Abstract
A rough set, an extension of a crisp set, is a mathematical tool to understand and model uncertainty without much prior information, additional adjustments or pre-defined membership functions. To manipulate the subjectivity and vagueness of decision-making problems, rough models provide more objective description of given information using upper and lower approximations. In this research paper, we study the absurdity and falsity of existing definition of rough graph. Based on rough relations, we introduce the concepts of rough graphs and rough digraphs and establish certain formulae, lower and upper bounds of color energy of rough graphs. Using D numbers, rough weights and rough entropy weights, we develop rough D-TOPSIS method which incorporates the capability to analyze uncertain and vague information without additional assumptions. We study the importance of rough information for the evaluation of water requirement in agricultural farming, investment analysis in organic and inorganic farming systems and illegal communication networks.
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