Advances in Project Scheduling

526 Pages·1989·8.956 MB·English

by  R. Slowinski and J. Weglarz (Eds.)

STUDIES IN PRODUCTION AND ENGINEERING ECONOMICS
Edited by Professor R. W. Grubbstrom, Department of Production Economics, Linkoping Institu-
te of Technology, S-58182 Linkoping, Sweden.
Vol. 1 Production Control and Information Systems for Component-Manufacturing Shops
(Bertrand and Wortmann)
Vol. 2 The Economics and Management of Inventories. Proceedings of the First International
Symposium on Inventories, Budapest, September 1-5,1980. Part A: Inventories in the
National Economy.
Part B. Inventory Management; Mathematical Models of Inventories (Chikan, Editor)
Vol. 3 New Results in Inventory Research. Proceedings of the Second International Symposium
on Inventories, Budapest, August 23-27,1982 (Chikan, Editor)
Vol. 4 Production Economics-Trends and Issues (Grubbstrom and Hinterhuber, Editors)
Vol. 5 Hierarchical Spare Inventory Systems (Petrovic, Senborn, Vujosevic)
Vol. 6 Inventory in Theory and Practice (Chikan, Editor)
Vol. 7 The Economics of Inventory Management (Chikan and Lovell, Editors)
Vol. 8 Multiple Criteria Decision Making in Industry (Tabucanon)
Vol. 9 Advances in Project Scheduling (Stowihski andWeglarz, Editors)
T
ADVANCES IN
PROJECT SCHEDULING
edited by
ROMAN StOWINSKI
and
JAN WEGLARZ
Institute of Control Engineering
Technical University of Poznari
Poznah, Poland
ELSEVIER
AMSTERDAM - OXFORD - NEW YORK -TOKYO 1989
ELSEVIER SCIENCE PUBLISHERS B.V.
Sara Burgerhartstraat 25,
P.O. Box 1991,1000 BZ Amsterdam, The Netherlands
Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC.
655 Avenue of the Americas,
New York, NY, 10010, U.S.A.
Library of Congress Cataloglng-in-Publication Data
Advances in project scheduling / edited by Roman Siowinski and Jan
Wgg larz.
p. cm. — (Studies in production and engineering economics ;
9)
ISBN 0-444-87358-9
1. Scheduling (Management) I. S-fowinski, Roman. II. Weglarz,
Jan. III. Series.
TS157.5.A38 1989
658.5'3—dc19 89-1269
CIP
ISBN 0-444-87358-9 (Vol. 9)
ISBN 0-444-41963-2 (Series)
© Elsevier Science Publishers B.V., 1989
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, electronic, mechanical, photocopying, recording
or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers
B.V./Physical Sciences and Engineering Division, P.O. Box 1991,1000 BZ Amsterdam, The Nether-
lands.
Special regulations for readers in the U.S.A. - This publication has been registered with the
Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained
from the CCC about conditions under which photocopies of parts of this publication may be
made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A.,
should be referred to the copyright owner, Elsevier Science Publishers B.V., unless otherwise
specified.
No responsibility is assumed by the publisher for any injury and/or damage to persons or
property as a matter of products liability, negligence or otherwise, or from any use or operation
of any methods, products, instructions or ideas contained in the material herein.
Printed in The Netherlands
V
PREFACE
The main goal of this book is to make an overview of main
recent approaches and methods for modelling and solving project
scheduling problems. Methodological aspects are stressed, such as
interpretation of assumptions made in the models, properties of
schedules and algorithms for finding them, interconnections with
other problems of scheduling Ci.e. machine, production]) and
resource allocation C static or dynamic!). It does not mean that
practical aspects are ignored. It simply means that we do not
present arbitrarily or heuristically chosen methods for solving
more or less efficiently some specific cases, but we rather show a
methodology of choosing methods Cexact, heuristic and even
analytical!) which are most adequate and efficient for solving a
given class of problems. In this v/ay, the book gives a background
for building methodologically correct decision support systems for
a large class of scheduling problems.
The material is divided into three parts: two dealing with
deterministic models, and one with stochastic ones. The first part
is devoted to models which are basic in the sense that they
preserve some traditional assumptions made in classical project
scheduling models Cnon-preempti ve schedules, discrete resource
requirements!). However, they considerably exceed the classical
framework, e.g. by considering different categories of resources
Crenewable, nonrenewable and doubly constrained!), many alternative
modes of performing activities and many project performance
measures considered in single- or multi-objective optimization.
In the second part, there are gathered chapters either relaxing
the basic assumptions Cpreemptive scheduling with the minimization
of the number of preemptions, continuous resource requirements!) or
lying near to the conventional border of project scheduling from
the side of machine/production scheduling or resource allocation
problems. The goal of this part is on the one hand to characterize
methods for solving non-classical models which are important in
some modern practical applications Ce.g. computer scheduling), and
on the other hand, to show advantages of relaxing the basic
vi Preface
assumptions mentioned above. It may lead to polynomial Cincluding
linear programming} algorithms for finding non-preemptive schedules
in some cases, or even to analytical solutions when resource
requirements can be treated as continuous, i.e. arbitrary within
given intervals. The third part deals with basic concepts
concerning stochastic and GERT networks.
The potential readership of the book follows from its
methodological character, however strictly related to real-world
problems and emphasizing algorithmic aspects of the methods
presented. Thus, it includes researchers and graduate or advanced
undergraduate students within the following disciplines: Business
Administration, Management Science, Operations Research, Industrial
Engineering, System Analysis and Control, Computer Science and
Applied Mathematics. It should be also useful for decision makers,
executive managers, consultants and practitionners in business
corporations, service industries, governmental organisations etc.
We should like to thank all the authors for their contributions
and the fruitful cooperation, as well as other colleagues in the
field whose results and/or opinions stimulated us to undertake and
perform this editorial job.
Roman Siowihski and Jan Weglarz
Poznah, September 1988
Advances in Project Scheduling, edited by R. Stowinski and J. Weglarz
Elsevier Science Publishers B.V., Amsterdam, 1989 - Printed in The Netherlands 3
PART I
Chapter 1
AN ALGORITHM FOR A GENERAL CLASS OF PRECEDENCE AND RESOURCE
CONSTRAINED SCHEDULING PROBLEMS
J.H. PATTERSON1, R. SLOWINSKI2, F. B. TALBOT3, J. WEGLARZ2
±
Indiana University, Bloomington, IN. 47401 , U.S.A.
Technical University of Poznah, 60-965 Poznan, Poland
3University of Michigan, Ann Arbor, MI. 48109, U.S.A.
1 INTRODUCTION
This chapter presents a simple, yet general, backtracking
algorithm for heuristically or optimally solving precedence- and
resource-constrained scheduling problems. In order to illustrate
the logic and mechanics of this procedure, the discussion will
focus on the solution of nonpreemptive resource constrained,
project-scheduling problems. As indicated previously Crefs. 1,2D ,
this problem is the generic form of a class of scheduling problems
which includes resource-constrained job shop and assembly line
balancing problems. Thus, the algorithm to be introduced can be
used, with appropriate modifications, to solve these and related
scheduling problems.
Simply stated, the problem addressed is how to schedule
precedent-related and resource-constrained activities in a project
in order to accomplish a given managerial objective. Over the past
twenty years, a number of techniques have been developed to help
project managers answer this question, the applicability of each
technique being a function of project characteristics and
managerial objectives (see, for example, Davis Cref. 3D,
Elmaghraby Cref. 4D and Patterson Cref. 5D for comparisons of the
various techniques previously investigated!) . The current paper
introduces a scheduling technique that is capable of heuristically
or optimally solving most of the nonpreemptive forms of project
scheduling problems previously examined in the literature. This
includes, but is not restricted to, simple time-based, time-cost
trade-off, time-resource trade-off, and resource constrained
projects. In addition, the proposed algorithm permits the
scheduling of activities where activity performance can increase
as well as decrease the availability of resources such as cash.
4
The latter situation often occurs in multiproject environments
where the cash flow generated by the completion of one project
supports the continuation of others. It is also observed within
single project supports where performance (progress) payments are
based upon the satisfactory completion of key activities
CmilestonesD . These payments in turn facilitate the completion of
other activities in the project.
The specific form of the scheduling problem examined here
permits each activity to be performed in one of several ways
called operating modes, or simply modes. Each mode represents a
different way of combining resources to accomplish a given
activity. The duration and resource requirements of each activity
mode are known a priori , and thus we can speak about
resource-duration interactions. In this model a variety of
multiple resource-duration interactions, such as using different
technologies or types of labor to accomplish the same activity,
can be expressed and evaluated.
Following the scheme suggested by Slowihski Crefs. 6,73 and
Weglarz Crefs. 8,9D resources are assumed to be renewable,
nonrenewable, or doubly constrained. Renewable resources are used
and constrained on a period-by-period basis. For example, skilled
labor would be considered a renewable resource if it is used each
day of a project and if it is also available in limited quantities
each day. This is the category of resource which has been most
frequently modeled by researchers in the past Csee, for example
Crefs. 1,2,10—1610. Nonrenewable resources are available and
consumed on a total-project basis. For example, money may be
considered a nonrenewable resource if only X units of money can be
spent to support all the activities in a project. Doubly
constrained resources are simultaneously constrained on a period
and project basis. Money would be doubly constrained if both
perperiod cash flow and total project expenditures are restricted.
These three categories permit the evaluation of a variety of
common resource restrictions.
Resource-duration interactions in project scheduling were first
considered in time-cost tradeoff problems where it is assumed that
activity duration can be affected only by expenditure levels; that
is, by varying one nonrenewable resource: money. The allocation of
renewable resources with resource-duration interactions has been
examined by Elmaghraby Cref. 4, p.l73D.
The first attempts to model and solve scheduling problems with
5
resource-duration interactions where activity operating modes are
possible combinations of renewable, nonrenewable and doubly
constrained resources where made by Weglarz Crefs. 8,93 and
Slowihski Cref. 63 for the preemptive case. A multieriteria
version of these problems has been examined by Slowihski Cref. 73.
The nonpreemptive case has more recently been investigated by
Talbot Cref. 173. A synthetic presentation of these new models and
procedures for scheduling on machines under multiple-category
resource constraints has been made in Cref. 183.
This chapter proposes a broader interpretation of nonrenewable
resources, which specifically permits the decrease or increase of
these resources as a function of activity status. This will allow,
for example, modeling of progress payments which can increase cash
flow for use by other activities or projects. In addition, this
notion of cash inflows and outflows can better capture the
importance and timing of large cash transactions in a project.
Given the problem of selecting and scheduling activity modes in
a project under the various resource conditions posed above, two
scheduling objectives will be considered in detail in this
chapter: minimizing project completion time and maximizing project
net present value. In section 2, these problems will be defined
formally as zero-one integer programs. Several other objective
functions will also be discussed. Section 3 of the chapter
introduces a backtracking algorithm as a heuristic and as an
optimizing procedure for solving the project completion time
minimization problem. In section 4 modifications of the algorithm
to solve the net present value criterion problem are presented.
Some relaxations of the model assumptions are discussed in
section 5. Solved numerical examples illustrating both of these
problems are given in section 6, with computational considerations
discussed in section 7. Summary comments are given in section 8.
2 FORMULATION OF THE PROJECT SCHEDULING PROBLEM
It is assumed that a project can be depicted as an acyclic
network such as shown in Fig. 1 Ccf. ref. 4, p. 1793. Activities
are represented by integer-labeled nodes, such that the label of a
node is always greater then the labels of all its immediate
predecessor nodes. Arcs represent precedence relationshipsbetween
activities. Unique starting and ending dummy activities with zero
duration are appended to each network. Without loss of generality,
all model variables and parameters are assumed to be
6
TABLE 1
Definition of terms.
Symbol Definition
Clisted alphabetically])
oiCa' ) the ready time Cdue dateD of activity j
j j
B the i-th activity number in the ordered set of
l
activity numbers representing the current best
solution
B. the completion time assignment of the i-th
activity in B.
c the cash flow of activity j mode m in its d-th
jmd
period in process Cd = 1 , . . . , D D;
jm
if c < 0 there is a cash withdrawal;
jmd
if c > 0 there is a cash inflow
jmd
*
c a single non-negative cash flow v periods
j mv • = ■ • =- r-
after the completion of activity j mode m
Cv > ID
C a non-negative integer variable indicating the
net cash position in period t; C is the cash
available at the beginning of the project
D the duration of mode m of job j: D > 0,
jm j™
except D = 0, and D =0
j± n
E. CL.D initially calculated as the earliest ClatestO
j J
critical-path-based completion time for the
shortest Clongest]) mode of activity j.
J the identifying number of the unique dummy
terminal activity in the project without
successors. Activity J has one mode with zero
duration and it consumes no resources;
however, a positive delayed cash flow may
occur after it is assigned
k,K k identifies a specific renewable resource; K
is the number of renewable resources; hence,
k = 1 , . . . , K
M. the number of modes associated with activity i
Cm = 1 M)
j
77C.D the number of elements in a set
7
TABLE 1 - continued
Symbol Definition
•* * ■
N and N N is the number of immediate descendants of
J j j
activity j currently in Y.; N. is the position
index in Y of the descendant of i that is to
j
be evaluated for assignment next
P CSD the set of all immediate predecessor
J J
Csuccessor!) activities of activity j.
PCSD the set of a ll pairs of immediate predecessor
CsuccessorD activities; Ca,bDeP indicates that
activity a is an immediate predecessor of
activity b
R t he amount of renewable resource k currently
available in period t (R > CO
^ kt
the amount of renewable resource k required by
jmk
mode m of activity j each period m is in
process Cr > CD
jmk
a due date for the project
a single-payment, present-value discount
multiplier for period t at interest I
"t -fr-H
f i
w
oo. a weight attributed to activity j
x a zero-one variable which equals zero unless
jmt
mode m of activity j is assigned a completion
time in period t; then, x =1
jmt
Y the set of all immediate descendants of
j
activity j in the precedence tree for current
partial solution; Y is the activity number
J9
of the g-th immediate descendant of activity j
contained in set Y for the current partial
j
solution
Z. the i-th activity number in the ordered set of
activities that have a feasible assignment in
the current partial solution Ci = 1,...,JD
Z. the completion time of the i-th activity in Z.