Program Evaluation and Review Technique (PERT)

Adapted from Course materials ( VI1.D Student Activity Sheet 11-12) for AMDM developed under the leadership of the Charles A. Dana Center, in collaboration with the Texas Association of Supervisors of Mathematics and with funding from Greater Texas Foundation.

You are in charge of organizing the senior class party. The following is your estimate of the time required to perform all the necessary activities:

Activity and Time in Hours
Activity	Time (in hours)
Plan music playlist	.5
Download music	1.0
Buy groceries and decorations	2.0
Bake a cake and prepare food	1.5
Set up	2.0

Since there are several classmates helping, some of these tasks can be performed at the same time. For instance, people can begin setting up the decorations while the cake is baking. However, you cannot bake the cake or set it up until after the shopping has taken place. And you cannot begin downloading music until you know what songs you want to download. The following activity graph can be used to help organize this information:

activity graph example

Activity graphs are a very powerful tool for visualizing and analyzing complicated organizational situations. Based on this analysis, business managers can decide how to best allocate limited resources to a project.

The numbers represent the time it takes, in hours, to complete each activity.

Why is there an arrow going from Activity A to Activity B, but not from Activity A to Activity C?

Activity B cannot begin until you finish Activity A, while you can begin Activity C without having finished Activity A.

Beginning at Start, there are several paths through the graph (following the arrows) that end at Finish. For each path, calculate the total time required to perform all the activities along the path.

Path I: Start → Activity A → Activity B → Finish takes 1.5 hours.

Path II: Start → Activity C → Activity D → Finish takes 3.5 hours.

Path III: Start → Activity C → Activity E → Finish takes 4 hours.

What is the minimum amount of time required to perform all five activities? The minimum time required is 4 hours. To complete all five activates in 4 hours, you must begin activities A and C right away. When activity SA is finished start immediately on Activity B. When C is finished start immediately Activity D and E.
Which path corresponds to this minimum time? Which activities are along this path? Path III ( the longest) corresponds to 4 hours. It includes Activities c and E.
Which activities could take a little longer to complete without affecting the total completion time? Activities A, B, and D could each take a little longer without affecting the total completion time of 4 hours.

Scheduling classes in college can be very similar to the previous scenario. Over four years, there are certain classes that you must take, and many classes have prerequisites—classes that must be taken first. Suppose you need to take the following classes with the identified prerequisites.

Class and Prerequisite
Class	Prerequisite
Calculus I	none
Calculus II	Calculus I
Physics I	Calculus I
Physics II	Physics I
Psychology I	None
Speech	None
Argument and Debate	Speech

Below is a constructed activity graph for this situation using the following rules:

Create Start and Finish squares.
Any activity that can be performed right away is connected to Start.
Activity A is connected to Activity B by an arrow only when Activity A needs to be performed directly before Activity B.
Any activity that does not precede any other activity can be connected to Finish.

Identify the longest path from Start to Finish. How long is this path?

The longest path is Start → Calculus I → Physics I → Physics II → Finish, which is three classes long or, if you count Start and Finish, it is five steps long.

If each class is a semester long, how many semesters are needed to take all these classes? It will take longer than 3 semesters
If you want to finish these classes as soon as possible, which classes should you not delay taking? You should not delay taken the classes in the longest path- Calculus I, Physics 1 and Physics II.
How long could you wait to take Psychology I without delaying your overall program of classes? You could wait two semesters and take Psychology 1 in the third semester.
How long could you wait to take Speech without delaying your overall program of classes? You could wait one semester and take speech in the second semester, which would mean taking Argument and Debate in the third semester.
Given any activity graph like the previous ones, explain how you would determine the minimum time required to perform all activities. The minimum time required to complete all the activities corresponds to the time it take to complete all activities in the longest path from start to finish.
Activities that cannot be delayed without increasing the minimum time for completion are called critical activities. Given any activity graph like the previous ones, explain how you would determine which activities are critical activities. Any activity that is along the longest path is critical.

For the following activity graph, the times given for each activity are in hours.

Determine the minimum time required to complete all the activities shown in the graph.

The longest path has a length of 10 (Start → Activity A → Activity C → Activity E → Finish), so the minimum time is 10 hours.

Which activities are critical activities? All the activities along the longest path - activities A, C, and E are critical .
How long could Activity F be delayed without affecting the overall completion time? The longest path through Activity F ( Start > Activity A 111. Activity C> Activity F> Finish) has a length of 9 hours, so Activity F could be delayed 1 hour ( or could take an extra hour to complete), and all activities could be still completes in 10 hours.
How long could Activity D be delayed without affecting the overall completion time? The longest path through Activity D ( Activity B > Activity D> Activity E has a length of 9 hours, so activity D could be delayed 1 hour without affecting the overall complete there.
What if Activities F and D were both delayed? if activity F was delayed by 1 hour and activity D by 2 hours, the minimum complete would change to 11 hours. Alternatively , activity D and F could both be delayed 1 hour and the minimum completion would still be 10 hours.

Sometimes the time to complete an activity is given by two numbers: the estimate for a minimum completion time and the estimate for a maximum completion time.

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