UNIVERSITI TEKNOLOGI MARA

MULTIASPECT SOFT SETS AND ITS GENERALIZATIONS

NOR HASHIMAH BINTI SULAIMAN

Thesis submitted in fulfillment of the requirements for the degree of

Doctor of Philosophy

Faculty of Computer and Mathematical Sciences

February 2016

CONFIRMATION BY PANEL OF EXAMINERS

I certify that a panel of examiners has met on 4 December 2015 to conduct the final examination of Nor Hashimah Binti Sulaiman on her Doctor of Philosophy thesis entitled "Multiaspect Soft Sets and its Generalizations" in accordance with Universiti Teknologi MARA Act 1976 (Akta 173). The panel of Examiners recommends that the student be awarded the relevant degree. The panel of Examiners was as follows:

Zainab Abu Bakar, PhD Professor Faculty of Computer and Mathematical Sciences Universiti Teknologi MARA (Chairman)

Ajab Bai Akbarally, PhD Associate Professor Faculty of Computer and Mathematical Sciences Universiti Teknologi MARA (Internal Examiner)

Dato' Abdul Razak Salleh, PhD Professor Faculty of Mathematical Sciences Universiti Kebangsaan Malaysia (External Examiner)

Nairn Cagman, PhD Professor University of Gaziomanpasa, Turkey (External Examiner)

SITI HALIJJAH SHARIFF, PhD Associate Professor Dean Institute of Graduate Studies Universiti Teknologi MARA Date: 3rd February 2016

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AUTHOR'S DECLARATION

I declare that the work in this thesis was carried out in accordance with the regulations

of Universiti Teknologi MARA. It is original and the results of my own work, unless

otherwise indicated or acknowledged as referenced work. This thesis has not been

submitted to any other academic institution or non-academic institution for any degree

or qualification.

I, hereby, acknowledge that I have been supplied with the Academic Rules and

Regulations for Post Graduates, Universiti Teknologi MARA, regulating the conduct

of my study and research.

Name of Students

Student I.D Number

Programme

Faculty

Thesis Title

Signature of Student

Date

Nor Hashimah binti Sulaiman

2010127701

Doctor of Philosophy (Mathematics) - CS990

Computer and Mathematical Sciences

Mulnaspect Soft Sets and its Generalizations

February 2016

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ABSTRACT

The theory of soft sets introduced in 1999 by Molodtsov is an alternative mathematical tool for dealing with uncertainties. It basically deals with information representations of objects characterized by parameters which are defined over a single common universal set. Combinations of the theory with fuzzy sets and interval-valued fuzzy sets have resulted in the so-called fuzzy soft sets and interval-valued fuzzy soft sets. Various theoretical studies on these theories and the variants have been made, and applications of the theories in various areas particularly in the area of decision making are continuously explored. Soft sets, fuzzy soft sets and interval-valued fuzzy soft sets have greater potential in information representation should the universe sets of elements not be restricted to only a common universal set. Real life situations may involve descriptions of objects, situations or entities based on certain characteristics or attributes which may be associated with different sets of elements of different types of universal sets. In this thesis, we introduce the concepts of multiaspect soft set (MASS), multiaspect fuzzy soft set (MAFSS) and multiaspect interval-valued fuzzy soft set (MAIVFSS) which are generalizations of soft sets, fuzzy soft sets and interval-valued fuzzy soft sets, respectively. These concepts provide platforms for information representations that allow elements from different universal sets be taken into consideration in the description of a particular object, item or entity. MASS is defined for crisp data representation while MAFSS and MAIVFSS are respectively defined for fuzzy data representation with single and interval-valued membership degrees. For each concept, the set operations are established and the algebraic properties are studied. The concepts of mapping for multiaspect soft classes, multiaspect fuzzy soft classes and multiaspect interval-valued fuzzy soft classes are presented. In addition, we put forward the axiomatic definitions of distance, distance-based similarity measures and entropy for MAFSS and MAIVFSS. We introduce weighted and non-weighted distances and similarity measures based on the Hamming distance and the Euclidean distance. Relationships between the three measures are investigated. In the final part of the thesis, we highlight the applicability of some of the introduced concepts in solving group decision making problem under MAFSS and MAIVFSS environment.

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TABLE OF CONTENTS

Page

CONFIRMATION BY PANEL OF EXAMINERS ii

AUTHOR'S DECLARATION iii

ABSTRACT iv

ACKNOWLEDGEMENT v

TABLE OF CONTENTS vi

LIST OF TABLES ix

LIST OF ABBREVIATIONS x

CHAPTER ONE: INTRODUCTION 1

1.1 Research Background 1

1.2 Problem Statement 7

1.3 Research Objectives 9

1.4 Thesis Organizations 9

CHAPTER TWO: FUNDAMENTAL CONCEPTS 11

2.1 Introduction 11

2.2 Fuzzy Sets 11

2.2.1 Some Basic Definitions and Operations on Fuzzy Sets 12

2.2.2 Extension Principle 14

2.2.3 Aggregation Operators Of Fuzzy Sets 14

2.3 Interval-Valued Fuzzy Sets 16

2.4 Soft Sets 20

2.4.1 Basic Operations and Properties of Soft Sets 21

2.4.2 Mapping on Soft Classes 23

2.5 Fuzzy Soft Sets 24

2.6 Interval-Valued Fuzzy Soft Sets 27

2.7 Reduct Fuzzy Soft Set 27

2.8 Hurwicz Criterion 28

CHAPTER THREE: MULTIASPECT SOFT SETS 30

3.1 Introduction 30 vi

ABSTRACTTABLE OF CONTENTSCHAPTER ONE: INTRODUCTION1.1 RESEARCH BACKGROUND1.2 PROBLEM STATEMENT1.3 RESEARCH OBJECTIVES1.4 THESIS ORGANIZATIONS

CHAPTER TWO: FUNDAMENTAL CONCEPTS2.1 INTRODUCTION2.2 FUZZY SETS2.2.1 Some Basic Definitions and Operations on Fuzzy Sets2.2.2 Extension Principle2.2.3 Aggregation Operators of Fuzzy Sets

2.3 INTERVAL-VALUED FUZZY SETS2.4 SOFT SETS2.4.1 Basic Operations and Properties of Soft Sets2.4.2 Mapping on Soft Classes

2.5 FUZZY SOFT SETS2.6 INTERVAL-VALUED FUZZY SOFT SETS2.7 REDUCT FUZZY SOFT SET2.8 HURWICZ CRITERION

CHAPTER THREE: MULTIASPECT SOFT SETS3.1 INTRODUCTION3.2 MAIN DEFINITIONS3.3 OPERATIONS ON MULTIASPECT SOFT SETS3.4 MAPPING ON MULTIASPECT SOFT CLASSES3.5 SIMILARITY MEASURE FOR MULTIASPECT SOFT SETS3.6 SUMMARY

CHAPTER FOUR: MULTIASPECT FUZZY SOFT SETS AND MULTIASPECT INTERVAL-VALUED FUZZY SOFT SETS4.1 INTRODUCTION4.2 MAIN DEFINITIONS4.3 OPERATIONS ON MULTIASPECT INTERVAL-VALUED FUZZY SOFT SETS4.3.1 Union, Intersection and Complement

4.4 'AND' AND 'OR' OPERATIONS4.5 AGGREGATION OPERATIONS4.6 MAIVFSS WITH HURWICZ CRITERION4.7 MAPPING ON MULTIASPECT INTERVAL-VALUED FUZZY SOFT CLASSES4.8 SUMMARY

CHAPTER FIVE: DISTANCE, SIMILARITY MEASURES AND ENTROPY FOR MULTIASPECT INTERVAL-VALUED FUZZY SOFT SETS AND MULTIASPECT FUZZY SOFT SETS5.1 INTRODUCTION5.2 DISTANCE, SIMILARITY MEASURES AND ENTROPY FOR MAIVFSS AND MAFSS5.2.1 Distance and Similarity Measures for MAIVFSS5.2.2 Entropy for MAIVFSSs5.2.3 Similarity Measure Based on Entropy

5.3 SUMMARY

CHAPTER SIX: GROUP DECISION MAKING IN MAFSS AND MAIVFSS ENVIRONMENT WITH SOME ILLUSTRATIVE APPLICATIONS6.1 INTRODUCTION6.2 ENTROPY WEIGHT FOR MAIVFSS6.3 GROUP DECISION MAKING WITH MAIVFSS AND MAFSS6.4 SUMMARY

CHAPTER SEVEN: CONCLUSION AND FUTURE WORKS7.1 CONCLUSIONS7.2 FUTURE WORKS

REFERENCESAUTHOR'S PROFILE