Fuzzy Preference Ordering of Interval Numbers in Decision Problems Book

Chapter 1 Introduction 1

1.1 Conventional distinction between uncertainty and imprecision 3

1.2 Imprecise data representation: Preliminaries on Fuzzy set and genesis of Interval Numbers as imprecise data 4

1.3 Interval Numbers: A better tool to represent imprecision and uncertainty 10

1.4 Interval Arithmetic: Notation and relevant preliminaries 12

1.5 The motivation and organization of the book 16

1.6 References 19

Chapter 2 On Comparing Interval Numbers: A Study on Existing Ideas 25

2.1 Introduction 25

2.2 Criteria for comparing interval numbers 26

2.3 Interval comparing schemes 28

2.3.1 Set theoretic approaches 28

2.3.2 Probabilistic approaches 30

2.4 References 35

Chapter 3 Acceptability Index and Interval Linear Programming 39

3.1 Introduction 39

3.2 The Acceptability Index 41

3.2.1 Illustrative example 46

3.3 A satisfactory crisp equivalent system of Ax≥B 48

3.3.1 Tong's approach 48

3.3.2 Discussion 49

3.3.3 A satisfactory crisp equivalent system of Ax≥B based on A- index 50

3.4 An Interval Linear Programming Problem and its Solution 52

3.4.1 Solution to the problem stated in Section 3.1 55

3.5 Conclusion 56

3.6 References 57

Chapter 4 Fuzzy Preference ordering of Intervals 59

4.1 Introduction 59

4.2 The strength and weakness of A-index 60

4.3 Preference ordering for a pessimistic DM for a maximization problem 61

4.4 Choice of the DMs with different degrees of pessimism 64

4.4.1 Illustrative example 69

4.5 Preference pattern for a minimization problem 72

4.6 Comparative advantage of the Fuzzy Preference Ordering over the other interval ordering schemes 75

4.6.1 A note on some recent rankingschemes 82

4.7 References 89

Chapter 5 Solving the Shortest Path Problem with Interval Arcs 91

5.1 Introduction 91

5.2 Choosing a preferred minimum from a set of intervals 92

5.2.1 Numerical illustration of the procedure 95

5.3 Shortest Path Problem 99

5.4 Large-Scale numerical example 103

5.5 Conclusion 105

5.6 References 109

Chapter 6 Travelling Salesman problem with Interval Cost Constraints 111

6.1 Introduction 111

6.1.1 The problem 112

6.2 Algorithm for Interval-valued Traveling Salesman Problem 112

6.3 Solution to the numerical example 6.1.1 114

6.4 Conclusion 118

6.5 References 119

Chapter 7 Interval Transportation Problem with Multiple Penalty Factors 121

7.1 Introduction 121

7.2 Problem formulation 122

7.3 The scope of an Interval-valued Objective Function in a real decision set up 123

7.3.1 A numerical example 127

7.3.2 A discussion on Chanas & Kuchta (1996b)'s approach and a comparative analysis with our approach 128

7.4 A numerical example of ITPMPF 132

7.5 Conclusion 135

7.6 References 135

Chapter 8 Fuzzy Preference based TOPSIS for Interval multi-criteria Decision Making 139

8.1 Introduction 139

8.2 Relevant preliminaries 141

8.2.1 The original TOPSIS 141

8.2.2 A note on Give (2002)'s Bag based Interval-TOPSIS 142

8.3 Fuzzy Preference Ordering of Interval Attributes in I-TOPSIS 146

8.4 A numerical example 151

8.5 Conclusion 153

8.6 References 154

Chapter 9 Concluding Remarks and the Future Scope 155

9.1 Introduction 155

9.2 Chapter Summary and Conclusion 156

9.3 Future Research Agenda 158

Index 161

List of Figures 163

List of Tables 165