Table Of ContentNORTH-HOLLAND MATHEMATICS STUDIES 132
Annals of Discrete Mathematics (31)
General Editor: Peter L. HAMMER
Rutgers University, New Brunswick, NJ, U.S .A.
Advisory Editors
C. BERGE, Universite de Paris, France
M. A. HARRISON, University of California, Berkeley, CA, U.S.A.
V: KLEE, University of Washington, Seattle, WA, U.S.A.
J. -H. VAN LINT, California Institute of Technology,P asadena, CA, U.S.A.
G. -C.R OTA, Massachusetts Institute of Technology,C ambridge, MA, U.S .A.
NORTH-HOLLAND -AMSTERDAM NEW YORK 'OXFORD 'TOKYO
SURVEYS IN
COM BI NATORIAL OPTIMIZATION
Edited by
Silvano MARTELLO
DEIS - University of Bologna
Italy
Gilbert LAPORTE
Ecole des H. E. C.
Montreal, Quebec, Canada
Michel MINOUX
CNET
Paris, France
Celso RlBElRO
Catholic University of Rio de Janeiro
Brazil
1987
NORTH-HOLLAND -AMSTERDAM .NEW YORK 'OXFORD ,TOKYO
Elsevier Science Publishers E.V., 1987
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ISBN: 0 444 70136 2
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ilrr:~r\ (4( ',tllgrcr\( 'ai;illigiiig-iii-Pl,blicatinll h a
Surveys in corbinatorinl optimizntion.
(iiorth-IIolland mathematics fitdies ; 13 ) (Anrmls
of discrete rrnti~entitics ; 31)
"2ased on a series of tutorial lectures i4vt-n at the
School on Con'ciiatorial Qtirnizat.icr1, held at the Federal
"rivrrsity cf riio de Janeiro, nrezil, July L-19, 190i"--
CIP fwd.
1. Coirb-liiatorial oFtici;x.ticn. 1. Martello, Silvano.
11. Series. 111. Gerics: Aiinals rif discrete
nathematics ; 31.
C$4OU.5.S..5 lC,~'j 519 ;I -. 41.
ICBK 0-444-701j
PRINTED IN THE NETHERLANDS
V
FOREWORD
This book is based on a series of tutorial lectures given at the School on Combi-
natorial Optimization. held at the Federal University of Rio de Janeiro, Brazil,
July 8 - 19, 198.5.
Over 100 participants benefitted from the high quality of the tutorial lectures
and technical sessions. This event was the first of its kind held in Latin America
and undoubtedly contributed to the diffusion of knowledge in the field of Com-
binato rial Opt im iza tion.
I would like to take this opportunity to acknowledge with pleasure the efforts
and assistance provided by Professors Ruy Eduardo Campello, Gerd Finke, Peter
Hammer, Gilbert Laporte, Silvuno Martello, Michel Minoux and Celso Ribeiro in
the organization of the School. I am also very grateful to CNPq Conselho
-
Nacional de Desenvolvimento Cientifico e Tecnoldgico and to FINEP Financia-
-
dora de Estudos e Projetos of the Brazilian Ministry of Science and Technology,
who provided financial support for this stimulating meeting.
Nelson Maculan, chairman
Rio de Janeiro, 1985
Vii
PREFACE
Ever since the field of Mathematical Programming was born with the discovery
of the Simplex method by G.B. Dantzig in the late 1940s, researchers have
devoted considerable attention to optimization problems in which the variables
are constrained to take only integer values, or values from a finite or discrete
set of numbers.
After the first significant contributions of R.E. Gomory to the field of Integer
Programming, discrete optimization problems were to attract growing interest,
partly because of the fact that a large number of practical applications of Opera-
tions Research techniques involved integrality constraints, and partly because
of the theoretical challenge.
During the same period, Graph Theory had been developing, providing a natural
and effective conceptual framework for expressing and sometimes solving combi-
natorial problems commonly encountered in applications, thus giving rise to
network flow theory, matroid theory and complexity theory.
What is nowadays referred to as Combinatorial Optimization therefore derives
from the combination and cross-fertilization of various research streams, all
developed somewhat independently over the past two or three decades (such as
Graph Theory, Integer Programming, Combinatorics and Discrete Mathematics).
The present volume is an attempt to provide a synthetic overview of a number
of important directions in which Combinatorial Optimization is currently deve-
loping, in the for& of a collection of survey papers providing detailed accounts of
recent progress over the past few years.
A number of these papers focus more specifically on theoretical aspects
and fundamental tools of Combinatorial Optimization: ((Boolean Programming))
and ((Probabilistic Analysis of Algorithms)). From the computational point of
view, a good account of recent work on the use of vector processing and parallel
computers for implementing algorithms will be found in the paper on ((Parallel
Computer Models and Combinatorial Algorithms)).
In addition, substantial space has been allocated to a number of well-known
problems, which have some relation with applications, but which are addressed
here as prototypes of hard-to-solve combinatorial problems, and which, as such,
have traditionally acted as stimulants for research in the field: we refer, in parti-
cular, to the papers on ((The Linear Assignment Problem ((The Quadratic
>),
Assignment Problem)), ((The Knapsack Problem)) and ((The Steiner Problem in
Graphs)).
Besides these, it also seemed important that the wide applicability of the
techniques and algorithmic tools developed in the field of Combinatorial Optimi-
Viii Prefoce
zation be illustrated with a number of more complicated models chosen for their
technical, industrial or economic importance. Accordingly, the reader will find
a number of applications oriented papers devoted to combinatorial problems
arising in such practical contexts as: communication networks (((Network Synthe-
sis and Dynamic Network Optimization))), location and routing (((Single Facility
Location Problems on Networks)) and ((The Vehicle Routing Problem))), manufac-
turing and planning in production systems (((Scheduling Problems))).
All the survey papers included herein have been written by well-known specia-
lists in the field, with particular emphasis on pedagogical quality (in this respect,
special mention should be given to the bibliography which contains almost
1000 titles) and, as far as possible, completeness.
The Editors wish to thank the participants of the School on Combinatorial
Optimization for their valuable comments and criticisms on preliminary versions
of the papers. The financial support of the Federal University of Rio de Janeiro,
the National Research Council of Italy and the ficole des Hautes etudes Commer-
ciales de MontrCal is gratefully acknowledged.
We hope that this volume will be considered as a milestone in the rich and
fast evolving field of Combinatorial Optimization.
Silvano Martello
Gilbert Laporte
Michel Minoux
Celso Ribeiro
Annals of Discrete Mathematics 31 (1987) 1 - 60
0 Elsevier Science Publishers B.V. (North-Holland)
SELECTED TOPICS IN SCHEDULING THEORY
Jacek BZAZEWICZ
1. Introduction
This study is concerned with deterministic problems of scheduling tasks on
machines (processors), which is one of the most rapidly expanding areas of
combinatorial optimization. In general terms, these problems may be stated as
follows. A given set of tasks is to be processed on a set of available processors,
so that all processing conditions are satisfied and a certain objective function
is minimized (or maximized). It is assumed, in contrast to stochastic scheduling
problems, that all task parameters are known a priori in a deterministic way.
This assumption, as will be pointed out later, is well justified in many practical
situations. On the other hand, it permits the solving of scheduling problems
having a different practical interpretation from that of the stochastic approach.
(For a recent survey of stochastic scheduling problems see [117, 1181). This
interpretation is at least a valuable complement to the stochastic analysis and
is often imposed by certain applications as, for example, in computer control
systems working in a hard -real -time environment and in many industrial applica-
tions. In the following we will emphasize the applications of this model in com-
puter systems (scheduling on parallel processors, task preemptions). However, we
will also point out some other interpretations, since tasks and machines may
also represent ships and dockyards, classes and teachers, patients and hospital
equipment, or dinners and cooks. To illustrate some notions let us consider
the following example.
Let us assume that our goal is to perform five tasks on three parallel identical
processors in the shortest possible time. Let the processing times of the tasks
be 3, 5, 2, 1, 2, respectively, and suppose that the tasks are to be processed
without preemptions, i.e. each task once started must be processed until com-
pletion without any break on the same processor. A solution (an optimal schedu-
le) is given in Fig. 1. I, where a Gantt chart is used to present the assignment of
the processore to the tasks in time. We see that the minimum processing time
for these five tasks is 5 units and the presented schedule is not the only possible
solution. In this case, the solution has been obtained easily. However, this is not
generally the case and, to construct an optimal schedule, one has to use special
algorithms depending on the problem in question.
1
2 J. BIaiewicz
Processors
~
PI
T2
I
p2 T2
Idle time
p3 TI
c
I I
In this paper we would like to present some important topics in scheduling
theory. First, and interpretation of its assumptions and results, especially in
computer systems, is described and some other applications are given. Then a
general approach to the analysis of scheduling problems is presented in detail
and illustrated with an example. In the next two sections, the most important
results, in our opinion, concerning the classical scheduling problems with parallel
machines and in job systems, are summarized. These include "-hardness results
and optimization as well as approximation algorithms with their accuracy evalua-
tion (mean or in the worst case). Because of the limited space we have not been
able to describe all the existing results. Our choice has been motivated first
by the importance of a particular result, e.g. in the case of algorithms, their
power to solve more than one particular scheduling problem. This is, for example,
the case with the level algorithm, network flow or linear programming approaches.
Secondly, our choice has been influenced by the relevance of the results present-
ed to a computer scheduling problem. Hence, not much attention has been
paid to enumerative methods. However, in all the considered situations we refer
to the existing surveys which cover the above issues in a very detailed way. New
directions in scheduling theory are then presented, among which special attention
is paid to scheduling under resource constraints and scheduling in microprocessor
systems. An Appendix for the notations of scheduling problems can be found at
the end of this study.
2. Basic Notions
2.1. Problem formulation
In general, we will be concerned with two sets: a set of n tasks ."7 =
= { T,, T,, . . . , T,,} and a set of m processors 9={P I,P z, . . . , P,}. There are two
general rules to be followed in classical scheduling theory. Each task is to be
processed by at most one processor at a time and each processor is capable of
processing at most one task at a time. In Section 6 we will show some new applica-
tions in which the first constraint will be removed.
We will now characterize the processors. They may be either parallel, perform-
Selected topics in scheduling theory 3
ing the same functions, or dedicated, i.e. specialized for the execution of certain
tasks. Three types of parallel processors are distinguished depending on their
speeds. If all processors from set 9h ave equal task-processing speeds, then we call
them identical processors. If the processors differ in their speeds, but the speed
of each processor is constant and does not depend on the tasks of cTt,h en they
are called uniform. Finally, if the speeds of the processors depend on the partic-
lar task which is processed, then they are called unrelated.
In the case of dedicated processors, there are three modes of processing:
flow shop, open shop and job shop systems. In a flow shop system each task
from set cym ust be processed by all the processors in the same order. In an
open slzop system each task must also be processed by all processors, but the
order of processing is not given. In a job shop system the subset of processors
which are to process a task and the order of the processing are arbitrary, but must
be specified a priori.
We will now describe the task set in detail. In the case of parallel processors
each task can be processed by any given processor. When tasks are to be processed
5
on dedicated processors, then each E y i s d ivided into operations Oil, Oiz, . . . ,
Ojkj, each of which may require a different processor. Moreover, in a flow shop
system, the number of operations per task is equal to m and their order of pro-
<,
cessing is such that Oil is processed on processor Ojz on 4,a nd so on. Moreover,
the processing of Oj,i-, must always precede the processing of Oii, j = 1, 2, . . . , n,
i = 2,. . . , m. In the case of an open shop system, each task is also divided into
m operations with the same allocation of the operations to processors, but the
order of processing operations is not specified. In ajob shop system, the number
of operations per task, their assignment to processors and the order of their
processing, are arbitrary but must be given in advance.
Task E ,?is characterized by the following data.
1. A vector of processing times - fi = [pjl,p jz,. . . ,pi,], where pii is the
q.
time needed by processor P, to complete In the case of identical processors
we have pji = pi, i = 1, 2, . . . , m. If the processors in 9 are uniform then pji =
= pi/bi, i = 1, 2, . . . , m, where pi is a standard processing time (usually on the
4..
slowest processor) and bi is a processing speed factor of processor
2. An arrival time (a ready time) - ri, which is the time at which 1; is ready
for processing. If for all tasks from .Fthe arrival times are equal, then it is assum-
ed that rj = 0,j = 1, 2, . . . , n.
3. A due-date -di. If the processing of must be completed at time di,
then dj is called the deadline.
q.
4. A weight (priority) - wj,w hich expresses the relative urgency of
Below, some definitions concerning task preemptions and precedence con-
straints among tasks are given.
The mode of processing is called preemptive if each task (each operation in
the case of dedicated processors) may be preempted at any time and restarted
later at no cost, perhaps on another processor. If the preemption of any task is not
4 .I. Blaiewicz
allowed we will call the scheduling nonpreemptive.
In set Fprecedence constraints among tasks may be defined. 7; < 7; means
T q
that the processing of must be completed before can be started. In other
words, set cyis ordered by a binary relations <. The tasks in set Fare called
dependent if at least two tasks in cTa re ordered by this relation. Otherwise,
the tasks are called independent. A task set ordered by the precedence relation
is usually represented as a directed graph (a digraph) in which nodes correspond
to tasks and arcs to precedence constraints. An example of a dependent task
set is shown in Fig. 2.1 (nodes are denoted by ?/pi). Let us notice that in the
case of dedicated processors (except in the open shop system) operations that
constitute a task are always dependent, but tasks can be either independent or
dependent. It is also possible to represent precedence constraints as a task-on-arc
graph, which is also called an activity network representation. Usually, the first
approach is used, but sometimes the second presentation may be useful and we
q
will mention these cases in what follows. Task will be called available at
moment t if ri S t and all its predecessors (with respect to the precednece con-
traints) have been completed at time t.
Now we will give the definitions concerning schedules and optimality criteria.
A schedule is an assignment of processors from set 9 to tasks from set 9 in
time such that the following conditions are satisfied:
- at every moment each processor is assigned to at .most one task and each
task is processed by at most one processor;
- task q is processed in time interval [ri, DO);
- all tasks are completed;
- for each pair T,7 ;, such that q. < Ti, the processing of ';. is started after the
completion of T, ;
- in the case of nonpreemptive scheduling no task is preempted (the schedule
is called nonpreemptive), otherwise the number of preemptions of each task is
Fig. 2.1. An example of a task set.
