Uncorrected Author Proof Journal of Intelligent & Fuzzy Systems xx (20xx) x–xx DOI:10.3233/JIFS-18017 IOS Press 1 Multi-criteria decision-making methods under soft rough fuzzy knowledge 1 2 Muhammad Akram ∗ and Fariha Zafar 3 Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan 4 Abstract. Due to the uncertainty of real world problems and the limitation of human’s knowledge to understand the complex problems, it is very difficult for one to apply only a single type of uncertainty method to cope with such problems. One can develop a more powerful new model to solve decision making problems by incorporating the advantages of many other different theories of uncertainty. Fuzzy sets, soft sets and rough sets are very useful mathematical models for dealing with uncertainty. Combinations of these models result into several useful hybrid models. In view of this, in this research paper, the concepts and methods of rough soft sets and fuzzy sets are used to construct a new soft rough fuzzy set model. We employ the concept of soft rough fuzzy sets to graphs and investigate some properties of this model. We apply this new model to describe and resolve some multi-criteria decision-making problems. 5 6 7 8 9 10 11 12 Keywords: Soft rough fuzzy relation, soft rough fuzzy digraph, decision-making Mathematics Subject Classification 2010: 03E72, 68R10, 68R05 13 14 1. Introduction 15 Vagueness arises in several complex issues of 16 engineering, science and many other fields. These 17 issues cannot be resolved using crisp mathematical 18 methods. Many well-known theories, including 19 probability theory, fuzzy set theory (FST), intu- 20 itionistic fuzzy set theory (IFST),vague set theory 21 (VST), interval mathematics theory (IMT) and 22 rough set theory (RST) have been established to 23 explain uncertainty. Molodtsov [22] pointed out the 24 limitations of these theories. To overcome these 25 limitations, he initiated the idea of soft sets, which 26 can be regarded as a useful mathematical model 27 for coping with vagueness. Soft set theory (SST) 28 seems to be free from the limitations affecting the 29 existing methods. SST has very useful applications in 30 many fields, including operational research, Perron 31 integration, probability theory, and measurement 32 ∗ Corresponding author. Muhammad Akram, Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan. E-mail: makrammath@yahoo.com theory. Due to practical applications of SST, many 33 researchers worked on it and applied it to different 34 decision-making problems. Maji et al. [19] discussed 35 various operations on soft sets and also gave the 36 idea of hybrid structures, including fuzzy sets and 37 soft sets. They introduced the concept of fuzzy soft 38 sets (FSSs), which can be considered as a fuzzy 39 generalization of soft sets. Majumdar and Samanta 40 [20] modified the definition of FSSs and introduced 41 the notion of generalized fuzzy soft set (GFSS). 42 Yang et al. [31, 32] introduced the notion of interval 43 valued fuzzy soft sets (IVFSSs) by combining 44 interval-valued fuzzy set (IVFS) and soft set models 45 (SSMs) and also discussed multi fuzzy soft sets 46 (MFSSs) with applications in DM. 47 48 RST was introduced by Pawlak [26]. The idea of 49 RST is a generalization of crisp set theory (CST) 50 to study the intelligence systems containing incom- 51 plete, inexact or uncertain information. It is an 52 effective drive for bestowal with vagueness and 53 incomplete information. RST is an important math- 54 ematical approach to imprecise knowledge. RST 55 1064-1246/18/$35.00 © 2018 – IOS Press and the authors. All rights reserved