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universiteit gent
faculteit economie en bedrijfskunde
Academiejaar 2012–2013
an accuracy study and improvementof a time-dependent earned value model
using monte carlo simulation
Masterproef voorgedragen tot het bekomen van de graad van
Master of Science in de
Toegepaste Economische Wetenschappen: Handelsingenieur
Pieter Beeckman en Kenny Vanleeuwen
onder leiding van
Prof.dr. Mario Vanhoucke en Mathieu Wauters
universiteit gent
faculteit economie en bedrijfskunde
Academiejaar 2012–2013
an accuracy study and improvementof a time-dependent earned value model
using monte carlo simulation
Masterproef voorgedragen tot het bekomen van de graad van
Master of Science in de
Toegepaste Economische Wetenschappen: Handelsingenieur
Pieter Beeckman en Kenny Vanleeuwen
onder leiding van
Prof.dr. Mario Vanhoucke en Mathieu Wauters
Permission
‘The authors give the authorization to make this thesis available for consultation and to
copy parts of it for personal use. Any other use is subject to the limitations of copyright,
in particular with regard to the obligation to explicitly mention the source when quoting
results from this thesis.’
Pieter Beeckman Kenny Vanleeuwen
Abstract
English
In the world of project management, the Earned Value Management (EVM) theory
was introduced to help project managers to accomplish their goal, which is keeping the
duration and cost of their projects under control. An important feature of EVM is that
it enables project managers to make predictions of the final project duration and cost.
In this thesis a thorough analysis is done concerning the added value of a new model,
introduced by R.D.H Warburton in his paper ‘A time-dependent earned value model
for software projects’ [21]. In this paper Warburton suggests that the incorporation of
time dependency in the EVM theory can lead to more accurate predictions of the final
project duration and cost, based on the data that is available in an early stadium of
project completion. The goal is to critically analyze this new model, benchmark its
performance against the existing EVM techniques, and improve the suggested model.
To conduct this research, a Monte Carlo simulation was set up that makes it possible
to test the model in different scenarios which a project managers might be confronted
with.
In general, the results show that the Warburton model is not a valuable addition to the
EVM theory to forecast the final project duration, and further research remains to be
done for this. However, the model did deliver very promising results to forecast the final
project cost. After improvement, it could be concluded the Warburton model might well
be a valuable addition to a project manager’s toolkit to forecast the final project cost.
Abstract
Nederlands
In de wereld van project management werd de Earned Value Management (EVM) theorie
geıntroduceerd om project managers te helpen bij het bereiken van hun doel, namelijk
de duurtijd en kosten van hun projecten te controleren en binnen aanvaardbare grenzen
te houden. Een belangrijke meerwaarde van EVM is de mogelijkheid om voorspellingen
te doen van de finale project duurtijd en kost.
In deze thesis wordt onderzoek gevoerd naar de meerwaarde van een nieuw model,
voorgesteld door R.D.H. Warburton in zijn verhandeling ‘A time-dependent earned value
model for software projects’ [21]. In deze verhandeling houdt Warburton een pleidooi
voor de incorporatie van tijdsafhankelijkheid in de EVM theorie om meer accurate voor-
spellingen van de finale project duurtijd en kost te bekomen, en dit op basis van data die
beschikbaar is in een vroeg stadium van het project. Het doel is om de methode kritisch
te analyseren, de prestatie ervan af te wegen tegen de bestaande EVM methodes, en
indien mogelijk te verbeteren. Om dit onderzoek te voeren werd een Monte Carlo simu-
latie opgezet die het mogelijk maakte het nieuwe model te onderzoeken in verschillende
scenario’s waarmee een project manager mee geconfronteerd kan worden.
Uit de resultaten kon worden besloten dat het model van Warburton geen meerwaarde
brengt voor de EVM theorie inzake het voorspellen van de finale project duurtijd. Inzake
het voorspellen van de finale project kost, echter, leverde het model zeer belovende
resultaten. Na verbetering bleek dat het model wel degelijk een meerwaarde kan zijn
voor een project manager inzake het voorspellen van de finale kost.
Preface
This thesis is the result of hard work done by two friends and the valuable contributions
of a few people we would therefore like to thank.
First, we would like to express our gratitude to our promoter, prof.Dr. Mario Vanhoucke,
for introducing us with great enthusiasm to the fascinating world of project management
and the Earned Value Management theory. His convincing way of teaching the course
‘Project Management’ excited our interest and made the decision to choose a thesis
concerning this subject easy.
Also, a very big thank you goes to Mathieu Wauters for his open door and the valuable
feedback he provided during the process of writing this thesis. He was always prepared
to take the time to read our work in progress, answer our questions and suggest some
interesting ideas for extra research that could be an added value for our thesis.
Thank you, Mario and Mathieu!
i
Contents
Preface i
List of Abbreviations vii
List of Figures ix
List of Tables xiii
I INTRODUCTION 1
1 General Introduction 2
2 Introduction to Earned Value Management 8
2.1 Definition and purpose of EVM . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 EVM Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Performance Measures . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Shortcomings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Earned Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Making predictions of the future with EVM . . . . . . . . . . . . . . . . . 13
2.4.1 Estimated duration at Completion (EAC(t)) . . . . . . . . . . . . 14
2.4.2 Estimated cost at Completion (EAC) . . . . . . . . . . . . . . . . 15
II WARBURTON’S MODEL 19
3 A review of the time-dependent Earned Value model 20
3.1 Reasons for and goal of the time-dependent earned value model . . . . . 21
3.1.1 Origin of the idea for developing the model . . . . . . . . . . . . . 21
3.1.2 Goal of Warburton’s model . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Parameters and set-up of the time-dependent earned value model . . . . 23
3.2.1 Definitions of the parameters of Warburton’s model . . . . . . . . 23
ii
3.2.2 Set-up of Warburton’s model and estimation of parameter values . 25
3.2.3 Making predictions of the final project cost and duration with
Warburton’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Differences with the traditional EVM method . . . . . . . . . . . . . . . 30
3.3.1 Functional time dependence . . . . . . . . . . . . . . . . . . . . . . 30
3.3.2 Early project data . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Example Project 31
4.1 The Warburton method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.2 End of project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.3 Impact of each parameter on the model . . . . . . . . . . . . . . . . . . . 37
4.3.1 Total amount of labor N . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.2 Time of the labor peak T . . . . . . . . . . . . . . . . . . . . . . . 38
4.3.3 Reject rate of activities r . . . . . . . . . . . . . . . . . . . . . . . 38
4.3.4 Cost overrun of rejected activities c . . . . . . . . . . . . . . . . . 39
4.3.5 Repair time of rejected activities τ . . . . . . . . . . . . . . . . . . 40
4.3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Critical analysis of the model and set-up duration forecasting methods 42
5.1 Shortcomings of Warburton’s model . . . . . . . . . . . . . . . . . . . . . 43
5.1.1 Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1.2 Critical path . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1.3 Forecasting the final project duration . . . . . . . . . . . . . . . . 44
5.1.4 Cumulative values . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Calculation methods for parameter T . . . . . . . . . . . . . . . . . . . . . 46
5.2.1 Calculation T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.2 Calculation T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 Methods to forecast the final project duration using Warburton’s model . 47
5.3.1 Eight methods to forecast the final project duration . . . . . . . . 47
5.3.2 Set-up of the methods to forecast the final project duration . . . . 52
III SPECIFIC CHALLENGES AND METHODOLOGY OF THESIMULATION STUDY 53
6 Specific challenges, Research questions and Hypotheses 54
6.1 Specific challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.1.1 General applicability and forecast accuracy . . . . . . . . . . . . . 54
6.1.2 Comparison with traditional EVM methods . . . . . . . . . . . . . 55
6.1.3 Improve the initial Warburton model . . . . . . . . . . . . . . . . . 55
6.2 Research questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2.1 Necessary input for Warburton’s model . . . . . . . . . . . . . . . 56
6.2.2 Forecast accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6.2.3 Project completion stage . . . . . . . . . . . . . . . . . . . . . . . 57
6.2.4 Topological structure . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2.5 Linear, convex and concave time/cost-relationship . . . . . . . . . 58
7 Methodology of the simulation study 59
7.1 Project data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.1.1 Topological network indicators . . . . . . . . . . . . . . . . . . . . 60
7.1.2 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.2 Project scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.3 Project execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.3.1 Monte Carlo simulation . . . . . . . . . . . . . . . . . . . . . . . . 63
7.3.2 Triangular distributions and scenarios . . . . . . . . . . . . . . . . 64
7.3.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.4 Project monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
IV ACCURACY STUDY OF WARBURTON’S MODEL 69
8 Necessary input for Warburton’s model 70
8.1 Ratio values of methods EAC(t)w5, EAC(t)w6 and EAC(t)w7 . . . . . . . 71
8.2 Parameter T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
8.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
9 Forecast accuracy 77
9.1 Accuracy of the final project duration forecasts . . . . . . . . . . . . . . . 77
9.1.1 Accuracy using the MAPE . . . . . . . . . . . . . . . . . . . . . . 77
9.1.2 Direction of the forecasting error using the MPE . . . . . . . . . . 82
9.2 Accuracy of the final cost forecasts . . . . . . . . . . . . . . . . . . . . . . 83
9.2.1 Accuracy using the MAPE . . . . . . . . . . . . . . . . . . . . . . 83
9.2.2 Direction of the forecasting error using the MPE . . . . . . . . . . 85
9.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
10 Project completion stage 87
10.1 Accuracy of the final duration forecasts per completion stage . . . . . . . 88
10.2 Accuracy of the final cost forecasts per completion stage . . . . . . . . . . 90
10.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
11 Topological structure 93
11.1 The influence of the serial/parallel indicator (SP) . . . . . . . . . . . . . . 94
11.1.1 Impact of SP factor on EAC(t)w methods to forecast final duration 94
11.1.2 Impact of SP factor on the EACw method to forecast final cost . . 95
11.2 The influence of the activity distribution (AD) . . . . . . . . . . . . . . . 97
11.2.1 Impact of AD factor on EAC(t)w methods to forecast final duration 97
11.2.2 Impact of AD factor on the EACw method to forecast the final
project cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
11.3 The influence of the length of arcs (LA) and topological float (TF) . . . . 100
11.3.1 Impact of LA and TF factor on the EAC(t)w methods to forecast
the final project duration . . . . . . . . . . . . . . . . . . . . . . . 100
11.3.2 Impact of LA and TF factor on the EACw method to forecast the
final project cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
12 Time/cost relationship 103
12.1 Time/cost relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
12.1.1 Linear time/cost relationship . . . . . . . . . . . . . . . . . . . . . 104
12.1.2 Convex time/cost relationship . . . . . . . . . . . . . . . . . . . . . 105
12.1.3 Concave time/cost relationship . . . . . . . . . . . . . . . . . . . . 105
12.2 Impact of time/cost relationship on the EAC methods . . . . . . . . . . . 106
12.2.1 Forecast accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
12.2.2 Project completion stage - Late projects . . . . . . . . . . . . . . 109
12.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
V IMPROVEMENT OF WARBURTON’S MODEL 115
13 Shortcomings of Warburton’s model and set-up of improved model 116
13.1 Shortcomings of Warburton’s model . . . . . . . . . . . . . . . . . . . . . 117
13.1.1 Accelerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
13.1.2 Other shortcomings . . . . . . . . . . . . . . . . . . . . . . . . . . 117
13.2 Set-up of the new and improved Warburton model . . . . . . . . . . . . . 118
13.2.1 Modification of the parameters . . . . . . . . . . . . . . . . . . . . 118
13.2.2 Modification of Warburton’s curves . . . . . . . . . . . . . . . . . . 120
13.3 Example project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
13.3.1 Schedule delay (τ > 0) . . . . . . . . . . . . . . . . . . . . . . . . . 124
13.3.2 Schedule acceleration(τ < 0) . . . . . . . . . . . . . . . . . . . . . 126
13.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
14 Accuracy study of the new Warburton model and comparison with the
initial Warburton model 130
14.1 Necessary input for the new Warburton model . . . . . . . . . . . . . . . 131
14.1.1 Ratio values of methods EAC(t)w5, EAC(t)w6 and EAC(t)w7 . . . 131
14.1.2 Parameter T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
14.2 Forecast accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
14.2.1 Hypotheses regarding forecast accuracy of the new Warburton model133
14.2.2 Accuracy of the final duration forecasts . . . . . . . . . . . . . . . 134
14.2.3 Accuracy of the final cost forecasts . . . . . . . . . . . . . . . . . . 137
14.3 Project completion stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
14.3.1 Early projects (Scenario 1 and 2) . . . . . . . . . . . . . . . . . . . 143
14.3.2 Average over all scenarios . . . . . . . . . . . . . . . . . . . . . . . 146
14.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
VI FINAL REFLECTIONS 151
15 Final conclusions 152
15.1 Performance and added value of Warburton’s model . . . . . . . . . . . . 153
15.1.1 Forecasting the final project duration . . . . . . . . . . . . . . . . 153
15.1.2 Forecasting the final project cost . . . . . . . . . . . . . . . . . . . 155
15.1.3 Recommendations for practitioners . . . . . . . . . . . . . . . . . . 157
15.2 Limitations and guidelines for future research . . . . . . . . . . . . . . . . 159
Bibliography 161
A Tables based on Measuring Time settings 164
A.1 Nine scenarios of Measuring Time . . . . . . . . . . . . . . . . . . . . . . 164
A.2 Six new defined scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
B Concave time/cost function: Mathematical derivation 166
C Summary tables initial and new Warburton model 168
C.1 Summary table: Parameters of the initial and New Warburton model . . . 168
C.2 Summary table: Warburton curves of the initial and new Warburton model168
List of Abbreviations
A
AC Actual Cost
AC Actual Cost according to Warburton’s model
AD Actual Duration
AT Actual Time
B
BAC Budget At Completion
BCRW Budgeted Cost of Remaining Work
C
CPI Cost Performance Index
CV Cost Variance
E
EAC Estimate At Completion (cost)
EACw Estimate At Completion (cost) based on Warburton’s model
EAC(t) Estimate At Completion (time)
EAC(t)w Estimate At Completion (time) based on Warburton’s model
ED Earned Duration
ES Earned Schedule
ESS Earliest Start Schedule
EV Earned Value
EV(t)w Earned Value according to Warburton’s model
EVM Earned Value Management
vii
L
LA Length of Arcs
M
MAPE Mean Absolute Percentage Error
MPE Mean Percentage Error
P
PD Planned Duration
PDw0 Planned Duration based on Warburton’s model
PDWR Planned Duration of Work Remaining
PF Performance Factor
PV Planned Value
PV(t)w Planned Value according to Warburton’s model
PVR Planned Value Rate
R
RD Real Duration
S
SCI Schedule Cost Index
SP Serial Parallel
SPI Schedule Performance Index
SV Schedule Variance
T
TV Time Variance
TF Topological Float
List of Figures
1.1 Overview parts and chapters . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 EVM: key parameters, performance measures, and forecasting indicators
[15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 EVM: key metrics for early and late projects with cost under- and over-
runs, with amount of weeks on the x-axis and budget on the y-axis[19]. . . 11
2.3 The Earned Schedule (ES) for a fictitious project example, with amount
of weeks on the x-axis and budget on y-axis[18]. . . . . . . . . . . . . . . . 13
3.1 Example of the instantaneous curves (left) and the cumulative curves
(right) of Warburton’s model. . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Example project: Activity-on-the-node representation. . . . . . . . . . . . 31
4.2 Example project: Baseline schedule. . . . . . . . . . . . . . . . . . . . . . 32
4.3 Example project: (Fictitious) real project execution until 30 % of BAC is
earned. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Example project: Instantaneous Warburton-curves based on available
data after 30 % project completion. . . . . . . . . . . . . . . . . . . . . . . 34
4.5 Example project: Cumulative Warburton-curves based on available data
after 30% project completion. . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.6 Example project: (Fictitious) real project execution until 100% of BAC
earned. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.7 Example project: pv(t)w-, ev(t)w- and ac(t)w-curve after doubling the
value of N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.8 Example project: pv(t)w-, ev(t)w- and ac(t)w-curve after doubling the
value of T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.9 pv(t)w-, ev(t)w- and ac(t)w-curve after after bringing r to a value of 1. . . 39
4.10 pv(t)w-, ev(t)w- and ac(t)w-curve after bringing c to a value of 1. . . . . . 39
4.11 pv(t)w-, ev(t)w- and ac(t)w-curve after bringing τ to a value of 3. . . . . . 40
7.1 Overview of the methodology of the simulation study. . . . . . . . . . . . 59
ix
7.2 Parameters of triangular distributions [16]. . . . . . . . . . . . . . . . . . 65
7.3 Parameter values of triangular distributions for each scenario used in the
simulation study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
8.1 Average ratio values (y-axis) for methods EAC(t)w5 and EAC(t)w6 for
each scenario (x-axis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
8.2 Average ratio values (y-axis) for method EAC(t)w7 for each scenario (x-
axis). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8.3 Average ratio values for methods EAC(t)w5 and EAC(t)w6 per SP-factor. 73
8.4 Average ratio values for method EAC(t)w7 per SP-factor. . . . . . . . . . 73
8.5 MAPE values of final duration forecasters based on Warburton’s model. . 75
9.1 MPE values for the traditional PV2, ED2 and ES1 forecasting methods
per scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.2 MPE values for the EAC(t)w1, EAC(t)w2 and EAC(t)w3 method per sce-
nario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.3 MPE values of the best EVM forecasting methods for the final project cost. 85
9.4 MPE values of the Warburton’s forecasting method for the final project
cost. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.1 MAPE values for early projects (scenario 1 and 2) of the forecasting meth-
ods for final project duration, per project completion stage (early, middle,
late). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.2 MAPE values for on time projects (scenario 3 and 4) of the forecasting
methods for final project duration, per project completion stage (early,
middle, late). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.3 MAPE values for late projects (scenario 5 and 6) of the forecasting meth-
ods for final project duration, per project completion stage (early, middle,
late). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.4 MAPE values for early projects (scenario 1 and 2) of the forecasting meth-
ods for final project cost, per project completion stage (early, middle, late). 91
10.5 MAPE values for on time projects (scenario 3 and 4) of the forecasting
methods for final project cost, per project completion stage (early, middle,
late). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
10.6 MAPE values for late projects (scenario 5 and 6) of the forecasting meth-
ods for final project cost, per project completion stage (early, middle,
late). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
11.1 Influence of the SP factor (x-axis) on the time forecast error (y-axis) of
methods EAC(t)w1, EAC(t)w2 and EAC(t)w3 under the 6 scenarios. . . . 94
11.2 Influence of the SP factor (x-axis) on the time forecast error (y-axis) of
the EACw method to forecast the final project cost under the 6 scenarios. 96
11.3 Influence of the AD factor (x-axis) on the time forecast error (y-axis) of
methods EAC(t)w1, EAC(t)w2 and EAC(t)w3 under the 6 scenarios. . . . 97
11.4 Influence of the AD factor (x-axis) on the time forecast error (y-axis) of
the EACw method to forecast the final project cost under the 6 scenarios. 99
11.5 Influence of the LA factor (x-axis) on the time forecast error (y-axis) of
methods EAC(t)w1, EAC(t)w2 and EAC(t)w3 under the 6 scenarios. . . . 100
11.6 Influence of the TF factor (x-axis) on the time forecast error (y-axis) of
methods EAC(t)w1, EAC(t)w2 and EAC(t)w3 under the 6 scenarios. . . . 101
12.1 Linear, convex and concave time/cost relationship. . . . . . . . . . . . . . 104
12.2 MAPE values for late projects (scenario 5 and 6) of the forecasting meth-
ods for final project cost, per completion stage (early, middle, late) under
the assumption of a linear time/cost relationship. . . . . . . . . . . . . . . 109
12.3 MAPE values for late projects (scenario 5 and 6) of the forecasting meth-
ods for final project cost, per 10 % of the work completed under the
assumption of a linear time/cost relationship. . . . . . . . . . . . . . . . . 110
12.4 MAPE values for late projects (scenario 5 and 6) of the forecasting meth-
ods for final project cost, per completion stage (early, middle, late) under
the assumption of a convex time/cost relationship. . . . . . . . . . . . . . 110
12.5 MAPE values for late projects (scenario 5 and 6) of the forecasting meth-
ods for final project cost, per 10 % of the work completed under the
assumption of a convex time/cost relationship. . . . . . . . . . . . . . . . 111
12.6 MAPE values for late projects (scenario 5 and 6) of the forecasting meth-
ods for final project cost, per completion stage (early, middle, late) under
the assumption of a concave time/cost relationship. . . . . . . . . . . . . . 112
12.7 MAPE values for late projects (scenario 5 and 6) of the forecasting meth-
ods for final project cost, per 10 % of the work completed under the
assumption of a concave time/cost relationship. . . . . . . . . . . . . . . . 113
13.1 Overview of differences between the initial Warburton model and the new
Warburton model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
13.2 Example project: (Fictitious) real project execution (τ > 0, delay) until
30 % of BAC is earned. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
13.3 Example project: (Fictitious) real project execution when project is ac-
celerated (τ < 0) until 30 % of BAC is earned. . . . . . . . . . . . . . . . 127
13.4 Example project: (Fictitious) real project execution for accelerated project
until 100 % of BAC is earned. . . . . . . . . . . . . . . . . . . . . . . . . . 128
14.1 MAPE values of final duration forecasts using methods based on the new
Warburton model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
14.2 Accuracy of the EACw method based on the initial Warburton model and
on the new Warburton model under the assumption of a linear time/cost
relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
14.3 Accuracy of the EACw method based on the initial Warburton model and
on the new Warburton model under the assumption of a convex time/cost
relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
14.4 Accuracy of the EACw method based on the initial Warburton model and
on the new Warburton model under the assumption of a concave time/cost
relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
14.5 Forecast error (MAPE) of the EAC methods for late projects under the
assumption of a linear time/cost relationship for early projects (scenario
1 and 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
14.6 Forecast error (MAPE) of the EAC methods for late projects under the
assumption of a convex time/cost relationship for early projects (scenario
1 and 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
14.7 Forecast error (MAPE) of the EAC methods for late projects under the
assumption of a concave time/cost relationship for early projects (scenario
1 and 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
14.8 Forecast error (MAPE) of the EAC methods under the assumption of a
linear time/cost relationship averaged over all six scenarios. . . . . . . . . 146
14.9 Forecast error (MAPE) of the EAC methods under the assumption of a
convex time/cost relationship averaged over all six scenarios. . . . . . . . 147
14.10Forecast error (MAPE) of the EAC methods under the assumption of a
concave time/cost relationship averaged over all six scenarios. . . . . . . . 147
List of Tables
2.1 EAC(t) Forecasting Methods . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 EAC Forecasting Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Parameters of Warburton’s model . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 Example project: Planned, earned and actual cost values at each time
instance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Summary table of the effect of an increase in each parameter on Warbur-
ton’s model with ∼ ↑ indicating a very small increase. . . . . . . . . . . 41
5.1 Formulas for calculating the eight methods to forecast the final duration. 52
8.1 Average ratio values for time forecasting methods EAC(t)w5, EAC(t)w6and EAC(t)w7 per SP factor. . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.1 Forecasting accuracy (MAPE) of final project duration using the tradi-
tional EVM methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.2 Forecasting accuracy (MAPE) of final project duration using the methods
based on Warburton’s model. . . . . . . . . . . . . . . . . . . . . . . . . . 78
9.3 Forecasting accuracy (MAPE) of the final project cost. . . . . . . . . . . . 83
12.1 Forecasting accuracy (MAPE) of the final project cost under the assump-
tion of a linear time/cost relationship. . . . . . . . . . . . . . . . . . . . . 106
12.2 Forecasting accuracy (MAPE) of the final project cost under the assump-
tion of a convex time/cost relationship. . . . . . . . . . . . . . . . . . . . 107
12.3 Forecasting accuracy (MAPE) of the final project cost under the assump-
tion of a concave time/cost relationship. . . . . . . . . . . . . . . . . . . . 108
13.1 Example project: Comparison of the results of the initial and new War-
burton model when τ > 0 (delay). . . . . . . . . . . . . . . . . . . . . . . 126
13.2 Example project: Comparison of the results of the initial and new War-
burton model when τ < 0 (acceleration). . . . . . . . . . . . . . . . . . . 128
xiii
14.1 Average ratio values for time forecasting methods EAC(t)w5, EAC(t)w6and EAC(t)w7 for the new Warburton model. . . . . . . . . . . . . . . . . 132
14.2 Forecasting error (MAPE) of final project duration using the traditional
EAC(t) methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
14.3 Forecasting error (MAPE) of final project duration using the EAC(t)wmethods based on the initial Warburton model. . . . . . . . . . . . . . . . 135
14.4 Forecasting error (MAPE) of final project duration using the EAC(t)wmethods based on the new Warburton model. . . . . . . . . . . . . . . . . 135
14.5 Forecasting error (MAPE) of the final project cost under the assumption
of a linear time/cost relationship. . . . . . . . . . . . . . . . . . . . . . . . 137
14.6 Forecasting error (MAPE) of the final project cost under the assumption
of a convex time/cost relationship. . . . . . . . . . . . . . . . . . . . . . . 139
14.7 Forecasting error (MAPE) of the final project cost under the assumption
of a concave time/cost relationship. . . . . . . . . . . . . . . . . . . . . . . 141
A.1 Average forecasting accuracy (MAPE) of the time EVM methods for the 9
scenarios of Measuring Time ([12], pg. 68), assumption concerning project
completion as in Measuring Time . . . . . . . . . . . . . . . . . . . . . . . 164
A.2 Average forecasting accuracy (MAPE) of the time EVM methods for the
our 6 scenarios, assumption concerning project completion as in Measur-
ing Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
C.1 Summary Table: Parameters, Initial and New Warburton Model . . . . . 169
C.2 Summary Table: Warburton Curves, Initial and New Warburton Model . 170
Part I
INTRODUCTION
1
Chapter 1
General Introduction
Topic and goal
In the world of project management, the Earned Value Management (EVM) theory,
introduced in 1967, makes it possible to measure the performance of a project in terms
of cost and time during its execution. EVM systems have been developed to provide
project managers with crucial information concerning the performance of their projects
through the interaction of three project management elements: time, cost and scope.
EVM also makes it possible to provide project managers with early warning signals for
poor performance, which indicate it might be useful to take corrective actions. In EVM,
actual and budgeted costs are compared to the earned value.
Recently, an article was published that introduced an interesting yet unproven method
for including time dependency into Earned Value Management. This novelty was pro-
posed by Roger D.H. Warburton in his paper ‘A time-dependent earned value model
for software projects’ [21]. The presented model requires three parameters which map
directly to the fundamental triple constraint of scope, cost and schedule: the reject rate
of activities, the cost overrun parameter, and the time to repair the rejected activities.
Time dependent expressions for planned value, earned value and actual cost are derived,
along with the cost performance index and schedule performance index. In this paper,
R.D.H. Warburton applied the model to a software dataset which demonstrated how the
2
Chapter 1. General Introduction 3
estimate of the project’s final cost converged faster to the correct answer and with less
variability than the standard Estimate at Completion (EAC) calculation with the Cost
Performance Index (CPI) as the performance factor, which is based on the EVM theory.
The aim of our thesis is to thoroughly investigate this new concept. We thought it was
an intriguing challenge to investigate whether the model can be a valuable addition to
a project manager’s toolkit or not.
Scope and limitations
A first and crucial point is that the model proposed by R.D.H. Warburton was only
tested for its usefulness and performance on a single software project. As mentioned, we
accepted the challenge to thoroughly investigate this new model. In our opinion, three
specific challenges are relevant to do this. First, it is of vital importance to investigate
the general applicability and forecast accuracy of the model concerning the final project
duration and cost, and this in different settings and for different project networks. A
second main goal is to benchmark the performance of the model against the existing
EVM forecasting methods for final project duration, which were already thoroughly
investigated by Vanhoucke in Measuring Time [12], and final project cost. Finally, a
third challenge is to learn from the accuracy study conducted on Warburton’s model and
see whether we can make adjustments to improve the forecast accuracy of the model.
The scope of this thesis is restricted to the forecasting indicators for final project duration
and cost, and does not include a thorough investigation of the performance measures,
being the Cost Performance Index (CPI) and the Schedule Performance Index (SPI). The
reason for this is that, in our opinion, the biggest opportunities of the model proposed
by Warburton lie in the forecasting indicators. However, at the end of this thesis, some
recommendations for future research concerning the performance measures based on
Warburton’s model will be discussed.
Chapter 1. General Introduction 4
Method
A first step towards conducting the research to meet the challenges mentioned above was
to bring clarity in the new parameter definitions and formulas of Warburton’s model.
This is done by means of a comprehensive example project which is used to illustrate
the meaning, calculation and use of the underlying parameters and formulas. With this
example, we seek to provide a simple way for the reader to get familiarized with the
basic concepts of Warburton’s model. Also, before starting the actual accuracy study
of the model, we take a critical look at the model to see if there are already interesting
features that are worth discussing.
In order to conduct the actual accuracy and benchmarking study, we decided not to
restrict ourselves to the use of a few real project examples. Instead, a Monte Carlo
simulation was set up which will be applied to multiple datasets which contain a wide
variety of diverse project network structures. This approach was chosen in order to be
able to test the general applicability of the model and generalize the results from our
simulation study. To conduct this Monte Carlo simulation, we made use of ‘P2 Engine’,
which is a command line utility tool based on the LUA scripting language and developed
by OR-AS [9].
Structure
This thesis consists of 15 chapters, divided over six parts, which are visualized in figure
1.1 on page 7 and which can be briefly summarized along the following lines.
Together with this general introduction, Chapter 2 completes part I of this thesis.
In this chapter, an overview of the Earned Value Management (EVM) theory is given
that mainly contains the elements that will be needed further along this thesis. The
basic components of the EVM philosophy are the Planned Value (PV), Actual Cost
(AC) and Earned Value (EV). Based on these basic components, EVM can be used to
forecast the final project duration and cost. The most important formulas to do this are
also discussed in this chapter.
Chapter 1. General Introduction 5
Part II of this thesis contains a thorough discussion of the model proposed by
R.D.H. Warburton. Chapter 3 gives an overview of the parameters and time-dependent
expressions for planned value, earned value and actual cost, proposed by Roger D.H.
Warburton. The set-up of the model is discussed together with how the model can be
used to make predictions. Also, the main differences with the traditional EVM method
are addressed. To clearly distinguish the model from other methods, the integration of
time-dependence will be explicitly indicated by adding ‘(t)’ to all quantities, and also
a subscript ‘w’ which refers to Warburton will be added. Furthermore, a lower case
notation will be used for instantaneous values, while an upper case notation indicates
cumulative values. In chapter 4, an example project is presented that will help to
demonstrate the interpretation of the parameters and the use of the model. In chapter
5, a thorough look is taken at the model and some shortcomings found during this
analysis are discussed.
Part III of this thesis discusses the challenges and methodology of the accuracy
study. Chapter 6 more thoroughly discusses the specific challenges that were stated
above under ‘Scope and limitations’. As mentioned, a first challenge is to investigate
the general applicability and forecast accuracy of the model. Second, the performance
of the model is benchmarked against the existing EVM methods. The third challenge
is to learn from the accuracy study and look for opportunities to improve Warburton’s
model. The research questions and hypotheses linked to these specific challenges are
also discussed in this chapter. Chapter 7 contains an overview of the methodology of
the simulation study. This includes a discussion of the used datasets, the Monte Carlo
simulation, the used scenarios, the underlying assumptions of the study and the project
monitoring measures.
Part IV of this thesis handles the accuracy study of the Warburton model. In
chapter 8, prior research is done to determine some unknown factors that are needed as
input for the accuracy study. In chapter 9, the actual accuracy study starts. The meth-
ods for forecasting the final project duration and final project cost based on Warburton’s
Chapter 1. General Introduction 6
model are tested on accuracy and benchmarked against the existing EVM forecasting
methods. As a second part of the study, the relation between the forecast accuracy of
Warburton’s model and the project completion stage is investigated in chapter 10. Con-
trary to chapter 9, the performance of the model will be investigated in different stages
of project completion rather than looking at the average performance along the whole
project lifetime. Chapter 11 handles the relation between the topological structure of
the project and the forecast accuracy of the time and cost methods based on Warbur-
ton’s model. Finally, in chapter 12, the influence of different time/cost relationships
on the cost forecast accuracy of Warburton’s model is analyzed. For this research, the
relationship between the duration and the cost of an activity is simulated under three
different settings: a linear, convex and concave time/cost relationship.
Part V of this thesis includes the lessons learned from the accuracy study and
discusses how we improved Warburton’s model. In chapter 13, the insight gained by the
study of part 4 with regard to the shortcomings of the model will be discussed. With
these lessons learned, we saw an opportunity to modify and improve the model. A new
and improved model, to which we will refer to as ‘the new Warburton model’, is set up.
This entails a thorough discussion of the modifications of Warburton’s model introduced
in his paper [21], to which we will refer to as ‘the initial Warburton model’. To end this
chapter, the functioning of the new Warburton model and the difference with the initial
Warburton model is illustrated using the example project of chapter 4. In chapter 14,
a similar accuracy study will be conducted as in part 4 of this thesis, but this time for
the new Warburton model. The goal of this chapter is to discuss the performance of the
new model and benchmark it against the initial Warburton model.
Finally, in part VI of this thesis, chapter 15 contains the main conclusions from
part IV and V, together with some personal recommendations concerning the use of
Warburton’s model together with some guidelines for future research.
Chapter 1. General Introduction 7
PART I: Introduction
Chapter 1: General introduction Chapter 2: Introduction to EVM
PART 2: Warburton’s model
Chapter 3: A review of the time-
dependent Earned Value model Chapter 5: Critical analysis of the
model and set-up duration forecasting
Chapter 4:Example project
PART 3: Specific challenges and Methodology of simulation study
Chapter 6: Specific challenges,
Research questions and Hypotheses
Chapter 7: Methodology of the
simulation study
PART 4: Accuracy Study Warburton’s model
Chapter 8: Necessary input for Warburton’s model
Chapter 9:
Forecast accuracy
Chapter 10:
Project completion
stage
Chapter 11:
Topological structure
Chapter 12: Time/cost relationship
PART 5: Improvement of Warburton’s model
Chapter 13: Shortcomings of Warburton’s model and set-up of new improved model
Chapter 14: Accuracy study of the new Warburton model and
comparison with the initial Warburton model
PART 6: Final reflections
Chapter 15: Final conclusions
Figure 1.1: Overview parts and chapters
Chapter 2
Introduction to Earned Value
Management
In this chapter an overview of the Earned Value Management (EVM) theory is given
that mainly contains the elements that will be needed further along this thesis. This
means it should not be viewed as an exhaustive overview. For this, the interested reader
is referred to the earned value bibliography [4].
2.1 Definition and purpose of EVM
Earned Value Management (EVM) is a theory introduced in 1967 that makes it possible
to measure the performance of a project in terms of cost and time during its execution.
EVM systems have been developed to provide project managers with crucial information
concerning the performance of their projects through the interaction of three project
management elements: time, cost and scope. EVM also makes it possible to provide
project managers with early warning signals for poor performance, which indicate it
might be useful to take corrective actions. In EVM, actual and budgeted costs are
compared to the earned value.
In the following section, an overview is given of the basic EVM components, as de-
8
Chapter 2. Introduction to Earned Value Management 9
scribed by several authors such as Henderson [5], Anbari [1], and Vanhoucke [14]. These
components are summarized in figure 2.1.
Figure 2.1: EVM: key parameters, performance measures, and forecasting indicators [15].
2.2 EVM Components
The EVM technique is based on the management of three key parameters: Planned Value
(PV), Earned Value (EV) and Actual Cost (AC). The Planned Value is the scheduled
cost for the work done at that point in time, thus if the project would be executed
according to schedule. The planned value for the whole project, calculated as the sum
of the PVs of all project activities, is called the Budget At Completion (BAC=∑
PV).
This is the budget needed to complete the project if every activity is executed according
to plan. However, due to unforeseen events this might not be the case. Activities may
take more time to complete than estimated because of machine breakdowns, strikes, etc.
Activities may also start late, for example because of preceding activities ending late.
For these and many other reasons the actual cost AC, which is the real cost of the work
done at a certain point in time, may deviate from the planned value PV.
Chapter 2. Introduction to Earned Value Management 10
This is where Earned Value is introduced. During the execution of a project, value is
acquired or earned. This value is expressed as a portion of the BAC. To calculate EV,
we need to know what percentage of work has already been completed. This portion is
called the Percentage Completed (PC).
EV = PC ∗BAC (2.1)
2.2.1 Performance Measures
Cost Variance (CV) and Cost Performance Index (CPI)
CV = EV −AC (2.2)
CPI =EV
AC(2.3)
CV and CPI are measurements to evaluate the cost performance of a project. If, at a
certain point in time, EV is greater than AC, more value was earned than the real costs
that have been incurred. In this case the project is running under budget, CV will be
positive and CPI will be greater than 1. On the other hand, when EV is smaller than
AC, the project is running over budget, CV will be negative and CPI will be smaller
than 1.
Schedule Variance (SV) and Schedule Performance Index (SPI)
SV = EV − PV (2.4)
SPI =EV
PV(2.5)
SV and SPI are measurements to evaluate the time progress of a project. If, at a certain
point in time, EV is greater than PV, more value was earned than budgeted at that
moment. In this case the project is running ahead of schedule, SV will be positive
and SPI will be greater than 1. If, on the other hand, EV is smaller than PV, the
project is running late, SV will be negative and SPI will be smaller than 1. Combining
both measurements for schedule and cost performance results in four possible project
scenarios, shown in figure 2.2.
Chapter 2. Introduction to Earned Value Management 11
Figure 2.2: EVM: key metrics for early and late projects with cost under- and overruns, with
amount of weeks on the x-axis and budget on the y-axis[19].
Chapter 2. Introduction to Earned Value Management 12
2.2.2 Shortcomings
In literature, several authors have discussed the shortcomings of the basic EVM technique
([5],[8], etc.). First, SV expresses schedule performance in monetary terms and not in
time units. Second, if a project starts earlier than planned, PV will be equal to zero,
which means SPI cannot give a measurement for schedule performance. Third, at the
end of the project, the Percentage Completed is 100 % and EV will equal PV. As a
result, SPI will always converge to 1 at the end of the project. When observing values of
0 and 1 for SV and SPI respectively, one can wonder whether the project is completed
or whether the project is performing perfectly according to plan.
2.3 Earned Schedule
To cope with the problems mentioned in section 2.2.2, the Earned Schedule (ES) indica-
tors were introduced by Lipke [8]. These indicators are time-based instead of cost-based
which makes them easier to understand when considering schedule performance. On
top, the ES indicators are reliable during the whole time horizon of the project as the
adapted formula for schedule performance does not necessarily converge to 1 towards
the end. ES is calculated as follows:
Find time t such that EV ≥ PVt and EV < PVt+1
ES = t+EV − PVtPVt+1 − PVt
, (2.6)
The Earned Schedule is thus a translation of the EV into time units by determining
when this EV should have been earned in the baseline schedule. The calculation of ES
is illustrated in figure 2.3, in which one can see that the earned value at time instance
7 (AT) should have been earned at time instance 5.14 (ES) for the project to be on
schedule. The Actual Time (AT) indicates the time that has passed from the beginning
of the project until the present moment of calculation. Because the value of ES is
expressed in time units, it can be compared to the Actual Time (AT), which results in
Chapter 2. Introduction to Earned Value Management 13
Figure 2.3: The Earned Schedule (ES) for a fictitious project example, with amount of weeks
on the x-axis and budget on y-axis[18].
two new schedule performance measures.
SV (t) = ES −AT (2.7)
SPI(t) =ES
AT(2.8)
SV(t) will be positive and SPI(t) greater than 1 when the project is running ahead of
schedule. When the project is running behind, SV(t) will be negative and SPI(t) smaller
than 1.
2.4 Making predictions of the future with EVM
EVM can be used to forecast the final project duration and cost. In this section, an
overview of the most important forecasters is given. In part IV of this thesis, the perfor-
mance of Warburton’s model, which is the subject of this thesis, will be benchmarked
against the performance of these forecasters.
Chapter 2. Introduction to Earned Value Management 14
2.4.1 Estimated duration at Completion (EAC(t))
As EAC(t) is a forecast of the final project duration, the schedule performance measures
needs to be translated from monetary units to time units. In literature, three methods
have been proposed and are evaluated extensively by Vanhoucke [14]. In this section,
we summarize the parts that are most relevant for this thesis.
The general formula for EAC(t) is based on the sum of the Actual Duration (AD) of the
project at the time instance of calculation and the Planned Duration of Work Remaining
(PDWR). Note that the Actual Duration is the same as the Actual Time used in the
Earned Schedule calculation.
EAC(t) = AD + PDWR (2.9)
Planned Value Method of Anbari
The PV method proposed by Anbari [1], is based on the traditional EVM metrics as
described in section 2.2 and proposes additional metrics. The Planned Value Rate (PVR)
is calculated as follows:
PV R =BAC
PD(2.10)
The Planned Duration (PD) equals the scheduled duration of the project which can be
derived from the baseline schedule. PVR indicates what amount of value is expected to
be earned on average per scheduled time unit of the project. With the Time Variance
(TV) concept, proposed by Anbari [1], PVR is used to translate the Schedule Variance
(SV) into time units. The PV method does not directly give an estimate for the Planned
Duration of Work Remaining (PDWR). Instead, the EAC(t) is based on the TV. De-
pending on whether PDWR performs according to plan, follows the current SPI trend
or the current SCI trend (CPI x CPI), three different formulas are proposed as can be
seen in table 2.1 on page 17. TV is calculated as follows:
TV =SV
PV R(2.11)
Chapter 2. Introduction to Earned Value Management 15
Earned Duration method of Jacob and Kane
Jacob and Kane [7] propose a second method. This method uses the Earned Duration
(ED) concept and is calculated as follows:
ED = AD ∗ SPI (2.12)
In this method, SV and SPI are also translated into time units. For the calculation of
EAC(t), the unearned remaining duration is corrected by the Performance Factor (PF),
after which it is added to the Actual Duration. The resulting formulas proposed by this
method can be found in table 2.1 on page 17.
Earned Schedule method of Lipke
In section 2.2.2, Lipke criticizes the use of SV and SPI as schedule performance measures
because of their unreliability near the end of a project. Two new schedule performance
measures, SV(t) and SPI(t), were developed to cope with this and are also directly
expressed in time units. Lipke calculates EAC(t) as the sum of AD and the unearned
remaining duration corrected by the Performance Factor (PF). Table 2.1 on page 17
displays the resulting formulas for this method.
2.4.2 Estimated cost at Completion (EAC)
In general, the EAC is formulated as the sum of the Actual Cost at the considered time
instance and the Budgeted Cost of Remaining Work (BCRW). To make more accurate
estimations of the final project cost, assumptions should be made about the performance
of the work that still has to be done. For example, if a project is running above budget,
one could assume that the remaining work will still be performed within budget or that
the remaining work is likely to follow the past trend of performing above budget. To
incorporate these assumptions into the EAC, a Performance Factor (PF) is used to adjust
the Budgeted Cost of Remaining Work:
EAC = AC +BCRW
PF(2.13)
Chapter 2. Introduction to Earned Value Management 16
If it can be assumed that the rest of the project will be executed according to plan, the
Performance Factor (PF) is equal to 1. If there are reasons to believe that the rest of
the project will be executed at the same level of cost performance as the work that has
already been executed, the CPI factor can be used as PF. The same reasoning can be
used for schedule performance by using the SPI as PF. If it is desirable not only to take
the current cost performance into account, but also the current schedule performance,
the Performance Factor can be expressed as the Schedule Cost Index (SCI), which equals
CPI * SPI. Finally, one can also use the SPI(t) as correcting factor, or the SCI(t) which
equals CPI * SPI(t). On the website of ‘PM Knowledge Center’ [17], two more options
are given where the PF is a weighted average of the SPI or SPI(t) and CPI. An overview
can be found in table 2.2 on page 17.
Chapter 2. Introduction to Earned Value Management 17
Tab
le2.1
:E
AC
(t)
Fore
cast
ing
Met
hod
s
Per
form
ance
Pla
nn
edV
alu
eE
arn
edD
ura
tion
Ear
ned
Sch
edu
le
Anb
ari
Jac
ob(a)
Lip
ke
acco
rdin
gto
pla
nE
AC
(t) PV1
EA
C(t
) ED1
EA
C(t
) ES1
=PD−TV
=PD
+AD∗(1−SPI)
=AD∗(PD−ES)
acco
rdin
gto
curr
ent
EA
C(t
) PV2
EA
C(t
) ED2
EA
C(t
) ES2
tim
ep
erfo
rman
ce=
PD
SPI
=PD
SPI
=AD
+PD−ES
SPI(t
)
acco
rdin
gto
curr
ent
EA
C(t
) PV3
EA
C(t
) ED3
−EAC
(t) ES3
tim
e/co
stp
erfo
rman
ce=
PD
SCI
=AD
+PD−ED
CPI∗SPI
(b)
=AD
+PD−ES
CPI∗SPI(t
)
(c)
(a)In
situationswheretheproject
work
isnotyet
completed(i.e.when
AD>
PD
andSPI<
1),
thePD
willbe
substitutedby
theAD
(b)This
forecastingform
ula
doesn’t
appearin
Jacob[6]andhasbeen
added
byVandevoo
rdeandVanhoucke[11]
(c)This
forecastingform
ula
doesn’t
appearin
Lipke
[8]andhasbeen
added
byVandevoo
rdeandVanhoucke[11]
Chapter 2. Introduction to Earned Value Management 18
Tab
le2.2
:E
AC
Fore
cast
ing
Met
hod
s
Per
form
ance
SP
IS
PI(
t)
acco
rdin
gto
pla
nE
AC1
EA
C1
=AC
+BCRW
=AC
+BCRW
acco
rdin
gto
curr
ent
EA
C2
EA
C2
cost
per
form
ance
=AC
+BCRW/CPI
=AC
+BCRW/CPI
acco
rdin
gto
curr
ent
EA
C3
EA
C4
tim
ep
erfo
rman
ce=AC
+BCRW/SPI
=AC
+BCRW/SP(t)
acco
rdin
gto
curr
ent
EA
C5
EA
C6
tim
e/co
stp
erfo
rman
ce=AC
+BCRW/SCI
(d)
=AC
+BCRW/SCI(t)(e)
acco
rdin
gto
wei
ghte
dE
AC7
EA
C8
tim
e/co
stp
erfo
rman
ce=AC
+BCRW
0.8∗C
PI+0.2∗S
PI
=AC
+BCRW
0.8∗C
PI+0.2∗S
PI(t)
(d)withSCI=SPI∗CPI
(e)withSCI(t)=SPI(t)∗CPI
Part II
WARBURTON’S MODEL
19
Chapter 3
A review of the time-dependent
Earned Value model
In the paper ‘A time-dependent earned value model for software projects’ [21] a formal
method for including time dependence into Earned Value Management is proposed by
Roger D.H. Warburton. This model is based on three essential parameters which are
directly related to the fundamental triple constraint of scope, cost and schedule. These
parameters are respectively the reject rate of activities, the cost overrun parameter and
the time to repair the rejected activities. The model is built on the well-established
Putnam-Norden-Rayleigh labor rate profile [10], and therefore requires another two pa-
rameters, being the total amount of labor to be done and the time of the labor peak.
Time-dependent formulas were derived for the planned value, earned value and actual
cost, along with the CPI and SPI.
In what follows, an overview of the parameters and time-dependent expressions for
planned value, earned value and actual cost, proposed by Roger D.H. Warburton, is
given. In section 3.1 the reason for and the intended use of the model is discussed. In
section 3.2 the parameters and set-up of the model are discussed, together with how the
model can be used for predictions. Finally, in section 3.3, the main differences with the
traditional EVM method are addressed.
20
Chapter 3. A review of the time-dependent Earned Value model 21
To clearly distinguish the original model presented in this chapter from the variations
that are introduced later in this thesis , we will from now on refer to this original model of
section 3.2 as “Warburton’s (initial) model”. The integration of time-dependence in the
model will be explicitly indicated by adding ‘(t)’ to all quantities, and to avoid confusing,
also a subscript ‘w’ which refers to Warburton will be added. Furthermore, a lower case
notation will be used for instantaneous values, while an upper case notation indicates
cumulative values. This means that for example ev(t)w refers to the instantaneous time-
dependent earned value at moment t of Warburton’s model, while ev would simply refer
to the traditional EVM instantaneous earned value.
3.1 Reasons for and goal of the time-dependent earned
value model
3.1.1 Origin of the idea for developing the model
The starting point is the well established EVM theory which is able to provide project
managers with early warning signals for when the project is in trouble, as introduced in
chapter 2. As discussed by R.D.H. Warburton [21], ‘somewhat overlooked in the EVM
theory is what we will refer to as “instantaneous” values for the CPI. Less than 10 %
of contracts have 3-month stable CPI values, which means that almost all measured
CPI values were found to change, and are thus unstable, when continually recomputed
over short 3-month intervals. Less than one third of projects have stable 6-month CPIs.
Christensen [2] did establish that the continually updated 3-month averages provided
the most reliable estimate of the final cost, despite the variability. According to R.D.H.
Warburton, these ideas suggest that the instantaneous CPI changes over time and, as
Christensen and Payne [3] observed, only stabilizes because of its cumulative nature.’
R.D.H. Warburton notices [21] ‘that one should carefully distinguish two types of vari-
ation in the CPI. First, there are the statistical fluctuations, being the inherent uncer-
tainty and variation in project data. Second, there is the existence of a functional time
Chapter 3. A review of the time-dependent Earned Value model 22
dependence, and it is this kinds of variation that is rarely discussed and is integrated into
Warburton’s model. A functional time-dependence is important because the CPI and
its determining factors AC en EV are used to determine the EAC, and a changing CPI
implies a changing EAC. R.H.D. Warburton suggests, however, that the EAC should not
change over time for it to be a useful concept. He notices that project managers want
to know the final budget, and it is understandable they might be upset by a continually
changing EAC.’ Warburton’s model wants to demonstrate that there are several reason-
able time-dependent shapes for the CPI and the underlying actual cost and earned value
curves, but that the resulting EAC is in fact a constant.
3.1.2 Goal of Warburton’s model
The often overlooked issue of functional time-dependence in the EVM theory was the
basis for Warburton’s model. Although the paragraph above mainly discusses the CPI,
Warburton’s model was set up with the goal of improving the theory of EVM by including
time-dependence into the definitions of all quantities. By doing this, Warburton’s model
eyes at delivering precise estimates of the project’s final cost and duration. The CPI
discussion is merely a way of illustrating where the opportunity for improvement is
situated. On top, Warburton’s model eyes at establishing these precise estimates of the
final cost and duration of a project in an early stage of the project, and is not meant to
be constantly updated like the traditional EVM method. The model wants to make use
of data available in an early stage of the project, the so-called ‘early project data’, to
determine the parameter values, set up the curves and make accurate predictions. The
reason for this is that warning signals concerning project performance are considered to
be most useful for project managers in early stages of the project.
R.D.H. Warburton thinks that, despite the preliminary nature of the model and much
research remains to be done, the model could provide a useful contribution to a project
manager’s toolbox by providing more reliable estimates of a project’s final cost overrun
and schedule delay in an early stage.
Chapter 3. A review of the time-dependent Earned Value model 23
3.2 Parameters and set-up of the time-dependent earned
value model
In this section, a discussion of the parameters and set-up of Warburton’s model will
be given together with a further explanation of how the model is intended to be used.
For a detailed discussion and deduction of the formulas and the underlying Putnam-
Norden-Rayleigh (PNR) labor rate profile, the reader is referred to the original paper of
R.D.H. Warburton [21]. In chapter 4, an example project is presented that will help to
demonstrate the interpretation of the parameters and the use of the model.
3.2.1 Definitions of the parameters of Warburton’s model
As the PNR curve, further discussed in 3.2.2, is at the basis of the model, parameter N
is a first parameter which is inherent to the model. N is defined as the total amount of
labor needed to finish a project. In other words, N represents the total amount of value
to be earned during a project and thus equals the well-known BAC. Another parameter
directly linked to the PNR labor profile is T, which represents the time of the labor
peak. This is the moment in a project at which the amount of labor finished is at its
highest level, or in other words when the most value is earned.
Next to these two parameters, inherent to the model because of the PNR labor curve,
there are three parameters which really characterize Warburton’s model. The first is the
reject rate of activities, represented by the letter r. This is the fraction of the activities
which are finished at the time of calculation, but took longer than initially planned. In
other words, it is the fraction of the activities that are not finished within their planned
duration. Second, the cost overrun parameter, represented by the letter c, equals the
fractional extra cost of the planned cost it takes to finish a rejected activity. Third, the
repair time of rejected activities, represented by τ , is the amount of extra time units on
top of the planned amount of time units needed to finish a rejected activity. Notice that
this parameter is defined in absolute time units, and is not a fractional (%) value such
as parameters r and c. A summary is given in table 3.1.
Chapter 3. A review of the time-dependent Earned Value model 24
Table 3.1: Parameters of Warburton’s model
Parameter Meaning
N Total amount of labor The total amount of value to be earned (equals BAC)
T Time of the labor peak Time instance of the project at which most value is earned
r Reject rate of activities Fraction of the activities that were rejected,
meaning they took longer than planned
c Cost overrun Average extra cost to finish a rejected activity,
as a fraction of the planned cost of the activity
τ Repair time of rejected Average additional time needed to finish a
activities rejected activity
Parameters N and T can be determined at the start of the project, while parameters r, c
and τ will be determined with early project data, being data available after some part of
the project is completed. As a result of the parameter formulation, all these parameters
have a value greater than or equal to zero and are assumed to be constant over the life
of the project when making predictions.
Chapter 3. A review of the time-dependent Earned Value model 25
3.2.2 Set-up of Warburton’s model and estimation of parameter values
As mentioned, the model is built on the PNR labor rate profile, which makes use of
parameters N and T as discussed in section 3.1.1 . The formula can be seen in equation
3.1. In this section it is discussed how the time-dependent planned value, earned value
and actual cost curves are generated.
Planned Value
As stated by R.D.H. Warburton [21], ‘the planned value curve is generated at the start
of a project, in the planning phase. When a project is planned, the time-phased budget
is developed by summing the time-phased labor contributions of the scheduled activities,
which is the labor rate curve. When one is in the planning phase, the rate of completion
of activities is simply the planned labor over time, pv(t)w.’ For Warburton’s model, this
means that the instantaneous planned value pv(t)w is the work rate, or the PNR-curve,
while the cumulative PV(t)w is the cumulative sum of the instantaneous pv(t)w:
pv(t)w = PNR(t) =Nt
T 2exp(− t2
2T 2
)(3.1)
PV (t)w =
t∫0
pv(s)w ds = N[1− exp
(− t2
2T 2
)](3.2)
To plot these curves, the parameter values for N and T are needed. As we are in the
planning phase, these two parameter values have to be determined at the start of the
project. As discussed in section 3.2.1, N is equal to the determined BAC. The time of
the labor peak T is somewhat less obvious to determine. Two approaches, both making
use of the traditional EVM planned value curve, can be followed. The first possibility
to determine T is by looking at which time unit t 40 % of the BAC, or N, is planned
to be finished according to the traditional EVM PV curve. The second possibility is to
determine which time instance t minimizes the squared deviation between Warburton’s
pv(t)w-curve and the traditional pv-curve. A more thorough discussion concerning these
two methods to determine T, and which of the two is preferably used, will be held in
chapter 8.
Chapter 3. A review of the time-dependent Earned Value model 26
Earned Value
The ev(t)w-curve is intended to be generated after some part of the project is already
completed, contrary to the pv(t)w-curve which is established before the start of the
project. Warburton’s model is intended to set up this curve for the whole project by only
using early project data, contrary to the traditional EVM ev-curve which is constantly
updated at every time instance until the end of the project.
R.D.H. Warburton [21] states that ‘in each interval, a fraction r of the activities were
not completed as expected, were therefore rejected and should not be considered as
having earned value. The other activities, being a fraction of 1-r, were completed and
have earned value. Warburton’s model says that in the beginning of the project, in the
interval t < τ , the earned value is simply the planned value of the fraction of activities
that were successfully completed. For t > τ , the earned value consists of two contributing
parts. There’s the fraction of activities that were successfully completed at time t, and
those from t-τ that were delayed and are now complete.’ This way the ev(t)w-curve is
developed, and the EV(t)w-curve follows by integrating the instantaneous values:
ev(t)w =
(1− r)pv(t)w, t ≤ τ
(1− r)pv(t)w + r.pv(t− τ)w, t > τ
(3.3)
EV (t)w =
N(1− r)
[1− exp
(− t2
2T 2
)], t ≤ τ
N −N(1− r)exp(− t2
2T 2
)− r.N.exp
(− (t−τ)2
2T 2
), t > τ
(3.4)
For a large t it is clear that EV(t → ∞)w=N=BAC, which simply means that at the
end of the project all activities are completed.
To plot these curves, four parameter values are needed. N and T were already determined
at the start of the project, as addressed above. Parameters r and τ however, will be
determined with early project data, which is data available after some part of a project
Chapter 3. A review of the time-dependent Earned Value model 27
is completed. This can be, for example, when 30 % of N is earned. To determine r, one
should look at the early project data and determine the amount of finished activities.
Then one should determine which fraction of these completed activities have incurred
a delay and were thus not completed within their planned duration. The value for τ is
then simply the average delay of these rejected activities.
r =# finished activities with real duration > planned duration
# finished activities(3.5)
τ =
∑(real duration− planned duration)of each rejected activity
#rejected activities(3.6)
Actual Cost
As stated by R.D.H. Warburton [21], ‘the actual cost curve is, similar to the ev(t)w-
curve, intended to be generated using early project data. Warburton’s model assumes
that, initially, the labor was executed according to plan, but a fraction r of the activities
were not completed because the foreseen amount of labor was not enough. Therefore
the actual cost is assumed to be equal to the planned value for t < τ . The extra cost to
finish the rejected activities will be incurred in the future. That is why for t > τ , the
instantaneous ac(t)w values include the cost for both the work performed on activities
that were successfully completed at time t, as well as the cost for completing the rejected
activities of time instance t−τ that are now completed. As defined above, the fractional
extra cost is presented with parameter c.’ This leads to the formulas for both the ac(t)w-
and AC(t)w-curve:
ac(t)w =
pv(t)w, t ≤ τ
pv(t)w + r.c.pv(t− τ)w, t > τ
(3.7)
AC(t)w =
N[1− exp
(− t2
2T 2
)], t ≤ τ
N[1− exp
(− t2
2T 2
)]− r.c.N
[1− exp
(− (t−τ)2
2T 2
)], t > τ
(3.8)
Chapter 3. A review of the time-dependent Earned Value model 28
In the above formulas, r.pv(t-τ)w represents the fraction of the previously rejected ac-
tivities, while c accounts for the additional cost to finish these activities.
To plot these curves, the already discussed parameters N, T, r and τ are again needed,
of which the first two are determined at the beginning of the project, while the latter
two are determined using early project data. Similar to r and τ , the value for c is again
determined by looking at the available early project data at the moment of establishing
the ac(t)w-curve. The value for c is calculated as folows:
c =
∑((actual cost− planned cost)/ planned cost)of each rejected activity
#rejected activities(3.9)
An overview of the parameters and formulas of Warburton’s model is given in the ap-
pendix in tables C.1 and C.2 on pages 169 and 170 respectively, under ‘the initial War-
burton model’. Furthermore, a sketch of how these curves of Warburton’s model look
like is given in the figures below.
Figure 3.1: Example of the instantaneous curves (left) and the cumulative curves (right) of
Warburton’s model.
Chapter 3. A review of the time-dependent Earned Value model 29
3.2.3 Making predictions of the final project cost and duration with
Warburton’s model
This section discusses how the model of section 3.2.2 can be used to make estimations
of the final project cost EACw and duration EAC(t)w, based on early project data.
Predicting the final cost
Warburton’s model is able to predict the final cost of a project, using early project data,
by looking for the final value of the cumulative AC(t)w-curve:
EACw = AC(t→∞)w = N(1 + r ∗ c) (3.10)
This means that once the values of parameters N, r and c are determined, the final
project cost can be estimated. As addressed, N is already known at the beginning of the
project, while r and c are estimated by looking at the early project data.
An important note concerning this prediction is that it is constant, as it only depends
on the fundamental project parameters of Warburton’s model, which are assumed to
be constant over the lifetime of the project. Therefore, it is possible to generate rather
stable final cost predictions with an early determination of the parameters r and c, which
is an advantage for project managers.
Predicting the final duration
Although R.D.H. Warburton suggests the model can be used to predict the final duration
of the project, no specific method for this is brought forward. One could assume the final
duration can be found by looking for the duration after which the project is completed,
or with other words, when the total value of N or the BAC is earned:
EAC(t)w = time unit t at which EV (t)w equals the BAC (3.11)
However, this is an inaccurate way of approaching the problem as the final duration will
be highly overestimated due to the long tail of the ev(t)w curve. This is considered to
Chapter 3. A review of the time-dependent Earned Value model 30
be a first disadvantage of the current model, which will be further discussed in Chapter
5 where we will take a look at some shortcomings of Warburton’s model and how we can
approach the problem of forecasting the final duration.
3.3 Differences with the traditional EVM method
3.3.1 Functional time dependence
The often overlooked functional time dependence which was discussed in section 3.1
is integrated in Warburton’s model. This leads to the possibility of establishing con-
stant forecasts of the final project duration (EAC(t)w) and cost (EACw) based on early
project data, instead of constantly changing EAC and EAC(t) values as with the tradi-
tional EVM method. Remember that only for EACw a method was brought forward in
Warburton’s model, while the method for EAC(t)w is not specified.
3.3.2 Early project data
Important to stress is that the time-dependent curves of Warburton’s model will be
established using so-called early project data. This means that the necessary parameter
values to establish the curves will be determined after the project has started and already
some percentage, e.g. 30 %, of the BAC is earned. All the real project execution data
one has available at that moment, e.g. all the data available once 30 % of the project is
completed, is the so-called early project data.
Once the parameter values are determined, Warburton’s model can establish all the
necessary curves to make predictions about the final project cost and duration. So,
contrary to the traditional EVM method which constantly updates its earned value
and actual cost curves with real project execution data, Warburton’s model eyes at
establishing reliable earned value and actual cost curves for the whole project, only
using early project data, as it is at this early stage of a project that warning signals for
project managers are most useful.
Chapter 4
Example Project
In order to demonstrate Warburton’s model that was discussed in chapter 3 and the
impact of each parameter, a comprehensive example project is presented. Figure 4.1
contains the activity-on-the-node representation of the network with for each of the 12
non-dummy activities its planned duration and resource cost per time unit. Here a linear
cost of 20 cost units per time unit is assumed. Note that the start and end node are
dummy activities to which no duration or costs are allocated.
Figure 4.1: Example project: Activity-on-the-node representation.
31
Chapter 4. Example Project 32
For reasons of simplicity, no resource constraints were taken into account. Therefore
each activity of the project can be scheduled as soon as possible, resulting in a baseline
schedule with a duration of 13 time units (figure 4.2) and a budget at completion of 740.
Figure 4.2: Example project: Baseline schedule.
4.1 The Warburton method
Warburton’s pv(t)w-curve can be generated at the start of the project. Two parameter
values are needed for this: N and T. According to the definition of N, the value is
simply equal to the BAC. So in our example N equals 740. To determine T, one can
look for the value of this parameter that leads to the smallest squared deviation, or
least squares, between the Warburton pv(t)w-curve and the traditional EVM pv-curve.
A simple solver in Excel gives us a value of 5.2. By filling in these two values in the
aforementioned formula, the instantaneous pv(t)w and cumulative PV(t)w curve can be
generated.
The Warburton model is designed to make predictions about the final cost and duration
of a project based on data that is available in the early stages of a project, the so-
called early project data. In this example we will define early project data as all the
Chapter 4. Example Project 33
data concerning project execution that is available once 30 % of the total planned value
(BAC) is earned, which equals 222 (740 * 30 %) in our example. The representation of
the (fictitious) real execution of the project up until 30 % of the BAC is earned can be
found in figure 4.3.
Figure 4.3: Example project: (Fictitious) real project execution until 30 % of BAC is earned.
It is easy to see that a value of 222 is earned just over a period of 4 time units. After 4
time units, activities 1, 2 and 3 are completed, which adds up to an earned value of 160
(40 + 60 + 60). On top, activities 4 and 5 are partially completed, leading to earned
values of 20 and 40 respectively. This means that after four time units a value of 220
is earned and after 4.036 time units a value of 222 is earned, or 30 % of the project is
completed.
With this available data at this point in the project we can calculate the three remaining
parameters r, c and τ . Parameter r equals the fraction of the completed activities at time
instance 4 which weren’t completed in time and needed extra work. After 4 time units
activities 1, 2 and 3 are finished of which 1 and 3 both had one time unit delay. This
means r equals 0.67 (2/3). For parameter c, we calculate the average extra cost relative
to the planned cost of the activities that were delayed, being activities 1 and 3. In this
Chapter 4. Example Project 34
example parameter c equals 0.42 ( ((60-40)/40 + (80-60)/60)/2) ). Finally, parameter τ
equals the average extra duration of these activities, which is 1 (((3-2)+(4-3))/2 ). By
filling in these values in the aforementioned formulas, the Warburton curves for earned
value and actual cost can be generated, as displayed in figure 4.4 (instantaneous values)
and in figure 4.5 (cumulative values).
Figure 4.4: Example project: Instantaneous Warburton-curves based on available data after
30 % project completion.
Figure 4.5: Example project: Cumulative Warburton-curves based on available data after 30%
project completion.
As clearly shown in the figures above, the mathematical model has the disadvantage of
smoothing out with very long tails, leading to neglectable values towards the end of the
Chapter 4. Example Project 35
project. For the forecast of the final cost this doesn’t impose a problem, as we can simply
use the aforementioned formula N*(1+rc). In this example, this leads to an EACw of
946 ( 740*(1+0.67*0.42) ). However, this kind of formula is not available for the forecast
of the final duration EAC(t)w. This imposes a problem. The time until no more value
is added and the total BAC is earned in this example equals 198 time units, which is
indicated in table 4.1 on page 36, but not in the figures to maintain relevant graphs.
A way to cope with this is looking at the time at which the cumulative EV(t)w-curve
stabilizes on figure 4.5, which would be after about 19 time units, which could then be
the forecast of the final duration. However, this is not an accurate estimation as it is
subject to discussion of where it really stabilizes. This is further discussed in section 4.4
Remarks and observations.
4.2 End of project
To determine how good the forecasts of Warburton’s model actually are, the real final
duration and cost of the project are needed. The (fictitious) real project execution is
displayed below in figure 4.6.
Figure 4.6: Example project: (Fictitious) real project execution until 100% of BAC earned.
In this example project, the final cost is 880 and the real final duration is 18 time units.
Chapter 4. Example Project 36
Table 4.1: Example project: Planned, earned and actual cost values at each time instance.
Time unit pv(t)w ev(t)w ac(t)w PV(t)w EV(t)w AC(t)w
1 26.87 8.96 26.87 13.56 4.52 13.56
2 50.83 34.85 58.29 52.76 26.62 56.52
3 69.51 57.06 83.63 113.45 72.99 128.10
4 81.43 73.49 100.74 189.52 138.81 221.03
5 86.18 83.02 108.80 273.91 217.65 326.56
6 84.39 85.59 108.33 359.70 302.51 435.78
7 77.41 82.06 100.86 440.96 386.78 540.87
8 67.04 73.96 88.55 513.39 465.10 635.88
9 55.08 63.06 73.70 574.52 533.77 717.13
10 43.07 51.08 58.37 623.54 590.86 783.13
11 32.13 39.42 44.09 661.02 636.03 834.22
12 22.91 29.06 31.83 688.38 670.14 872.00
13 15.63 20.48 21.99 707.49 694.75 898.70
14 10.22 13.83 14.56 720.27 711.75 916.79
15 6.40 8.95 9.24 728.46 723.00 928.53
16 3.85 5.55 5.63 733.49 730.14 935.84
17 2.22 3.31 3.29 736.46 734.48 940.21
18 1.23 1.89 1.85 738.15 737.03 942.72
19 0.66 1.04 1.00 739.07 738.46 944.11
20 0.34 0.55 0.52 739.55 739.23 944.84
21 0.17 0.28 0.26 739.79 739.63 945.22
22 0.08 0.14 0.12 739.90 739.83 945.40
23 0.04 0.06 0.06 739.96 739.92 945.49
24 0.02 0.03 0.03 739.98 739.97 945.53
25 0.01 0.01 0.01 739.99 739.99 945.54
26 0.00 0.01 0.00 740.00 739.99 945.55
27 0.00 0.00 0.00 740.00 739.997816526582 945.55
28 0.00 0.00 0.00 740.00 739.999185243261 945.55
29 0.00 0.00 0.00 740.00 739.999706962054 945.56
30 0.00 0.00 0.00 740.00 739.999898414050 945.56
... ... ... ... ... ... ...
198 0.00 0.00 0.00 740.00 740.000000000000 945.56
Chapter 4. Example Project 37
4.3 Impact of each parameter on the model
In order to gain some insight in the impact of each parameter before starting the sim-
ulation study, the impact of each parameter on the shape of the Warburton curves and
on the EAC(t)w and EACw metrics is illustrated by using the example project. The
initial parameters are N=740, T=5.2, r=0.67, c=0.42 and τ=1, as calculated above.
The pv(t)w-, ev(t)w- and ac(t)w-curve were displayed in figure 4.4 on page 34. Each
parameter will now be adjusted separately: N and T will be doubled in value, r and c
will be brought to 1, and τ will be adjusted to a value of 3. A summary table about the
impact of each parameter on EAC(t)w and EAC is given at the end of this section.
4.3.1 Total amount of labor N
When doubling the value of N, which thus means doubling the BAC, all pv(t)w, ev(t)w
and ac(t)w values are doubled. Also EACw will be doubled to 1892, as this is calculated
by N*(1+r*c), while EAC(t)w will not be affected. It can be said that N doesn’t cause a
structural change in the shape and position of Warburton’s curves, except for a change
in the size of the values (as seen on the y-axis).
Figure 4.7: Example project: pv(t)w-, ev(t)w- and ac(t)w-curve after doubling the value of N.
Chapter 4. Example Project 38
4.3.2 Time of the labor peak T
When doubling the value of the time of the labor peak T, more time elapses before the
project is ended (as seen on the x-axis). Due to the shift to the right of the ev(t)w-
curve, EAC(t)w increases to 395 time units, which is the time at which EV(t)w reaches
the BAC. Although the distribution of the actual costs is changed, the value for EACw
is not affected as total costs remain the same.
Figure 4.8: Example project: pv(t)w-, ev(t)w- and ac(t)w-curve after doubling the value of T.
4.3.3 Reject rate of activities r
The graph on figure 4.9 illustrates the limited change in the pv(t)w- and ev(t)w-curve
when increasing the value of parameter r to 1, which means that all activities are rejected
at the time of calculation. The ac(t)w-curve, however, is lying higher as the rejected
activities require extra work and therefore cost more than planned. The EACw also
increases according to the formula N*(1+r*c) to a value of 1050.8 cost units. The
forecast for the final project duration, EAC(t)w, increases only to a very limited extent
and is rather irresponsive to a change in the parameter r.
Chapter 4. Example Project 39
Figure 4.9: pv(t)w-, ev(t)w- and ac(t)w-curve after after bringing r to a value of 1.
4.3.4 Cost overrun of rejected activities c
When increasing the value of parameter c to a value of 1, which means that on average
the total cost of a rejected activity has been doubled, figure 4.10 shows that only ac(t)w
is considerably affected and lies much higher now. Also EACw increases according to
the formula N*(1+r*c) to 1235.8 cost units. The forecast for the final project duration,
EAC(t)w, is not affected by a change in the parameter c, as the formulas for ev(t)w and
EV(t)w do not contain the parameter c.
Figure 4.10: pv(t)w-, ev(t)w- and ac(t)w-curve after bringing c to a value of 1.
Chapter 4. Example Project 40
4.3.5 Repair time of rejected activities τ
When increasing the value of parameter τ to a value of 3, which means that a rejected
activity has incurred an average delay of 3 time units, only the beginning of the curves
shows a different pattern, as can be seen in figure 4.11. Similar to r, EAC(t)w increases
but again only to a very limited extent. This is a first indication that EAC(t)w is rather
irresponsive to the parameters r and τ . Although EACw is not explicitly affected by
the parameter τ , it is by parameter c which changes hand in hand with parameter τ as
activity costs are determined by the duration of an activity.
Figure 4.11: pv(t)w-, ev(t)w- and ac(t)w-curve after bringing τ to a value of 3.
4.3.6 Conclusion
Table 4.2 on page 41 provides a summary of the effect of an increase in each parameter
on Warburton’s model. Chapter 5 will handle the shortcomings of Warburton’s model
that were already found by taking a critical look at the mathematical formulation of the
model and the example illustrated in this chapter.
Chapter 4. Example Project 41
Table 4.2: Summary table of the effect of an increase in each parameter on Warburton’s model
with ∼ ↑ indicating a very small increase.
Parameter Effect on Effect on Main influence on
EAC(t)w EACw
N ↑ = ↑ Magnitude of the curve values
T ↑ ↑ = Position of the peak in the curves
r ↑ ∼ ↑ ↑ Project scope
c ↑ = ↑ Cost
τ ↑ ∼ ↑ = Time
Chapter 5
Critical analysis of the model and
set-up duration forecasting
methods
Before starting the accuracy study, a thorough look is taken at the theoretical model
of Warburton that was described and illustrated in chapters 3 and 4. In section 5.1,
shortcomings of the model that were found after a critical analysis are discussed. After
the accuracy study conducted in part 4, we will come back on this in chapter 14 and see
if any additional remarks can be made then.
As discussed in chapter 3 and 4, no specific method has been brought forward in R.D.H.
Warburton’s paper to calculate parameter T and to forecast the final duration. That is
why we looked for possible methods to do this, so these can then be used in the accuracy
study of part 4. Section 5.2 will handle the calculation of parameter T. In section 5.3,
seven own developed EAC(t)w-methods to forecast the final duration of a project, using
Warburton’s model, are introduced.
42
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 43
5.1 Shortcomings of Warburton’s model
5.1.1 Accelerations
After taking a look at the definitions of the parameters, it quickly becomes clear that
the model does not take into account schedule accelerations or, in other words, the
impact of activities that are finished earlier than planned. The definition of parameter
r, the reject rate of activities, says this is the fraction of the activities that are not
finished within their planned duration and are therefore rejected and need extra work.
However, Warburton’s model does not foresee an adjustment of activities that meet
their predefined goal earlier than planned, or are with other words accelerated and are
ahead of schedule. This is a crucial shortcoming of the model, considering the other
basic parameters c and τ won’t be adjusted either. Parameters c and τ , respectively
the cost overrun and time to repair the rejected activities, are both calculated based
on the rejected activities determined by parameter r. R.D.H. Warburton also explicitly
refers to the shortcoming of the mathematical model to cope with negative values for τ
(schedule accelerations) because of the time delay terms in the formulas, e.g. pv(t-τ)w,
in his recommendations for future research [21].
This shortcoming becomes of more critical importance for the accuracy of Warburton’s
model when the proportion of activities that are accelerated increases. Therefore, the
performance of Warburton’s model in early projects is expected to be highly affected by
the lack of parameter adjustments, as a high proportion of the activities have a shorter
real duration than initially foreseen and will not be taken into account when calculating
the fundamental parameters. In contrast, in late projects most of the activities will be
delayed, meaning the shortcoming for activity accelerations is not expected to have a
big impact on the performance of Warburton’s model in these late projects.
This shortcoming can also be illustrated with the example project of chapter 4. Looking
at figures 4.2 and 4.3, we can see that activity 2 was initially planned to take 3 time units,
but only took 2 time units in reality. However, the calculation of r, c and τ doesn’t take
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 44
into account this acceleration as they only look at the activities that incurred a delay.
In part 5, once the accuracy study of Warburton’s initial model is finished and discussed,
we will take a look at how we can deal with this shortcoming and improve the model.
5.1.2 Critical path
Because of the mathematical formulas that determine the pv(t)w-, ev(t)w- and ac(t)w-
curves, Warburton’s model doesn’t make a distinction between critical and non-critical
activities. A delay in a critical activity has an immediate impact on the final duration
of the project, while this is not necessarily true for a non-critical activity. Warburton’s
model is not capable of taking this into account when determining its parameters r, c
and τ . Although final conclusions about the performance of Warburton’s model can only
be made after the accuracy study of part 3, the absence of critical path dependency is a
remarkable feature of the model.
5.1.3 Forecasting the final project duration
As addressed in section 3.2.2, R.D.H. Warburton suggests the model can be used to
predict the final duration of the project, but no specific method for this is brought
forward. One could assume the final duration can be found by looking for the duration
after which the project is completed, or with other words, when the total value of N or
the BAC is earned. We will refer to this method as EAC(t)w0:
EAC(t)w0 = time unit t at which EV (t)w equals the BAC (5.1)
However, because of the mathematical formulation of the model, which works with
exponential factors and has to deal with long tails in its curves, we expect this method
of forecasting the final duration to lead to large overestimations. Indeed, towards the
end of the project, the value of ev(t)w and the increase of EV(t)w will be very small,
and a lot of time units go by until the project is completed, i.e. N is reached. This can
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 45
be made clear when looking at equations 3.1 and 3.3 of chapter 3:
ev(t)w =
(1− r)pv(t)w, t ≤ τ
(1− r)pv(t)w + r.pv(t− τ)w, t > τ
pv(t)w = PNR(t) =Nt
T 2exp(− t2
2T 2
)
Here, it can be clearly seen that the further we go in time and the larger t, the smaller
pv(t)w becomes as the exponential factor, by which Nt/T2 gets divided, increases with
t. This means that, towards the end of the project, very small pv(t)w- and ev(t)w-values
will occur, leading to long tails in these curves. This problem is also illustrated in the
example project of chapter 4, where the time until no more value is added and the total
BAC is earned equals 198 time units, which is a large overestimation of the final duration
of 18 days. Notice that this means that both the planned duration as real duration are
overestimated because of these long tails. Also notice that the EACw method to forecast
the final project cost is not affected by this long tail problem, as EAC equals N(1+rc).
In an attempt to handle this shortcoming, seven own developed EAC(t)w-methods are
introduced in section 5.3.
5.1.4 Cumulative values
Finally, a rather surprising remark can be made concerning the relation of the instan-
taneous and cumulative values of Warburton’s model. As illustrated by the example
project in table 4.1 on page 36, the cumulative values at a certain time instance do not
equal the exact sum of the instantaneous values up until this time instance. This is a
direct result of the mathematical formulation of the model.
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 46
5.2 Calculation methods for parameter T
No specific way of determining parameter T was brought forward by R.D.H. Warburton.
Two ways of calculating parameter T, to which we will refer as T1 and T2, are presented
here.
5.2.1 Calculation T1
The first way of calculating parameter T is by filling in t=T in the cumulative PV(t)w
formula:
PV (T )w = N[1− exp
(− T 2
2T 2
)]= N
[1− exp
(−1
2
)]= N(1− 0.606) ∼ 0.40N (5.2)
This means, at t=T, the project is at the 40 % completion point or, with other words,
40 % of the BAC (=N) is earned. Now it is known that the time of the labor peak, T,
is reached once 40 % of the project is completed and can be determined by looking for
the time unit t that corresponds to this value in the traditional PV-curve:
T = time t at which the traditional PV equals 0.40N (5.3)
5.2.2 Calculation T2
The second way of calculating parameter T is on the basis of a solver which uses the least
squares method to determine the best possible fit between Warburton’s pv(t)w-curve and
the traditional pv-curve:
T = time unit t which minimizes
PD∑t=0
(pv − pv(t)w)2 (5.4)
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 47
5.3 Methods to forecast the final project duration using
Warburton’s model
Except for the EAC(t)w0-method that was introduced in section 5.1.3 above, no specific
method to forecast the final duration with Warburton’s model exists so far. That is why
seven own developed EAC(t)w methods are presented in this section that can be used in
the accuracy study of part 4. Also, each method is illustrated using the example project
of chapter 4.
5.3.1 Eight methods to forecast the final project duration
Before getting into the different methods, we should first agree on some terminology. PD
and RD are used to refer to the traditional planned and real duration. EAC(t)wx is used
to refer to the forecasting method for the final duration based on Warburton’s model,
with ‘w’ referring to Warburton and ‘x’ referring to one of the eight methods discussed
below. PDw0 is used to refer to the planned duration according to Warburton’s model,
which is the time at which PV(t)w equals the BAC or N. Although the ev(t)w-curve is
the most important curve in this section, the pv(t)w-curve also plays a prominent role in
some of the proposed methods as it forms the basis of the ev(t)w-curve, as can be seen
in equation 3.3.
Method 0
The EAC(t)wo -method was already discussed in section 5.1.3.
EAC(t)w0 = time unit t at which EV (t)w equals the BAC (5.5)
This method is expected to lead to large overestimations of the RD because of the long
tails of the model. In the project example of chapter 4, EAC(t)w0 was equal to 198 time
units.
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 48
Method 1
This method uses the long tail-problem of the pv(t)w-curve as a starting point. It
uses the ratio of the traditional PD to Warburton’s PDw0 to correct the EAC(t)w0 for
overestimation because of the long tails:
EAC(t)w1 = EAC(t)w0 ∗PD
PDw0(5.6)
In the example project of chapter 4, PD is equal to 13 time units, as can be seen in the
baseline schedule of figure 4.2, while PDw0 can be found by looking for the t at which
PV(t)w equals the BAC, being 740. In our example this is at t=44. By consequence
EAC(t)w1 would be 198 * 13/44 = 59 time units.
Method 2
This method uses a totally different approach to forecast the final duration. First, the
PV(PD)w-value is determined, which is the cumulative planned value of Warburton’s
model that is reached at de traditional planned duration of the project. Once this value
is known, EAC(t)w2 is determined by looking for the time t at which EV(t)w reaches
this value:
EAC(t)w2 = time t at which EV (t)w equals the value of PV (PD)w (5.7)
In the example project, the PD equals 13 time units. Using this value for t in the
PV(t)w-curve gives us a value of 707.4867. By consequence, EAC(t)w2 equals the time
unit t at which EV(t)w reaches a value of 707.4867, which is after 14 time units, as can
be seen in table 4.1 on page 36.
Method 3
This method is based on the so-called ‘schedule slip’, which is de absolute deviation
between the planned and real duration of a project. Applying this concept to the War-
burton model means the deviation between the time units t at which the PV(t)w- and
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 49
EV(t)w-curve reache a value equal to the BAC or N. In other words, this equals the devi-
ation between respectively PDw0 and EAC(t)w0. The reasoning behind this is that both
time units will be overestimations of respectively the planned and real project duration,
as they both have to deal with the problem of the long tails, mentioned in section 5.1.3.
To determine EAC(t)w3, the traditional PD of the baseline schedule is used as a basis
and will be added with the schedule slip of Warburton’s model:
EAC(t)w3 = PD + schedule slipw (5.8)
with schedule slipw = EAC(t)w0 − PDw0 (5.9)
In our example project the schedule slipw equals 198-44 = 154 time units. By conse-
quence EAC(t)w3 = 13+154 = 167.
Method 4
This method is closely linked with the calculation of T1, discussed in section 5.2. This
method assumes it is possible to determine the planned duration and forecast the real
duration with the model once the peak in the labor curve, T, is known. As shown, at
t=T1, the project is at the 40 % completion point. Therefore, one could say the planned
duration according to Warburton’s model equals 2.5*T1 time units. To determine the
estimated final duration, the PV(t)w that is reached after 2.5*T1 time units should
be determined, after which the time t at which this value is reached in Warburton’s
EV(t)w-curve can be found:
EAC(t)w4 = time t at which EV (t)w equals the value of PV (2.5 ∗ T1)w (5.10)
In the example project, it was determined that the peak of labor occurred at t=5.2 time
units. This means we should determine the PV(t)w-value that is reached after 2.5*5.2=
13 time units. As can be seen in table 4.1 on page 36, this gives us a value of 707.
Therefore EAC(t)w4 equals the time t at which the EV(t)w-curve has reached a value of
707, which is after 14 time units.
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 50
Method 5
Contrary to methods 1 to 4, methods 5 to 7 make use of a corrective factor that first
needs be determined by prior investigation. We will refer to these corrective factors
as ratiow5, ratiow6 and ratiow7 respectively, and their values will be determined in our
simulation study of part 3, chapter 8. Method 5 multiplies the average ratio of the real
project duration with EAC(t)w0:
EAC(t)w5 = EAC(t)w0 ∗ ratiow5 (5.11)
with ratiow5 =1
#projects n∗
n∑i=project 1
RD
EAC(t)w0∗ 100 (5.12)
In other words, the ratiow5 is intended to adjust the forecast of the final duration, which
is expected to be overestimated by EAC(t)w0 because of the long tails, to the actual
real duration. Of course the real duration of an individual project can’t be known in
advance, which is why a simulation will determine the value for ratiow5 by taking the
average of these ratio values of multiple fictitious project executions.
In the example project, the real duration is 18 time units, while EAC(t)w0 is 198 time
units, leading to a ratiow5 of 18/( 198) *100 = 9.09 %, leading to a EAC(t)w5 = 198*9.09
% = 18. This way of determining ratiow5 is not correctly conducted, as we used the RD
of this example project which in real life can’t be known in advance.
Method 6
The first step to calculate EAC(t)w6 is determining the value that the EV(t)w-curve
reaches when time unit t is equal to the real project duration RD. Ratiow6 is introduced
to make this possible. It is the average of a simulation of multiple fictitious project
executions in which the ratio of the EV(RD)w to the BAC or N is calculated. Once
this value is known, EAC(t)w6 equals the time t at which this value is reached in the
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 51
EV(t)w-curve:
EAC(t)w6 = time t at which EV (t)w equals the value of N ∗ ratiow6 (5.13)
with ratiow6 =1
#projects n∗
n∑i=project 1
EV (RD)wN
∗ 100 (5.14)
Ratiow6 is the average taken over multiple fictitious project executions during a prior
simulation study, as again, the individual EV(RD)w can’t be determined in advance.
In the example project, N equals 740 and RD is 18 at which the EV(t)w-curve reaches
a value of 737, as can be seen in table 4.1 on page 36. This leads us to a ratiow6 of
737/(740 )*100 = 99.59 %. So, EAC(t)w6 equals the time t at which the EV(t)w reaches
a value of 740*99.59 %=737, which is 18. Again, the way of determining ratiow6 in the
example is of course not correct due to the same reason as explained for method 5.
Method 7
The first step to calculate EAC(t)w7 is again determining the value that the EV(t)w-
curve reaches when time unit t is equal to the real project duration RD. Ratiow7 is
introduced to make this possible. It is the average of a simulation of multiple fictitious
project executions in which the deviation between EV(RD)w and the BAC or N is
calculated. Once this value is known, EAC(t)w7 again equals the time t at which this
value is reached in the EV(t)w-curve. So, method 7 is very similar to method 6, except
that it has another way of approaching the EV(RD)w value as ratiow7 has the same
elements as ratiow6, but rather than dividing them, the absolute deviation is calculated:
EAC(t)w7 = time t at which EV (t)w equals the value of (N − ratiow7) (5.15)
with ratiow7 =1
#projects n∗
n∑i=project 1
(N − EV (RD)w) (5.16)
In the example project this would lead to a ratiow7 of 740 - 737= 3, which means
EAC(t)w7 equals the time t at which EV(t)w reaches the value of 740-3 = 737, which
Chapter 5. Critical analysis of the model and set-up duration forecasting methods 52
is after 18 time units. Once again, the way of determining ratiow7 in the example is of
course not correct due to the same reason as explained for method 5.
5.3.2 Set-up of the methods to forecast the final project duration
The set up each method described in section 5.3.1 is somewhat different, but a main
distinction can be made between methods 1-4 and methods 5-7. Methods 1-4 all can
be applied at a certain point in the project using the available data (such as PD, PDw
and PV(PD)w). Methods 5-7, on the other hand, need a prior research to determine the
discussed ratios w5, w6 and w7, as these methods would otherwise require the unknown
RD of a project. A summary of the eight methods for forecasting the final project
duration is given in table 5.1.
Table 5.1: Formulas for calculating the eight methods to forecast the final duration.
Prior research? Method Formula
EAC(t)w0 = t at which EV(t)w equals BAC
EAC(t)w1 = EAC(t)w0 * PDPDw
No EAC(t)w2 = t at which EV(t)w equals the value of PV(PD)w
EAC(t)w3 = PD + schedule slipw
with schedule slipw = EAC(t)w0 - PDw
EAC(t)w4 = t at which EV(t)w equals the value of PV(2.5*T1)w
EAC(t)w5 = EAC(t)w0* ratiow5
with ratiow5 = 1projects n *
∑ni=project1
RDEAC(t)w0
∗ 100
Yes EAC(t)w6 = t at which EV(t)w equals the value of N*ratiow6
with ratiow6 = 1projects n *
∑ni=project1
EV (RD)wN ∗ 100
EAC(t)w7 = t at which EV(t)w equals the value of (N-ratiow7)
with ratiow7 = 1projects n *
∑ni=project1 (N-EV(RD)w)
Part III
SPECIFIC CHALLENGES AND
METHODOLOGY OF THE
SIMULATION STUDY
53
Chapter 6
Specific challenges, Research
questions and Hypotheses
6.1 Specific challenges
A first and crucial point is that the model proposed by R.D.H. Warburton was only tested
for its usefulness and performance on a single software project. A first challenge is to
investigate the general applicability and forecast accuracy of the model, as discussed
in section 6.1.1. Second, the performance of the model is benchmarked against the
existing EVM methods as described in section 6.1.2. Both of these challenges will be
investigated on the basis of the research questions of section 6.2 with the simulation
methodology of chapter 7. A third challenge is to learn from the accuracy study and
look for opportunities to improve the model.
6.1.1 General applicability and forecast accuracy
As thoroughly discussed in section 3.1, Warburton’s model is set up with the goal of
improving the theory of EVM by including time-dependence into the definitions of all
quantities. By integrating time-dependence into the planned value, earned value and
actual cost, Warburton’s model eyes at delivering precise estimates of the project’s final
54
Chapter 6. Specific challenges, Research questions and Hypotheses 55
cost and duration. Moreover, Warburton’s model also eyes at establishing these precise
estimates of the final cost and duration of a project in an early stage of the project.
Although the model originally was developed for software projects, it might well be
applicable in other situations also. The first big goal is to investigate whether this is the
case or not. The model will be tested in different scenarios, from early to late, and with
different project network structures. Specifics on how this will be done are discussed in
chapter 7 in which the methodology of the simulation study is addressed.
6.1.2 Comparison with traditional EVM methods
A second main goal is to benchmark the performance of Warburton’s model in terms
of forecasting accuracy of the final project cost and duration against the existing EVM
methods. These traditional forecasting methods were thoroughly addressed in chapter
2, of which an overview can be found in table 2.1 (time) on page 17 and table 2.2 (cost)
on page 18. The comparison will be done for various project network structures and
scenarios, which are defined in chapter 7.
6.1.3 Improve the initial Warburton model
A third goal is to look for opportunities to improve Warburton’s model after it was
thoroughly analyzed during the accuracy study of part 4, as some shortcomings will be
revealed. This will be the subject of part 5, in which new hypotheses will be stated for
the new Warburton model and a similar accuracy study as for the initial Warburton
model in part 4 will be done.
6.2 Research questions
In this section a short overview of the different issues that will be investigated in part 4
is given. For each research question the goal of the research is given and, when relevant,
an alternative hypothesis is stated.
Chapter 6. Specific challenges, Research questions and Hypotheses 56
6.2.1 Necessary input for Warburton’s model
Ratio values of the EAC(t)w5, EAC(t)w6 and EAC(t)w7 forecasting methods
First, the ratio values for forecasting methods 5, 6 and 7, defined section 5.2.1, have to
be determined by a prior simulation study. For the calculation of ratiow5 , ratiow6 and
ratiow7 , the reader is referred to equation 5.12, 5.14 and 5.16 respectively. In the first
section of chapter 8, the results will be presented.
Parameter T
Second, the best way to determine parameter T has to be investigated. In section 5.2,
two calculation methods for T were discussed, referred to as T1 and T2. In the second
section of chapter 8, it is investigated which way is the best choice.
6.2.2 Forecast accuracy
In chapter 9, the actual accuracy study starts. The methods that are used to forecast
the final project duration and final project cost based on Warburton’s model are tested
on accuracy and benchmarked against existing EVM forecasting methods. Remember
there are now eight methods to determine EAC(t)w, as discussed in section 5.3.1, and
one method to determine EACw, as discussed in section 3.2.3.
This research question investigates the average forecast accuracy across the whole project
duration, by taking the average of the forecast error values along different project comple-
tion stages. In each completion stage, parameters r, c and τ are updated or recalculated
using all the available data at that moment, resulting in a new EAC(t)w and EACw.
As Warburton’s model has the goal to improve the EVM theory by integrating time-
dependence and is designed to make predictions based on early project data, we could
expect the model to perform better than the existing EVM techniques in early stages of
the project, but probably not during the whole project lifetime. This is why we state
the hypothesis here that we don’t expect Warburton’s model to outperform any of the
existing EVM techniques with regard to the average forecast accuracy over the project
Chapter 6. Specific challenges, Research questions and Hypotheses 57
lifetime. On top, because of the shortcomings concerning accelerations, as discussed in
section 5.1.1, we expect Warburton’s model to perform significantly worse for projects
that end earlier than initially planned.
Hypothesis 1: On average, traditional EVM methods will be more accurate than War-
burton’s methods for forecasting both final duration and cost.
Hypothesis 2: Warburton’s model will perform significantly worse for projects that end
early compared to the forecast accuracy levels of the on time and late scenarios.
6.2.3 Project completion stage
As a second part of the accuracy study, the relation of the performance of Warburton’s
model and the project completion stage will be investigated in chapter 10. Contrary
to chapter 9, the forecast accuracy of the model will be investigated in different stages
of project completion rather than looking at the average performance along the whole
project execution. Similar to chapter 9, the forecasting methods based on Warburton’s
model will again be benchmarked against the performance of the traditional EVM meth-
ods. As mentioned in section 6.2.2 above, we expect the model to perform better than
the existing EVM techniques in early stages of the project.
Hypothesis 3: Forecasts for the final project duration and cost based on Warburton’s
model will be more accurate than the traditional EVM forecasting methods in the early
stages of project completion.
6.2.4 Topological structure
Project networks can have a different topological structure, based on their Serial or
Parallel (SP), Activity Distribution (AD), Length of Arcs (LA) indicator and Topological
Float (TF) indicator. These indicators are more thoroughly discussed in chapter 7 which
handles the methodology. Chapter 11 will handle the investigation of the influence of
each of these factors on the performance of Warburton’s model.
Chapter 6. Specific challenges, Research questions and Hypotheses 58
As the model was originally developed for software projects and does not differentiate
non-critical from critical activities, we expect the model to perform better for serial
project structures rather than for parallel ones when it comes to forecasting the final
duration. This is because the number of non-critical activities decreases for increasing
SP values which means a delay of an activity in serial projects is more likely to be a
delay of a critical activity. If so, it will have a direct impact on the final duration of a
project, and this is also more in line with the formulation of Warburton’s model. For
the other indicators, AD, LA and TF, we expect no clear impact on the final project
duration in Warburton’s model.
Concerning the final cost forecasts, we don’t expect the model to be influenced by the
topological structure. Also not by the SP-factor, as deviations in the cost of each activity
have an equal impact on the final project cost, irrespective of being a critical or non-
critical activity.
Hypothesis 4: Higher values of the SP indicator goes along with an improving forecast
accuracy of the final project duration.
Hypothesis 5: The forecast accuracy of the final project cost is rather insensitive to-
wards all four topological indicators (SP, AD, LA and TF).
6.2.5 Linear, convex and concave time/cost-relationship
During the research done concerning the forecasting accuracy in chapter 9, 10 and 11,
a linear time/cost-relationship will be assumed. This means that when, for example, an
activity takes twice as long as planned, the actual cost will also be twice as much as the
planned cost. However, interesting to know is whether a convex of concave time/cost-
relationship would lead to different results of the accuracy of Warburton’s cost forecast.
This will be the subject of chapter 12.
Chapter 7
Methodology of the simulation
study
In order to test the forecast accuracy of Warburton’s model and benchmark it against
the traditional EVM techniques, a simulation study is set up. Figure 7.1 provides an
overview of the methodological approach that is followed in this study. Every aspect of
this figure will be explained in detail during this chapter.
Figure 7.1: Overview of the methodology of the simulation study.
59
Chapter 7. Methodology of the simulation study 60
7.1 Project data
To test the general applicability and forecast accuracy of Warburton’s model, the sim-
ulation study will be applied to a set of random network structures with a different
topological structure. To do this a random network generator was used, called RanGen2
[13].
7.1.1 Topological network indicators
The topological structure of a network can be measured by the combination of four
topological indicators. These are the Serial or Parallel (SP), the Activity Distribution
(AD), the Length of Arcs (LA) and Topological Float (TF) indicator. We will shortly
explain the meaning of each indicator. For an extensive study and calculation of each
indicator, the reader is referred to ‘Measuring Time’ [12].
Serial or parallel indicator: SP
As described by Vanhoucke in ‘Measuring Time’ ([12], p.57), ‘the SP indicator measures
the closeness of the project network to a parallel or a serial network and SP ∈ [0,1].
When SP=0 then all activities are in parallel, when SP=1 then all activities lie in a
straight line sequence and the project is completely serial. SP values between these two
extremes represent networks close to a serial or parallel network’.
Activity distribution: AD
As described by Vanhoucke in ‘Measuring Time’ ([12], p.57),‘the AD indicator measures
the distribution of project activities along the levels of the project and AD ∈ [0,1]
and takes the width of each level (the number of activities at that level) into account.
When AD=0, all levels contain a similar number of activities and results in an uniformly
distribution of the activities over the levels. When AD=1, one level contains a maximal
number of activities as the other levels contain all a single activity’. The AD indicator
serves as a measure for the workload variability.
Chapter 7. Methodology of the simulation study 61
Length of arcs: LA
As described by Vanhoucke in ‘Measuring Time’ ([12], p.57), ‘the LA indicator, ∈ [0,1],
measures the length of each precedence relation (i,j) in the network as the difference
between the level of the end activity j and the level of the start activity i. When LA=0,
the network has many precedence relations between two activities on levels far from
each other and hence the activity can be shifted further in the network. When LA=1,
many precedence relations have a length of one, resulting in activities with immediate
successors on the next level of the network, and hence little freedom to shift’.
Topological float: TF
As described by Vanhoucke in ‘Measuring Time’ ([12], p.57), ‘the TF indicator, ∈ [0,1],
measures the topological float of a precedence relation as the number of levels each
activity can shift without violating the maximal level of the network (as defined by
SP). Hence, TF=0 when the network structure is 100 % dense and no activities can be
shifted within its structure with a given SP value. A network with TF=1 consists of one
chain of activities without topological float (they define the maximal level and hence,
the SP value) while the remaining activities have a maximal float value (which equals
the maximal level, defined by SP, minus 1)’.
7.1.2 Datasets
To investigate the forecast accuracy and the influence of the four topological indicators,
four different sets of networks have been generated using a random network generator
called ‘Rangen2’ [13]. Each network consists of 30 activities which have a duration of
maximum 20 time units. Furthermore, no resource constraints are taken into account.
The settings of each set are summarized below.
Set 1: Serial or parallel network (SP)
SP = { 0.1; 0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8; 0.9 } and random values for AD, LA, TF from interval [0,1].
Chapter 7. Methodology of the simulation study 62
Set 2: Activity distribution (AD)
Subset 2.1: AD = { 0.2; 0.4; 0.6; 0.8} with SP=0.2 and random values for LA en TF from interval [0,1].
Subset 2.2: AD = { 0.2; 0.4; 0.6; 0.8} with SP=0.5 and random values for LA en TF from interval [0,1].
Set 3: Length of arcs (LA)
Subset 3.1: LA = { 0.2; 0.4; 0.6; 0.8} with SP=0.2 and random values for AD en TF from interval [0,1].
Subset 3.2: LA = { 0.2; 0.4; 0.6; 0.8} with SP=0.5 and random values for AD en TF from interval [0,1].
Subset 3.3: LA = { 0.2; 0.4; 0.6; 0.8} with SP=0.8 and random values for AD en TF from interval [0,1].
Set 4: Topological float (TF)
Subset 4.1: TF = { 0.2; 0.4; 0.6; 0.8} with SP=0.2 and random values for AD en LA from interval [0,1].
Subset 4.2: TF = { 0.2; 0.4; 0.6; 0.8} with SP=0.5 and random values for AD en LA from interval [0,1].
Subset 4.3: TF = { 0.2; 0.4; 0.6; 0.8} with SP=0.8 and random values for AD en LA from interval [0,1].
For each SP factor of dataset 1, 100 project network instances have been generated, re-
sulting in a total of 900 project network instances. For each AD factor and its subsequent
SP factor of dataset 2, 100 project network instances have been generated, resulting in a
total of 2 * 400 = 800 project network instances. For each LA factor and its subsequent
SP factor of dataset 3, 100 project network instances have been generated, resulting in
a total of 3 * 400 = 1200 project network instances. And for each TF factor and its
subsequent SP factor of dataset 4, 100 project network instances have been generated,
resulting in a total of 3 * 400 = 1200 project network instances. In total, 4100 project
baseline schedules have been generated.
For the accuracy study in chapters 9 and 10, dataset 1 will be used without splitting up
the results per SP-factor. That way, a dataset of completely random project network
structures can be used for this research. In chapter 11, which handles the topological
structure, all four datasets will be used.
Chapter 7. Methodology of the simulation study 63
7.2 Project scheduling
Using the networks of the generated datasets, the planning for project execution will
be set up as is done at the start of the project. On the one hand, this will be done
with Warburton’s model. As discussed in chapter 3, this means determining the values
of parameters N and T, which can be calculated at the start of a project. With these
parameters the pv(t)w- and PV(t)w- curve can be generated. On the other hand, tradi-
tional baseline scheduling will be done. The Earliest Start Schedule (ESS) based on the
critical path-method will be set up which leads to the traditional pv- and PV-curve.
7.3 Project execution
Using a Monte Carlo simulation, 100 fictitious real life executions will be generated for
each baseline project schedule that was set up in the project scheduling step. Variation in
the duration and cost of activities will be simulated on the basis of six scenarios: a normal
and more extreme case for an early (PD<RD), on time (PD=RD) and late (PD>RD)
scenario. To set up these scenarios, triangular distributions are used, which are discussed
in section 7.3.2. Important assumptions that are at the basis of the simulation study
are discussed in section 7.3.3.
7.3.1 Monte Carlo simulation
As described by Ricardo Viana Vargas ([20], p.7), ‘Monte Carlo simulation is a method
in which the distribution of possible results is produced from successive recalculations
of the data of the project, allowing the construction of multiple scenarios. In each one
of the calculations, new random data is used to represent a repetitive and interactive
process. The combination of all these results creates a probabilistic distribution of the
results.’
In this study, Monte Carlo simulations are used to evaluate the accuracy of the forecast-
ing methods. As described above, a Monte Carlo simulation is a repetitive process of
Chapter 7. Methodology of the simulation study 64
multiple runs. Each simulation run generates a fictitious real duration for each activity
of the project, given its predefined uncertainty profile. More information about the dis-
tribution functions and possible deviations between fictitious real and planned activity
duration is given in section 7.3.2 which handles the triangular distributions.
During each simulation run that imitates fictitious project progress, the forecasts based
on Warburton’s model for final project duration (EAC(t)w methods, discussed in section
5.3.1) and cost (EACw method, discussed in section 3.2.3) can be calculated, together
with the forecasts based on the traditional EVM methods which were discussed in section
2.4.
After each Monte Carlo simulation run the fictitious real project status, being the real
project duration (RD) and real project cost (AC), will be known. These can of course
be different from the planned duration and planned cost. This means that after each
simulation run, the forecast accuracy of the different methods can be determined. This
is done by calculating the MAPE and MPE ratios, which are discussed in section 7.4,
Project Monitoring.
7.3.2 Triangular distributions and scenarios
Each project that is executed in real life is characterized by an inherent uncertainty
concerning multiple factors such as duration and cost. This is the reason why single
point estimates for project duration often lead to unrealistic project estimates, as they
don’t include risk and, above that, the estimate used for the duration of each project is
the one of the baseline schedule. Therefore, we opt to make use of triangular distribution
functions that will lead to a more realistic analysis of each project run, as the progress
of each run will include changes of activity durations compared to the original point
estimates of the baseline schedule.
Thus, in this study, a Monte Carlo simulation with triangular distributions is set up.
As mentioned by the Project Management Knowledge Center [16], ‘the triangular dis-
Chapter 7. Methodology of the simulation study 65
tribution is a continuous probability distribution which can easily be used by estimating
three parameters: a lower limit ‘a’, an upper limit ‘b’ and a mode ‘m’. The lower limit a
indicates the probability of being early, while the upper limit b does the same for being
late. The mode m is simply an indicator for the probability of being on time. This
means that triangular distributions can be used to express risk as a degree of skewness
in which an activity is subject to risk within a certain range. A triangular distribution
skewed to the left means an activity is more likely to be early than late, while a skew-
ness to the right indicates the opposite. A symmetric distribution means the probability
of incurring duration a or b are symmetric below and above the average m.’ This is
represented in figure 7.2.
Figure 7.2: Parameters of triangular distributions [16].
During this simulation study, the upper limit a and b are calculated based on the planned
duration of an activity. This is done by multiplying this planned duration by a factor
smaller than 1 for activity accelerations and bigger than 1 for activity delays. These
multiplying factors are fixed for all activities of the same network that is ran in a certain
scenario. Remark that this does not mean that a and b will have the same values for all
activities, as they depend on the planned duration of each activity.
Using these triangular distributions, six scenarios have been created that will be used
throughout the simulation study. In figure 7.3 ‘di’ stands for ‘planned duration of activity
i’, which matches the factor m that is discussed above. Remember this planned duration
di is maximum 20 time units in the datasets used for this study. Further, the lower and
Chapter 7. Methodology of the simulation study 66
upper limits of the factors by which di is multiplied are indicated.
Figure 7.3: Parameter values of triangular distributions for each scenario used in the simulation
study.
7.3.3 Assumptions
Resources
It is assumed that resources are available at all times, so no limitations on the project
scheduling is put as a consequence of resources. This means all activities can be started
earlier or later than planned, depending on the progress of preceding activities.
Time/cost relationship
In the study conducted in part 4 of this thesis, a linear relationship between costs and
the duration of each activity is assumed, with a variable cost of 20 cost units per time
unit. The formulation of the applied linear time/cost relationship is defined below. In
chapter 12 it is investigated whether a convex or concave time/cost relationship would
Chapter 7. Methodology of the simulation study 67
lead to different results.
Activity cost = V C ∗AD (7.1)
with
VC= variable cost factor=(20 cost units/time unit)
AD= activity duration
Activity duration
As discussed in the section above, we assume that the duration of an activity is dis-
tributed according to a triangular distribution with parameters dependent on the sce-
nario, as displayed in figure 7.3.
Corrective actions
It is assumed that, once a project is started, no corrective actions can be taken to get a
project that is running behind schedule. We realize that corrective actions are common
in real-life project executions, but it is difficult to include this in the model and outside
the scope of this thesis. We would like to refer to the literature for a more elaborate
discussion of the possible effects of corrective actions on project execution [22].
7.4 Project monitoring
To measure the forecast accuracy and investigate the performance of Warburton’s model
compared to EVM, all relevant data will be stored in a database to be analyzed. This
includes data such as the instantaneous planned value, earned value and actual cost val-
ues, the EAC and EAC(t) forecasts, and the Mean Absolute Percentage Error (MAPE)
and Mean Percentage Error (MPE) values.
Chapter 7. Methodology of the simulation study 68
The Mean Absolute Percentage Error (MAPE) and Mean Percentage Error (MPE) calcu-
late respectively the absolute and relative deviations between the periodic time (EAC(t))
and cost (EAC) predictions and the final project duration (RD) and cost (AC). Both
the MAPE as the MPE are calculated for each fictitious project execution, from which
average MAPE and MPE values can be deducted.
Time Forecast Error
MPEtime =1
T
T∑time=1
EAC(t)−RDRD
∗ 100 (7.2)
MAPEtime =1
T
T∑time=1
|EAC(t)−RD|RD
∗ 100 (7.3)
Cost Forecast Error
MPEcost =1
T
T∑time=1
EAC −ACAC
∗ 100 (7.4)
MAPEcost =1
T
T∑time=1
|EAC −AC|AC
∗ 100 (7.5)
with
T: Number of periodic reviews (= number of EAC and EAC(t) values)
EAC(t): Time forecast (at each periodic review period)
EAC: Cost forecast (at each periodic review period)
RD: Real duration (known upon completion of each simulation run)
AC: Actual cost (known upon completion of each simulation run)
Part IV
ACCURACY STUDY OF
WARBURTON’S MODEL
69
Chapter 8
Necessary input for Warburton’s
model
The research done in this chapter must be seen as prior research before the accuracy
study can be started. This prior research is necessary to determine some unknown
factors that are needed in the model. First, in section 8.1, the ratio values for forecasting
methods 5, 6 and 7 of section 5.3 will be determined. Second, in section 8.2, the best
way to calculate parameter T is determined.
70
Chapter 8. Necessary input for Warburton’s model 71
8.1 Ratio values of methods EAC(t)w5, EAC(t)w6 and EAC(t)w7
The ratio values for methods EAC(t)w5, EAC(t)w6 and EAC(t)w7 are, as explained in
section 5.3, calculated as follows:
ratiow5 =1
#projects n∗
n∑i=project 1
RD
EAC(t)w0∗ 100 from eq. 5.12
ratiow6 =1
#projects n∗
n∑i=project 1
EV (RD)wN
∗ 100 from eq. 5.14
ratiow7 =1
#projects n∗
n∑i=project 1
(N − EV (RD)w) from eq. 5.16
As discussed, it is not possible to know the real duration of a project on beforehand,
though it is possible to determine the average values of the ratios by simulating multiple
fictitious project executions. Notice that, as described in section 5.2, there are two
ways of calculating the parameter T, to which we refer as T1 and T2. As the research
concerning whether T1 or T2 amounts to the best results can only be done once the
necessary ratio values of EAC(t)w5, EAC(t)w6 and EAC(t)w7 are known, both T1 and
T2 are used for each method. The average ratio values for each scenario can be found
in figures 8.1 and 8.2.
Figure 8.1: Average ratio values (y-axis) for methods EAC(t)w5 and EAC(t)w6 for each scenario
(x-axis).
Chapter 8. Necessary input for Warburton’s model 72
Figure 8.2: Average ratio values (y-axis) for method EAC(t)w7 for each scenario (x-axis).
The figures show that the ratio values differ per scenario. In particular, the ratio values
for early scenarios 1 and 2 deviate quite strongly from those of the other scenarios. The
reason for this lies in the inability of Warburton’s model to cope with accelerations, as
discussed in section 5.1.1. Because it is in real life impossible for a project manager
to know in which scenario the project will end up, we obviously can’t say “in scenario
x, one should apply ratio value y”. However, to deal with the distorted ratio values of
scenarios 1 and 2, they are excluded from the calculation of the average ratio values so
they don’t affect the accuracy of the model for the other scenarios.
Influence of network structure
Contrary to the scenario a project will end up in, it is possible to determine the topologi-
cal structure of a project in advance. That’s why the impact of the topological structure
on the ratio values was investigated. It became clear that the SP-factor, discussed in
section 7.1.1, had a clear impact, as displayed on figures 8.3 and 8.4 on the next page.
These figures display that an increasing SP-factor (more serial network) leads to lower
ratio values for all three methods, especially for EAC(t)w6 and EAC(t)w7. Moreover, it is
possible to determine the SP-factor in advance, which is why one can say “for SP-factor
x, one should use ratio value y”. This led to table 8.1 on page 74 which represents the
ratio values per SP-factor for each of the three methods.
Chapter 8. Necessary input for Warburton’s model 73
Figure 8.3: Average ratio values for methods EAC(t)w5 and EAC(t)w6 per SP-factor.
Figure 8.4: Average ratio values for method EAC(t)w7 per SP-factor.
Chapter 8. Necessary input for Warburton’s model 74
Table 8.1: Average ratio values for time forecasting methods EAC(t)w5, EAC(t)w6 and
EAC(t)w7 per SP factor.
Average ratio values
SP EAC(t)w5 EAC(t)w6 EAC(t)w7
T1 T2 T1 T2 T1 T2
0.1 57.8% 43.3% 99.7% 99.1% -10.3 -31.2
0.2 55.1% 42.0% 99.6% 98.0% -14.5 -64.2
0.3 54.5% 41.6% 99.0% 96.6% -32.6 -111.7
0.4 49.4% 35.2% 98.7% 95.3% -43.6 -155.6
0.5 48.0% 36.1% 98.8% 94.4% -39.6 -185.7
0.6 44.2% 30.6% 98.3% 93.7% -55.2 -210.8
0.7 38.7% 27.8% 97.8% 93.1% -71.3 -224.6
0.8 34.6% 26.7% 97.5% 91.9% -82.7 -267.0
0.9 32.4% 26.5% 97.0% 91.7% -99.2 -272.0
Chapter 8. Necessary input for Warburton’s model 75
8.2 Parameter T
In this section it is determined which calculation method should be used for the accuracy
study. Remember two possible calculation methods for parameter T, T1 and T2, were
brought forward in section 5.2.
MAPE values using T1 and T2
T1 and T2 are compared by looking at the accompanying MAPE values concerning the
forecast accuracy of the final project duration. The EAC(t)w methods were discussed in
section 5.3. Each method is now executed once with T1 and once with T2. The MAPE
values concerning the forecast accuracy of the final project cost are not included in this
investigation, as the formula N(1+rc) is not influenced by parameter T. Although the
MAPE values are used here to assess the forecast accuracy, it is not the intention to
compare and discuss the forecast accuracy of the different methods against each other,
but to decide which calculation of parameter T leads to the best results.
Figure 8.5: MAPE values of final duration forecasters based on Warburton’s model.
Methods EAC(t)w0 and EAC(t)w6 were excluded from figure 8.5 because they led to
extremely high MAPE values which distorted the graphs. For these two methods, T1
led to better results than T2. As can be seen in the graph, it doesn’t make much
difference if T1 or T2 is used for methods EAC(t)w1, EAC(t)w2 and EAC(t)w3. For
Chapter 8. Necessary input for Warburton’s model 76
methods EAC(t)w4, EAC(t)w5 and EAC(t)w7, however, T2 clearly dominates T1.
As the EAC(t)w0 and EAC(t)w6 methods are the only methods where T1 leads to better
results than T2, and these are also the two methods with the worst forecasting accuracy
or highest MAPE values, it can be decided that the T2 calculation method for parameter
T is preferred. For the accuracy study that is conducted in the following chapters,
parameter T will by consequence always determined with the T2 calculation method
that is described in section 5.2.
8.3 Conclusion
In this chapter, research prior to the accuracy study was done to determine some un-
known factors that are needed for Warburton’s model. First, the ratio values for the
EAC(t)w5, EAC(t)w6 and EAC(t)w7 methods were determined and displayed in table
8.1 on page 74. Second, it was determined that the T2 calculation method will be used
for parameter T.
Chapter 9
Forecast accuracy
Here the actual accuracy study starts. The methods for forecasting the final project
duration and cost based on Warburton’s model are tested on accuracy and benchmarked
against the existing EVM forecasting methods. Section 9.1 handles the forecast accuracy
of the final duration, while section 9.2 does the same for forecasting the final cost. Both
sections are subdivided in a part where the accuracy in terms of the MAPE is discussed
and a part where the direction of the error in the forecasts is investigated with the MPE.
Remember there are now eight methods to determine EAC(t)w, as discussed in section
5.2.1, and one method to determine EACw, as discussed in section 3.2.3. The traditional
EVM forecasting methods were discussed in section 2.4.
9.1 Accuracy of the final project duration forecasts
9.1.1 Accuracy using the MAPE
Table 9.1 on the next page presents the MAPE values of the traditional EVM methods
to forecast the final project duration, while table 9.2 does the same for the methods
based on Warburton’s model. In all forecasting tables, the results per scenario as well
as the average results over all 6 scenarios can be found, the best performing methods
are indicated in bold and the best method is marked in grey.
77
Chapter 9. Forecast accuracy 78
Table 9.1: Forecasting accuracy (MAPE) of final project duration using the traditional EVM
methods.
TIME EAC(t)
Scenario PV1 PV2 PV3 ED1 ED2 ED3 ES1 ES2 ES3
1 19.6% 12.7% 28.5% 22.4% 12.7% 18.5% 19.6% 11.7% 21.1%
2 7.0% 6.0% 12.1% 7.7% 6.0% 8.5% 6.6% 7.0% 11.1%
3 4.6% 6.4% 10.2% 4.7% 6.4% 8.6% 4.4% 9.7% 12.2%
4 1.8% 2.5% 3.9% 1.8% 2.5% 3.4% 1.8% 6.8% 7.8%
5 4.9% 4.7% 8.9% 5.2% 4.7% 6.6% 4.4% 8.1% 11.8%
6 11.4% 8.2% 18.1% 11.6% 7.7% 12.0% 10.1% 8.3% 17.7%
Average 8.2% 6.7% 13.6% 8.9% 6.7% 9.6% 7.8% 8.6% 13.6%
Table 9.2: Forecasting accuracy (MAPE) of final project duration using the methods based on
Warburton’s model.
TIME EAC(t)
Scenario w0 w1 w2 w3 w4 w5 w6 w7
1 373.4% 36.0% 36.8% 34.6% 46.2% 64.7% 346.9% 41.2%
2 289.2% 11.7% 13.1% 10.6% 26.0% 38.0% 267.5% 20.9%
3 253.6% 5.7% 6.5% 5.7% 19.8% 27.4% 234.0% 15.0%
4 250.5% 2.2% 3.1% 2.5% 18.4% 26.3% 231.1% 13.7%
5 221.7% 8.1% 7.2% 9.6% 15.4% 19.8% 203.9% 12.8%
6 186.6% 18.1% 16.6% 18.9% 17.2% 17.0% 170.8% 18.5%
Average 262.5% 13.6% 13.9% 13.6% 23.8% 32.2% 242.4% 20.3%
Chapter 9. Forecast accuracy 79
An important remark has to be made concerning table 9.1. In Measuring Time [12]
a study concerning the forecasting accuracy of these traditional EVM methods was
already done. However, there are three important reasons for also running these methods
through the simulation model of this thesis and set up table 9.1 instead of simply using
the results of Measuring Time ([12], p. 68). First, the study of this thesis doesn’t work
with the 9 scenarios of Measuring Time, but instead uses 6 scenarios that are based on
the triangular distributions described in section 7.3.2. To provide a fair and accurate
benchmark to compare Warburton’s model with, it is necessary to work with the same
6 scenarios for the traditional EVM methods as well. Second, a different assumption
concerning stage of completion is made in this thesis. In Measuring Time, the progress
of a project is expressed in percentage of the final duration. For example “the project
is at the 10 % completion stage” means that the time instance at which the project
passes 10 % of the final duration is reached. In this thesis however, the same expression
means 10 % of the BAC is earned, meaning 10 % of the work to be done is finished. In
other words, in this thesis, project completion percentage equals EV/BAC *100. We
opted for this approach because, in our opinion, it makes sense to express the project
completion in terms of work that is done instead of time that has passed. Moreover, this
approach is closely linked to the way Warburton’s model works, namely with the early
project data. A third reason, linked with this, is that the average MAPE values in this
thesis are based on the values reached every time 10 % of the project is completed. In
Measuring Time, however, the average MAPE values are based on the values of every
time unit of the real duration of a project.
The conclusions concerning performance mainly remain the same, except for EAC(t)ES2
which performs worse in our simulation study compared to the results of Measuring
Time. Especially the different assumption concerning the expression of project comple-
tion is the reason for this difference. To illustrate this, table A.1 and table A.2 were
generated and included in the appendix on page 164 and 165 respectively. For the first
table, the 9 scenarios and the same assumption concerning project completion of Mea-
Chapter 9. Forecast accuracy 80
suring Time were used. In this table the same results as in Measuring Time are reached,
which confirms the correctness of our simulation. For the second table, the 6 scenarios
of this thesis are used but still with the Measuring Time assumption concerning project
completion. In this table the ES2-method remains the most accurate method, which
confirms that the different approach of the term project completion stage is the main
reason for the difference in table 9.1. In what follows, a thorough discussion and com-
parison of the performance of the different methods is held, once for the average of the
6 scenarios and once for each scenario separately.
Average accuracy over the 6 scenarios
Looking at the average MAPE over the 6 scenarios, table 9.1 shows that traditional
methods ED2 and PV2 perform the best with both an average MAPE of 6.7 %. From
table 9.2, it is clear that the EAC(t)w0 method performs very bad with an average
MAPE of 262.5 %. This confirms our expectation concerning overestimations because
of the long tail problem, discussed in section 5.1.3. Methods EAC(t)w1, EAC(t)w2 and
EAC(t)w3 are the clear winners among these methods with a respective average MAPE
of 13.6 %, 13.9 % and 13.6 %.
On average, when taking the complete project execution into account and the average
results over all 6 scenarios, none of the methods based on Warburton’s model outperform
any of the traditional EVM methods. Only EAC(t)w1 and EAC(t)w3 perform equally
well as the PV3 and ES3 method. The best methods based on Warburton’s model, being
EAC(t)w1, EAC(t)w2 and EAC(t)w3, have average MAPE values of about two times as
high as the best traditional EVM methods.
Before continuing to the results per scenario, remark that these results do show that,
except for EAC(t)w6, the methods that we have developed in section 5.1 do already
account for a remarkable improvement compared to the original EAC(t)w0 method con-
cerning time forecasting. Further, we also mainly expect an added value of Warburton’s
model in the early stages of project execution, while here the whole project execution is
Chapter 9. Forecast accuracy 81
taken into account. This will be handled in chapter 10. Also, the poor performance for
the early scenarios elevates the averages over all 6 scenarios.
Average accuracy per scenario
For the early scenarios 1 and 2, none of the methods based on Warburton’s model
perform better than any of the traditional EVM methods. In the more extreme scenario
1, the best MAPE of 34.6 % is reached by EAC(t)w3, which still is about 3 times as
high as the best MAPE of 11.7 % that is reached by the ES2 method. For the less
extreme scenario 2, EAC(t)w3 reaches a MAPE of 10.6 %, which is better than methods
PV3 and ES3, but is still a higher error than the best MAPE of 6 % that is reached by
methods PV2 and ED2. These results for the early scenarios are also in line with our
expectations. As discussed in section 5.1.1, Warburton’s model is not capable of taking
accelerations in activity durations into account, and it is exactly in these early scenarios
that a lot of these accelerations take place.
For the on time scenarios 3 and 4, methods EAC(t)w1, EAC(t)w2 and EAC(t)w3 perform
remarkably better. In the more extreme scenario 3, EAC(t)w1 and EAC(t)w3 both lead
to a MAPE of 5.7 %, which is already a good forecast of the final project duration. This
is an on average better performance than the traditional PV2, PV3, ED2, ED3, ES2 and
ES3 method. The traditional ES1 method delivers the best result in this scenario with
a MAPE of 4.4 %. In scenario 4, the EAC(t)w1 method in particular performs better
with a MAPE of 2.2 %, which is very close to the results of the traditional methods and
is only slightly higher than the best MAPE of 1.8 % that is reached by the PV1 as well
as the ED1 and ES1 method.
For both of the late scenarios 5 and 6, EAC(t)w2 leads to the best results for the meth-
ods based on Warburton’s model with respective MAPE values of 7.2 % and 16.6 %.
With these results the EAC(t)w2 method only dominates the traditional PV3 and ES3
methods. The best MAPE values come from the ES1 method with a value of 4.4 % for
scenario 5, and from the ED2 method with a value of 7.7 % for scenario 6.
Chapter 9. Forecast accuracy 82
9.1.2 Direction of the forecasting error using the MPE
Using the MPE values, the direction of the forecasting errors can be discussed. In other
words, it is discussed whether the forecasting methods lead to an over- or underestimation
of the final duration of a project, and this for every scenario. The results are presented
in figures 9.1 and 9.2 below. To keep these figures accessible and relevant, only the best
performing methods are presented for the traditional methods as well as for the methods
that are based on Warburton’s model.
Figure 9.1: MPE values for the traditional PV2, ED2 and ES1 forecasting methods per sce-
nario.
Figure 9.2: MPE values for the EAC(t)w1, EAC(t)w2 and EAC(t)w3 method per scenario.
Chapter 9. Forecast accuracy 83
The figures show that the direction of the forecasting error is the same for the methods
based on Warburton’s model as it is for the traditional forecasting methods. For the
early scenarios 1 and 2 the EAC(t)w1, EAC(t)w2 and EAC(t)w3 method lead to an over-
estimation of the final duration. Also the MPE values are the biggest for these scenarios.
This is in line with our expectations, as Warburton’s model doesn’t take into account
accelerations of activities. For the late scenarios 5 and 6, the EAC(t)w1, EAC(t)w2 and
EAC(t)w3 methods lead to an underestimation of the final project duration.
9.2 Accuracy of the final cost forecasts
9.2.1 Accuracy using the MAPE
Table 9.3 presents the results of the cost forecasting accuracy for the traditional EVM
forecasting methods as well as for the Warburton method. In the table the results per
scenario as well as the average results over all 6 scenarios are presented.
Table 9.3: Forecasting accuracy (MAPE) of the final project cost.
COST Traditional EVM methods Warburton
Scenario EAC1 EAC2 EAC3 EAC4 EAC5 EAC6 EAC7 EAC8 EACw
1 17.5% 7.2% 8.3% 8.9% 15.7% 17.4% 7.2% 7.4% 38.5%
2 5.8% 3.0% 3.7% 5.1% 6.8% 8.8% 3.1% 3.3% 12.9%
3 2.4% 3.8% 4.4% 7.3% 6.8% 9.7% 3.8% 4.3% 7.4%
4 0.9% 1.4% 1.7% 5.0% 2.5% 5.9% 1.4% 2.0% 2.1%
5 3.7% 2.4% 3.0% 6.1% 5.2% 9.4% 2.4% 2.6% 3.0%
6 8.8% 3.7% 4.8% 6.1% 10.1% 14.5% 3.8% 3.7% 5.5%
Average 6.5% 3.6% 4.3% 6.4% 7.9% 10.9% 3.6% 3.9% 11.6%
Chapter 9. Forecast accuracy 84
Average accuracy over the 6 scenarios
Looking at the average MAPE over the 6 scenarios, table 9.3 shows that the traditional
methods EAC2 and EAC7 perform the best on average with both a MAPE of 3.6 %. The
EAC3 and EAC8 method are also very close to this accuracy level. Warburton’s method
to forecast the final cost, EACw, on average over the 6 scenarios and the complete project
duration, never outperforms any of the traditional EVM methods. The MAPE of EACw
is 11.6 %, which is about 3 times as high as the MAPE of the EAC2 and EAC7 method.
However, we again have to make the remark that we mainly expect an added value of
Warburton’s model in the early stages of project execution, while here the whole project
execution is taken into account. Also, the poor performance for the early scenarios
elevates the average over all 6 scenarios.
Average accuracy per scenario
For the early scenarios 1 and 2 Warburton’s method again performs much worse than
for the other scenarios with a MAPE of respectively 38.5 % and 12.9 %, and does not
come close to the accuracy level of the best traditional EVM methods which reach a
MAPE of 7.2 % with methods EAC2 and EAC7. The poor performance of the EACw
method is again partly because of the inability of the model to take accelerations into
account, as discussed in section 5.1.1.
For the more extreme on time scenario 3, Warburton’s EACw method still performs
rather poor compared to the traditional methods. In the on time scenario 4, however, it
outperforms the EAC4, EAC5 and EAC6 method with a MAPE of 2.1 %. The traditional
EAC1 method delivers the lowest MAPE value of 0.9 %.
For the late scenarios 5 and 6, Warburton’s method delivers a MAPE of 3 % and 5.5
% respectively. This is very close to the performance of the traditional methods. The
EAC2, EAC7 and EAC8 method deliver the lowest MAPE values of about 2.4 % for
scenario 5 and 3.7 % for scenario 6.
Chapter 9. Forecast accuracy 85
9.2.2 Direction of the forecasting error using the MPE
Using the MPE values, the direction of the forecasting errors can again be discussed. It
is discussed whether the forecasting methods lead to an over- or underestimation of the
final cost of a project, and this for every scenario. The results are presented in figures
9.3 and 9.4. Again only the best performing methods are presented in the graph.
Figure 9.3: MPE values of the best EVM forecasting methods for the final project cost.
Figure 9.4: MPE values of the Warburton’s forecasting method for the final project cost.
The figure shows that the direction of the forecasting error is the same for Warburton’s
method as it is for the traditional EAC7 method, while it is the opposite for the EAC2,
EAC3 and EAC8 method. This is true for all scenarios, except for scenario 5 and
6. For the early and on time scenarios 1, 2, 3 and 4, Warburton’s method leads to
an overestimation of the final cost. Again the MPE values for the early scenarios 1
Chapter 9. Forecast accuracy 86
and 2 are a lot higher because of the fact that Warburton’s model doesn’t take into
account accelerations. In the late scenarios 5 and 6, Warburton’s method leads to a
little underestimation of the final project cost.
9.3 Conclusion
In this chapter, the forecast accuracy across the whole project duration of the traditional
methods and the methods based on Warburton’s model were benchmarked against each
other, and this for final duration and cost. After the discussion held above, the hy-
potheses that were stated in section 6.2.2 can be evaluated. Hypothesis 1, which states
that one could expect the traditional forecasting methods to outperform Warburton’s
methods on average over all scenarios for both final duration and cost, is confirmed. Also
Hypothesis 2, which expresses the expectation of Warburton’s model to perform notably
worse for projects that end early compared to on time and late projects, is confirmed.
Chapter 10
Project completion stage
As a second part of the accuracy study, the relation between the forecast accuracy
of Warburton’s model and the project completion stage is investigated. Contrary to
chapter 9, the accuracy of the model will be investigated in different stages of project
completion rather than looking at the average performance along the whole project
lifetime. Similar to chapter 9, the methods based on Warburton’s model will again be
benchmarked against the performance of the traditional EVM methods. Section 10.1
handles the forecasting methods for the final project duration, while section 10.2 does the
same but for the final project cost. To conduct this investigation, each of the traditional
EVM forecasting methods and the forecasting methods based on Warburton’s model are
applied at different completion stages of the project. In this simulation study a 10 %
interval of project completion is used. As mentioned in chapter 9, we define the project
completion percentage as the amount of value that is earned relative to the BAC:
Project completion =EV
BAC∗ 100 (%) (10.1)
So each time 10 % more value is earned relative to the BAC, the parameters r, c and τ
are recalculated , the forecasting methods are applied and their respective MAPE values
are calculated. Furthermore, using this data, the forecasting accuracy is separated for
three stages of project completion that are defined as follows:
87
Chapter 10. Project completion stage 88
� Early stage = MAPE values of the 10 %, 20 % and 30 % project completion stage
� Middle stage = MAPE values of the 40 %, 50 % and 60 % project completion stage
� Late stage = MAPE values of the 70 %, 80 % and 90 % project completion stage.
10.1 Accuracy of the final duration forecasts per comple-
tion stage
In figures 10.1, 10.2 and 10.3 on the next page, the results for the forecasting methods
of the final project duration can be found for early, on time and late projects respec-
tively. Only the best performing methods are displayed to keep the figures accessible
and relevant.
As shown on all three figures, the forecasting accuracy of all three EAC(t)w methods
that are based on Warburton’s model almost doesn’t change at all for different amounts
of early project data. In all three stages, the accuracy of the EAC(t)w methods remains
as good as the same. These results are rather unexpected, as one could expect lower
MAPE values, or thus more accurate forecasts, when more early project data is used to
determine parameters r, c and τ and set up Warburton’s model. However, in section 4.3
there already was an indication that adjustments to the parameter values had a limited
impact on the EAC(t)w forecasts, and the reason for this lack of change is the long tail
problem that was discussed in section 5.1.3. As can be seen on the figures above, the
traditional EVM methods do change and become more accurate in later stages.
As Warburton’s model has the goal to improve the EVM theory and is designed to
make predictions based on early project data, we could expect the model to perform
better than the existing EVM techniques in early stages of the project. This is only the
case for the on time projects, though the ES1 method remains superior. For early and
late projects, none of the methods based on Warburton’s model outperform any of the
displayed traditional methods, neither in the early, middle or late stage.
Chapter 10. Project completion stage 89
Figure 10.1: MAPE values for early projects (scenario 1 and 2) of the forecasting methods for
final project duration, per project completion stage (early, middle, late).
Figure 10.2: MAPE values for on time projects (scenario 3 and 4) of the forecasting methods
for final project duration, per project completion stage (early, middle, late).
Figure 10.3: MAPE values for late projects (scenario 5 and 6) of the forecasting methods for
final project duration, per project completion stage (early, middle, late).
Chapter 10. Project completion stage 90
10.2 Accuracy of the final cost forecasts per completion
stage
In figures 10.4, 10.5 and 10.6 on the next page, the results for the forecasting methods
of the final project cost can be found, for early, on time and late projects respectively.
For the early scenarios 1 and 2, figure 10.4 shows that the accuracy of the EACw method
again doesn’t change across the different completion stages and the MAPE values are
high. As explained in section 5.1.1, the reason for this is that parameter r stays very
small as it does not take into account accelerated activities. So, for early scenarios 1
and 2, Warburton’s method is dominated by all traditional methods across all project
completion stages.
For the on time scenarios 3 and 4, Warburton’s EACw method performs remarkably
well in the early stage of a project with a MAPE of 4.4 %. With this fairly accurate
estimation of the final cost, the EACw method does better than the traditional EAC3,
EAC4, EAC5, EAC6 and EAC8 method, and about equally good as the EAC2 and EAC7
method. Only the EAC1 method dominates Warburton’s EACw method with a MAPE
of 2.3 % in the early stage. For the middle and late stages, the traditional methods again
dominate Warburton’s method, as expected. We repeat that Warburton’s model has the
goal to improve the EVM theory in the early stage and is designed to make predictions
based on early project data.
For the late scenarios 5 and 6 in the early stage, the EACw method leads to an average
MAPE of 7.18 % which is a better accuracy level than the one reach by the EAC1,
EAC4, EAC5 and EAC6 method, and almost as good as the EAC3 method . Only the
traditional methods EAC2, EAC7 and EAC8 do better with a MAPE of about 5 %. As
Warburton’s model does adjust for activity delays, contrary to activity accelerations,
the same conclusions can be taken for the middle and late stage for late projects.
Chapter 10. Project completion stage 91
Figure 10.4: MAPE values for early projects (scenario 1 and 2) of the forecasting methods for
final project cost, per project completion stage (early, middle, late).
Figure 10.5: MAPE values for on time projects (scenario 3 and 4) of the forecasting methods
for final project cost, per project completion stage (early, middle, late).
Figure 10.6: MAPE values for late projects (scenario 5 and 6) of the forecasting methods for
final project cost, per project completion stage (early, middle, late).
Chapter 10. Project completion stage 92
Contrary to the EAC(t)w methods to forecast the final project duration, the forecast
error for the final project cost decreases when more early project data becomes available
in the on time and late scenarios. This is in line with our expectations. The reason can
be found in the fact that, contrary to the EAC(t)w methods to forecast final duration,
the EAC method which states that EAC = N(1+r*c) is not affected by the long tail
problem that was discussed in section 5.1.3.
10.3 Conclusion
In this chapter, the forecast accuracy in different stages of completion of both the tradi-
tional as well as Warburton’s methods were benchmarked against each other, and this
for the final duration and cost. After the discussion held above, Hypothesis 3 of section
6.2.3, which states that ‘forecasts for final project duration and cost based on Warbur-
ton’s model will be more accurate than the traditional EVM forecasting methods in the
early stages of project completion’, can be evaluated. For final project duration fore-
casts, this is only confirmed for on time projects, where the EAC(t)w methods perform
about equally well as the best traditional method. For final project cost forecasts, this
hypothesis is confirmed for both on time and late projects, although the EACw method
is never the absolute best method.
Furthermore, it was found that the amount of early project data doesn’t have an impact
on the accuracy of the EAC(t)w methods to forecast the final duration because of the
long tail problem inherent to Warburton’s model. The EAC(t)w1, EAC(t)w2, as well
as the EAC(t)w3 method don’t have changing MAPE values across the different stages.
The EACw method to forecast the final project cost, however, is not effected by the
long tail problem and does give more accurate forecasts with more early project data.
This is true for both the on time and the late scenarios, but not for the early scenarios
because of the inability of Warburton’s model to incorporate accelerations. It can thus
be concluded that Hypothesis 3 is only confirmed for the EACw method to forecast final
cost in the on time and late scenarios.
Chapter 11
Topological structure
As discussed in section 7.1.1, project networks can have a different topological structure
based on their SP, AD, LA and TF factor. In this chapter, the relation between the
topological structure of the project and the forecast accuracy of the time and cost meth-
ods based on Warburton’s model is investigated. During this study, the focus lies on
the methods with the best forecasting accuracy, as determined in chapter 9. These are
methods EAC(t)w1, EAC(t)w2 and EAC(t)w3 for the final project duration and EACw
for the final project cost.
Section 11.1 will handle the influence of the SP factor, while section 11.2 does the same
for the AD factor, 11.3 for the LF factor and 11.4 for the TF factor. Each section will
handle the impact on the methods to forecast the final duration and on the methods to
forecast the final cost, and this for each scenario. Because of the fact that Warburton’s
model doesn’t take into account accelerations, as discussed in section 5.1.1, and this
is a big problem mainly for scenario 1 and 2 as confirmed in chapter 9, somewhat less
importance will be given to the results of these scenarios. Also, as the goal is to compare
the performance of the forecast accuracy across different topological indicators, we are
less interested in the direction of the forecast error, which is why MAPE is used as the
forecast error evaluation metric.
93
Chapter 11. Topological structure 94
11.1 The influence of the serial/parallel indicator (SP)
11.1.1 Impact of SP factor on EAC(t)w methods to forecast final du-
ration
Figure 11.1 displays the MAPE values for the project networks with varying SP values
ranging from 0.1 (indicating a more parallel network) to 0.9 (indicating a more serial
network) in steps of 0.1. A graph is given for each of the 6 scenarios. We remark that
the y-axis of the graphs might have different scales in order to fit all the data and keep
visibility.
Figure 11.1: Influence of the SP factor (x-axis) on the time forecast error (y-axis) of methods
EAC(t)w1, EAC(t)w2 and EAC(t)w3 under the 6 scenarios.
Chapter 11. Topological structure 95
The results concerning the influence of the SP factor can be summarized as follows.
First, for all scenarios the differences in MAPE values across the various SP factors are
significant, but minor and often not intuitive for all three methods. Only in scenario 1,
the SP indicator has a big impact on the forecast accuracy, where more accurate results
are found with lower SP values. Second, for projects that finish on schedule (scenario 3
and 4), higher values of the SP indicator (more serial networks) result in more accurate
forecasts of the final project duration. Finally, for projects that end behind schedule
(scenario 5 and 6), the SP indicator has a less clear influence.
In chapter 9, it was already determined that methods EAC(t)w1, EAC(t)w2 and EAC(t)w3
outperformed the other methods based on Warburton’s model in terms of forecast ac-
curacy. Now, it is also clear that these methods are rather stable and robust towards
fluctuations in the SP indicator as, in general, no major differences in MAPE values
were found across different SP values.
11.1.2 Impact of SP factor on the EACw method to forecast final cost
Similar as for the forecasting methods of the final duration, the differences in the MAPE
value across the various SP factors are significant but minor for the EACw method,
as displayed in figure 11.2 on the next page. Keeping this in mind, the results can
be summarized as follows. First, for projects that finish ahead of schedule (scenario 1
and 2) and for projects that finish on schedule (scenario 3 and 4), the results reveal
that a higher SP factor goes along with less accurate estimations of the final project
cost. Second, for projects that finish behind schedule (scenario 5 and 6), a higher SP
value (more serial project) leads to more accurate forecasts of the final cost although
the differences stay small. The main improvement in accuracy takes place between an
SP value of 0.1 and 0.5.
It can be concluded that the EACw method is rather stable and robust towards fluc-
tuations in the SP indicator as, in general, no major differences in MAPE values were
found across different SP values.
Chapter 11. Topological structure 96
Figure 11.2: Influence of the SP factor (x-axis) on the time forecast error (y-axis) of the EACw
method to forecast the final project cost under the 6 scenarios.
Chapter 11. Topological structure 97
11.2 The influence of the activity distribution (AD)
11.2.1 Impact of AD factor on EAC(t)w methods to forecast final du-
ration
Figure 11.3 displays the MAPE for the project networks with varying AD values ranging
from 0.2, indicating that the activities are more spread out over the network, to 0.8,
indicating that the activities are less spread out over the network, in steps of 0.2.
Figure 11.3: Influence of the AD factor (x-axis) on the time forecast error (y-axis) of methods
EAC(t)w1, EAC(t)w2 and EAC(t)w3 under the 6 scenarios.
The results concerning the influence of the AD factor can be summarized as follows.
First, for all scenarios except scenario 1, some of the differences in MAPE values across
the various AD factors are significant, but remain minor and are often not intuitive for
Chapter 11. Topological structure 98
all three methods. Second, methods EAC(t)w1 and EAC(t)w3 are the most reliable in
this context as they show only small changes across all AD values in all scenarios, in
contrast with method EAC(t)w2 that shows an outlier in the results for an AD factor
of 0.8. Third, for these two reliable methods, the following conclusions can be made.
For projects that finish ahead of schedule (scenario 1 and 2), higher values of the AD
indicator result in a less accurate forecast of the final project duration. For projects
that finish on schedule (scenario 3 and 4), the AD indicator has little to no influence
and for projects that finish behind schedule (scenario 5 and 6), higher values of the AD
indicator result in a more accurate forecast of the final project duration.
In chapter 9, it was already determined that methods EAC(t)w1, EAC(t)w2 and EAC(t)w3
outperformed the other methods based on Warburton’s model in terms of forecast ac-
curacy. Now, it can also be stated that EAC(t)w1 and EAC(t)w3 are stable and robust
towards fluctuations in the AD indicator, contrary to the EAC(t)w2 method.
11.2.2 Impact of AD factor on the EACw method to forecast the final
project cost
Similar as for the forecasting methods of the final duration, the differences in the MAPE
values across the various AD factors are significant but minor for the EACw method,
as displayed in figure 11.4 on the next page. The results can be summarized as follows.
First, for projects that finish ahead of schedule (scenario 1 and 2), the forecast accuracy
is rather insensitive towards different AD values. Although some values are significantly
different, the differences are very small. Second, projects that finish on schedule (scenario
3 and 4) show a little more accurate forecasting results as the AD value goes up. Third,
for projects that finish behind schedule (scenario 5 and 6), we see the opposite effect of
more inaccurate results as the AD value goes up.
It can be stated that the EACw method is rather stable and robust towards fluctuations
in the AD indicator as, in general, no major differences in MAPE values were found
across different AD values.
Chapter 11. Topological structure 99
Figure 11.4: Influence of the AD factor (x-axis) on the time forecast error (y-axis) of the EACw
method to forecast the final project cost under the 6 scenarios.
Chapter 11. Topological structure 100
11.3 The influence of the length of arcs (LA) and topolog-
ical float (TF)
11.3.1 Impact of LA and TF factor on the EAC(t)w methods to forecast
the final project duration
The results of both the LA and TF factors say that these indicators don’t always have a
significant effect on the performance of the methods based on Warburton’s model. The
results are displayed in figures 11.5 and 11.6.
Figure 11.5: Influence of the LA factor (x-axis) on the time forecast error (y-axis) of methods
EAC(t)w1, EAC(t)w2 and EAC(t)w3 under the 6 scenarios.
Chapter 11. Topological structure 101
Figure 11.6: Influence of the TF factor (x-axis) on the time forecast error (y-axis) of methods
EAC(t)w1, EAC(t)w2 and EAC(t)w3 under the 6 scenarios.
For both variations in the LA and TF factor, some of the differences in the MAPE
values across the various LA factors are significant, but all of them remain minor and
are often not intuitive for all three methods, and for all scenarios. Except for scenario
1, the fluctuations in the MAPE values are not bigger than 1 % for all scenarios. It
can be concluded that the methods EAC(t)w1, EAC(t)w2 and EAC(t)w3 are stable and
their performance does not depend on the network structure measured by the LA or TF
indicator.
Chapter 11. Topological structure 102
11.3.2 Impact of LA and TF factor on the EACw method to forecast
the final project cost
Also here, the results of both the LA and TF factors show that these indicators don’t
always have a significant effect on the performance of the methods based on Warburton’s
model. All MAPE values across different LA values were not significantly different from
each other, except in scenario 6, and even in this scenario the maximum difference was
no more than 0.5 % and is neglectable. This is why these results are not displayed here.
It can be concluded that the EACw method is stable and the performance does not
depend on the network structure measured by the LA or TF indicator.
11.4 Conclusion
In general, none of the topological indicators have a big impact, meaning the EAC(t)w1,
EAC(t)w2 and EAC(t)w3 method to forecast the final duration and the EACw method to
forecast the final cost are rather stable and their performance does not heavily depend on
the network structure measured by these indicators. It can be concluded that Hypothesis
4 of section 6.2.4, that states that higher values of the SP indicator go along with an
improving forecast performance of the final project duration, is only true for on time
projects (scenario 3 and 4), although the impact remains minor. For the EACw method
to forecast the final cost, Hypothesis 5 which states that the forecast accuracy of the
final project cost is rather insensitive towards all four topological indicators (SP, AD,
LA and TF) is proven to be true. Only the SP factor had a visible influence, but also
here the differences in MAPE values are small.
Chapter 12
Time/cost relationship
In this chapter, the influence of different time/cost relationships on the cost forecast
accuracy of Warburton’s model is analyzed. No attention will be paid to the forecast
accuracy of the final duration, as this aspect is not affected by the different time/cost
relationships. For this research, the relationship between the duration and the cost of
an activity is simulated under three different settings: a linear, convex and concave
time/cost relationship. These are defined in section 12.1. Section 12.2 handles the
impact of the different time/cost relationships on the performance of both the forecasting
method for final cost of Warburton’s model and of the traditional EVM cost methods.
A discussion similar to chapter 9 will be held concerning the forecast accuracy under the
different settings, followed by a discussion similar to chapter 10 concerning the forecast
accuracy in the different stages of completion.
For the discussion of the impact on Warburton’s model, the main focus lies on the late
scenarios 5 and 6 and partly on the on time scenarios 3 and 4. No attention is paid
to the early scenarios 1 and 2 as they highly suffer from the disability of Warburton’s
model to take activity accelerations into account. As a result, Warburton’s cost method
will always be dominated by the other EVM cost methods in scenario 1 and 2 and that
for all three types of cost profiles. For the evaluation of the impact of the cost profiles
on the traditional EVM methods, all scenarios will be involved.
103
Chapter 12. Time/cost relationship 104
12.1 Time/cost relationships
Until now, a linear relationship was assumed between the duration of each activity and
the corresponding activity cost by means of a variable cost of 20 cost units per time unit,
as mentioned in chapter 7 which handles the methodology of this study. Furthermore,
the duration of the activities used in the study can range from a minimum of 1 to a
maximum of 20 time units. Here, two additional time/cost relationships are introduced,
a convex and concave one. The three time/cost relationships are illustrated on figure
12.1 and discussed below.
Figure 12.1: Linear, convex and concave time/cost relationship.
12.1.1 Linear time/cost relationship
In the first and original setting that was used in part 4 of this thesis, the actual cost
function is set as a linear accrue of the unit cost per time unit. The formulation of the
applied linear cost function was defined as follows:
Activity cost = V C ∗AD from eq. 7.1
with
VC= variable cost factor=20 cost units/time unit
AD= activity duration
Chapter 12. Time/cost relationship 105
12.1.2 Convex time/cost relationship
In the second setting, the time/cost relationship of an activity follows a convex pattern.
Here, a slower initial cost increase per time unit is in place. For the concave cost accrue,
a quadratic function of the form y=ax2 is used. The applied function is defined as
follows:
Activity cost = V C ∗ AD2
m(12.1)
with
m= maximum activity duration = 20 time units
12.1.3 Concave time/cost relationship
In the third setting, the time/cost relationship of an activity follows a concave pattern
which can be seen as the negative of the convex function. For the mathematical deriva-
tion of this function, the reader is referred to appendix B. Here the cost per time unit
initially increases more and has a descending increase in cost per time unit over time.
The applied function is defined as follows:
Activity cost =V C√
1V C
∗√AD ∗mV C
(12.2)
Chapter 12. Time/cost relationship 106
12.2 Impact of time/cost relationship on the EAC methods
12.2.1 Forecast accuracy
Linear time/cost relationship
Table 12.1: Forecasting accuracy (MAPE) of the final project cost under the assumption of a
linear time/cost relationship.
LINEAR Traditional EVM methods Warburton
Scenario EAC1 EAC2 EAC3 EAC4 EAC5 EAC6 EAC7 EAC8 EACw
1 17.5% 7.2% 8.3% 8.9% 15.7% 17.4% 7.2% 7.4% 38.5%
2 5.8% 3.0% 3.7% 5.1% 6.8% 8.8% 3.1% 3.3% 12.9%
3 2.4% 3.8% 4.4% 7.3% 6.8% 9.7% 3.8% 4.3% 7.4%
4 0.9% 1.4% 1.7% 5.0% 2.5% 5.9% 1.4% 2.0% 2.1%
5 3.7% 2.4% 3.0% 6.1% 5.2% 9.4% 2.4% 2.6% 3.0%
6 8.8% 3.7% 4.8% 6.1% 10.1% 14.5% 3.8% 3.7% 5.5%
Average 6.5% 3.6% 4.3% 6.4% 7.9% 10.9% 3.6% 3.9% 11.6%
The final project cost forecasting accuracy of the traditional EVM methods and War-
burton’s method for projects with a linear time/cost relationship was already thoroughly
discussed in section 9.2. However, it is interesting to briefly repeat the main takeaways
relevant for this chapter to clearly indicate the impact of the other time/cost relation-
ships. As was shown in table 12.1 (which is a repetition of table 9.3), the EACw method
was dominated in all scenarios by the four best performing traditional methods, which
were EAC2, EAC3, EAC7 and EAC8. The scenarios in which the accuracy of the EACw
method was closest to the accuracy of the best EVM method was in the on time and
late scenarios, and in particular in scenario 5.
Chapter 12. Time/cost relationship 107
Convex time/cost relationship
Table 12.2: Forecasting accuracy (MAPE) of the final project cost under the assumption of a
convex time/cost relationship.
CONVEX Traditional EVM methods Warburton
Scenario EAC1 EAC2 EAC3 EAC4 EAC5 EAC6 EAC7 EAC8 EACw
1 30.6% 19.8% 14.4% 13.6% 26.5% 27.2% 18.4% 18.4% 74.3%
2 9.0% 13.2% 6.6% 7.3% 16.8% 17.7% 12.0% 12.1% 25.4%
3 6.6% 15.0% 8.2% 9.8% 16.9% 17.4% 13.9% 14.0% 13.2%
4 3.6% 10.9% 4.2% 6.3% 11.6% 11.8% 9.9% 10.0% 4.3%
5 11.1% 12.7% 8.6% 7.9% 12.6% 11.6% 12.0% 11.9% 6.9%
6 20.8% 15.2% 14.9% 12.8% 16.1% 15.5% 14.9% 14.7% 12.2%
Average 13.6% 14.5% 9.5% 9.6% 16.7% 16.9% 13.5% 13.5% 22.7%
When projects are characterized by a convex time/cost relationship, the forecast accu-
racy results for all methods change significantly, as presented in table 12.2. The best
performing methods over all the scenarios are methods EAC3 (PF=SPI) and EAC4
(PF=SPI(t)). For the on time scenarios 3 and 4, EAC1 outperforms all other methods
with a MAPE of 6.6 % and 3.6 % respectively, including Warburton’s EACw method with
a MAPE of 13.2 % and 4.3 % respectively. However, when looking at the performance
in the late scenarios 5 and 6, the remarkable statement can be made that Warburton’s
EACw method outperforms all traditional EAC methods. In case of a convex time/cost
relationship, the best performing traditional method in scenarios 5 and 6 is EAC4 with
a MAPE of respectively 7.9 % and 12.8 %. In the same setting, Warburton’s EACw
method is more accurate with a MAPE of respectively 6.9 % and 12.2 %.
Chapter 12. Time/cost relationship 108
Concave time/cost relationship
Table 12.3: Forecasting accuracy (MAPE) of the final project cost under the assumption of a
concave time/cost relationship.
CONCAVE Traditional EVM methods Warburton
Scenario EAC1 EAC2 EAC3 EAC4 EAC5 EAC6 EAC7 EAC8 EACw
1 10.9% 7.3% 7.1% 8.8% 10.4% 11.8% 5.9% 5.4% 19.1%
2 5.0% 7.5% 3.6% 4.9% 6.6% 5.2% 5.9% 5.0% 6.2%
3 3.1% 7.7% 4.4% 7.1% 8.6% 9.0% 6.4% 6.0% 3.9%
4 2.5% 7.7% 2.8% 5.3% 7.7% 8.0% 6.3% 6.0% 1.0%
5 1.8% 7.8% 4.6% 9.3% 11.8% 17.9% 6.8% 7.8% 1.4%
6 2.8% 7.9% 7.0% 11.9% 17.9% 25.0% 7.4% 8.4% 2.5%
Average 4.3% 7.7% 4.9% 7.9% 10.5% 12.8% 6.5% 6.4% 5.7%
When projects are characterized by a concave time/cost relationship, the best performing
methods over all the scenarios are methods EAC1 (PF=1) and EAC3 (PF=SPI), with a
MAPE of 4.3 % and 4.9 % respectively. Remarkable is that, despite taking the results
for the early scenarios 1 and 2 into the average, Warburton’s EACw method performs
remarkably well here, with a MAPE of 5.7 %, and takes the third place in the ranking
of the best performing methods over all the scenarios. Looking at the results for the
scenarios separately, again some remarkable statements can be made. In on time scenario
3, the EAC1 method is the most accurate with a MAPE value of 3.1 %, but is closely
followed by Warburton’s EACw method with a MAPE of 3.9 %, which outperforms all
other traditional EAC methods. More important, Warburton’s EACw method offers the
highest forecast accuracy in on time scenario 4, and also again in late scenarios 5 and 6,
with respective MAPE values of 1 %, 1.4 % and 2.5 %. The best performing traditional
method in these scenarios is the EAC1 method with respective MAPE values of 2.5 %,
1.8 % and 2.8 %.
Chapter 12. Time/cost relationship 109
12.2.2 Project completion stage - Late projects
Because of Warburton’s inability to deal with activity accelerations and the fact that the
most promising results can be found in the late scenarios 5 and 6 for all three time/cost
relationships, we have decided to only include the results for late projects in this section.
Figures 12.2, 12.4 and 12.6 on the next pages present the forecast error of the different
EAC methods for late projects along the early, middle and late stage, and this for a
linear, convex and concave time/cost relationship respectively. The definitions of the
stages are the same as described for the first time in Chapter 10 on page 87. Figures
12.3, 12.5 and 12.7 on the next pages respectively give a more detailed graph which
represents the forecast error for each 10 % interval of project completion to determine
for which early data percentages Warburton’s method performs very well compared to
the other methods, and this for every time/cost relationship.
Linear time/cost relationship
Figure 12.2: MAPE values for late projects (scenario 5 and 6) of the forecasting methods for
final project cost, per completion stage (early, middle, late) under the assumption
of a linear time/cost relationship.
Although the results of figure 12.2 were already thoroughly discussed in chapter 10, it
is again useful to summarize the main takeaways from this chapter to see the impact of
the different time/cost relationships. For late projects, the traditional methods EAC2,
EAC3, EAC7 and EAC8 all provide a higher forecast accuracy than Warburton’s EACw
Chapter 12. Time/cost relationship 110
method in every stage. However, Warburton’s EACw method does provide good fore-
casts, as its MAPE is only 2 % higher in the early stage and less than 1 % higher in the
middle stage compared to the best EVM method. Between a project completion of 25
% and 60%, Warburton’s accuracy is close to the best performing methods and can be
considered as a good forecasting method during this interval for late projects. This can
be seen on figure 12.3.
Figure 12.3: MAPE values for late projects (scenario 5 and 6) of the forecasting methods for
final project cost, per 10 % of the work completed under the assumption of a
linear time/cost relationship.
Convex time/cost relationship
Figure 12.4: MAPE values for late projects (scenario 5 and 6) of the forecasting methods for
final project cost, per completion stage (early, middle, late) under the assumption
of a convex time/cost relationship.
Chapter 12. Time/cost relationship 111
Under the setting of a convex time/cost relationship, again promising statements can be
made. In the early stage, Warburton’s EACw method and the traditional EAC4 method
offer the highest accuracy with a MAPE of 15 %, which is considerably lower than all
other methods which all have a MAPE of about 25 % in this stage, except for EAC3
which has a MAPE value of 17.5 %. In the middle stage, Warburton’s EACw method
delivers a MAPE of 7.91 % and outperforms all traditional EAC methods which all have
a MAPE of 10 % or higher in this stage. In the late stage, almost all methods perform
equally well. The same conclusions can be drawn from the more detailed figure 12.5 in
which we see that Warburton’s EACw method offers the highest accuracy of all methods
between a project completion of 20 % and 70 %. At the very end of a project, the
traditional EAC methods logically go to a forecast error of 0 %, while this is not the
case for Warburton’s model as a logic result of the underlying model specifications.
Figure 12.5: MAPE values for late projects (scenario 5 and 6) of the forecasting methods for
final project cost, per 10 % of the work completed under the assumption of a
convex time/cost relationship.
Chapter 12. Time/cost relationship 112
Concave time/cost relationship
Figure 12.6: MAPE values for late projects (scenario 5 and 6) of the forecasting methods for
final project cost, per completion stage (early, middle, late) under the assumption
of a concave time/cost relationship.
Under the setting of a concave time/cost relationship, Warburton’s EACw method per-
forms exceptionally well as it offers the highest forecast accuracy of all methods in the
early, middle and late stage of a project with a MAPE of 3.49 %, 1.49 % and 0.97 %
respectively. Of the traditional methods, only the EAC1 method is close to Warburton’s
accuracy. In the more detailed figure 12.7, it is shown that Warburton’s EACw method
offers the highest accuracy of all methods between a project completion of 10 % and 90
%. Moreover, together with EAC1, EACw offers fairly robust and stable results over the
project life span compared to the other methods. At a project completion of 10 %, the
MAPE of the EACw and EVM1 method are already lower than 5 %, while the MAPE
values of the other traditional methods range from 10 % to 75%.
Chapter 12. Time/cost relationship 113
Figure 12.7: MAPE values for late projects (scenario 5 and 6) of the forecasting methods for
final project cost, per 10 % of the work completed under the assumption of a
concave time/cost relationship.
12.3 Conclusion
In this chapter, the impact of different types of time/cost relations on the forecast
accuracy for final cost of Warburton’s model was investigated and compared to the
impact on the traditional methods. Three different settings were simulated: a linear,
convex and concave time/cost relationship.
The results concerning the average forecast accuracy averaged over the complete project
execution and all 6 scenarios can be summarized as follows. First, in case of a linear
time/cost relationship, EAC2 (with PF= CPI) offers the highest forecast accuracy or,
in other words, the statement ‘future performance is expected to follow the current cost
performance’ fits best here. Second, in case of a convex time/cost relationship, EAC3
(with PF= SPI) and EAC4 (with PF= SPI(t)) offer the highest forecast accuracy or, in
other words, the statement ‘future performance is expected to follow the current time
performance’ fits best here. Third, in case of concave time/cost relationship, EAC1
(with PF= 1) offers the highest forecast accuracy or, in other words, the statement
‘future performance is expected to follow the baseline schedule’ fits best here.
Chapter 12. Time/cost relationship 114
So, when looking at the accuracy averaged over the complete project execution and
all 6 scenarios, Warburton’s EACw method is always dominated. However, it is clear
by now that the inability of Warburton’s model to take accelerations into account, as
discussed in section 5.1.1, leads to very poor performances in early scenarios 1 and 2,
leading to a higher MAPE value for the average over all 6 scenarios. When looking at the
performances in the separate scenarios, Warburton’s EACw method is very promising,
especially for the late scenarios. In these late scenarios, Warburton’s EACw method
performs almost as good as the traditional EAC methods in case of a linear time/cost
relationship and, more important, outperforms all traditional EAC methods in case of
a convex and concave time/cost relationship.
When looking at the forecast accuracy for the final cost along the different comple-
tion stages of late projects under all three time/cost relationships, Warburton’s EACw
method again leads to some promising results. For a linear time/cost relationship, War-
burton’s accuracy is close to the best performing traditional EAC methods and can be
considered as a good forecasting method between a project completion of 25 % and 60
%. For a convex time/cost relationship, Warburton’s EACw method offers the highest
accuracy of all methods between a project completion of 20 % and 70 %. Finally, for a
concave time/cost relationship, Warburton’s EACw method offers the highest accuracy
of all methods between a project completion of 10 % to 90 %.
To conclude, we would like to shortly refer to part 5 of this thesis, in which we attempted
to improve Warburton’s model by making it possible to cope with accelerations, based
on the lessons learned from this study. A similar discussion will be held for the improved
Warburton model, with the expectation that the promising results that were found here
for late projects, will then also stand for early projects. If so, the model could very well
be a valuable addition to a project manager’s toolkit.
Part V
IMPROVEMENT OF
WARBURTON’S MODEL
115
Chapter 13
Shortcomings of Warburton’s
model and set-up of improved
model
In chapter 5, some of the shortcomings of Warburton’s model that could be identified
prior to the accuracy study of part 4 were discussed. In this chapter, the insights gained
by the study of part 4 with regard to these shortcomings will be discussed in section
13.1. With these lessons learned, we saw an opportunity to modify and improve the
model in order to deal with the shortcoming for accelerations. In section 13.2, our
improved model, to which we will refer to as ‘the new Warburton model’, is set up. This
entails a thorough discussion of the modified model parameters and the refined formulas
for the calculation of the earned value and cost curves. To end this chapter, section
13.3 will illustrate the functioning of the improved model and the difference with ‘the
initial Warburton model’ using the example project of chapter 4. An overview of the
parameters and curves of the initial Warburton model versus the new Warburton model,
as discussed in this chapter, can be found in appendix C in tables C.1 and C.2 on page
169 and 170 respectively. In chapter 14, the performance of this new Warburton model
will be benchmarked against the performance of the initial Warburton model.
116
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 117
13.1 Shortcomings of Warburton’s model
13.1.1 Accelerations
As already introduced in section 5.1.1, an important shortcoming in Warburton’s model
is that it doesn’t take into account schedule accelerations in the duration of an activity
when calculating the fundamental parameters of the model r, c and τ . The definition
of parameter r, the reject rate of activities, tells us this is the fraction of the activities
that are not finished within their planned duration and are therefore rejected and need
extra work. However, Warburton’s model does not foresee an adjustment for activities
that meet their predefined goal earlier than planned, or are in other words accelerated
and ahead of schedule. This is because the initial Warburton model is not able to cope
with negative values of τ (schedule accelerations) because of the time delay terms in the
formulas, e.g. pv(t-τ)w, as mentioned in R.D.H. Warburton’s paper ([21], pg. 8). This is
a crucial shortcoming of the model, considering the other basic parameters c and τ won’t
be adjusted either. Parameters c and τ , respectively the cost overrun and time to repair
the rejected activities, are both calculated based on the rejected activities determined by
parameter r. As explained, the importance of this shortcoming is especially crucial for
early scenarios 1 and 2, as it is in these scenarios that a large proportion of the activities
have a shorter real duration than initially planned and are thus accelerated. As confirmed
in part 4 of this thesis, the initial Warburton model performs consistently worse for these
scenarios, and it is this shortcoming that is handled with the new Warburton model.
13.1.2 Other shortcomings
As discussed in section 5.1.2, the initial Warburton model doesn’t make a distinction
between critical and non-critical activities. Also, as discussed in section 5.1.3, the initial
Warburton model has to deal with long tails in its curves, and the accuracy study of
part 4 showed that the EAC(t)w methods to forecast the final duration are insensitive
towards changes in the parameter r and τ because of this problem. One could wonder
why the new Warburton model is only adjusted for the shortcoming of accelerations and
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 118
not for these shortcomings. This is because no immediate solutions were found.
First, because of the mathematical formulation of the Warburton curves, it is hard to
comprehend what the distinction of critical and non-critical activities would do for the
model and if this would actually improve the model. Also, this would only impact the
time aspect and not the cost aspect as deviations in the cost of each activity have an
equal impact on the final project cost, irrespective of being a critical or non-critical
activity. Second, we already attempted to handle the long tail problem before even
starting the accuracy study by developing the seven extra EAC(t)w methods to forecast
the final project duration. These were thoroughly discussed in section 5.3.1. However,
part 4 showed that despite the effort, the EAC(t)w methods don’t adjust enough to
provide more accurate forecasts along the project completion stage because of the long
tail problem.
13.2 Set-up of the new and improved Warburton model
To incorporate the capability to deal with accelerations, it is needed to change both
the formulation of parameters r, c and τ and to introduce additional formulas for the
calculation of the ev(t)w-, EV(t)w-, ac(t)w- and AC(t)w- curve. No changes are needed
for parameters N and T, as they are determined before the actual execution of the
project. As a result, the formula for the calculation of pv(t)w and PV(t)w remains
untouched.
13.2.1 Modification of the parameters
As discussed in section 3.2, the parameters of the initial Warburton model all have to
be non-negative. This constraint is embedded in the formulation of the model, and as
a consequence only delays and cost surpluses are taken into account in the calculation
of the parameter values. In the setup of the improved model, τ and c should be able to
assume both positive and negative values, as the latter represents respectively a project
acceleration and a cost underrun.
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 119
Parameter r: The deviation rate of activities
In Warburton’s initial model, r stood for the fraction of the activities at a given time
instance that require extra work and therefore take longer than initially foreseen. In
the new Warburton model, both the activities that incur a delay or an accelaration in
its duration should have an impact on the model. Therefore, the redefined parameter
r stands for the fraction of the actitivies of which the real duration deviates from its
planned duration. This deviation can be either an acceleration or a delay. The redefined
r will be referred to as ‘the deviation rate of activities’, as this better reflects the new
idea behind r compared to the initial name ‘the reject rate of activities’. In the new
Warburton model, r is calculated as follows:
r =# finished activities with real duration 6= planned duration
# finished activities(13.1)
Parameter τ : The time deviation of deviated activities
In Warburton’s initial model, τ is defined as the average delay of the rejected activities.
Because of the initial definition of r, the reject rate of activities, the accelerated activities
are also not taken into account for the calculation of τ . As a consequence the value for
τ is always bigger than zero. In the new Warburton model, parameter τ is calculated
based on the deviated activities, as determined by the redefined r as discussed above.
Here, the new τ stands for the average deviation of the planned duration of the deviated
activities. This means τ can be either positive, negative or zero which would mean the
deviated activities are respectively on average delayed, accelerated or on time. The value
for the new τ is calculated as follows:
τ =
∑(real duration− planned duration)of each deviated activity
#deviated activities(13.2)
Parameter c: The cost overrun or underrun
In Warburton’s initial model, parameter c is defined as the average fractional extra cost
of the planned cost to finish a rejected activity. Again, because of the initial definition
of r, only cost overruns are incorporated in this initial model as accelerated activities
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 120
are not taken into account. In the new Warburton model, parameter r accounts for both
activity accelerations and delays, which means parameter c should be able to reflect both
an activity cost overrun or underrun. This is done by calculating the value for c on the
basis of the deviated activities instead of the rejected activities as in the initial model.
This means c can take three diffferent values. First, c can be positive and represents
an on average fractional cost overrun if, on average, the fraction r of deviated activities
are delayed and thus τ > 0. Second, c can be equal to zero and represents no average
fractional cost deviation if, on average, the fraction r of deviated activities are on schedule
and thus τ = 0. Third, c can be negative and represents an on average fractional cost
underrun if, on average, the fraction r of deviated activities are accelerated and thus
τ < 0. It is clear by now that a positive (negative) value of τ goes along with a positive
(negative) value of c. In the new Warburton model, parameter c is calculated as follows:
c =
∑((actual cost− planned cost)/ planned cost)of each deviated activity
#deviated activities(13.3)
13.2.2 Modification of Warburton’s curves
Although Warburton’s initial model is based on the assumption that the model param-
eters are all positive, the model will work just as well when c is negative, representing a
cost underrun. So no immediate adjustment is needed here. However, we already noted
that a negative value of c goes along with a negative value for τ , meaning the model
must be able to cope with both negative values for c and τ . This is where the Wabur-
ton’s initial model comes up short, as it is not able to cope with schedule accelerations
(negative values of τ) because of the time delay terms, e.g. pv(t-τ)w. Therefore, there’s
a need to reformulate the model. This research question was also explicitly mentioned by
Warburton in his article [21], where he states that more research is required to determine
if the model can be reformulated to include schedule accelerations.
The first step towards solving this shortcoming is already taken with the reformulation
of the model parameters as they can now be negative and represent schedule acceler-
ations. As a second step, additional formulas for ev(t)w, EV(t)w, ac(t)w and AC(t)w
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 121
were constructed, especially to be able to cope with negative values of τ . In the new
Warburton model, an additional variant for the calculation of each of these curves was
generated, dependent on the value of parameter τ (positive or negative). Only one out
of the two variants will be applied to a certain project, depending on its τ value. Figure
13.1 gives a visual overview of the fundamental differences between the new and initial
model.
Figure 13.1: Overview of differences between the initial Warburton model and the new War-
burton model.
In case the value of τ is positive (schedule delay) or equal to zero (on schedule) of
a project, the same formulas for the calculation of the ev(t)w-, EV(t)w-, ac(t)w- and
AC(t)w- curve as in Warburton’s initial model can be applied. Important to mention
here is that this does not mean that the values for the earned value and actual cost
curves in the new Warburton model will be equal to these of Warburton’s initial model
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 122
as the underlying parameters r, c and τ are now calculated differently, as discussed in
section 13.2.1.
In case the value of τ is negative (schedule accelerations), the formulas for the calculation
of the ev(t)w-, EV(t)w-, ac(t)w- and AC(t)w- curve used in Warburton’s initial model
become unusable. Therefore, a set of additional formulas has been generated. These
additional formulas for when τ < 0 are presented and discussed below. For the formulas
to be used when τ ≥ 0 and a discussion of the reasoning behind these formulas, the reader
is referred to chapter 3 and R.D.H. Warburton’s paper [21]. It is not our intention to
discuss the formulas in detail again, but to show where and how adjustments were made
in case τ < 0.
Earned value curves ev(t)w and EV(t)w for projects with τ < 0
In the new Warburton model, the earned value curves for projects with a negative τ value
are generated as follows. At time instance t=1, the instantaneous earned value consist of
two contributing parts. First, there’s the value earned by the fraction of activities, 1-r,
that were executed according to plan. Second, there’s the value that was earned τ time
units earlier than planned which equals fraction r, or in other words, the activities that
are accelerated and already finished at time instance t=1. For t>1, the earned value
consists of the same two contributing parts as in Warburton’s initial model that was
applied when t> τ . The only difference here is that τ is negative. There’s the fraction
of activities that were successfully completed at time t, and those from t-τ that incurred
an acceleration and are completed earlier than foreseen. This way, the ev(t)w-curve is
developed, and the EV(t)w-curve follows by integrating the instantaneous values. Also
here EV(t→∞)w equals N. When τ <0:
ev(t)w =
(1− r)pv(t)w +
∑−τ+1i=1 r.pv(i)w, t = 1
(1− r)pv(t)w + r.pv(t− τ)w, t > 1
(13.4)
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 123
EV (t)w =
N(1− r)
[1− exp
(− t2
2T 2
)]+∑−τ+1
i=1 r.N[1− exp
(− i2
2T 2
)], t = 1
N −N(1− r)exp(− t2
2T 2
)− r.N.exp
(− (t−τ)2
2T 2
), t > τ
(13.5)
Actual curves ac(t)w and AC(t)w for projects with τ < 0
In the new Warburton model, the actual cost curve for projects with a negative τ value
is generated as follows. At time instance t=1, the instantaneous actual cost consists of
two parts. First there are the costs incurred by the activities that are executed according
to plan. Second, there’s the sum of the reduced costs that goes along with the fraction
r of the pv(t)w that is scheduled from time instance t=1 until time instance t=−τ + 1.
The reasoning behind this equation is that, within this time interval, a fraction r of the
activities have incurred an acceleration of on average τ time units, which reduces the
total actual project cost. As a negative value for τ always goes along with a negative
value for parameter c, this second value of the equation is always negative and reduces
the actual cost. For t>1, the actual cost consists of the same two contributing parts as in
Warburton’s initial model that was applied when t> τ . The only difference here is that
τ is negative now. There’s the fraction of activities that were successfully completed at
time t, and those from t-τ that incurred an acceleration and are completed under budget.
This way the ac(t)w-curve is developed, and the AC(t)w-curve follows by integrating the
instantaneous values. When τ <0:
ac(t)w =
pv(t)w +
∑−τ+1i=1 r.c.pv(i)w, t = 1
pv(t)w + r.c.pv(t− τ)w, t > 1
(13.6)
AC(t)w =
N[1− exp
(− t2
2T 2
)]+∑−τ+1
i=1 r.c.N[1− exp
(− i2
2T 2
)], t = 1
N[1− exp
(− t2
2T 2
)]− r.c.N.
[1− exp
(− (t−τ)2
2T 2
)], t > τ
(13.7)
As in the Warburton’s initial model, EACw= AC(t→∞)w = N(1+r*c), which represents
the forecasted actual project cost.
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 124
For an overview of the parameters and curves of the initial Warburton model versus the
new Warburton model, the reader is referred to tables C.1 and C.2 on page 169 and 170
in appendix C.
13.3 Example project
Using an example project, the use of the modified parameters and formulas together
with the impact on the time and cost forecast accuracy is illustrated. The same baseline
schedule and real life execution of the example project of chapter 4 will be used. On
top, an extra real life execution of the project will be generated to demonstrate the
difference between a project with a schedule acceleration (negative τ) and a schedule
delay (positive τ), which allows us to clearly demonstrate the added value of the new
Warburton model. The baseline schedule can be found in figure 4.2 on page 32.
13.3.1 Schedule delay (τ > 0)
To apply the new Warburton model to a project with a positive τ (Schedule delay), the
same real life execution of the example project is used as in chapter 4. The representation
of this (fictitious) real execution of the project up until 30 % of the BAC is earned, which
is at time unit 4, can be found on figure 13.2, which is the same figure as figure 4.3 on
page 33. With the available data at this point in the project, the calculation of the
modified parameters r, c and τ can be illustrated. These parameters can have different
values compared to Warburton’s initial model, because of the incorporation of activities
that are accelerated.
Parameter r, the deviation rate of activities, equals the fraction of the completed activ-
ities at time instance 4 which weren’t completed according to their planned duration.
After 4 time units activities 1, 2 and 3 are finished of which activity 1 and 3 both had
one time unit delay and activity 2 was finished one time unit ahead. This means r equals
1 (3/3). Note that this value equaled 0.67 (2/3) for Warburton’s initial model, as the
acceleration of activity 2 wasn’t taken into account. For parameter c, one calculates the
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 125
Figure 13.2: Example project: (Fictitious) real project execution (τ > 0, delay) until 30 % of
BAC is earned.
average of the extra or reduced cost relative to the planned cost of the activities that
belongs to the fraction r. In this example parameter c equals 0.17 (((60-40)/40 + (40-
60)/60 + (80-60)/60)/3)) as the variable cost was set at 20 cost units/time unit. This
positive value of c means that, on average, the project has a cost overrun. Parameter τ
equals the average extra or shorter duration of these activities, which is an extra 0.33
time units (((3-2)+(2-3)+(4-3))/3) in this example.
The curves for earned value and actual cost can be generated by filling in these values in
the appropriate formulas, which are the same formulas used in Warburton’s initial model
as τ has a positive value in this fictitious project execution (project delay). However, the
values of the curves will be different, as the underlying parameters have changed. This
also leads to other forecasts. The fact that the parameters r, c and τ are now more able
to be close to reality by not only taking into account delays, we expect this to lead to
better forecasts of the final project duration and cost, based on the same early project
data. With the new Warburton model, the EACw, which is determined by N(1+r*c),
equals 863 (740*(1+1*0.17). For EAC(t)w we can, as done in chapter 4, look where
the EV(t)w-curve stabilizes, which is again after 19 time units. Of course we now have
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 126
the different EAC(t)w-methods of section 5.1.3 to forecast final duration, but these were
already illustrated once and would lead us too far in this example.
To determine how good the forecasts of the new Warburton model in this example are,
the real final duration and cost of the project are needed. The same (fictitious) real
project execution as in chapter 4 is used and is displayed on figure 4.6 on page 35. In
this real life execution of the project, the final cost was 880 cost units and the final
duration was 18 time units. The results for both the initial and new Warburton model
applied to this example project are displayed in table 13.1.
Table 13.1: Example project: Comparison of the results of the initial and new Warburton
model when τ > 0 (delay).
Delay N T r c τ EAC(t)w MAPEtime EACw MAPEcost
Initial Model 740 5.2 0.67 0.42 1 19 5.6% 945 7.4%
New Model 740 5.2 1 0.17 1 19 5.6% 863 1.9%
With the use of the new model, EACw lies closer to the real project cost. This improve-
ment is easily seen in the lower cost MAPE value, which decreases from 7.4 % to 1.9
% . No significant changes are found in the prediction of the final project duration, as
EAC(t)w is rather insensitive towards changes in the parameter r, which increased to its
maximum value of 1. This insensitivity of EAC(t)w for changes in r and τ is because of
the long tail problem that was discussed in section 5.1.3.
13.3.2 Schedule acceleration(τ < 0)
To illustrate the difference in forecast accuracy between the initial and new Warburton
model for accelerated projects with a negative τ , another real life execution of the same
baseline schedule of the example project of chapter 4 is constructed. The representation
of this (fictitious) real project execution up until 30 % of the BAC is earned can be
found on figure 13.3. At this point, activities 1, 2 and 3 are finished of which activity
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 127
1 finished right on schedule and activity 2 and 3 both were accelerated with one time
unit. Applying the new Warburton model, one can easily calculate the values for the
modified parameters r, c and τ the same way as illustrated in section 13.3.1. These are
respectively equal to 0.67, -0.22 and -1.
Figure 13.3: Example project: (Fictitious) real project execution when project is accelerated
(τ < 0) until 30 % of BAC is earned.
The adjusted earned value and actual cost curves can now be generated by filling in
these values in the newly developed formulas for when τ < 0, as presented in section
13.2.2. With the new Warburton model, forecasts can again be made the same way as
in section 13.3.1, leading to an EAC(t)w and EACw of 630 and 19 respectively.
To determine how good the forecasts of Warburton’s model actually are, the real final
duration and cost of the project are needed. Another real project execution at 100 %
completion is displayed on figure 13.4. In this example project, the final cost is 640 and
the final duration is 12 time units. Using the MAPE formulas, this leads to respective
absolute deviations of 1.6 % and 58.3% of the forecasts of Warburton’s model.
It’s interesting to go one step further here and to see what happens if we would apply
Warburton’s initial model to the 30 % early data of figure 13.3. The parameters r, c
and τ would all be equal to zero, as none of the activities incurred a delay after 30 %
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 128
Figure 13.4: Example project: (Fictitious) real project execution for accelerated project until
100 % of BAC is earned.
project completion. Warburton’s curves for earned value and actual cost can then be
generated by filling in these values in Warburton’s original formulas. The final project
cost and time forecast equals 740 cost units and 19 time units respectively, which leads
to respective absolute deviations of 15.6 % and 58.3 % of the forecasts of Warburton’s
initial model. A summary of these results can be found in table 13.2.
Table 13.2: Example project: Comparison of the results of the initial and new Warburton
model when τ < 0 (acceleration).
Delay N T r c τ EAC(t)w MAPEtime EACw MAPEcost
Initial Model 740 5.2 0 0 0 19 58.3% 740 15.60%
New Model 740 5.2 0.67 -0.22 -1 19 58.3% 630 1.60%
Comparing both models, one can easily see that the forecast error of the final project
cost dropped substantially by using the new model. This is a direct result of the fact that
the underlying parameters better reflect the current project performance. No significant
differences are found in the prediction of the final project duration for this example
project, as EAC(t)w is rather insensitive towards the model parameters r and τ.
Chapter 13. Shortcomings of Warburton’s model and set-up of improved model 129
13.4 Conclusion
This chapter presented the improved version of Warburton’s initial model. Two fun-
damental and promising advantages of the new model can be distinguished compared
to the initial model. First, the modified parameters better reflect the current project
performance as they take into account both activities with a shorter and a longer real
activity duration than planned. The parameters in Warburton’s initial model only took
into account finished activities that had incurred a delay, which resulted in distorted
parameter estimations. Second, the new Warburton model is able to cope with negative
values of τ (schedule accelerations), as the earned value and cost curves functions have
been adjusted.
Thanks to these improvements in the model, we expect an increase of the applicably of
the model as it is expected now to not only perform well in on time and late scenarios,
but also in early scenarios. In the example project, the new Warburton model dominated
the initial model concerning the forecast accuracy of the final project cost. However,
an example of course doesn’t provide a solid base to draw general conclusions about
the performance of the new model. This is why another Monte Carlo simulation was
conducted to benchmark the performance of our new Warburton model against Warbur-
ton’s initial model. The results of this study is the subject of chapter 14. The aim is
to quantify the improvements we made on the initial model and benchmark the forecast
accuracy of the new Warburton model against the initial one.
Chapter 14
Accuracy study of the new
Warburton model and comparison
with the initial Warburton model
A similar accuracy study was conducted as in part 4 of this thesis, but this time for the
new and improved Warburton model. The goal of this chapter is to discuss the accuracy
of the new model and benchmark it against the initial Warburton model. An oversight
will be given of the most relevant and most important results of this accuracy study.
The methodology is entirely the same as described in part 3 and used in part 4 of this
thesis.
Section 14.1 is similar to the study conducted in chapter 8. Here the necessary input
to set up the new model is determined. First, the ratio values necessary for forecasting
methods 5, 6 and 7 that were described in section 5.2.1 are recalculated. Next, it is
determined if the T2 calculation of section 5.2 for parameter T remains the best method,
similar to section 8.2.
Section 14.2 is similar to the study conducted in chapter 9. The methods that are used
to forecast the final project duration and final project cost based on the new Warbur-
130
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 131
ton model are tested on accuracy and benchmarked against existing EVM forecasting
methods. Because of the promising results for the convex and concave time/cost rela-
tionships, these relations are also included for the EACw method that is based on our
new Warburton model. As described in chapter 13, the new Warburton model is now
able to cope with accelerations in activity duration. This is why we expect the model
to perform better, especially in the early scenarios, compared to the initial Warburton
model. These expectations and hypotheses are discussed in the beginning of this section.
Finally, section 14.3 is similar to the study conducted in chapter 10. Here, the relation of
the performance of the new Warburton model and the project completion stage will be
investigated. Because of the ambiguous results concerning the influence of the topological
structure and small differences in MAPE values, it was decided not to include these
results in this chapter.
14.1 Necessary input for the new Warburton model
14.1.1 Ratio values of methods EAC(t)w5, EAC(t)w6 and EAC(t)w7
The new ratio values per SP factor can be found in table 14.1 on the next page. A
notable difference with the values of the initial Warburton model presented in table 8.1
on page 74 is that the values for the new Warburton model in table 14.1 are average
values over all 6 scenarios and don’t exclude early scenarios 1 and 2, as the new model
is now able to deal with accelerations.
14.1.2 Parameter T
Based on the results displayed in figure 14.1 on the next page, it can be concluded
that T2 is also the better calculation method for parameter T for the new Warburton
model. For the accuracy study that is conducted in the following sections of this chapter,
parameter T will by consequence always be determined with the T2 calculation method
that is described in section 5.2.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 132
Table 14.1: Average ratio values for time forecasting methods EAC(t)w5, EAC(t)w6 and
EAC(t)w7 for the new Warburton model.
Average ratio values
SP EAC(t)w5 EAC(t)w6 EAC(t)w7
T1 T2 T1 T2 T1 T2
0.1 63.4% 42.7% 99.7% 98.5% -8.7 -49.7
0.2 54.2% 39.9% 99.2% 96.7% -27.0 -106.8
0.3 53.6% 39.7% 98.1% 94.5% -60.9 -180.4
0.4 46.3% 33.1% 97.5% 92.4% -82.7 -251.9
0.5 44.5% 33.5% 97.4% 90.9% -86.6 -300.8
0.6 40.8% 28.3% 96.7% 89.8% -110.0 -341.8
0.7 35.7% 25.7% 95.7% 88.9% -137.3 -361.3
0.8 32.0% 24.7% 95.1% 87.3% -163.9 -415.9
0.9 29.9% 24.4% 94.0% 87.0% -194.4 -424.7
Figure 14.1: MAPE values of final duration forecasts using methods based on the new War-
burton model.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 133
14.2 Forecast accuracy
The intention here is to benchmark the forecast accuracy of the new Warburton model,
both in terms of final duration and final cost, against the accuracy of the initial War-
burton model and the traditional EVM forecasting methods. The results for the latter
two were simulated in chapter 9 of this thesis. Remember there are eight methods to
determine EAC(t)w, as discussed in section 5.3, and one method to determine EACw,
as discussed in section 3.2.3. The results for forecasting the final duration are discussed
in section 14.2.2. The same is done for the final cost in section 14.2.3, in which also
the three time/cost relationships (linear, convex and concave) are included. But before
getting into these results, our expectations for the new model are formulated in section
14.2.1.
14.2.1 Hypotheses regarding forecast accuracy of the new Warburton
model
As described in chapter 13, the new Warburton model is now able to cope with accelera-
tions in activity duration with possible negative values for parameters τ and c, contrary
to Warburton’s initial model. This is why we expect the new model to perform better,
especially in the early scenarios 1 and 2 as it is in these scenarios that the probability
for activity accelerations is the highest.
Because of the problem of the long tails and small changes in the forecasting accuracy of
the EAC(t)w methods, as concluded from part 4 of this thesis, we don’t expect the fore-
cast accuracy of the final project duration to improve a lot, despite of the incorporation
of activity accelerations in the model. However, concerning the forecast accuracy of the
final project cost, we do expect a big improvement, as changing parameters r and c have
a big influence on the final project cost prediction. The biggest improvement should
be a reduced forecast error in the early scenarios as these were affected the most by
the distorted parameters and the non-negativity restriction. The following alternative
hypotheses apply to the performance of the new model:
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 134
Hypothesis 6: Both for the time and cost forecast accuracy, the new Warburton model
will mainly be an improvement for the early scenarios, while only small improvements
are expected for on time scenarios and almost no change is expected for late scenarios 1.
Hypothesis 7: The overall forecast accuracy of the final project duration will only
improve to a very limited extent.
Hypothesis 8: The overall forecast accuracy of the final project cost will highly improve.
14.2.2 Accuracy of the final duration forecasts
Table 14.2 presents the MAPE values for the traditional EVM methods to forecast the
final project duration, while table 14.3 on the next page does the same for the EAC(t)w
methods based on Warburton’s initial model. These two tables are simply a repetition
of the results of section 9.1. For the new Warburton model, the results are presented in
table 14.4 on the next page. In each table, the results per scenario as well as the average
results over all 6 scenarios can be found.
Table 14.2: Forecasting error (MAPE) of final project duration using the traditional EAC(t)
methods.
EVM EAC(t)
Scenario PV1 PV2 PV3 ED1 ED2 ED3 ES1 ES2 ES3
1 19.6% 12.7% 28.5% 22.4% 12.7% 18.5% 19.6% 11.7% 21.1%
2 7.0% 6.0% 12.1% 7.7% 6.0% 8.5% 6.6% 7.0% 11.1%
3 4.6% 6.4% 10.2% 4.7% 6.4% 8.6% 4.4% 9.7% 12.2%
4 1.8% 2.5% 3.9% 1.8% 2.5% 3.4% 1.8% 6.8% 7.8%
5 4.9% 4.7% 8.9% 5.2% 4.7% 6.6% 4.4% 8.1% 11.8%
6 11.4% 8.2% 18.1% 11.6% 7.7% 12.0% 10.1% 8.3% 17.7%
Average 8.2% 6.7% 13.6% 8.9% 6.7% 9.6% 7.8% 8.6% 13.6%
1This hypothesis states where the main improvements are expected to be, not how big these improve-
ments might be, as this is stated in Hypothesis 7 and 8.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 135
Table 14.3: Forecasting error (MAPE) of final project duration using the EAC(t)w methods
based on the initial Warburton model.
INITIAL EAC(t)
Scenario w0 w1 w2 w3 w4 w5 w6 w7
1 373.4% 36.0% 36.8% 34.6% 46.2% 64.7% 346.9% 41.2%
2 289.2% 11.7% 13.1% 10.6% 26.0% 38.0% 267.5% 20.9%
3 253.6% 5.7% 6.5% 5.7% 19.8% 27.4% 234.0% 15.0%
4 250.5% 2.2% 3.1% 2.5% 18.4% 26.3% 231.1% 13.7%
5 221.7% 8.1% 7.2% 9.6% 15.4% 19.8% 203.9% 12.8%
6 186.6% 18.1% 16.6% 18.9% 17.2% 17.0% 170.8% 18.5%
Average 262.5% 13.6% 13.9% 13.6% 23.8% 32.2% 242.4% 20.3%
Table 14.4: Forecasting error (MAPE) of final project duration using the EAC(t)w methods
based on the new Warburton model.
NEW EAC(t)
Scenario w0 w1 w2 w3 w4 w5 w6 w7
1 368.1% 34.2% 33.8% 29.7% 44.6% 53.0% 326.4% 29.6%
2 286.3% 10.8% 12.0% 8.2% 25.6% 29.3% 252.2% 14.6%
3 252.1% 5.5% 6.1% 5.7% 19.7% 21.5% 221.0% 13.7%
4 249.7% 2.1% 2.8% 2.7% 18.4% 20.7% 218.9% 13.0%
5 221.4% 8.2% 7.3% 10.0% 15.4% 17.7% 193.1% 16.8%
6 186.6% 18.2% 16.7% 19.0% 17.3% 17.7% 161.5% 24.6%
Average 260.7% 13.2% 13.1% 12.5% 23.5% 26.6% 228.8% 18.7%
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 136
Comparison of initial and new Warburton model
As can be seen in table 14.4, methods EAC(t)w1, EAC(t)w2 and EAC(t)w3 are the
clear winners among the methods based on the new Warburton model with a respective
average MAPE over all scenarios of 13.2 %, 13.1 % and 12.5 %. For the initial Warburton
model (table 14.3), a similar statement was made but with MAPE values of 13.6 %, 13.9
% and 13.6 % respectively. Compared to the initial model, the forecast error is thus
lowered, although only to a very limited extent. The biggest improvement in forecast
accuracy is found in the early scenarios, where the EAC(t)w3 method performs the best.
In the on time scenarios, the forecast accuracy remains stable compared to the initial
model, with EAC(t)w1 as the best method. For the late scenarios, the forecast accuracy
also remains stable, although there is a very small increase in the MAPE values. This
can be explained as follows. As Warburton’s initial model already underestimated the
final project duration in the late scenarios, as seen on figure 9.2 on page 82, a lower
value of τ as a result of also taking accelerated activities into account will only further
steer this trend. The EAC(t)w2 method remains the best one for late scenarios.
Benchmarking against the traditional EAC methods
When looking at the comparison of the average accuracy over and within the 6 scenarios,
the conclusions remain the same as discussed in section 9.1.1, which is why this discussion
is not repeated here. It can be stated that the same traditional EAC(t) forecasting
methods stay superior to the EAC(t)w methods based on the new Warburton model.
This should not come as a surprise, as the new Warburton model deals with the problem
of accelerations, but has not been adjusted for the long tail problem. No immediate
solution was found to cope with this problem, as explained in section 13.1.2.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 137
14.2.3 Accuracy of the final cost forecasts
In this section, the accuracy of the new Warburton model for predicting the final project
cost will be investigated under three different time/cost relationships (linear, convex and
concave) which were discussed in section 12.1. The results for the initial Warburton
model were discussed in section 9.2 and section 12.2.1. To remind the reader, War-
burton’s EACw formula equals N(1+rc). In the remainder of this section EACw initial
(EACw new) will be used to refer to the EACw forecasts based on the initial Warburton
model (new Warburton model).
14.2.3.1 Linear time/cost relationship
Table 14.5: Forecasting error (MAPE) of the final project cost under the assumption of a linear
time/cost relationship.
LINEAR Traditional EVM methods Warburton
Scenario EAC1 EAC2 EAC3 EAC4 EAC5 EAC6 EAC7 EAC8 EACw initial EACw new
1 17.5% 7.2% 8.3% 8.9% 15.7% 17.4% 7.2% 7.4% 38.5% 9.4%
2 5.8% 3.0% 3.7% 5.1% 6.8% 8.8% 3.1% 3.3% 12.9% 4.2%
3 2.4% 3.8% 4.4% 7.3% 6.8% 9.7% 3.8% 4.3% 7.4% 4.4%
4 0.9% 1.4% 1.7% 5.0% 2.5% 5.9% 1.4% 2.0% 2.1% 1.3%
5 3.7% 2.4% 3.0% 6.1% 5.2% 9.4% 2.4% 2.6% 3.0% 3.3%
6 8.8% 3.7% 4.8% 6.1% 10.1% 14.5% 3.8% 3.7% 5.5% 5.9%
Average 6.5% 3.6% 4.3% 6.4% 7.9% 10.9% 3.6% 3.9% 11.6% 4.7%
Average accuracy over the 6 scenarios
As discussed in section 9.2.1, the traditional EAC2, EAC3, EAC7 and EAC8 method offer
the highest forecast accuracy and EACw initial is dominated by all traditional methods.
Comparing the performance of EACw initial to EACw new (table 14.5), the forecast error
decreases from 11.6 % to 4.7 %. The incorporation of activity accelerations into the
model is thus successful in this regard. The EACw new method now dominates four of
the traditional EVM methods (EAC1, EAC4, EAC5 and EAC6) and comes close to the
accuracy of the best methods, EAC2 and EAC7, which have a forecast error of 3.6 %.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 138
Average accuracy per scenario
For the early scenarios 1 and 2, the new Warburton model is a big improvement con-
cerning the forecast accuracy of the final project cost. Compared to EACw initial, the
forecast error dropped from 38.5 % to 9.4 % and from 12.9 % to 4.2 % respectively
for EACw new. These results confirm that the new model is able to work with sched-
ule accelerations and has more realistic values for the underlying parameters r and c.
The EACw new method now outperforms methods EAC1, EAC5 and EAC6 in the early
scenarios 1 and 2, and also EAC4 in scenario 2.
For the more extreme on time scenario 3, the EACw method improved from 7.4 % to
4.4 %, and outperforms traditional methods EAC4, EAC5 and EAC6. In the on time
scenario 4, the forecast error of the EACw dropped from 2.1 % to 1.3 % and is only
dominated by EAC1 with the lowest MAPE of 0.9 %. So, also here it is confirmed that
EACw new leads to better forecasts of the final cost compared to EACw initial.
For the late scenarios 5 and 6, the forecast error of the EACw new method showed a
minimal absolute increase in the MAPE values of 0.3 % and 0.4 % respectively. This
means the EACw new method is still dominated by traditional methods EAC2, EAC3,
EAC7 and EAC8. This little increase may seem unexpected but is a logical consequence
of the incorporation of accelerations into the model together with the characteristics of
the model that were discovered in the accuracy study of part 4: In chapter 9, the cost
MPE values revealed that EACw initial leads to an underestimation of the final project
cost in the late scenarios. As parameter c is now probable of having a slightly lower value
in the new model by incorporating activity accelerations, this will result in a slightly
larger underestimation of the final project cost. We say a slightly lower value as the
probability for activity accelerations is low in these late scenarios. However, as the EACw
formula equals N*(1+rc) and parameter r will have a slightly higher value compared to
the initial model by incorporating activity accelerations, the underestimation of the final
cost is again lowered. In total, it seems that the increase in underestimation because of
a lower c slightly outweighs the decrease because of the higher r. As the total difference
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 139
is minor, it can be stated that the accuracy of EACw new is comparable to EACw initial
concerning the late scenarios. Figure 14.2 visually illustrates the improvement of the
EACw accuracy of the new Warburton model for all scenarios.
Figure 14.2: Accuracy of the EACw method based on the initial Warburton model and on the
new Warburton model under the assumption of a linear time/cost relationship.
14.2.3.2 Convex time/cost relationship
Table 14.6: Forecasting error (MAPE) of the final project cost under the assumption of a
convex time/cost relationship.
CONVEX Traditional EVM methods Warburton
Scenario EAC1 EAC2 EAC3 EAC4 EAC5 EAC6 EAC7 EAC8 EACw initial EACw new
1 30.6% 19.8% 14.4% 13.6% 26.5% 27.2% 18.4% 18.4% 74.3% 18.3%
2 9.0% 13.2% 6.6% 7.3% 16.8% 17.7% 12.0% 12.1% 25.4% 9.0%
3 6.6% 15.0% 8.2% 9.8% 16.9% 17.4% 13.9% 14.0% 13.2% 9.6%
4 3.6% 10.9% 4.2% 6.3% 11.6% 11.8% 9.9% 10.0% 4.3% 3.0%
5 11.1% 12.7% 8.6% 7.9% 12.6% 11.6% 12.0% 11.9% 6.9% 7.6%
6 20.8% 15.2% 14.9% 12.8% 16.1% 15.5% 14.9% 14.7% 12.2% 12.7%
Average 13.6% 14.5% 9.5% 9.6% 16.7% 16.9% 13.5% 13.5% 22.7% 10.0%
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 140
Average accuracy over the 6 scenarios
As discussed in section 12.2.1, the traditional EAC3 and EAC4 methods offer the highest
forecast accuracy in this setting and EACw initial is dominated by all traditional methods.
Comparing the performance of EACw initial and EACw new (table 14.6), the forecast error
decreased from 22.7 % to 10 %. Thanks to this improvement, EACw new now becomes
nearly as accurate as the best traditional methods EAC3 and EAC4, which both have a
forecast error of about 9.6 %.
Average accuracy per scenario
For the early scenarios 1 and 2, the new Warburton model is again a big improvement
concerning the forecast accuracy of the final project cost. The cost forecast error of
EACw initial dropped from 74.3 % to 18.3 % and from 25.4 % to 9 % for EACw new
respectively. The EACw new method now outperforms all traditional EAC methods
except for EAC3 and EAC4.
For the more extreme on time scenario 3, the EACw method improved from 13.2 % to 9.6
%, resulting in the fact that EACw new now outperforms all traditional EAC methods
except for EAC1 and EAC3. In the on time scenario 4, the forecast error of EACw initial
dropped from 4.3 % to 3.0 % for EACw new, which is more accurate than all traditional
EAC methods.
For the late scenarios 5 and 6, EACw new outperforms all other methods, despite of the
fact that the forecast error slightly increased with absolute values of 0.7 % and 0.5 %
respectively, compared to the EACw initial. The reason for this little increase in forecast
error was already explained in the section for the linear time/cost relationship above.
On the next page, figure 14.3 again displays the improvement of the EACw accuracy
under a convex time/cost relationship.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 141
Figure 14.3: Accuracy of the EACw method based on the initial Warburton model and on the
new Warburton model under the assumption of a convex time/cost relationship.
14.2.3.3 Concave time/cost relationship
Table 14.7: Forecasting error (MAPE) of the final project cost under the assumption of a
concave time/cost relationship.
CONCAVE Traditional EVM methods Warburton
Scenario EAC1 EAC2 EAC3 EAC4 EAC5 EAC6 EAC7 EAC8 EACw initial EACw new
1 10.9% 7.3% 7.1% 8.8% 10.4% 11.8% 5.9% 5.4% 19.1% 4.8%
2 5.0% 7.5% 3.6% 4.9% 6.6% 5.2% 5.9% 5.0% 6.2% 2.0%
3 3.1% 7.7% 4.4% 7.1% 8.6% 9.0% 6.4% 6.0% 3.9% 2.1%
4 2.5% 7.7% 2.8% 5.3% 7.7% 8.0% 6.3% 6.0% 1.0% 0.6%
5 1.8% 7.8% 4.6% 9.3% 11.8% 17.9% 6.8% 7.8% 1.4% 1.5%
6 2.8% 7.9% 7.0% 11.9% 17.9% 25.0% 7.4% 8.4% 2.5% 2.7%
Average 4.3% 7.7% 4.9% 7.9% 10.5% 12.8% 6.5% 6.4% 5.7% 2.3%
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 142
Average accuracy over the 6 scenarios
As discussed in section 12.2.1, the traditional EAC1 and EAC3 methods offer the highest
forecast accuracy and EACw initial is only dominated by these two methods. With the
EACw new method (table 14.7), the forecast error decreased from 5.7 % to 2.3 %. Thanks
to this improvement, EACw new now becomes the most accurate forecasting method for
final cost, when looking at the average MAPE values over all 6 scenarios and over the
complete project execution.
Average accuracy per scenario
For the early and on time scenarios, the new Warburton model is again a big improvement
concerning the forecast accuracy of the final project cost. The biggest improvement can
again be found in scenario 1 where the MAPE dropped from 19.1 % for EACw initial to
4.8 % for EACw new. Again for the late scenarios a slight increase of maximum 0.2 % is
found.
The remarkable statement can be made here that EACw new now dominates all tradi-
tional EAC methods in all 6 scenarios under the assumption of a concave time/cost
relationship. Figure 14.4 displays the improvement of the EACw accuracy of our new
Warburton model under a concave time/cost relationship.
Figure 14.4: Accuracy of the EACw method based on the initial Warburton model and on the
new Warburton model under the assumption of a concave time/cost relationship.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 143
14.3 Project completion stage
As no impressive improvements were accomplished for any of the three time/cost rela-
tionships for the forecast accuracy of the final project duration of the EAC(t)w methods
based on the new Warburton model, the results concerning project completion stage will
also be very similar to those of the initial Warburton model as discussed in section 10.1.
That is why section 14.3 will not incorporate the EAC(t)w methods based on the new
model.
Furthermore, as discussed in section 14.2 above, the biggest improvements were as ex-
pected realized for the early scenarios and only slight differences for the late scenarios
were found for all three time/cost relationships. This means the results for the new
Warburton model for the late scenarios are also similar to those discussed in section 10.2
and section 12.2.2, and will not be discussed again. Instead, the focus of section 14.3.1
lies on the relation of the cost forecast accuracy of EACw new and the project completion
stage, and this for the early scenarios 1 and 2. In section 14.3.2 the same is done but
averaged over all 6 scenarios.
14.3.1 Early projects (Scenario 1 and 2)
Figures 14.5, 14.6 and 14.7 on the next two pages present the results of the traditional
EVM methods, EACw initial and EACw new, and this under the assumption of respec-
tively a linear, convex and concave time/cost relationship.
In chapters 10 and 12, it was shown that the accuracy of EACw initial doesn’t change
across the different completion stages and that the MAPE values were very high for every
time/cost relationship. For EACw new however, the forecast accuracy does improve along
a further stage of completion, and this for all three time/cost relationships. The results
for the early stage are thoroughly discussed. The results for the middle and late stage
are similar.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 144
Figure 14.5: Forecast error (MAPE) of the EAC methods for late projects under the assump-
tion of a linear time/cost relationship for early projects (scenario 1 and 2).
Figure 14.6: Forecast error (MAPE) of the EAC methods for late projects under the assump-
tion of a convex time/cost relationship for early projects (scenario 1 and 2).
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 145
Figure 14.7: Forecast error (MAPE) of the EAC methods for late projects under the assump-
tion of a concave time/cost relationship for early projects (scenario 1 and 2).
Already in the early stage, it is shown that EACw new is a big improvement compared
to EACw initial. On average, the forecast error decreased from 25 % to 11.5 % in a
linear time/cost setting, from 50 % to 21 % in a convex time/cost setting and from
12.5 % to 6.2 % in the concave time/cost setting. Overall, it can be stated that the
forecast error of EACw initial is reduced by half for EACw new in the early stages of a
project. Contrary to EACw initial, the forecasting accuracy improves when going in a
further stage of completion for EACw new, which means this improvement in accuracy
only becomes bigger, leading to a forecast error of EACw new of about one fifth of the
error of EACw initial in the late stage, and this for all three time/cost relationships.
For a linear time/cost setting, EACw new now dominates traditional methods EAC1,
EAC4, EAC5 and EAC6 and performs only slightly worse than all other traditional
methods in the early stage, while EACw initial was dominated by all traditional EAC
methods in this setting. In the early stage with a convex time/cost setting, the EACw new
dominates all traditional methods except for EAC3 and EAC4, while EACw initial was
again dominated by all traditional methods. In the early stage with a concave time/cost
setting, EACw new now impressively dominates all traditional EAC methods. For the
middle and late stage, the same conclusions can be stated, with that difference that for
all methods, except for the EACw initial, lower MAPE values are reached.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 146
14.3.2 Average over all scenarios
To discuss the general applicability of the new Warburton model to forecast the final
cost, the average results over all six scenarios between a project completion stage of 10
% and 100% with intervals of 10% are displayed in figures 14.8, 14.9 and 14.10 for a
linear, convex and concave time/cost relationship respectively.
One can see that EACw new follows a more stable pattern and offers more robust results
for forecast accuracy during the project lifetime than most of the traditional EAC meth-
ods as there is a smaller adjustment of the MAPE values, and this for all three time/cost
settings.
Figure 14.8: Forecast error (MAPE) of the EAC methods under the assumption of a linear
time/cost relationship averaged over all six scenarios.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 147
Figure 14.9: Forecast error (MAPE) of the EAC methods under the assumption of a convex
time/cost relationship averaged over all six scenarios.
Figure 14.10: Forecast error (MAPE) of the EAC methods under the assumption of a concave
time/cost relationship averaged over all six scenarios.
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 148
Early stage of a project (0%-30%)
Especially in the early stage of a project most of the traditional EAC methods have a
substantial higher forecast error than EACw new, and this for all three time/cost rela-
tionships.
In case of a linear time/cost relationship (figure 14.8), EAC1, EAC4, EAC5 and EAC6
have a high forecast error during the early stage of a project, while the accuracy of
EACw new already lies in the region of the best performing EVM methods: EAC2, EAC3,
EAC7 and EAC8. After 20 % of the value earned, EACw new has a MAPE of 7.2 %,
which is not much worse than the best method EAC2 with a MAPE of 5.9 %. Once 30 %
of the value is earned, the accuracy of EACw new with a MAPE of 5.6 % is comparable
to the best performing method EAC2 with a MAPE of 4.7 %.
In case of a convex time/cost relationship (figure 14.9), EACw new offers a lower forecast
error than all other EVM methods, except for method EAC3 and EAC4. After 20 %
of the value is earned, EACw new has a MAPE of 14.3 %, which is fairly equal to the
MAPE of EAC3 and EAC4. After 30 % of the value is earned, EACw new offers the
lowest forecast error of all methods, with a MAPE of 11.4 %.
In case of a concave time/cost relationship (figure 14.10), EACw new clearly offers the
lowest forecast error of all methods during the early stage of a project. After 20 % (30
%) of the value earned, EACw new has a MAPE of 3.7 % (2.7 %), which is a lot lower
than the MAPE value of 6.5 % (5.6 %) for the best traditional method EAC1.
Middle stage of a project (30 %-70 %)
Although R.D.H. Warburton states the focus lies on the early stage of a project, it’s
interesting to look at the performance of the new Warburton model during the middle
stage of a project while updating the parameters r and c each time an additional 10 % of
the total value is earned. For a linear cost function, it can be seen that EACw new offers
a forecast accuracy comparable to the best performing EVM methods during the middle
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 149
stage of a project. For a convex time/cost relationship and especially for a concave cost
time/cost relationship, EACw new keeps offering the highest forecast accuracy during the
middle stage. So the model could be a valuable addition, not only in the early but also
the middle stage.
Late stage of a project (70 %-100 %)
Once the project has entered the late stage (after 70 % of the value is earned), updating
Warburton’s parameters becomes of less value as they do not change a lot anymore.
Moreover, as the EVM methods always converge towards a forecast error of zero at the
end of the project, it becomes hard for EACw new to outperform or even give as accurate
results in the late stage of a project as the traditional EVM methods. This is especially
the case when a linear or convex cost function is applied, but not for a concave cost
function. There, EACw new still outperforms all other methods even after a completion
stage 90 % due to the fact it provides a very high forecast accuracy.
14.4 Conclusion
In this chapter, the new Warburton model that was introduced in chapter 13 was the
subject of similar accuracy study that was done in part 4 of this thesis for the initial
Warburton model. In section 14.1, the ratio values for the EAC(t)w5, EAC(t)w6 and
EAC(t)w7 methods were determined and displayed in table 14.1 on page 132. Also, it
was found that the T2 calculation method also outperforms the T1 calculation method
for the new Warburton model. With this information, the hypotheses stated in section
14.2 could be tested.
Hypothesis 6, which states the main improvements are expected to be found for the
early scenarios, while small improvements are expected for the on time scenarios and
almost no change in accuracy is expected in the late scenarios, was confirmed for both the
forecasts of the final duration and cost. So the expectations concerning where the biggest
improvements would be are fulfilled. However, for the EAC(t)w methods to forecast the
Chapter 14. Accuracy study of the new Warburton model and comparison with theinitial Warburton model 150
final project duration, these improvements remained very small, which is a confirmation
of Hypothesis 7, and by consequence the new Warburton model is also not a valuable
addition to the EVM theory to forecast final project duration. On the other hand, very
promising results were found for the new Warburton model concerning the forecasting
method EACw new for forecasting the final project cost. Hypothesis 8, which states that
the new Warburton model will lead to a big improvement of the forecast accuracy of the
final project cost, was confirmed under the assumption of a linear, as well as a convex and
concave time/cost relationship. This is clearly illustrated on figures 14.2, 14.3 and 14.4
on page 139,141 and 142 respectively. For the linear time/cost relationship, EACw new
now comes close to the accuracy of the traditional methods, but doesn’t deliver the best
accuracy in any of the scenarios. For the convex time/cost relationship EACw new is
the best method in scenarios 4, 5 and 6 and is as good as the best traditional method
averaged over all scenarios. The most remarkable results are reached for the concave
time/cost relationship, where EACw new now dominates all traditional EAC methods in
all 6 scenarios.
When looking at the results per project completion stage, it can be stated that the
forecast error of EACw initial is reduced by half for EACw new in the early stages of a
project. Contrary to EACw initial, the forecasting accuracy improves when going in a
further stage of completion for EACw new, which means this relative improvement in
accuracy only becomes bigger, leading to a forecast error of EACw new of about one
fifth of the forecast error of EACw initial, and this for all three time/cost relationships.
Although R.D.H Warburton stated the model is mainly intended to improve the EVM
theory in the early stages of the project, the results suggest that it might be useful to
recalculate EACw new along the middle and sometimes the late stages of a project. By
doing this, EACw new is able to offer a comparable or even higher forecast accuracy than
the traditional methods during the project execution.
Part VI
FINAL REFLECTIONS
151
Chapter 15
Final conclusions
In the world of project management, Earned Value Management (EVM) systems have
been developed to provide project managers with crucial information concerning the
performance of their projects through the interaction of three project management ele-
ments: time, cost and scope. EVM makes it possible to provide project managers with
early warning signals for poor performance, which indicate it might be useful to take
corrective actions. Therefore, one of the important contributions of EVM is its ability
to forecast the final project duration and cost.
Recently, an article was published that presents an interesting yet unproven method for
including time dependence into Earned Value Management. This novelty was proposed
by Roger D.H. Warburton in his paper ‘A time-dependent earned value model for soft-
ware projects’ [21]. The model was set up with the goal of improving the theory of EVM
by including time-dependence into the definitions of all quantities and, by doing this,
delivering precise estimates of the project’s final cost and duration. The model makes
use of data available in an early stage of the project, the so-called ‘early project data’,
as it is in this stage of a project the warning signals are considered to be most crucial
for project managers.
The aim of this thesis was to thoroughly investigate this new concept. In particular,
three specific challenges were handled. First, the forecast accuracy of the model for final
152
Chapter 15. Final conclusions 153
project duration and cost was investigated, and this in different settings and for different
project networks. Second, the performance of the model was benchmarked against the
existing EVM forecasting methods for final project duration and cost. And third, with
the lessons learned from the accuracy study, the model proposed by R.D.H Warburton
was improved. To avoid confusing, we use ‘the initial Warburton model’ to refer to the
model as it was proposed in Warburton’s paper [21], and ‘the new Warburton model’ to
refer to the model after it was improved by us.
In what follows, the main conclusions of this study are reviewed, together with some
recommendations concerning the practical use of the model for project managers and
some guidelines for future research.
15.1 Performance and added value of Warburton’s model
The forecasting methods based on Warburton’s model for final project duration are
referred to as the EAC(t)w methods, while the forecasting method based on Warburton’s
model for final project cost is referred to as the EACw method. When we are talking
about methods based on the new Warburton model, also a subscript ‘new’ is added. The
performance of these methods was discussed based on their forecasting error measured
by the Mean Absolute Percentage Error (MAPE).
15.1.1 Forecasting the final project duration
The initial Warburton model
Although R.D.H. Warburton suggested the presented model can be used to predict the
final duration of the project, no specific method for this is brought forward. Therefore,
seven own developed EAC(t)w methods based on the Warburton model were introduced
and evaluated.
On average, when taking the complete project execution into account and the average
results over all scenarios (early, on time and late), none of the EAC(t)w methods based
Chapter 15. Final conclusions 154
on Warburton’s model outperform any of the traditional EAC(t) methods of the EVM
theory. The best EAC(t)w methods have average MAPE values of about two times as
high as the best traditional EVM methods. When looking at the scenarios separately,
it was clear that Warburton’s model performed especially poor for projects that end
early. Furthermore, when taken a look at the performance of the best EAC(t)w methods
across the project completion stages (early, middle and late) of a project, it was shown
that the forecast accuracy of these methods almost doesn’t change across the different
stages of completion. This means that the amount of early project data used doesn’t
have an impact on the accuracy of the EAC(t)w methods. These results are rather
unexpected, as one could expect more accurate forecasts when more early project data
is used to determine the parameters to set up Warburton’s model. In general, the
underlying topological structure of the project doesn’t have a big impact on any of these
statements, which means the EAC(t)w methods are rather stable and their performance
does not heavily depend on the network structure.
During this analysis, two crucial shortcomings of Warburton’s model were shown. First,
the initial Warburton model is not able to cope with accelerations in a project because
of its parameter definitions and underlying formulas. This means it doesn’t take into
account the impact of activities that are finished quicker than initially planned. This
shortcoming becomes of more critical importance for the accuracy of Warburton’s model
when the proportion of activities that are accelerated increases. This explains the re-
markable poor performance of the model for the early scenarios. Second, because of the
exponential factors in the Warburton formulas, long tails for the instantaneous curves
are inherent to the model. This means that, towards the end of the project, the increase
in planned and earned value in Warburton’s model is very small, and a lot of time units
go by until the project is completed, i.e. the BAC is reached. This also explains the lack
of change in the EAC(t)w forecast accuracy across the different completion stages.
Chapter 15. Final conclusions 155
The new Warburton model
Two fundamental and promising advantages of the new model can be distinguished
compared to the initial model. First, the modified parameters better reflect the cur-
rent project performance as they take into account both activities that end earlier and
later than planned. The parameters in Warburton’s initial model only took into account
finished activities that had incurred a delay, which resulted in distorted parameter esti-
mations. Second, the new Warburton model is able to cope with schedule accelerations
because of our modifications of the formulas for the earned value and cost curves, based
on our modified parameters. Thanks to these improvements in the model, the shortcom-
ing of accelerations is handled. However, no immediate solution was found for the long
tail problem.
On average, the forecast error concerning final project duration is lowered compared to
the initial Warburton model, but only to a very limited extent. The biggest improvement
in forecast accuracy is found in the early scenarios. When looking at the comparison
of the average accuracy over all scenarios and per scenario separately, the conclusions
concerning the performance compared to the traditional EVM methods remain the same
as for the initial Warburton model. It can be stated that, despite the improvement for
accelerations, the same traditional EAC time forecasting methods stay superior to the
EAC(t)w new methods. The reason for this can thus be found in the long tail problem
for which the new Warburton model was not adjusted. In general, it can be stated that
the new Warburton model is not a valuable addition to the EVM theory to forecast final
project duration. Further research remains to be done, as discussed in section 15.2.
15.1.2 Forecasting the final project cost
The initial Warburton model
To forecast the final project cost, the EACw method was introduced by R.D.H Warbur-
ton. This method also deals with the shortcoming of the inability to incorporate activity
accelerations, just as the EAC(t)w methods for final project duration. However, the long
Chapter 15. Final conclusions 156
tail problem inherent to Warburton’s model does not affect the EACw method, as this
problem only causes many time units to go by until the BAC is reached but has no affect
on the final cost estimation itself. The performance of the EACw method was studied
under the assumption of a linear, convex and concave relationship between the activity
duration and activity cost.
When looking at the accuracy averaged over the complete project execution and all
scenarios (early, on time and late), Warburton’s EACw method is always dominated.
However, the mentioned inability of Warburton’s model to take accelerations into ac-
count leads to very poor performances in early scenarios, leading to a higher MAPE
for the average over all scenarios. When looking at the performance in the separate
scenarios, the EACw method is very promising, especially for the late scenarios. In
these late scenarios, Warburton’s EACw method performs almost as good as the best
traditional EAC methods in case of a linear time/cost relationship and, more important,
outperforms all traditional EAC methods in case of a convex and concave time/cost
relationship. Furthermore, when looking at the forecast accuracy for the final cost along
the different completion stages of late projects under all three time/cost relationships,
Warburton’s EACw method again leads to some promising results. For a linear time/-
cost relationship, Warburton’s accuracy is close to the best performing traditional EAC
methods and can be considered as a good forecasting method between a project com-
pletion of 25 % and 60 %. For a convex time/cost relationship, Warburton’s EACw
method offers the highest accuracy of all methods between a project completion of 20
% and 70 %. Finally, for a concave time/cost relationship, Warburton’s EACw method
offers the highest accuracy of all methods between a project completion of 10 % and
90 %. In general, the underlying topological structure of the project doesn’t have a big
impact on any of these statements, which means the EACw method is rather stable and
its performance does not heavily depend on the network structure.
Chapter 15. Final conclusions 157
The new Warburton model
With the new Warburton model, it was attempted to extend the promising results for
forecasting the final project cost beyond the restriction of late scenarios. As the new
Warburton model is able to cope with schedule accelerations, the goal was to attain
similar promising results for the early scenarios as for the late scenarios, as discussed for
the initial Warburton model.
The expectation that the new Warburton model would lead to a big improvement of the
overall forecast accuracy of the final project cost was confirmed under the assumption of
a linear, as well as a convex and concave time/cost relationship. For the linear time/cost
relationship, EACw new now comes close to the accuracy of the traditional methods, but
still doesn’t deliver the best accuracy in any of the scenarios. For the convex time/cost
relationship, however, the remarkable statement can be made that EACw new is the best
method in late scenarios and is as good as the best traditional methods, averaged over
all scenarios. The most remarkable results are reached for the concave time/cost rela-
tionship, where EACw new now dominates all traditional EAC methods in all scenarios
(early, on time and late). Furthermore, when looking at the results per project com-
pletion stage, it can be stated that the forecast error of EACw initial is reduced by half
for EACw new in the early stages of a project. Contrary to EACw initial, the forecasting
accuracy improves when going in a further stage of completion for EACw new, which
means this relative improvement in accuracy only becomes bigger, leading to a forecast
error of EACw new of about one fifth of the forecast error of EACw initial, and this for
all three time/cost relationships.
15.1.3 Recommendations for practitioners
As discussed, it can be stated that the Warburton model does not seem a valuable
addition to the EVM theory to forecast final project duration as it is dominated by
the traditional EAC methods, even after the improvement of the model. That is why
we would not recommend for practitioners to use the Warburton model to forecast the
Chapter 15. Final conclusions 158
final duration. However, because of the promising results of the EACw new method, the
new and improved Warburton model might well be a valuable contribution to a project
manager’s toolbox to forecast the final project cost, especially under the assumption of
a convex or concave time/cost relationship. The use of the EACw new method is also
straightforward and fairly easy to apply for the project manager, as it only requires three
steps.
First, the project manager has to determine the Budget At Completion (BAC) at the
start of the project, which is referred to with N in Warburton’s model. Second, two
parameter values have to be determined based on early project data which is available
after some part of the project is completed. These are the deviation rate of activities
‘r’, which is simply calculated as the fraction of the actitivies of which the real duration
deviates from its planned duration, and the cost over- or underrun ‘c’, which is simply
calculated as the average cost over- or underrun of these activities. With these values,
the EACw new can simply be calculated with the formula N.(1+rc), which gives the
forecasted final project cost based on the early project data.
Although R.D.H Warburton stated the model is mainly intended to improve the EVM
theory in the early stages of the project, our results suggest that it might be useful
for the project manager to recalculate this forecasting indicator along the middle and
sometimes the late stages of a project, by recalculating parameters r and c with the
available project data. The term ‘early project data’ thus has to be interpreted somewhat
broader than suggested by R.D.H Warburton. Following this approach, EACw new is able
to offer a comparable or even higher forecast accuracy than the traditional EAC methods
during the project execution. Under the assumption a convex time/cost relationship, the
EACw new method delivers the most accurate forecast of the final project cost up until
about 60 % of the project is completed. Under the assumption of a concave time/cost
relationship, EACw new still outperforms all other methods even after a completion stage
of 90 %. In these situations, the project manager might benefit from using the Warburton
model to forecast the final project cost.
Chapter 15. Final conclusions 159
15.2 Limitations and guidelines for future research
As mentioned, no specific method was introduced by R.D.H. Warburton to forecast the
final project duration based on the model. In this thesis it was attempted to find a
solution for this by developing and evaluating the eight EAC(t)w methods which were
discussed in section 5.3.1. However, the accuracy study of part 4 showed that, despite
the effort, the EAC(t)w methods don’t adjust enough to provide more accurate forecasts
along the project completion stage. The reason for this lies in the long tails that are
inherent to Warburton’s model. Towards the end of the project the increase in earned
value is neglectable, and a lot of time units go by until the project is completed, i.e. the
BAC is reached. Further research is needed to see how Warburton’s model can be used
to accurately forecast the final project duration.
Also, the scope of this thesis is restricted to the forecasting indicators for final project
duration and cost, and does not include a thorough investigation of the performance
measures Cost Performance Index (CPI) and Schedule Performance Index (SPI). That is
because, in our opinion, the biggest opportunities of the model proposed by Warburton
lie in the forecasting indicators, and especially for the final project cost. The reason
for this is that the Warburton model is intended to be set up using so-called early
project data, with which the necessary parameter values can be determined to make
predictions about the future. This is contrary to the performance measures, which
aim at given a ratio value which indicates how the project is doing at the time of the
calculation of these measures, using the traditional key indicators: Planned Value (PV),
Earned Value (EV) and Actual Cost (AC). However, R.D.H. Warburton suggests that
by incorporating time dependence, as is done in the model, it is possible to better
understand the behavior of the performance measures throughout the project execution.
In particular, ‘it would help to distinguish between changes due to the time dependence
and genuine changes in project performance that are a result of significant managerial
actions’ [21]. In this regard, future research concerning the performance measures based
on Warburton’s model might be useful.
Chapter 15. Final conclusions 160
Finally, it was shown that the (improved) Warburton model might very well be a valuable
addition to a project manager’s toolkit to forecast the final project cost. However, the
study of this thesis only confirmed this statement in the theoretical setting of a Monte
Carlo simulation. It could be interesting to test the forecast accuracy of the final project
cost of the (improved) model on various real-life project examples.
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Appendix A
Tables based on Measuring Time
settings
A.1 Nine scenarios of Measuring Time
Table A.1: Average forecasting accuracy (MAPE) of the time EVM methods for the 9 scenarios
of Measuring Time ([12], pg. 68), assumption concerning project completion as in
Measuring Time
TIME EAC(t)
Scenario PV1 PV2 PV3 ED1 ED2 ED3 ES1 ES2 ES3
1 34.5% 18.7% 37.0% 40.4% 18.7% 20.3% 32.5% 12.1% 21.5%
2 31.8% 19.1% 23.3% 35.1% 19.1% 18.0% 29.3% 13.2% 15.5%
3 8.2% 24.7% 165.1% 6.5% 24.7% 110.0% 7.7% 33.0% 135.3%
4 2.1% 5.3% 25.9% 1.5% 5.3% 15.0% 1.8% 5.9% 15.4%
5 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0%
6 6.0% 22.3% 163.6% 4.3% 22.3% 106.8% 5.8% 30.8% 132.8%
7 4.2% 7.6% 37.1% 3.4% 7.5% 21.3% 3.4% 7.9% 21.4%
8 21.6% 16.5% 19.2% 17.3% 11.8% 14.4% 15.4% 7.6% 12.6%
9 20.9% 14.3% 26.8% 16.7% 9.4% 16.8% 14.6% 5.4% 20.3%
164
Appendix A. Tables based on Measuring Time settings 165
A.2 Six new defined scenarios
Table A.2: Average forecasting accuracy (MAPE) of the time EVM methods for the our 6
scenarios, assumption concerning project completion as in Measuring Time
TIME EAC(t)
Scenario PV1 PV2 PV3 ED1 ED2 ED3 ES1 ES2 ES3
1 19.5% 12.7% 24.8% 22.1% 12.7% 14.4% 17.6% 9.6% 16.5%
2 7.0% 5.8% 10.5% 7.6% 5.8% 6.9% 5.9% 5.4% 8.7%
3 4.4% 5.6% 8.9% 4.5% 5.6% 7.4% 4.0% 7.4% 9.5%
4 1.7% 2.2% 3.5% 1.8% 2.2% 2.9% 1.6% 4.3% 5.2%
5 4.9% 4.6% 8.1% 5.0% 4.5% 6.0% 3.9% 5.4% 8.6%
6 11.5% 8.8% 17.1% 10.9% 7.7% 11.3% 8.7% 6.4% 14.9%
Average 8.2% 6.6% 12.1% 8.7% 6.4% 8.1% 7.0% 6.4% 10.6%
Appendix B
Concave time/cost function:
Mathematical derivation
166
Appendix B. Concave time/cost function: Mathematical derivation 167
If we suppose y(x)=activity cost and activity duration AD = x, we can reformulate the
convex cost function formula (see equation 12.1 on page 105):
y(x) = V C ∗ x′2
m
As the concave cost function is the negative of the convex function, x and y are substi-
tuded by each other, and a constant value a is added:
x = a ∗ V C ∗ y(x)2
m
⇔ y(x) = a ∗√x ∗mV C
To find the value of a, the activity cost at time instance t=20 (which equals m) should
be equal to m*VC=400, as this value is also reached by a linear and convex cost function
after 20 time units:
y(m) = a ∗√m2
V C= m ∗ V C
⇒ a =m ∗ V C√
m2
V C
⇔ a =V C√
1V C
y(x) is than equal to:
y(x) =V C√
1V C
∗√x ∗mV C
with
y(x)=activity cost at time instance x
x= activity duration AD
m= maximum activity duration = 20 time units
VC= variable cost factor=20 cost units/time unit
which is the same formula as defined in equation 12.2 on page 105.
Appendix C
Summary tables initial and new
Warburton model
C.1 Summary table: Parameters of the initial and New
Warburton model
C.2 Summary table: Warburton curves of the initial and
new Warburton model
with
PADr = Planned Activity Duration of a rejected/deviated activity
RADr = Real Activity Duration of a rejected/deviated activity
PACr = Planned Activity Cost of a rejected/deviated activity
AACr = Actual Activity Cost of a rejected/deviated activity
168
Appendix C. Summary tables initial and new Warburton model 169
Tab
leC
.1:
Su
mm
ary
Tab
le:
Para
met
ers,
Init
ial
an
dN
ewW
arb
urt
on
Mod
el
Para
mete
rsIN
ITIA
LW
arb
urt
on
Model
NE
WW
arb
urt
on
Model
Cate
gory
Sym
bol
Unit
Main
Infl
uence
on
Nam
eForm
ula
Dom
ain
Nam
eForm
ula
Dom
ain
Determinedbeforestartproject
NC
ost
Covera
ge
Tota
lN
=B
AC
[0,
+∞
]T
ota
lN
=B
AC
[0,+∞
]
unit
sW
arb
urt
on
am
ount
am
ount
curv
es
of
lab
or
of
lab
or
TT
ime
Posi
tion
peak
Tim
eof
tim
eunit
t]0
,+∞
]T
ime
of
tim
eunit
t]0
,+∞
]
unit
st
Warb
urt
on
the
Lab
or
whic
hm
inim
izes
the
Lab
or
whic
hm
inim
izes
curv
es
peak
∑ PD
t=
0(pv−pv(t
) w)2
peak
∑ PD
t=
0(pv−pv(t
) w)2
Determinedbasedon(early)projectdata
r%
Scop
eT
he
reje
ct
#finished
act.w
ith
RD
>P
D#
finished
activities
[0,
1]
The
devia
tion
#finished
act.w
ith
RD6=
PD
#finished
activities
[-1,+
1]
rate
of
rate
of
acti
vit
ies
acti
vit
ies
c%
Cost
The
cost
∑ ((A
ACr−
PA
Cr)/P
ACr)
#rejected
activities
[0,
1]
The
cost
∑ ((A
ACr−
PA
Cr)/P
ACr)
#deviated
activities
[-1,+
1]
overr
un
overr
un
or
onderr
un
τT
ime
Dura
tion
The
repair
∑ (R
AD
r−
PA
Dr)
#rejected
activities
[0,
+∞
]T
he
tim
e
∑ (R
AD
r−
PA
Dr)
#deviated
activities
[-∞
,+∞
]
unit
st
tim
eof
devia
tion
of
fracti
on
rfr
acti
on
r
Appendix C. Summary tables initial and new Warburton model 170
Tab
leC
.2:
Su
mm
ary
Tab
le:
Warb
urt
on
Cu
rves
,In
itia
lan
dN
ewW
arb
urt
on
Mod
el
NEW
Warb
urtonModel
INIT
IAL
Warb
urtonModel
AdditionalForm
ulasNEW
Model
Ifτ≥
0Ifτ<
0
pv(t) w
=Nt
T2exp( −
t2
2T
2
)pv(t) w
=Nt
T2exp( −
t2
2T
2
)
ev(t) w
=
(1−r)pv(t) w,
t≤τ
(1−r)pv(t) w
+r.pv(t−τ) w,
t>τ
ev(t) w
=
(1−r)pv(t) w
+∑ −τ
+1
i=1
r.pv(i) w,
t=
1
(1−r)pv(t) w
+r.pv(t−τ) w,
t>
1
ac(t)w
=
pv(t) w,
t≤τ
pv(t) w
+r.c.pv(t−τ) w,
t>τ
ac(t)w
=
pv(t) w
+∑ −τ
+1
i=1
r.c.pv(i) w,
t=
1
pv(t) w
+r.c.pv(t−τ) w,
t>
1
PV(t) w
=N[ 1−exp( −
t2
2T
2
)]PV(t) w
=N[ 1−exp( −
t2
2T
2
)]
EV(t) w
=
N(1−r)[ 1−exp( −
t2
2T
2
)] ,t≤τ
N−N(1−r)exp( −
t2
2T
2
) −r.N.exp( −(
t−τ)2
2T
2
) ,t>τ
EV(t) w
=
N(1−r)[ 1−exp( −
t2
2T
2
)] +∑ −τ
+1
i=1
r.N[ 1−exp( −
i2
2T
2
)] ,t=
1
N−N(1−r)exp( −
t2
2T
2
) −r.N.exp( −(
t−τ)2
2T
2
) ,t>τ
AC(t) w
=
N[ 1−exp( −
t2
2T
2
)] ,t≤τ
N[ 1−exp( −
t2
2T
2
)] −r.c.N[ 1−exp( −(
t−τ)2
2T
2
)] ,t>τ
AC(t) w
=
N[ 1−exp( −
t2
2T
2
)] +∑ −τ
+1
i=1
r.c.N[ 1−exp( −
i2
2T
2
)] ,t=
1
N[ 1−exp( −
t2
2T
2
)] −r.c.N.[ 1−exp( −(
t−τ)2
2T
2
)] ,t>τ