This is the final lesson of the first part of
the course. Here we combine the results of previous lessons to
obtain estimates of the capital cost for a project.
The estimates explicitly consider risk. The WBS model
is adapted to include probability distributions
for cost components. Using formulas from probability
theory we compute the mean and variance of the system costs.
With these results we can compute several measures that
assist decision-makers in risky circumstances.
For most situations the results obtained are approximate
in that the normal distribution is used to model the capital
cost. When this approximation is unacceptable, Monte-Carlo simulation
is used to obtain results.
Because of time limitations, this lesson concentrates
on total capital costs. The Estimate add-in
uses the same methods to find moments when time is considered
and net present worth is the measure of interest. The same methods,
with some variations, are used to find moments for life-cycle
Assume the task costs are modeled
using independent normal distributions. Make point
estimates and probability statements about the total
Assume the task costs are modeled
using the uniform, triangular, beta or normal distribution.
Assuming independence, find the mean and variance
of the total capital cost. Approximating the distribution
of the total cost with the normal distribution, make
probability statements about the cost.
Use the Estimate and Random
to define a project with non-normal distributions for
task costs. Make point estimates of the total capital
cost with mean values, percentiles and simulation.
Use the Simulation RV feature of the Random
Variables add-in to estimate capital cost and
make probability statements without the normality assumption.
The textbook does not include this
We return to our assembly line example from the
Capital Cost lesson (Lesson 3), but now we add probability
distribution information to the model. The Estimate add-in
has an option to construct models with random variables specified
for task costs. The Random Variables add-in is used
during the construction process, so it must be installed in Excel
to define and analyze models with risk.
below shows part of a worksheet constructed with this option.
(The add-in uses the word activity instead of task; we will use the two terms as equivalents. In addition, the add-in defines activity costs as WP costs. In the text that follows, task costs are synonomous with the WP costs in the add-in.) Adjacent
to each task are columns that show the distribution name and
parameters. The example shows that a triangular distribution
has been assigned to each task indicated by the letters "Tri".
The next three columns hold the parameters.
The example uses the data given in the Capital Cost lesson as
the most likely values, m, for the general task.
The minimum values are 0.8m, and the maximum values
are 1.4m. The distributions are skewed to the right
with a longer right-hand tail than left-hand tail.
The data in the figure are only illustrative. The
methods described here allow any continuous or discrete distribution
included with the Random Variables add-in. The important
assumption of this lesson and throughout this course, is that
the random variables defined for the tasks must be statistically
To continue, we must compute the mean and variance
of each task cost. Although we can do this with the formulas
provided in the Estimation with Risk lesson, the Random
Variables add-in computes them automatically. Three columns
to the right of the distribution information display
the mean, variance and standard deviation of the each cost.
Click the icon below to see the larger area of the worksheet.
The arrows in the picture show the Excel functions that compute
the mean and variance. The standard deviation is the square root
of the variance.
The worksheet also provides a point estimate
for each cost, here in column L. We have chosen the distribution
mean for the point estimate. Because each distribution is skewed
to the right, the mean is greater than the most likely value.
The percentile or a simulated value could also have been selected. These options are discussed below.
Distribution of the Capital Cost
The total capital cost is the sum of the task costs.
We assume that the task costs are independent
random variables. Then we can use two results from probability
theory to derive the moments of the capital cost.
mean of the sum of random variables
is equal to the sum of the means of the
variance of the sum of independent random variables
is equal to the sum of the variances of the random
For our model, we compute the mean and variance of the capital
cost by the following equations.
The summary provided by the add-in performs these
computations and reports the moments for each level 2 task and
the total capital cost.
The point estimate of the capital cost of the project
is at the extreme right of the figure. This is simply the sum
of the mean value estimates,
or $328,533. Recall that our earlier analysis (in the Capital
Cost lesson) estimated
the capital cost as $308,000. This estimate summed the
most likely values of the triangular distributions. Because
these distributions are skewed to the right, the sum of the mean
values is greater than the sum of the most likely values.
In order to continue the analysis we will make
an approximating assumption based on the Central Limit Theorem of
Central Limit Theorem states that the sum of many independent,
identically distributed random variables is approximately
When the summed random variables have normal distributions
a stronger result applies.
sum of independent, normally distributed
random variables is normally distributed.
For practical instances, it will rarely be true
that the conditions in these two boxes are satisfied. Nevertheless,
we will find it convenient
to make probability statements about the project cost using the
normal distribution. When this assumption is not warranted we
use simulation as discussed below. In the statement below, C is
the the random variable that represents the capital cost for the project.
This formula may be used, for example, to compute the probability
of getting a lower value than the value determined assuming the
most-likely cost for all tasks. Using the standard normal
distribution we calculate that there is only
about a 11% chance that the capital cost will be that low.
Textbooks often provide tables for the approximate
evaluation of the standard normal distribution. The link following shows
such a table. The result just calculated can be found approximately
from this table for z = -1.25 instead of -1.235.
When working in Excel it is not necessary to use
the standard normal distribution for the calculations. The figure below shows
two approaches for the example. The one on the left uses the
built-in NORMDIST function. The one on the right uses the Random
Variables add-in function RV_prob.
When we know its distribution, we
can make interval statements about the project cost. The
interval estimate is a range of the random variable that covers
a given probability. The equations below describe a two-sided
confidence interval with equal probabilities on both sides of
the range. One-sided intervals compute one or the other limit
without dividing the confidence level by 2. The other limit is
When the project cost is normally distributed,
the interval is found with the inverse normal distribution
function. When the project cost is approximated by a normal distribution
or when the probabilities are determined by simulation, the coverage
level should not be too high. The approximation of the tails
of the distribution may be poor and the interval may not be accurate.
The equations below use the inverse of the standard normal.
For the example, we will now find the interval
with 0.9 coverage. Given the estimates and assuming
normality, there is a 0.9 probability that the project cost will
fall within this range; that is,
These results were computed with
Excel functions. Approximate results can be found with the
standard normal inverse table.
Excel provides a function called NORMINV to compute
the inverse of the normal distribution.
The bounds on the left in the following figure are computed with this function. The Random
Variables add-in uses the function RV_inverse to compute
inverse probabilities as shown at the right.
We earlier used the mean values
of the task estimates to estimate the project costs, but
there are other possibilities. In the last lesson, Estimation
with Risk, we mentioned
the mode, median, percentile, and simulation as ways to make point
estimates. The Estimate add-in allows the mean, percentile,
and simulation methods. The median is the 50th percentile. To
choose a different method, click the Change button at
the top of the Estimate worksheet. We
consider here using the percentile.
Say all the persons estimating
the task costs are conservative. They estimate costs
that cannot be exceeded more than 40% of the time. That is, they
estimate the task costs at the 60th percentile. The Random
computes percentiles with the inverse probability function. Click
the icon to see the results for the example problem.
The figure shows that with the normal distribution
assumption for the project cost, the probability that this
cost is less than or equal to the sum of the estimates is 66%.
Although the individual task estimates are at the 60th percentile,
the project estimate is at the 66th percentile. The sum of conservative
estimates results in an even more conservative project cost estimate.
The actual project cost, when realized, is the sum
of independent random variables. Some will be greater than the
mean and some will be less than the mean. There is a 66% chance
that this sum will be less than the value obtained as the sum of
the 60th percentile estimates.
When the task costs are not normally
distributed it is not accurate to assume the summed task
costs are normally distributed. Although, in many cases the normality
assumption may yield fairly accurate answers, the simulation
approach does not require normality. Simulation is a very
powerful tool that can be used in many contexts. Here we apply
it to estimate the project cost.
First, click the Change button
to make simulation the estimation method for task
costs. Each point estimate on the worksheet will be replaced
by a function that simulates a value every time the worksheet
is recalculated. This occurs automatically when a data item
is changed and also when the user initiates a recalculation
command. For the
Windows OS, the "Ctrl-=" causes recalculation.
For the Mac OS, the "Command-=" causes
recalculation. Click the icon below to see the worksheet with simulated
Column L holds the
simulated values. The sum of the values is just below the column
in L25. The same value is included the summary in cell X14. Excel
changes all the simulated values every time the worksheet is
recalculated. The sum that
represents the project cost changes with each recalculation.
We have gathered five observations in column X with the mean
and standard deviation of these five numbers immediately below.
The five numbers show the variability of simulated results. The
variability represents the uncertainty in the project cost estimate.
So what information does this provide? One simulated
value or even five gives no meaningful information. By simulating
a large number of observations, however, one learns a great deal
about the distribution of the project cost or any function
of the simulated values. A single simulated value is
one realization of the population of all project costs. To learn
more about the population we use the Random
to simulate 1000 observations.
The window that opens when you click on the next icon
explains how to use the Estimate and Random Variables add-ins
for simulation. The text describes several steps that become visible when you scroll down the page.
The figure below shows the results of a sample
of 1000 simulated observations. The numbers are slightly different
than obtained in the summary because they are from a different
simulation run. The bottom three numbers in column Z are statistics computed
from the 1000 observations. The simulated mean and standard deviation
are nearly those computed from probability theory. The skewness
is positive indicating that the distribution of project cost
is not symmetric as it would be if the project cost were actually
normally distributed. Since our assumed distributions have a
positive skew, this is expected.
The Random Variables add-in also provides a histogram of the
distribution of the observations. It
is skewed to the right.
Based on the statistics collected during the simulation,
the add-in constructs a new random variable, named SimX14 for
the example. The name comes from the location of the simulated
cell on the worksheet. Using this distribution, the RV_prob and
RV_inverse functions can be used to obtain the probability
and interval estimates described earlier. In contrast to the
earlier calculations, the results below do not depend on the normality
assumption for the project cost. They are obtained by repeatedly
simulating the task costs and collecting data on the project
cost. The next figure gives the estimates that were obtained from the histogram constructed from the simulation runs.
The results themselves are random variables. If the system were
simulated again, the estimates that we see here would change.
Simulation is a valuable analytical tool. It is
used in almost every field in which uncertainty is modeled with random
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