Project Simulation

In this section we can take a different approach to managing risk. Specifically, we build on the probabilistic foundation established in the previous section and use simulation to handle the arithmetic as well as help us to understand the consequences of uncertainty. Simulation analysis can provide insight into the range and distribution of project com­pletion times. To illustrate this, once again we use the project activities listed in Table 5-4 and the corresponding network diagram shown in Figure 5-14.

For the purpose of this example, we will make one simplifying assumption. As in Chapter 4, we will assume that the activity times may be reasonably represented by a beta distribution. The spreadsheet shown in Table 5-6 was created to simulate the project.

Having opened Excel^® and Crystal Ball^® (CB), we devoted a column to each activity (i.e., columns A through H) and these will be our assumption variables. We then devoted one column (columns I through L) to each path through the network. Finally, we calculate the network Completion Time in column M by taking the maximum value of columns I—L, and this cell becomes our forecast cell. This gives us a very simple model for a complex problem. The hardest part of constructing this model is to identify all paths through the network. On small networks like our example, this is easy. On large networks it may be quite difficult.[1] Using an MSP PERT network to identify paths can be very helpful.

Reminding the reader that it is easier to follow our instructions if you have the soft­ware running, we can now enter data into what becomes Table 5-6.

1. First enter the proper column labels in the first two rows of the spreadsheet, and then enter the most likely time for activity a (from Table 5-4) in cell A3—10 days. CB won’t let you define an assumption cell for an empty cell.
2. Click on Define Assumption in the CB ribbon.

3. Click on BetaPERT in the Distribution Gallery and then the OK button.
4. In the BetaPERT Distribution Minimum and Maximum text boxes enter the pessi­mistic and optimistic times. Note that when you entered the most likely time in step 1, that time already appeared in the BetaPERT Distribution Likeliest text box when it was displayed. Note also that the distribution box assumed pessimistic and optimis­tic times. Simply click on those boxes and change the times to the ones shown in Table 5-4. When you have done this, click Enter and then OK.
5. Enter the rest of the activity times. Note that activity time for d is a constant. Simply enter “6” in cell D3 and do not use any distribution for the cell.
6. We trust you found the four paths through the network. Identify each path as shown in Figure 5-14 or Figure 5-15, and enter the path duration formulas that simply sum the activity times for each path. (Caution: Recall that the Excel^® copy/paste com­mand will not work, though may appear to do so. Enter the data manually for best results or use the CB copy/paste command.)
7. Now enter the project duration formula in cell M3, that is, = MAX(I3:L3). Click on the cell and then on Define Forecast in the CB ribbon. Label the forecast textbox “Project Completion Time” or whatever you fancy. The Units are “Days.” Click OK. Save the spreadsheet you have created before proceeding with the next step. We will use this spreadsheet again in Chapter 6.
8. If the “Trials” box is not set to 1000, adjust it and then click on the “Start” green arrow. After the run, you will see a statistical distribution similar to Figure 5-17. To view summary statistics, click on the “View” menu option in the Forecast dialog box (see Figure 5-17) and the click on “Statistics” to see a table similar to Table 5-7.

Table 5-7 shows that we made 1000 trials and that the mean project completion time was 47.83 days (in the run made for this writing), the shortest completion time was 41.02 days, and the longest was 55.59 days. Recall that the mean completion time with the analytic approach using the beta distribution was 47 days. The different assumptions about the distribution of the activity times may account for some of the difference between the two estimates of completion time, but some of the difference is also due to randomness. As usual, you can determine the probability that the actual result of the project will have a duration above or below any given time by entering those times in the plus or minus Infinity boxes of the distribution of times; see Figure 5-17.

1. Incorporating Costs into the Simulation Analysis

In the previous chapter we addressed the topic of budgeting. As you can likely imagine, the cost to complete a project is driven in large part by the time required to complete the project. The longer it takes to complete a task the more resources that are required which in turn leads to greater costs. Furthermore, in a similar fashion to activity durations being uncertain, the cost to complete an activity can be uncertain. For example, the cost to complete an activity may depend on which resources as well as how many resources are assigned to it. Therefore, it stands to reason that we can get a better understanding of the uncertainty surrounding the project’s budget by incorporating both schedule and cost Risk uncertainty into our analysis. This is perhaps best illustrated with an example.

For the purpose of our discussion, we extend the analysis conducted in the previous example. More specifically, we supplement the information provided in Table 5-4, with the cost information shown in Table 5-8. As shown in Table 5-8, optimistic, most likely, and pessimistic cost rates have been estimated for each activity. These cost rates are multiplied by the activity duration to determine the cost of the activity. For example, activity A’s most likely duration is 10 days (see Table 5-4). At a normal cost rate then, we expect activity A to cost $750 (10 day x $75/day).

Based on the cost estimates provided in Table 5-8, the simulation model shown in Table 5-6 can be enhanced to develop a distribution of potential project costs in addition to the distribution of project completion times as shown in Table 5-9. To develop the enhanced model shown in Table 5-9, the same steps discussed earlier were used to define new Assumption Cells and a new Forecast Cell. More specifically, Assumption Cells for the cost rate of each activity were added in row 7. For the purpose of this example, the cost rates were assumed to follow a triangular distribution. Next, formulas to calculate the cost of each activity were added in row 11. Again, the cost of completing an activity was computed by multiplying its duration in row 3 by its cost rate in row 7. Note that in our enhanced model, we have two sources of uncertainty, namely, the activity durations and cost rates for the activities (except for Activity D, the duration of which is known with certainty). To complete the simulation model, a formula that calculates the cost of the project was added as the sum of activity costs A through H in Cell 111, and then this cell was specified to be a Forecast Cell.

The distribution of the total cost of the project based on simulating the project 1,000 times is shown in Figure 5-18. Likewise, summary statistics for the total cost of the pro­ject are shown in Table 5-10. As can be observed from Table 5-10, the analysis suggests that the project is expected to cost $3,606.76 with a range of $2,646.95 to $4,338.51.

2. Traditional Statistics versus Simulation

At many conferences and in the literature of project management, there are sessions devoted to risk management. The participants at one recent conference, all highly expe- Risk rienced PMs from several different industrial sectors, were not of a single mind about the optimum way to handle the problems we have discussed in this chapter. Some favored the traditional statistical approach, some favored simulation, and some favored adding sizable contingency allowances in both the budget and the schedule, and then ignoring uncertainty. A decade ago, the group that ignored uncertainty would have been in the majority, but they were the smallest of the three groups at this discussion. Dealing with risk is no longer the esoteric interest of a few statisticians, but as we discuss throughout this book, one of the primary roles of the PM.

The major difficulty is making sure that the risk analyst understands risk analysis. Irrespective of how the arithmetic is performed, by human or machine, the analyst must understand the nature of the calculations, what they mean and what they do not mean.

Conducting a risk analysis by examining the many paths through a large network, and finding those that may turn out to be critical or near-critical, can be an overwhelming task. There may be hundreds or thousands of activities and dozens of paths to be exam­ined. Simply identifying the paths is a daunting job.

Some Common Issues There are a number of issues associated with either the sta­tistical approach or simulation. With an important exception we note just below, both procedures assume that task durations are statistically independent. As we noted above, this might not be the case when resources are shared across the tasks, but the problem can be handled by re-estimating task durations based on the altered resource. Second, with the same exception, both procedures assume that the paths are independent.[2] Even when paths have tasks in common, the common task durations (and variances) are the same for every path in which they are an element during any given estimate of path dura­tion by whatever procedure. It is as if a constant is being added to each path.

The exception we mention is that simulation has a direct way of circumventing the assumption of statistical independence if the assumption is not realistic. With simula­tion, one simply includes the activity or path dependencies as a part of the model. The dependencies are modeled by expressing the functional relationship between the activi­ties or paths along with its distribution, mean, and variance so that it may become a vari­able in the simulation. While this can be done in the statistical approach also, it is much more difficult to handle.

Using the statistical process, the analyst must find the T_E and variance for each path. Using simulation, the computer selects a sample from the distribution of activity times for each activity and then calculates the path duration for each path enumerated.

On the other hand, no matter which method is used, it is rarely necessary to evaluate every path carefully. In a large network, many paths will have both short duration and low variance when compared to high duration paths. Even when it is technically possible for one of the short paths to be critical, it is often very unlikely. For example, consider the path a-b-d-f in our example. It shares activities a, b, and d with a-b-d-g-h, the path with the longest T_E. Activity f has a pessimistic time of 10 days. Activities g and h have optimistic times of 5 + 4 = 9 days. What is the probability that f will take on its maxi­mum value at the same time that both g and h take on their minimum values so that a-b-d-f will be longer than a-b-d-g-h, not to mention simultaneously longer than the remaining two paths? Given that these estimates were made at the ±3o limit, the prob­ability of g or h being at or below their optimistic estimates is (1 — .9987) = .0013 for each. It is the same for f being at or above its pessimistic estimate. These three things must happen at the same time for a-b-d-f to be longer than a-b-d-g-h. The probability is .0013³ = .000000002. That probability is not zero, but most PMs will not spend time and effort worrying about it. Even when estimates of the optimistic and pessimistic times, a and b, are made at the 90 percent level, the chance that the activities on these particu­lar paths will simultaneously take on their high or low extreme values is about one in a thousand.

The PM will discover the duration of each activity when, and only when, the activ­ity is completed, which is to say after the fact, regardless of the method used to estimate and calculate project duration. Dealing with activity duration as certain does not make it so. We cannot know which of the paths will take the longest time to complete until the project is actually completed. And because we cannot determine before the start of the project which path will be critical, we cannot determine how much slack the other paths will have. We can, however, often make reasonable estimates. We can put our managerial efforts on the activities and paths most likely to require our efforts.

Simulation Because of the computational effort required by the statistical method, we recommend simulation as the preferred tool for risk analysis—after, and only after, the analyst understands the underlying theory of the analysis.

We have included Crystal Ball® with this book because it is user-friendly software that can be used to facilitate and enhance the simulation of the project networks as well as simulate a wide variety of other types of problems. In addition to the uses we have demonstrated, CB can also facilitate the task of selecting the best distributions to be used to model alternative activities if historical data are available for these activities. Likewise, CB can determine the best distribution to use to characterize project comple­tion times and other outputs of the simulation analysis. Furthermore, it has a compre­hensive set of probability distributions available and the selection of these distributions is facilitated by graphically showing the analyst the shape of the distribution based on the parameters specified. This capability allows the user to interact with the software when specifying the parameters of a distribution. The analyst can immediately assess the impact that alternative parameter settings have on the shape of the distribution. Another powerful feature of Crystal Ball® is its ability to quickly calculate the probabil­ity associated with various outcomes such as the probability that the project can be completed by a specified time. In addition, CB can display the results in a variety of formats including frequency charts, cumulative frequency charts, and reverse cumula­tive frequency charts. It also provides all relevant descriptive statistics as was illus­trated earlier.

Using the sample problem, risk analysis is carried out by a simulation using Crystal Ball®. Each step in the process is described. Conclusions similar to those reached in the statistical procedure of Section 5.2 are reached through simulation. The two pro­cedures are compared by examining the assumptions on which they are based as well as the problems encountered in using them. The computational effort and assump­tions required by the traditional statistical approach lead us to the conclusion that simulation is the preferred technique for carrying out risk analysis.

Source: Meredith Jack R., Mantel Jr. Samuel J., Shafer Scott M., Sutton Margaret M. (2017), Project Management in Practice, John Wiley & Sons, Inc. 3th Edition.