Certain Types of Covering-Based Multigranulation (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi mathvariant="normal">ℐ</a:mi> <a:mo>,</a:mo> <a:mi mathvariant="script">T</a:mi> </a:math>)-Fuzzy Rough Sets with Application to Decision-Making
MA Jue, Mohammed Atef, S.I. Nada, Ashraf S. Nawar
Abstract
As a generalization of Zhan’s method (i.e., to increase the lower approximation and decrease the upper approximation), the present paper aims to define the family of complementary fuzzy <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"> <a:mi>β</a:mi> </a:math> -neighborhoods and thus three kinds of covering-based multigranulation ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"> <c:mi mathvariant="normal">ℐ</c:mi> <c:mo>,</c:mo> <c:mi mathvariant="script">T</c:mi> </c:math> )-fuzzy rough sets models are established. Their axiomatic properties are investigated. Also, six kinds of covering-based variable precision multigranulation ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M4"> <g:mi mathvariant="normal">ℐ</g:mi> <g:mo>,</g:mo> <g:mi mathvariant="script">T</g:mi> </g:math> )-fuzzy rough sets are defined and some of their properties are studied. Furthermore, the relationships among our given types are discussed. Finally, a decision-making algorithm is presented based on the proposed operations and illustrates with a numerical example to describe its performance.