PERT or the Program Evaluation and Review Technique is a method that analyzes the time required to complete each task and its associated dependencies, and to determine the minimum amount of time required to complete a certain project. The process takes into consideration three different time estimates:
- Optimistic Time (To): The minimum amount of time required to complete the project, assuming everything goes better than expected.
- Pessimistic Time (Tp): The maximum time required to complete the task, assuming things go wrong.
- Most Likely Time (Tm): The most likely amount of time required to complete the tasks, assuming everything goes alright.
Now, let’s get started with the PERT Analysis. Before we get into the PERT Analysis process, we must talk about some important concepts: Events and Activities. Let’s understand these terms with the help of a network diagram (which is the final output of the method).
Event
A circle represents events and will occur at the start and end of an activity. Event 1 is the tail event, and Event 2 is the head event. In the case of our example, node 1 will be referred to as the tail event, and 2 will be referred to as the head event.
Activity
Activities represent action and consumption of resources like time, money, and energy required to complete the project. In the case of our example, A, B, C, D, and E represent the activities taking place between their respective events.
Dummy Activity
A dummy activity represents a relationship between two events. In the case of the example below this, the dotted line represents a relationship between nodes 3 and 2.
The activity between these nodes will not have any value.
Other rules that need to be considered are:
- The network should have a unique starting and ending node.
- No activity can be represented by more than a single arc (the line with an arrow connecting the events) in the network.
- No two activities can have the same starting and ending node.
The PERT Analysis Method
In the question here, we have three objectives:
1) Draw the network diagram.
2) Find the mean and variance.
3) Find the critical path and estimated time of completion.
Now, let’s draw the network diagram.
First, let’s look at the activities and their immediate predecessors.
We can see that activities A, B, and C don’t have any immediate predecessors. This means that we can draw individual arcs to each of them. Let’s draw the nodes for the first activity, activity A. We can see that activity A acts as the immediate predecessor for the activity D.
Similarly, activities B and C don’t have any immediate predecessors and hence, can be directly connected to node 1. Node B acts as an immediate predecessor for E, while node C acts as the immediate predecessor for activities F and G. Let’s go ahead, and draw that.
Let’s have a look at activity D. This activity is the immediate predecessor for activity A. This means that we can directly draw an arc from node 2.
Now, we’ve drawn activities A, B, C, and D as part of the PERT analysis. Now, looking at activity E, it acts as the immediate predecessor to activity H along with activity F. Since it’s preceded only by activity B, we can directly connect it to node 3.
Now, for activity F. If we have a look at the table, we can see that a combination of the activities E and F act as immediate predecessors for activity H. This means that activities E and F need to come together at node 6.
Next up, let’s have a look at activity G. It is immediately preceded by activity C, and acts as an immediate predecessor for activity J, along with activity H. Since it’s an independent activity, we can draw it like so:
For activity H, we can see that it and G act as immediate predecessors for activity J. This means that nodes 6 and 7 need to be connected.
And finally, we activities I and J. These activities don’t act as immediate predecessors for any other activity. This means that they’ll connect directly to the final node.
Now that we’ve created the network diagram, let’s move ahead. Next, as part of the PERT analysis, let’s have a look at how to determine the mean and variance.
The mean, which is also the estimated time can be determined using the formula:
We can calculate the variance using this formula:
Let’s apply the formula to each activity.
For activity A,
The mean will be: (To + 4*Tm + Tp) /6 = (6 + 4*7 + 8) /6 = 7
For activity B,
The mean will be: : (To + 4*Tm + Tp) /6 = (3 + 4*5 + 7) /6 = 5
For activity C,
The mean will be: : (To + 4*Tm + Tp) /6 = (4 +4*7 +10) /6 = 7
For activity D,
The mean will be: : (To + 4*Tm + Tp) /6 = (2 + 4*3 +4) /6 = 3
For activity E,
The mean will be: : (To + 4*Tm + Tp) /6 = (3 + 4*4 + 11) /6 = 5
For activity F,
The mean will be: : (To + 4*Tm + Tp) /6 = (4 + 4*8 + 12) /6 = 8
For activity G,
The mean will be: : (To + 4*Tm + Tp) /6 = (3 + 4*3 + 9) /6 = 4
For activity H,
The mean will be: : (To + 4*Tm + Tp) /6 = (6 + 4*6 + 12) /6 = 7
For activity I,
The mean will be: : (To + 4*Tm + Tp) /6 = (5 + 4*8 + 11) /6 = 7
For activity J,
The mean will be: : (To + 4*Tm + Tp) /6 = (3 + 4*3 + 9) /6 = 4
This mean can be applied to the network, to each of the activities.
Now, let’s find the variance for each of these activities.
2 = [(Tp - To) /6]2
For activity A:
2 = [(Tp - To) /6]2= 2 = [(8 - 6) /6]2= 0.11
For activity B:
2 = [(Tp - To) /6]2= 2 = [(7 - 3) /6]2= 0.44
For activity C:
2 = [(Tp - To) /6]2= 2 = [(10- 4) /6]2= 1
For activity D:
2 = [(Tp - To) /6]2= 2 = [(4 - 2) /6]2= 0.11
For activity E:
2 = [(Tp - To) /6]2= 2 = [(11 - 3) /6]2= 1.77
For activity F:
2 = [(Tp - To) /6]2= 2 = [(12 - 4) /6]2= 1.77
For activity G:
2 = [(Tp - To) /6]2= 2 = [(9 - 3) /6]2= 1
For activity H:
2 = [(Tp - To) /6]2= 2 = [(12 - 6) /6]2= 1
For activity I:
2 = [(Tp - To) /6]2= 2 = [(11 - 5) /6]2= 1
For activity J:
2 = [(Tp - To) /6]2= 2 = [(9 - 3) /6]2= 1
Now, for the third part of the PERT analysis. We need to find the critical path and the estimated time.
For this, we’ll need to find two values, Earliest Start Time (Es) and Latest Completion Time (Lc).
The process of determining the Es for all events is called a forward pass.
The process of determining the Lc for all events is called a backward pass.
Let’s get into the forward pass. For this first, we must create boxes at all nodes. We then divide these into two. The lower half of the box represents the earliest start time of the node, while the lower half represents the latest completion time.
Your network diagram should look something like this.
For this, we’ll be using the formula, Esj = max (Esi + Dij)
Which when simplified, the earliest start time for the second node (head node), is the maximum of the combination of the earliest start time of the tail node and the duration between the two nodes.
So, for node 1, the earliest start time is always zero.
For node 2,
Es2 = 0 (Es1) + 7(D1-2) = 7
Next node 3.
Es3 = 0(Es1) + 5(D1-3) = 5
Now, for node 4.
Es4 = 0(Es1) + 7(D1-4) = 7
Next, we have node 5.
Es5 = 7(Es2) + 3(D2-5) =10
Now for node 6.
Since there are two arcs connecting to the node, we need to choose the maximum of the two options available.
Es6 = 5(Es3) + 5(D3-6) = 10 or
Es6 = 7(Es4) + 8(D4-6) = 15
We must choose the maximum of the two, so we’ll select 15.
Next, we have node 7. Since there are two nodes connecting to it; we need to choose the maximum among the two options.
Es7 = 15(Es6) + 7((D6-7) = 22 or
Es7 = 7(Es4) + 4(D4-7) = 11
We’ll need to choose the maximum, and we’ll choose 22.
And finally, we’ll need to find the earliest start time for node 8.
Es8 = 10(Es5) + 7(D5-8) = 17 or
Es8 = 22(Es7) + 4(D7-8) = 26
Since we need to choose the maximum value, we’ll choose 26.
And like that, the forward pass is complete. Now, for the second part of the PERT Analysis. Let’s take up the backward pass. For that, we will be using the following formula.
Lci = min(Lcj - Dij)
This, when put simply, means the latest completion time of the tail node is equal to the latest completion time of the head node minus the distance between the two.
Let’s start from the final node, number 8.
The Lc for this node will always be equal to its Es.
So, Lc8 = 26
Now let’s go to node 7. Since it’s an independent node, we’ll directly apply the formula.
Lc7 = 26(Lc8) - 4(D7-8) = 22
Next up, let’s take a look at the latest completion time for node 6. Again, since it’s an independent node, we can directly apply the formula.
Lc6 = 22(Lc7) - 7(D6-7) = 15
Now, for node 5.
Node 5 is an independent node. We’ll directly apply the formula here.
Lc5 = 26(Lc8) - 7(D5-8) = 19
The network diagram as part of the PERT Analysis will look like so.
Now that we’re done with node 5, let’s go to node 4.
Here, we can see that two arcs connect it to nodes 6 and 7. We need to choose the minimum latest completion time from these two nodes.
Applying the formula,
Lc4 = 22(Lc7) - 4(D4-7) = 18 or
Lc4 = 15(Lc6) - 8(D4-6) = 7
Since we have to choose the minimum, we’ll choose 7.
Next, we have node 3. Since it’s an independent node with a single connection, we can directly apply the formula to it.
Lc3 = 15(Lc6) - 5(D3-6) = 10
Now for node 2.
We can directly apply the formula to node 2.
Lc2 = 19 (Lc5) - 3(D2-5) = 16
And finally, we have node 1. Since there are multiple nodes connected to node1, we’ll have to choose the minimum latest completion time.
Lc1 = 16(Lc2) - 7(D1-2) = 9 or
Lc1 = 10(Lc3) - 5(D1-3) = 5 or
Lc1 = 7(Lc4) - 7(D1-4) = 0
Since we need to choose the minimum, we’ll choose 0.
And that’s the backward pass, complete in the PERT Analysis.
Now, for the ultimate step of the critical path method. To determine the critical path, there are three major criteria that need to be satisfied.
Esi = Lci
Esj = Lcj
Esj - Esi = Lcj - Lci = Dij
From the diagram, we can see that nodes that satisfy the requirements are:
1 - 4 - 6 - 7 - 8 or C - F - H - J
The estimated time is: 7 + 8 + 7 + 4 = 26 days.
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What’s Next?
Now that you know the PERT Analysis Method, you can go into some more complex concepts of project management. For that, we recommend signing up for the PMP Certification Training Course, which prepares you for your PMP certification exam. When you use effective tools like Microsoft Excel templates, you can raise your career to a whole new level.
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