There are many kinds of uncertainty, ranging from "I have no idea" to "I can estimate probabilities about uncertain quantities." We use the latter case, which is sometimes called risk. We assume that although activity durations cannot be known with certainty, we can provide probability distributions that indicate the likelihood of various values. For a discussion of probability see the article in the ORMM/Supplements/Models/Probability section. The PDF document about Continuous Probability Distributions is particularly relevant.

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. If times, resource use and costs are recorded, the project manager will know how the project is progressing. At times after the project has begun but before it is complete, the manager may take action to change all features of the model that are not yet completed, make 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 first critical path. Estimates made before the project begins must surely be adjusted to account for the reality of results. The ability to make adjustments after the project has begun partially helps with the problem with uncertainty. Plans can change. Decisions may shift as the future is revealed by the passage of time.

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, all of which can affect activity duration and cannot be accurately assessed. Projects involving creative effort such as software development depend on the intellect and insights of the participants, certainly difficult to predict. The remainder of this page and the next will consider the effects of uncertainty using continuous probability distributions. The treatment of the times as random variables is an explicit recognition of uncertainty. The CPM model, using only one estimate, completely neglects the effect. PERT with three estimates, recognizes uncertainty, but does not prescribe probability distributions. This makes it impossible to use the results of probability theory except in an approximate way or to use simulation as part of the analysis. Simulation is considered on a later page.

The triangular and Beta distributions are the most often used in project management studies because both can be characterized by the minimum, most likely, and maximum parameters. As we will see later, other distributions are also available.

The triangular distribution is used in our first example. An illustration of the triangular distribution is below. The example has a minimum of 0, a most likely value (or mode) of 0.3 and a maximum of 1. The distribution is shown by the heavy line and the cumulative distribution is shown by the dotted line. Triangular distributions with other parameters are shifted and/or spread out. The area under the distribution curve is always 1. The cumulative distribution shows the probability that the random variable is less than the value given on the horizontal axis. For the example, the probability that the random variable is less than 0.4 is about 0.5. In general this is called the median. The mean of this example is 0.433. If this were an activity time distribution, project management analysis will use the mean value of 0.433 as the single time estimate for the activity time. The variance is also used for the analysis, and for the example pictured, the variance is 0.0439. The mean and variance values are computed by the Random Variables add-in. The figure below was constructed by the add-in.

To create a model with random activity times click the Distribution button in the Activity Times frame of the definition dialog. Two distributions are available in the Distribution frame, Triangular and Beta. We use the Pump example considered earlier, but now the activity times will have triangular distributions.

The worksheet model is constructed as below. We have hidden some columns to simplify the explanation. We provide only two columns for predecessors and use no resources for this example. The model has already been solved and the critical activities are colored red in column B. The critical predecessor relations are colored green in columns E and F. The mean values computed for the triangular distribution are used in the Time column (N). The only apparent difference in this worksheet and the worksheet for the traditional model is the column containing the letters "Tri". These letters identify the triangular distribution. The letters "Fix" specify a fixed or constant distribution for the start and end activities.

It is interesting to compare the results with the triangular distribution with the results using the traditional PERT formulas for computing mean and variance. We show below the columns with mean and variance. The distribution parameters are as above. The triangular distribution results are on the left and the traditional results are on the right. The mean and variance values for the triangular distribution are computed by functions provided by the Random Variables add-in.

As expected, the mean, variance and standard deviation values for each activity are different because the traditional analysis uses empirical formulas for computing these quantities and the Random Variables add-in computes the accurate values for the triangular distribution. Since, the activity times used for the analysis are the mean values, the results shown at the top of the page are different for the two cases. In both cases we choose the due date of 59. The critical path, which is the same for both cases, has an expected value of 57 for the traditional method and the project has a slack of 2. For the triangular distribution, the critical path has an expected value of 61 and the project is late by 2. The variance and standard deviation values are also different leading to a different estimate of the standard deviation for the project. Of course the probabilities of completing the project by the due date are quite different.

Which analysis is more accurate? Probably neither is accurate. The estimates of the parameters are usually the biggest source of error and the assumption about probability distributions less important. Although the traditional analysis is unclear about the distribution, there is really no reason to believe the triangular distribution represents reality. The example does point out that the numerical results such as the expected value of the critical path and the probability of meeting the due date, are sensitive to distribution assumptions and are thus suspect regarding validity. The principal value of using a known probability distribution is that Monte-Carlo simulation is then possible. Analysis with simulation may lead to useful results as described later.

Another distribution often used for project analysis is the Beta distribution described on the next page.

Time Estimates

To this point an activity time has been estimated using the mean value of the probability distribution. We see the figure below the columns representing the distribution parameters in columns J, K and L. The distribution means, standard deviations in columns are computed in columns M, N and O. The estimates of the activity times in column P are the same as the distribution means. Cells Q3 computes the sum of the means for the activities on the critical path. Cell Q4 computes the sum of the variances for the activities on the critical path, and Q5 is the square root of that value. Assuming the normal distribution for activity times, cell Q6 computes the probability will be completed before the due date.

The times in column P are point estimates of the activity times. They are important because they are used for all the other analyses on the project worksheets. We have used the mean for the point estimates, but other point estimates are possible. Cell J8 holds the identity of the point estimate, Mean, for the example.

To change the point estimate, click the Change button at the top of the page. The change dialog shows the two other options, Simulate and Percentile. We choose the latter first and indicate a 60% as the percentile level.

The third kind of estimate is the simulated estimate. The percentile has no meaning in this case.