Table Of ContentLecture Notes in Economics and Mathematical Systems
(Vol. 1-15: Lecture Notes in Operations Research and Mathematical Economics, Vol. 16-59: Lecture
Notes in Operations Research and Mathematical Systems)
Vol. 1: H. BOhlmann. H. Loeffel. E. Nievergelt. EinfOhrung in die Vol. 30: H. Noltemeier. Sensitivitiitsanalyse bei diskreten linearen
Theorie und Praxis der Entscheidung bei Unsicherheit. 2. Auflage. Optimierungsproblemen. VI. 102 Seiten. 1970.
IV. 125 Seiten. 1969. Vol. 31: M. KOhlmeyer. Die nichtzentrale t-Verteilung. II. 106 Sei·
Vol. 2: U. N. Bhat. A Study of the Queueing Systems M/G/l and ten. 1970.
GI/M/1. VIII. 78 pages. 1968. Vol. 32: F. Bartholomes und G. Hotz. Homomorphismen und Re
Vol. 3: A Strauss. An Introduction to Optimal Control Theory. duktionen Ii nearer Sprachen. XII. 143 Seiten. 1970. OM 18.-
Out of print Vol. 33: K. Hinderer. Foundations of Non·stationary Dynamic Pro
Vol. 4: Branch and Bound: Eine EinfOhrung. 2 .• geanderte Auflage. gramming with Discrete Time Parameter. VI. 160 pages. 1970.
Herausgegeben von F. Weinberg. VII. 174 Seiten. 1973.
Vol. 34: H. Sttlrmer. Semi-Markoll-Prozesse mit endlich vielen
Vol. 5: L. P. Hyvarinen. Information Theory for Systems Engineers. Zustiinden. Theorie und Anwendungen. VII. 128 Seiten. 1970.
VII. 205 pages. 1968.
Vol. 35: F. Ferschl. Markovketten. VI. 168 Seiten. 1970.
Vol. 6: H. P. KOnzi. O. MOiler. E. Nievergelt. EinfOhrungskursus in
die dynamische Programmierung. IV. 103 Seiten. 1968. Vol. 36: M. J. P. Magill. On a General Economic Theory of Motion.
VI. 95 pages. 1970.
Vol. 7: W. PoPP. EinfOhrung in die Theorie der Lagerhaltung. VI.
173 Seiten. 1968. Vol. 37: H. MOlier-Merbach. On Round-Oil Errors in Linear Pro
gramming. V. 48 pages. 1970.
Vol. 8: J. Te ghem. J. Loris-Teghem. J. P. Lambotte. Modeles
d·Attente M/GiI et GI/M/I a Arrivees et Services en Groupes. III. Vol. 38: Statistische Methoden I. Herausgegeben von E. Walter.
53 pages. 1969. VIII. 338 Seiten. 1970.
Vol. 9: E. Schultze. EinfOhrung in die mathematischen Grundlagen Vol. 39: Statistische Methoden II. Herausgegeben von E. Walter.
der Informationstheorie. VI. 116 Seiten. 1969. IV. 157 Seiten. 1970.
Vol. 10: D. Hochstadter. Stochastische Lagerhaltungsmodelle. VI. Vol. 40: H. Drygas. The Coordinate-Free Approach to Gauss
269 Seiten. 1969. Markov Estimation. VIII. 113 pages. 1970.
Vol. 11/12: Mathematical Systems Theory and Economics. Edited Vol. 41 : U. Ueing. Zwei Ltlsungsmethoden fiir nichtkonvexe Pro
by H. W. Kuhn and G. P. Szegtl. VIII. III. 486 pages. 1969. grammierungsprobleme. IV. 92 Seiten. 1971.
Vol. 13: Heuristische Planungsmethoden. Herausgegeben von Vol. 42: A V. Balakrishnan. Introduction to Optimization Theory in
F. Weinberg und C. A. Zehnder. II. 93 Seiten. 1969. a Hilbert Space. IV. 153 pages. 1971.
Vol. 14: Computing Methods in Optimization Problems. V. 191 pages. Vol. 43: J. A Morales. Bayesian Full Information Structural Analy
1969. sis. VI. 154 pages. 1971.
Vol. 15: Economic Models. Estimation and Risk Programming: Vol. 44:-G. Feichtinger. Stochastische Modelle demographischer
Essays in Honor of Gerhard Tintner. Edited by K. A. Fox. G. V. L. Prozesse. IX. 404 Seiten. 1971.
Narasimham and J. K. Sengupta. VIII. 461 pages. 1969.
Vol. 45: K. Wendler. Hauptaustauschschritte (Principal Pivoting).
Vol. 16: H. P. KOnzi und W. Oettli. Nichtlineare Optimierung: 11.64 Seiten. 1971.
Neuere Verfahren. Bibliographie. IV. 180 Seiten. 1969.
Vol. 46: C. Boucher. Lec;ons sur la theorie des automates ma
Vol. 17: H. Bauer und K. Neumann. Berechnung optimaler Steue thematiques. VIII. 193 pages. 1971.
rungen. Maximumprinzip und dynamische Optimierung. VIII. 188
Vol. 47: H. A Nour Eldin. Optimierung linearer Regelsysteme
Seiten. 1969.
mit quadrati scher Zielfunktion. VIII. 163 Seiten. 1971.
Vol. 18: M. WollI. Optimale Instandhaltungspolitiken in einfachen
Systemen. V. 143 Seiten. 1970. Vol. 48: M. Constam. FORTRAN fOr Anfanger. 2. Auflage. VI.
148 Seiten. 1973.
Vol. 19: L. P. Hyvarinen. Mathematical Modeling for Industrial Pro
Vol. 49: Ch. SchneeweiB. Regelungstechnische stochastische
cesses. VI. 122 pages. 1970.
Optimierungsverfahren. XI. 254 Seiten. 1971.
Vol. 20: G. Uebe. Optimale Fahrpliine. IX. 161 Seiten. 1970.
Vol. 50: Unternehmensforschung Heute - Obersichtsvortriige der
Vol. 21: Th. M. Liebling. Graphentheorie in Planungs-und Touren ZOricher Tagung von SVOR und DGU. September 1970. Heraus
problemen am Beispiel des stadtischen StraBendienstes. IX. gegeben von M. Beckmann. IV. 133 Seiten. 1971.
118 Seiten. 1970.
Vol. 51: Digitale Simulation. Herausgegeben von K. Bauknecht
Vol. 22: W. Eichhorn. Theorie der homogenen Produktionsfunk und W. Nef. IV. 207 Seiten. 1971.
tion. VIII. 119 Seiten. 1970.
Vol. 52: Invariant Imbedding. Proceedings 1970. Edited by R. E.
Vol. 23: A Ghosal. Some Aspects of Queueing and Storage
Bellman and E. D. Denman. IV. 148 pages. 1971.
Systems. IV. 93 pages. 1970.
Vol. 24: G. Feichtinger. Lernprozesse in stochastischen Automaten. Vol. 53: J. Rosenmiiller. Kooperative Spiele und Markte. iii. 152
V. 66 Seiten. 1970. Seiten. 1971.
Vol. 25: R. Henn und O. Opitz. Konsum-und Produktionstheorie I. Vol. 54: C. C. von Weizsiicker. Steady State Capital Theory. III.
11,124 Seiten. 1970. 102 pages. 1971.
Vol. 26: D. Hochstadter und G. Uebe. Okonometrische Methoden. Vol. 55: P. A. V. B. Swamy. Statistical Inference in Random Coef·
XII. 250 Seiten. 1970. ficient Regression Models. VIII. 209 pages. 1971.
Vol. 27: I. H. Mufti. Computational Methods in Optimal Control Vol. 56: Mohamed A. EI-Hodiri. Constrained Extrema. Introduction
Problems. IV. 45 pages. 1970. to the Differentiable Case with Economic Applications. III. 130
Vol. 28: Theoretical Approaches to Non-Numerical Problem Sol pages. 1971.
ving. Edited by R. B. Banerji and M. D. Mesarovic. VI. 466 pages. Vol. 57: E. Freund. Zeitvariable MehrgrtlBensysteme. VIII.160 Sei
1970. ten. 1971.
Vol. 29: S. E. Elmaghraby. Some Network Models in Management Vol. 58: P. B. Hagelschuer. Theorie der linearen Dekomposition.
Science. III. 176 pages. 1970. VII. 191 Seiten. 1971.
continuation on page 271
Lectu re Notes
in Economics and
Mathematical Systems
Managing Editors: M. Beckmann and H. P. KGnzi
Operations Research
172
Klaus Neumann
Ulrich Steinhardt
GERT Networks
and the Time-Oriented Evaluation of Projects
Springer-Verlag
Berlin Heidelberg New York 1979
Editorial Board
H. Albach· A. V. Balakrishnan· M. Beckmann (Managing Editor)
P. Dhrymes . J. Green· W. Hildenbrand· W. Krelle
H. P. KOnzi (Managing Editor) . K. Ritter· R. Sato . H. Schelbert
P. Schonfeld
Managing Editors
Prof. Dr. M. Beckmann Prof. Dr. H. P. KOnzi
Brown University Universitat ZOrich
Providence, RI 02912/USA 8090 ZOrich/Schweiz
Authors
Prof. Dr. Klaus Neumann Dr. Ulrich Steinhardt
Institut fOr Wirtschaftstheorie BroichstraBe 17
und Operations Research 0-5300 Bonn 3
Universitat Karlsruhe
KaiserstraBe 12
0-7500 Karlsruhe
AMS Subject Classifications (1970): 90-02, 90815
ISBN-13: 978-3-540-09705-1 e-ISBN-13: 978-3-642-95363-7
001: 10.1007/978-3-642-95363-7
This work is subject to copyright. All rights are reserved, whether the whole or part
of the material is concerned, specifically those of translation, reprinting, re-use
of illustrations, broadcasting, reproduction by photocopying machine or similar
means, and storage in data banks. Under § 54 of the German Copyright Law where
copies are made for other than private use, a fee is payable to the publisher, the
amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1979
PREFACE
Towards the end of the fifties methods for planning, scheduling,and control of proj
ects were developed on the basis that the evolution of a project can be associated
with a special weighted directed graph, called activity network. In this association,
the individual activities of the project correspond to the arcs, the so-called proj
ect events (beginning or termination of activities) correspond to the nodes, and
the durations of the activities correspond to the weights of the respective arcs
of the directed graph. 1) Contiguous arcs are assigned to activities which succeed
one another immediately. The event corresponding to a node occurs exactly at the
time at which all activities which are associated with the arcs leading into the
node are terminated. After the occurrence of an event all those activities are be
gun which correspond to the arcs emanating from the respective node. This implies
especially that the evolution of the project has to be uniquely determined before
hand, that every activity and every event are realized exactly once during the exe
cution of the project, and that "feedback" (corresponding to cycles in the asso
ciated network) is not permitted.
Many projects, for example most R&D projects and projects in the area of production
p1 a nni ng, do not sa ti sfy the foregoi ng res tri cti ons. In these cases it can happen
that (after the occurrence of an event) some activities are not carried out with
certainty, but only with a probability less than one. It can happen furthermore that
an event occurs already at the time at which not all but perhaps only one of the
activities corresponding to the arcs leading into the respective node is terminated.
In addition, in the course of the execution of the project, it is possible to return
to events which have already occurred once before (feedback).
For these reasons, attempts have been made since about the middle sixties to modify
the above so-called "classical" activity networks in a way which will take into
account the three particular properties just mentioned. EISNER made the first sug
gestions in this direction. Because it is impossible to specify uniquely the evolu
tion of many R&D projects beforehand, EISNER introduced so-called decision boxes
within the networks, with the convention that activities emanating from such deci
sion boxes may be carried out by choice (that is, no longer need all activities
1) Activity network methods, as e.g. MPM, in which the activities correspond to the
nodes of a directed graph will not be considered in this monograph.
2 Preface
necessarily be carried out). The decisive step was made by ELMAGHRABY and thereafter
by PRITSKER and HAPP. They introduced, in addition to the two possible "node exits"
defined by EISNER, three different "node entrances" (corresponding to the logical
operations "and", "inclusive-or", and "exclusive-or"). As to the weights of the
arcs, they added the execution probability of the respective activity to its (sto
chastic) duration. ELMAGHRABY called his concept (which originally considered only
deterministic durations of activities) GAN ("Generalized Activity Networks"), where
as PRITSKER and HAPP called their method for the evaluation of such activity net
works (in the sense of time planning of the associated projects) GERT ("Graphical
Evaluation and Review Technique"). These activity networks, which are considerably
more genera 1 than the "class i ca 1" networks, are ca 11 ed stoahastia aativity networks
or, in accordance with PRITSKER, HAPP, and WHITEHOUSE (see WHITEHOUSE [25], chapter
8) , GERT networks.
Those GERT networks which contain the so-called STEaR node (with stochastic exit
and exclusive-or entrance) as the only node type can be evaluated by MASON's rule,
which was originally developed for linear transmission systems. PRITSKER and others
have written several simulation programs for the evaluation of more general GERT
networks. In this monograph, however, we shall use a different approach to evaluat
ing GERT networks. Already ELMAGHRABY has pointed out that a homogeneous semi-MARKOV
process can be associated with every STEaR network (see ELMAGHRABY [3], chapter 4,
and [4] , chapter 5). But neither ELMAGHRABY nor PRITSKER and his school have ex
ploited this finding for the development of a methodology. Starting with the fact
that a MARKOV renewal process corresponds to every STEaR network, we will show that
the quantities of interest in time planning can be computed in a simple manner from
the renewal functions of the respective process. We shall indicate a very effective
algorithm for determining the renewal functions. This approach has the following
advantages:
(1) It provides a method for evaluating STEaR networks which requires considerably
less computational effort than MASON's rule or simulation.
(2) It enables more insight into the structure of these networks and the connected
stochastic processes than the methods known so far.
(3) It can be generalized, in contrast to MASON's rule, to certain GERT networks
which also contain "non-STEaR nodes" (so-called GERT networks with "basic ele
ment structures").
The particular contents of the present monograph are as follows: In chapter 1, we
give an exact definition of the concept "GERT network" and an enumeration of those
assumptions that have to be stipulated because of practical or methodological con-
Preface 3
siderations. We then make precise what is to be understood by the evaluation of a
GERT network (in the sense of time planning). As a first class of GERT networks we
treat STEOR networks in chapter 2, and we explain the already mentioned evaluation
method based on 'results of the theory of MARKOV renewal processes. The MASON rule
is also explained for purposes of comparison. In chapter 3, we investigate to what
extent the results obtained for STEOR networks can be applied to certain GERT net
works which contain non-STEOR nodes, too. In chapter 4, a method for evaluating
general GERT networks is given.
Cycles represent an important new element within GERT networks as compared to clas
sical activity networks. At first it suggests itself to assume that only STEOR nodes
which are relatively simple to handle are present in cycles. Those new points of
view that result when this assumption is waived are discussed in chapter 5. The case
where the probabilities of carrying out activities, and the distributions of the
duration of activities, depend on the times of occurrence of the initial events of
the respective activities (which is especially important for cost planning) is
treated in chapter 6.
The simulation of GERT networks is dealt with in chapter 7. This last chapter also
contains a universal evaluation method which comprises the simulation and the proce
dures treated in the chapters 2, 3 as well and which represents a very effective
algorithm for evaluating general GERT networks. Finally, some formulas from proba
bility theory and some concepts and theorems from the theory of stochastic processes
are summarized in the appendix.
GERT networks are of great practical importance (not only in the framework of project
planning). This aspect is illustrated by numerous examples of applications for the
different types of GERT networks that are treated in this monograph.
The cost planning of projects to which GERT networks are assigned leads to very inter
esting optimization problems in the areas of control theory and stochastic dynamic
programming. This will be investigated in a subsequent volume.
We should like to thank all the members of the Institut fUr Wirtschaftstheorie und
Operations Research, University of Karlsruhe, who have helped generously during the
writing of this book. Special thanks are due to Wolfgang Fix and Wolfram Nicolai
who read the manuscript and suggested numerous significant improvements. We are
especially indebted also to Eginhard J. Muth, University of Florida, Gainesville,
for reading the manuscript, making valuable comments, and producing the English
Preface
version of this monograph. All errors and misunderstandings of the English version,
however, are the authors' sole responsibility. Finally, we wish to thank '1rs. Inge
Toelstede, who did an excellent typing job. We are also grateful to Friedrich
Allendorf, who drew the numerous figures.
Karl s ruhe Klaus Neumann
July 1979 Ulrich Steinhardt
CONTENTS
List of Symbols 8
Summary of Assumptions 10
Chapter 1 Basic Concepts 11
1.1 Directed Graphs and Activity Networks 11
1.2 GERT Networks 18
1.3 Assumptions Required for GERT Networks 25
1.4 Evaluation of GERT Networks 30
1.5 Subnetworks of GERT Networks 37
Chapter 2 STEOR Networks 45
2.1 Stochastic Processes Connected with STEOR Networks 45
2.1.1 STEOR Networks and MARKOV Renewal Processes 45
2.1.2 GERT Networks with Only EOR Nodes 49
2.2 The MRP Method for the Evaluation of STEOR Networks 53
2.2.1 The Activation Functions and Activation Numbers 53
2.2.2 Special Cases and Examples 59
2.3 The Numerical Implementation of the MRP Method 65
2.4 The MASON Rule 75
2.5 Earliest and Latest Times in STEOR Networks 82
2.6 Applications 93
2.6.1 Time Planning for R&D Projects 94
2.6.2 Production Planning 98
2.6.3 Legislation and Administration of Justice 101
2.6.4 Evaluation of Linear Transmission Systems 103
2.6.5 Description and Analysis of Queueing Models 108
Chapter 3 GERT Networks with Basic Element Structures 116
3.1 Nodes Which Belong Together 117
3.2 Basic Elements 120
3.3 Basic Element Structures 124
3.3.1 Definition and Properties of a Basic Element Structure 124
3.3.2 BES Networks 129
3.3.3 Examples 137
3.4 Evaluation of Admissible Basic Element Structures 140
Contents
6
3.5 Determination of Admissible Interior Basic Element Structures 146
3.5.1 Determination of Nodes Belonging Together (Labeling Process) 147
3.5.2 Construction of a Possible Interior Basic Element Structure 149
3.5.3 Testing of a Possible Interior Basic Element Structure 153
3.5.4 The BES Method 156
3.6 Applications 160
3.6.1 Rendezvous of Two Space Vehicles 161
3.6.2 Production of a Television Program 161
3.6.3 Introduction of a New Industrial Product 167
3.6.4 Reliability Problems 169
Chapter 4 Evaluation of General GERT Networks 172
4.1 Cycle Reduction 172
4.2 Evaluation of an Acyclic GERT Network N 175
4.2.1 Construction of the Sequence (N~) of Subnetworks of N 177
4.2.2 Determination of the Conditional Probabilities in (4.2.2) 180
4.2.3 Test of the Assumptions A6 and A7 186
4.2.4 Example 187
4.3 Applications 193
4.3.1 Construction of a Turbine 193
4.3.2 Development of a Camera 196
4.4 Replacement of Non-genuine lOR Nodes 199
Chapter 5 Multiple Activations of Non-STEOR Nodes 204
5.1 Generalized GERT Networks and Closed Subnetworks 204
5.2 Assumptions Required for Generalized GERT Networks 208
5.3 Evaluation of Generalized GERT Networks 212
5.3.1 A Method for Evaluating Admissible Generalized GERT Networks 212
5.3.2 Test of the Assumptions for a Generalized GERT Network 214
5.4 Applications 215
5.4.1 Introduction of a New Product 215
5.4.2 Overhaul of a Generator 218
Chapter 6 GERT Networks with Time-dependent Arc Weights 220
6.1 Basic Concepts 220
6.2 STEOR Networks 222
6.3 GERT Networks with Basic Element Structures 225
6.4 General GERT Networks 231
Contents 7
Chapter 7 Simulation 233
7.1 GERTS Networks and Equivalent GERT Networks 233
7.2 GERTS Networks Which Do Not Have Corresponding GERT Networks 236
7.3 Simulation of GERTS Networks 238
7.4 Generalized GERTS Networks 242
7.5 A Universal Method for Evaluating Admissible GERT Networks 244
Appendix 250
A.l Some Formulas from Probability Theory 250
A.2 Stochastic Processes 252
A.3 Precise Formulation of Assumption A3 257
References 260
Index 262
