lib.ugent.be · abstract english in the world of project management, the earned value management...

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


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


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


11.6 Influence of the TF factor (x-axis) on the time forecast error (y-axis) of


12.1 Linear, convex and concave time/cost relationship. . . . . . . . . . . . . . 104


ods for final project cost, per completion stage (early, middle, late) under

the assumption of a linear time/cost relationship. . . . . . . . . . . . . . . 109


ods for final project cost, per 10 % of the work completed under the

assumption of a linear time/cost relationship. . . . . . . . . . . . . . . . . 110



the assumption of a convex time/cost relationship. . . . . . . . . . . . . . 110



assumption of a convex time/cost relationship. . . . . . . . . . . . . . . . 111



the assumption of a concave time/cost relationship. . . . . . . . . . . . . . 112



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


on the new Warburton model under the assumption of a convex time/cost

relationship. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141


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


assumption of a convex time/cost relationship for early projects (scenario

1 and 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144


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


tion of a convex time/cost relationship. . . . . . . . . . . . . . . . . . . . 107


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


of a convex time/cost relationship. . . . . . . . . . . . . . . . . . . . . . . 139


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.


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.


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


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.


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.


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.


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].


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


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.


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)


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)


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.


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]


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


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.


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.


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.


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.


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


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)


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.


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


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


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


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


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.


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


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.


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.


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.


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.


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


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


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.


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)


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.


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


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.


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


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


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


∑ni=project1

EV (RD)wN ∗ 100

EAC(t)w7 = t at which EV(t)w equals the value of (N-ratiow7)


∑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.


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


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.


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.


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].


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].



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].



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.


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


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-


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


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


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.


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).


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.


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.


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


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


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%


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-


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


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.


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.


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%



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.


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.


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


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.


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).


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.


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).


final project cost, per project completion stage (early, middle, late).


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.


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.


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.


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


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


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.


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.


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



Figure 11.6: Influence of the TF factor (x-axis) on the time forecast error (y-axis) of methods


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.


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


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)


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


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.


Convex time/cost relationship


convex time/cost relationship.

CONVEX Traditional EVM methods Warburton


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 %.


Concave time/cost relationship


concave time/cost relationship.

CONCAVE Traditional EVM methods Warburton


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 %.


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


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


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.


final project cost, per 10 % of the work completed under the assumption of a

linear time/cost relationship.

Convex time/cost relationship



of a convex time/cost relationship.


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.





Concave time/cost relationship



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%.





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.


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


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.


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


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


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


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)


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.


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


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


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


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 %


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.


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.


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.


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:


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)


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.


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)


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)


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%


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.


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%


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 %.



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


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 Traditional EVM methods Warburton


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%



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 %.


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.



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 Traditional EVM methods Warburton


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%



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.


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.


new Warburton model under the assumption of a concave time/cost relationship.


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.


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).


tion of a convex time/cost relationship for early projects (scenario 1 and 2).



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.


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.


Figure 14.9: Forecast error (MAPE) of the EAC methods under the assumption of a convex


Figure 14.10: Forecast error (MAPE) of the EAC methods under the assumption of a concave



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


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


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


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.


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


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.


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


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.


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.


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)


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)


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

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