Studies in Fuzziness and Soft Computing

By: Atanu Sengupta, Tapan Kumar Pal

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Hardcover | 13 March 2009

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180 Pages

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23.5 x 15.88 x 0.64

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In conventional mathematical programming, coefficients of problems are usually determined by the experts as crisp values in terms of classical mathematical reasoning. But in reality, in an imprecise and uncertain environment, it will be utmost unrealistic to assume that the knowledge and representation of an expert can come in a precise way. The wider objective of the book is to study different real decision situations where problems are defined in inexact environment. Inexactness are mainly generated in two ways - (1) due to imprecise perception and knowledge of the human expert followed by vague representation of knowledge as a DM; (2) due to huge-ness and complexity of relations and data structure in the definition of the problem situation. We use interval numbers to specify inexact or imprecise or uncertain data. Consequently, the study of a decision problem requires answering the following initial questions: How should we compare and define preference ordering between two intervals?, interpret and deal inequality relations involving interval coefficients?, interpret and make way towards the goal of the decision problem?

The present research work consists of two closely related fields: approaches towards defining a generalized preference ordering scheme for interval attributes and approaches to deal with some issues having application potential in many areas of decision making.

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"This research monograph is concerned with modeling fuzzy preference ordering in the frame- work of decision-making. ... The fuzzy preference - based TOPSIS is discussed as well. Overall, the book is a concise and useful treatment of preference handing in uncertain decision-making problems. The wealth of the illustrative numeric material is a useful feature of the book." (Witold Pedryez, Zentralblatt MATH, Vol. 1169, 2009)

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis Computing & I.T.Computer Science Artificial Intelligence Applied Mathematics Graphical & Digital Media Applications Computer-Aided Design CAD Engineering & Technology Technology in General

Maths for Engineers Engineering in General

Introduction	p. 1
Conventional distinction between uncertainty and imprecision	p. 3
Imprecise data representation: Preliminaries on Fuzzy set and genesis of Interval Numbers as imprecise data	p. 4
Interval Numbers: A better tool to represent imprecision and uncertainty	p. 10
Interval Arithmetic: Notation and relevant preliminaries	p. 12
The motivation and organization of the book	p. 16
References	p. 19
On Comparing Interval Numbers: A Study on Existing Ideas	p. 25
Introduction	p. 25
Criteria for comparing interval numbers	p. 26
Interval comparing schemes	p. 28
Set theoretic approaches	p. 28
Probabilistic approaches	p. 30
References	p. 35
Acceptability Index and Interval Linear Programming	p. 39
Introduction	p. 39
The Acceptability Index	p. 41
Illustrative example	p. 46
A satisfactory crisp equivalent system of Ax≥B	p. 48
Tong's approach	p. 48
Discussion	p. 49
A satisfactory crisp equivalent system of Ax≥B based on A- index	p. 50
An Interval Linear Programming Problem and its Solution	p. 52
Solution to the problem stated in Section 3.1	p. 55
Conclusion	p. 56
References	p. 57
Fuzzy Preference ordering of Intervals	p. 59
Introduction	p. 59
The strength and weakness of A-index	p. 60
Preference ordering for a pessimistic DM for a maximization problem	p. 61
Choice of the DMs with different degrees of pessimism	p. 64
Illustrative example	p. 69
Preference pattern for a minimization problem	p. 72
Comparative advantage of the Fuzzy Preference Ordering over the other interval ordering schemes	p. 75
A note on some recent ranking schemes	p. 82
References	p. 89
Solving the Shortest Path Problem with Interval Arcs	p. 91
Introduction	p. 91
Choosing a preferred minimum from a set of intervals	p. 92
Numerical illustration of the procedure	p. 95
Shortest Path Problem	p. 99
Large-Scale numerical example	p. 103
Conclusion	p. 105
References	p. 109
Travelling Salesman problem with Interval Cost Constraints	p. 111
Introduction	p. 111
The problem	p. 112
Algorithm for Interval-valued Traveling Salesman Problem	p. 112
Solution to the numerical example 6.1.1	p. 114
Conclusion	p. 118
References	p. 119
Interval Transportation Problem with Multiple Penalty Factors	p. 121
Introduction	p. 121
Problem formulation	p. 122
The scope of an Interval-valued Objective Function in a real decision set up	p. 123
A numerical example	p. 127
A discussion on Chanas & Kuchta (1996b)'s approach and a comparative analysis with our approach	p. 128
A numerical example of ITPMPF	p. 132
Conclusion	p. 135
References	p. 135
Fuzzy Preference based TOPSIS for Interval multi-criteria Decision Making	p. 139
Introduction	p. 139
Relevant preliminaries	p. 141
The original TOPSIS	p. 141
A note on Give (2002)'s Bag based Interval-TOPSIS	p. 142
Fuzzy Preference Ordering of Interval Attributes in I-TOPSIS	p. 146
A numerical example	p. 151
Conclusion	p. 153
References	p. 154
Concluding Remarks and the Future Scope	p. 155
Introduction	p. 155
Chapter Summary and Conclusion	p. 156
Future Research Agenda	p. 158
Index	p. 161
List of Figures	p. 163
List of Tables	p. 165
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Studies in Fuzziness and Soft Computing

At a Glance

Hardcover

Industry Reviews

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