Download Probabilistic Model for Cost Contingency

Probabilistic Model for Cost Contingency
Ali Touran, M.ASCE1
Abstract: This paper proposes a probabilistic model for the calculation of project cost contingency by considering the expected number
of changes and the average cost of change. The model assumes a Poisson arrival pattern for change orders and independent random
variables for various change orders. The probability of cost overrun for a given contingency level is calculated. Typical input values to the
model are estimated by reviewing several U.S. Army Corps of Engineers project logs, and numerical values of contingency are calculated
and presented. The effect of various parameters on the contingency is discussed in detail.
DOI: 10.1061/共ASCE兲0733-9364共2003兲129:3共280兲
CE Database subject headings: Probabilistic methods; Models; Cost control; Project management.
Introduction
In many construction projects, the owner plans for unexpected
events that may affect project cost by adding a contingency to the
estimated cost. Depending on the owner’s organization and level
of sophistication, this contingency is calculated in various ways.
One of the most simple and common methods is to consider a
percent of estimated cost, such as 10%, based on previous experience with similar projects. Such an approach will not quantify
the degree of confidence that the contingency will provide against
cost overruns.
In this paper we introduce a probabilistic model that incorporates uncertainties in project cost and calculates the contingency
based on the level of confidence specified by the owner. The
model’s input parameters are few, and the application is straightforward.
Model
Let us assume that events causing cost adjustments 共change orders兲 occur randomly in time. Assuming that these events happen
according to a Poisson process, we have
e ⫺␭ ␭ x
P 关 X⫽x 兴 ⫽
; x⫽0,1,2,...
(1)
x!
where X is a random variable denoting the number of incidents
during the project, and ␭ is the mean of the distribution and is
calculated from
␭⫽␣T
(2)
where ␣ is the mean rate of occurrence and T is the estimated
project duration. We further assume that each incident will affect
the project cost. Specifically, the cost of change order i is assumed
to be C I , which is a random variable. The total cost of changes
C ch will be
1
Associate Professor, Dept. of Civil and Environmental Engineering,
Northeastern Univ., 400 Snell Engineering Center, Boston, MA 02115.
Note. Discussion open until November 1, 2003. Separate discussions
must be submitted for individual papers. To extend the closing date by
one month, a written request must be filed with the ASCE Managing
Editor. The manuscript for this paper was submitted for review and possible publication on January 25, 2002; approved on May 28, 2002. This
paper is part of the Journal of Construction Engineering and Management, Vol. 129, No. 3, June 1, 2003. ©ASCE, ISSN 0733-9364/2003/3280–284/$18.00.
x
C ch ⫽
兺 Ci
(3)
i⫽1
The probability mass function 共PMF兲 of total cost of changes can
now be computed.
Derivation of Distribution for PMF of C ch
We consider two cases: in the first case we assume that change
orders are distributed according to a normal distribution; in the
second case this assumption is relaxed.
Special Case
If one assumes that cost of change orders are independent and
identically distributed normal variables, then
C i ⬃N 共 ␮ C ,␴ C2 兲
(4)
In the above equation, ␮ C and ␴ C are the mean and standard
deviation of change order cost, respectively. Total cost of changes
is normally distributed because it is a linear sum of x normal
variables, where x is itself a random variable, as it represents the
number of incidents causing cost adjustments in a construction
project
n
for x⫽n⇒C ch 兩 x⫽n ⫽
兺 Ci
(5)
i⫽1
From the above equation, one can calculate conditional distributions for C ch and the expected value of C ch as follows:
C ch 兩 x⫽n ⬃N 共 n␮ C ,n␴ C2 兲
冋兺 册
(6)
x
E 关 C ch 兴 ⫽E
i⫽1
C i ⫽E 关 X 兴 ␮ C
Now, referring to Poisson distribution for X, we have
e ⫺␭ ␭ n
P 关 X⫽n 兴 ⫽ P 关 C ch ⫽N 共 n␮ C ,n␴ C2 兲兴 ⫽
n!
(7)
(8)
General Case
At this point we can relax the requirement that all costs are identical; the equations presented above will change but the approach
remains the same
C i ⬃N 共 ␮ C i ,␴ C i 兲
(9)
280 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003
Table 1. Projects’ Cost and Schedule Data
Description
Original duration
共cda兲
Original budget
共$兲
Number of changes
Cost of changes
共$兲
Regional finance center
Concrete repairs
Replace gate system
Construct seepage control
Meadow brook restoration
Flood damage reduction
730
150
360
730
120
730
4,664,369
139,283
1,127,000
16,360,130
413,954
6,774,400
37
4
12
10
7
25
1,382,721
71,132
172,320
712,761
364,456
1,159,187
Project code
DACA51-98-C-0033
DACW33-00-C-0014
DACW33-95-C-0037
DACW33-97-C-0018
DACW33-97-C-0019
DACW33-97-C-0021
a
Calendar days.
冉兺
n
C ch 兩 x⫽n ⬃N
i⫽1
n
␮ Ci,
兺
i⫽1
␴ C2
i
冊
(10)
We now relax the requirement that all change costs are normally distributed. Eq. 共10兲 still holds because even if C i ’s are not
normally distributed, according to the central limit theorem the
sum of these will be normal, given that i is sufficiently large.
Several references discuss the magnitude of i. References in applied statistics such as Devore 共2000兲 recommend values larger
than 30, but suggest that if individual variables are close to bellshaped, even a small value for i will suffice. Others, mostly engineering sources 关such as Moder et al. 共1983兲兴, suggest that with
i being as small as four, still the assumption of normality is approximately acceptable. For this specific application, the number
of changes for a project with a duration of between 12 and 24
months would be somewhere between two cases. The main problem with the ‘‘general case’’ is that it would be difficult to estimate parameters of n delays and n cost distributions with any
degree of certainty.
It should be noted that the cost increase calculated above is
based on a specific ␭, the total number of change orders. If project
changes cause an increase in project duration, the expected number of changes will increase accordingly and will impact the cost
of change. In this model, this increase in cost is disregarded partially because the contingency is estimated at the beginning of the
project as a function of original budget.
Calculation of Contingency As Percentage of Original
Cost
It is convenient to express contingency as a percentage of the
original estimate or budget. We express contingency ␩ as the ratio
of total contingency ␤ over original budget C.
Consider the following: C⫽original project cost estimate
共budget excluding contingency兲; T⫽original project duration estimate 共duration excluding contingency兲; and f C ⫽coefficient of
variation for change in cost due to change orders.
␴ C⫽ f C␮ C
Calculation of Contingency
Now that we have the distribution of cost overruns, we can
specify a desired contingency value. Here, for simplicity, we
again assume that changes are identical and independent. If the
owner desires a confidence level of p against cost overruns 共that
is, the probability of not having a cost overrun being p兲, he or she
would need a contingency of ␤ such that
P 关 C ch ⭐␤ 兴 ⭓p
(11)
⬁
P 关 C ch ⭐␤ 兴 ⫽
兺
x⫽0
⬁
⫽
P 关 C ch ⭐␤ 兩 X⫽x 兴 P 关 X⫽x 兴
兺⌽
x⫽0
冋冑 册
␤⫺x␮ C e ⫺␭ ␭ x
⭓p
x!
x␴ 2
O i⫽
(13)
冉
2 2
␮ C f C␮ C
Ci
⇒O i ⬃N
,
C
C
C2
O ch 兩 x⫽n ⬃N
冉
n␮ C n f C ␮ C
,
C
C2
2
2
冊
冊
(14)
(15)
In the above equations, O i denotes the cost of a change order
as a ratio of total original project cost, and O ch is the total cost
adjustment as a result of changes in the project. Using the following equations, we can calculate percent contingencies for the budget. Note that changes are assumed to be identical and independent
(12)
P 共 O ch ⭐␩ 兲 ⭓p
C
(16)
Table 2. Projects’ Input Values and Statistical Parameters
Project code
DACA51-98-C-0033
DACW33-00-C-0014
DACW33-95-C-0037
DACW33-97-C-0018
DACW33-97-C-0019
DACW33-97-C-0021
Mean of
change order
共$兲
Standard
deviation of
change order
共$兲
Coefficient
of variation
of change order
Mean of
change as ratio
of original
estimate 共%兲
Cost of
changes as ratio
of original
estimate 共%兲
Rate of change
per month
Level of significance
for exponential
distribution
共%兲
53,400
62,800
21,200
72,800
52,100
43,000
131,000
40,800
28,100
70,200
63,200
50,900
2.44
0.65
1.32
0.97
1.21
1.19
1.14
45.1
1.9
0.4
12.5
0.6
28
51
15
4
88
17
1.5
0.8
1
0.4
1.8
1
9
⬍1
⬍1
1
30
⬍1
JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003 / 281
Fig. 1. Distribution of time between changes for projects studied
⬁
P 共 O ch ⭐␩ 兲 ⫽
兺
x⫽0
⬁
⫽
more, the expected value of O ch can be calculated. It can be
shown that if a random variable can be modeled as the sum of X
random variables, where X is a random variable itself, we have
共Benjamin and Cornell 1970兲
P 关 O ch ⭐␩ 兩 X⫽x 兴 P 关 X⫽x 兴
兺⌽
x⫽0
冋冑 册
␩⫺
x␮ C
C
x f C2 ␮ C2
X
e ⫺␭ ␭ x
x!
⭓p
Y⫽
(17)
C2
Eq. 共17兲 gives the probability of the total cost of changes remaining below any assumed contingency percentage ␩. Further-
兺 Zi
(18)
i⫽1
E 关 Y 兴 ⫽E 关 X 兴 •E 关 Z 兴
(19)
Var关 Y 兴 ⫽Var关 Z 兴 •E 关 X 兴 ⫹ 兵 E 关 Z 兴 其 2 •Var关 X 兴
(20)
From the equations above one can calculate parameters of O ch
282 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003
E 关 O ch 兴 ⫽E 关 X 兴 •E 关 O i 兴 ⫽E 关 X 兴 •
␮c
␮c
⫽␭
C
C
Var关 O ch 兴 ⫽Var关 O i 兴 •E 关 X 兴 ⫹ 兵 E 关 O i 兴 其 •Var关 X 兴 ⫽
2
f C2 ␮ C2
C2
(21)
␭⫹
␮ C2
␭
C2
(22)
Examination of Assumptions and Calculation of
Input Values
To develop numerical values for a contingency, one needs to establish typical ranges for the input variables; and model assumptions also need to be evaluated. The main statistical assumptions
in the model are the following: change orders arrive according to
a Poisson process, and the cost of change orders are independently and identically distributed normal variables. If the number
of changes is sufficiently large, say, larger than 10 共Miller 1983兲,
then the assumption of normality for individual change orders can
be relaxed. Regardless of the distribution of changes, the total
cost of change will have an approximately normal distribution.
In order to evaluate the Poisson assumption made in the development of the model, we examined data for several projects
obtained form the U.S. Army Corps of Engineers, New England
Division. The Corps provided us with project change logs of 34
projects that had either cost or schedule changes. Of the 34
projects investigated, only 6 were selected for statistical analysis.
The reasons for eliminating the others were that 7 of the projects
had no change orders, 13 had fewer than four change orders
共mostly projects of very limited scope and duration兲, and 8 had
some important information missing, such as ‘‘notice to proceed’’
or the value of some of the change orders 共Davul 2001兲. The 6
projects that were examined in detail consisted of building, dredging, and drainage projects 共Table 1兲. We also used this data to
calculate typical values for inputs to the model such as average
change size, coefficient of variation for changes, and number of
changes.
Fig. 2. Probability of cost overrun given contingency of 15% and
various values of f c and ␮ c (␭⫽12)
a sufficient number of changes for the assumption of normality
for the total cost of changes. The main inputs to the model are
mean change size (␮ c ); coefficient of variation ( f c ); and total
number of change orders 共␭兲.
Reasonable ranges for these variables were selected based on
observed data. For example, for the Army Corps of Engineers
共ACE兲 projects, one may consider a range of 0.5 to 2.5 for the
coefficient of variation, 0.5 to 2% of the original project estimate
for average change, and a rate of 0.5 to 1.5 changes per month
共Table 2兲. With the model calibrated, one can perform various
sensitivity analyses and investigate the effect of each variable on
potential cost overruns and required contingencies. As an example, Fig. 共2兲 shows that the probability of a cost overrun is
much more sensitive to variability of average change size compared to f c . In fact, one may be able to make a case by stating
that f c may be taken as a fixed value; a sufficient contingency can
be estimated based on two random factors: ␮ c and ␭ 共estimated
number of change orders during project execution兲.
Statistical Analysis
We assumed that changes happen according to a Poisson process,
which means that the time between changes is distributed according to an exponential distribution. The test of chi-square goodness
of fit was performed on the 6 projects, and the results are provided in Table 2. As can be seen, the level of significance for 3 of
the projects is less than 1%, which shows that while an exponential distribution may not be a good fit for presentation of data in
all cases, it is appropriate in a good proportion of cases 共in this
experiment, in 3 out of 6 projects兲. Fig. 1 gives the histogram for
the time between changes for all 6 projects. As can be seen,
although the test results were not encouraging for all cases, the
exponential distribution in general seems to provide a reasonable
visual fit for the data.
Input Data for Model
A convenient way of using the model developed in this study is to
provide a tabular or graphical solution to Eq. 共17兲. In order to
achieve this, a MATLAB 共Hanselman and Littlefield 1999兲 routine was created whose results are reported later in this paper.
Inputs to the model were selected based on parameters presented
in Table 2. For this case, we have limited the application to
projects with durations between 12 and 24 months, which ensured
Application and Analysis of Results
Using the ACE data, the model is calculated and the probability
of a cost overrun for various assumed contingency levels is calculated. For this experiment, a fixed f c ⫽1.0 is selected because,
as shown earlier, the probability of a cost overrun is less sensitive
to variations of f c . The result of the analysis for a 12-month
project is presented in Table 3.
Assuming that a confidence level of 75% is desired with respect to contingency allocation, Table 3 highlights the combinations of ␮ and ␣ 共rate of change per month兲 that result in a cost
overrun 共insufficiency of contingency兲.
Fig. 3 shows the result of the analysis graphically. The expected value of the total change can be calculated using Eq. 共21兲
as (12␣)␮ c /C. As an example, if the average change order is 1%
of the original estimate, then the expected value of cost of
changes, given one change per month, would be 12⫻1⫻1%
⫽12%. This means that a contingency of 12% should be adequate about 50% of the time. While one can use average values
and perform a deterministic analysis for the contingency, the
model provides confidence levels when the amount of contingency deviates from the expected values. Increasing the contingency will reduce the probability of a cost overrun.
JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003 / 283
Table 3. Probability of Cost Overrun with 15% Contingencya
␣
0.5
0.6
0.7
0.8
0.9
1.0
a
␮ c/C ⫽0.5%
␮ c/C ⫽1%
␮ c/C ⫽1.5%
␮ c/C ⫽2%
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.03
0.07
0.11
0.18
0.26
0.13
0.22
0.32
0.43
0.54
0.64
0.31
0.43
0.56
0.66
0.75
0.82
Project duration is assumed to be 12 months.
Finally, to see the effect of contingency variation on the probability of a cost overrun, the value of contingency was ranged and
the probability of a cost overrun was calculated as a function of a
contingency percentage 共Fig. 4兲. This is equivalent to the graphical solution of Eq. 共17兲. Using Fig. 4, the owner can choose a
level of contingency commensurate with a risk acceptable to him
on her. In this figure, it is assumed that the coefficient of variation
f C ⫽1 and ␮ C ⫽1%.
Fig. 4. Probability of cost overrun for various values of contingency
for various number of change orders (␮ C ⫽0.01 and f C ⫽1)
Future Work
A probabilistic model for the calculation of contingency is presented and its potential demonstrated by applying typical input
values; a similar model can be developed for schedule contingency. Interaction of schedule delays and cost increases is another
area that deserves further research. Also, an extensive survey of
various project types can be conducted to calculate typical input
values for specific types of projects. As an example, a transit
agency is usually engaged in specific types of construction
projects. By reviewing the historical data of a specific transit
agency, one can calculate rates of changes, size and distribution
of changes, and times between changes for similar projects and
prepare risk profiles or cumulative probability curves for various
values of contingency. The outcome can be used at the budgeting
phase of a new project to ensure that consideration is given to
potential cost overruns after the project starts.
Acknowledgments
The writer would like to thank Paul Cooper of the Army Corps of
Engineers, New England Division, for his assistance in providing
project cost data. Atilla Davul, a graduate student in the Civil
Engineering Department, Northeastern University, helped in data
analysis.
References
Fig. 3. Probability of cost overrun given contingency budget of 15%
of original estimate and T⫽12 months for various rates of change
orders per month
Benjamin, J. R., and Cornell, C. A. 共1970兲. Probability, statistics, and
decision for civil engineers, McGraw-Hill, New York.
Davul, A. 共2001兲. ‘‘Analysis of the construction cost data.’’ MS thesis
submitted to Northeastern Univ., Boston.
Devore, J. L. 共2000兲. Probability and statistics for engineering and the
sciences, 5th Ed., Duxbury, Pacific Grove, Calif.
Hanselman, D., and Littlefield, B. 共1999兲. MATLAB user’s guide, Mathworks Inc., Natick, Mass.
Miller, R. W. 共1983兲. Schedule, cost, and project control with PERT,
McGraw-Hill, New York.
Moder, J. J., Phillips, C. R., and Davis, E. W. 共1983兲. Project management with CPM, PERT and precedence diagramming, Van Nostrand
Reinhold, New York.
284 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / MAY/JUNE 2003