Engineering Finance


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.

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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 add-in to perform the analysis with both the traditional and simulation methods.
9.11 Dealing with Uncertainty
Probability Distributions

Our examples model uncertainty with the Random Variables add-in, which permits the use of most named probability distributions. Functions provided by the add-in compute moments, such as the mean and variance, compute probabilities and inverse probabilities, and perform Monte-Carlo simulation. Many of these funcitons are used by the analysis on this page.

Random Variables Add-in

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.

Triangular Distribution

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.

Beta Distribution

Other Distributions

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

Other Distributions

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 add-in.

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
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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 add-in. The linked page shows how to use the add-in for the analysis of uncertainty concerning activity times.

. Oil Collection Problem
Project Management Add-in
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 add-in. The probability that the project will be completed in 14 days is 0.968. The 60th percentile estimate is 12.274.

If the add-in 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 add-in. 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 three-point estimate.

Three-Point Estimates

To perform simulation, the Random Variables add-in 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 top-left 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 Add-in

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

. Oil Collection Problem
Project Management Add-in Simulation
. Browser Page
Uncertainty Summary

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Engineering Finance
by Paul A. Jensen
Copyright 2005 - All rights reserved

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