Williams Logic and Integer Programming

1. Auflage 2009
ISBN: 978-0-387-92280-5
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 130, 200 Seiten

Reihe: International Series in Operations Research & Management Science

Paul Williams, a leading authority on modeling in integer programming, has written a concise, readable introduction to the science and art of using modeling in logic for integer programming. Written for graduate and postgraduate students, as well as academics and practitioners, the book is divided into four chapters that all avoid the typical format of definitions, theorems and proofs and instead introduce concepts and results within the text through examples. References are given at the end of each chapter to the more mathematical papers and texts on the subject, and exercises are included to reinforce and expand on the material in the chapter. Methods of solving with both logic and IP are given and their connections are described. Applications in diverse fields are discussed, and Williams shows how IP models can be expressed as satisfiability problems and solved as such.

H.P.(Paul) Williams is Professor of Operational Research at the London School of Economics. He has a degree in Mathematics from Cambridge University and a PhD in Mathematical Logic from Leicester University (having studied under the late Professor R.L.Goodstein). His research work has been primarily in Linear and Integer Programming. This proposed book combines his knowledge in all these areas.He worked for IBM on developing software for and helping clients model and solve problems in Linear and Integer Programming. This work was continued in a number of academic posts. He has held chairs at Edinburgh and Southampton Universities and published many papers in these areas (listed on his web site). He is particularly well known for his book Model Building in Mathematical Programming (Wiley) first published in 1978 and now in a fourth edition. It has been translated into a number of other languages. 

Autoren/Hrsg.

Williams, H. Paul

Inhaltsverzeichnis

1;Preface;7
2;Contents;9
3;1 An Introduction to Logic ;12
3.1; The Purpose of Logic: Philosophical: Computational;12
3.2; Logical Inference and Consistency;13
3.3; The Propositional Calculus;14
3.3.1; Connectives and Truth Tables;14
3.3.2; Equivalent Statements;16
3.3.3; Disjunctive and Conjunctive Normal Forms;17
3.3.4; Complete Sets of Connectives;21
3.3.5; The Calculus of Indications;22
3.3.6; Venn Diagrams;24
3.4; The Predicate Calculus;25
3.4.1; The Use of Quantifiers;26
3.4.2; Prenex Normal Form;27
3.5; Decidable Fragments of Mathematics;29
3.5.1; The Theory of Dense Linear Order;29
3.5.2; Arithmetic Without Multiplication;31
3.6; References and Further Work;33
3.7; Exercises;33
4;2 Integer Programming ;36
4.1; Linear Programming;36
4.1.1; The Dual of an LP Model;39
4.1.2; A Geometrical Representation of a Linear Programme;41
4.2; Integer Programming;46
4.2.1; The Branch-and-Bound Algorithm;49
4.2.2; The Convex Hull of an IP;54
4.3; The Use of 0--1 Variables;60
4.3.1; Expressing General Integer Variables as 0--1 Variables;60
4.3.2; Yes/No Decisions;61
4.3.3; The Facility Location Problem;61
4.3.4; Logical Decisions;62
4.3.5; Products of 0--1 Variables;64
4.3.6; Set-Covering, Packing and Partitioning Problems;64
4.3.7; Non-linear Problems;66
4.3.8; The Knapsack Problem;69
4.3.9; The Travelling Salesman Problem;69
4.3.10; Other Problems;71
4.4; Computational Complexity;71
4.4.1; Problem Classes and Instances;71
4.4.2; Computer Architectures and Data Structures;72
4.4.3; Polynomial and Exponential Algorithms;73
4.4.4; Non-deterministic Algorithms and Polynomial Reducibility;75
4.4.5; Feasibility Versus Optimisation Problems;76
4.4.6; Other Complexity Concepts;77
4.5; References and Further Work;77
4.6; Exercises;79
5;3 Modelling in Logic for Integer Programming;82
5.1; Logic Connectives and IP Constraints;82
5.2; Disjunctive Programming;86
5.2.1; A Geometrical Representation;86
5.2.2; Mixed IP Representability;89
5.3; Alternative Representations and Tightness of Constraints;95
5.3.1; Disjunctive Versus Conjunctive Normal Form;97
5.3.2; The Dual of a Disjunctive Programme;100
5.4; Convexification of an IP Model;102
5.4.1; Splitting Variables;103
5.5; Modelling Languages Based On Logic;107
5.5.1; Algebraic Languages;107
5.5.2; The `Greater Than or Equal' Predicate;109
5.6; References and Further Work;113
5.7; Exercises;113
6;4 The Satisfiability Problem and Its Extensions ;115
6.1; Resolution and Absorption;116
6.2; The Davis--Putnam--Loveland (DPL) Procedure;119
6.3; Representation as an Integer Programme;119
6.4; The Relationship Between Resolution and Cutting Planes;121
6.5; The Maximum Satisfiability Problem;123
6.6; Simplest Equivalent Logical Statement;126
6.7; Horn Clauses: Simple Satisfiability Problems;129
6.8; Constraint Logic Programming;133
6.8.1; Modelling in CLP;134
6.8.2; Solving CLP Models;136
6.8.3; Hybrid CLP and IP systems;137
6.9; Solving Integer Programmes as Satisfiability Problems;139
6.10; Applications;144
6.10.1; Electrical Circuit Design Using Switches;144
6.10.2; Logical Net Design Using Gates;145
6.10.3; The Logical Analysis of Data (LAD);147
6.10.4; Chemical-processing networks;149
6.10.5; Other Applications;150
6.11; References and Further Work;151
6.12; Exercises;152
7;References;155
8;Index;160