Soft Rough Graphs

Symmetry

Fuzzy sets, rough sets and soft sets are different tools for modeling problems involving uncertainty. Graph theory is another powerful tool for representing the information by means of diagrams, matrices or relations. A possible amalgamation of three different concepts rough sets, soft sets and graphs, known as soft rough graphs, is proposed by Noor et al. They introduced the notion of vertex, edge induced soft rough graphs and soft rough trees depending upon the parameterized subsets of vertex set and edge set. In this article, a new framework for studying the roughness of soft graphs in more general way is introduced. This new model is known as the modified soft rough graphs or MSR-graphs. The concept of the roughness membership function of vertex sets, edge sets and of a graph is also introduced. Further, it has been shown that MSR-graphs are more robust than soft rough graphs. Some results, which are not handled by soft rough graphs, can be handled by modified soft rough graphs. The notion of uncertainty measurement associated with MSR-graphs is introduced. All applications related to decision makings are only restricted to the information of individuals only, not their interactions, using this technique we are able to involve the interactions (edges) of individuals with each other that enhanced the accuracy in decisions.

Open Mathematics

Soft set theory and rough set theory are two new tools to discuss uncertainty. Graph theory is a nice way to depict certain information. Particularly soft graphs serve the purpose beautifully. In order to discuss uncertainty in soft graphs, some new types of graphs called soft covering based rough graphs are introduced. Several basic properties of these newly defined graphs are explored. Applications of soft covering based rough graphs in decision making can be very fruitful. In this regard an algorithm has been proposed.

Neutrosophic sets (NSs) handle uncertain information while fuzzy sets (FSs) and intuitionistic fuzzy sets (IFs) fail to handle indeterminate information. Soft set theory, neutrosophic set theory, and rough set theory are different mathematical models for handling uncertainties and they are mutually related. The neutrosophic soft rough set (NSRS) model is a hybrid model by combining neutrosophic soft sets with rough sets. We apply neutrosophic soft rough sets to graphs. In this research paper, we introduce the idea of neutrosophic soft rough graphs (NSRGs) and describe different methods of their construction. We consider the application of NSRG in decision-making problems. In particular, we develop efficient algorithms to solve decision-making problems.

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 2019

Soft set theory and rough set theory are two newer tools to discuss uncertainty. Soft graphs are a nice way to depict certain information. In order to discuss uncertainty in soft graphs, a new type of graphs called neighborhood based soft covering rough graphs is introduced. We have discussed the uncertainty measures associated with neighborhood based soft covering rough graphs such as roughness measure, entropy measure and granularity. Some important properties of these uncertainty measures are investigated and the relationships between such measures are established. These properties will help to understand the essence of uncertainty measurement and in measuring the quality of a decision rule.

Rough membership function defines the degree of relationship between conditional and decision attributes of an information system. It is defined by µ R X : U → [0, 1] where X is the subset of U under the relation R where U is the universe of discourse. It can be expressed in different forms like cardinality form, probabilistic form etc. In cardinality form, it is expressed as µ R X = |[x] R ∩X| |[x] R | where as in probabilistic form it can be denoted as P (x ∈ X| [x] R) = P ([x] R ∩X) P ([x] R) where [x] R is the equivalence class of x with respect to R. This membership function is used to measure the value of uncertainty. In this paper we have introduced the concept of graphical representation of rough sets. Rough graph was introduced by He Tong in 2006. In this paper, we propose a novel method for the construction of rough graph through rough membership function ω F G (f). We propose that there is an edge between vertices if max(ω F G (v i) , ω F G (v j)) > 0. The rough graph is being constructed for an information system; here objects are considered as vertices. Rough path, rough cycle, rough ladder graph are introduced in this paper. We develop the operations on rough graph and also extend the properties of rough graph.

Annals of Pure and Applied Mathematics, 2017

Granulate the similar things is an essential part in multivalued information system. Rough set theory plays a vital role to solve imprecise problem. In Particular, multigranular rough set is an efficient tool to work on multivalued information system. Soft set theory is also deal uncertainty. In this paper we propose multigranular rough soft set and its properties.

International Journal of Computational Intelligence Systems, 2018

Fuzzy rough set theory is a hybrid method that deals with vagueness and uncertainty emphasized in decision-making. In this research study, we apply the concept of fuzzy rough sets to graphs. We introduce the notion of fuzzy rough digraphs and describe some of their methods of construction. In particular, we consider applications of fuzzy rough digraphs. We also present algorithms to solve decision-making problems regarding selection of a city for treatment and identification of best location in a department to set mobile phone Jammer.

Information Sciences, 2007

In this article, we present some extensions of the rough set approach and we outline a challenge for the rough set based research.

1998

The present state of rough set theory and its applications is presented by articles in this collection as well as by research papers listed in APPENDIX 1 to which we refer the reader. We would like to discuss here some directions for further research as well as to point to some recent results not mentioned earlier which seem to us to be of importance for development of rough set theory and its applications.

International Journal of Fuzzy Logic System, 2021

Rough set theory and its hybrid models are new emerging techniques to deal with uncertainties, impreciseness and ambiguities. In past few years, researchers are taking keen interest in rough set theory, which is an important area of meteorological field especially in weather forecasting and in artificial intelligence. In the present communication basic concepts and some aspects of rough set theory are defined and explained. Its hybrid models namely fuzzy rough and rough fuzzy sets are described in detail. Fuzzy rough information measure is characterized and its application is studied with illustration. Approximate equalities using rough fuzzy and rough intuitionistic fuzzy sets are also given with the conclusion in the end.