Rough Fuzzy Substructures of Quantale Module Under Soft Relations and Corresponding Decision-Making Methods

This research paper has developed a way of roughness of fuzzy substructures by using soft relations for developing rough fuzzy substructures in Quantale module. Thus, an innovative concept of fuzzy substructures of Quantale module under rough environment by soft relations, is presented. The lower and upper approximations of fuzzy subsets of quantale module are defined by aftersets and foresets. This relationship leads to various characterizations of the rough fuzzy substructures of quantale modules. Besides of more comprehensive results, soft compatible and soft complete relations are required with foresets and aftersets. Soft relations are further being used to determine upper (lower) approximation of fuzzy subsets of quantale module using foreset and afterset. Moreover, several characterizations of rough fuzzy substructures of quantale module are investigated. Furthermore, the algebraic relations of upper (lower) approximations of fuzzy quantale submodule and fuzzy quantale submodule ideals are studied with the help of soft relations under weak quantale module homomorphism. To illustrate that the suggested approach is superior to the given methods, examples are provided. At last, we describe decision-making methods by using rough fuzzy substructures of Quantale module under soft relations to deal with uncertainties in the real-world problems. To demonstrate the validity, applicability, and efficacy of the suggested method, a detailed example of the decision-making process is provided.