Estimation
with Risk
This lesson adds measures of risk to the estimation
process. Uncertainty of estimation is always present and in our view, it is better to account for it
explicitly rather than to simply ignore it. The methods for dealing
with risk introduced here will be used throughout the course.
The textbook describes a variety of types
of risks associated with projects, but our analytic methods
concentrate on the uncertainties inherent in the
estimates computed by the WBS and CBS models. This lesson uses
probability distributions to model the uncertainties of cost
and revenue estimates. In the next lesson, the distributions are
combined to produce moments for the capital cost model. Both
the Random
Variables addin
and several tables are used for the computations.
The lesson assumes that students have some background
in probability and statistics. Students should be able to compute
simple statistics such as the mean and standard deviation of
data and be sufficiently familiar with the continuous distributions
used in this section to accomplish the goals
listed below. 

Goals 



List the sources of uncertainty associated
with a project.

Use probability distributions to make estimates that
explicitly measure risk.

Be familiar with four distributions
used in this lesson:
Uniform, Triangular, Beta and Normal. For each case,
know how the distribution parameters affect the shape
of the distribution.

Use the Random Variables addin
to compute for a given distribution the mean and standard
deviation. Be able to evaluate probability and inverse
probability statements. Be able to simulate random
variables.

Use the Random Variables addin
to make point estimates using the expected value, the
kth percentile and simulation.

For the normal distribution, use tables
to make point estimates using the expected, the kth
percentile and simulation. Also be able to evaluate
probability and inverse probability statements.




Text 

There are many sources of risk and
uncertainty in engineering projects. The referenced sections
in the text provide some general remarks regarding the source
and management of risk. Be sure to read them to find the variety
of reasons why the success of a project is uncertain and ways
to reduce or respond to risk during project planning. The lesson
concentrates on uncertainties in costs or revenues, but students
should be aware of the other sources as well.


1.3.2 Risk
and Uncertainty 




Probability Distributions 

We use continuous probability for estimates that
admit uncertainty. With probability distributions, we have
a mathematical basis for analyzing the effects of decisions on
risk. Probability theory and simulation procedures provide many useful tools for the analyst and decision
maker.
With a probability distribution representing a
cost, the actual cost that will be observed is a random variable,
indicated as x below.
For this discussion we use continuous random variables. The
discussion refers to cost, but revenue items are similarly estimated.
We also use the inverse cumulative function.
Although we focus on
four named distributions, the Random Variables addin
allows a variety of named continuous and discrete distributions.
The link below opens a page in the ORMM site that lists the distributions
available.
Several continuous probability distributions, specifically
identified by the Estimate addin, are listed below.
Others also may be used.
Range Estimate 

Here the estimator provides
lower and upper bounds for the cost. The lower
bound is a and the upper bound is b.
The range specification provides no information about the
distribution of values within the range so we use the uniform
distribution. This assumes all values are equally likely.

ThreePoint Estimate 

A traditional way to get subjective estimates
from an expert is to ask three questions: what is the smallest you think
the cost can be? what is the most likely value? what is
the largest value? The triangular distribution is defined
exactly by these three measures identified respectively
by the letters, a for the smallest, m for
the most likely, and b for the largest. The most
likely value for a distribution is called the mode.

Beta Estimate 

This is an upgrade to both the range and
threepoint estimates. The Beta distribution is defined
by its lower and upper bounds, a for the lower
bound and b for the upper bound, as well as two
parameters defining its shape and .
A variety of shapes can be defined. The mode is a function
of the parameters.

Normal Estimate 

The normal distribution has the traditional
bellshape appropriate for many measurements. The estimate
is defined by two parameters the mean, ,
and the standard deviation, .
When we estimate a cost based on the average of a large
number of observations. The statistical mean and standard
deviation provide estimates of the distribution parameters.

Other Distributions 

The Random Variables addin provides a variety
of other distributions that might be useful in modeling
the variability of estimates.

Using a probability distribution to represent the uncertainty
of an estimate complicates the estimation problem because rather
than a single point estimate, there are two or more estimates
necessary for the parameters of the distribution. The additional
effort is well rewarded by the many useful results that can be
obtained from probability theory. 

Information from Distributions 

Once a distribution is specified with its parameters,
several useful quantities may be computed using mathematical
formulas or numerical methods. This course uses the Random
Variables addin. The addin uses numerical methods for
computations. For this lesson, the student should be able to
define random variables and use the Excel functions listed in
the table below to discover information about the distribution.




The figure below shows a random variable representing
the cost of a work package in a WBS. The definition of the random
variable, called WP1, is placed on the worksheet by the addin.
The definition has been placed in the range of cells D5:D9. The
definition consists of the name of the distribution (Beta) and
its parameters, alpha, beta, lower and upper.
The cells below the definition hold functions that are provided
by the addin to compute information about the distribution.
Although in this case the functions are placed below the definition,
they can be placed anywhere on the worksheet. Be aware that the notation
of the addin uses e rather than w in the figure.
The quantities computed should be familiar to a
student with a background in probability theory. They are described
in the following table with their Excel function. These functions are available
only when the Random Variable addin is installed.
Quantity 
Purpose 
Excel Function 
Mean Value 
The expected value of the random variable. This is often
used as a point estimate. 
=RV_MEAN(WP1) 
Variance 
Represents the variability of the random variable. Another
useful quantity is the standard deviation that is
the square root of the variance. 
=RV_VAR(WP1) 
Probability 
The probability that the random variable falls within
the range c to d. If c left blank, the lower limit is negative
infinity. If d left blank, the upper limit is positive
infinity.
P(c < x < d) 
=RV_PROB(WP1,c,d) 
Inverse Probability 
Computes the value of the random variable
such that the probability that the random variable is less
than that value is equal to w.
The median is RV_INVERSE(WP1,0.5)
The 90th percentile is RV_INVERSE(WP1,0.9) 

Simulation 
Simulates the random variable. The function
name ending in V indicates that this function is volatile.
Every time the worksheet is computed a new value is simulated.
This is useful for MonteCarlo simulation. 
=RV_SIMV(WP1) 
In the next lesson we learn how these functions
are used to compute project cost estimates. 

Simulation 

There are many situations that are affected by
more than one random variable. For all but the simplest,
probability models will be too difficult to construct and evaluate. Rather,
it is necessary to simulate the several random variables
that describe the situation and observe the results. By running
a simulation model many times, statistical analysis provides
the estimates we need. In this lesson we describe the reverse
transformation method for simulating a single random variable.
In the next lesson we use simulation to estimate the capital
costs of a project.
To simulate a random variable with a known distribution,
choose a random number from a uniform distribution bounded between
zero and one. Find the value of the random variable for which
the cumulative distribution is equal to the random number. This
is the same as finding the inverse cumulative value. Report that
value as the simulated value. Mathematically, this is the process
shown below.
The process is illustrated below for a normal distribution
with mean 10 and standard deviation 3. The figure shows the cumulative
distribution. To simulate an observation, choose a random number.
For the illustration we choose r as 0.73. Set that number
to the value of F(x) on the vertical axis.
Project over to the cumulative distribution and then down to
the xaxis to find the simulated value. For r equal
to 0.73, the value of x is 11.83.
For
a few simple distributions an explicit form of the inverse
cumulative is available, and simulation can be done analytically.
For those that cannot numerical methods are used. TheRandom
Variables addin uses the RV_SIMV(WP1) function. The argument
is the name of a distribution defined on the worksheet by the
addin.
There are many sources for random numbers that
may be used for simulation. Excel has a random number generator.
The link below opens a table of random variables generated by
Excel.


Normal Distribution 

To illustrate the methods of this
lesson, it is helpful to consider the normal distribution. We
use this distribution to describe capital cost estimates. We
provide tables below that are useful for these purposes. The
tables are constructed for Standard
Normal distribution with mean 0 and standard deviation 1.
For the more general normal variant we must transform the results.

For solving problems regarding the normal distribution we
provide tables of the cumulative and inverse cumulative functions
for the Standard Normal. Click on the icons to see the tables.


Standard Normal
Cumulative 




Point Estimate 

Say you have chosen a distribution.
Your boss doesn't really understand this newfangled concept of
probability and he proclaims, "Give me a number. Just give
me a number." He
wants a point estimate, so what should you tell him?
The most familiar, and often the most reasonable
response is the expected value or the mean of
the distribution. Another possibility is the median or
the middle value. Another measure of central tendency is the
mode. The mode is the value with the highest value of
the density function. If you are feeling conservative and want
to estimate a value greater than the middle value you might give
him the kth percentile
with k greater
than 50%. If you think he's looking for a low cost estimate you
might give him an aggressive estimate as the kth percentile
with k less than 50%. The formulas below describe the
mean, median and percentile estimates using probability notation.
We use x' as the value of the point estimate.
We could always simulate a value using the MonteCarlo
method and report that value as the point estimate. Your boss
might find it disconcerting, however, that you keep changing
your estimate every time he asks, but one simulated value is
as good as any other. Perhaps giving your boss a point estimate
that is simulated is not a good policy, but simulation is very
useful for estimating a quantity that involves several random
variables.
For some probability distributions the point estimates
can be determined analytically. The linked document has the available
formulas for the four distributions of this page.
When analytical estimates are inconvenient or impossible,
the numerical procedures of the Random Variables addin
provide all of these point estimates except the mode. 

Summary 



Estimation
with Risk Summary 



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