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Construction Research Congress 2014 ©ASCE 2014
A Mathematical Structure for Modeling Uncertainty in Cost, Schedule, and
Escalation Factor in a Portfolio of Projects
Ali TOURAN, Ph.D., P.E., F. ASCE1
1
Professor, Northeastern University, Department of Civil and Environmental
Engineering, 400SN, 360 Huntington Avenue, Boston, MA 02115. Phone: (617) 3735508; Fax: (617) 373-4419; email: [email protected]
ABSTRACT
The past decade has witnessed a surge in the application of formal
probabilistic risk assessment on cost, schedule, or both in major capital projects.
Depending on the level of detailed information required the sophistication of risk
assessment approach ranges from simply considering some important variables to
fully integrated cost/schedule risk models. The most common modeling approach is
to model cost components as random variables and calculate total cost distribution.
For projects or portfolios spanning several years, the effect of cost escalation on
budget is profound. Because of this, the variability of the escalation factor should be
considered in the conduct of the risk assessment. This paper provides a mathematical
framework for modeling of cost uncertainty in a portfolio of large infrastructure
projects with a multi-year duration. This framework considers the randomness of cost
and escalation factor at the project level. Relevant equations are presented that
consider various degrees of probabilistic modeling from basic to complex. The paper
can be used for understanding major drivers of uncertainty in portfolio budget and to
evaluate the effect of escalation variability on project costs. The concepts and the
suggested approach are explained using a numerical example.
INTRODUCTION
This paper provides an overview of a mathematical framework for modeling
of uncertainty in a portfolio of projects. A portfolio is defined as a collection of
multiple projects managed by a single management team or organization. The
projects can be interrelated and part of a program, or independent from each other but
under the supervision of a single organization. The proposed overview is intended for
large infrastructure portfolios (such as transportation) where project development can
take several years. Under this scenario, the uncertainty may be considered and
modeled in three areas: cost, schedule, and escalation factor. Table 1 gives an
overview of probabilistic modeling of these three factors.
Table 1. Various modeling levels for probabilistic estimating
Factors modeled as
General approach and input needed
random variables
(1) Cost, schedule,
escalation factor
Joint density function of the three variables; or
marginals and correlations
(2) Cost, schedule
Joints density function of the two variables; or
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marginals and correlations
(3) Cost, esc. factor
Joints density function of the two variables; or
marginals and correlations
(4) Cost
Distribution of cost components
(5) Schedule
Distribution of schedule elements or activities
In the most general case, cost, schedule, and escalation factor are all modeled as
random variables (Case (1) in Table 1). Because there is interaction between the
values of these variables, the only way to model these correctly is to develop the joint
density for these variables, something that is not achievable under general conditions
and given the limited available data. With some simplifying assumptions, a
multinormal distribution can be considered for modeling the factors however,
correlations among these variables would be required. In Case (2) the problem is
somewhat simpler but still the joint density function of cost and schedule is not
necessarily obtainable. The cost and duration variability is quite interactive; longer
delays increase the cost, and larger overruns are usually an indication of project
delays. Simpler approaches can be used where the cost can be modeled as the sum of
direct and indirect costs and then assumed that indirect costs are a linear function of
duration. From there, distributions of total cost given various durations could be
obtained. Another common simplification is to ignore the correlation among these
variables and model these as independent random variables. Case (3) considers the
randomness of cost and the escalation factor. The current paper treats this case under
the special condition that cost and escalation factor are assumed to be independent.
The last two cases are the traditional cases of probabilistic cost estimating and
probabilistic scheduling (for example PERT). Each of these cases would need to be
aggregated at the portfolio level when dealing with multiple projects.
In this paper, the effect of cost escalation will be explicitly considered because
in large construction programs that span several years, cost escalation can be a major
source of uncertainty. In this paper, distributions of cost and escalation factor are
assumed to be statistically independent. Cost estimate components are costs of
materials and labor at present time given a specified scope of work. The uncertainty
in cost is a function of variations in this scope and the increase in labor and material
costs due to inflation. Escalation factor measures the movement of material and labor
costs in the future. In order to assure the separation of these two elements, the process
of risk assessment should consider risk factors such as productivity issues, ground
conditions, probability of permits and funding approvals for cost risk analysis. The
risks associated with labor rates and material prices should not be considered when
identifying cost risk factors. These risks will be considered when modeling escalation
factor as a random variable. This way the distribution of cost and escalation factor
will remain more or less independent.
Figure 1 below shows the components of the construction portfolio. The
portfolio consists of n projects each with an estimated construction phase duration of
di and an estimated cost of Ci. Ci is the current dollar estimate for Project i. In other
words, the estimates are in terms of Year 0 (current) dollars. In the most general case,
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both the durations and costs are modeled as random variables represented by
appropriate probability distributions.
In this paper, the project cost Ci is assumed to follow a normal distribution
and variance
(Eq. 1). The project cost is the sum of a number of
with mean
cost items and the assumption of independent cost components will result in a normal
distribution for the total project cost.
(1)
Also, the time of start of each project is si .
ti = si + di/2
(2)
ti is the time to the midpoint of project i. In order to consider the effect of escalation,
an escalation factor r should be considered. Two cases are assumed: (1) r is a fixed
value, and (2) r is a random variable. For the second case, r is modeled as a normal
random variable (Touran and Lopez 2005). Mean and standard deviation of rj or
escalation factor for period j can be calculated from historical data. These are
represented with
and in Eq. 3.
C1
C2
Projects
C3
d3
Cn
s3
Time
t3 = s3 + d3/2
Figure 1. Portfolio of projects
(3)
With the above definitions, we can calculate the cost of project i in the year of
expenditure (YOE), Ci YOE. Note that each project has a distribution of expenditure
which is assumed to be known (Fig. 2). So the current estimate would be:
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(4)
In Eq (4), the planned expenditure in current dollars for project i for year j of
that project is Cij. It is assumed that the expenditure for each year occurs at the start of
the year to simplify the calculations.
di
Cost
2
ci3
ci
ci1
cidi
Si
Time
Figure 2. Project planned expenditures (in current dollars)
ESCALATED PROJECT COST
Escalated project cost for a project as depicted in Fig. (2) will be:
(5)
In Eq (5), r can be modeled as a fixed value or a random variable. Much work
has been done to characterize the shape of the cumulative expenditure curve (S-curve)
on construction projects (Perry 1970; Drake 1978; Peer 1982; Miskawi 1989;
Bhurisith 2000; Touran et al 2004). Based on research by Bhurisith (2000), it is
suggested to use a Beta distribution with shape factors α = 1.70 and β = 1.85 for
modeling project expenditure. The beta distribution is presented in Eq (6):
a<x<b
(6)
where B is the beta function, and the start and end point for the duration is a and b,
respectively (Ang and Tang 2007). Figure (3) shows a beta distribution with a = 0, b
= 1, α = 1.70, and β = 1.85.
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f(x)
x
Figure 3. Beta Distribution with a=0, b=1, α=1.70 and β=1.85
So for example, in a 4-yr project in Fig (3), Ci1would represent the area under the
curve between 0 and 0.25, Ci2 would represent the area under the curve between 0.25
and 0.50, and so on. For year j of a project’s duration, and assuming that the beta
curve starts at 0, the expenditure can be estimated using Eq.(7):
(7)
Using Eqs (5) and (7),
can be calculated. Note that if any other distribution is
deemed superior to what is suggested here, the process of substituting the cost
will be similar to what was described.
distribution and calculating
Midpoint of Project Approach
One common approach for simplifying the calculations is to assume that all
project expenditures are expensed at the midpoint of the project. This is a common
industry practice and will not result in errors as long as project cost expenditures are
symmetrical with respect to its midpoint. However, when one is considering
uncertainties and using a probabilistic approach, the midpoint of project approach will
result in an underestimation of total cost variance.
Fixed Escalation Factor
If escalation factor is assumed to be a fixed value r, then total project cost
expressed in the year of expenditure dollars will be:
(8)
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Escalation Factor as a Random Variable
If escalation factor is modeled as a random variable, then it will be different
for different periods and the total project cost expressed in YOE dollars will be:
(9)
In the above equation, si is the period of the start of project i. If the mid-year of
can be calculated from Eq. (10).
expenditure approach is used,
(10)
As was discussed at the beginning of the paper, a normal distribution is used
for modeling escalation factor (Eq 3). Furthermore, it is assumed that values of
escalation factors are independent of each other in various years. This approach has
proved more accurate compared to the approach that incorporated correlations
between consecutive periods. Zhang (2013) verified that the correlation coefficient
among the escalation rates calculated based on ENR index values was small among
consecutive years. Also Zhang (2013) verified that using the independence
assumption between values of cost escalation in consecutive years resulted in better
accuracy compared to the case where autocorrelations were considered among
consecutive years.
is the product of several normal
Referring to Eq (10), one can see that
random variables and so it should follow a lognormal distribution. However, because
of the shear magnitude of
compared to other multipliers (i.e., (1+rj)’s) , the
distribution of
remains close to normal. The parameters (mean and variance)
of this lognormal distribution can be calculated from Eqs (11) and (12), as long as ti is
assumed to be deterministic. Also, the assumption is that all the random variables
involved are independent.
Construction Research Congress 2014 ©ASCE 2014
TOTAL PORTFOLIO COST
Having calculated various
using Eq. (13).
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, one can calculate total program cost, CYOE
(13)
will be very close to
From Eq (13), one can conclude that the distribution of
normal as long as the number of projects in the portfolio is sufficiently large and the
correlation among various project costs can be neglected. It should be noted that the
assumption of independence among project costs is not conservative. This assumption
would be more accurate if the portfolio consists of projects that are in different
geographical regions. It is evident that in the proposed framework, the randomness of
costs and the escalation factor all play a role in the randomness of the total program
cost,
.
NUMERICAL EXAMPLE
In this hypothetical example, it is assumed that durations are fixed and costs
and escalation factor are modeled as random variables. The portfolio consists of four
projects as depicted in Figure 4.
1
2
3
4
5
6
7
Year
8
9
10
11
12
13
Cost / Budget
Project 1
$100,000,000
Project 2
50,000,000
Project 3
$80,000,000
Project 4
$60,000,000
Figure 4. Portfolio barchart
Project statistics are given in Table 2. Current portfolio total is $290M. The
first project is scheduled to start one year from today and total duration for the four
projects is 11 years. The escalated project cost is $351,632,000. Project costs follow a
normal distribution with standard deviations equal to 20% of their means. These
distributions are generally obtainable by conducting risk assessment on each project,
identifying the risk factors, and summing up these risk factors to arrive at these
distributions (Molenaar et al 2010). In the risk register, risks associated with labor
and material cost variations are not included because those risks are captured through
the distribution for escalation factor. The escalation factor follows a normal
distribution with a mean of 3% per year and a standard deviation of 1.5%. The
midpoint of construction approach has been used for calculation of CYOE. Values of
Column (4) are calculated using Eq. (11) and values of Column (5) are calculated
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using Eq. (12). Total cost distribution is close to a normal distribution with the
calculated mean and standard deviations. As a matter of validating the analysis, the
process is simulated and simulation results are shown in Column (6).
Table 2. Comparison of simulated vs calculated results
Project
Mean of Ci
Std Dev of
Ci
Mean of
CiYOE
(1)
(2)
(3)
(4)
1
$100,000,00
0
$20,000,00
0
2
$50,000,000
3
Calculate
d Std Dev
of CiYOE
Simulated
Std Dev of
CiYOE
(5)
(6)
$109,272,70
0
$22,034,602
$22,022,66
5
$10,000,00
0
$56,275,441
$11,378,575
$11,351,69
6
$80,000,000
$16,000,00
0
$101,341,60
7
$20,710,875
$20,774,98
5
4
$60,000,000
$12,000,00
0
$83,054,032
$17,107,682
$17,085,97
2
TOTA
L
$290,000,00
0
$351,632,00
0
$36,559,660
Figure 5. Results of simulation analysis for total portfolio costs
Figure 5 shows the total cost distribution obtained through simulation. If the
escalation factor had been modeled as a fixed value, then the total mean would have
remained the same. The variance for the case of deterministic escalation factor would
Construction Research Congress 2014 ©ASCE 2014
be smaller and could be calculated by summing project variances; total standard
deviation for this case would have been $36,038,000.
SUMMARY
This paper provided an overview of modeling options for treating uncertainty
in project portfolio cost, schedule, and escalation. It provided a practical solution to
the case of probabilistic modeling of project cost and escalation factor. A numerical
example was used to demonstrate the approach, and the accuracy of the analytical
approach was verified by simulation.
REFERENCES
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John Wiley and Sons, New York, NY.
Bhurisith, I. (2000). "A cash flow model for construction project financing." MS
thesis, Northeastern Univ., Boston.
Drake B. E. (1978). "A mathematical model for expenditure forecasting post
contract." Proc., CIB W-65 Symp. on Organization and Management of
Construction, Haifa, Israel, 11-163-11-183.
Miskawi, Z. (1989). "An S-curve equation for project control." Constr. Manage.
Econom., 7, 115-124.
Molenaar, K., S. Anderson, and C. Schexnayder. (2010). “Guidebook on risk analysis
tools and management practices to control transportation project costs.” NCHRP
Report 658, Transportation Research Board, Washington, D.C.
Peer, S. ( 1982). "Application of cost-flow forecasting models." J. Constr. Div.,
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Perry, W. W. (1970). "Automation in estimating contractor earnings." Mil. Eng., 410,
393-395.
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Projects.” J. of Construction Engrg. & Manangement, ASCE, 132 (8), 853-860.
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Transportation Prompt Pay Provisions.” J. of Construction Engrg. &
Manangement, ASCE, 130 (5), 719-725.
Zhang, Y.(2013). “A Portfolio Management Decision Support System for Transit
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