PERT Charts - Project Management

Often large projects with many inter-dependencies can seem quite challenging to plan and execute. It can seem as if there is so much work to be done. While we often hear the advice to "prioritize tasks", what does that actually mean? Experienced project managers (especially those with PMP certifications from the Project Management Institute) will be familiar with Gantt and PERT charts.

Let's create a sample model to explain the methodology of project management:

• Assume that you have five mile stones to achieve A, B, C, D and E.
• Milestone A is your starting point
• Milestone B is dependent on A and requires 2 hours of work to achieve
• Milestone C is also dependent on A and requires 4 hours of work
• Milestone D is dependent on Milestone B and take 5 hours of work
• Milestone E is dependent on Milestone B and requires 2 hours of work, requires 1 hour of work after Milestone C and finally requires 3 hours of work after Milestone D.
  

The line highlighted in red is the critical path. Assuming that you have enough resources that can work in parallel (and that you cannot add resources to reduce job times) the critical path (the path that determines the minimum time require to finish the job) is:

• A --> B (2 hours)
• B --> D (5 hours)
• D --> E (3 hours)

For a total critical path time of 10 hours. This is because there is no other path from A to E which requires more time. Note that A --> B --> E has a slack time (or float) of 6 hours (10 hours - 4 hours). This means that this path can be delayed as much as 6 hours without an adverse effect on the project as a whole. Similarly, for A --> C --> E which has a slack time of 5 hours (10 - 5 hours).

However, this becomes more complicated if resources can be moved from one task to another to influence (shorten) the job time. In this case it is possible to use graph theory and Dijkstra's algorithm to determine which paths are relatively over capacity (shortest paths) so that they can contribute resources to the critical path to reduce over all project time.

In the example above, resources could be taken from the following tasks and reallocated to the slowest task (B --> D) with the goal being to reduce systematic slack time across any path to 0.

• A --> C
• C --> E
• B --> E
  

In this case, it becomes more appropriate to describe the tasks work in a quantity of man hours.

SAT Based Example: It takes 4 people 1 hour to paint a house. How many man hours are required for 16 people to paint 8 houses?

Answer: 1 house takes 4 people one hour so each house is 4 man hours of work. 8 houses would be 32 man hours (4 man hours per house x 8 houses) of work.

32 man hours divided amongst 16 people evenly is 2 hours.

Obviously, the examples provided in this post assume that workers are interchangeable, that there are no diminishing returns of labour etc. However, even without these assumptions, if there are large unbalances in the project paths, there is an obvious benefit to moving resources from one area to another to improve the overall project completion time.