Program Evaluation and Review Technique PERT

The PERT system was developed for the Polaris Missile Program in the 1950s. At that time there was a lot of pressure on the U.S. Navy to complete the Polaris Missile Program. The Cold War was raging, and the United States needed a deterrent that would discourage the threat of nuclear weapons by Russia. A mobile missile that could be carried aboard a submarine and launched from beneath the surface of the sea would be a formidable weapon.

The problem for the U.S. Navy was that there were two separate projects to be done. One was to develop a submarine that could launch these missiles. The second project was to develop a missile that could be launched from a submarine. The durations of the project plan activities had a great deal of uncertainty in them. The navy needed a method to predict the project schedule with better reliability than was possible in the past. PERT was developed to assist in analyzing projects where there was uncertainty in the duration of the activities.

The normal probability distribution relates the event of something happening to the

Figure 2-13. Critical path method.

Select Site

Fri 10/13/00

pproval Thu 10/19/00

Mon 7/10/00

Thu 7/13/00

Fri 10/13/00

Thu 10/19/00

Mon 10/9/00

Thu 10/12/00

Mon 10/9/00

Thu 10/12/00

Develop Project Deliverables

Approval from Stakeholders

Evaluate and Select Vendors

Mon 6/12/00

Fri 6/30/00

w

Mon 7/3/00

Fri 7/7/00

w

Mon 7/10/00

Thu 7/13/00

Mon 6/12/00

Fri 6/30/00

Mon 7/3/00

Fri 7/7/00

Wed 8/23/00

Mon 8/28/00

z

Test Hardware

Integrate

Wed 7/19/00

Tue 8/1/00

W

Fri 9/15/00

Thu 10/12/00

Fri 9/1/00

Thu 9/14/00

Fri 9/15/00

Thu 10/12/00

Purchase Hardware

Fri 7/14/00

Tue 7/18/00

Tue 8/29/00

Thu 8/31/00

Design Software

Write Code

Test

Mon 7/10/00

Fri 7/28/00

W

Mon 7/31/00

Fri 9/8/00

w

Mon 9/11/00

Thu 9/14/00

Mon 7/10/00

Fri 7/28/00

Mon 7/31/00

Fri 9/8/00

Mon 9/11/00

Thu 9/14/00

probability that it will occur. It turns out that by experiment, the normal distribution describes many phenomena that actually occur. The duration as well as the estimated cost of project activities comes close to matching a normal distribution. In reality, another distribution, called the beta distribution, fits these phenomena better, but the normal curve is close enough for practical purposes.

Suppose we have a scheduled activity that has an expected completion time of thirty-five days. In Figure 2-14, the curve shows the probability of any other day occurring. Since thirty-five days is the expected value of the activity, it follows that it would have the highest probability of all of the other possibilities. Another way of saying this is that, if all of the possibilities are shown, then they represent 100 percent of the possibilities and 100 percent of the probability.

If it were possible for this project to be done thousands and thousands of times, sometimes the time to do the activity would be 35 days, other times it would be 33 days, and still other times it would be 37 days. If we were to plot all of these experiments we would find that 35 days occurred most often, 34 days occurred a little less often, 30 days even less, and so on. Experimentally, we could develop a special probability distribution for this particular activity. The curve would then describe the probability that any particular duration would occur when we really decided to do the project and that activity. In the experiment, if 35 days occurred 134 times and the experiment was performed 1,000 times, we could say that there is a 13.4 percent chance that the actual doing of the project would take 35 days. All 1,000 of the activity times were between 20 and 50 days.

It is impractical to do this activity a thousand times just to find out how long it will take when we schedule it. If we are willing to agree that many phenomena, such as schedule durations and cost, will fit the normal probability distribution, then we can avoid doing the experiment and instead do the mathematics. To do this we need only have a simple way to approximate the mean and standard deviation of the phenomena.

The mean value is the middle of the curve along the x-axis. This is the average or expected value. A good approximation of this value can be obtained by asking the activity estimator to estimate three values instead of the usual one. Ask the estimator to estimate the optimistic, the pessimistic, and the most likely. (The estimator is probably doing this anyway.) The way people perform the estimating function is to think about what will happen if things go well, what will happen if things do not go well, and then what is likely to really happen. This being the case, the three values we need are free for the asking. These are the optimistic, the pessimistic, and the most likely values for the activity duration.

If we have these three values, it becomes simple to calculate the expected value and the standard deviation. For the expected value we will take the weighted average:

Expected value = [Optimistic + Pessimistic + (4 X Most likely)] / 6

Figure 2-14. Schedule probability.

35 Days

Standard deviations can be calculated using the formula:

Standard deviation = (Pessimistic - Optimistic) / 6

As can be seen in Table 2-2, with these two simple calculations we can calculate the probability and a range of values that the dates for the completion of the project will have when we actually do the project. For the purpose of ease of calculation, if we were to decide that 95.5 percent probability would be sufficient for our purposes, then it turns out that this happens to be the range of values that is plus or minus 2 standard deviations from the mean value.

If the expected value of the schedule is 93 days and the standard deviation is 3 days, we could make this statement: This project has a probability of 95 percent that it will be finished in 87 to 99 days.

For example, suppose we use the same example we used earlier. This time we have probabilistic dates instead of the specific ones that we had before. We have collected estimates on the duration of each of the activities and show the optimistic, pessimistic, and most likely values in the table. The expected value is from the formula:

EV = [Optimistic + Pessimistic + (4 X Most likely)] / 6

The standard deviations can be calculated using the formula:

Standard deviation = (Pessimistic - Optimistic) / 6

One thing must be pointed out here. Unlike cost estimating, where the cost of every activity in the project must be added up to get the total cost, the sum of the time it will take to do the project is the sum of the expected value of the items that are on the critical

Table 2-2. PERT exercise.

Most CP

Table 2-2. PERT exercise.

Most CP

Activity

Description

Optimistic

Pessimistic

Likely

EV

SD

Variance

CP EV

Variance

1

Develop project deliverables

13

16

15

14.83

0.50

0.2500

14.83

0.2500

2

Approval from stakeholders

4

6

5

5.00

0.33

0.1111

5.00

0.1111

3

Site selection

4

4

4

4.00

0.00

0.0000

4

Evaluate and select vendor

4

5

4

4.17

0.17

0.0278

5

Purchase hardware

3

3

3

3.00

0.00

0.0000

6

Design software

14

17

15

15.17

0.50

0.2500

15.17

0.2500

7

Write code

24

33

30

29.50

1.50

2.2500

29.50

2.2500

8

Test software

4

4

4

4.00

0.00

0.0000

4.00

0.0000

9

Test hardware

9

11

10

10.00

0.33

0.1111

10

Integrate hardware and

software

20

23

20

20.50

0.50

0.2500

20.50

0.2500

11

Install and final acceptance

5

5

5

5.00

0.00

0.0000

5.00

0.0000

Sum

=

94.00

3.1111

sq. rt

:. var.

= SD

1.763834

path only. Other activities in the project do not contribute to the length of the project, because they are done in parallel with the critical path.

The sum of the durations for the critical path items is 18.3 days. The standard deviation is 2.3 days. We can say that there is a 95 percent probability that the project will be finished in 13.7 days to 22.9 days.

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Project Management Made Easy

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