# CPM diagram

The critical path method or CPM diagram (Critical Path Method) is an algorithm based on network theory that allows calculating the minimum time to complete a project.

This method uses deterministic intervals, unlike others like PERT that are based on probabilities.

This means that it is expected that, under identical conditions, the result of a process will be the same. Therefore, in this case the times are known a priori.

## Origin of CPM diagram

The origin of the CPM diagram was in an operations center that developed it for the firms Dupont and Remington Rand. The date of its creation is considered the interval between December 1956 and February 1959.

The objective was to control the completion times and with it, the costs involved. As a curiosity, it was created a year before the PERT method.

Morgan Walker of Dupont and James E. Kelley of Remington Rand, engineer and mathematician, managed to have this time management system ready (in a short period of time). The objective was to optimize the costs involved in the different projects. In this case, as mentioned, the times are known a priori.

## The critical path in the CPM diagram

To calculate it, you have to know two basic rules. The first is that each activity must be identified by two nodes, one at the beginning and one at the end. The second is that, if two activities go to the same end node, use a dummy one that is represented by an arc of points.

To know the critical path it is necessary to follow a series of steps.

- First, you have to make a table with the activities, their priorities and duration.
- The CPM diagram is then created with the dummy activities if they are required.
- The three time indicators are calculated. Going through the network from left to right and vice versa, the earliest times (T1), the latest times (T2) and the slack times (H) are obtained as the difference of both. We will see it better in the example.
- The critical path will be the one with clearances equal to zero. Sometimes there can be more than one route that has this condition and they are all valid.

## CPM diagram example

Let's look at a simple example, which is similar to a PERT chart. Let's imagine a company that has four activities: A, B, C and D. The last one (D) receives from B and C, therefore, we create a fictitious one (Fb) that does not consume time or resources. This only serves to meet the basic requirements of the diagram.

Now we fill in the earliest times (T1) starting from zero in A and adding that of the previous node to the next task. When two tasks arrive at the same node, the one with the highest T1 is chosen. The last one will be the sum of the previous tasks. Now we calculate T2 starting from node 4 and subtracting the times instead of adding. If two arrive, we take the smallest of them.

As the last step in the CPM diagram we calculate the clearances (H) as the difference between T1 and T2. As we can see, at the beginning the times will be zero and the maximum and minimum execution time (which are equal) is reflected in the last node. The critical path (dark blue) will be the one in which the nodules have no slack (H = 0).

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