If all the tasks were strung out in series and just added together, the errors would tend to "average out." We might be safe in assuming that errors in estimating would be high and low and would cancel each other out. If only real projects were that convenient.
Project activities aren't usually strung out like a string of pearls. They look more like Figure 1, one of the classic project plan segments used in descriptions of PERT.
Figure 1. Classic PERT segment
We'll play with this to get a sense of the difference between PERT predictions and Monte Carlo simulation .
There are six activities in the plan, each with an optimistic, most-likely, and pessimistic estimate. Figure 2 shows a spreadsheet used to estimate the results using the PERT formula. The yellow cells are the input and the green cells are the output. The three paths (C, E, F) to the final milestone are each additive, but the finish date is the maximum of those three paths, in this case 221 days.
Figure 2. PERT Estimate
By way of comparison, Figure 3 is the histogram and cumulative curve we get from the same inputs converted into shapes and run through a 400-trial simulation.
Figure 3. Monte Carlo Simulation
To do this right, the shape associated with each task would be derived from historical data, expert estimations, or both. For this exercise, however, I wanted to keep the same data as was used for the PERT calculations. I used the three numbers as the min, mode, and max of a triangular distribution with 400 samples. This is fairly coarse and a good way to signal that we don't know much about the actual shape. It also tends to give too much weight to values close to the mode, so the resulting confidence intervals will be tighter than they would likely be with real data. In any case, the triangular shape has a firm minimum and maximum, which is an advantage over other distributions that might be used here.
The first thing Figure 3 tells us is that the 90% confidence interval runs from about 215 days up to about 255 days. The next thing is that the 221-day PERT finish is at the 25% cumulative probability mark; the odds are 3:1 against meeting that estimate. Even money, where we're as likely to be early as late, is at 230 days.
A lot depends on the client's risk tolerance, but if there's a need to synchronize with other activities, it's likely that we're going to plan for 250 days, six weeks later than the PERT estimate.
This example involves a very small project. If we apply this to a large project, particularly one with a lot of parallel paths, the PERT errors get very interesting.
If you'd like to play with this, there's an Excel spreadsheet at http://smpro.ca/crunch/pertsim.xls
The chart is interactive, so if you change the inputs (in yellow) and press F9, you'll see the data tables and chart change. It may be slow, so wait for it. If you find an input combination that makes the PERT calculation more pessimistic than the Monte Carlo, please let me know because, so far, I don't think it's possible.