Statistical Analysis of Project Duration

Worksheet 25. Statistical Analysis of Project Duration

Lower 95% Upper 95% Path

Estimate Estimate

Project Due Date

Expected Project Completion Date

Standard Deviation of Project Completion

95% Lower Confidence Interval (expected

completion date minus two standard deviations) 95% Upper Confidence Interval (expected

completion date plus two standard deviations) Probability of Meeting Due Date

If the probability of not meeting the due date is high, the deadline should be discussed with the project sponsor. In some cases it is necessary to modify the project’s charter, assign additional resources, etc.

Using Simulation to Analyze Project Schedules The analysis conducted above involves a number of simplifying assumptions and

approximations. In most cases, the accuracy of the results will be more than adequate. However, it is possible to conduct a more precise analysis using computer simulation.

The example here uses the program Crystal Ball 22 (CB).

22 Available from Decisioneering, Inc., www.decisioneering.com.

In the earlier example, the calculations assumed that the task durations could be modeled using the beta distribution. If, during the historical analysis phase, we located sufficient data on similar tasks, we would not have needed to make this assumption. Instead, we could use software to determine the best-fit statistical distribution for each task. However, if historical data are not available, we can still use CB to model our project schedule using the duration estimates. In the figure below, data are entered for activity A, assuming that optimistic = 0.1% probability, most likely = 50%, and pessimistic = 99.9%. CB will use a beta distribution with these parameters in the simulation. (A wide variety of other distributions could also be used.) Data for other activities were entered in a similar manner.

Figure 15. Computer Screen for Entering Task Duration Data

Project duration is defined in the spreadsheet as the maximum of the critical or the noncritical path duration. CB runs as many simulations as desired; for the example, 1,000 project schedules were simulated. The results are shown in Figure 16.

Figure 16. Results of Simulation for Example

Y FL AM TE

Team-Fly 64

Discussion of Results As expected, the distribution of project completion times appears to be approximately

normal. The mean and range of results are also quite close to what we obtained from the statistical estimates. However, the simulation allows us to quickly explore a variety of questions. For example, the second chart in Figure 16 shows us that the critical path is not always the critical path! This chart is a histogram of the difference between the critical path and the so-called noncritical path. It can be seen that about 1.5% of the time the difference is negative, indicating that the noncritical path took longer to complete than the critical path. This has obvious project management implications, such as not to ignore tasks that are not on the critical path.

The ease of asking such what-if questions is a major benefit of simulation software. In addition, many people (even some project sponsors!) are unfamiliar with statistics: assumptions, such as the approximate normality of project durations, make them uneasy. For these people, simulation can serve as a valuable confirmation of these assumptions. The truth is, more than a few statisticians breathe a sigh of relief when simulation results match analytical predictions.

The simulation can be used to answer other questions as well. For example, the probability of the project being completed by its 34-day due date is shown in Figure 17. The simulation prediction of 95.9% probability of success is very close to the 96.6% predicted by the analytical approach used earlier.

Figure 17. Simulation Results: Probability of Meeting Due Date

Due date duration

Calculating the Cost of a Schedule The cost of following a particular schedule should be evaluated carefully. It often

happens that a cost savings can be achieved by using a schedule other than the schedule based on the most likely or weighted average duration estimates. As activity durations are compressed, the time it takes to complete the project will decline, while the direct costs of completing the project will increase. Conversely, indirect costs such as overhead generally decrease when projects take less time to complete. When the indirect costs are added to the direct costs, total costs will tend to decline to a minimum for a particular schedule, which we will call a cost-optimized schedule.

Identifying a cost-optimized schedule involves these steps: 23

1. Ask those assigned to each activity to estimate the direct and indirect costs of completing their activity in the duration of the optimistic, most likely, and pessimistic estimates.

2. Create a spreadsheet of the above cost and time estimates.

3. Compute the total cost of the schedule, including direct and indirect costs.

4. Create a column showing the cost per unit of time saved for each activity. E.g., if an activity can be completed in four weeks at a cost of $2000 or in two weeks at a cost of $4000, then the cost per week saved is $1000.

5. Rank-order the activities in ascending order by cost per unit of time saved, i.e., put the lowest cost per unit of time saved at the top of the list and the highest cost per unit of time saved at the bottom of the list.

6. Assuming that critical path activities with the lowest cost per unit of time saved were completed in the optimistic duration,

a. Recalculate the schedule duration.

b. Recalculate the cost of the schedule.

7. If the cost of the new schedule is lower than that of the previous schedule,

a. Recalculate the critical path for the new schedule.

b. Return to step 5. Else the cost of the new schedule is higher than or equal to that of the previous