In the present paper we introduce soft topological spaces which are defined over an initial universe with a fixed set of parameters. The notions of soft open sets, soft closed sets, soft closure, soft interior points, soft neighborhood of a point and soft separation axioms are introduced and their basic properties are investigated. It is shown that a soft topological space gives a parametrized family of topological spaces. Furthermore, with the help of an example it is established that the converse does not hold. The soft subspaces of a soft topological space are defined and inherent concepts as well as the characterization of soft open and soft closed sets in soft subspaces are investigated. Finally, soft T"i-spaces and notions of soft normal and soft regular spaces are discussed in detail. A sufficient condition for a soft topological space to be a soft T"1-space is also presented.

On soft topological spaces

Soft set theory and uni–int decision making

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Dubois and Prade investigated the problem of combining fuzzy sets with rough sets. Soft set theory was proposed by Molodtsov as a general framework for reasoning about vague concepts. The present paper is devoted to a possible fusion of these distinct but closely related soft computing approaches. Based on a Pawlak approximation space, the approximation of a soft set is proposed to obtain a hybrid model called rough soft sets. Alternatively, a soft set instead of an equivalence relation can be used to granulate the universe. This leads to a deviation of Pawlak approximation space called a soft approximation space, in which soft rough approximations and soft rough sets can be introduced accordingly. Furthermore, we also consider approximation of a fuzzy set in a soft approximation space, and initiate a concept called soft---rough fuzzy sets, which extends Dubois and Prade's rough fuzzy sets. Further research will be needed to establish whether the notions put forth in this paper may lead to a fruitful theory.

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

In this paper, we introduce the concept of picture fuzzy sets (PFS), which are direct  extensions of the fuzzy sets and the intuitonistic fuzzy sets. Then some operations on PFS with some properties are considered. The following sections are devoted to the Zadeh Extension Principle, picture fuzzy relations and picture fuzzy soft sets. Here,  the basic preliminaries of PFS theory are presented.

https://vjs.ac.vn/index.php/jcc/article/download/5032/5249

Picture fuzzy sets

In this work, we define soft matrices and their operations which are more functional to make theoretical studies in the soft set theory We then define products of soft matrices and their properties We finally construct a soft max-min decision making method which can be successfully applied to the problems that contain uncertainties

Soft matrix theory and its decision making

Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. In this paper, we first point out that several assertions (Proposition 2.3 (iv)-(vi), Proposition 2.4 and Proposition 2.6 (iii), (iv)) in a previous paper by Maji et al. [P.K. Maji, R. Biswas, A.R. Roy, Soft set theory, Comput. Math. Appl. 45 (2003) 555-562] are not true in general, by counterexamples. Furthermore, based on the analysis of several operations on soft sets introduced in the same paper, we give some new notions such as the restricted intersection, the restricted union, the restricted difference and the extended intersection of two soft sets. Moreover, we improve the notion of complement of a soft set, and prove that certain De Morgan's laws hold in soft set theory with respect to these new definitions.

On some new operations in soft set theory

Soft sets and soft rough sets

An adjustable approach to fuzzy soft set based decision making

The existing definition of generalized intuitionistic fuzzy soft sets (GIFSSs) is clarified and reformulated as a combination of an IFSS over the universe of discourse and an intuitionistic fuzzy set in the parameter set. With this new perspective, two different types of generalized intuitionistic fuzzy soft subsets and various new operations are developed for GIFSSs. The upper and lower substitutions for both intuitionistic fuzzy sets and GIFSSs are defined as well. Some existing notions and results are improved by virtue of these new concepts. Using the expectation score function, two binary relations are proposed for comparing intuitionistic fuzzy values. An algorithm is designed for coping with multiattribute decision making (MADM) problems with a combined use of the GIFSS, the extended intersection operation, the intuitionistic fuzzy weighted averaging operator, and other related notions. A case study concerning a faculty appointment problem is conducted to illustrate the proposed algorithm. Moreover, a comparative analysis between our approach and other related works is given to demonstrate the effectiveness and advantages of the newly proposed method.

Another View on Generalized Intuitionistic Fuzzy Soft Sets and Related Multiattribute Decision Making Methods

Intuitionistic fuzzy multiple attribute decision making deals with the issue of ranking alternatives based on the decision information quantified in terms of intuitionistic fuzzy values. Lexicographic orders can serve as efficient and indispensable tools for comparing intuitionistic fuzzy values. This paper introduces a number of lexicographic orders by means of several measures such as the membership, non-membership, score, accuracy and expectation score functions. Some equivalent characterizations and illustrative examples are provided, from which the relationships among these lexicographic orders are ascertained. We also propose three different compatible properties of preorders with respect to the algebraic sum and scalar product operations of intuitionistic fuzzy values, and apply them to the investigation of compatible properties of various lexicographic orders. In addition, a benchmark problem regarding risk investment is further explored to give a comparative analysis of different lexicographic orders and highlight the practical value of the obtained results for solving real-world decision-making problems.

https://www.mdpi.com/2227-7390/7/2/166/pdf

Xiaoyan Liu

Papers

On some new operations in soft set theory

Soft sets and soft rough sets

An adjustable approach to fuzzy soft set based decision making

Another View on Generalized Intuitionistic Fuzzy Soft Sets and Related Multiattribute Decision Making Methods

Lexicographic Orders of Intuitionistic Fuzzy Values and Their Relationships