Mathematical Modeling and Optimization: An Essay for the Design of Computer-Based Modeling Tools

322 Pages·1999·12.233 MB·English

by  Tony Hürlimann (auth.)

Mathematical Modeling and Optimization
Applied Optimization
Volume 31
Series Editors:
Panos M. Pardalos
University of Florida. U.S.A.
Donald Hearn
University of Florida. U.S.A.
Mathelllatical Modeling and
Optilllization
An Essay for the Design of
Computer-Based Modeling Tools
by
Tony Hiirlimann
Institute for Informatics,
University of Fribourg,
Switzerland
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-1-4419-4814-4 ISBN 978-1-4757-5793-4 (eBook)
DOI 10.1007/978-1-4757-5793-4
Printed on acid-free paper
AII Rights Reserved
© 1999 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1999
Softcover reprint of the hardcover 1s t edition 1999
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
Denna bok ar tillagnad
mina d6ttrar Malika Nora och Selma Aina
Table of Contents
1. Introduction I
1.1. MODELS AND THEIR FUNCTIONS 2
1.2. THE ADVENT OF THE COMPUTER 4
1.3. NEW SCIENTIFIC BRANCHES EMERGE 5
104. MATHEMATICAL MODELING - THE CONSEQUENCES 8
1.5. COMPUTER-BASED MODELING MANAGEMENT 10
1.6. ABOUT THIS BOOK 1 2
2. What is Modeling? 1 7
2.1. MODEL: A DEFINITION 1 7
2.1. MATHEMATICAL MODELS 23
2.2. MODEL THEORY 26
2.3. MODELS AND INTERPRETATIONS 29
204. RELATED CONCEPTS 30
2.5. DECLARATIVE VERSUS PROCEDURAL KNOWLEDGE 33
3. The Modeling Life Cycle 37
3.1. STAGE 1: SPECIFICATION OF THE REAL PROBLEM 39
3.2. STAGE 2: FORMULATION OF THE MATHEMATICAL MODEL 39
3.3. STAGE 3: SOLUTION OF THE MODEL 43
304. STAGE 4: VALIDATION OF THE MODEL AND ITS SOLUTION 44
304.1. Logical Consistency 50
3.4.2. Data Type Consistency 51
304.3. Unit Type Consistency 51
30404. User Defined Data Checking 52
304.5. Simplicity Considerations 53
304.6. Solvability Checking 53
3 04.7. Numerical Stability and Sensitivity Analysis 53
304.8. Checking the Correspondence 55
3.5. STAGE 5: WRITING A REPORT 57
3.6. TWO CASE STUDIES 58
4. Model Paradigms 67
4.1. MODEL TYPES 67
4.1.1. Optimization Models 68
4.1.2. Symbolical - Numerical Models 68
4.1.3. Linear - Nonlinear Models 68
4.] A. Continuous - Discrete Models 69
4.1.5. Deterministic - Stochastic Models 69
4.1.6. Analytic - Simulation Models 70
4.2. MODELS AND THEIR PURPOSES 71
4.3. MODELS IN THEIR RESEARCH COMMUNITIES 72
viii
4.3.1. Differential Equation Models 72
4.3.2. Operations Research 73
4.3.3. Artificial Intelligence 75
4.4. MODELING UNCERTAINTY 80
4.4.1. Mathematical Models and Uncertainty 80
4.4.2. General Approaches in Modeling Uncertainty 82
4.4.3. Classical Approaches in OR 84
4.4.4. Approaches in Logical Models 86
4.4.5. Fuzzy Set Modeling 92
4.4.6. Outlook 96
5. Problems and Concepts 101
5.1. PRESENT SITUATION IN MMS 102
5.2. WHAT MMS IS NOT 102
5.3. MMS, WHAT FOR? 106
5.4. MODELS VERSUS PROGRAMS 108
6. An Overview of Approaches 115
6.1. SPREADSHEET 1 15
6.2. RELATIONAL DATABASE SYSTEMS 119
6.3. GRAPHICAL MODELING 126
6.4. CONSTRAINT LOGIC PROGRAMMING LANGUAGES (CLP) 1 31
6.5. ALGEBRAIC LANGUAGES 139
6.5.1. AIMMS 140
6.5.2. AMPL 146
6.5.3. Summary 148
6.6. GENERAL REMARKS ] 50
6.6.1. Structured Modeling 151
6.6.2. Embedded Language Technique 152
6.6.3. Multi-view Architecture 153
6.6.4. A Model Construction and Browsing Tool 154
6.6.5. Conclusion 154
7. A Modeling Framework 1 55
7.1. THE REQUIREMENTS CATALOGUE 155
7.1.1. Declarative and Procedural Knowledge 156
7.1.2. The Modeling Environment 157
7.1.3. Informal Knowledge 159
7.1.4. Summary 159
7.2. THE MODELING LANGUAGE 161
7.2.1. The Adopted Approach 162
7.2.2. The Overall Structure of the Modeling Language 167
7.2.3. Entities and Attributes 172
7.2.4. Index-sets 176
7.2.5. Expression 182
7.2.6. The Instruction Entities 184
ix
7.2.7. The Complete Syntax Specification 185
7.2.8. Semantic Interpretation 187
7.2.8. Summary 188
7.3. FOUR EXAMPLES 189
7.4. MODELING TOOLS 201
7.4.1. A Textual-Based Tool 202
7.4.2. A Tool Based on Graphs 203
7.5. OUTLOOK 204
8. The Definition of the Language 209
8.1. INTRODUCTION 209
8.2. AN OVERVIEW OF THE LPL-LANGUAGE 210
8.2.1. The Entities and the Attributes 210
8.2.2. Index-Sets 212
8.2.3. Data 213
8.2.4. Expressions 219
8.2.5. Logical Modeling 220
8.3. THE BACKUS-NAUR SPECIFICATION OF LPL 232
9. The Implementation 235
9.1. THE KERNEL 235
9.2. THE ENVIRONMENT (USER INTERFACE) 236
9.3. THE TEXT BROWSER 238
9.4. THE GRAPHICAL BROWSER 243
10. Selected Applications 249
10.] . GENERAL LP-, MIP-, AND QP-MODELS 249
10.2. GOAL PROGRAMMING 260
10.3. LP'S WITH LOGICAL CONSTRAINTS 263
10.5. PROBLEMS WITH DISCONTINUOUS FUNCTIONS 281
]0.6. MODELING UNCERTAINTY 284
11. Conclusion 289
References 295
Glossary 307
Index 311
Model Examples
Example 2-1: The Intersection Problem 19
Example 3-1: The Frustum Problem 40
Example 3-2: Theory of Learning 42
Example 3-3: The 3-jug Problem 48
Example 3-4: Cooling a Thermometer 58
Example 3-5: A Production Problem (Product.lpl) 60
Example 4-1: A Letter Game Problem 77
Example 4-2: A PSAT Problem 88
Example 4-3: Probabilistic Entailment 89
Example 4-4: PSAT versus ATMS 90
Example 4-5: Modeling Dynamic Systems 94
Example 4-6: Fuzzy LPs 95
Example 5-1: An Energy Import Problem (Import.lpl) III
Example 6-1: A Portfolio Problem (Invest.lpl) 115
Example 6-2: A Transhipment-Assignment Problem (Location.lpl) 120
Example 6-3: The Transportation Problem 127
Example 6-4: The Car Sequencing Problem (Car.lpl) 136
Example 6-5: The Cutting Stock Problem 141
Example 6-6: The Diet Problem 146
Example 7-1: The n-Queen Problem (8-queen.lpl) 189
Example 7-2: The Cutting Stock Problem (again) 192
Example 7-3: The n-Bit-Adder 194
Example 7-4: A Budget Allocation Problem (Budget.lpl) 199
Example 10-1: Detennine Workforce Level (prod.1pl) 249
Example 10-2: "Rhythmed" Flow-Shop (Rflowsh.lpl) 253
Example 10-3: Assignment of Players to Teams (soccerJpl) 255
Example 10-4: Portfolio Investment (Quadl.lpl) 259
Example 10-5: Book Acquirements for a Library (library.lpl) 260
Example 10-6: The Intersection Problem (Intersec.lp\) 264
Example 10-7: A Satisfiability Problem (SAT) (Satl.lpl) 268
Example 10-8: How to Assemble a Radio (Radio.lpl) 270
Example 10-9: An Energy Import Problem (Import.lpl) 271
Example 10-10: Finding Magic Squares (Magicsq.lpl) 277
Example 10-11: The Capacitated Facility Location Problem (Capafac.lpl) 278
Example 10-12: A Two-Persons Zero-Sum Game (Game.lpl) 281
Example 10.13. The weighted Tardiness Problem (wtl.lpl) 283
Example 10.14. A Stochastic Model 284
Example 10.15. A Model Using Fuzzy-sets 286
Figures
Figure 0-1: The Birth ofLPL (Kohlas J.) xxi
Figure 2-1: El torro 18
Figure 2-2: The Intersection Problem 19
Figure 2-3: Topological Deformation 20
Figure 2-4: Solution to the Intersection Problem 20
Figure 2-5: The Intersection Problem, an Interpretation 24
Figure 2-6: Similarities between Theories 30
Figure 2-7: Model Structure versus Model Instance 32
Figure 3-1 : The Model Life Cycle 38
Figure 3-2: The Frustum Problem 41
Figure 3-3: The Frustrum Problem, an Intermediate Step 41
Figure 3-4: The Frustrum Problem, the Solution 42
Figure 3-5: A Learning Model based on Markov Chain 43
Figure 3-6: Validation of a Model 46
Figure 3-7: The 3-jug Problem 49
Figure 3-8: The 3-jug Problem, the Search Path 49
Figure 3-9: Sensitivity Analysis 55
Figure 3-10: The Correspondence Problem 56
Figure 3-11: The Temperature Plot 59
Figure 4-1: Different Optimizing Criteria 73
Figure 4-2: The Search Path of the Letter Game Problem 78
Figure 4-3: Fuzzy Sets for small, medium, and tall 94
Figure 4-4: The Inverted Pendulum 95
Figure 5-1: Different Model Representations 107
Figure 5-2: Different Representations with many Links 107
Figure 6-1: A Plot of the Reinvestment Flow 117
Figure 6-2: A Spreadsheet for the Portfolio Model 118
Figure 6-3: Product Flow in the Transhipment-Assignment Problem 122
Figure 6-4: Netform of a Transportation Model 129
Figure 6-5: Aggregated Netform for the Transportation Model 130
Figure 6-6: Multi-view Architecture 153
Figure 7-1: An Architecture for Modeling Tools 160
Figure 7-2: Modeling Language Embedding 162
Figure 7-3: An Index-tree 178
Figure 7-4: An Index-tree with Collapsing Paths 179
Figure 7-5: An Index-tree Viewed as a Compound Index-set 179
Figure 7-6: A tagged Index-tree 180
Figure 7-7: A Marked Index-tree 182
Figure 7-8: A Solution for the 4- and the 8-Queen Problem 190
Figure 7-9: The n-Bit-Adder 194
Figure 7-10: AND-, OR-, NOT-Gates 194
Figure 7-11: The XOR-Gate 195
Figure 7-12: A Half-Adder 195
Figure 7-13: A Full-Adder 196
Figure 7-14: A 3-bit Adder 197