Probability distribution fitting of schedule overruns in construction ...

Probability distribution fitting of scheduleoverruns in construction projectsPED Love

1�, C-P Sing1, X Wang

1, DJ Edwards

2and H Odeyinka

3

1Curtin University, Perth, Australia;

2Birmingham City University, Birmingham, UK; and

3University of Ulster, Belfast, UK

The probability of schedule overruns for construction and engineering projects can be ascertained usinga ‘best fit’ probability distribution from an empirical distribution. The statistical characteristics ofschedule overruns occurring in 276 Australian construction and engineering projects were analysed.Skewness and kurtosis values revealed that schedule overruns are non-Gaussian. Theoretical probabilitydistributions were then fitted to the schedule overrun data; including the Kolmogorov–Smirnov,Anderson–Darling and Chi-Squared non-parametric tests to determine the ‘Goodness of Fit’. A FourParameter Burr probability function best described the behaviour of schedule overruns, provided thebest overall distribution fit and was used to calculate the probability of a schedule overrun beingexperienced. The statistical characteristics of contract size and schedule overruns were also analysed, andthe Wakeby (oAU$1m and AU$11–50m), Three Parameter Log-logistic (AU$1–A$10m) and Beta(AU$51–A$100m and 4AU$101m) models provided the best distribution fits and were used tocalculate schedule overrun probabilities by contract size.

Journal of the Operational Research Society (2013) 64, 1231–1247. doi:10.1057/jors.2013.29

Published online 13 March 2013

Keywords: Australia; schedule overrun; distribution fitting; probability; probability distribution

Introduction

Clients’ demands for early completion to minimize finance

costs and increase return on investment to satisfy investors

and stakeholders can lead to over-optimistic schedules

being produced (Mansfield et al, 1994; Kog et al, 1999; Luu

et al, 2009). As a consequence, the likelihood of schedule

overruns being experienced increases. According to the

Building Cost Information Service of the Royal Institute of

Chartered Surveyors, 48% of projects experience schedule

overruns (Kennett, 2009). Schedule overruns can adversely

influence the organizational performance and profitability

of clients, contractors and key stakeholders. Well-known

Australian projects that have attracted the attention of the

popular press for experiencing schedule overruns include

Perth Arena, Victorian Desalination Plant, Southern Cross

Railway Station, Sydney Cross City Tunnel, RiverCity

Motorway and the M7 Clem Jones Tunnel (CLEM7). Such

projects have also incurred significant cost overruns.

Schedule and cost overruns often arise simultaneously,

although projects can experience a cost overrun and be

delivered ahead of schedule or vice versa (Love, 2002;

Ashan and Gunawan, 2010).

To determine the duration of construction projects,

several models that extend the seminal works of Bromilow

(1974) have been developed (eg, Ireland, 1983; Kaka and

Price, 1991; Chan, 1999; Skitmore and Ng, 2003; Love et

al, 2005; Ogunsemi and Jagboro, 2006). Fundamentally,

the developed models aim to predict a project’s duration

based on its cost and/or size (ie, number of floors, gross

floor area). However, while project duration can be predic-

ted with a high degree of accuracy using the developed

models (Love et al, 2005; Ogunsemi and Jagboro, 2006),

they are unable to accommodate the potential for a sche-

dule overrun. A plethora of reasons are proffered within

the extant literature to justify the occurrence of schedule

overruns including change orders, rework, unforeseen

ground conditions and delays in design information

(eg, Chan and Kumaraswamy, 1997; Kaming et al, 1997;

Kog et al, 1999; Aibinu and Odeyinka, 2006; Ashan and

Gunawan, 2010; Jergeas and Ruwanpura, 2010; Love et al,

2011). Yet despite the accumulation of knowledge pertain-

ing to the causal nature of schedule overruns, there has

been limited research that quantifies their empirical and

statistical distributions (eg, Anastasopoulos et al, 2009;

Anastasopoulos et al, 2010; Anastasopoulos et al, 2012).

Planning tools such as the critical path method (CPM)/

programme evaluation review and technique (PERT)

and Work Breakdown Structure estimates, which take a

Journal of the Operational Research Society (2013) 64, 1231–1247 © 2013 Operational Research Society Ltd. All rights reserved. 0160-5682/13

www.palgrave-journals.com/jors/

�Correspondence: PED Love, School of Built Environment, Curtin

University, GPO Box U1987, Perth, Western Australia 6845, Australia.

bottom-up approach, are often used to determine schedule

risks (Mulholland and Christian, 1999; Galway, 2004).

According to Galway (2004), an alternative strategy is to

take a retrospective, top-down approach by reviewing past

projects in order to determine how long they took to

complete. The empirical relationship between the estimated

and actual time taken to complete a project can be then

used to determine schedule risk.

A schedule overrun, which is akin in nature to a cost

overrun, can be classified as a ‘random continuous vari-

able’, as it can take an infinite range of values (Jahren and

Ashe, 1990; Anastasopoulos et al, 2009; Love et al, 2013).

Typically, the probability density function (PDF) of the

Normal distribution (otherwise known as Gaussian) is used

in risk analysis to determine schedule overruns. A Normal

distribution is symmetric about its mean value, and there-

fore cannot be used to accurately model left or right

skewed data. Even if schedule overrun data is symmetric by

nature, it is possible that it is best described using heavy

tailed distribution models such as a Cauchy. Fitting an

empirical distribution to data can be a difficult task consi-

dering the array of statistical distribution choices that are

available. The selection of an inappropriate statistical distri-

bution can produce incorrect probabilities, which can

adversely affect decision making and therefore lead to

negative outcomes (Bedford and Cooke, 2001; Anastaso-

poulos et al, 2009). Against this contextual backdrop,

this paper uses data from 276 Australian construction

and engineering projects to determine the ‘best fit’ pro-

bability distribution and enable a realistic estimate of

possible schedule overrun probabilities to be determined.

Such probability distributions can be used to provide an

estimation of the uncertainty of the risk and its possible

impact of a construction project’s schedule. Moreover, the

developed probability distributions can be used to devise

suitable risk management strategies to reduce and amelio-

rate risk.

Quantification of schedule overruns

Schedule overruns are often referred to as ‘delays’, ‘sche-

dule growth’ or ‘time overruns’. A schedule overrun occurs

when the original contract period specified at contract

award is extended beyond what was agreed before the

commencement of works on-site (Mansfield et al, 1994;

Kog et al, 1999; Assaf and Al-Hejji, 2006; Luu et al, 2009).

Likewise, Anastasopoulos et al (2009) use the term ‘time

delay’ and define it as the difference between a project’s

planned and its actual duration. Jahren and Ashe (1990)

revealed that larger projects are prone to schedule over-

runs due to the correlation between size and complexity.

Bhargava et al (2010) also found projects with longer

planned durations experience higher cost overruns. The

converse is also true, as Odeck’s (2004) research, for

example, demonstrated that larger cost overruns were

experienced in smaller projects. Thus, in this instance,

Odeck (2004) suggests that larger projects are better

managed and that longer completion times provide an

opportunity to make adjustments to facilitate better plann-

ing and scheduling to minimize delays that may be caused

by factors such as inclement weather and change orders.

A study by Bromilow (1969) of 309 Australian con-

struction projects undertaken revealed that 88% experi-

enced a schedule overrun. Over 40 years later, Ashan and

Gunawan (2010) revealed almost identical findings and

observed that 86% of development projects conducted

experienced a schedule overrun. The amount by which

a project experienced a schedule overrun was found to vary

significantly. Assaf and Al-Hejji (2006) found that most

contractors and consultants indicated that average schedule

overruns range from 10 to 30% of original contract period.

Ashan and Gunawan (2010) reported that the average

schedule overrun was found to be 33.37%, although this

varied between different countries. For example, the mean

schedule overrun in Bangladesh was 34.41%, China

13.63%, India 55.69% and Thailand 32.71%. Projects

experiencing a schedule underrun on average experienced a

US$79 million cost underrun (19% of planned cost). Of the

86% of projects completed within budget or under budget,

29% experienced an average schedule overrun of 16

months (Ashan and Gunawan 2010). Anastasopoulos

et al (2009) examined time delays in 1722 highway projects

using random parameter statistical models to examine the

factors that contribute to the likelihood of encountering an

increase in a project’s planned duration. Anastasopoulos

et al (2009) found that 87% of projects sampled experi-

enced time delays with a mean delay of 97 days and a

standard deviation of 115 days.

Research approach

The data sets presented in Love et al (2009a) for Australian

construction and engineering projects are used to develop

‘best fit’ statistical distributions in order to determine

probabilities for schedule overruns at contract award. For

the purpose of clarity, the method of data collection and

data set characteristics will be presented again. The resea-

rch primarily aims to determine the ‘best fit’ probability

distribution so as to assess at contract award the likelihood

that a construction project will be completed on time.

Questionnaire survey

The developed questionnaire survey was used to extract

schedule overrun information as well as cost overrun and

rework costs and causes. A stratified random sampling

procedure was used to select the study sample. The process

of sampling involves any procedure that uses a small

1232 Journal of the Operational Research Society Vol. 64, No. 8

number of items or parts of the whole population to make

conclusions regarding the entire population. The sample

needs to be representative of the population to produce a

result of theoretical and practical value, that is, the results

obtained from the sample must approximate to those that

would be obtained if it was possible to survey the entire

population. Thus, in this research, a stratified random

sampling technique was used to select the study sample due

to the homogenous and fragmented nature of the popula-

tion; an innate feature of the construction and engineering

industry. In addition to increasing the representativeness of

samples, stratified random sampling is a useful technique

for making general statements about the portions of the

population. In this research, it was necessary to establish

whether there were significant differences between respon-

dent groups estimated schedule overruns for the projects

that they had selected.

The stratified random sampling procedure was used

to divide the population into sub-populations called strata

(singular stratum); random samples were drawn from

each of the strata identified. In this case, strata were

architects, contractors, mechanical and electrical engineers,

project managers, quantity surveyors and structural engi-

neers. There are two general approaches that are used to

determine how many elements should be drawn from each

stratum (Leedy and Omrod, 2010). One approach is to

draw equal-sized samples from each stratum. The second

approach is to draw elements for the sample on a

proportional basis. Leedy and Omrod (2010) suggests that

the sample size is largely dependent on the degree to which

the sample population approximates the qualities and

characteristics of the general population. However, it is

difficult, and perhaps impossible, to determine the number

of organizations involved with the procurement of con-

struction projects in Australia. Not all firms are registered

with professional bodies, such as ‘The Royal Australian

Institute of Architects’, ‘The Institute of Engineers,

Australia’, ‘Australian Institute of Building’ or ‘Australian

Institute of Project Managers’. In fact, when approached

for addresses for the survey, these professional bodies were

reluctant to provide details of the organizations that were

registered with them. Consequently, the Yellow Pages were

used to select firms for the research.

Two main benefits were derived from using a stratified

sample:

(1) It ensured the acquisition of adequate and representa-

tive respondents within each subgroup under study.

(2) It also ensured that respondents within the same group

were homogeneous.

Before the sample size for the main study could

be determined, a pilot survey was completed with 30

building contractors and 20 civil engineering con-

tractors. As the survey of building contractors was

undertaken first, it was considered to be reliable, and

then used to pilot the civil engineering sample. The firms

sampled comprised design and engineering consultants,

project managers and contractors. The rationale was to

test the questionnaire’s suitability, clarity and compre-

hensibility, as well as measure the response rate. Partic-

ipating firms were contacted by telephone to inform

them of the research aims and objectives and to assure

them that all responses would remain strictly confiden-

tial; although generalizations of the findings would be

made available to all participants.

Upon participant consent, questionnaires were mailed

to the sample, with a stamped addressed return envelope

enclosed. Participants were invited to critically review the

questionnaire’s design and structure by annotating com-

ments onto the document itself in order to provide

constructive feedback. Comments received were positive

and therefore the questionnaire remained largely unaltered

for the main surveys; although a few minor layout changes

were made to increase clarity. A total of 25 responses were

received in the building project pilot survey, giving an 83%

response rate. For the civil engineering project survey, a

total of 17 responses were received, giving an 85% response

rate. These high response rates were attributed to the fact

that prior consent to support the work was obtained from

all survey participants.

In the case of building projects, 60 questionnaires (ie,

420 in total) were each mailed to architects, quantity

surveyors, structural, mechanical, and electrical engineers,

contractors and consultant project managers. For the civil

engineering survey, 100 questionnaires (ie, 300 in total)

were mailed to each of the following engineering con-

sultants, consultant project managers and contractors. As

there were no fundamental changes required to either of

the pilot questionnaire surveys, they were added to the

main survey samples. In all, 161 and 115 responses were

received, respectively, for the building and civil engineering

projects, representing a total consolidated response rate of

38%, which is within an acceptable range for a survey with

industry practitioners (Alreck and Settle, 1985).

Data reliability

Data reliability relates to data source and the identification

of the position held by the respondent completing the

questionnaire (Oppenheim, 1992). It was critically impor-

tant that only selected senior personnel who had sufficient

knowledge and experience about the procurement pro-

cesses associated with a project answered the questionnaire.

From the total responses gathered, 133 respondents pro-

vided information relating to their individual job position

and title, and subsequent analysis revealed that most

respondents held senior positions within their organi-

zations. On the basis of this finding, the direct mailing to

individuals in organizations seemed to have achieved its

PED Love et al—Probability distribution fitting of schedule overruns 1233

objective of reaching senior staff with knowledge and

experience of the procurement processes. Furthermore,

because the research design mailed questionnaires to

organizations in different States in Australia, the risk of

duplicating projects was minimized.

Procedure

Descriptive statistics, such as the mean (M), standard

deviation (SD) and inter-quartile, were calculated to obtain

measures of central tendency and distribution. A one-way

Analysis of Variance (ANOVA) was used to determine

whether cost overruns significantly varied between con-

struction and engineering projects and original contract

value at a 0.05 significance level. PDFs were developed

using the software EastFit 5. A PDF for a continuous

distribution can be expressed in terms of an integral

between two points:

Zb

a

f ðxÞdx ¼ PðapXpbÞ ð1Þ

A cumulative distribution functions (CDF) was also pro-

duced. For theoretical continuous distributions, the CDF

is expressed as a curve and denoted by:

FðxÞ ¼Zx

�1

f ðtÞdt ð2Þ

The empirical CDF, which is displayed as a stepped

discontinuous line and dependent on the number of bins, is

represented by:

FnðxÞ ¼1

n� ½Number of observationspx� ð3Þ

The PDF, CDF and distribution parameters (a, b, g, m,k, m, s, x) for continuous distributions such as Beta, Burr,

Cauchy, Error, GumbelMax/Min, Johnson SB, Normal and

Wakeby were examined using their respective estimation

methods of Maximum Likelihood Estimates. Using the

software StatAssist 5.5, the ‘best fit’ distribution was then

determined using the following ‘Goodness of Fit’ tests,

which measure the compatibility of a random sample with

a theoretical probability distribution:

K Kolmogorov–Smirnov statistic (D): On the basis of the

largest vertical difference between the theoretical and

empirical CDF:

D ¼ max1pipn

FðxiÞ �i � 1

n;i

n� FðxiÞ

� �ð4Þ

K Anderson–Darling statistic (A2): A general test to

compare the fit of an observed CDF with an expected

CDF. The test provides more weight to a distribution’s

tails than the Kolmogorov–Smirnov test. The Anderson–

Darling statistic is defined as:

A2 ¼ �n� 1

n

Xni¼1ð2i � 1Þ � ½InFðxiÞ þ Inð1� Fðxn�iþ1ÞÞ� ð5Þ

K Chi-squared statistic (w2): Determines whether a sample

derives from a population with a specific distribution.

The Chi-squared statistic is defined as:

w2 ¼Xki¼1

ðOi � EiÞ2

Ei; ð6Þ

whereOi is the observed frequency for bin i and Ei is the

expected frequency bin i calculated by:

Ei ¼ Fðx2Þ � Fðx1Þ ð7Þ

Here F is the CDF of the probability distribution being

tested, and x1, x2 the limits for the bin i.

The above ‘Goodness of Fit’ tests were used to test the

null (H0) and alternative hypotheses (H1) that the data sets:

H0—follow the specified distribution; and H1—do not

follow the specified distribution. The hypothesis regarding

the distributional form is rejected at the chosen significance

level (a), if the statistic D, A2, w2 is greater than the critical

value. For this research, a 0.05 significance level was used

to evaluate the null hypothesis. The p-value, in contrast

to fixed a values, is calculated based on the test statistic

and denotes the threshold value of significance level in the

sense that H0 will be accepted for all values of a less than

the p-value. Once the ‘best fit’ distribution was identi-

fied, the probabilities for cost overruns were calculated

using the CDF. Then, to simulate the sample’s randomness

and derive cost overrun probabilities, a Mersenne Twister,

which is a pseudorandom number generating algorithm,

was used to generate a sequence of numbers that approxi-

mated the sample to 1000 (Matsumoto and Nishimura,

1998).

Results

Data from a total of 276 construction (n¼ 161) and

civil engineering (n¼ 115) projects were obtained. The

construction projects ranged from banks, to hospitals and

hotels, while the civil engineering projects ranged from

tunnelling, to road construction and sewer treatment

plants. The summary statistics revealed that the mean

original contract value was AU$23142 486 (SD¼A$41171

772; minimum¼AU$132 347; maximum¼A$390 million)

and the mean actual contract value was AU$25455 372

(SD¼AU$45090 928; minimum¼AU$136671; maximum

¼AU$420 million).


To better understand the sample composition, an

examination was completed for both the construction

and civil engineering samples, of respondents’ strati-

fication, geographical dispersion and company turnover.

In terms of respondent stratification, 45% were design

consultants (architects and quantity surveyors, as well

as structural, mechanical and electrical engineers), 31%

were contractors and 24% comprised project managers.

With regards to geographical dispersion, organiza-

tions were situated across states: Victoria (45%), New

South Wales (17%), Queensland (27%), South Australia

(9%) and Western Australia (2%). The analysis revea-

led that the average annual turnover of the organiza-

tions sampled varied: o AU$1m¼ 14%; AU$1–AU$10m

¼ 37%; AU$11–AU$50m¼ 20%; AU$51–AU$250m¼13%; and 4 AU$250m¼ 16%.

Table 1 provides the descriptive and percentile statistics

for the schedule overruns that were incurred in the sampled

projects. The mean overall project schedule overrun for the

sample was 11.42% (SD¼ 14.24). Mean schedule overruns

were also determined for the two project type groupings:

K Construction 13.07% (SD¼ 16.89%) or 8.88 weeks

(SD¼ 13.51 weeks)

K Civil engineering 11.13% (SD¼ 11.96%) or 4.27weeks

(SD¼ 5.50 weeks)

The spread of the schedule overruns incurred in each

project was plotted against the original contract value

(Figure 1).

An ANOVA test was used to determine whether there

were significant differences between the schedule overruns

experienced in the construction and engineering projects

(p¼ 0.05). The analysis revealed that there were no signifi-

cant differences between the schedule overruns experienced

between construction and civil engineering projects

[F (1, 274)¼ 0.413, p¼ 0.21]. These reported findings are

contrary to the held belief that schedule overruns vary with

project size, type and procurement method. According to

Shenhar et al (2001), risk impacts and probabilities vary

with the nature of the project, for example, its complexity,

size and reliance on new technology. Naturally, with new

emerging technologies such as Building Information Model-

ling (BIM), schedule risk will be reduced and therefore

influence risk profiles of projects. However, the introduction

of BIM will require a plethora of additional risks to be

considered. Noteworthy, BIM was not implemented in any

of the 256 construction and engineering projects sampled.

Distribution fitting: probability of schedule overruns

The construction and engineering data sets were combined

and the ‘best fit’ probability distribution was examined

using the ‘Goodness of Fit’ tests: Kolmogorov–Smirnov and

Anderson–Darling. The results of the ‘Goodness of Fit’

tests revealed that Burr Four Parameter (4P) distribution

provided the ‘best fit’ for the data set (Table 2). The

Kolmogorov–Smirnov test revealed a D statistic of 0.08758

with a p-value of 0.2725 and the H0 was accepted at

a¼ 0.02 and a¼ 0.01 levels. The Anderson–Darling

statistic A2 was revealed to be 3.0801 and the H0 was

accepted at a¼ 0.02 and a¼ 0.01 levels. The Chi-squared

(w2) statistic was found to be 1.8008 with a p-value of

0.0.87596 and the H1 was accepted.

A Burr distribution is a continuous probability distri-

bution for a non-negative random variable (Burr, 1942).

It has a flexible shape, and controllable scale and location,

and is sometimes considered as an alternative to a Normal

distribution when data demonstrate positive skewness.

Table 1 Descriptive statistics for schedule overruns

Statistic Value (%)

Range 99Mean 11.42Variance 203Standard deviation 14.24Coefficient of variation 1.247Standard error 0.857Skewness 0.389Excess kurtosis 1.325Min �40.545% �1010% 025% (Quartile 1) 2.35750% (Median) 10.0875% (Quartile 3) 19.38790% 30.20795% 38.772Max 58.46

Figure 1 Schedule overrun in relation to original contractvalue.


The parameters of the Burr (4P) are all continuous: k is a

shape parameter (k40), a is a shape parameter (a40), b is

a scale parameter and g is a location parameter (g�0 yields

the 4P Burr distribution). The PDF is expressed as:

FðxÞ ¼ak x�y

b

� �a�1

b 1þ x�yb

� �a� �kþ1 ð8Þ

The CDF is expressed as:

FðxÞ ¼ 1� 1þ x� y

b

� �a� ��kð9Þ

The distribution parameters for the sampled con-

struction and engineering projects were found to be:

k¼ 0.19541, a¼ 2.1247Eþ 8, b¼ 5.6456Eþ 8, g¼�5.6456Eþ 8 (Figure 2). Figures 2 and 3 present the PDF and

CDF based on the calculated distribution parameters. The

calculated probabilities of a schedule overrun being expe-

rienced are presented in Tables 3 and 4. The probabi-

lity of experiencing a schedule overrun of 410% is 45%

(P(X4X1), 0.45). Delimiters were also used to provide

probabilities of schedule overruns within ranges. The

probability of a project experiencing between 1 and 10%

schedule overrun, for example, is 36% (Figure 3). For

a mean schedule overrun of 15.91%, the likelihood that a

project’s duration is extended by this amount is approxi-

mately 60% (P(XoX1)¼ 0.60).

Project size

An ANOVA sought to determine whether schedule over-

runs varied with different contract values (p¼ 0.05). The

homogeneity of variance assumptions was violated for

schedule overrun comparisons, and hence the results of

a t test with equal variances not assumed are reported

(p¼ 0.05). There were no significant differences for con-

tract value and project schedule overruns t (272)¼�1.11,p¼ 0.121. Figure 1 reveals that as contract value increased,

the schedule overrun tended to decrease. Table 5 indi-

cates that construction projects ranging from AU$51 to

AU$100m experienced a mean schedule overrun of 18.25

weeks. Likewise for civil engineering projects, a mean of

11.57 weeks was experienced.

Table 2 Goodness of Fit tests for construction and engineering projects

Distribution type andcontract range(AU$ million)

N Significant alevel

Kolmogorov–Smirnov (D)critical value

Anderson–Darling (A2)critical value

Chi-squared(w2) critical

value

Burr (4P)(Construction and engineering projects)

276 0.2 0.0645 1.3749 11.030.1 0.0736 1.9286 13.3620.05 0.0817 2.5018 15.5070.02 0.0913 3.2892 18.1680.01 0.0980 3.9074 20.09

Wakebyo1

14 0.2 2.7481 1.3749 1.64240.1 0.3141 1.9286 2.70550.05 0.3489 2.5018 3.84110.02 0.3497 3.2892 5.41190.01 0.4172 3.9074 6.6349

Log-logistic (3P)1–10

131 0.2 0.0937 1.3749 9.80320.1 0.1068 1.9286 12.0170.05 0.1186 2.5018 14.0670.02 0.1326 3.2892 16.6220.01 0.1423 3.9074 18.475

Wakeby11–50

102 0.2 0.1062 1.3749 8.55810.1 0.1211 1.9286 10.6450.05 0.1344 2.5018 12.5920.02 0.1503 3.2892 15.0330.01 0.1613 3.9074 16.812

Beta51–100

17 0.2 0.250 1.374 3.2180.1 0.286 1.928 4.6050.05 0.317 2.501 5.9910.02 0.355 3.289 7.8240.01 0.380 3.907 9.210

Beta4100

12 0.2 0.295 1.374 1.6420.1 0.338 1.928 2.7050.05 0.375 2.501 3.8410.02 0.419 3.289 5.4110.01 0.449 3.907 6.634


The construction and engineering projects were com-

bined to create a large data set in order to determine the

‘best fit’ distribution for each range of schedule overrun

experienced within a range of contract value classifications.

Only construction projects that had contract values

o AU$1m were not combined, as no engineering project

Probability Density Function

% Schedule Overrun160140120100806040200–20

Pro

babi

lity

of S

ched

ule

Ove

rrun

0.044

0.04

0.036

0.032

0.028

0.024

0.02

0.016

0.012

0.008

0.004

0

Delimiters

X1=1% X2=10%

Burr (0.19541; 2.1247E+8; 5.6456E+8; –5.6456E+8)

Figure 2 Burr (4P): PDF for schedule overruns.Parameters: k¼ 0.19541, a¼ 2.1247Eþ 8, b¼ 5.6456Eþ 8, g¼�5.6456Eþ 8.

Cumulative Distribution Function

% Schedule Overrun

50403020100–10–20–30–40

Prob

abili

ty o

f Sc

hedu

le O

verr

un

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0% Schedule Overrun Burr (4P)

Figure 3 Burr (4P): CDF for schedule overruns.

Table 3 Generic discrete probabilities for schedule overruns

Probability scheduleoverrun (%)

P(X o X1) P(X 4 X1)

5 0.36 0.6410 0.55 0.4515 0.60 0.4020 0.78 0.2225 0.85 0.1530 0.9 0.10

Table 4 Generic range of probabilities for schedule overruns

Probability of a scheduleoverrun between (%)

P(X1 o X o X2)

1 and 5 0.176 and 10 0.1911 and 15 0.1416 and 20 0.0921 and 25 0.0726 and 30 0.05


had a contract value less than this figure. There were only

two construction projects that had a value 4AU$200m.

As a result, these two projects were combined into a

classification of 4AU$101m. The ‘best fit’ probability

distribution for each range of classification was examined

using the ‘Goodness of Fit’ tests: Kolmogorov–Smirnov,

Anderson–Darling and Chi-Squared (Table 2).

For the contract rangesoAU$1m and AU$11–A$50m,

the best fitting distribution was a Wakeby (Table 2).

For projects oAU$1m, the Kolmogorov–Smirnov test

revealed a D statistic of 0.1535 with a p-value of 0.84831

for the sample of 14 projects. The Anderson–Darling

statistic A2 was revealed to be 0.49548. The Chi-squared

(w2) statistic was found to be 0.01745 with a p-value of

0.8949. For AU$11–A$50m projects, the Kolmogorov–

Smirnov test revealed a D statistic of 0.0402 with a

p-value of 0.96379 for the sample of 102 projects.

The Anderson–Darling statistic A2 was revealed to be

0.16489. The Chi-squared (w2) statistic was found to

be 1.027 with a p-value of 0.98457. The ‘Goodness of Fit’

tests all accepted the H0 for both samples distribution

‘best fit’.

The Wakeby is a form of Generalized Extreme Value

distribution. The parameters of a Wakeby, a, b, g, d, x, areall continuous. The domain for this distribution is expre-

ssed as xpx, if dX and g40, xpxpa/b�g/d if do0 or

g¼ 0. The distribution parameters for the rangeoAU$1m

were a¼ 33.005, b¼ 0.29475, g¼ 0, d¼ 0, x¼�5.3672, and

Table 5 Project size and average schedule overrun

Project type Project sizeAU$ million

N Mean originalcontract

value AU$

Standarddeviation

contract value

Meanschedule overrun

(weeks)

Standard deviationschedule overrun

(weeks)

Construction o1 14 513546 A$344668 5.61 6.191–10 70 3 874368 2 830 183 5.00 10.4211–50 60 21 860966 10 099 633 8.43 14.6851–100 12 66 883333 15 062 888 18.25 22.92101–200 6 162 666666 22 339 800 9.50 21.094200 2 376 000000 19 798 989 2.00 1.41Sub-total 161 25 521927 51 957 899 8.88 13.51

Civil engineering 1–10 61 8 136604 3 464 888 2.5 3.1811–50 43 24 906930 10 487 326 5.33 6.1551–100 7 77 285714 12 499 523 11.57 8.204101 4 165 250000 25 051 613 7.75 8.57Sub-total 115 22 092965 33 897 118 4.27 5.50Total 276 23 142486 41 171 772 6.96 11.11

Percentage of Schedule Overrun48444036322824201612840

Prob

abili

ty o

f Sc

hedu

le O

verr

un

0.3

0.28

0.26

0.24

0.22

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

% Schedule Overrun Wakeby

Figure 4 Wakeby: PDF for projects with contract values o AU$1m.Parameters: a¼ 33.005, b¼ 0.29475, g¼ 0, d¼ 0, x¼�5.3672.


AU$11–AU$50m were a¼ 242.27, b¼ 9.6408, g¼ 21.159,

d¼�0.29463, x¼�33.129. The probability distribu-

tions for each of the contract ranges are presented in

Figures 4–7.

Using the Wakeby PDF function, the probabilities for

the schedule overruns within the ranges oAU$1m and

AU$11–A$50m are presented in Table 6. The Wakeby

distribution is defined by the quantile function (inverse

CDF):

xðFÞ ¼ xþ ab

1� ð1� FÞb� �

� gd

1� ð1� FÞ�d� �

ð10Þ

A Log-logistic Three Parameter (3P) was found to be the

‘best-fit’ distribution for projects with contract values rang-

ing from AU$1 to AU$10 million as identified (Table 2).

The Kolmogorov–Smirnov test revealed a D statistic

of 0.03981 with a p-value of 0.98036 for the sample of

131 projects. The Anderson–Darling statistic A2 was

revealed to be 0.37216. The Chi-squared (w2) statistic

was found to be 1.5292 with a p-value of 0.98128. The

‘Goodness of Fit’ tests all accepted the H0 for the sample

distribution’s ‘best fit’.

A Log-logistic 3P is a continuous probability distri-

bution for a non-negative random variable and is often


Percentage of Schedule Overrun

48444036322824201612840

Prob

abili

ty o

f Sc

hedu

le O

verr

un

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0% of Schedule Overrun Wakeby

Figure 5 Wakeby: CDF for projects with contract values o AU$1m.


484032241680–8–16–24

Prob

abili

ty o

f Sc

hedu

le O

verr

un

0.28

0.26

0.24

0.22

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

% Schedule Overrun Wakeby

Figure 6 Wakeby: PDF for projects with contract values AU$11–AU$50m.Parameters: a¼ 242.27, b¼ 9.6408, g¼ 21.159, d¼�0.29463, x¼�33.129.


used in survival analysis as a parametric model (Bennett,

1983; Ashker and Mahdi, 2006). Within the specified

contract ranges, projects with values of between AU$1 and

AU$5 million were found to have greater schedule

overruns than those between AU$5 and AU$10 million.

The aforementioned projects reside within the heavy tail of


Percentage of Schedule Overrun484032241680–8–16–24

Prob

abili

ty o

f Sc

hedu

le O

verr

un

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 % Schedule Overrun Wakeby

Figure 7 Wakeby: CDF for projects with contract values AU$11–AU$50m.

Table 6 Discrete probabilities for project schedule overruns by contract value

Contract value(AU$ million)

Probability ofschedule overrun (%)

P(X o X1) P(X 4 X1) P(X1o X o X2) P(X o X2) P(X 4X2)

Wakebyo1

1 and 5 0.18 0.82 0.1 0.28 0.726 and 10 0.31 0.69 0.11 0.39 0.61

11 and 15 0.41 0.59 0.09 0.49 0.5116 and 20 0.51 0.49 0.08 0.58 0.4221 and 25 0.59 0.41 0.07 0.65 0.3526 and 30 0.67 0.33 0.06 0.73 0.27

Log-logistic (3P)1–10

1 and 5 0.32 0.68 0.19 0.38 0.626 and 10 0.47 0.52 0.24 0.63 0.37

11 and 15 0.63 0.37 0.16 0.79 0.2116 and 20 0.76 0.24 0.09 0.88 0.1221 and 25 0.86 0.14 0.05 0.94 0.0626 and 30 0.92 0.08 0.03 0.97 0.03

Wakeby11–50

1 and 5 0.37 0.63 0.12 0.36 0.646 and 10 0.52 0.47 0.18 0.61 0.39

11 and 15 0.64 0.36 0.17 0.76 0.2416 and 20 0.74 0.26 0.12 0.85 0.1521 and 25 0.82 0.18 0.07 0.89 0.1126 and 30 0.88 0.12 0.04 0.93 0.07

Beta51–100

1 and 5 0.31 0.69 0.06 0.38 0.686 and 10 0.40 0.60 0.07 0.47 0.63

11 and 15 0.49 0.51 0.08 0.57 0.4316 and 20 0.59 0.41 0.09 0.69 0.3121 and 25 0.71 0.29 0.11 0.83 0.1726 and 30 0.86 0.14 — — —

Beta4101

1 and 5 0.30 0.70 0.03 0.33 0.676 and 10 0.34 0.66 0.03 0.37 0.63

11 and 15 0.38 0.62 0.03 0.42 0.5816 and 20 0.42 0.58 0.03 0.46 0.5421 and 25 0.47 0.53 0.03 0.51 0.4926 and 30 0.52 0.48 0.04 0.56 0.44


the distribution. The parameters of Log-logistic (3P) a, b, gare all continuous. a is the shape parameter (a40), b is a

scale parameter (b40) and g is a location parameter (g�0yields the two parameter Log-logistic distribution). The

domain for this distribution is expressed gpxoþN. The

PDF for a Log-logistic (3P) is defined as:

f ðxÞ ¼ ab

x� gb

� �a�11þ x� g

b

� �a� ��2ð11Þ


FðxÞ ¼ 1þ bx� g

� �a� ��1ð12Þ

The PDF and CDF presented in Figures 8 and 9 has

distribution parameters for the contract value range of

A$1–AU$10 million of a¼ 116.32, b¼ 903.14 and g¼�896.46. Using the Log-logistic PDF, the probabilities for


32241680–8–16–24–32

Prob

abili

ty o

f Sc

hedu

le O

verr

un

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

% Schedule Overrun Log-Logistic (3P)

Figure 8 Log-logistic (3P): PDF for projects with contract values AU$1–AU$10m.Parameters: a¼ 116.32, b¼ 903.14, g¼�896.46.


Percentage of Schedule Overrun32241680–8–16–24–32

Prob

abili

ty o

f Sc

hedu

le O

verr

un

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0Log-Logistic (3P)% Schedule Overrun

Figure 9 Log-logistic (3P): CDF for projects with contract values AU$1–AU$10m.


schedule overruns for projects ranging from AU$1–AU$10

million are calculated and presented in Table 6.

A Beta was found to be the ‘best-fit’ distribution for

projects with contract values ranging from AU$51 to

AU$100m and 4AU$101m as identified in Table 2. For

projects ranging from AU$51 to AU$100m, the Kolmo-

gorov–Smirnov test revealed a D statistic of 0.11676 with a

p-value of 0.95353 for the sample of 17 projects. The

Anderson–Darling statistic A2 was revealed to be 1.441.

The Chi-squared (w2) statistic was found to be 1.0577

with a p-value of 0.58927. For projects 4AU$101m, the

Kolmogorov–Smirnov test revealed a D statistic of 0.11502

with a p-value of 0.9917 for the sample of 12 projects. The

Anderson–Darling statistic A2 was revealed to be 1.729.

The Chi-squared (w2) statistic was found to be 0.3384 with

a p-value of 0.56096. The ‘Goodness of Fit’ tests all

accepted the H0 for both samples distribution ‘best fit’.

The domain of a Beta distribution can be viewed

as a probability and can be used to describe the distri-

bution of an unknown probability. It can be used to model

events that are constrained within an interval and maxi-

mum value. Thus, the Beta distribution is often used in

PERT and CPM to describe the time to complete a task

(Grubbs, 1962; Williams, 1992; Keefer and Verdini, 1993).

The Beta distribution is defined by the following para-

meters, which are all continuous: a1, a2 and a, b. The shape

parameters are a1 (a140) and a2 (a240), with a, b the

boundary parameters (aob). The domain for this distri-

bution is expressed as apxpb. The PDF for a Beta

distribution is defined as:

f ðxÞ ¼ 1

Bða1; a2Þðx� aÞa1�1ðb� xÞa2�1

ðb� aÞa1þa2�1ð13Þ


FðxÞ ¼ IZða1; a2Þ ð14Þ

where

z � x� a

b� að15Þ

B is the Beta Function and IZ is the Regularized

Incomplete Beta Function.

The PDF and CDF presented in Figures 10–13 are used

to determine the distribution parameters for the contract

value ranges from AU$51 to AU$100m and4AU$100m,

which are a1¼ 1.0483, a2¼ 0.69063, a¼�20.251, b¼ 28.42

and a1¼ 0.53671, a2¼ 0.42258, a¼�16.25, b¼ 39.84,

respectively.

Using the Beta PDF, the probabilities for schedule over-

runs for projects ranging from AU$51 to AU$100 million

and 4 AU$100m are calculated and presented in Table 6.

Managing schedule risk

Determining the likelihood of a project to experience a

schedule overrun from contract award is a risk that clients

and construction organizations generally fail to consider.

Instead, emphasis has generally focused on the potential

for cost increases by adding a contingency into the contract

value. However, there is a proclivity for schedule contin-

gency to be contained within a project’s construction

programme as ‘float’ or ‘slack’. This is the amount of time

that a task in a project network can be delayed without

causing a delay to subsequent tasks (free float) and/or its

completion date (total float).

% Schedule Overrun

2824201612840–4–8–12–16

Prob

abili

ty o

f Sc

hedu

le O

verr

un

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

% Schedule Overrun Beta

Figure 10 Beta: PDF for projects with contract values AU$51–AU$100m.Parameters: a1¼ 1.0483, a2¼ 0.69063, a¼�20.251, b¼ 28.42.


Tasks are often delayed by rework, which can signifi-

cantly contribute to schedule overruns being experienced

and negatively impact construction-related tasks (eg, pro-

ductivity), completion dates and resourcing (Love, 2002).

Hegazy et al (2011, p 1052) identified three specific cases

where rework can impact activities and cause a delay on:

(1) A single activity without resource constraints: If the

affected activity is non-critical, the activity float can be

used to consume time needed to complete the rework.

Contrastingly, if the activity affected is critical, then

any additional time taken to rectify work can delay the

project if no acceleration is undertaken.

(2) A single activity with resource constraints: If the activity

is non-critical then the amount of time needed to

rectify work will prolong an activity’s duration and

could lead to a resource over-allocation, leading to a

project delay as well as increased costs.

(3) Multiple activities: For example, when a reinforced

block wall requires rework due to poor set-out, other

intrinsic and interrelated activities such as reinforce-

ment and concreting also become necessary. This event

% Schedule Overrun

4036322824201612840–4–8–12–16

Prob

abili

ty o

f Sc

hedu

le O

verr

un

0.26

0.24

0.22

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

% Schedule Overrun Beta

Figure 12 Beta: PDF for projects with contract values 4 AU$100m.Paramters: a1¼ 0.53671, a2¼ 0.42258, a¼�16.25, b¼ 39.84.



2824201612840–4–8–12–16

Prob

abili

ty o

f Sc

hedu

le O

verr

un

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 % Schedule Overrun Beta

Figure 11 Beta: CDF for projects with contract values AU$51–AU$100m.


may adversely impact schedule, hinder progress, and

therefore require acceleration of other activities. Thus,

rework becomes a documented progress event that

combines with all others to impact the time, cost and

resource use in a project.

While ‘float’ has been conventionally used as a risk

management mechanism to accommodate for client

initiated change orders, clients and construction organiza-

tions have tended to place emphasis on assessing risk they

can control by establishing deterministic estimates of

duration and cost. This approach invariably anchors

project teams to early optimistic estimates. Assumptions,

omissions and qualifications, particularly for the determi-

nation of projects’ schedule duration, are made to establish

a basis for the project from a tactical and operational

perspective. It is often assumed that risks, particularly with

delays and schedule overruns, will reduce with time and

progress. When assessing schedule risk ‘attention needs to

be paid to the knock-on effects of risks arising from

complex chains of outcomes as the risks themselves’

(Ackermann et al, 2007, p 40). A systemic approach to

risk is required to ensure that compounding effects are

considered (Mulholland and Christian, 1999; Ackermann

et al, 2007). It has been demonstrated that the compound-

ing effects can result in vicious cycles that produce self-

sustaining failures (Williams et al, 1995a, b; Eden et al,

2000; Howick and Eden, 2004). By taking a systemic view

of risk, its causal nature needs to be analysed and managed

and there also needs to be a specific focus on those risks

that can cause significant disruptions to a project and its

supply chain (Ackermann et al, 2007).

The volatility of risks and their impact will vary over

time, particularly those of a strategic nature (Westney

and Dobson, 2006), and specifically in the case of design

errors, as they enter a period of incubation, and ‘when’

or ‘if’ discovered may have disastrous consequences on

a project’s schedule or a facility’s structural integrity.

Strategic risks are often risks that organizations have to

take in order to expand, and even continue in the long

term. For example, construction contractors may accept

a lump sum contract with a low margin in an attempt to

secure future work with a client. Cioffi and Khamooshi

(2009) state that good project teams will invariably

identify relevant risks and then estimate their probabi-

lities and impacts. Yet often, risks, such as errors and

omissions in contract documentation, are ignored despite

knowing that they will occur (Love et al, 2006). Failure to

recognize risks may result in the uncertainty surrounding

schedule risk being understated (Westney and Dobson,

2006). For example, Figure 14 assumes that the operational

risks, such as those associated with errors in contract



4036322824201612840–4–8–12–16

Prob

abili

ty o

f Sc

hedu

le O

verr

un

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 % Schedule Overrun Beta

Figure 13 Beta: CDF for projects with contract values 4 AU$100m.

Figure 14 Nature of risk in projects.Source: Westney (2008a, b).


documentation, change orders and the procurement of

materials significantly reduce as a project’s schedule time

reduces. It is frequently assumed by practitioners that any

risks not considered in the deterministic estimate and

covered by contingency during the design development

process will not be taken into consideration at contract

award.

Yet failure to deliver projects on schedule generates

widespread negative publicity. For example, in the private

sector poor schedule performance is unacceptable when

capital markets desire predictability and strong returns.

Projects that underperform, perhaps due to rework, are

often explained away as being an isolated instance of

unfortunate circumstance and considered beyond normal

practice (Love et al, 2009b); an outlier event within projects

(Figure 14). In spite of its outlier status, explanations and

justifications for its occurrence, after the fact, are often

made by organizations in an attempt to make the event

explainable and predictable. Many organizations are reluc-

tant to admit to existing problems within their systems and

processes for fear of being judged irresponsible by their

stakeholders (Love et al, 2009b).

The reluctance to recognize and embrace problems is the

prevailing nemesis for many clients, consultants and con-

tractors. It is imperative that within the agreed construc-

tion programme, all risks are taken into account including

rework. Mapping the empirical distribution of schedule

overruns to the ‘best fit’ probability distribution provides

the underlying impetus to consider operational risks. The

aim of the research was not to identify what these specific

risks are, but rather provide generic probabilities of sche-

dule overruns based on real-life project experiences.

Conclusion

To manage and control the risk of schedule overruns, its

probability of occurrence must be determined. Using data

obtained from 276 construction and engineering projects,

the statistical characteristics of schedule overruns were

analysed. Using the contract award as a reference point,

schedule overruns from 276 construction and engineering

projects were calculated. The mean schedule overrun was

revealed to be 11.42% from contract award. No significant

differences for schedule overruns were found between pro-

curement method, project type and contract size. The empi-

rical distributions for schedule overruns were found to be

non-Gaussian and non-parametric ‘Goodness of Fit’ tests

were used to select the ‘best fit’ probability distribution.

A 3P Burr probability function was found to provide

the best overall distribution fit to calculate the probability

of schedule overruns. As a line of enquiry, the statistical

characteristics of contract size and schedule overruns were

also analysed. The Wakeby (oAU$1m and AU$11–

AU$50m), Log-logistic (AU$1–10m, oAU$101m) and

Beta (AU$11–$AU50m and 4AU$101m) were found to

provide the best distribution fits, and therefore used to

calculate schedule overrun probabilities by contract size.

It is suggested that distribution fitting of empirical distri-

butions is necessary to produce reliable and realistic sche-

dule overrun probabilities and as a result improve decision

making. The research has consequently provided an initial

platform to examine the probability of schedule overruns.

Determining the ‘best fit’ distribution is pivotal to calcula-

ting realistic schedule overrun probabilities. Project mana-

gers are often confronted with having to make decisions

based on an imperfect and incomplete knowledge of future

events. This is particularly the case at contract award when

more often than not contract documentation is incomplete

and contains errors. One approach to improving manage-

rial decision making is to quantify uncertainties using pro-

bability. Further research, however, is required to extend

the data set and test the reliability of probabilities that have

been produced. Rather than focusing on the contract

award as reference point for determining schedule risk, it is

suggested that future research should focus on developing

probabilities and impacts of risk that arise from the design

and documentation process of a project, particularly the

interaction between risks. The use of probabilistic network

models, such Bayesian networks, are a suitable tool for

measuring and managing schedule risk in projects due to

their ability to take into account causal relations. More-

over, the inherent heterogeneity and limited availability of

data about design error and omissions and subsequent

rework, for example, makes the modelling of operational

risk more problematic than strategic risk.

Acknowledgements—The authors would like to thank the threeanonymous reviewers for their constructive comments, which havehelped improve the quality of this manuscript.

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Received December 2011;accepted January 2013 after two revisions


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