Location, Scheduling, Design and Integer Programming

227 Pages·1996·6.26 MB·English

by  Manfred Padberg

LOCATION, SCHEDULING, DESIGN
and INTEGER PROGRAMMING
INTERNATIONAL SERIES IN
OPERATIONS RESEARCH & MANAGEMENT SCIENCE
Frederick S. Hillier, Series Editor
Department of Operations Research
Stanford University
Stanford, California
Saigal, Romesh.
The University of Michigan
LINEAR PROGRAMMING: A Modern Integrated Analysis
Nagurney, Annal Zhang, Ding
University of Massachusetts @ Amherst
PROJECI'ED DYNAMICAL SYSTEMS AND VARIATIONAL INEQUALITIES
WITH APPLICATIONS
LOCATION, SCHEDULING, DESIGN
and INTEGER PROGRAMMING
Manfred Padberg
Professor and Research Professor
of Statistics and Operations Research
New York University
New York, USA
•
Minendra P Rijal
Lecturer
Tribhuvan University
Kathmandu, NEPAL,
and Visiting Assistant Professor
New York University
New York, USA
KLUWER ACADEMIC PUBLISHERS
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Library of Congress Cataloging-in-Publication Data
c.1.
A P. Catalogue record for this book is available from the Library of
Congress.
ISBN-13: 978-1-4612-8596-0 e-ISBN-13: 978-1-4613-1379-3
DOl: 10.1007/978-1-4613-1379-3
Copyright ~ 1996 by Kluwer Academic Publishers
Softcover reprint of the hardcover 1s t edition 1996
All rights reserved. No part of this publication may be reproduced, stored in
a retrieval system or transmitted in any form or by any means, mechanical,
photo-copying, recording, or otherwise, without the prior written permission of
the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park,
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Printed on acid-free paper.
PREFACE
Location, scheduling and design problems are assignment type problems with
quadratic cost functions and occur in many contexts stretching from spatial
economics via plant and office layout planning to VLSI design and similar prob
lems in high-technology production settings. The presence of nonlinear inter
action terms in the objective function makes these, otherwise simple, problems
NP hard. In the first two chapters of this monograph we provide a survey
of models of this type and give a common framework for them as Boolean
quadratic problems with special ordered sets (BQPSs). Special ordered sets
associated with these BQPSs are of equal cardinality and either are disjoint
as in clique partitioning problems, graph partitioning problems, class-room
scheduling problems, operations-scheduling problems, multi-processor assign
ment problems and VLSI circuit layout design problems or have intersections
with well defined joins as in asymmetric and symmetric Koopmans-Beckmann
problems and quadratic assignment problems. Applications of these problems
abound in diverse disciplines, such as anthropology, archeology, architecture,
chemistry, computer science, economics, electronics, ergonomics, marketing,
operations management, political science, statistical physics, zoology, etc. We
then give a survey of the traditional solution approaches to BQPSs. It is an
unfortunate fact that even after years of investigation into these problems, the
state of algorithmic development is nowhere close to solving large-scale real
life problems exactly. In the main part of this book we follow the polyhedral
approach to combinatorial problem solving because of the dramatic algorith
mic successes of researchers who have pursued this approach. In particular, we
define and utilize in Chapters 4 and 5 the concept of a "locally ideal" lineariza
tion to obtain improved linear programming formulations of these problems. A
locally ideal linearization is a linearization that yields an ideal, i.e., minimal
and complete, linear description of each pair or certain sets of pairs of variables
in the quadratic interaction terms of the objective function. In a way, using
this concept of formulating BQPSs is analogous to investigating thoroughly a
few threads of a cobweb as a starting point for a full-fledged study of the entire
cobweb. In Chapter 6 we compare alternative formulations of some schedul
ing problems analytically and give some results on the facial structure of their
associated polytopes. Chapter 7 deals with the affine hull and the dimension
v
VI LOCATION, SCHEDULING, DESIGN
of quadratic assignment polytopes and their symmetric relatives. Chapter 8
reports some very preliminary computational results.
By comparison to traveling salesman problems and other combinatorial opti
mization problems where we know a lot about the facial structure of the associ
ated polytopes - knowledge that has been put to use in the actual optimization
of large-scale problems - little such operational knowledge has been accumu
lated so far for quadratic assignment problems. We hope that this monograph
will help focus interest and provoke more work along polyhedral lines of inves
tigation into the fascinating world of location, scheduling and design problems.
We are confident that following this line of work and implementing a proper
branch-and-cut algorithm will push the limits of exact computation far beyond
the current ones. Due to space and time limitations we have not included a sur
vey about the polyhedral/polytopal methods that we employ in the main part
of this book. There are now several texts available where the reader can find
the pertaining material covered in detail. In particular, any unexplained termi
nology can be found in Chapters 7 and 10 of M. Pad berg's Linear Optimization
and Extensions (Springer-Verlag, Berlin, 1995).
The writing of this monograph has been made possible in part by the financial
support that Professor Karla Hoffman of George Mason University and Pro
fessor Padberg have received from ONR. We would like to thank Dr. Donald
Wagner of the Office of Naval Research for his continued support.
New York City Manfred Pad berg
Minendra P Rijal
CONTENTS
Preface v
List of Figures
IX
List of Tables
XI
1 LOCATION PROBLEMS 1
1.1 A Modified KB Model 5
1.2 A Symmetric KB Model 8
1.3 A Five-City Plant Location Example 14
1.4 Plant and Office Layout Planning 20
1.5 Steinberg's Wiring Problem 26
1.6 The General Quadratic Assignment Problem 31
2 SCHEDULING AND DESIGN PROBLEMS 35
2.1 Traveling Salesman Problems 35
2.2 Triangulation Problems 36
2.3 Linear Assignment Problems 38
2.4 VLSI Circuit Layout Design Problems 39
2.5 Multi-Processor Assignment Problems 44
2.6 Scheduling Problems with Interaction Cost 47
2.7 Operations-Scheduling Problems 50
2.8 Graph and Clique Partitioning Problems 52
2.9 Boolean Quadric Problems and Relatives 56
2.10 A Classification of Boolean Quadratic Problems 57
VII
Vlll LOCATION, SCHEDULING, DESIGN
3 SOLUTION APPROACHES 59
3.1 Mixed zero-one formulations of QAPs 61
3.2 Branch-and-bound algorithms for QAPs 65
3.3 Traditional cutting plane algorithms 72
3.4 Heuristic procedures 75
3.5 Polynomially solvable cases 76
3.6 Computational experience to date 77
4 LOCALLY IDEAL Lp·FORMULATIONS I 79
4.1 Graph Partitioning Problems 82
4.2 Operations Scheduling Problems 88
4.3 Multi-Processor Assignment Problems 95
5 LOCALLY IDEAL LP FORMULATIONS II 105
5.1 VLSI Circuit Layout Design Problems 105
5.2 A General Model 111
5.3 Quadratic Assignment Problems 117
5.4 Symmetric Quadratic Assignment Problems 122
6 QUADRATIC SCHEDULING PROBLEMS 133
6.1 Alternative Formulations of the OSP 133
6.2 Quadratic Scheduling Polytopes 144
7 QUADRATIC ASSIGNMENT POLYTOPES 151
7.1 The Affine Hull and Dimension of QAPn 151
7.2 Some Valid Inequalities for QAPn 157
7.3 The Affine Hull and Dimension of SQPn 161
8 SOLVING SMALL QAPs 167
A FORTRAN PROGRAMS FOR SMALL SQPs 173
REFERENCES 205
INDEX 217
LIST OF FIGURES
1.1 A 5 x 5 plant-location assignment example 2
1.2 United States plant-location assignment example 15
1.3 Reduction of T in the U.S. example to increase sparsity 17
1.4 Section of the backboard of a Univac Solid-State Computer 27
2.1 A layout of a small condition-code circuit made up completely
of standard cells 39
2.2 Example of a sea-of-cells master 40
2.3 A 5 x 3 circuit layout design example 41
2.4 Cell placement in the sea-of-cells technology 42
2.5 A 5 x 3 task-processor assignment example 45
2.6 A 5 x 3 activity-facility assignment example 48
2.7 A 5 x 3 work-center assignment of operations example 51
2.8 A 6-node graph and its 2-partition 53
2.9 A 6-node complete graph and its 2-partition 55
2.10 A classification of BQPSs 58
4.1 Traditional and locally ideal linearizations of the BQP 80
5.1 The locally ideal linearization of CLDPs 107
5.2 The locally ideal linearization of the general model 113
5.3 The locally ideal linearization of QAPs 118
6.1 The partitioning of the inequalities (6.10), ... , (6.12) 137
7.1 The matrix F used in the proof of Proposition 7.3 163
7.2 Summary of the construction of the proof of Proposition 7.3 165
IX
LIST OF TABLES
=
1.1 Data for a Koopmans-Beckmann problem with n 5 U.S.
cities 16
1.2 The equation system (1.9) of size 44 x 95 for the U.S. example
=
with a 6 18
1.3 The inequality system (1.11) of size 50 x 95 for the U.S. ex-
=
ample with a 6 19
1.4 Reduction in problem size and LP values for the U.S. example 20
1.5 The 19 facilities, their functions and optimal locations 22
1.6 Distance and flow matrix for 19 facilities 23
1.7 Reduction in problem size and LP values for the hospital
layout example 24
1.8 Connection matrix and distance matrix in Manhattan-norm
for the wiring problem 29
1.9 Reduction in problem size and LP values for the wiring prob-
lem 30
= =
2.1 Data for a circuit layout design problem with m 5, n 3 43
= =
2.2 Data for a multi-processor problem with m 5, n 3 46
= =
2.3 Data for a class-room scheduling problem with m 5, n 3 49
= =
2.4 Data for an operations-scheduling problem with m 5, n 3 52
4.1 The feasible 0-1 vectors of the local polytope P of GPP 83
4.2 The feasible 0-1 vectors of the local polytope P of OSP 90
4.3 The feasible 0-1 vectors of the local polytope P of MPP 96
5.1 The feasible 0-1 vectors of the local polytope P of SQP 123
7.1 All cut inequalities needed for a complete description of QAP4 162
8.1 Computational results for super sparse QAPLIB problems 169
8.2 Computational results for some selected QAPLIB problems 170
Xl