Table Of ContentLecture Notes in Economics
and Mathematical Systems 508
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Springer-Verlag
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Klaus Neumann
Christoph Schwindt
JUrgen Zimmermann
Project Scheduling
with Time Windows
and Scarce Resources
Temporal and Resource-Constrained
Project Scheduling
with Regular and Nonregular
Objective Functions
Springer
Authors
Prof. Dr. Klaus Neumann
Dr. Christoph Schwindt
Dr. JUrgen Zimmermann
Institute for Economic Theory and
Operations Research
University of Karlsruhe
KaiserstraBe 12
76128 Karlsruhe, Germany
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Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Neumann, Klaus:
Project scheduling with time windows and scarce resources: temporal and
resource constrained project scheduling with regular and nonregular
objective functions I Klaus Neumann; Christoph Schwindt; liirgen
Zimmermann.
(Lecture notes in economics and mathematical systems; 508)
ISBN 978-3-540-42463-5 ISBN 978-3-662-22341-3 (eBook)
DOI 10.1007/978-3-662-22341-3
ISSN 0075-8450
ISBN 978-3-540-42463-5
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Preface
A project is a unique undertaking or endeavor to be accomplished that can
be divided into individual subtasks or activities each of which requires time
and scarce resources for its completion. Also there is a desired objective to
be attained (for example, minimization of project duration or of variation
of resource utilization, or maximization of net present value of the project).
Moreover, there are given precedence relationships among activities prescrib
ing the order in which activities must be carried out. Project scheduling, in
its basic form, consists of finding start times for all activities such that pre
scribed resource and precedence constraints are satisfied and an objective
function is optimized.
Since the concept of a project can be interpreted quite broadly, project
scheduling problems arise in a great variety of practical situations. These in
clude construction work, the development and introduction of new products,
service systems, or software packages, strategic long-term planning in manu
facturing and the service sector, emergency planning, and even the conduct
ing of political campaigns. Recently, project scheduling has been successfully
applied to production and operations management, e.g. make-to-order pro
duction in manufacturing and batch production in process industries.
Since the late 1950's, network-based planning methods for project sched
uling problems have been developed, cf. Elmaghraby (1977) and Moder et al.
(1983). Many solution methods, however, pertain only to temporal project
scheduling, that is, they do not allow for the limited availability of resources
in practice. Efficient procedures for solving large resource-constrained proj
ect scheduling problems have been proposed only recently.
This monograph grew out of two courses on project scheduling given by
one of the authors at the University of Karlsruhe, ~ermany, in recent years. It
reports on the state of the art of deterministic resource-constrained project
scheduling with time windows. Most results appearing here are published
in book form for the first time. The following features differentiate this
monograph from other texts on project scheduling published recently (for
example, Kolisch, 1995, Hartmann, 1999, W~glarz, 1999, or Klein, 2000):
1. In addition to minimum time lags between project activities (which are
a generalization of precedence relationships), maximum time lags are
considered from the beginning. In other words, (schedule-dependent)
vi Preface
time windows for activities are given. We model projects with mini
mum and maximum time lags by (cyclic) activity-on-node networks.
Maximum time lags have been discussed in the literature only rarely.
However, they are needed for modelling a variety of practical require
ments. These include deadlines for subprojects or individual activities,
simultaneous or non-delay execution of several activities, prescribed
time windows for resources, and the processing of perishable chemical
substances within prescribed time intervals.
2. In addition to the minimization of project duration, also known as
makespan, many other objective functions that are important in prac
tice are studied. Most of the objective functions are nonregular, i.e. not
increasing in the start times of activities. For instance, we discuss max
imizing the net present value of the project or minimizing: the earliness
plus tardiness cost of activities with prescribed due dates, the cost of
resources purchased or rented, or the variation of resource utilization
over time. Exact algorithms and different types of heuristic solution
methods are proposed, which can solve large problem instances.
3. An order-based structural analysis of the feasible region of the sched
uling problems considered turns out to be very useful for studying dif
ferent kinds of schedules. We also classify regular and nonregular ob
jective functions and develop efficient schedule generation schemes for
project scheduling problems with different types of objective functions
exploiting the structural analysis of the feasible region.
4. Some interesting applications of resource-constrained project schedul
ing to production and operations management, in particular, to make
to-order and batch production, are investigated. To aid in modelling
and solving such problems, we discuss renewable resources (such as
machines, manpower, or equipment), nonrenewable resources (e.g. cap
ital or energy), and so-called cumulative resources (storage facilities)
as well as multi-mode project scheduling and the case of calendariza
tion, that is, the existence of specific time intervals, like weekends or
holidays, where certain resources are not available.
The contents of this book are as follows: In Chapter 1, we introduce
some basic concepts (for example, minimum and maximum time lags and
activity-on-node project networks) and deal with temporal project schedul
ing. Chapter 2 is devoted to resource-constrained project scheduling with
minimization of project duration and other regular objective functions. After
discussing the structure of the feasible region and different sets of schedules,
a branch-and-bound algorithm and several heuristic solution procedures,
as well as an experimental performance analysis, are presented. Also, ap
plications to production and operations management and special features
needed for such applications (e.g. cumulative resources, sequence-dependent
Preface vii
changeover times, calendarization, and multi-mode project scheduling) are
studied. Chapter 3 is concerned with resource-constrained project scheduling
with nonregular objective functions. A classification of objective functions
is proposed, and we discuss which types of schedules (and the correspond
ing special points of the feasible region) may represent possible optimal so
lutions to project scheduling problems with different nonregular objective
functions. After that, exact and heuristic procedures for solving such project
scheduling problems are proposed and an experimental performance analysis
is presented.
The monograph assumes a knowledge of elementary mathematical pro
gramming and some basic concepts from the theory of graphs and networks.
Most of these concepts are alluded to before they are used, which makes the
book self-contained.
The monograph contains much research that was supported, in part, by
the Deutsche Forschungsgemeinschaft. Company SAP AG was gracious to
provide support for the research on batch scheduling in process industries.
We would also like to thank our colleagues at Karlsruhe: Iris Lechleiter,
Thomas Hartung, Roland Heilmann, Thomas Selle, and Norbert Trautmann,
who read the manuscript and suggested valuable improvements. In addition,
we like to acknowledge the advice and support of Sigrid Knust, Peter Brucker,
Bajis Dodin, Ulrich Dorndorf, Andreas Drexl, Salah E. Elmaghraby, Rolf
Mohring, Erwin Pesch, Siegfried Schaible, Frederik Stork, and Marc Uetz.
Lembach/ Alsace Klaus Neumann
May 2001 Christoph Schwindt
Jiirgen Zimmermann
Contents
1 Temporal Project Scheduling 1
1.1 Minimum and maximum time lags. 1
1.2 Activity-on-node project networks 6
1.3 Temporal project scheduling computations 9
1.4 Orders in the set of activities ...... . 15
2 Resource-Constrained Project Scheduling - Minimization
of Project Duration 21
2.1 Formulation of the problem .............. . 24
2.2 Cycle structures in activity-on-node project networks. 26
2.3 Properties of the feasible region . . . . . . . . . . . . . 29
2.3.1 Strict orders and order polyhedra ....... . 30
2.3.2 Forbidden sets and resolution of resource conflicts 33
2.4 Different types of shifts and sets of schedules ...... . 38
2.5 Branch-and-bound and truncated branch-and-bound methods 45
2.5.1 Enumeration scheme 45
2.5.2 Preprocessing ........ . 53
2.5.3 Lower bounds ........ . 59
2.5.4 Branch-and-bound algorithm 62
2.5.5 Truncated branch-and-bound methods 65
2.5.6 Alternative enumeration schemes ... 69
2.5.7 Alternative preprocessing and constraint propagation 71
2.5.8 Alternative lower bounds. 74
2.6 Priority-rule methods. . . . . . 80
2.6.1 Direct method ..... . 81
2.6.2 Decomposition methods 82
2.6.3 Priority rules . . . . . . 83
2.6.4 Serial generation scheme 84
2.6.5 Parallel generation scheme 90
2.7 Schedule-improvement procedures . 93
2.7.1 .. Genetic algorithm . . . . . . 94
2.7.2 Tabu search ........ . 97
2.8 Experimental performance analysis · 101
2.8.1 Random generation of projects · 101
2.8.2 Computational experience . . . · 105
x Contents
2.9 Application to make-to-order production in manufacturing in-
dustry . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 111
2.10 Regular objective functions different from project duration . 117
2.11 Calendarization . . . . . . . . . . . . . . . . . . . . . . .. . 120
2.12 Project scheduling with cumulative resources. . . . . . .. . 127
2.13 Project scheduling with sequence-dependent changeover times 136
2.14 Multi-mode project scheduling problems . . . . . . 146
2.14.1 Problem formulation and basic properties. . . . 147
2.14.2 Solution methods . . . . . . . . . . . . . . . . . 153
2.15 Application to batch production in process industries . 163
2.15.1 Case study. . . . . . . . . . 164
2.15.2 Project scheduling model. . 168
2.15.3 Enumeration scheme . . . . 171
3 Resource-Constrained Project Scheduling - Minimization
of General Objective Functions 175
3.1 Different objective functions . . . . . . . . . . . . 177
3.2 Additional types of shifts and sets of schedules . . 185
3.3 Classification of objective functions . . . . . . . . 196
3.3.1 Separable and resource-utilization dependent
objective functions ........... . · 197
3.3.2 Class 1 of regular objective functions .. · 199
3.3.3 Class 2 of antiregular objective functions .200
3.3.4 Class 3 of convex objective functions . . · 201
3.3.5 Class 4 of binary-monotone objective functions .202
3.3.6 Class 5 of quasiconcave objective functions ... .203
3.3.7 Class 6 of locally regular objective functions .. .205
3.3.8 Class 7 of locally quasiconcave objective functions · 210
3.4 Time complexity of time-constrained project scheduling. · 214
3.5 Relaxation-based approach for function classes 1 to 5 . . · 223
3.5.1 General enumeration scheme ........... . .223
3.5.2 Branch-and-bound algorithm for the net present value
problem . . . . . . . . . . . . . . . . . 226
3.5.3 Branch-and-bound algorithm for the
earliness-tardiness problem ..... .243
3.6 Tr:ee-based approach for function classes 6 and 7 . .246
3.6.1 General enumeration scheme ....... . .247
3.6.2 Branch-and-bound algorithms for resource investment,
resource levelling, and resource renting problems . . 254
3.6.3 Experimental performance analysis . . 264
3.7 Priority-rule methods. . . . . . . . . . . . . . . . 265
3.7.1 Time-constrained project scheduling . . . 266
3.7.2 Resource-constrained project scheduling . 277
3.7.3 Experimental performance analysis . . . . 286
