Soft multi-rough set topology with applications to multi-criteria decision-making problems
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METHODOLOGIES AND APPLICATION
Soft multi-rough set topology with applications to multi-criteria decision-making problems Muhammad Riaz1
· Faruk Karaaslan2 · Iqra Nawaz1 · Mahwish Sohail1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract 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. Keywords Soft multi-set · Soft multi-rough set (SMRS) · SMR-approximation space · SMR-topology · MCDM
1 Introduction In the modern era, the technology we are using to study the collected data has developed immensely. Whereas the complexity of the collected data has also grown widely. This complexity may occur due to uncertainties in the given information or human’s incomplete knowledge. Mostly, we are Communicated by V. Loia.
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Muhammad Riaz [email protected] Faruk Karaaslan [email protected] Iqra Nawaz [email protected] Mahwish Sohail [email protected]
1
Department of Mathematics, University of the Punjab, Lahore, Pakistan
2
Department of Mathematics, Çankiri Karatekin University, Çankiri, Turkey
unable to find the accurate solutions in challenging mathematical models like decision- making in daily life problems. Many scientific fields such as biology, economics, engineering, environment science, ecology, medical science and social science contain various types of uncertainties. To evaluate the hidden facts from the uncertain data, researchers have proposed various types of mathematical theories. These theories include fuzzy set theory (Zadeh 1965), intuitionistic fuzzy set theory (Atanassov 1986), rough set theory (Pawlak 1982) and bipolar fuzzy
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