On hybrid structures involving rough sets generalized fuzzy structures and graphs

Abstract

Vague and incomplete information poses many challenges to the researchers in newlinescience and engineering. Among the various techniques developed for tackling newlinethese burdensome constraints, fuzzy set (FS) theory and rough set (RS) theory are newlineprominent directions which have produced multiple extensions and hybridizations. newlineEven though RS theory was originally developed as the outcome of an indiscernibility newlinerelation, which can be identified mathematically as an equivalence relation, soon newlinethe shift was towards similarity or coverings as well as their fuzzy counterparts. newlineMany hybrid models were proposed towards this aim. With this exhaustive process, newlinethe distance between those ideas became shorter. Although fuzzy set theory is a newlinevalid form to express the uncertain evaluation information, it is still inefficient in newlinesolving some complex situations in real life. In practice, other types of uncertain newlineand complex evaluations will be inevitably given by decision makers. Atanassov s newlineintuitionistic fuzzy set theory expanded the scope of fuzzy set theory with the addition newlineof a degree of indeterminacy to fuzzy sets. Real situations sometimes impose a newlineneutral component in addition to indeterminacy. For modeling situations like this, newlinepicture fuzzy sets were designed. They have positive, neutral and negative or refusal newlinemembership functions. In a different line of thought, situations where the decision newlinemakers hesitate among various possible values appeal to hesitant fuzzy sets and its newlinehybrid models. newlineA Graph is a symmetric binary relation on a set. It is a fundamental tool in newlinemathematical modeling and has applications in almost all branches of Science and newlineEngineering. Many of the real life problems were solved through mathematical newlinemodeling with the help of graph theory. Graph theory, where objects are represented newlineby vertices and relations by edges, is a convenient way of representing information involving relationship between objects newline