Abdul Razak Salleh

Possibility Fuzzy Soft Set

We introduce the concept of possibility fuzzy soft set and its operation and study some of its properties. We give applications of this theory in solving a decision-making problem. We also introduce a similarity measure of two possibility fuzzy soft sets and discuss their application in a medical diagnosis problem.

Soft Expert Sets

In 1999, Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Many researchers have studied this theory, and they created some models to solve problems in decision making and medical diagnosis, but most of these models deal only with one expert. This causes a problem with the user, especially with those who use questionnaires in their work and studies. In our model, the user can know the opinion of all experts in one model. So, in this paper, we introduce the concept of a soft expert set, which will more effective and useful. We also define its basic operations, namely, complement, union intersection AND, and OR. Finally, we show an application of this concept in decision-making problem.

Possibility Intuitionistic Fuzzy Soft Set

Possibility intuitionistic fuzzy soft set and its operations are introduced, and a few of their properties are studied. An application of possibility intuitionistic fuzzy soft sets in decision making is investigated. A similarity measure of two possibility intuitionistic fuzzy soft sets has been discussed. An application of this similarity measure in medical diagnosis has been shown.

Fuzzy Soft Multiset Theory

In 1999 Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Alkhazaleh et al. in 2011 introduced the definition of a soft multiset as a generalization of Molodtsovs soft set. In this paper we give the definition of fuzzy soft multiset as a combination of soft multiset and fuzzy set and study its properties and operations. We give examples for these concepts. Basic properties of the operations are also given. An application of this theory in decision-making problems is shown.

Relations between complex vague soft sets

Display Omitted Complex vague soft sets (CVSSs for short) are a hybrid model of complex fuzzy sets and vague soft sets and the study of CVSSs was initiated by Selvachandran et al. (2016).In this paper, the Cartesian product and the composition of CVSSs are defined and these are used to introduce the relations between CVSSs.This relations are then classified into symmetric, transitive, reflexive and equivalence relations and the algebraic properties and the relationships between these relations are then studied.The relations between CVSSs introduced here have the ability to: (i) Describe the degree of interaction and the phase between the elements in two sets by means of an interval-based membership structure. (ii) Capture the information pertaining to the time frame of the intersection between the elements in two sets, which cannot be accomplished by using other relations such as relations between fuzzy sets, soft sets or complex fuzzy sets.An application of the relations between CVSSs in the area of economics is then presented through the discussion of a real-life problem involving the influence of the financial indicators of the Chinese economy on the financial indicators of the Malaysian economy. 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.

FREE CROSSED RESOLUTIONS OF GROUPS AND PRESENTATIONS OF MODULES OF IDENTITIES AMONG RELATIONS

We give formulae for a module presentation of the module of identities among relations for a presentation of a group, in terms of information on 0- and 1-combings of the Cayley graph. This is seen as a special case of extending a partial free crossed resolution of a group given a partial contracting homotopy of its universal cover.

Complex vague soft sets and its distance measures

In this paper, we propose the concept of complex vague soft sets which are vague soft sets defined in a complex setting. Based on this new concept we define some concepts related to this notion as well as some basic operations namely the complement, union, intersection, AND and OR. The basic properties and relevant laws pertaining to this concept such as the De Morgan’s laws are also verified. We introduce the axiomatic definition of the distance function between two complex vague soft sets and subsequently define several distance measures between complex vague soft sets. Finally some of the algebraic properties of these distance measures are verified.

Linguistic variable, hedges and several distances on complex fuzzy sets

Complex fuzzy set CFS is a fuzzy set characterized by complex-valued membership function consisting of amplitude and phase terms, whose range of membership function extended to the unit circle in the complex plane. We often use words to describe variables in our daily life. Hence, the definition of linguistic variables and linguistic hedges on CFS are introduced. The most commonly used hedges are obtained. This paper also illustrates priority needs and advantages to introduce several distance measures in the case of complex fuzzy sets. Solutions to some problems in different fields are suggested through complex fuzzy distance measure and several distance measures between two complex fuzzy sets are introduced. The boundary for each distance of complex fuzzy sets is obtained.

Complex Atanassov's Intuitionistic Fuzzy Relation

This paper presents distance measure between two complex Atanassovs intuitionistic fuzzy sets (CAIFSs). This distance measure is used to illustrate an application of CAIFSs in solving one of the most core application areas of fuzzy set theory, which is multiattributes decision-making (MADM) problems, in complex Atanassovs intuitionistic fuzzy realm. A new structure of relation between two CAIFSs, called complex Atanassovs intuitionistic fuzzy relation (CAIFR), is obtained. This relation is formally generalised from a conventional Atanassovs intuitionistic fuzzy relation, based on complex Atanassovs intuitionistic fuzzy sets, in which the ranges of values of CAIFR are extended to the unit circle in complex plane for both membership and nonmembership functions instead of [0, 1] as in the conventional Atanassovs intuitionistic fuzzy functions. Definition and some mathematical concepts of CAIFS, which serve as a foundation for the creation of complex Atanassovs intuitionistic fuzzy relation, are recalled. We also introduce the Cartesian product of CAIFSs and derive two properties of the product space. The concept of projection and cylindric extension of CAIFRs are also introduced. An example of CAIFR in real-life situation is illustrated in this paper. Finally, we introduce the concept of composition of CAIFRs.

Fuzzy Parameterized Soft Expert Set

In 2011 Alkhazaleh and Salleh introduced the concept of soft expert set and gave an application in a decision-making problem. In this paper we introduce the concept of fuzzy parameterized soft expert set by giving an important degree for each element in the set of parameters. We also study its properties and define its basic operations, namely, complement, union, intersection, AND, and OR. Finally, we give an application in decision making.