handbhandbook of modeling for discert optimizationook of modeling for discert optimization
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HANDBOOK ON MODELLING
FOR DISCRETE OPTIMIZATION
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Recent titlesin the
INTERNATIONAL SERIES IN
OPERATIONS RESEARCH & MANAGEM ENT SCIENCE
Frederick S. Hillier, Series Editor,Stanford University
M a r o s / COMPUTATIONAL TECHNIQUES OF THE SIMPLEX METHOD
Harr i son , Lee & Neale /
THE PRACTICE OF SUPPLY CHAIN
MANAGEMENT:
Where
Theory and
Application C onverge
Shanth ikumar , Yao & Zi jnV STOCH ASTIC MODELING AND OPTIMIZATION OF
MANUFACTURING SYSTEMS AND SUPPLY
CHAINS
N abrzysk i , S chop f & W ?g l a rz / GRID RESOURCEMANAGEMENT: State of the Art and Fu ture
Trends
T hi s sen & H erde r /
CRITICA L INFRASTR UCTU RES: State of the Art in Research and Application
Carl sson , Fedr i zz i , & Ful l e r /
FUZZY LOGIC IN MANAGEMENT
S oye r , M azz uch i & S i ngpu rw a l l a / MATHEMATICALRELIABILITY:An Expository Perspective
C hakrava r t y & E l i a shbe rg / MANAGING BUSINESS
INTERFACES: Marketing,
Engineering, and
Manufacturing P erspectives
Tal lur i & van Ryzin /
THE THEORY AND PRACTICE OF REVENUE MANAGEMENT
K avad i a s & Loch/PROJECT SELECTION UNDER UNCERTAINTY:Dynamically Allocating
Resources to Maximize Value
Brand eau , Sa infor t & Pierska l la / OPERATIONS RESEARCH AND
HEALTHCARE:
A
Handbook of
Methods and Applications
Cooper , Se i ford & Zhu/ HANDBOOK OF
DATA
ENVELOPMENT
ANALYSIS:
Models and
Methods
L uenbe rge r / LINEAR AND NONLINEAR PROGRAMMING,
2'"^
E d
S he rb rooke / OPTIMAL INVENTORY MODELING OFSYSTEMS: Multi-Echelon Techniques,
Second E dition
C hu , L eung , H u i & C heung / 4th PARTY
CYBER
LOGISTICS FO R AIR CARGO
Simchi -Levi , Wu &
Sh^nl HANDBO OK OF QUANTITATfVE SUPPLY CHAIN
ANALYSIS:
Modeling in the E-Business E ra
G ass & A ssad / AN ANNOTATED TIMELINE OF OPERATIONSRESEARCH:A n Informal History
G reenbe rg / TUTORIALS ON EMERGING METHODOLO GIES AND APPLICATIONS IN
OPERATIONS RESEARCH
W e b e r /
UNCERTAINTY IN THE ELECTRIC POWERINDUSTRY: Methods and Models for
Decision Support
Figuei ra , Greco & Ehrgot t / MULTIPLE CR ITERIA DECISION
ANALYSIS:
State of the Art
Surveys
Revel io t i s / REAL-TIME MANAGEMENT O F RESOURCE ALLOCATIONS
SYSTEMS:
A D iscrete
Event Systems Approach
Kai l & Mayer /
STOCHASTIC LINEAR PROGRAM MING: Models, Theory, and Computation
Seth i , Yan & Zhang/
INVENTORY AND SUPPLY CHAIN MANAGEMENT WITH FORECAST
UPDATES
C o x /
QUANTITATTVE
HEALTH RISK ANALYSIS
METHODS:
Modeling the Human Health Impacts
of Antibiotics Used in Food Animals
C hi ng & Ngf MARKOV
CHAINS:
Models, Algorithms and Applications
Li & Sun/NONLINEAR INTEGER PROGRAM MING
K al i szew sk i / SOFT
COMPUTING
FOR COMPLEX MULTIPLE CRITERIA DECISIONMAKING
Bouyssou e t a l /EVALUATION AND D ECISION MODELS
WITH
MULTIPLE CR ITERIA:
Stepping stone s for the analyst
Blecker & Fr i edr i ch / MASSCUSTOMIZATION: Challenges and Solutions
*A list of the early publications in the series is at the end of the book *
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HANDBOOK ON MODELLING
FOR DISCRETE OPTIMIZATION
Edited by
GAUTAM APPA
Operational Research Department
London School of Economics
LEONIDAS PITSOULIS
Department of Mathematical and Physical Sciences
Aristotle University of Thessaloniki
H.PAUL
W ILLIAMS
Operational Research Department
London School of Economics
Sprin
ger
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Gautam Appa Leonidas Pitsoulis
London School of Econ om ics Aristotle Unive rsity of Thessa loniki
United Kingdom Greece
H. Paul W illiams
London School of Economics
United Kingdom
Library of Congress Control Number: 2006921850
ISBN -10: 0-387-32941-2 (HB) ISBN-10: 0-387-32942-0 (e-book)
ISBN-13:
978-0387-32941-3 (HB) ISBN -13: 978-0387-32942-0 (e-book)
Printed on acid-free paper.
2006 by Springer Science+Business Media, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without
the written permission of the publisher (Springer Science + Business Media, Inc., 233
Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with
reviews or scholarly analysis. Use in connection with any form of information storage
and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now know or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks and similar terms,
even if the are not identified as such, is not to be taken as an expression of opinion as to
whether or not they are subject to proprietary rights.
Printed in the United States of America.
9 8 7 6 5 4 3 2 1
springer.com
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Contents
List of Figures ix
List of Tables xiii
Con tributing Authors xv
Preface xix
Acknowledgments xxii
Part I Me thods
1
The Form ulation and Solution of Discrete Optimisation Models 3
H. Paul Williams
1.
The App licability of Discrete Optimisation 3
2.
Integer Programming 4
3.
The Uses of Integer Variables 5
4.
The Modelling of Comm on Conditions 9
5.
Reformulation Techniques 11
6. Solution Methods 22
References 36
2
Con tinuous App roaches for Solving Discrete Optimization Problem s 39
Panos M Pardalos,
Oleg
A Prokopyev and Stanislav B usy gin
1.
Introduction 39
2.
Equivalence of Mixed Integer and Com plementarity Problem s 40
3.
Continuous Formulations for 0-1 Programming Problems 42
4.
The Maximum Clique and Related Problems 43
5. Th e Satisfiability Problem 48
6. The Steiner Problem in Graphs 51
7.
Semidefinite Program ming Approaches 52
8. Minimax Approaches 54
References 55
3
Logic-Based Modeling 61
John N H ooker
1.
Solvers for Logic-Based Constraints 63
2.
Good Formulations 64
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vi HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
3.
Prepositional Logic 69
4. Cardinality Formulas 77
5.
0-1 Linear Inequalities 83
6. Cardinality Rules 85
7.
Mixing Logical and Con tinuous Variables 87
8. Add itional Global Constraints 92
9. Conclusion 97
References 99
4
Mo delling for Feasibility - the case of Mutually Orthogonal Latin Squares 103
Problem
Gautam Appa, Dimitris Mag os, loannis M ourtos and Leonidas P itsoulis
1.
Introduction 104
2.
Definitions and notation 106
3.
Formulations of the fcMOL S problem 108
4.
Discussion 122
References 125
129
129
130
133
134
137
139
141
144
146
148
6
Modeling and Optimization of Vehicle Routing Problem s 151
Jean-Francois Cordeau an d Gilbert Laporte
1. Introduction 151
2.
The Vehicle Routing Problem 152
3.
The Chinese Postman Problem 163
4.
Constrained Arc Rou ting Problem s 168
5.
Conc lusions 181
References 181
Part II Applications
7
Radio Resource Managem ent 195
Katerina Papad aki and
Vasilis
Friderikos
1. Introduction 196
2. Problem Definition 199
;wc
ugl
1.
2.
3.
4.
5.
)rk Modelling
as
R.
Shier
Introduction
Transit Networks
Amplifier Location
Site Selection
Team Elimination ir
1
Sports
6. Reasoning in Artificial Intelligence
7.
Ratio Com parisons in Decision Analysis
8. DNA Sequencing
9. Computer Memory Managem ent
References
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Contents
vii
3.
Myopic Problem Formulations 203
4.
The dynamic downlink problem 208
5.
Concluding Remarks 222
References 224
Strategic and tactical planning models for supply chain: an application of 227
stochastic mixed mteger programming
Gautam M itra, C handra Poojari and Suvrajeet Sen
1.
Introduction and Background 228
2.
Algorithms for stochastic mixed integer program s 234
3.
Supply chain planning and management 237
4.
Strategic supply chain planning : a case study 244
5.
Discussion and conclusions 259
References 260
9
Log ic Inference and a Decomposition Algorithm for the Resource-C onstrained 265
Scheduling of Testing Tasks in the Development of New Pharmaceu
tical and Agrochemical Products
Christos
T,
Maravelias and Ignacio E. Grossmann
266
266
268
271
277
281
281
282
283
284
284
285
10
A Mixed-integer Non linear Program ming Approach to the Optimal Plan- 291
ning of Ons ho re Oilfield Infrastructures
Susara
A,
van den Heever and Ignacio E. Grossmann
291
294
295
301
306
309
311
312
314
11
Radiation Treatment Plann ing: Mixed Integer Program ming Form ula- 317
tions and Approaches
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Introduction
Motivating E xample
Model
Logic Cuts
Decomposition Heuristic
Computational Results
Example
Conclusions
Nomenclature
Acknowledgment
References
Appendix: Example Data
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
Problem Statement
Model
Solution Strategy
Example
Conclusions and Future Work
Acknowledgment
Nomenclature
References
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viii HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
Michael
C.
Ferris, Robert
R .
Meyer and
Warren D 'Souza
1.
Introduction 318
2. Gamma Knife Radiosurgery 321
3.
Brachytherapy Treatment Planning 327
4.
IMRT 331
5.
Conclusions and Directions for Future Research 336
References 336
12
Multiple Hypothesis Correlation in Track-to-Track Fusion Management 341
Aubrey B
Poore,
Sabino M Gad aleta and Benjamin J Slocumb
1.
Track Fusion Architectures 344
2.
The Frame-to-Frame Matching Problem 347
3.
Assignment Problems for Frame-to-Frame Matching 350
4.
Com putation of Cost Coefficients using a Batch Methodology. 360
5.
Summary 368
References 369
13
Com putational Molecular Biology 373
Giuseppe Lancia
1. Introduction 373
2.
Elemen tary Molecular Biology Concepts 377
3.
Alignment Problems 381
4. Single Nucleotide Polymorphisms 401
5.
Genom e Rearrangements 406
6. Genomic Mapping and the TSP 412
7. Applications of Set Covering 415
8. Conclusions 417
References 418
Index 427
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List of Figures
1.1 A piecewise linear approximation to a non-linear function 8
1.2 The convex hull of a pure IP 12
1.3 The convex hull of a mixed IP 13
1.4 Polytopes with different recession directions 15
1.5 A cutting pattern 20
1.6 Optimal pattern 21
1.7 An integer programm e 23
1.8 An integer programm e with Gom ory cuts 27
1.9 Possib le values of an integer variable 28
1.10 The first branch of a solution tree 29
1.11 Solution space of the first branch 29
1.12 Final solution tree 30
3.1 Conversion ofF to CNF without additional variables. A
formula of the form (if
A
/) V G is regarded as having
t h e f o r m G V ( i J A / ) . 7 2
3.2 Linear-time conversion to CN F (adapted from [21]). The
letter
C
represents any clause. It is assumed that
F
does
not contain variables x i , X 2 ,. .. . 73
3.3 The resolution algorithm applied to clause setS 74
3.4 The cardinality resolution algorithm applied to card inal
ity formula set 5 81
3.5 The 0-1 resolution algorithm applied to set 5 of
0-1
inequalities 85
3.6 An algorithm, adapted from [40], for generating a con
vex hull formulation of the cardinality rule (3.26). It is
assumed that a^,bj G {0,1} is part of the formulation.
The cardinality clause {a^} > 1 is abbreviated a . The
procedure is activated by calling it with (3,26) as the ar
gument. 86
5.1 A transit system Gwith 6 stops 131
5.2 The time-expanded network G 132
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HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
5.3 Bipartite flow network 136
5.4 Bipartite flow network associated with Team 3 138
5.5 A constraint graph 141
5.6 Network for assessing probabilities 142
5.7 Revised network for assessing probabilities 144
5.8 DNA sequencing network 145
6.1 The Konigsberg bridges problem 165
6.2 Exam ple for the Frederickson's heuristic does not yield
an optimal solution. 169
6.3 Illustration of procedure SHORTEN 171
6.4 Illustration of procedure DR OP 172
6.5 Illustration of procedure AD D 172
6.6 Illustration of procedure 2-OPT 173
6.7 Illustration of procedure PASTE 177
6.8 Illustration of procedure CU T 178
7.1 Feasible region for two users 211
7.2 System events in the time domain for the original state
variable and pre-decision state variable in time periods
t
a n d t + 1 214
7.3 Geom etrical interpretation of the heuristic used for the
embedded IP optimization problem for user
i.
The next
feasible rate tov iis r/" u{si + ai) 218
7.4 Com putational complexity of the LAD P, textbook DP,
and exhaustive search in a scenario where the outcome
space consist of an eight state Markov channel, the ar
rivals have been truncated to less than twelve packets per user 219
7.5 Com putational times of the LA DP algorithm in terms of
CPU-time as a function of the number of mobile users 222
8.1 A scenario tree 231
8.2 Hierarchy of the supply chain planning 238
8.3 A strategic supply chain network 239
8.4 Supply chain systems hierarchy (source- Shapiro, 1998) 242
8.5 SCHUM ANN Models/Data Flow 243
8.6 Influence of time on the strategic decisions 248
8.7 The Lagrangian algorithm 254
8.8 Pseudo Code
1
256
8.9 Best hedged-value of the configuration 258
8.10 The frequency of the configuration selected 258
8.11 The probability weighted objective value of the configuration 259
9.1 Motivating exam ple 267
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Listo f Figures
xi
9.2 Different cycles for four tests 273
9.3 Branch & bound tree of motivating example 274
9.4 Incidence matrix of constraints 278
9.5 Decomposition heuristic 280
10.1 Configuration of fields, well platforms and production
platforms (Iyer
et al
, 1998) 292
10.2 Logic-based OA algorithm 303
10.3 Iterative aggregation/disaggregation algorithm 306
10.4 Results 308
10.5 The fina l configuration 309
10.6 The optimal investement plan 310
10.7 Production profile over six years 310
11.1 The Gam ma Knife Treatment Unit. A focusing helmet is
attached to the frame on the patient's head. The patient
lies on the couch and is moved back into the shielded
treatment area 321
11.2 Underdose of target regions for (a), (c) the pre-treatment
plan and (b), (d) the re-optimized plan, (a) and (b) show
the base plane, while (c) and (d) show the apex plane 332
12.1 Diagram s of the (a) hierarchical architecture without feed
back, (b) hierarchical architecture w ith feedback, and (c)
fully distributed architecture. S-nodes are sensor/tracker
nodes, while F-nodes are system/fusion nodes 344
12.2 Diagram showing the sensor-to-sensor fusion process 346
12.3 Diagram showing the sensor-to-system fusion process 347
12.4 Illustration of source-to-source track correlation 349
12.5 Illustration of frame-to-frame track correlation 350
12.6 Illustration of two-dimensional assignm ent problem for
frame-to-frame matching 352
12.7 Illustration of three-dimensional assignment problem for
frame-to-frame matching 354
12.8 Illustration of multiple hypo thesis, multiple frame, cor
relation approach to frame-to-frame matching 357
12.9 Detailed illustration of multiple hypo thesis, multiple frame,
correlation approach to frame-to-frame matching 358
12.10 Illustration of sliding window for frame-to-frame matching 360
12.11 Illustration of the batch scoring for frame-to-frame matching 365
13.1 Schematic DNA replication 378
13.2 (a) A noncrossing matching (alignment), (b) The di
rected grid. 382
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HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
13.3 Graph of RNA secondary structure 3 91
13.4 (a) An unfolded protein, (b) After folding, (c) The con
tact map graph. 395
13.5 An alignment of value 5 396
13.6 A chromosom e and the two haplotypes 401
13.7 A SNP matrix M and its fragment conflict graph 403
13.8 Evolutionary events 407
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List of Tables
3.1 Prim e implications and convex hull formulations of some
simple propositions. The set of prime implications of a
proposition can serve as a consistent formulation of that
proposition. 78
3.2 Catalog of logic-based constraints 98
3.3 Advantages of consistent and tight formulations 99
4.1 A pair of OLS of order 4 107
5.1 A system of routes and stops 131
5.2 Poss ible locations for amplifiers 134
5.3 Revenues
Vij
and costs Q 135
5.4 Current team rankings 137
5.5 Gam es remaining to be played 138
5.6 History of requests allocated to frames 147
5.7 An optimal assignment of requests 148
5.8 Another optimal assignment of requests 148
6.1 Solution values produced by several TS heuristics on
the fourteen CM T instances. Best known solutions are
shown in boldface. 164
7.1 Analysis on the performance of different algorithm s 223
8.1 App lications of SM IPs 233
8.2 Configuration generation 249
8.3 Dim ension of the strategic supply chain model 251
8.4 The stochastic metrics 257
9.1 Model statistics of test problem s 272
9.2 Addition of cycle-breaking cuts: Num ber of constraints
and LP relaxation 272
9.3 Testing data for the motivating exam ple 273
9.4 Income data for the motivating example. 273
9.5 Preprocessing algorithm (PPRO CA LG) 274
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HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
9.6 Precedence implications and cuts: No. of constraints
and LP relaxation 276
9.7 Solution statistics 277
9.8 Decomposition algorithm (DECA LG ) 280
9.9 Solution statistics of the exam ples 281
9.10 Solution statistics of the exam ples 282
9.11 Optimal solution 282
9.12 Heuristic solution 282
9.13 Resource assignment of heuristic solution 283
9.A.1 Testing data of product A 286
9.A.2 Testing data of product B 287
9.A.3 Testing data of product C 288
9.A.4 Testing data of product D 288
9.A.5 Incom e data for products 289
11.1 Target coverage of manual pre-plan vs optimized pre
plan vs OR re-optimized plan 331
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Contributing Authors
Gautam Appa
Department of
Operational
Research
London Schoo l of Economics, London
United Kingdom
Vasilis Friderikos
Centre for Telecommunications Research
King'sCollegeLondon
United Kingdom
Stanislav Busygin
Department of Industrial and Systems
Engineering, University of
Florida
303
WeilHall,
Gainesville FL 32611
USA
busygin @uf
I.edu
Sabino M. Gadaleta
Numeric a
PO Box 271246
Fort Collins, CO 805 27-1246
USA
Jean-Fran9ois Cordeau
Canada R esearch Ch air in Distribution
Management and G ERAD, HEC Montreal
3000 chemin de la Cdte-Sainte-Catherine
Montreal, Canada H3T2A7
cordeau @crt.umontreal.ca
Ignacio E. Grossmann
Department of Chemical Engineering
Carnegie Mellon
University,
Pittsburgh
PA 15213,
USA
Warren D'Souza
University of Maryland Scho ol of Medicine
22 South Green Street
Baltimore, MD 21201
USA
Susara A. van den H eever
Department ofChemicalEngineering
Carnegie Mellon University, Pittsburgh
PA 15213,
USA
Michael C. Ferris
Com puter Sciences Department
University of
Wisconsin
1210
West
Dayton
Street,
Madison
Wl53706,
USA
John N. H ooker
Graduate School of Industrial Adm inistration
Carnegie Mellon U niversity, PA 1 5213
USA
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XVI
HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
Giusseppe Lancia
Dipartimento di M atematica e Informatica
Universita di Udine
Viad elle Scienze 206 , 33100 Udine
Italy
Gilbert Laporte
Canada Research Chair in Distribution
Management and GERAD , HEC Montreal
3000 chemin de la Cbte-Sainte-Catherine
Montreal, Canada H3T2A7
Dimitris Magos
Department of Informatics
Technological
Ed ucational Institute of Athens
12210
Athens, Greece
Christos T. Maravelias
Department ofChemicaland B iological
Engineering, U niversity ofWisconsin
1415 Engineering D rive,
Madison, WI53706-1691,
USA
Robert R. Meyer
Com puter Sciences Department
University of
Wisconsin
1210
West
Dayton
Street,
Madison
WI53706,
USA
Gautam Mitra
CARISMA
School of Information Systems,
Computing and Mathematics,
Brunei University, London
United Kingdom
loannis M ourtos
Department of Economics
University ofPatras, 26500
Rion,
Patras
Greece
Katerina Papadaki
Department of
Operational
Research
London School of Economics, London
United Kingdom
Panos Pardalos
Department of Industrial and Systems
Engineering, University of Florida
303
WeilHall,
Gainesville FL 32611
USA
Leonidas Pitsoulis
Department o f Mathematical and
Physical Sciences, Aristotle U niversity
of Thessaloniki, 54124 Thessaloniki,
Greece
Chandra Poojari
CARISMA
School of Information Systems,
Computing and Mathematics
Brunei University, London
United Kingdom
Aubrey Poore
Department of Mathematics
Colorado State U niversity
Fort Co llins, 80523,
USA
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Contributing Authors
xvn
Oleg A Prokopyev
Department o f Industrial and Systems
Engineering, University of Florida
303 Weil
Hall,
Gainesville FL 32611
USA
Suvrajeet Sen
Department of Systems and Industrial
Engineering , University of Arizona,
Tuscon,AZ 85721
USA
Douglas R. Shier
Department of Mathem atical Sciences
Clemson University, Clemson,
SC 29634-0975
USA
Benjamin J, Slocumb
Num eric a
PO Box 271246
Fort Collins, CO 80527-12 46
USA
H. Paul Williams
Department of
Operational
Research
London School of Econom ics, London
United Kingdom
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Preface
The primary reason for producing this book is to demonstrate and commu
nicate the pervasive nature of Discrete O ptimisation. It has applications across
a very w ide range of activities. Many of the applications are only known to
specialists. Our aim is to rectify this.
It has long been recognized that ''modelling" is as important, if not more
important, a mathematical activity as designing algorithms for solving these
discrete optimisation problem s. Nevertheless solving the resultant models is
also often far from straightforward. Although in recent years it has become
viable to solve many large scale discrete optimisation problems some problem s
remain a challenge, even as advances in mathematical methods, hardware and
software technology are constantly pushing the frontiers forward.
The subject brings together diverse areas of academic activity as well as di
verse areas of applications. To date the driving force has been Operational R e
search and Integer Programming as the major extention of the well-developed
subject of Linear Program ming. However, the subject also brings results in
Computer Science, Graph Theory, Logic and Combinatorics, all of which are
reflected in this book.
We have divided the chapters in this book into two parts, one dealing with
general methods in the modelling of discrete optimisation problems and one
with specific app lications. The first chapter of this volume, written by Paul
Williams, can be regarded as a basic introduction of how to model discrete
optimisation problems as Mixed Integer Programmes, and outlines the main
methods of solving them.
Chapter 2, written by Pardalos et al., deals with the intriguing relationship
between the continuous versus the discrete approach to optimisation problems.
The authors in chapter 2 illustrate how many well known hard discrete optimi
sation problems can be modelled and solved by continuous methods, thereby
giving rise to the question of whether or not the discrete nature of the problem
is the true cause of its computational complexity or the presence of noncon-
vexity.
Another subject of great relevance to modelling is Logic. This is covered in
chapter 3. The author, John Hooker, describes the relationship with an alter-
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XX HANDBOOK ONMODELLING FORDISCRETE OPTIMIZATION
native solution (and modelling) approach known as Constraint Satisfaction or,
as it is sometimes called, Constraint Logic Programming. This approach has
emerged more from Computer Science than Operational Research. However,
the possibility of "hybrid methods" based on combining the approaches is on
the horizon, and has been realized with some problem specific implementa
tions.
In chapter 4 Appa et al. illustrate how discrete optimisation modelling and
solution methods can be applied to answer questions regarding a problem aris
ing from combinatorial mathematics. Specifically the authors present various
optimisation formulations of the mutually orthogonal latin squares problem,
from constraint programming (which is covered in detail in chapter 3) to mixed
integer programming formulations and matroid intersection, all of which can
be used to answer existence questions for the problem.
It has long been established that Networks can model most of today's com
plex systems such as transportation systems, telecommunication systems, and
computer networks to name a few, and network optimisation has proven to be
a valuable tool in analyzing the behavior of these systems for design purposes.
Chapter 5 by Shier enhances further the applicability of network modelling by
presenting how it can also be applied to less apparent systems ranging from
genomics, sports and artificial intelligence.
Chapter 6 is the last chapter in the methods part of the book, w here Cordeau
and Laporte discuss a class of problems known as vehicle routing problems.
Vehicle routing problems enjoy a plethora of applications in the transportation
and logistics sector, and the authors in chapter 6 present the state of the art with
respect to exact and heuristic methods for solving them.
In the second part of the book various real life applications are presented,
most of them formulated as mixed integer linear or nonlinear programming
problem s. Chapter 7 by Papadaki and Friderikos, is concerned with the so
lution of optimization problems arising in resource management problems in
wireless cellular systems by employing a novel approach, the so called approx
imate dynamic programming.
Most of the discrete optimisation models presented in this book are of de
terministic nature, that is the values of the input data are assumed to be known
with certainty. There are however real life applications where such an assump
tion is inapplicable, and stochastic m odels need to be considered. This is the
subject of chapter 8, by Mitra et al. where stochastic mixed integer program
ming models are discussed for supply chain management problems.
In chapters 9 and 10 Grossmann et al. present how discrete optimisation
modeling can be efficiently applied to two specific application areas. In chap
ter 9 mixed integer linear programming models are presented for the problem
of scheduling regulatory tests of new pharmaceutical and agrochemical prod
ucts, while in chapter 10 a mixed integer nonlinear model is presented for the
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PREFACE xxi
optimal planning of offshore oilfield infrastructures. In both chapters the au
thors also present solution techniques.
Optimization models to radiation therapy for cancer patients is the subject
discussed in chapter 11 by Ferris and Meyer. They show how the problem of
irradiating patients for treatment of cancerous tumors can be formulated as a
discrete optimisation problem and can be solved as such.
In chapter 12 the data association problem that arises in target tracking is
considered by Poore et al. The objective in this chapter is to partition the data
that is created by multiple sensors observing multiple targets into tracks and
false alarms, which can be formulated as a multidimensional assignment prob
lem, a notoriously difficult integer programming problem which generalizes
the well known assignment problem.
Finally chapter 13 is concerned with the life sciences, and Lancia shows
how some challenging problems of Computational Biology can now be solved
as discrete optimisation models.
Assem bling and planning this book has been much m ore ofachallenge than
we at first envisaged. The field is so active and diverse that it has been difficult
covering the whole subject. Moreover the contributors have themselves been so
deeply involved in practical applications that it has taken longer than expected
to complete the volume.
We are aware that, w ithin the limits of space and the time of contributors we
have not been able to cover all topics that we would have liked. For example we
have been unable to obtain a contributor on Com puter Design, an area of great
importance, or similarly on Computational Finance and Air Crew Scheduling.
By way of mitigation we are pleased to have been able to bring together some
relatively new application areas.
We hope this volume proves a valuable work of reference as well as to stim
ulate further successful applications of discrete optimisation.
GAUTAM APPA , LEONIDAS PITSOULIS, H. PAUL WILLIAMS
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xxii
HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
Acknowledgments
We would like to thank the contributing authors of this volume, and the
many anonym ous referees who have helped us review the chapters all of which
have been thoroughly refereed. We are also thankful to the staff of Springer,
in particular Gary Folven and Carolyn Ford, as well as the series editor Fred
Hillier.
Paul Williams acknowledges the help which resulted from Leverhulme Re
search Fellowship RF& G/9/RFG/2000/0174 and EPSR C Grant EP/C530578/1
in preparing this book.
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I
METHODS
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Chapter 1
THEFORMULATIONANDSOLUTIONOFDISCRETE
OPTIMISATION MODELS
H. Paul W illiams
Department of
Operational
Research
London Scho ol of Econom ics, London
United Kingdom
Abstract
This introductory chapter first discusses the applicability of Discrete Optimisa
tion and how Integer Programming is the most satisfactory method of solving
such problems. It then describes a number of modelling techniques, such as
linearisng products of variables, special ordered sets of variables, logical condi
tions,
disaggregating constraints and variables, column generation etc. The main
solution methods are described, i.e. Branch-and-Bound and Cutting Planes. Fi
nally alternative methods such as Lagrangian Relaxation and non-optimising
methods such as Heuristics and Constraint Satisfaction are outlined.
Keywords:
Integer Program ming, Global Optima, Fixed Costs,Convex Hull, Reformu lation,
Presolve, Logic, Constraint Satisfaction.
1.
The Applicability of Discrete Optimisation
The purpose of this introductory chapter is to give an overview of the scope
for discrete optimisation models and how they are solved. Details of the mod
elling necessary is usually problem specific. Many applications are covered in
other chapters of this book. A fuller coverage of this subject is given in [25]
together with many references.
For solving discrete optimization models, when formulated as (linear) In
teger Programmes (IPs), much fuller accounts, together with extensive refer
ences,
can be found in Nemhauser and Wolsey [19] and Williams [24]. Our
purpose, here, is to make this volume as self contained as possible by describ
ing the main methods.
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4
HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
Limiting the coverage to linear IPs should not be seen as too restrictive in
view of the reformulation possibilities described in this chapter.
It is not possible, totally, to separate modelling from the solution meth
ods. Different types of model will be appropriate for different methods. With
some m ethods it is desirable to modify the model in the course of optimisation.
These tw o considerations are addressed in this chapter.
The modelling of many
physical
systems is dominated by
continuous
(as
opposed todiscrete)mathem atics. Such models are often a simplification of
reality, but the discrete nature ofthereal systems is often at a microscopic level
and continuous modelling provides a satisfactory simplification. What's more,
continuous mathematics is more developed and unified than discrete mathe
matics. The calculus is a powerful tool for the optimisation of many contin
uous problems. There are, however, many systems where such models are
inappropriate. These arise with physical systems (e.g. construction problems,
finite element analysis etc) but are much more common in decision making
(operational research) and information systems (computer science). In many
ways,
we now live in a 'discrete world'. Digital systems are tending to replace
analogue systems.
2. Integer Programm ing
The most common type of model used for discrete optimisation is an
Inte
ger Programm e
(IP) although Constraint Logic Programmes (discussed in the
chapter by Hooker) are also applicable (but give more emphasis to obtaining
feasible rather than optimal solutions). An IP model can be written:
Maximise/Minimise c 'x - i - d 'y (1.1)
Subject to: A x + B y 10
Xi + X2 > 10
xi + 2x4 > 23
(1.81)
(1.82)
(1.83)
X1,
X2,
X3,X4 > 0 and integer
is to leave out some of the conditions or constraints on a (difficult) IP model
and solve the resultant (easier) model. By leaving out constraints we may
well obtain a 'better' solution (better objective value), but one which does not
satisfy all the original constraints. This solution is then used to advantage in
the subsequent process.
For the above example the optimal solution of the LP Relaxation is
. 3 o 5 _ , . . . 1
xi = 4- ,X 2 = 3 , Objective =
12'
6
We have obtained the optimal fractional solution at C. The next step is to de
fine acutting plane which will cut-off the fractional solution at C, without
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24
HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
removing any of the feasible integer solutions (represented by bold dots). This
is known as the
separation
problem. Obviously a number of cuts are possi
ble. The facet defining constraints, are possible cuts as are "shallower" cuts
which are not facets. A major issue is to create a
systematic
way of gener
ating those cuts (ideally facet defining) which separate C from the feasible
integer solutions. Before we discuss this, however, we present a major result
due to Chvatal [6]. This is that all the facet defining constraints (for a PIP) can
be obtained by a finite number of repeated applications of the following two
procedures.
(i)
Addtogethercon straints
in suitable multiples (when all in the same form
eg " < " or " > " ) and add or subtract "= " constraints in suitable m ultiples.
(ii) Divide through the coefficients by their greatest common divisor and
round the right-hand-side coefficient
up (in the case of " > " constraints)
or down (in the case of " < " constraints).
We illustrate this by deriving all the facet defining constraints for the exam
ple above. However, we emphasise that our choice of which constraints to add,
and in what multiples, is ad-hoc. It is a valid procedure but not systematic.
This aspect is discussed later.
1. Add - x i + 2 x 2 < 7
-xi < 0
to give 2^1 + 2x2
0, (2.6)
0
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44 HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
problem asks to find a maximum clique. Its cardinality is called the clique
number
of the graph and is denoted by a;(G).
Along with the maximum clique problem, we can consider the maximum
independent set problem,
A subset / C
V
is called an
independent set
if the
edge set of the subgraph induced by / is empty. The maximum independent
set problem is to find an independent set of maximum cardinality. The inde
pendent number a{G)
(also called the
stability number)
is the cardinality of a
maximum independent set of
G,
It is easy to observe that / is an independent set of G if and only if / is a
clique of G. Therefore, maximum independent set problem for some graph G
is equivalent to solving m aximum clique problem in the complementary graph
G
and vice versa.
Next, we may associate with each vertex i V of the graph a positive
number
wi
called the vertex
weight.
This way, along with the adjacency matrix
AQ,
we consider the vector of vertex weights
w
G M^. The total weight of a
vertex subset S C.V will be denoted by
ies
The
maximum weight clique problem
asks for a clique
Q
of the largest
W{Q)
value. This value is denoted by u;(G,
w).
Similarly, we can define
maximum
weight independentsetproblem.
The maximum cardinality and the maximum weight clique problems along
with maximum cardinality and the maximum weight independent set problems
are A^P-hard [22], so it is considered unlikely that an exact polynomial time
algorithm for them exists. Approximation of large cliques is also hard. It was
shown in [36] that unless NP == ZPP no polynomial time algorithm can
approximate the clique number within a factor of
n^~^
for any e > 0. In [42]
this margin was tightened to
n/2^^^^^^ ~\
The approaches to the problem offered include such comm on combinatorial
optimization techniques as sequential greedy heuristics, local search heuris
tics,
methods based on simulated annealing, neural networks, genetic algo
rithms, tabu search, etc. However, there are also methods utilizing various
formulations of the clique problem as a continuous optimization problem. An
extensive survey on the maximum clique problem can be found in [5].
The simplest integer formulation of the maximum weight independent set
problem is the following so called edge formulation:
n
maxf(x) =Y2 i i
(2-8)
subject to
Xi+ Xj < 1, V(ij) eE, xe {0 ,l}"". (2.9)
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Continuous Approaches
45
In [60] Shor considered an interesting continuous formulation for the max
imum weight independent set by noticing that edge formulation (2.8)-(2.9) is
equivalent to the following multiquadratic problem:
n
m a x / ( x ) ^
^^WiXi,
(2.10)
subject to
XiXj - 0 , V ( i , j ) eE, Xi^ -Xi-^Q, i=- l , 2 , . . . ,n . (2.11)
Applying dual quadratic estimates, Shor reported good computational results
[60]. Lagrangian bounds based approaches for solving some related discrete
optimization problem on graphs is discussed in [61].
Next two continuous polynomial formulations were proposed in [34, 35].
T H E O R E M
6
Ifx'^ is the solution of the following (continuous)quadraticpro
gram
n
m a x / ( x )
= 2_\^i ~ y^ ^i^j = ^^^ 1/2X^AGX
subject to
0 < ^i 0 (2.13)
iev
is
A recent direct proof of this theorem can be found in [1]. Similar formula
tions with some heuristic algorithms were proposed in [8, 24].
In [7], the formulation was generalized for the maximum weight clique
problem in the following natural way. Let
Wuim
be the smallest vertex weight
existing in the graph and a vector d
G
R^ be such that
W i
Consider the following quadratic program:
m a x / ( x ) =
X^{AG
+ d iag ( ( i i , , . .
^dn))x
(2.15)
subject to
^Xi l,
x>0. (2.16)
T H E O R E M 9
The global optimum value oftheprogram (2.15)-(2.16) is
1 - ^ ^ .
(2.17)
Furthermore, for each maximum weight clique
Q*
of the graph
G(V,
E) there
is a global maximizer
x*
of
the
program (2,15, 2.16) such that
^* _ /
Wi/uj{G,w), ifi e
Q*
"" ~ " \ 0,
ifieV\Q\
^^-^^^
Obviously, when all
Wi=
1, we have the original M otzk in-Straus formulation.
Properties of maximizers of the Motzkin-Straus formulation were investi
gated in [25]. Generally, since the program (2.12) is not concave, its arbitrary
(local) maximum does not give us the clique number. Furtherm ore, if x* is
some maximizer of (2.12), its nonzero components do not necessarily corre
spond to a clique ofthegraph. However, S. Busygin showed in [7] that nonzero
components of any global maximizer of (2.15)-(2.16) correspond to a complete
multipartite subgraph of the original graph and any m aximal clique of this sub
graph is the maximum weight clique of the graph. Therefore, it is not hard to
infer a maximum weight clique once the maximizer is found.
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Continuous Approaches
47
Performing the variable scaling Xi >y/wiXi, one may transform the for
mulation (2.15H2.16) to
mSixf{x) = x^A^^^x
(2.19)
subject to
z^x
-= 1, X
>
0, (2.20)
whereAQ^ {a\^^)nxn is such that
if = {
V^ ^
'^HhJ)^E
(2.21)
0, ifzT^ j a n d ( i , j )
^
;,
zeR"" : Zi= y i (2.22)
and
is the vector
of
square roots of the vertex weights. The attractive property
is this rescaled formulation
is
that maximizers corresponding to cliques of
the same weight are located at the same distance from the zero point. Mak
ing use of this, S. Busygin developed a heuristic algortihm for the maximum
weight clique problem, called QUALEX-MS, which is based on the formula
tion (2.19)-(2.20) and shows a great practical efficiency [7]. Its main idea con
sists in approximating the nonnegativity constraint by a spherical constraint,
which leadsto atrust region problem known to be polynomially solvable.
Then, stationary points of the obtained trust region problem show correlation
with maximum weight clique indicators with a high probability.
A good upper bound on uj{G^w) can be obtained using semidefinite pro
gramming. The value
i}{G,w)= m axz Xz, (2.23)
s.tXij O iiJ) eE, tr(X)-l,
where
z
is defined by (2.22), 5 ^ is the cone of positive sem idefinite
nxn
ma
trices, and tr(X)
XlILi
^ii
denotes the
trace
of the matrix
X
is known as
the
(weighted) Lovdsz number
(T^-function) of a graph. It bounds from above
the (weighted) independence number of the graph. Hence, considering it for
the complem entary graph, we obtain an upper bound on u;(G,
w).
It was shown
in [9] that unless
P NP,
there exists no polynom ial-time computable upper
bound on the independence number provably better than the Lovasz number
(i.e., such that whenever there wasagap between the independence number
and the Lovasz number, that bound would be closer to the independence num
ber than the Lovasz num ber). It implies that the Lovasz num ber of the comple
mentary graph is most probably the best possible approximation from above
for the clique number that can be achieved in polynomial time in the worst
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48
HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
case.
For a survey on the Lovasz number and its remarkable properties we
refer the reader to [44].
It is worth m entioning that the semidefinite program (2.23) can be also used
for deriving large cliques ofthegraph. Burer, Monteiro, and Zhang employed a
low-rank restriction upon this program in
ih^xx Max-AO
algorithm and obtained
good computational results on a wide range of the maximum clique problem
instances [6].
5. The Satisfiability Problem
Thesatisfiabilityproblem (SATproblem)
is one of the classical hard com bi
natorial optimization problems. This problem is central in mathematical logic,
computing theory, artificial intelligence, and many industrial application prob
lems. It was the first problem proved to be A/^P-complete [12, 22 ]. More
formally, this problem is defined as follows [32]. Let x i , . . . , x ^ be a set of
Boolean variables whose values are either 0 (false), or 1 (true), and let xi de
note the negation of
x^.
A
literal
is either a variable or its negation. A
clause
is an expression that can be constructed using literals and the logical operation
or
(V). Given a set of
n
clauses Ci, . . . ,
Cn,
the problem is to check if there
exists an assignment of values to variables that makes a Boolean formula of
the following
conjunctive normal form (CNF)
satisfiable:
C i
A
C2 A . . .
A
Cn,
where A is a logical and operation.
SAT problem can be treated as a constrained decision prob lem. Another
possible heuristic approach based on optimization of a non-convex quadratic
programming problem is described in [40, 41]. SAT problem was formulated
as an integer programm ing feasibility problem of the form:
B^w < b,
(2.24)
we{-l,ir, (2.25)
whereB e W^"^, b e R'^ md w e R"".It iseasy to observe that this integer
programming feasibility problem is equivalent to the following non-convex
quadratic programming problem:
m ax w^w (2.26)
B^w < b,
(2.27)
-e
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49
A^w
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HANDBOOK ON MODELLING FOR DISCRETE OPTIMIZATION
The correspondence between x and y is defined as follows (for
1
< i < m):
r 1 i f ^ i - l
Xi=
0 gap recognition and
a(G)-upper bounds,
ECCC Report TR03-052
(2003).
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in Com binatorial Optimization, in: D.-Z. Du, PM . Pardalos, eds.,Minimax
and App lications,K luwer Academic Publishers, 1995, pp. 269-29 2.
[11] X. Cheng, Y. Li, D.-Z. Du, H.Q. Ngo , Steiner Trees in Industry, in: D.-Z.
Du, P.M. Pardalos,eds.,
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Suppl.
Vol. B, pp. 193-216, 2005.
[12] S. Cook. The complexity of theorem-proving procedures, in: Proc. 3rd
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Association for Computing
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[15] D.-Z. Du, J. Gu, PM. Pardalos, eds..
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demic Publishers, 1995.
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[31] J. Gu, Parallel Algorithms for Satisfiability (SAT) Problem, in: RM.
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19-151.
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Algorithms on Strings, Trees, and Sequences: Computer
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Chapter 3
LOGIC-BASED MODELING
John N Hooker
Graduate School of Industrial A dministration
Carnegie Mellon
University,
Pittsburgh, PA 15213
USA
Abstract Logic-based modeling can result in decision mod els that are more natural and
easier to debug. The addition of logical constraints to mixed integer program
ming need not sacrifice computational speed and can even enhance it if the con
straints are processed correctly. They should be written or automatically refor
mulated so as to be as nearly consistent or hyperarc consistent as possible. They
should also be provided with a tight continuous relaxation. This chapter shows
how to accomplish these goals for a number of logic-based cons traints: formu
las of propositional logic, cardinality formulas, 0-1 linear inequalities (viewed as
logical formulas), cardinality rules, and mixed logical/linear constraints. It does
the same for three global constraints that are popular in constraint programming
systems: the all-different, element and cumulative constraints.
Clarity is as important as computational tractability when building scien
tific m odels. In the broadest sense, models are descriptions or graphic rep
resentations of some phenomenon. They are typically written in a formal or
quasi-formal language for a dual purpose: partly to permit computation of the
mathematical or logical consequences, but equally to elucidate the conceptual
structure of the phenomenon by describing it in a precise and limited vocabu
lary. The classical transportation model, for example, allows fast solution with
the transportation simplex method but also displays the problem as a network
that is easy to understand.
Optimization modeling has generally emphasized ease of computation more
heavily than the clarity and explanatory value of the model. (The transportation
model is a happy exception that is strong on both counts.) This is due in part
to the fact that optimization, at least in the context of operations research, is
often more interested in prescription than description. Practitioners who model
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a manufacturing plant, for example, typically want a solution that tells them
how the plant should be run. Yet a succinct and natural model offers several
advantages: it is easier to construct, easier to debug, and more conducive to
understanding how the plant works.
This chapter explores the option of enriching mixed integer programming
(MILP) models with logic-based constraints, in order to provide more natural
and succinct expression of logical conditions. D ue to formulation and solution
techniques developed over the last several years, a modeling enrichment of this
sort need not sacrifice computational tractability and can even enhance it.
One historical reason for the de-emphasis of perspicuous models in opera
tions research has been the enormous influence of linear programming. Even
though it uses a very small number of primitive terms, such as linear inequal
ities and equations, a linear programming model can formulate a remarkably
wide range of problems. The linear format almost always allows fast solution,
unless the model is truly huge. It also provides such analysis tools as reduced
costs, shadow prices and sensitivity ranges. There is therefore a substantial
reward for reducing a problem to linear inequalities and equations, even when
this obscures the structure of the problem.
When one moves beyond linear models, however, there are less compelling
reasons for sacrificing clarity in order to express a problem in a language
with a small number of primitives. There is no framework for discrete or dis
crete/continuous models, for exam ple, that offers the advantages of linear pro
gramming. The mathematical programming community has long used MILP
for this purpose, but MILP solvers are not nearly as robust as linear program
ming solvers, as one would expect because MILP formulates NP-hard prob
lems. Relatively small and innocent-looking problems can exceed the capabil
ities of any existing solver, such as the market sharing problems identified by
Williams [35] and studied by Comuejols and Dawande [16]. Even tractable
problems may be soluble only when carefully formulated to obtain a tight lin
ear relaxation or an effective branching scheme.
In addition MILP often forces logical conditions to be expressed in an un
natural way, perhaps using big-M constraints and the like. The formulation
may be even less natural if one is to obtain a tight linear relaxation. Current
solution technology requires that the traveling salesman p roblem, for exam ple,
be written with exponentially many constraints in order to represent a simple
all-different condition. MILP may provide no practical formulation at all for
important problem classes, including some resource-constrained scheduling
problems.
It is true that MILP has the advantage of a unified solution approach, since
a single branch-and-cut solver can be applied to any MILP model one might
write. Yet the introduction of logic-based and other higher-level constraints no
longer sacrifices this advantage.
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The discussion begins in Section 3.1 with a brief description of how solvers
can accommodate logic-based constraints: namely, by automatically convert
ing them to MILP constraints, or by designing a solver that integrates MILP
and constraint programm ing. Section 3.2 describes what constitutes a good
formulation for M ILP and for an integrated solver. The rem aining sections de
scribe good formulations for each type of constraint: formulas of propositional
logic, cardinality formulas, 0-1 linear inequalities (viewed as logical formu
las),
cardinality rules, and mixed logical/linear constraints. Section 3.8 briefly
discusses three global constraints that are popular in constraint programming
systems: the all-different, element and cumulative constraints. They are not
purely logical constraints, but they illustrate how logical expressions are a
special case of a more general approach to modeling that offers a variety of
constraint types. The chapter ends with a summary.
1. Solvers for Logic-Based Constraints
1.1 Two Approaches
There are at least two ways to deal with logic-based constraints in a unified
solver.
One possibility is to provide automatic reformulation of logic-based con
straints into MIL P constraints, and then to apply a standard M ILP solver
[26,
28]. The reformulation can be designed to result in a tight linear
relaxation. This is a viable approach, although it obscures some of the
structure of the problem.
A second approach is to design a unified solution method for a diversity
of constraint types, by integrating MILP with constraint programming
(CP). Surveys of the relevant literature on hybrid solvers may be found
in [11, 18 ,21 ,22 ,27 ,3 6] .
Whether one uses automatic translation or a CP/MILP hybrid approach,
constraints must be written or automatically reformulated with the algorith
mic implications in mind. A good formulation for M ILP is one with a tight
linear relaxation. A good formulation for CP is as nearly "consistent" as pos
sible. A consistent constraint set is defined rigorously below, but it is roughly
analogous to a convex hull relaxation in MILP. A good formulation for a hy
brid approach should be good for both MILP and CP whenever possible, but
the strength of a hybrid approach is that it can benefit from a formulation that
is good in either sense.
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1.2 Structured Groups of Constraints
Very often, the structure of a problem is not adequately exploited unless
constraints are considered in groups. A group of constraints can generally
be given a MILP translation that is tighter than the combined translations of
the individual constraints. The consistency-maintenance algorithms of CP are
more effective when applied to a group of constraints whose overall structure
can be exploited.
Structured groups can be identified and processed in three ways.
Automatic detection.
Design solvers to detect special structure and process
it appropriately, as is commonly done for network constraints in MILP
solvers. However, since modelers are generally aware of the problem
structure, it seems more efficient to obtain this information from them
rather than expecting the solver to rediscover it.
Hand
coding.
Ask modelers to exploit structured constraint groups by hand.
They can write a good MILP formulation for a group, or they can write
a consistent formulation.
Structurallabeling.
Let modelers label specially structured constraint groups
so that the solver can process them accordingly. The CP comm unity
implements this approach with the concept of a
global
constraint, which
represents a structured set of more elemen tary constraints.
As an exam ple of the third approach, the global constraint all-different(a,
h,
c)
requires that a,handctake distinct values and thus represents three inequations
a ^ h, a ^ c and h
i=-
c. To take another example, the global constraint
cnf(a V 6, a V -i6) can alert the solver that its argum ents are propositional
formulas in conjunctive normal form (defined below). This allows the solver
to process the constraints with specialized inference algorithms.
2. Good Formulations
2,1 Tight Relaxations
A good formulation of an MILP model should have a tight linear relaxation.
The tightest possible formulation is a
convex hull formulation,
whose continu
ous relaxation describes the convex hull of
the
model's feasible set. T he convex
hull is the intersection of all half planes containing the feasible set. At a min
imum it contains inequalities that define all the facets of the convex hull and
equations that define the affine hull (the smallest dimensional hyperplane that
contains the feasible set). Such a formulation is ideal because it allows one to
find an optimal solution by solving the linear programming relaxation.
Since there may be a large number of facet-defining inequa lities, it is com
mon in practice to generate
separating
inequalities that are violated by the
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optimal solution of the current relaxation. This must be done during the so
lution process, however, since at the modeling stage one does not know what
solutions will be obtained from the relaxation. Fortunately, som e comm on con
straint types may have a convex hull relaxation that is simple enough to analyze
and describe in advance. Note, however, that even when one writes a convex
hull formulation of each constraint or each structured subset of constraints, the
model as a whole is generally not a convex hull formulation.
It is unclear how to measure the tightness of a relaxation that does not de
scribe the convex hull. In practice, a "tight" relaxation is simply one that pro
vides a relatively good bound for problems that are commonly solved.
2.2 Consistent Form ulations
A good formulation for CP is
con sistent,
meaning that its constraints explic
itly rule out assignments of values to variables that cannot be part of a feasible
solution. No te that consistency is not the same as satisfiability, as the term
might suggest.
To make the idea more precise, suppose that constraint set S contains vari
a b l e s x i , . . . ,Xn. Each variable Xj has a domain Dj, which is the initial set
of values the variable may take (perhaps the reals, integers, etc.) Let SLpartial
assignment
(known as a
compound label
in the constraints community) specify
values for some subset of the variables. Thus a partial assignment has the form
(^ji ^"'^^3k) = i^h^"'^ ^jk)^ where eachji e D j^ (3.1)
A partial assignment (3.1) is
redundant
for
S
if it cannot be extended to a
feasible solution ofS. That is, every complete assignment
(x i , ...,Xn)='{vi,..., Vn),
where each
j
G
Dj
that is consistent with (3.1) violates some constraint in
S.
By convention, a partial assignment violates a constraint Conly if it assigns
some value to every variable in C Thus if Di = D2 = 71,the assignment
x i ==
1
does not violate the constraint xi + x^ > 0 since
X2
has not been
assigned a value. The assignment is redundant, however, since it cannot be
extended to a feasible solution of
xi + x^ >
0. No value in X2's domain will
work.
A constraint set
S
is
consistent
if every redundant partial assignment vi
olates some constraint in
S.
Thus the constraint set
{xi
+ ^2 > 0} is not
consistent.
It is easy to see that a consistent constraint set
S
can be solved without
backtracking, provided S is satisfiable. First assign x i a value vi GDi that
violates no constraints inS (which is possible because S is satisfiable). Now
assignX2 a valueV2 GD2such that ( x i , 2:2) = ('? i,^ 2)violates no constraints
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the smallest finite domains. Bounds consistency is generally maintained by
interval arithmetic and by specialized algorithms for global constraints with
numerical variables.
Both hyperarc and bounds consistency tend to reduce backtracking, because
CP systems typically branch by splitting a domain. If the domain is small or
narrow, less splitting is required to find a feasible solution. Small and narrow
domains also make domain reduction more effective as one descends into the
search tree.
2.3 Prime Implications
There is a weaker property than consistency, namely completeness, that can
be nearly as helpful when solving the problem. Com pleteness can be achieved
for a constraint set by generating its prime im plications, w hich, roughly speak
ing, are the strongest constraints that can be inferred from the constraint set.
Thus when it is impractical to write a consistent formulation, one may be able
to write the prime implications of the constraint set and achieve much the same
effect.
Recall that a constraint set
S
is consistent if any redundant partial assign
ment for
S
violates some constraint in
S. S
is
complete
if any redundant partial
assignment for
S
is
redundant
for some constraint in
S,
Checking whether a
partial assignment is redundant for a constraint is generally harder than check
ing whether it violates the constraint, but in many cases it is practical nonethe
less.
In some contexts a partial assignment is redundant for a constraint only if it
violates the constraint. In such cases com pleteness implies consistency.
Prime implications, roughly speaking, are the strongest constraints that can
be inferred from a constraint set. To develop the concept, suppose constraints
C and D contain variables m x (x i , . . . ,Xn ) . C impliesconstraint D if
all assignments to
x
that satisfy
C
also satisfy
D.
Constraints
C
and
D
are
equivalentif they imply each other.
Let H be some family of constraints, such as logical clauses, cardinality
formulas, or 0-1 inequalities (defined in subsequent sections). We assume H
is semantically finite, meaning that there are a finite number of nonequivalent
constraints in
R.
For example, the set of 0-1 linear inequalities in a given
set of variables is semantically finite, even though there are infinitely many
inequalities.
Let an
H-implication
of a constraint set 5 be a constraint in
H
that
S
im
plies. Constraint C isdi prime H-implication ofS'if C is a iJ-imp lication of 5 ,
and every /^-implication ofS that impliesCis equivalent toC. The following
is easy to show.
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L E M M A 3.1
Given
a sem anticallyfinite
constraint
set H, every H -implication
of a constraint set S is implied by some prime H-implication of S. Thus if
S C H, S is equivalent to the set ofitsprime H-implications.
For example, let
H
be the family of 0-1 linear inequalities in variables
xi, X2,
and let
S
consist of
^ 1 + 3:2 > 1
^ 1 3: 2 ^ 0
S has the single prime ^-im plic atio n x i > 1 (up to equivalence). Any 0-1
inequality implied by S is implied by xi > 1. Since S C H, S isequivalent
to {xi >1}.
Lemma 3.1 may not apply when H is semantically infinite. For instance,
let xi have domain [0,1] and H be the set of constraints of the form xi > a
for a e [0,1 ). Then all constraints in H are implied by 5' =: {xi > 1}
but none are implied by a prime i7-implication of
S,
since
S
has no prime
iJ-implications.
The next section states precisely how Lemma 3.1 allows one to recognize
redundant partial assignments.
2.4 Recognizing Redundant Partial Assignments
Let a clause in variables x i , . . . , x^i with domains D i , . . . , D^ be a con
straint of the form
{xj, ^ ^1) V . . . V{xj^ + v^) (3.2)
where each
2%
^ {I? ?^} and each f
.
G L>j.. A clause with zero disjuncts
is necessarily false, by convention. A constraint set H contains all clauses
for a set of variables if every clause in these variables is equivalent to some
constraint in
H,
L E M M A 3.2
If a semantically finite constraint set H contains all clauses for
the variables in constraint set S, thenanypartial assignm ent that is redundant
for S is redundant for someprime H -implication of S.
This is easy to see. If a partial assignment
{xj^
,...,
Xj^) =
( t ' l , . . . , ^p)
is redundant for
S,
then
S
implies the clause (3.2). By Lemma 3 .1, some
prime if-implication
P of S
implies (3.2). This means the partial assignment
is redundant for P.
As an example, consider the constraint setS consisting of
2x1+ 3x2> 4: (a)
(3 3)
3x1 + 2x2 < 5 (6)
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where the domain of eachXj is{0 ,1 ,2} . Since the only solutions of (3.3) are
X
= (1,1 ) and (0 ,2 ), there are two redundant partial assignments: x i
==
2 and
X2
0. Let
H
consist of all linear inequalities in xi,
X2,
plus c lauses in a;i, X2.
The //-prime implications of (3.3), up to equivalence, are (3.3b) and
xi +
2^2 > 3
-x i
+ 2x2 > 1 (3.4)
2 ^1 X2 < 1
It can be checked that each redundant partial assignment is redundant for at
least one (in fact two) of the prime implications. Note that the redundant par
tial assignment X2 == 0 is not redundant for either of the original inequalities
in (3.3); only for the inequalities taken together. Knowledge of the prime im
plications therefore makes it easier to identify this redundancy.
A constraint set
S
is
complete
if every redundant partial assignment for
S
is
redundant for some constraint inS. From Lemm a 3.2 we have:
C O R O L L A R Y
3 .3
If S contains all of
its
prime H-implications for some se-
man tically finite constraint set H that contains all clauses for the variables in
S, then S is complete.
Thus if inequalities (3.4) are added to those in (3.3), the result is a complete
constraint set.
3 . Prepositional Logic
3 ,1 Basic Ideas
A formal logic
is a language in which deductions are made solely on the
basis of the form of statements, without regard to their specific meaning. In
propositional logic,
a statement is made up of
atomic propositions
that can be
true or false. The form of
the
statement is given by how the atomic propositions
are joined or modified by such logical expressions as
not
(-i),
and
(A), and
inclusiveor(V).
A proposition defines a
prop ositional function
that maps the truth values
of its atomic propositions to the truth value of the whole proposition. For
example, the proposition
(a
V
-n6) A (-la
V
h)
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contains atomic propositions a,b and defines a function / given by the follow
ing truth table:
a b f{a,b)
0 0 1
0 1 0 (3.5)
1 0 0
1 1 1
Here 0 and 1 denotefalse and true respectively. One can define additional
symbols for implication (>), equivalence (= ) , and so forth. For any pair of
formulas A,
B
A-^
B
=def
-^Ay B
A = B
-def
{A-^B)A{B-^A)
Note that (3.5) is the truth table for equivalence. A
tautology
is a formula
whose propositional function is identically true.
A formula of propositional logic can be viewed as a constraint in which
the variables are atomic propositions and have domains {0,1}, where 0 and
1 signify false and true. The constraint is satisfied when the formula is true.
Many complex logical conditions can be naturally rendered as propositional
formulas, as for exam ple the following.
Alice will go to the party only a -^ c
if Charles goes.
Betty will not go to the party if (aV d) -^-^b
Alice or Diane goes.
Charles will go to the party
c^
{-^d
A
~ie)
unless Diane or Edward goes.
Diane will not go to the party d
>
(c
V
e)
unless Charles or Edward goes.
Betty and Charles never go to the
-^{bAc)
same party.
Betty and Edward are always b = e
seen together.
Charles goes to every party that 6
>
c
Betty goes to.
3 .2 Conversion to Clausal Form
It may be useful to convert formulas to clausal form, particularly since the
well-known resolution algorithm is designed for clauses, and clauses have an
obvious MILP representation.
In propositional logic, clauses take the form of conjunctions of
literals,
which are atomic propositions or their negations. We will refer to a clause
(3.6)
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Logic-Based Modeling 1
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of this sort as a
propositional clause,
A propositional formula is in
clausal
form, also known asconjunctive norm al form (CNF), if it is a conjunction of
one or more clauses.
Any propositional formula can be converted to clausal form in at least two
ways. The more straightforward conversion requires exponential space in the
worst case. It is accomplished by applying som e elementary logical equiva
lences.
De Morgan slaws
-^{FAG)
= -^Fy-^G
Distribution laws F A{GW H ) = {F AG) W {F A H)
FWIGAH)
- ( F V G ) A ( F V i f )
Dou ble negation -i-nF = F
(Of the two distribution laws, only the second is needed.) For example, the
formula
( a A - n6 ) V - n ( a V - i 6 )
may be converted to CNF by first applying De Morgan's law to the second
disjunct,
(a A -16) V (-la A b)
and then applying distribution,
(a V - la) A (a V 6) A (-16 V - ia) A {-^b V b)
The two tautologous clauses can be dropped. Similarly, the propositions in
(3.6) convert to the following clauses.
- la V c
- laV -16
- i c V - i d
ic
V ie
c V - d V e
^^'^^
-^bW -^c
-16 V e
6 V - i e
1 6 V c
The precise conversion algorithm appears in Fig. 3.1.
Exponential growth for this type of conversion can be seen in propositions
of the form
{aiAbi)V ,.,y{anAbn) (3.8)
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Let
F
be the formula to be converted.
SetA;= and 5 = 0.
R e p l a c e a l l s u b f o r m u l a s o f t h e f or m G~H w i t h {G ^ H) A {H ^ G) .
R e p l a c e a l l s u b f o r m u l a s o f t h e f or m
GDH
w i t h
->G W
H.
P e r f o r m C o n v e r t ( F ) .
T h e CNF f o rm i s t h e c o n j u n c t i o n o f c l a u s e s i n S.
F u n c t i o n C o n v e r t ( F )
If F i s a c l a u s e th e n a d d F to S.
Else if F h a s t h e f o r m - i -i G t he n p e r f o r m C o n v e r t ( G ) .
Else if F h a s t h e f o rm GAH then
p e r f o r m C o n v e r t ( G ) a n d C o n v e r t ( / / ) .
Else if F h a s t h e f o rm -^{G A H) th en p e r f o r m C o n v er t( -i G V - i / / ) .
Else if
F
h a s t h e f o rm
-^{G V H)
th en p e r f o r m C o n v er t( -i G A - i / / ) .
Else if F h a s t h e f o rm GV {H Al) then
P e r f o r m C o n v e rt (C V i f ) a n d C o n ve r t( G V / ) .
Figure 3.1. Convers ion of F to C N F wi thou t addi t ional variables. A form ula of the form
{H A I) V G IS
regarde d as having the form
GV {H A I).
The formula translates to the conjunction of 2^ clauses of the form Li V ... V
Ln,
where eachLj isaj or bj.
By adding new variables, however, conversion to CNF can be accom plished
in linear time. The idea is credited to Tseitin [34] but Wilson's more compact
version [38] simply replaces a disjunction F
V
G with the conjunction
(x i
VX2)
A
{-^xiV
F ) A (^^2
V
G),
where xi,
X2
are new variables and the clauses
-^xi
V
F
and -1x2 V
G
encode
implications
xi -^ F
and
X2> G,
respectively. For example, formula (3.8)
yields the conjunction,
n
(x i V . . .
V
Xn)
A
^
{-^Xj
V
aj