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103
What does it do? Most Probable Cost (m)
The communication of a range of values representing the
possible array of ultimate project costs creates a better under- Average Cost (always >m)
standing of estimate precision. The optimistic and pessimistic
Probability
values at the ends of the range do not necessarily represent the
very least or the very most that the project will cost, but typ-
Highest Credible Cost (p)
ically the most probable range of project costs. The size of the
range will be determined by the identified uncertainties. The
interpretation and use of the range depends on how aggres-
sive the agency is with the results.
Cost
When to use it? Lowest Possible Cost (o)
Ranges may be considered throughout project develop- Figure R3.4-1. Caltrans three-point estimate to
generate estimate range.
ment, but should be utilized on projects in earlier stages of
development to communicate the level of knowns and un-
knowns (risks) about the project. most optimistic situation, the work package will have a
cost that is greater than zero. At the other end of the scale,
there is no highest-possible cost. It is always possible to
How to use it? spend more money.
Federal planning regulations indicate that a three-point es-
timate or cost ranges/cost bands in the outer years of the met-
ropolitan transportation plan are acceptable. Therefore, sin- Tips
gle point estimates should be avoided before sufficient detail
about the project is known, when it is unrealistic to prepare a While estimate ranges transparently convey the uncer-
reasonably accurate single-point estimate. A three-point esti- tainty involved in a project, they can be misunderstood. The
mate is prepared at any point during this period by estimat- range theoretically shows the highest probable cost for a proj-
ing the lowest possible, the most likely, and the highest prob- ect. If people focus on the high end of the range, the project
able cost estimate based on a combination of available project can be slowed or stopped. The range should be used as part
data and informed judgment. of a comprehensive risk management plan. If the risks and
uncertainties that are driving the range can be understood,
they can likely be mitigated and the project can be completed
Example at the lowest possible cost.
Caltrans uses three-point estimates for some elements
of project costs and is planning to make wider use of this Resources
technique (Figure R3.4-1). Although the math may appear
Caltrans Office of Statewide Project Management Improvement (2007).
complex at first glance, it is easy to implement with a simple Project Risk Management Handbook: Threats and Opportunities,
spreadsheet. The three point estimating process uses these 2nd ed., May 2007, Caltrans, Sacramento, CA. http://www.dot.ca.
steps: gov/hq/projmgmt/guidance_prmhb.htm.
· Have subject matter experts develop three estimates for
R3.5 Estimate Ranges--
each item of work:
Monte Carlo Analysis
An optimistic estimate (o): the lowest credible cost assum-
ing that everything goes right. Expressing a cost estimate in terms of an estimate range
A most-likely estimate (m): the expert's best guess of transparently communicates the uncertainty associated with
the cost. an estimate. Monte Carlo analysis is part of a sophisticated
A pessimistic estimate (p): the highest credible cost, as- probabilistic model process that can be used to generate a
suming that virtually everything goes wrong. range estimate through simulation methods. The use of Monte
· The average cost of the item is (o+4m+p)/6. The average Carlo analysis is typically facilitated by experts in this field
is always greater than the most likely estimate. This is be- who work closely with estimators, project team members, and
cause there is a finite lowest-possible cost. Even in the subject matter experts.

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What is it? analysis can be used to determine project cost and schedule
ranges and the most likely values for each. Figure R3.5-1 shows
Monte Carlo analysis is a computerized probabilistic simu- typical probability outputs from a Monte Carlo analysis. The
lation modeling technique that uses random number genera- histogram is useful for understanding the mean and dispersion
tors to draw samples from probability distributions. Monte of the results. The cumulative chart is useful for determining
Carlo analysis uses repetitive trials to generate overall probabil- project budgets and contingency values at specific levels of cer-
ity distributions for project cost or schedule. It relies upon mul- tainty or confidence. In addition to graphically conveying in-
tiple inputs of probabilities for risk events and for uncertainty formation, Monte Carlo analyses produce numerical values for
in cost and duration of line items. A trial consists of the simula- common statistical parameters, such as the mean, standard de-
tion engine selecting a value for each of the line items based on viation, distribution range, and skewness.
their probabilities and generating a final estimate based on that
trial. This process is repeated many times (usually several thou-
sand) to generate a distribution for the total cost or schedule. When to use it?
Monte Carlo analysis is applied on complex projects and is
Why use it? used as the basis for a Type III risk analysis. The tool requires
that the project team be familiar with all project risks and be
Monte Carlo analysis has many advantages. It can determine able to quantitatively describe the risks. Application of Monte
risk effects for cost and schedule models that are too complex Carlo analysis requires knowledge and training for successful
for common analytical methods. The output of a Monte Carlo implementation. Input to Monte Carlo analysis requires the
simulation can provide a graphical distribution of project cost user to know and specify probability distribution information,
or schedule. This distribution can be used to generate an esti- mean, standard deviation, and distribution shape. While com-
mate range. It also can be used to calculate a contingency. plex and requiring significant modeling experience, Monte
Monte Carlo analysis can explicitly incorporate the risk knowl- Carlo analyses are the most common tool for quantitative risk
edge and judgment of the estimators, project team, and subject analysis because they provide detailed, illustrative information
matter experts for both cost and schedule risk events. The tech- about risk impacts on the project cost and schedule.
nique can reveal, through sensitivity analysis, the impact of
specific risk events on the project cost and schedule.
How to use it?
Monte Carlo analysis can be used to generate a number of
What does it do?
different decision-making tools for the project team. In order
The tool allows the project team to visualize the uncertainty to produce these tools, the input must be assessed to accu-
relating to the total project cost and schedule. Monte Carlo rately model project risks. Each risk can be given a different
Distribution for Total Project Costs Cumulative Total Project Costs
(Current $) (Current $)
0.020 1.00
Mean = 499.57
Mean = 499.57
0.015 0.75
0.010 0.50
0.005 0.25
0.000 0.00
400 500 600 700 400 500 600 700
5% 90% 5% 5% 90% 5%
437.98 566.93 437.98 566.93
Figure R3.5-1. Typical Monte Carlo output for total costs.

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105
Figure R3.5-2. Example tornado diagram output from a Monte Carlo analysis.
risk profile indicating where the most likely and least likely can be developed to produce simple Monte Carlo analyses.
values are. Among these different distributions are Triangular, For example, the WSDOT has developed its own Monte
Uniform, Normal, BetaPert, BetaPert modified, LogNormal, Carlo analysis package in Microsoft Excel using macros. Ad-
Discrete, Trigen, and any custom-defined distribution. ditionally, some stand-alone software exists to generate cost
In addition to the total cost ranges shown in Figure R3.5-1, and schedule Monte Carlo simulations. The most common
an additional output of a Monte Carlo analysis is a tornado stand-alone software is "Pertmaster."
diagram. The tornado diagram is a graphic depiction of a sen-
sitivity analysis. The tornado diagram can be used to show
Example
which risks will have the greatest positive or negative effect on
project cost and schedule. Figure R3.5-2 indicates the corre- WSDOT has developed a risk-based approach to cost esti-
lation that project risks have to the total project schedule. The mating in CEVP. CEVP is used to convey project cost through
risks with the longest bars have the largest impact on the over- estimate ranges. Figure R3.5-3 provides an example of how
all cost or schedule variability. CEVP is used to convey an estimate range. The project rep-
Several commercial software packages exist to help teams resented has a 10 percent chance of being completed for
run Monte Carlo analyses. As well as software that integrates $651 million or less, while there is a 90 percent chance that the
within existing spreadsheet programs, spreadsheet macros project will cost $693 million or less. However, there is a chance
10% Cost Mean Cost
$651 $668
0.14
0.12 90% Cost
$693
0.10
Probability
0.08
0.06
0.04
0.02
720
700
710
690
670
680
650
660
640
Distribution for Cost to Completion (2002 $ million)
Figure R3.5-3. Example of an estimate range.