Possibility neutrosophic soft sets and PNS-decision making method

In this paper, we introduce concept of possibility neutrosophic soft set and define some related concepts such as possibility neutrosophic soft subset, possibility neutrosophic soft null set, and possibility neutrosophic soft universal set. Then, based on definitions of n-norm and n-conorm, we define set theoretical operations of possibility neutrosophic soft sets such as union, intersection and complement, and investigate some properties of these operations. We also introduce AND-product and OR-product operations between two possibility neutrosophic soft sets. We propose a decision making method called possibility neutrosophic soft decision making method (PNS-decision making method) which can be applied to the decision making problems involving uncertainty based on AND-product operation. We finally give a numerical example to display application of the method that can be successfully applied to the problems.     

Soft ideals and soft filters of soft ordered semigroups

It is shown that soft ordered semigroups over an ordered semigroup are actually soft ordered subsemigroups of (S,A). Soft idealistic semigroups over an ordered semigroup are soft ideals of soft ordered semigroup (S,A). The concept of soft filters of a soft ordered semigroup is introduced. It is shown that restricted complement of a soft filter is a soft prime ideal of (S,A).     

Transitive closures and orderings on soft sets

In this paper an attempt is made to extend some standard results in set theory on the basis of soft set relations. Antisymmetric relation and transitive closure of a soft set relation are introduced and an analogue of Warshall’s algorithm is proposed for calculating the transitive closure of a soft set relation. Ordering on a soft set is defined and some set theoretical results based on this are proved.     

Some algebraic applications of soft sets

The paper presents an algebraic construction of the basic notions of soft set theory. Here we introduce the definition of bijective soft groups, dependent and independent soft groups. Some of their properties are investigated. Also we investigate the relationships between bijective soft groups and classical groups. In addition, we express some applications of bijective soft groups on coding theory.     

Soft sets and soft rough sets

In this study, we establish an interesting connection between two mathematical approaches to vagueness: rough sets and soft sets. Soft set theory is utilized, for the first time, to generalize Pawlak’s rough set model. Based on the novel granulation structures called soft approximation spaces, soft rough approximations and soft rough sets are introduced. Basic properties of soft rough approximations are presented and supported by some illustrative examples. We also define new types of soft sets such as full soft sets, intersection complete soft sets and partition soft sets. The notion of soft rough equal relations is proposed and related properties are examined. We also show that Pawlak’s rough set model can be viewed as a special case of the soft rough sets, and these two notions will coincide provided that the underlying soft set in the soft approximation space is a partition soft set. Moreover, an example containing a comparative analysis between rough sets and soft rough sets is given.     

The normal parameter reduction of soft sets and its algorithm

This paper is concerned with the reduction of soft sets and fuzzy soft sets. Firstly, the problems of suboptimal choice and added parameter set of soft sets are analyzed. Then, we introduce the definition of normal parameter reduction in soft sets to overcome these problems. In addition, a heuristic algorithm of normal parameter reduction is presented. Two new definitions, parameter important degree and decision partition, are proposed for analyzing the algorithm of normal parameter reduction. Furthermore, the normal parameter reduction is also investigated in fuzzy soft sets.     

Relations between complex vague soft sets

Complex vague soft sets are essentially vague soft sets characterized by an additional parameter called the phase term which is defined over the set of complex numbers. In this study, we introduce and discuss the relations between complex vague soft sets. We present the definitions of the Cartesian product of complex vague soft sets and subsequently that of complex vague soft relations. The definition of the composition of complex vague soft sets is also provided. The notions of symmetric, transitive, reflexive and equivalence complex vague soft relations are then proposed and the algebraic properties of these concepts are verified. The relation between complex vague soft sets is then discussed in the context of a real-life problem: the relation between the financial indicators of the Chinese economy which are characterized by their degrees of influence on the financial indicators of the Malaysian economy, and the time required for the former to affect the latter. Interpretations of the results obtained from this example are then proposed by relating them to recent significant real-life events in the Chinese and Malaysian economies.     

Computing generating sets of lattice ideals and Markov bases of lattices

In this article, we present an algorithm for computing generating sets of lattice ideals or equivalently for computing Markov bases of lattices. Generating sets of lattice ideals and Markov bases of lattices are essentially equivalent concepts. In contrast to other existing methods, the algorithm in this article computes with projections of lattices. This algorithm clearly outperforms other algorithms in our computational experience. Two areas of application for generating sets of lattice ideals and Markov bases lattices are algebraic statistics and integer programming.     

多值直觉模糊软集

将多值直觉模糊集和软集相结合,提出了多值直觉模糊软集的概念。给出了多值直觉模糊软集的补、交、并“、且”“、或”运算的概念,并基于这些概念研究了若干性质。     

双极值模糊软集

提出了双极值模糊软集的概念,给出双极值模糊软集的补、并、交、"且"及"或"运算,并讨论了它们的性质。     

犹豫模糊软集

犹豫模糊集是对模糊集的一种推广,它是一类关于域中每个元素所含隶属度的集合,常应用于群决策中,但由于其本身在参数工具上的缺乏使得难于处理不确定数据。为了提高决策的精确性,将软集与犹豫模糊集结合起来,提出犹豫模糊软集的概念,并给出犹豫模糊软集的基本运算法则和性质。     

Soft rough fuzzy sets and soft fuzzy rough sets

Fuzzy set theory, soft set theory and rough set theory are mathematical tools for dealing with uncertainties and are closely related. Feng et al. introduced the notions of rough soft set, soft rough set and soft rough fuzzy set by combining fuzzy set, rough set and soft set all together. This paper is devoted to the further discussion of the combinations of fuzzy set, rough set and soft set. A new soft rough set model is proposed and its properties are derived. Furthermore, fuzzy soft set is employed to granulate the universe of discourse and a more general model called soft fuzzy rough set is established. The lower and upper approximation operators are presented and their related properties are surveyed.     

Intuitionistic fuzzy soft modules

Molodtsov (1999) initiated the concept of soft sets in [1]. Maji et al. (2003) defined some operations on soft sets in [22]. Akta? and Ça?man (2007) generalized soft sets by defining the concept of soft groups in [16]. After them, Sun et al. (2008) gave soft modules in [21]. In this paper, the concept of an intuitionistic fuzzy soft module is introduced and some operations on intuitionistic fuzzy soft sets are given. Finally, some of its basic properties are studied.     

A novel soft rough set: Soft rough hemirings and corresponding multicriteria group decision making

In this paper, we investigate the relationships among rough sets, soft sets and hemirings. The concept of soft rough hemirings is introduced, which is an extended notion of a rough hemiring. It is pointed out that in this paper, we first apply soft rough sets to algebraic structure-hemirings. Further, we first put forward the concepts of C-soft sets and CC-soft sets, which provide a new research idea for soft rough algebraic research. Moreover, we study roughness in hemirings with respect to MSR-approximation spaces. Some new soft rough operations over hemirings are explored. In particular, lower and upper MSR-hemirings (k-ideal and h-ideal) are investigated. Finally, we put forth an approach for multicriteria group decision making problem based on modified soft rough sets and offer an actual example.     

Soft sets and soft groups

Molodtsov introduced the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. In this paper we introduce the basic properties of soft sets, and compare soft sets to the related concepts of fuzzy sets and rough sets. We then give a definition of soft groups, and derive their basic properties using Molodtsov’s definition of the soft sets.     

On a novel uncertain soft set model: Z-soft fuzzy rough set model and corresponding decision making methods

In this paper, a kind of novel soft set model called a Z-soft fuzzy rough set is presented by means of three uncertain models: soft sets, rough sets and fuzzy sets, which is an important generalization of Z-soft rough fuzzy sets. As a novel Z-soft fuzzy rough set, its applications in the corresponding decision making problems are established. It is noteworthy that the underlying concepts keep the features of classical Pawlak rough sets. Moreover, this novel approach will involve fewer calculations when one applies this theory to algebraic structures. In particular, an approach for the method of decision making problem with respect to Z-soft fuzzy rough sets is proposed and the validity of the decision making methods is testified by a given example. At the same time, an overview of techniques based on some types of soft set models is investigated. Finally, the numerical experimentation algorithm is developed, in which the comparisons among three types of hybrid soft set models are analyzed.     

Vague软集的一些代数性质

基于Vague集和软集现有理论以及Vague集思想与软集思想之间的联系,初步提出了Vague软集的狭义交、狭义并、相对补运算的概念,并在此基础上分别研究了这些运算的若干代数性质。     

Enhancing PROMETHEE method with intuitionistic fuzzy soft sets

The notion of intuitionistic fuzzy soft sets (IFSSs) provides an effective tool for solving multiple attribute decision making with intuitionistic fuzzy information. The most crucial issue in decision making based on IFSSs is how to derive the ranking of alternatives from the information quantified in terms of intuitionistic fuzzy values. In this study, we propose a new extension of the preference ranking organization method for enrichment evaluation (PROMETHEE), by taking advantage of IFSSs. In addition to presenting a myriad of new notions, such as intuitionistic fuzzy membership (or nonmembership) deviation matrices, intuitionistic fuzzy membership (or nonmembership) preference matrices, and aggregated intuitionistic fuzzy preference matrices, we put more emphasis on the construction of three distinct preference structures and related utility functions on the corresponding weakly ordered sets by considering the positive, negative, and net flows of the alternatives based on the aggregated intuitionistic fuzzy preference matrix. We present a new algorithm for solving multiple attribute decision-making problems with the extended PROMETHEE method based on IFSSs. Moreover, a benchmark problem concerning risk investment is investigated to give a comparative analysis and show the feasibility of our approach.     

软集合和软MV-代数

通过将Molodtsov在1999年提出的软集合理论应用于MV-代数理论,提出了软MV-代数和软MV-子代数的概念,并且得到了相关的一些性质。     

Applications of soft sets in ideal theory of BCK/BCI-algebras

The notion of soft ideals and idealistic soft BCK/BCI-algebras is introduced, and several examples are given. Relations between soft BCK/BCI-algebras and idealistic soft BCK/BCI-algebras are provided. The intersection, union, “AND” operation, and “OR” operation of soft ideals and idealistic soft BCK/BCI-algebras are established.