Uncertainty
Uncertainty is an inherent aspect of project
management. When the model is constructed and the critical path
is determined, all activities are future events. Inputs to the
project management analysis include for each activity estimates
of the time duration, estimates of resource usage and estimates
of costs. Of course, there may also be uncertainty with regard
to the activities that comprise the project and the precedence
relations among them. In many cases everything about the
project is to some extent uncertain, but in this lesson we consider
only the uncertainty of activity durations.
There are many human and political factors that
bring to question the time estimates provided by persons associated
with a project. We will not discuss them here, but there is a
good discussion in the book Critical Chain by
Eliyahi Goldratt (North River Press, 1997). Putting the human
and political issues aside, the duration of something that has
yet to occur is at best a random variable. It cannot be known
with certainty. Construction projects depend on weather, labor,
and material availabilities. Projects involving creative effort,
such as software development, depend on the intellect and insights
of the participants  something difficult to factor into our estimates.
In a project environment, all uncertainty will
eventually be revealed, assuming that the project is actually
finished. The revelation occurs as time passes and as activities
are started, performed and finished. At times after the project
has begun but before it is complete, the manager may take action
to change many features of the model that are not yet completed,
perform new analyses, and take corrective action where appropriate.
Project management is far more than determining the precedence
relations, estimating resources and costs, and computing the
critical path. Estimates made before the project begins must
surely be adjusted to account for the realization of results. The
ability to make adjustments after the project has begun partially
helps with the difficulties associated with uncertainty. Plans can change. Decisions
may shift as the future is revealed by the passage of time.
This lesson models the effects of uncertainty using
continuous probability distributions. The treatment of activity
times as random variables is an explicit recognition of uncertainty.
Using several approximations with regard to the critical
path, the normal distribution is used to provide
probability estimates about the duration of the
project. Simulation is used when the normality
assumption is not appropriate. 

Goals 


For a project is given by a set of activities, precedence
relations, and activity times, our goals are as follows.

Describe the project using random variables
for activity times.

Find the critical path using mean values.

Compute the mean and standard deviation
of the project duration with the traditional approach.

Assuming normality, make probability statements regarding
the project duration.

Estimate the activity times using Monte Carlo simulation.

Interpret simulation results.

Use the Project Management addin to
perform the analysis with both the traditional and simulation methods.




Text 



9.11
Dealing with Uncertainty 




Probability Distributions 

Our examples model uncertainty with the Random
Variables addin, which permits the use of most named probability distributions.
Functions provided by the addin compute moments, such as the
mean and variance, compute probabilities and inverse probabilities,
and perform MonteCarlo simulation. Many of these funcitons are used
by the analysis on this page.
The traditional PERT method of project management recognizes
uncertainty by requiring three estimates of activity durations.
We describe this below along with the Triangular and Beta probability
distributions that are most often used in project management
studies.
Traditional PERT 

A traditional
way to get subjective estimates from an expert is to ask:
What is the smallest you think the time can be? What is
the most likely value? What is the largest value? The
most likely value for a distribution is called the mode.
PERT reflects uncertainty by allowing three estimates:
the minimum time, a, the most likely (or mode)
time, m,
and the maximum time, b. Based on these parameters,
empirical formulas approximate the mean and
variance of the random activity duration. Although, the
literature frequently refers to the distribution as a Beta
distribution, the moment expressions do not represent a
particular Beta distribution. The traditional method is
convenient for hand or spreadsheet calculations.

Triangular Estimate 

The triangular estimate uses the
same three parameters: minimum
time, a, most likely time, m, and
maximum time, b, but here we use the parameters
to define a triangular probability distribution. The mean
and variance can be calculated using formulas in the linked
document.

Beta Estimate 

The Beta distribution is defined
by its lower and upper bounds, as well as two parameters
defining its shape. A variety of shapes can be specified.
The mode is a function of the parameters.

Other Distributions 

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

For purposes of illustration, we assign triangular
distributions to the activity times. For convenience we choose
a symmetric distribution with the mode values the same as the oil collection
example given in the critical path lesson. When m is the
mode for the activity time, the lower bound is 0.5m and
the upper bound is 1.5m. The table below
shows the distribution parameters and also the mean, standard
deviation and variance values. The table was created by the Project
Management addin.
The column at the far right of the table gives
point estimates of the activities times. We use the mean value
for the example. Because the triangular distributions are all symmetric,
the mean values are the same as the modes.
Throughout this page we assume that the times for
the activities are independent random variables. 

Distribution of the Project Duration 

Our goal is to find the probability
distribution for the project completion time so we can answer
questions like: What is the probability that the project
will be completed by the due date? We will see that this
is not generally an easy task, but with a few assumptions approximate
answers can be obtained. The movie provides a summary of the
steps necessary for answering probability questions.


Uncertainty
in Activity Times
No Audio



When the activity times are random variables, the
project duration is also a random variable. A path
through the project network is a set of activities to be performed
in sequence. When we assume that as soon as one activity on the
path ends, the next one begins, the length of the path is
the total time to complete the activities on the path. Assuming
the times are independent random variables, the mean and variance
of the path duration is found by summing the means and variances
of the activities on the path.
The form of the path distribution depends on the
distributions assumed for the activity times. If they are all
normal distributions, the path duration is also normal. If they
are not normal, not much can be said about the path duration
distribution. In the following
we sometimes assume normality for the path duration so we can
make probability statements. Of course, the approximation is best
when the activity times come close to satisfying the conditions
for the central limit theorem. When normality is assumed, the
standard normal is available for probability statements. The
normal approximation also allows other estimations such as percentile
values and coverage intervals as discussed in earlier lessons.
So for any selected path from start to end we can
approximate the distribution of its duration, but what path
will govern the completion time of the project? One might suppose
that it is the critical path. When the activity times are
random variables, however, there is no single path through the
network that is guaranteed to be the critical one. Under
these conditions it is very difficult to find the exact distribution
of the project duration except for very simple networks.
In the following we take two approaches. The traditional
approach uses the critical path based on the mean values of the
activity times. The simulation approach does not make this assumption,
but through many simulated networks can make statistical estimates
of the distribution parameters. Both are implemented in the Project
Management addin. The linked page shows how to use the
addin for the analysis of uncertainty concerning activity times.


Project
Management Addin 




Traditional Approach 

This approach uses the critical
path method of Lesson 21 to find a single path and bases the
project duration distribution parameters on that path.
Use the mean values for the activity times. 
Find the critical path using the mean values. The path
is identified by the critical activities. 
Find the mean duration by summing
the activity times on the critical path. 
Find the variance of the duration by
summing the activity variances on the critical path. The
standard deviation is the square root of the variance. 
Assume the duration has a normal distribution. Make probability
statements regarding the duration using the cumulative standard
normal distribution. 
Applied to the example, the method considers only the critical
path determined by the mean values. The mean and standard deviation
are reported at the bottom of the figure.
The assumption of normality yields probability results. The
figure below shows the results obtained from the Random Variables addin.
The probability that the project will be completed in 14 days
is 0.968. The 60th percentile estimate is 12.274.
If the addin is not available, normal tables can be used in its place. These are available through the toolbox.
The figure below shows the example solved by the Project
Management addin.
The results at the top of the page in column R show the results
for the traditional method. We have chosen 14 as the due date.
The row labeled Critical holds the mean value
for the critical time with the variance and standard deviation
immediately below. The On Time Probability is the probability
that the critical time is less than 14. It is computed
using the normal distribution.
Click the icon below to see a series of tables that
show the solution of this example using the traditional threepoint
estimate.


Simulation 

To perform simulation, the Random
Variables addin must be installed, but it is
not necessary to interact with it. In
the figure below, the
point estimates of activity times (column Q) are replaced
by simulated values. The Change button accomplishes
this. Of course, when the times are simulated every observation
yields a different project duration, but as illustrated, this observation
results in a different critical path.
This figure below shows the critical path for the
simulated data (in green) along with the critical path based
on the mean values of the distribution (in red).
Clicking the Simulate button at the topleft
of the worksheet performs a series of simulation runs based on the logic in the box below. The program
accumulates statistics on the project duration and the number
of times each activity appears on the critical path. The activity
counts start at zero.
Do the following NSIM (the number of simulated observations)
times. 
 Simulate the activity times using the Monte Carlo method.
 Find the critical path. Compute the duration for the
path and store it in an array of simulated values.
 Observe the activities in the critical path and increase
the count for each critical activity.

Compute the mean, variance and standard deviation of the
simulated durations. 
Compute the proportion of the observations when each activity
was identified as critical. 
Report the results. 
When probability statements are desired, assume that the duration
has a normal distribution with the mean and standard deviation
determined by the simulation. Make probability statements
regarding the duration using the cumulative standard normal
distribution. 
The results of a simulation of 1000 observations
are reported below. The completion time is somewhat greater and
the variance is somewhat less than the amounts predicted from
the mean value critical path. This is not unexpected because
the critical path is the longest path through the network.
The important information here is that some activities,
not reported as critical in the mean value analysis, do have
a chance of becoming critical. Based on the simulation, the activities
on the critical path identified using mean values, all appeared
in at least 90% of the simulated critical paths. Activities I
and K are on 10% of the critical paths, while activities B,
E and N are on 1%. Of course, the results
of the simulation are also random, so a second simulation
would provide different percentages, but these results give probability
estimates. 

Project Management
Addin 

The Project Management addin performs
all the steps required for simulation automatically. The
following link opens the relevant page on the ORMM site.


Project
Management Addin Simulation 




Summary 


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