Soft multi-rough set topology with applications to multi-criteria decision-making problems

Rough set theory introduced by Pawlak (Int J Comput Inf Sci 11:341–356, 1982), multi-set theory proposed by Blizard (Notre Dame J Form Log 30:36–65, 1989) and soft set theory introduced by Molodtsov (Comput Math Appl 37(4–5):19–31, 1999) are fundamental concepts in computational intelligence, which have a myriad of applications in modeling uncertainties and decision making under uncertainty. In this paper, the idea of soft multi-rough set (SMRS) is introduced as a hybrid model of soft set, multi-set and rough set. The SMRS provides roughness of a multi-set in terms of soft multi-approximation space. The novel concept of soft multi-rough topology (SMR-topology) is defined to discuss topological structure of SMRSs by using pairwise SMR-approximations. The proposed models of SMRS and SMR-topology are suitable for modeling uncertainties in the real-life circumstances. SMR-topology is the generalization of crisp topology, soft topology and soft rough set topology. Some fundamental properties of SMR-topology and their related results are studied. Some algorithms for are developed for multi-criteria decision making based on soft multi-sets, soft multi-rough sets and soft multi-rough topology. Based on proposed algorithms, the applications of SMRSs and SMR-topology toward diagnosis of depression and diabetes are illustrated by the numerical examples. A comparison analysis of proposed methods with some existing methods is also given to justify their reliability, feasibility and flexibility.