Stochastic Project Networks: Temporal Analysis, Scheduling and Cost Minimization

249 Pages·1990·5.989 MB·English

by  Prof. Dr. Klaus Neumann (auth.)

Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and W. Krelle
344
Klaus Neumann
Stochastic Project Networks
Temporal Analysis, Scheduling and Cost Minimization
Springer-Verlag
Berlin Heidelberg New York London Paris Tokyo Hong Kong
Editorial Board
H. Albach M. Beckmann (Managing Editor)
P. Dhrymes G. Fandel G. Feichtinger J. Green W. Hildenbrand W. Krelle (Managing
Editor) H. P. Kunzi K. Ritter R. Sato U. Schittko P. Schonfeld R. Selten
Managing Editors
Prof. Dr. M. Beckmann
Brown University
Providence, RI 02912, USA
Prof. Dr. W. Krelle
Institut fur Gesellschafts-und Wirtschaftswissenschaften
der Universitat Bonn
Adenauerallee 24-42, D-5300 Bonn, FRG
Author
Prof. Dr. Klaus Neumann
Institut fur Wirtschaftstheorie und Operations Research
Universitat Karlsruhe
Kaiserstr. 12, D-7500 Karlsruhe 1
ISBN-13:978-3-S40-S2664-3 e-ISBN-13:978-3-642-61S1S-3
Dor: 1O.lO07/978-3-642-61S1S-3
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© Springer-Verlag Berlin Heidelberg 1990
2142/3140·543210 - Printed on acid-free paper
Preface
Project planning, scheduling, and control are regularly used in business and the service
sector of an economy to accomplish outcomes with limited resources under critical time
constraints. To aid in solving these problems, network-based planning methods have
been developed that now exist in a wide variety of forms, cf. Elmaghraby (1977) and
Moder et al. (1983).
The so-called "classical" project networks, which are used in the network techniques
CPM and PERT and which represent acyclic weighted directed graphs, are able to
describe only projects whose evolution in time is uniquely specified in advance. Here
every event of the project is realized exactly once during a single project execution and
it is not possible to return to activities previously carried out (that is, no feedback is
permitted).
Many practical projects, however, do not meet those conditions. Consider, for example,
a production process where some parts produced by a machine may be poorly
manufactured. If an inspection shows that a part does not conform to certain
specifications, it must be repaired or replaced by a new item. This means that we have
to return to a preceding stage of the production process. In other words, there is
feedback. Note that the result of the inspection is that a certain percentage of the parts
tested do not conform. That is, there is a positive probability (strictly less than 1) that
any part is defective. Hence the return to the preceding stage of the production process
generally occurs with a probability less than 1. Furthermore, the activities "repair the
part" and "order a new item" may be carried out several times. Aside from the
planning and supervision of production processes, many R&D projects possess certain
new properties, some of which have been mentioned above, that cannot be treated
within the framework of" classical" project networks.
To deal with these more general projects, stochastic project networks have been
introduced. They possess more general arc weights, several different node types
(containing stochastic elements), and cycles. Again the activity-on-arc representation
known from CPM and PERT is used. Stochastic project networks are also called GERT
networks, a name introduced by Pritsker who has done much pioneering work in this
area.
IV Preface
This monograph presents the state of the art of temporal analysis and cost
minimization of projects, as well as project planning under limited resources where the
projects are modelled by GERT networks. For a more detailed exposition of certain
advanced topics, the reader is referred to the extensive bibliography. We focus upon
analytical methods only. Thus, we do not deal with the method of simulation. The
monograph by Neumann and Steinhardt (1979a) gives a comprehensive survey of the
results known by 1979 on the time-oriented evaluation of GERT networks. The
chapters on temporal analysis in the present book represent an improved and expanded
version of the essential parts of Neumann and Steinhardt (1979a), with several new
results included. The chapters on cost minimization and scheduling with GERT
precedence constraints are completely new. This monograph does not contain any
applications. Numerous examples of applications of different types of GERT networks
can be found in Neumann and Steinhardt (1979a, 1979b) and in Foulds and Neumann
(1989). The book will be of interest to graduate students and researchers in project
planning and scheduling, machine scheduling, and stochastic networks. More generally,
the book is useful to potential readers who want to study stochastic models in
operations research.
The contents of this book are as follows: Chapter 1 defines GERT networks and deals
with some structural questions. The basic concepts and results from the temporal
analysis of GERT networks make up chapter 2. Chapters 3 and 4 are devoted to the
evaluation of special classes of GERT networks with regard to time planning. Chapter 3
discusses GERT networks all of whose nodes have exclusive-or entrance (so-called
EOR networks), which can be associated with Markov renewal processes. Results from
the theory of those stochastic processes can be exploited for the evaluation of EOR
networks. Chapter 4 focusses on GERT networks that may contain all kinds of GERT
nodes and can be reduced to EOR networks. Chapter 5 studies single-machine
scheduling problems with precedence constraints given by a GERT network. This
corresponds to the case where a single resource of capacity 1 is required to carry out
each activity of the respective project. Single-machine min-sum scheduling problems
with general GERT precedence constraints are solved here by dynamic programming.
Polynomial algorithms are presented for scheduling problems with EOR precedence
constraints, where the weighted expected flow-time or maximum expected lateness are
to be minimized. Chapter 6 discusses the cost minimization of projects modelled by
EOR networks, which leads to Markov renewal decision processes and can thus be
reduced to a stochastic dynamic programming problem. Chapter 7 deals with cost and
v
Preface
time minimization of so-called decision project networks, which represent a class of
project networks showing some affinity to GERT networks.
The monograph assumes a knowledge of elementary mathematical programming,
probability theory, and Markov chains. Some basic concepts from the theory of graphs
and networks, Markov chains and Markov renewal processes, deterministic scheduling,
and dynamic programming are summarized before they are needed. The coverage of
these topics makes the monograph self-contained. Proofs of certain of the theorems are
omitted when they are not of sufficient intrinsic interest or can be found in literature.
I would like to thank Matthias Bucker and Philipp Derr who read the entire manuscript
and suggested several improvements. I am especially indebted to Mrs. Christa Otto for
il.
the careful typing and patient retyping of the text in The drawings were done by
Bernd Hornung by means of Autosketch.
Conweiler, The Black Forest Klaus Neumann
December 1989
Contents
List of Symbols IX
Chapter 1 Basic Concepts 1
1.1 Directed Graphs and Project Networks 1
1.2 GERT Networks 11
1.3 Assumptions and Structural Problems 21
1.4 Complete and GERT Subnetworks 33
Chapter 2 Temporal Analysis of GERT Networks 37
2.1 Activation Functions and Activation Distributions 38
2.2 Evaluation of Admissible GERT Networks 41
2.3 Computation of Some Quantities Important to
Time Planning 48
2.4 Evaluation Methods for Admissible GERT Networks 53
Chapter 3 STEOR Networks and EOR Networks 56
3.1 Markov Chains and Markov Renewal Processes 56
3.2 STEOR Networks and Markov Renewal Processes 62
3.3 Basic Properties of Admissible EOR Networks 67
3.4 Coverings of Admissible EOR Networks 71
3.5 Properties and Computation of Activation Functions
and Activation Numbers 80
3.6 The MRP Method 90
Chapter 4 Reducible GERT Networks 95
4.1 STEOR-Reducible Subnetworks 95
4.2 Cycle Reduction 97
4.3 Nodes Which Belong Together 99
4.4 Basic Element Structures 102
4.5 BES Networks 107
4.6 Evaluation Methods for BES Networks and General
Admissible GERT Networks 111
VIII Contents
Chapter 5 Scheduling with GERT Precedence Constraints 116
5.1 Deterministic Single-Machine Scheduling 116
5.2 Stochastic Single-Machine Scheduling with GERT Precedence
Constraints: Basic Concepts 120
5.3 Stochastic Single-Machine Scheduling with GERT Precedence
Constraints: Optimality Criteria and Complexity 124
5.4 List Schedules and Sequences of Activity Executions 132
5.5 Minimum Flow-Time Scheduling in EOR Networks 135
5.6 A Flow-Time Scheduling Example 143
5.7 Minimizing the Maximum Expected Lateness in EOR Networks 149
5.8 Essential Histories and Scheduling Policies for Min-Sum
Problems in General GERT Networks 161
5.9 Elements of Dynamic Programming 168
5.10 Determination of an Optimal Scheduling Policy for the
General Min-Sum Problem 171
Chapter 6 Cost Minimization for STEOR and EOR Networks 176
6.1 STEOR Networks with Time-Dependent Arc Weights 177
6.2 Cost Minimization in STEOR Networks: Basic Concepts 179
6.3 A Dynamic Programming Approach 183
6.4 The Value-Iteration and Policy-Iteration Techniques 189
Chapter 7 Cost and Time Minimization for Decision Project Networks 198
7.1 Decision Project Networks 199
7.2 Cost Minimization 202
7.3 Randomized Actions 208
7.4 Multiple Executions of Projects 212
7.5 Time Minimization 219
References 227
Index 232
List of Symbols
Miscellaneous
. - Equal by definition
•
End of proof or of algorithm
lxJ greatest integer ~x
Set of real numbers
Set of nonnegative real numbers
IR + + Set of positive real numbers
IRn Set of n-tupels of real numbers
IN Set of positive integers (natural numbers)
INO: =lNu{ O} Set of nonnegative integers
o
Empty set
I I
M Number of elements of a finite set M
N!;M N is a subset of M
NcM N is a proper subset of M
I
M\N: ={ aEM a~N} Difference of the sets Ma nd N
IP (M) Power set (set of all subsets) of M
IP (M) : =IP (M) \ {0} Set of all nonempt y subsets of M
f : M~N Mapping (function) of Minto N
o
(f (n)) Landau's symbol" big oh": For f ,g: 1N-i1R + it holds that g (n) =0 (f (n)) if there
are a constant c>O and a positive integer nO such that g(n)~cf(n) whenever
n,nO
Probability
peA) Probability of random event A
P (A IB ) Conditional probability of event A given B
P(AUB) Probability of event" Ao r B"
p(AnB) Probability of event" Aa nd B"
IE(X) Expected value of random variable X
IE (X IA ) Conditional expectation of random variable X given event A
x
List of Symbols
Directed Graphs and Networks
V Node set of a directed graph or network (set of project events)
E Arc set of a directed graph or network (set of activities)
<i, j> Arc (activity) with initial node i and final node j
R Set of sources
S Set of sinks
P( i) Set of predecessors of node i
5( i) Set of successors of node i
1( i) Set of nodes reachable from node i
1(i) := 1(i)\{i}
1( i) Set of nodes from which node i is reachable
GERT Networks
Q Set of all possible project realizations (sample space)
p .. Execution probability of activity <i ,j>
1J
D. . Duration of activity <i ,j >
1J
F. . Distribution function of Di j
1J
A· Random event" node i is activated"
1
A· Random event" node i is not activated"
1
T. Time of activation of node i
1
qR' (qs') Probability that at least the sources iER' (sinks iES' ) are activated
qR,(qS') Probability that exactly the sources iER' (sinks iES') are activated
q. Probability that (at least) node i is activated
1
YV Activation function of set V
).V Activation distribution of set V' with respect to set V
U'
Y(S) :={YS' IS' Eif'(S)} Family of activation functions of sink set S
Y. Activation function of node j
J
y. . Activation function of node j given source i
1J
z· Activation number of node j
J
z· . Activation number of node j given source i
1J
D Project duration
Dskip
Skipping project duration
List of Symbols XI
GERT Scheduling (Chapter 5)
C. . Completion time of activity <i, j>
1J
f ij (t) Cost incurred when activity <i , j> is completed at time t
w· . Weighting factor of activity <i, j>
1J
o Operation
Lo Lateness of operation 0
o
Set of operations
Of Set of final operations
Q List schedule or precedence schedule
Q Set of precedence schedules
Qo Set of all precedence schedules whose last element is the final operation 0
X Set of all feasible arc-node sets
*
¢ Scheduling policy
H Minimum total cost function
Decision Networks (Chapter 7)
-
X· Entrance characteristic of node i
1
X- : Exit characteristic of node i
1
u· Node variable
1
w· . Arc variable
1J
w Project realization (deterministic action)
[ Set of admissible project realizations
c·· Cost of activity <i,j>
1J
Randomized action
11"
¢ Policy (sequence of randomized actions)
d. . Duration of activity <i ,j >
1J
t"'! Activation time of node i in project realization w
1
t~ Earliest possible activation time of node i
1