Fuzzy rough set theory

The notions of hesitant fuzzy left (resp., right, bi-, quasi-) ideals are introduced, and several properties are investigated. Relations between a hesitant fuzzy left (resp., right) ideal, a hesitant fuzzy bi-ideal and a hesitant fuzzy quasi-ideal are discussed. Characterizations of hesitant fuzzy left (resp., right, bi-, quasi-) ideals are considered.

In this paper we discuss three problems in Data Mining Sparse Decision Systems: the problem of short reduct calculation, discretization of numerical attributes and rule induction. We present algorithms that provide approximate solutions to these problems and analyze the complexity of these algorithms.

This study introduces a novel framework leveraging Rough Set Theory (RST)based feature selection-MLReduct, MLSpecialReduct, and MLFuzzyRoughSet-to enhance machine learning performance on uncertain data. Applied to a private cardiovascular dataset, our MLSpecialReduct algorithm achieves a peak Random Forest accuracy of 0.99 (versus 0.85 without feature selection), while MLFuzzyRoughSet improves accuracy to 0.83, surpassing our MLVarianceThreshold (0.72-0.77), an adaptation of the traditional VarianceThreshold method. We integrate these RST techniques with preprocessing (discretization, normalization, encoding) and compare them against traditional approaches across classifiers like Random Forest and Naive Bayes. The results underscore RST's edge in accuracy, efficiency, and interpretability, with MLSpecialReduct leading in minimal attribute reduction. Against baseline classifiers without feature selection and MLVari-anceThreshold, our framework delivers significant improvements, establishing RST as a vital tool for explainable AI (XAI) in healthcare diagnostics and IoT systems. These findings open avenues for future hybrid RST-ML models, providing a robust, interpretable solution for complex data challenges.

In this paper we have introduced the concept of fuzzy soft semigroup which is a generalization of soft semigroup and studied some basic properties. We have defined the product of two fuzzy soft sets over a semigroup, fuzzy soft semigroups and fuzzy soft ideals and investigated their basic properties. We have also discussed the notions of fuzzy soft quasi-ideals, bi-ideals, generalized bi-ideals and interior ideals defined over a semigroup. . (F,A) is said to be a fuzzy soft super set of (G,B), if (G,B) is a fuzzy soft subset of (F,A). We denote it by Definition 14: Two fuzzy soft sets (F,A) and (G,B) over a common universe U are said to be fuzzy soft equal if (F,A) is a fuzzy soft subset of (G,B) and (G,B) is a fuzzy soft subset of (F,A). Definition 15: [8] If (F,A) and (G,B) are two fuzzy soft sets over the same universe U then "(F,A) AND (G,B) " is a fuzzy soft set denoted by (F,A)∧(G,B) and is defined by ( )( ) ( ) ,,, FAGBHAB =× where, H((,))F()G() αβ=α∧β for all (,)AB αβ∈×. Here ∧ is the operation of fuzzy intersection of two fuzzy sets. Definition 16: [8] If (F,A) and (G,B) are two fuzzy soft sets over the same universe U then "(F,A) OR (G,B) " is a fuzzy soft set, denoted by (F,A)∨(G,B) and is defined by (F,A)(G,B)(O,AB) ∨=× where, O((,))F()G() αβ=α∨β for all (,)AB αβ∈×. Here ∨ is the operation of fuzzy union of two fuzzy sets. Definition 17: [8] Union of two fuzzy soft sets (F,A) and (G,B) over the common univers U is the fuzzy soft set (H,C), where C = A∪B and for all e∈C,

In this paper, we present some connections between graph theory and hyperstructure theory. In this regard, we construct a hypergroupoid by defining a hyperoperation on the set of degrees of vertices of a hypergraph and we call it a degree hypergroupoid. We will see that the constructed hypergroupoid is always anHv-group. We will investigate some conditions on a degree hypergroupoid to have a hypergroup. Further, we study the degree hypergroupoid associated with Cartesian product of hypergraphs. Finally, the fundamental relation and complete parts of a degree hypergroupoid are studied.

The main purpose of this paper is to investigate ordered 􀀀-semihypergroups in the general terms of ordered 􀀀-hyperideals. We intro-duce ordered (generalized) (m; n)-􀀀-hyperideals in ordered 􀀀-semihypergroups. Then, we characterize ordered 􀀀-semihypergroup by ordered (generalized) (0; 2)- 􀀀-hyperideals, ordered (generalized) (1; 2)-􀀀{hyperideals and ordered (general- ized) 0-minimal (0; 2)-􀀀-hyperideals. Furthermore, we investigate the notion of ordered (generalized) (0; 2)-bi-􀀀-hyperideals, ordered 0-(0; 2) bisimple ordered 􀀀-semihypergroups and ordered 0-minimal (generalized) (0; 2)-bi-􀀀-hyperideals in ordered 􀀀-semihyperoups. It is proved that an ordered 􀀀-semihypergroup S with a zero 0 is 0-(0; 2)-bisimple if and only if it is left 0-simple.

In this study, which focuses on the intersection of soft set theory and topological hyperrings, the concept of soft topological hyperrings is proposed and its relation with topological hyperrings is examined. Morever, some characterizations related to the family of soft topological hyperrings are obtained and the category of soft topological hyperrings is established. Finally, the concept of soft topological subhyperrings is described and several structural features are studied.

In [5] J. Jantosciak introduced several special types of subhyper-groups (invertible, closed, normal, re exive) of a general hypergroup and studied their relationship. In this article, the full description of such subhypergroups in hypergroups induced by quasiordered groups is given. Further, it is shown that there are no such non-trivial sub-hypergroups in quasiorder hypergroups.

Fully simple semihypergroups have been introduced in [9], motivated by the study of the transitivity of the fundamental relation β in semihypergroups. Here, we determine a transversal of isomorphism classes of fully simple semihypergroups with a right absorbing element. The structure of that transversal can be described by means of certain transitive, acyclic digraphs. Moreover, we prove that, if n is an integer ≥2, then the number of isomorphism classes of fully simple semihypergroups of size n + 1, with a right absorbing element, is the (n + 1)-th term of sequence A000712 in [20], namely, n k=0 p(k)p(n − k), where p(k) denotes the number of nonincreasing partitions of integer k.

We introduce a family of hypergroups, called weakly complete, generalizing the construction of complete hypergroups. Starting from a given group G, our construction prescribes the β-classes of the hypergroups and allows some hyperproducts not to be complete parts, based on a suitably defined relation over G. The commutativity degree of weakly complete hypergroups can be related to that of the underlying group. Furthermore, in analogy to the degree of commutativity, we introduce the degree of completeness of finite hypergroups and analyze this degree for weakly complete hypergroups in terms of their β-classes.

Hypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is the trivial group. However, very little is known about hypergroups that are neither 1-hypergroups nor belong to the first class. The goal of this work is to take a first step in classifying G-hypergroups, that is, hypergroups whose heart is a nontrivial group. We introduce their main properties, with an emphasis on G-hypergroups whose the heart is a torsion group. We analyze the main properties of the stabilizers of group actions of the heart, which play an important role in the construction of multiplicative tables of G-hypergroups. Based on these results, we characterize the G-hypergroups that are of type U on the right or cogroups on the right. Finally, we present the hyperproduct tables of all G-hypergroups of ...

Let I be the class of fully zero-simple semihypergroups generated by a hyperproduct. In this paper we study some properties of residual semihypergroup (H+, ⋆) of a semihypergroup (H, •) ∈ I. Moreover, we find sufficient conditions for (H, •) and (H+, ⋆) to be cyclic.

In this study, a Fuzzy Inference System is developed to create a knowledge-based for the diagnosis and detection of sepsis using Matlab's fuzzy logic toolbox. The FIS consists of expert-specified input membership functions, output membership function, and IF-THEN rules. To validate our system, we tested it with the domain (medical) expert knowledge by comparing its performance with at least 10 hypothetical scenarios. The KB-FIS gave good answers in 8/10 scenarios and generated rules for decision support. The answers are qualitatively correct but quantitatively small. The fuzzy inference system designed performs as well as the medical experts under controlled conditions. With the methodology used in this study, it is hoped that it can assist in enhancing the knowledge-base system for proper, effective and early diagnosis and detection of sepsis

In multi-instance learning, each learning object consists of many descriptive instances. In the corresponding classification problems, each training object is labeled, but its constituent instances are not. The classification objective is to predict the class label of unseen objects. As in traditional single-instance classification, when the class sizes of multi-instance data are imbalanced, classification is degraded. Many multi-instance classifiers have been proposed, but few take into account the possibility of class imbalance, which causes them to fail in this situation. In this paper, we propose a new type of classifier that embodies a solution to the multi-instance class imbalance problem. Our proposal relies on the use of fuzzy rough set theory. We present two families of classifiers respectively based on information extracted at bag-level and at instance-level. We experimentally show that our algorithms outperform state-of-theart solutions to multi-instance imbalanced data classification, evaluated by the popular metrics AUC and geometric mean.

In a network model, the evaluation information given by decision makers are occasionally of types: yes, abstain, no, and refusal. To deal with such problems, we use mathematical models based on picture fuzzy sets. The spherical fuzzy model is more versatile than the picture fuzzy model as it broadens the space of uncertain and vague information, due to its outstanding feature of vast space of participation of acceptable triplets. Graphs are a mathematical representation of networks. Thus to deal with many real-world phenomena represented by networks, spherical fuzzy graphs can be used to model different practical scenarios in a more flexible manner than picture fuzzy graphs. In this research article, we discuss two operations on spherical fuzzy graphs (SFGs), namely, symmetric difference and rejection; and develop some results regarding their degrees and total degrees. We describe certain concepts of irregular SFGs with several important properties. Further, we present an applicatio...

The multigranulation rough set (MGRS) is becoming a rising theory in rough set area, which offers a desirable theoretical method for problem solving under multigranulation environment. However, it is worth noticing that how to effectively extract decision rules in terms of multigranulation rough sets has not been more concerned. In order to address this issue, we firstly give a general ruleextraction framework through including granulation selection and granule selection in the context of MGRS. Then, two methods in the framework (i.e. a granulation selection method that employs a heuristic strategy for searching a minimal set of granular structures and a granule selection method constructed by an optimistic strategy for getting a set of granules with maximal covering property) are both presented. Finally, an experimental analysis shows the validity of the proposed rule-extraction framework in this paper.

The concept of convex ordered hyperrings associated with a strongly regular relation was investigated in this study. In this paper, we first studied hyperatom elements of ordered hyperrings and then investigated characterizations of quotient ordered rings. Is there a strongly regular relation θ on a convex ordered hyperring R for which R/θ is a convex ordered ring? This leads to an ordered ring obtained from an ordered hyperring.

The purpose of this paper is to construct Boolean rings from multirings. In this regards, a method to construct a multigroup(multiring) on a given non-empty set, are introduced and its properties has been investigated. Also, an equivalence relation on a multiring are introduced and it is extended to an smallest strongly regular  equivalence relation, such that its quotient  space be a commutative Boolean ring with identity. Finally, the transitivity of this relation based on complete parts are   proved.

The purpose of this paper is computing the fundamental relations and automorphism groups of very thin H v-groups. In this regards, we rst investigate some basic properties of H v-groups and then we show that any given group is isomorphic to the fundamental group of a nontrivial H v-group. The main properties of very thin H v-groups are also investigated and it is proved that every nite very thin H v-group is proper.

In this paper, we introduce the notion of reference point, lower and upper approximation with respect to reference point by a Lie algebra. We are concerned with some important properties of them. For a fuzzy Lie subalgebra μ of a Lie algebra L, t-level relation U(μ, t

One of the efficient tools to handle segregation of imbalanced data is support vector data description (SVDD). In contrast to support vector machine (SVM), enclosing target data in a hyper-sphere by SVDD leads to avoid biasing toward major data. SVDD can gain the best description of data when its free parameters are set to proper values. In this paper, we propose beliefbased chaotic krill herd algorithm for SVDD (BCKH-SVDD) with the aim of designing effective description of data. First, we introduce a new SVDD based on belief function theory, and then, we tune the free parameters by chaotic krill herd algorithm. Belief function theory is one of the best methods to enhance decision making for uncertain data. By adding a new belief-based weight, we can decide better about the data around the SVDD boundary and the classification will be more precise. Chaotic krill herd optimization algorithm introduces chaotic maps in the krill herd algorithm. With the help of chaotic maps, the two issues, namely local optima avoidance and convergence speed, can be overcome. Thus, chaotic krill herd algorithm is constructed based on chaotic functions and automatic switching between global and local searches of krill herd. To present the power of BCKH-SVDD, several experiments have been conducted based on tenfold cross-validation over real-world data sets from UCI repository. Experimental results show the superiority of the proposed algorithm to state-of-the-art methods in terms of classification accuracy, precision and recall measures.

Event handlers have wide range of applications such as medical assistant systems and fire suppression systems. These systems try to provide accurate responses based on the least information. Support vector data description (SVDD) is one of the appropriate tools for such detections, which should handle lack of information. Therefore, many efforts have been done to improve SVDD. Unfortunately, the existing descriptors suffer from weak data characteristic in sparse data sets and their tuning parameters are organized improperly. These issues cause reduction of accuracy in event handlers when they are faced with data shortage. Therefore, we propose automatic support vector data description (ASVDD) based on both validation degree, which is originated from fuzzy rough set to discover data characteristic, and assigning effective values for tuning parameters by chaotic bat algorithm. To evaluate the performance of ASVDD, several experiments have been conducted on various data sets of UCI repository. The experimental results demonstrate superiority of the proposed method over state-of-the-art ones in terms of classification accuracy and AUC. In order to prove meaningful distinction between the accuracy results of the proposed method and the leading-edge ones, the Wilcoxon statistical test has been conducted. Communicated by V. Loia.

Despite emerging of Web 2.0 applications and increasing requirements to well-behaved Web robots, malicious ones can reveal irreparable risks for Web sites. Regardless of behavior of Web robots, they may occupy bandwidth and reduce performance of Web servers. In spite of many prestigious researches trying to characterize Web visitors and classify them, there is a lack of concentration on feature selection to dynamically choose attributes used to describe Web sessions. On the other hand, depending on an accurate clustering technique, which can deal with huge number of samples in a reasonable amount of time, is practically important. Therefore, in this paper, a new algorithm, fuzzy rough set-Web robot detection (FRS-WRD), is proposed based on fuzzy rough set theory to better characterize and cluster Web visitors of three real Web sites. External evaluations show that in contrast to state-of-the-art algorithms, FRS-WRD achieves better results in terms of G-mean 95%, Jaccard 88%, entropy 0.36, and finally, purity 96%. Moreover, according to confusion matrixes, it can better detect malicious Web visitors.

Data mining plays a significant role in information acquisition and effective information utilisation from big data. Many techniques are available for data mining, In many disciplines, including health care services, data reduction using rough set theory is a commonly utilized mathematical technique. In this paper, Rough Set Theory (RST) is applied to a patient's satisfaction survey to identify sets of critical attributes which are responsible for the patient's satisfaction. RST is a useful tool for data mining, however the data it extracts lacks efficiency and precision. This study proposes a novel approach to tackle this problem, combining Rough Set Theory, Data Envelopment Analysis (DEA), and Artificial Neural Networks (ANN). The Data Envelopment Analysis (DEA) is the leading approach used to find the dependency of output variables on input variables. These powerful instruments Through the use of Artificial Neural Network (ANN) results, RST and DEA are utilised to determine the effectiveness of reductions. Based on cross-validation of ANN accuracy of forecasting is determined.Maximizing patient satisfaction is an important goal of Health care organizations. Patients' feedback helps them to identify the ways to improve their working methods which transforms into better care and happier patients.In this paper, Rough Set Theory is applied to a patient's satisfaction survey to identify sets of critical attributes which are responsible for the patient's satisfaction. DEA & ANN give the best set of critical attributes based on their efficiency.

This paper explores the defects in fuzzy (hyper) graphs (as complex (hyper) networks) and extends the fuzzy (hyper) graphs to fuzzy (quasi) superhypergraphs as a new concept.We have modeled the fuzzy superhypergraphs as complex superhypernetworks in order to make a relation between labeled objects in the form of details and generalities. Indeed, the structure of fuzzy (quasi) superhypergraphs collects groups of labeled objects and analyzes them in the form of the part to part of objects, the part of objects to the whole group of objects, and the whole to the whole group of objects at the same time. We have investigated the properties of fuzzy (quasi) superhypergraphs based on any positive real number as valued fuzzy (quasi) superhypergraphs, considering the complement of valued fuzzy (quasi) superhypergraphs, the notation of isomorphism of valued fuzzy (quasi) superhypergraphs based on the permutations, and we have presented the isomorphic conditions of (self complemented) valued fuzzy (quasi) superhypergraphs. The concept of impact membership value of fuzzy (quasi) superhypergraphs is introduced in this study and it is applied in designing the real problem in the real world. Finally, the problem of business superhypernetworks is presented as an application of fuzzy valued quasi superhypergraphs in the real world.

The development and selection of coatings or coating combinations is a complex and costly task. Numerical simulations provide a great help to analyze the behavior of coatings and layer interfaces under mechanical and thermal loading through the computation of stress and strain fields. They are also used to fine tune the geometrical configuration and define optimal thermo-mechanical properties related to the applied stress to enhance wear and crack resistance. Coating design has been based mainly on discrete layer models with abruptly changing properties at interface perfectly bonded to a substrate. These assumptions result in discontinuities in stress and temperature fields at the interface between successive layers. These models are based either on integral transform (classically Fourier transform) or finite element (FE) methods. The former cannot handle 3D thermo-mechanical problems. The latter demands huge calculation times, especially when considering thin layers with very small elements. The aim of this paper is to account for materials with continuously changing properties and to obtain an improved prediction of the resistance of such graded materials. This gradation may be linked to surface treatments like nitruration, thermal treatment, ... or to the deposition of coatings inducing diffuse boundaries between those dissimilar materials. The proposed model aims at providing valuable description and understanding of the behavior and resistance of such graded material. It is based on a second order finite difference (FD) formulation of the continuous thermal and elasticity equations. It can handle any kind of depth dependence of the material properties. Multigrid techniques are implemented to accelerate the convergence, reduce CPU time and thus permit the use of fine grids to accurately describe the variation in the material properties. The first step of this project consists in implementing the above-cited numerical techniques in a 2D plane strain model.

The development and selection of coatings or coating combinations is a complex and costly task. Numerical simulations provide a great help to analyze the behavior of coatings and layer interfaces under mechanical and thermal loading through the computation of stress and strain fields. They are also used to fine tune the geometrical configuration and define optimal thermo-mechanical properties related to the applied stress to enhance wear and crack resistance. Coating design has been based mainly on discrete layer models with abruptly changing properties at interface perfectly bonded to a substrate. These assumptions result in discontinuities in stress and temperature fields at the interface between successive layers. These models are based either on integral transform (classically Fourier transform) or finite element (FE) methods. The former cannot handle 3D thermo-mechanical problems. The latter demands huge calculation times, especially when considering thin layers with very small elements. The aim of this paper is to account for materials with continuously changing properties and to obtain an improved prediction of the resistance of such graded materials. This gradation may be linked to surface treatments like nitruration, thermal treatment, ... or to the deposition of coatings inducing diffuse boundaries between those dissimilar materials. The proposed model aims at providing valuable description and understanding of the behavior and resistance of such graded material. It is based on a second order finite difference (FD) formulation of the continuous thermal and elasticity equations. It can handle any kind of depth dependence of the material properties. Multigrid techniques are implemented to accelerate the convergence, reduce CPU time and thus permit the use of fine grids to accurately describe the variation in the material properties. The first step of this project consists in implementing the above-cited numerical techniques in a 2D plane strain model.

In this paper, we consider the notions of Q-algebras and hyper BCKalgebras, give some related results, introduce the relation β on them and let β∗ be the transitive closure of β. Then by considering the concept of strongly regular equivalence relation β∗ on hyper BCK-algebras, we define the notion of fundamental Q-algebra and we prove that any countable Q-algebra is a fundamental Q-algebra.

Since Pawlak defined the notion of rough sets in 1982, many authors made wide research studying rough sets in the ordinary case and the fuzzy case. This paper introduced a new style of rough fuzzy sets based on a fuzzy ideal ℓ on a universal finite set X. New lower and new upper fuzzy sets are introduced, and consequently, fuzzy interior and fuzzy closure operators of a rough fuzzy set are discussed. These definitions, if ℓ is restricted to ℓ • = {0}, imply the fuzzification of previous definitions given in the ordinary case, and moreover in the crisp case, we get exactly these previous definitions. The new style gives us a better accuracy value of roughness than the previous styles. Rough fuzzy connectedness is introduced as a sample of applications on the recent style of roughness.

Article Info Data-driven decision making is vital in credit risk assessment and other areas. Complex datasets are hard to rule. We use adaptive fuzzy network partitioning, rough set theory, and rule generation to improve data-driven credit risk assessment. An adaptive fuzzy network partitioning algorithm is used to cluster the dataset. Each cluster instance receives fuzzy membership degrees. Next, rough set-based attribute reduction identifies credit risk assessment attributes inside each cluster. Finally, attributes are used to build accurate and understandable credit risk assessment criteria. A loan application dataset is used to test the suggested method. The results show successful loan application clustering and the creation of credit risk criteria for each cluster. Accurate predictions and interpretable rules improve credit risk assessment comprehension and decisionmaking. By merging adaptive fuzzy network partitioning, rough set theory, and rule generation, the hybrid methodology overcomes classic technique constraints. These methods create a comprehensive framework for credit risk assessment criteria that improves accuracy and interpretability. Financial institutions and credit providers may benefit from the approach. The proposed approach can be tested in multiple domains and extended to handle increasingly complicated datasets. Evaluating the methodology on real-world datasets and comparing it to existing methods can also reveal its practicality and efficacy. This research generates accurate and interpretable rules for data-driven credit risk assessment using a hybrid method. Adaptive fuzzy network partitioning, rough set theory, and rule generation can improve decision-making across domains.

The notion of Intuitionistic fuzzy hypervector space has been generalized and a few basic properties on this concept are studied. It has been shown that the intersection and union of an arbitrary family of Intuitionistic fuzzy hypervector spaces are also Intuitionistic fuzzy hypervector space. Lastly, the notion of a linear transformation on a hypervector space is introduced and established an important theorem relative to Intuitionistic fuzzy hypervector spaces.

We give two su cient conditions for a hypergroupoid to be a feeble semi-hypergroupoid and a su cient condition to be a feeble hypergroup. We show that every hypergroup in the class of the k-quasi-Steiner hypergroupoids is feebly associative. Finally, we apply these algebraic results to study Steiner systems; in particular, we give a necessary and su cient condition for a Steiner system to be a projective plane.

Structuring the geostrategic landscape entails using integrated modeling methods to expand strategic horizons to forecast the development vectors of political and economic systems. The reasons for the barbarous war of the Moscow regime against the Ukrainian state have a significant basis both for theoretical study and further practical implementation of the obtained data into political and security practices. Examining the cause-effect complex of articulation of destructive paradigms has become the subject of many sciences. However, requiring their completed conceptualization within the political reality, individual phenomena are poorly studied from the standpoint of economic representation in the context of mathematical modeling of institutional implementing strategic national interests in the most crucial areas of life. Thus, public infrastructure policy is one of the reference areas. The development of the geostrategy of the modern Ukrainian state actualizes the task of elaborati...

The notions of fuzzy set (FS) and intuitionistic fuzzy set (IFS) make a major contribution to dealing with practical situations in an indeterminate and imprecise framework, but there are some limitations. Pythagorean fuzzy set (PFS) is an extended form of the IFS, in which degree of truthness and degree of falsity meet the condition 0≤Θ˘2(x)+K2(x)≤1. Another extension of PFS is a q´-rung orthopair fuzzy set (q´-ROFS), in which truthness degree and falsity degree meet the condition 0≤Θ˘q´(x)+Kq´(x)≤1,(q´≥1), so they can characterize the scope of imprecise information in more comprehensive way. q´-ROFS theory is superior to FS, IFS, and PFS theory with distinguished characteristics. This study develops a few aggregation operators (AOs) for the fusion of q´-ROF information and introduces a new approach to decision-making based on the proposed operators. In the framework of this investigation, the idea of a generalized parameter is integrated into the q´-ROFS theory and different genera...

The qualitative danger and threat assessment method based on the principle of the maximal allowable limits is proposed for the intelligent disaster decision support system. The proposed method uses the rough set based plausible disaster spreading model and the formal model of the territorial system, which allows us to consider the dynamics of natural disaster spreading discretely at the level of individual cells of the grid and describes disaster dynamics as moving a vague contour presented as a boundary region of a rough set on the certain terrain. Both the plausible disaster spreading model and the qualitative danger and threat assessment method were developed for wildfires as the most common class of natural disasters, and can be suitable for solving decision support tasks for protection against other natural disasters without loss of clarity and justification for the decision-maker.

In this paper, we define topological hyperrings and study their basic concepts which supported by illustrative examples. We show some differences between topological rings and topological hyperrings. Also, by the fundamental relation $Gamma^{*}$, we indicate the role of complete parts (saturated subsets) and complete hyperrings in topological hyperrings and specially we show that if every (closed) open subset is a complete part in a topological complete hyperring then its fundamental ring is a topological ring. Finally, we study the quotient topology induced by $Gamma^{*}$-relation on an associated Krasner hyperring obtained by a ring and show that it is isomorphic to a quotient space of the ring by its ideals.

This work presents a spatial model for the real-time GIS-based decision support systems based on dynamic fuzzy rough soft topology, which represents a spatial structure that contains a multitude of interacting processes, which evolve in space and time. The dynamics of destructive processes are modeled using the spread model. The area of interest is represented as an approximation by a grid of cubic cells. This allows taking into account the peculiarities of the initial information obtained using remote sensing techniques and having a significant uncertainty. As a result, boundaries of contours of spreading destructive processes are blurred using fuzzy rough soft topology. The proposed model reduces the computational complexity and provides the acceptable performance.

Stability analysis of rock slopes involves dealing with geometrical and geomechanical parameters that are approximate in nature. The geometrical parameters include dips and dip directions of discontinuities, which are conventionally selected based on drawing density contours on stereonet and then selecting the point with highest density. The density contours clearly indicate the approximation or "inexactness" in the geometrical parameters. Besides, approximations are also involved in expressing the geomechanical parameters such as cohesion and friction angle of the rock mass or the rock joints.
Theory of fuzzy sets is an ideal tool for dealing with variables of approximate nature. In this paper, an algorithm is proposed for stability analysis of rock slopes using theory of fuzzy sets. Appropriate fuzzy sets are assigned to geometrical and geomechanical parameters based on actual field and/or laboratory records and the final fuzzy safety factor is determined accordingly. A fuzzy reliability index is defined in order to measure reliability of safety factors applicable at different degrees of membership function. The procedure is illustrated using actual field records.

In this paper, we study relative ordered (m, n)-hyperideals in ordered semihypergroups. We also study relative (m, 0)-hyperideals and relative (0, n)-hyperideals as well as characterize regular ordered semihypergroups, and obtain some results based on these relative hyperideals. We prove that the intersection of all relative ordered (m, n)-hyperideals of S containing s is a relative ordered (m, n)-hyperideal of S containing s. Suppose that (S,•,≤) is an ordered semihypergroup, A ⊆ S and m,n are positive integers. We prove that if R(m,0) and L(0,n) be the set of all relative ordered (m,0)-hyperideals and the set of all relative ordered (0,n)-hyperideals of S, respectively. Then the following assertions are true: i) S is relative (m,0)-regular if and only if for all R ∈ R(m,0), R = (Rm •A]A. ii) S is relative (0,n)-regular if and only if for all L ∈ R(0,n), L = (A•Ln]A. Furthermore, suppose that (S, •, ≤) is an ordered semihypergroup and m, n are non-negative integers. Let A ⊆ S. Suppose that A(m,n) is the set of all relative ordered (m, n)-hyperideals of S. Then, we have the following: S is (m,n)-regular ⇐⇒ ∀A ∈ A(m,n),A = (Am •A•An]A.

The notion of a q-rung orthopair fuzzy soft rough set (q ROFSRS) appeared as an extension of q-rung orthopair fuzzy set (q ROFS) and q-rung orthopair fuzzy soft set (q ROFSS) with the aid of rough set (RS) definition. Thus, q ROFSRS and m-polar fuzzy set (m PFS) are convenient to deal with uncertain knowledge which helps us to solve many problems in decision making. In this paper, we define the soft rough q-rung orthopair m-polar fuzzy sets (q RO m PFS) and q-rung orthopair m-polar fuzzy soft rough sets (q RO m PFSRS) through crisp soft and q-rung orthopair (q-RO) m-polar fuzzy soft approximation space. The related characteristics of these models are also studied. Then, we construct two new algorithms for these models to solve MADM issues. The successful application and corresponding comparative analyses proves that our proposed models are rational and effective. INDEX TERMS q-rung orthopair fuzzy soft rough set, m-polar fuzzy set, soft rough q-rung orthopair m-polar fuzzy sets, q-rung orthopair m-polar fuzzy soft rough sets, multi-attribute decision making. AHMED MOSTAFA KHALIL received the M.Sc. degree in pure mathematics from Al-Azhar University, Egypt, and the Ph.D. degree in fundamental mathematics from Shaanxi Normal University, China. He is currently a Lecturer with the Department of Mathematics, Faculty of Science, Al-Azhar University. He has published several technical papers in refereed international journals,

In this research paper, we present a novel frame work for handling $m$-polar information by combining the theory of $m-$polar fuzzy  sets with graphs. We introduce certain types of edge regular $m-$polar fuzzy graphs and edge irregular $m-$polar fuzzy graphs. We describe some useful properties of edge regular, strongly edge irregular and strongly edge totally irregular $m-$polar fuzzy graphs. We discuss the relationship between degree of a vertex and degree of an edge in an $m-$polar fuzzy graph. We investigate edge irregularity on a path on $2n$ vertices and barbell graph $B_{n,n}.$We also present an application of $m-$polar fuzzy graph to decision making.