Multicriteria Scheduling : Theory, Models and Algorithms

369 Pages·2006·18.25 MB·English

by  Vincent T'kindt

Multicriteria Scheduling
Second Edition
Vincent T’kindt
Jean-Charles Billaut
Multicriteria
Scheduling
Theory,Models
and Algorithms
Translated from French by Henry Scott
Second Edition
with 138 Figures
and 15 Tables
123
Associate Professor Vincent T’kindt,
Professor Jean-Charles Billaut
Université François-Rabelais de Tours
Laboratoire d’Informatique
64 avenue Jean Portalis
37200 Tours
France
Translator
Henry Scott
www.hgs-scientific-translations.co.uk
Cataloging-in-Publication Data
Library ofCongress Control Number:2005937590
ISBN-10 3-540-28230-0 2nd ed. Springer Berlin Heidelberg New York
ISBN-13 978-3-540-28230-3 2nd ed. Springer Berlin Heidelberg New York
ISBN 3-540-43617-0 1st ed. Springer Berlin Heidelberg New York
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Preface to the second edition
It is a real pleasure for us to present the second edition of this book on multi-
criteria scheduling. In this preface we would like to introduce the reader with
the improvements made over the first edition. During the writing of the first
edition of this book we were focused on putting in it all the results, algorithms
and models necessary for the reader to tackle correctly the field of multicri-
teria scheduling, which is at the crossroad of several research domains: from
multicriteria optimisation to scheduling. Writing a second edition is a totally
different exercise since we concentrate more on refining, augmenting and, in
a sense, making growing the existing manuscript.
We received valuable comments that lead us to rewrite, more or less partially,
some chapters as Chapters 5 and 7. Besides, new significant research results
published since the first edition have been included into existing chapters of
that second edition. We review hereafter the most important changes.
Chapters 2 and 4 now include a survey on the complexity of counting and enu
meration optimisation problems with application to multicriteria scheduling.
These two chapters provide theoretical tools for evaluating the complexity of
the enumeration of the set of strict Pareto optima. Chapter 4 also includes
new real-life applications of multicriteria scheduling.
Chapter 5 has been drastically revised and now provides a general unified
framework for Just-in-Time scheduling problems. Besides, classic optimal
timing algorithms, which calculate optimal start times of operations when
the jobs order is fixed, are now presented.
At last, chapter 6 is a new chapter dealing with robustness in multicriteria
scheduling. This research area has been subject to a growing interest in the
literature since the last ten years, notably when considering a criterion of
flexibility or robustness in addition to a classic scheduling criterion. Hence
forth, the aim of some scheduling problems become to increase the robustness
of the calculated solution for its pratical use. Providing flexibility is a way to
ensure a certain robustness when unexpected events occur in the shop.
We hope that this new edition will become an important tool and a practical
guide for novice an senior researchers that work on multicriteria scheduling.
V. T'KINDT and J.-C. BILLAUT
Tours (Prance), October 15th 2005
Preface to the first edition
Prom Theory to Practice, there is a world, and scheduUng does not escape
this immutable rule.
For more than fifty years, theoretical researches on scheduling and complexity
theory have improved our knowledge on both a typology of academic prob
lems, mainly involving a single criterion, and on their solving. Though this
work is far from being completed, a few famous books have been a major
breakthrough. The typology will be all the more useful as it takes more and
more realistic constraints into account. This is just a matter of time.
The relevance of some single criteria, their equivalence and their conflict have
been studied...
Yet, numerous genuine problems, even outside the realm of scheduling, do not
square with these single criterion approaches. For example, in a production
shop, minimising the completion time of a set of jobs may be as interesting as
retaining a maximum fragmentation of idle times on an easily damaged ma
chine and minimising the storage of in-process orders. Moreover, even though
the optimal solutions to the F2\\Cmax yielded by S.M. Johnson's famous al
gorithm are numerous, they are far from appearing equivalent to the decision
maker when their structure is analysed. A genuine scheduling problem, in
essence, involves multiple criteria.
Besides, more general books on Decision Aid in a multicriteria environment
have been published and a pool of researchers have long tackled the problem.
Undoubtedly, a synthesis book offering a state-of-the-art on the intersection
of both the fields of Scheduling and Multicriteria Decision Aid and providing
a framework for tackling multicriteria scheduling problems is a must.
I am most happy to present this book. It is divided in four parts: - the first
one deals with research on scheduling, now an important branch of opera
tional research.
- the second one presents theories on Decision Aid and Multicriteria Optimi
sation as well as a framework for the resolution of multicriteria scheduling
problems.
VIII Preface
- the third and fourth parts involve a tremendous work since they contain
state-of-the-arts on multicriteria scheduhng problems. Numerous works and
resolution algorithms are detailed.
In my opinion, this book will become a reference book for researchers working
on scheduling. Moreover, I am convinced it will help PhD students suitably
and quickly embark on a fascinating adventure in this branch of Operational
Research. May they be numerous in joining us...
I very warmly thank MM. Vincent T'kindt and Jean-Charles Billaut for their
tenacity in writing this significant book, and Springer-Verlag publishing for
entrusting them.
Professor C. PROUST
Tours (Prance), february 22th 2002
The authors are very grateful to all the people who have directly or indirectly
contributed to the birth of this book. Professor Christian Proust is at the root
of this research and is undoubtedly the grandfather of this book. We would also
like to thank the members of the research team "Scheduling and Control" of the
Laboratory of Computer Science of the University of Tours for creating a friendly
environment and thus for having promoted the emergence of this book. In this
vein, all the technical and administrative persons of the E3i school have also to be
thanked.
At last, we would like to thank Professor Jacques Teghem of the "Faculte Poly tech
nique de Mons" for having provided excellent ideas and remarks which have helped
in improving this book.
Contents
1. Introduction to scheduling 5
1.1 Definition 5
1.2 Some areas of application 6
1.2.1 Problems related to production 6
1.2.2 Other problems 7
1.3 Shop environments 7
1.3.1 Scheduling problems without assignment 8
1.3.2 Scheduling and assignment problems with stages 8
1.3.3 General scheduling and assignment problems 9
1.4 Constraints 9
1.5 Optimality criteria 12
1.5.1 Minimisation of a maximum function: "minimax" cri
teria 13
1.5.2 Minimisation of a sum function: "minisum" criteria ... 13
1.6 Typologies and notation of problems 14
1.6.1 Typologies of problems 14
1.6.2 Notation of problems 16
1.7 Project scheduling problems 17
1.8 Some fundamental notions 18
1.9 Basic scheduling algorithms 21
1.9.1 Scheduling rules 21
1.9.2 Some classical scheduling algorithms 22
2. Complexity of problems and algorithms 29
2.1 Complexity of algorithms 29
2.2 Complexity of problems 32
2.2.1 The complexity of decision problems 33
2.2.2 The complexity of optimisation problems 38
2.2.3 The complexity of counting and enumeration problems 40
2.3 Application to scheduling 48
3. Multicriteria optimisation theory 53
3.1 MCDA and MCDM: the context 53
3.1.1 MultiCriteria Decision Making 54
X Contents
3.1.2 MultiCriteria Decision Aid 54
3.2 Presentation of multicriteria optimisation theory 55
3.3 Definition of optimality 57
3.4 Geometric interpretation using dominance cones 60
3.5 Classes of resolution methods 62
3.6 Determination of Pareto optima 64
3.6.1 Determination by convex combination of criteria 64
3.6.2 Determination by parametric analysis 70
3.6.3 Determination by means of the e-constraint approach . 72
3.6.4 Use of the Tchebycheff metric 76
3.6.5 Use of the weighted Tchebycheff metric 79
3.6.6 Use of the augmented weighted Tchebycheff metric ... 81
3.6.7 Determination by the goal-attainment approach 86
3.6.8 Other methods for determining Pareto optima 91
3.7 Multicriteria Linear Programming (MLP) 92
3.7.1 Initial results 93
3.7.2 AppHcation of the previous results 93
3.8 Multicriteria Mixed Integer Programming (MMIP) 94
3.8.1 Initial results 94
3.8.2 Application of the previous results 95
3.8.3 Some classical algorithms 97
3.9 The complexity of multicriteria problems 100
3.9.1 Complexity results related to the solutions 100
3.9.2 Complexity results related to objective functions 101
3.9.3 Summary 106
3.10 Interactive methods 107
3.11 Goal programming 108
3.11.1 Archimedian goal programming Ill
3.11.2 Lexicographical goal programming Ill
3.11.3 Interactive goal programming Ill
3.11.4 Reference goal programming 112
3.11.5 Multicriteria goal programming 112
4. An approach to multicriteria scheduling problems 113
4.1 Justification of the study 113
4.1.1 Motivations 113
4.1.2 Some examples 114
4.2 Presentation of the approach 118
4.2.1 Definitions 118
4.2.2 Notation of multicriteria scheduling problems 121
4.3 Classes of resolution methods 122
4.4 Application of the process - an example 123
4.5 Some complexity results for multicriteria scheduling problems 124
Contents XI
5. Just-in-Time scheduling problems 135
5.1 Presentation of Just-in-Time (JiT) scheduling problems 135
5.2 Typology of JiT scheduling problems 136
5.2.1 Definition of the due dates 136
5.2.2 Definition of the JiT criteria 137
5.3 A new approach for JiT scheduling 139
5.3.1 Modelling of production costs in JiT scheduling for
shop problems 141
5.3.2 Links with objective functions of classic JiT scheduling 145
5.4 Optimal timing problems 147
5.4.1 The l\di,seq\Fe{f'',E^) problem 147
5.4.2 The Poo\prec, fi convex\ ^^ fi problem 149
5.4.3 The l\fi piecewise linear\Fi{Y^^ fi^ ^. 7^) problem ... 153
5.5 Polynomially solvable problems 153
5.5.1 The l\di = d> Y.Vi\F(>{E,f) problem 153
5.5.2 The l\di = d unknown^nmit\F£{E^T^d) problem 155
5.5.3 The l\pi C [pijpj HN, di = d non restrictive\Fe(E,T,
CC"^) problem ^ ._^ 157
5.5.4 The P\di = d non restrictive^nmit\F£{E^T) problem . 157
5.5.5 The P\di = d unknown^ nmit\Fe{E^T) problem 159
5.5.6 The P\di = d unknown,pi = p,nmit\F(>{E, T^d)
problem 165
5.5.7 The R\pi^j € [Pi,j;Pij],cfi = d unknown\Fi{T,E,
CC"^) problem 169
5.5.8 Other problems 170
5.6 TVP-hard problems 173
5.6.1 The l\di, nmit\Fe(E'',T^)_pioblem 173
5.6.2 The F\prmu,di,nmit\Fe{E'^,T^) problem 176
5.6.3 The P\di = d non restrictive, nmit\fmax{E , T )
problem 178
5.6.4 Other problems 182
5.7 Open problems 188
5.7.1 The Q\di = d unknown, nmit\Fi{E,T) problem 188
5.7.2 Other problems 189
6. Robustness considerations 193
6.1 Introduction to flexibility and robustness in scheduling 193
6.2 Approaches that introduce sequential flexibility 195
6.2.1 Groups of permutable operations 195
6.2.2 Partial order between operations 197
6.2.3 Interval structures 199
6.3 Single machine problems 201
6.3.1 Stability vs makespan 201
6.3.2 Robust evaluation vs distance to a baseline solution... 202