Soft sets combined with fuzzy sets and rough sets: a tentative approach

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.

Soft rough fuzzy sets and soft fuzzy rough sets

Emerging trends in soft set theory and related topics.

In this study, we establish a connection between rough soft set (Shabir et al., Knowl Base Syst 40:72–80, 2013) and fuzzy set. Based on the novel granulation structure called modified soft rough approximation space, fuzzy rough soft set is introduced. The important basic properties of fuzzy rough soft set are studied and supported by illustrative examples. Moreover lattice theory is studied on fuzzy rough soft set. The definitions and propositions presented in this paper enrich the soft set theory, rough set theory and fuzzy set theory, and also extend their application scopes. The paper ends with conclusions having future investigations of the study.

Fuzzy set theory, rough set theory and soft set theory are all generic mathematical tools for dealing with uncertainties. There has been some progress concerning practical applications of these theories, especially, the use of these theories in decision making problems. In the present article, we review some decision making methods based on (fuzzy) soft sets, rough soft sets and soft rough sets. In particular, we provide several novel algorithms in decision making problems by combining these kinds of hybrid models. It may be served as a foundation for developing more complicated soft set models in decision making.

Rough set theory for the interval-valued fuzzy information systems

In this paper, we present a possible fusion of rough sets and multiple soft sets. According to the theory of soft rough sets we propose the concepts of the multi-granularity soft rough sets (MGSR-sets) and multi-granularity soft approximation space. Based on these, we consider the multi-granularity soft rough approximation operators and discuss some important properties of them by some illustrative examples. Finally, the jointly full soft set and multi-granularity soft rough relations are also introduced.

The classical soft rough set theory offers a formal theoretical framework for solving the uncertainty under a single granulation environment. However, it is noticeable that the soft rough set theory cannot be applied in the situation of multigranulation in the real world. To address this issue, in this paper we firstly present two types of multigranulation soft rough sets, which are the optimistic multigranulation soft rough set and the pessimistic multigranulation soft rough set. Then the basic and important properties of these multigranulation soft rough sets are studied. Finally, we investigate the relationships among single granulation soft rough set, namely classical soft rough set, optimistic multigranulation soft rough set and pessimistic multigranulation soft rough set. These results will enrich the soft set theory and multigranulation rough set theory, and also extend their application scopes.

Rough sets defined in terms of soft sets play a vital role in decision making problems. Covering-based soft rough sets and modified soft rough sets are some of the recently developing concepts. In this paper, for a given soft sets [Formula: see text] on a universe [Formula: see text] we define a novel rough set called as minimal soft rough sets using minimal soft description of the objects. The relation between modified soft rough set and minimal soft rough set is analyzed. The set of all minimal soft rough sets is proved to be a Poset with the inclusion relation having a GLB and LUB and hence is a lattice. An attempt is made in applying this concepts in medical diagnoses and also in analyzing the organizational culture system.

The concept of soft set is fundamentally important in almost every scientific field. Soft set theory is a new mathematical tool for dealing with uncertainties and is a set associated with parameters and has been applied in several directions. Since Molodtsov originated the idea of soft sets, some research on soft sets has been done in the literature. This theory represents a promising technique in imperfect data analysis which has found interesting extensions and various applications that handle imperfect knowledge, such as Bayesian inference, fuzzy set etc. In this paper, define the notion of soft sets, and the study that are interesting and valuable in the theory of soft sets, which emphasis on a series of applications especially in decision making problems. Also presents comprehensive study, development and survey of its existing literature. Keywords—BCI; BCK; FCM; Fuzzy Set; Fuzzy Soft Set; Soft Rough Set; Soft Semi Rings; Soft Set; Uncertainty.

Fuzzy sets, rough sets, and later on IF sets became useful mathematical tools for solving various decision making problems and data mining problems. Molodtsov introduced another concept soft set theory as a general frame work for reasoning about vague concepts. Since most of the data collected are either linguistic variable or consist of vague concepts so IF set and soft set help a lot in data mining problem. The aim of this paper is to introduce the concept of IF soft lower rough approximation and IF upper rough set approximation. Also, some properties of this set are studied, and also some problems of decision making are cited where this concept may help. Further research will be needed to apply this concept fully in the decision making and data mining problems.

Uncertainty about future developments constitutes the most important and most difficult challenge for mankind. Despite this fact, uncertainty is not a part of general science. General science assumes that the future development follows cause-effect relations which can be described by mathematical functions, where the argument represents the cause and the image represents the effect. Scientific theories have exactly this form and it is widely believed that these functions represent “truth”. Of course, this is nonsense as all the scientific theories are with certainty wrong and cannot describe the real evolution correctly. The inappropriate handling of uncertainty in science has produced a strange variety of “uncertainty theories” that causes confusion and helplessness. A friend of mine has expressed his confusion by the following words: ‘Crisp sets’, ‘fuzzy sets’, ‘rough sets’, ‘grey sets’, ‘fuzzy rough sets’, ‘rough fuzzy sets’, ‘fuzzy grey sets’, ‘grey fuzzy sets’, ‘rough grey sets’, ‘grey rough sets’, and now ‘affinity sets’. My goodness! Is there anybody around who can enlighten me, i.e., help me to see a clear pattern in this set of sets, allegedly providing powerful tools to model various kinds of uncertainty? This paper examines the role and the handling of uncertainty in quality control. How is uncertainty quantified in quality control for making decisions aiming at maintaining or improving quality of processes and products.KeywordsRandomnessIgnoranceKnowledgeFuzzynessQuantificationProbabilityCredibilityProbability spaceUncertainty spaceBernoulli space

Soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. The main purpose of soft rough set is reducing the soft boundary region by increasing the lower approximation and decreasing the upper approximation. This paper is devoted to the further generalization of the soft rough set theory by using the ideal notion to reduce the soft boundary region. A new soft rough set model, called soft ideal approximation space, is proposed and its properties are derived. The new soft ideal lower and upper approximation operators are presented and their related properties are surveyed.

A rough set approach to intuitionistic fuzzy soft set based decision making

Improving decision making approaches based on fuzzy soft sets and rough soft sets