set (RS) and the fuzzy set (FS) are constructed to accommodate the uncertainty in the data analysis. Linear Diophantine FS (LD-FS) as a novel approach to decision-making (DM), broadening the predominating theories of intuitionistic FS (IFS), Pythagorean FS (PFS), q-rung orthopair FS (q-ROFS) deals with uncertain and vague information by considering the control or reference parameters. Exploring RSs in the framework of LD-FS is a propitious direction in RS theory, where LD-FSs are approximated by Linear Diophantine fuzzy relation (LD-FR). The primary aim of this article is to develop a new linear Diophantine fuzzy RS (LDF-RS) model based on an LD-FR over dual universes. The notions of lower and upper approximations of an LD-FS are introduced by using an LD-FR, and several fundamental structural properties are explored. Moreover, a connection between LDF-RSs and linear Diophantine fuzzy topology (LDF-topology) is established. In addition, some similarity relations among LD-FSs based on their lower and upper approximations are studied. Finally, a DM approach is crafted for the ranking of alternatives using the notions of LDF-RS. Moreover, a numerical example is designed and compared with some existing techniques.

Uncertain data is a challenge to decision-making (DM) problems. Multi-criteria group decision-making (MCGDM) problems are among these problems that have received much attention. MCGDM is difficult because the existing alternatives frequently conflict with each other. In this article, we suggest a novel hybrid model for an MCGDM approach based on modified rough bipolar soft sets (MRBSs) using a well-known method of technique for order of preference by similarity to ideal solution (TOPSIS), which combines MRBSs theory and TOPSIS for the prioritization of alternatives in an uncertain environment. In this technique, we first introduce an aggregated parameter matrix with the help of modified bipolar soft lower and upper matrices to identify the positive and negative ideal solutions. After that, we define the separation measurements of these two solutions and compute relative closeness to choose the best alternative. Next, an application of the proposed technique in the MCGDM problem is introduced. Afterward, an algorithm for this application is developed, which is illustrated by a case study. The application demonstrates the usefulness and efficiency of the proposal. Compared to some existing studies, we additionally present several merits of our proposed technique. Eventually, the paper handles whether additional studies on these topics are needed.

Rough set (RS) and soft set (SS) theories are two successful mathematical approaches to dealing with uncertainty in data analysis. The classical soft rough set (SRS) theory proposed by Feng et al. (2011) offers a formal theoretical framework for solving the uncertainty under a single granulation environment. However, it is essential to note that the SRS theory cannot be applied in the context of multi-granulation in the real world. To address this issue, in this paper, we introduce the idea of soft multi-granulation RS (SMGRS) model based on two soft binary relations (S-BRs). Axiomatic operations, lower soft rough approximation space (lower SRA-space) and upper soft rough approximation space (upper SRA-space), are defined through after sets of soft relations. After that, the concept of SMGRSs is applied to a significant part of commutative algebra, group theory. In this respect, the primitive notions of SRA-spaces are defined with the help of two normal soft groups (NSGs). In groups, several important structural properties related to SMGRS are investigated in detail with illustrative examples. It is shown that SMGRS in groups may be influential in decision-making (DM) by some numerical examples. To demonstrate the flexibility, superiority, and effectiveness of the suggested technique, some comparative examples are given with some existing methods.

In this article, a new hybrid model named linear Diophantine fuzzy rough set (LDFRS) is proposed to magnify the notion of rough set (RS) and linear Diophantine fuzzy set (LDFS). Concerning the proposed model of LDFRS, it is more efficient to discuss the fuzziness and roughness in terms of linear Diophantine fuzzy approximation spaces (LDFA spaces); it plays a vital role in information analysis, data analysis, and computational intelligence. The concept of (,)-indiscernibility of a linear Diophantine fuzzy relation (LDF relation) is used for the construction of an LDFRS. Certain properties of LDFA spaces are explored and related results are developed. Moreover, a decision-making technique is developed for modeling uncertainties in decision-making (DM) problems and a practical application of fuzziness and roughness of the proposed model is established for medical diagnosis.

Rough set (RS) and fuzzy set (FS) theories were developed to account for ambiguity in the data processing. The most persuasive and modernist abstraction of an FS is the linear Diophantine FS (LD-FS). This paper introduces a resilient hybrid linear Diophantine fuzzy RS model (LDF-RS) on paired universes based on a linear Diophantine fuzzy relation (LDF-R). This is a typical method of fuzzy RS (F-RS) and bipolar FRS (BF-RS) on two universes that are more appropriate and customizable. By using an LDF-level cut relation, the notions of lower approximation (L-A) and upper approximation (U-A) are defined. While this is going on, certain fundamental structural aspects of LD-FAs are thoroughly investigated, with some instances to back them up. This cutting-edge LDF-RS technique is crucial from both a theoretical and practical perspective in the field of medical assessment.

Binary relations are most important in various fields of pure and applied sciences. The concept of linear Diophantine fuzzy sets (LDFSs) proposed by Riaz and Hashmi is a novel mathematical approach to model vagueness and uncertainty in decision-making problems. In LDFS theory, the use of reference or control parameters corresponding to membership and non-membership grades makes it most accommodating towards modeling uncertainties in real-life problems. The main purpose of this paper is to establish a robust fusion of binary relations and LDFSs, and to introduce the concept of linear Diophantine fuzzy relation (LDF-relation) by making the use of reference parameters corresponding to the membership and non-membership fuzzy relations. The novel concept of LDF-relation is more flexible to discuss the symmetry between two or more objects that is superior to the prevailing notion of intuitionistic fuzzy relation (IF-relation). Certain basic operations are defined to investigate some significant results which are very useful in solving real-life problems. Based on these operations and their related results, it is analyzed that the collection of all LDF-relations gives rise to some algebraic structures such as semi-group, semi-ring and hemi-ring. Furthermore, the notion of score function of LDF-relations is introduced to analyze the symmetry of the optimal decision and ranking of feasible alternatives. Additionally, a new algorithm for modeling uncertainty in decision-making problems is proposed based on LDFSs and LDF-relations. A practical application of proposed decision-making approach is illustrated by a numerical example. Proposed LDF-relations, their operations, and related results may serve as a foundation for computational intelligence and modeling uncertainties in decision-making problems.

Hybridization of soft sets and rough sets is an important way to deal with uncertainties. This paper aims to study the concept of roughness in soft sets over groups. In this regard, a pair of two soft sets, viz. soft lower and soft upper approximation spaces, are introduced by applying the normal soft groups corresponding to each parameter. Some important results related to these soft approximation spaces over groups are studied with examples. Furthermore, this paper presents a relationship between the soft approximation spaces based on the soft image and soft pre-image of a normal soft group via group homomorphisms. This work can be applicable in the field of information technology to connect two information systems.

In this research article, the notion of roughness in soft sets is investigated. Its applications are found on soft-intersection groups defined by Çağman et al. (Neural Comput Appl 21(1):S151–S158, 2012). The group homomorphism is manipulated to establish the connection between the approximation spaces of soft sets of groups.

The theory of rough sets is successfully applied in various algebraic systems (e.g. groups, rings, and modules). In this paper, the concept of roughness is introduced in modules of fractions with respect to its submodules. Hence, the notion of the lower and upper approximation spaces based on a submodule of the modules of fractions is introduced. Some fundamental results related to these approximation spaces are examined with examples. Moreover, this paper establishing several connections between the approximation spaces of two different modules of fractions with respect to the image and pre-image under a module homomorphism. This technique of building up a connection among the approximation spaces via module homomorphisms is useful to connect two information systems in the field of information technology.

In this paper, the conception of prime (semiprime) L-fuzzy soft bi- hyperideals, strongly prime L-fuzzy soft bi-hyperideals, irreducible (strongly irreducible) L-fuzzy soft bi-hyperideals of a semihypergroup S is introduced, where L is a complete bounded distributive lattice. Using the properties of these L-fuzzy soft bi-hyperideals some characterizations of regular and intra- regular semihypergroups are given.

In this paper, the notions of fuzzy zero-divisors and fuzzy integral domains are illustrated. Some fundamental properties of fuzzy integral domains are proved. Moreover, the notions of fuzzy regular element and fuzzy regular sequences are defined. It is shown that any permutation (resp. any positive integral power) of a fuzzy regular sequence is again a fuzzy regular sequence. At the end, fuzzy regular sequences of two fuzzy submodules are related with the help of fuzzy short exact sequences.

In this paper, L-fuzzy soft subsemihypergroups, L-fuzzy soft left (right, twosided) hyperideal of a semihypergroup S over U are defined, where L denotes a complete bounded distributive lattice. Furthermore, L-fuzzy soft prime (semiprime) hyperideals of S over U are studied. Some characterizations of those semihypergroups for which each hyperideal is prime (semiprime) are also given.