Restricted Union Research Articles

Possibility belief interval-valued soft set and its application in decision making

The aim of this article is to present the concept of restricted union and extended intersection of belief interval-valued soft sets, along with its properties. In addition, we propose the concept of possibility belief interval-valued soft set theory and investigate their properties. For suitability of possible applications, there are seven kinds of operations (e.g., union, intersection, restricted union, extended intersection, complement, soft max-AND, and soft min-OR) on the possibility belief interval-valued soft sets are defined and their basic theoretical are given. Then, we construct two algorithms by using soft max-AND and soft min-OR operations of possibility interval-valued soft sets for fuzzy decision-making problem. Lastly, we introduce an algorithm using a possibility interval-valued soft set to solve the decision-making problems and clarify its applicability by a numerical example.

Bipolar Hypersoft Sets

Hypersoft set theory is an extension of soft set theory and is a new mathematical tool for dealing with fuzzy problems; however, it still suffers from the parametric tools’ inadequacies. In order to boost decision-making accuracy even more, a new mixed mathematical model called the bipolar hypersoft set is created by merging hypersoft sets and bipolarity. It is characterized by two hypersoft sets, one of which provides positive information and the other provides negative information. Moreover, some fundamental properties relative to it such as subset, superset, equal set, complement, difference, relative (absolute) null set and relative (absolute) whole set are defined. Furthermore, some set-theoretic operations such as the extended intersection, the restricted union, intersection, union, AND-operation and OR-operation of two bipolar hypersoft sets with their properties are discussed and supported by examples. Finally, tabular representations for the purposes of storing bipolar hypersoft sets in computer memory are used.

RP-T-Fuzzy Soft Subrings and Ideals of Soft Rings

In this paper we introduce a concept which is called RP-T-fuzzysoft subring and examine some properties of the restricted intersection, therestricted union, the ∧-intersection and the product of their families. A con-dition to make the restricted union of RP-T-fuzzy soft subrings of a soft ringto be RP-T-fuzzy soft subring of this soft ring is determined. A correlationbetween the RP-T-fuzzy soft subring of a soft ring and α-level sets of this softring is demonstrated. The RP-T-fuzzy soft subrings under some binary opera-tions are investigated. Moreover, the image and pre-image of RP-T-fuzzy softsubrings under fuzzy soft homomorphisms is examined. Finally, we present theconcept of RP-T-fuzzy soft ideal and we investigate the analogue propertiesfor them.

A hybrid decision-making framework using rough mF bipolar soft environment

In this article, we present the notions of two novel hybrid multi-attribute decision-making models, namely m-polar fuzzy bipolar soft set (mFBSS, for short) model and rough mFBSS model. These mathematical models allow us to handle multipolar information under fuzziness with bipolar soft sets. We use these proposed models to handle complicated problems in which the degree of membership of an object in a given set uses m fuzzy values, to rank all the objects, and to find a suitable option. We illustrate our proposed concepts with examples. Furthermore, we investigate some useful properties, including complement, restricted union and intersection, extended union and intersection, AND and OR. We discuss two practical applications of mFBSSs and rough mFBSSs, that is, selecting a suitable employee for promotion and place for house construction, respectively. We develop efficient algorithms to solve multi-attribute decision-making problems. We also discuss a comparison analysis of the proposed approaches with some existing models.

Multi-Attribute Group Decision-Making Methods Based on Pythagorean Fuzzy N-Soft Sets

In this paper, by integrating Pythagorean fuzzy set with N-soft set, we propose a generalization of N-soft set called the theory of Pythagorean fuzzy N-soft set and explore some of related operations concerning this theory including the weak complement, extended union and intersection, restricted union and intersection. Then two algorithms are introduced to Pythagorean fuzzy N-soft sets for dealing with multi-attribute group decision-making problems. Finally, a practical example is provided to illustrate the validity and practicality of Pythagorean fuzzy N-soft sets in multi-attribute group decision-making problems. Compared with the existing models, we also elaborate the advantages of this model.

Generalized Picture Fuzzy Soft Sets and Their Application in Decision Support Systems

In this paper, a generalized picture fuzzy soft set is proposed, which is an extension of the picture fuzzy soft sets. We investigate the basic properties of picture fuzzy soft sets and define an F-subset, M-subset, extended union, extended intersection, restricted union, restricted intersection and also prove the De Morgan’s laws for picture fuzzy soft information. We investigate upper and lower substitution for both picture fuzzy sets and generalized picture fuzzy soft sets. Meanwhile, the related proofs are given in detail. Finally, we propose an algorithm to deal with generalized picture fuzzy soft information. To show the supremacy and effectiveness of the proposed technique, we illustrate a descriptive example using generalized picture fuzzy soft information. Results indicate that the proposed technique is more generalized and effective over all the existing structures of fuzzy soft sets.

Some results on lattice ordered double framed soft semirings

In this article the concept of lattice ordered double framed soft semirings are introduced and establish some results with examples by using basic operations like union, intersection, "AND" product, "OR" product, restricted union, restricted intersection. Furthermore, lattice ordered double framed soft ideals and lattice ordered idealistic double framed soft semirings are also given with some related examples.

On lattice ordered intuitionistic fuzzy soft sets

Lattices, soft sets, fuzzy sets and their generalizations have always been important for Mathematicians and the researchers working on uncertaities. In this paper our aim is to introduce the concept of lattice ordered intuitionistic fuzzy soft sets. After introducing extended union, extended intersection, AND-product, OR-product, basic union, basic intersection of intuitionistic fuzzy soft sets, in this paper the affects of lattice ordered intuitionistic fuzzy soft sets and anti-lattice ordered intuitionistic fuzzy soft sets on restricted union, restricted intersection, extended union, extended intersection,AND-product, OR-product, basic union, basic intersection of intuitionistic fuzzy sets are discussed. Further a decision making problem is solved by using these concepts.

AN APPLICATION OF SOFT SET IN DECISION MAKING

In this paper, the definition of soft set and a detailed theoretical study of basic operations of soft sets such as intersection, extended intersection, restricted intersection, union, restricted union, complement and relative complement, Null and universal soft set are given. With the aid of definition of AND operation of soft sets and tabular representation of soft set, we are able to show that soft set has vital and real life application in decision making. The main aim of this paper is to use the concept of AND operation to sort out two best candidates out of five applicants in an interview conducted by a certain bank. Also the identification of Idempotent Property of “AND” and “OR” operation of soft sets is given and proved.

ON SOME PROPERTIES OF SOFT α-IDEALS

The notion of soft α-ideals and α-idealistic soft BCI-algebras is introduced and their basic properties are discussed. Relations between soft ideals and soft α-ideals of soft BCI-algebras are provided. Also idealistic soft BCI-algebras and α-idealistic soft BCI-algebras are being related. The restricted intersection, union, restricted union, restricted difference and "AND" operation of soft α-ideals and α-idealistic soft BCI-algebras are established. The characterizations of (fuzzy) α-ideals in BCI-algebras are given by using the concept of soft sets. Relations between fuzzy α-ideals and α-idealistic soft BCI-algebras are discussed.

Interval-valued Hesitant Fuzzy Soft Sets and their Application in Decision Making

The soft set theory can be used as a newly mathematical tool to handle uncertainty. How- ever, the classical soft sets are not appropriate to deal wit h imprecise and fuzzy parameters. In this paper, we propose the interval-valued hesitant fuzzy soft sets which are a combination of the interval-valued hesitant fuzzy sets and soft sets. Then, th e complement, AND, OR, union, inter- section, restricted union, extended intersection, differ ence, average, and geometric operations are defined on the interval-valued hesitant fuzzy soft sets, and some basic properties are also discussed in detail. Finally, by means of TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) and the maximizing deviation method, an algorithm is presented, and a comprehensive sensitivity analysis is employed, and the effectiveness is proved by a numerical example.

Soft Set Theory Applied to General Algebras

Abstract The notions of a soft general algebra and a soft subalgebra are introduced and studied. The operations on them such as a restricted intersection, an extended intersection, a restricted union, a ∧-intersection, a ∨-union and a cartesian product are established.

Soft Vectorial MV–Algebras

In this paper the soft vectorial MV-algebra has been studied. In first, we defined the concept of soft vectorial MV-algebra and give some examples. Then we establish some basic properties and investigate the effect of the extended intersection, restricted intersection, ∧-intersection, extended union, restricted union, ∨-union and soft function on them. Keywords: Soft Set, Soft Vectorial MV-Algebra, Vectorial MV-Algebra

Characterization of Bipolar Fuzzy Soft Γ-Semigroups

This paper deals with the idea of bipolar fuzzy soft sets applied to the ideal theory of τ-semigroups. We have introduced the concept of bipolar fuzzy soft τ-subsemigroup and bipolar fuzzy soft τ-ideals in a τ-semigroup. It is proved that the extended union, extended intersection, restricted union and restricted intersection of two same kind bipolar fuzzy soft τ-ideals over a τ-semigroup produced a same kind's bipolar fuzzy soft τ-ideal. Also the AND and OR operations of two bipolar fuzzy soft A-ideals produced a same type's bipolar fuzzy soft τ-ideal. It is also proved that the collection of all bipolar fuzzy soft τ-ideals over a τ-semigroup forms a complete distributive lattice with these special unions and intersections.

Lattice Structure of Generalized Intuitionistic Fuzzy Soft Sets

Park et al. introduced the concept of generalized intuitionistic fuzzy soft sets, which can be seen as an effective mathematical tool to deal with uncertainties. In this paper, we introduce new operations such as restricted union and restricted intersection and study their basic properties, and deal with the algebraic structure of generalized intuitionistic fuzzy soft sets. The lattice structures of generalized intuitionistic fuzzy soft sets are constructed.

On Possibility Interval-Valued Fuzzy Soft Sets

Soft set theory, initiated by Molodtsov, can be used as a new mathematical tool for dealing with imprecise, vague, and uncertain problems. In this paper, the concepts of two types of possibil­ity interval-valued fuzzy soft sets are proposed. Their operations and basic properties are studied which are subset, equal, relative complement, union, intersection, restricted union, extended intersection, “AND”, “OR” and De Morgan Laws. Furthermore, an application of the new approach in decision making based on possibility interval-valued fuzzy soft set is illustrated.

The Impact of Act 10 on Public Sector Compensation in Wisconsin

After a protracted legal and political battle, on March 11, 2011 the Wisconsin state legislature passed Act 10, the Budget Repair Act, which increased public employee contributions toward pensions and health coverage and restricted union powers of collective bargaining and dues collection. This study analyzes public sector salaries and benefits in Wisconsin, with a particular focus on disentangling the risk-adjusted value of pension benefits offered in the public sector from accounting conventions that can understate the cost and value of defined benefit pension plans. We find that state and local government employees receive salaries roughly equal to those paid to private sector Wisconsin employees with similar education and experience or working in jobs with similar skill requirements. However, even following Act 10, pension benefits for Wisconsin public employees are roughly 4.5 times more valuable than private sector levels while health benefits are about twice as generous as those paid by larger private sector Wisconsin employers. This difference results in a combined salary-benefits compensation premium of around 22 percent for state workers over private sector workers, with varying but often larger pay advantages for local government employees.

On Some New Properties of Soft Set Operations

In this paper, we establish some new results related to distributive properties of AND ( )  and OR ( )  -operations with respect to operations of union, restricted union restricted intersection, extended intersection and restricted difference on soft sets, and provide some illustrative examples.

Semantic operations of multiple soft sets under conflict

Molodtsov initiated the concept of soft set theory, which can be used as a generic mathematical tool for dealing with uncertainty. Description Logics (DLs) are a family of knowledge representation languages which can be used to represent the terminological knowledge of an application domain in a structured and formally well-understood way. Nowadays, properties and semantics of ontology constructs mainly are determined by DLs. In this paper we investigate semantic operations of multiple standard soft sets by using domain ontologies (i.e., DL intensional knowledge bases). Concretely, we give some semantic operations such as complement, restricted difference, extended union, restricted intersection, restricted union, extended intersection, AND, and OR for (multiple) standard soft sets from a semantic point of view. Especially, we also present an approach to deal with conflict from a semantic point of view when we define these semantic operations. Moreover, the basic properties and implementation methods of these semantic operations under conflict are also presented and discussed.

On operations of soft sets

Soft set theory, proposed by Molodtsov, has been regarded as an effective mathematical tool to deal with uncertainties. In this paper, first we prove that certain De Morgan’s law hold in soft set theory with respect to different operations on soft sets. Then, we discuss the basic properties of operations on soft sets such as intersection, extended intersection, restricted union and restricted difference. Moreover, we illustrate their interconnections between each other. Also we define the notion of restricted symmetric difference of soft sets and investigate its properties. The main purpose of this paper is to extend the theoretical aspect of operations on soft sets.