# What is the critical path method?

The critical path method or CPM is a management tool that helps the project manager recognize where in the project schedule his management effort should be applied. The critical path method recognizes that the activities in the schedule that have zero float are the activities that cannot be delayed without delaying the completion of the project. These activities are called critical activities and should be identified as activities that require close supervision.

By this definition of critical activities, all the other activities must have some float. This means that those activities containing at least some float can be delayed without affecting the completion of the project and can be managed somewhat less closely than the critical path activities.

**Tell me more . . .**

The critical path method was developed by the Dupont Corporation in the 1950s at about the same time General Dynamics and the U.S. Navy were developing the program evaluation and review technique, which will be discussed next. The objective of CPM is that it can take projects that have reasonably well known task durations and identify the tasks that should be monitored more closely by management. CPM is therefore a management technique.

The critical path is really somewhat misleading since the critical activities do not necessarily have to fall on a path. The idea of a path comes from a more simple-minded approach to the critical path method where the critical path is determined by finding the longest path through the project network diagram, shown in Figure 5-7.

By using this technique you can find all of the possible ways of getting from the start end to the finish end of the network following the various branches of the interdependencies. This will give you every possible sequence of activities from the start of the project to the finish. When all the paths have been determined, you add up the durations for each possible path. When this is done, you declare that the critical path is the one with the highest total duration.

Obviously this will not simply produce the right answer if you have any kind of relationship in the network diagram other than a finish-start relationship or if you have leads and lags.

The proper method and the method that the computer software for project management uses is to determine the total float of each activity and identify those activities that have zero total float. These activities

Figure 5-7: **Network diagram with critical path** may sometimes not form a path through the network. They may even form two paths through the network.

To manage using the critical path, we use the float in each activity as the criterion for managing. Activities that have little or no float are managed much more closely than activities having more float. Since the activities having little or no float will cause project completion delays if they are late, they are obviously the ones upon which we must concentrate our management effort.

Of course, since this is a management tool and we are using it to determine where management effort should be concentrated, we should also consider the activities that have close to zero float since they are nearly critical. In the project management software, the number of days of free float allowed before an activity becomes critical is adjustable so that we can show activities with, say, three days or less of free float. To make this even easier, the software will highlight these activities by showing them in red (or a color of your choice).

# What is the PERT method?

The PERT method stands for program evaluation and review technique. It is a statistical approach to project schedules. Actually, it is a statistical way of predicting project completions when there is uncertainty about the project durations.

The PERT method was developed during the Polaris Missile program in the United States in the 1950s. At that time the United States was in the middle of the Cold War and had come up with the idea that ballistic nuclear missiles could be fired under water from a submarine. Of course this was a tremendous advantage in a nuclear war because the submarine could approach the coastline of the Soviet Union and fire the missiles before being detected. It seems no one told the submarine commander that as soon as the rocket took off, the submarine would be spotted and probably blown up. But that takes us a bit off the subject of PERT.

The difficulty for the U.S. Navy and General Dynamics was that they had two separate projects, the missile development project and the submarine development project. Because of the intensity of the Cold War, it would have been difficult to explain to Congress that the missile was ready for deployment when the submarine was not or that the submarine was ready to go on patrol but had no missiles. PERT was created to take project task durations that were uncertain and statistically estimate the amount of time that they would be expected to take and do that with a determined probability and range of values.

Each activity in a PERT analysis must have three different durations estimated for it. These are the optimistic, the pessimistic, and the most likely duration. The activity's expected duration and the activity's standard deviation are calculated from these three values by the following formulas:

**Expected Duration = (Optimistic + 4 X Most Likely + Pessimistic) / 6 Expected Standard Deviation = (Pessimistic — Optimistic) / 6**

These two values, the expected duration and the expected standard deviation, are approximations that allow us to predict the project completion date and a range of values that will give us the probability that the actual project will be completed within the range of values.

For example, if we predict that the expected value for the project completion is January 10 and that the expected standard deviation is four days, we could say that we have a 95 percent probability that the project will be completed between January 2 and January 18.

**Tell me more . . .**

The PERT method is used when there is uncertainty in the duration of the activities in a project.

Figure 5-8 shows what would be expected if you were to plot the probability distribution of the expected dates for completing a project. On the left side of the diagram, we have the optimistic completion date for the project. On the right side, we have the pessimistic completion date for the project. The optimistic and pessimistic dates for the project completion are the earliest and latest dates that are reasonable for the

Figure 5-8: **Skewed probability distribution**

project's completion; they should not be dates that are impossibly early or impossibly late.

Notice that the curve of the probability distribution is skewed to the right. This is because it is increasingly unusual for the project to be done earlier and earlier. As most of us have experienced, when things begin to go wrong in terms of project lateness, it seems that they get worse and worse or later and later. So, in this characteristic plot of projects we see more dates later than the most likely date and fewer dates earlier than the most likely date. This causes the PERT weighted average date to shift to a position later than the most likely date. The PERT weighted average is not the most likely date for finishing the project. It is shifted somewhat because the probability distribution is not symmetrical.

Figure 5-9 shows the range of values that is plus or minus two standard deviations from the PERT weighted average value. The standard deviation is always a positive number and is the distance from the expected value. In the case of PERT, it is the distance measured from the PERT weighted average.

The standard deviation of the project completion is the sum of the standard deviations of the durations of the activities that make up the critical path. Since the durations of the activities on the critical path are the only ones that should go into the total that is the project duration, only the critical path items' standard deviations should be used to determine the standard deviation of the project completion.

Figure 5-9: **Skewed probability distribution**

To the scheduling example we had been working with earlier, we now add estimated values for optimistic, pessimistic, and most likely. From these we calculate the expected value and the standard deviation (see Figure 5-10).

When we add up the standard deviations of the activities to get the standard deviation of the project, we must first square each one of the standard deviations of each activity, add them up, and then take the square root of the total.

In the example shown in Figure 5-11, the expected value for each activity is calculated by taking four times the most likely value of the duration for each activity and adding this to the optimistic and pessimistic values. The total of the three values is then divided by six. To calculate the standard deviation for each activity, we simply subtract the optimistic duration from the pessimistic duration and divide by six. Strictly speaking we should always take the absolute value of this, but since it is quite unusual for the pessimistic date to be earlier than the optimistic date, it is really not necessary.

To get the project total standard deviation, we add the square of the standard deviation of activities 1, 2, 4, 5, and 8 and then take the square root of the total. Since we are generally interested in a probability of 95 percent, we add and subtract two standard deviations to the expected value of the project duration to get the range of values for project duration that will give us a 95 percent probability of predicting the actual project completion date.

Figure 5-10: **PERT example**

Figure 5-11: **Example: PERT calculations**

Actually, in most projects we are interested in knowing whether the project will finish late or not, and we are not as concerned about whether the project will finish early. In other words we are interested in the project's finishing earlier than the date that is the PERT weighted average plus two standard deviations, and we are not concerned about the date that is the PERT weighted average minus two standard deviations. This raises the probability of the prediction to about 97 percent instead of 95 percent.

One of the problems that many people avoid talking about in PERT analysis is the problem of what happens when the critical path changes during the actual project. When the project is actually done, the tasks will only have one duration each, the actual duration. Since the duration of the task can be any value in the possible range of values for that task, the project could have a combination of durations that would cause the critical path to be different than originally predicted by the expected value durations for each task.

What we mean by this is that when the critical path is calculated in PERT analysis, there is only one critical path and that is determined by using only the expected value of the duration for each activity. Once these durations are found, the critical path is determined, and the project's expected duration is calculated by adding the expected durations of the activities on the critical path. The durations of all the other activities in the project are not added because they are being done in parallel with the critical path activities. The expected value of the duration of the project is then adjusted by adding and subtracting two standard deviations. This results in a range of values for the project. This actual project duration has a 95 percent probability of falling inside of this range.

But what if the durations of the actual project are such that a new critical path forms? To solve this problem we cannot practically solve the equations for all the possible values of all the durations of all the activities in the project. Instead we use computer simulation. This simulation is called Monte Carlo simulation and is discussed later in this chapter.