A directed graph can be used to represent an activity that consists of many smaller activities that all play a crucial role in forming the final product.
An activity is described as critical (critical activity), if it is delayed it will also delay the entire project.
Critical activities form part of the critical path. The critical path is the LONGEST path is a directed graph. An example of a critical path graph is shown below.
Essential Further Mathematics 4ed 2012 (Figure 24.6)
NOTE: Even though the critical path is the longest path, it represents the minimal time taken to complete the entire project.
In order to find the critical path there are two new terminologies that you must understand before we can continue.
This represents the earliest time an activity can start. When finding ESTs you are adding.
This represents the latest time an activity can be left for before it delays the entire project. You can find the LSTs by subtracting.
This is the time that represents that an activity can start later than its EST. It can be found by subtracting the LST from EST for each activity. NOTE: Activities on the critical path should have a slack time of 0.
The following diagrams have been obtained from VCAA Further Mathematics 2013 Exam 2 (p.34) and have been adapted to fit the purpose needed.
The diagram above shows the ESTs for each activity. The process to find the ESTs is as follows:
The above diagram shows the EST and LST (Right box) for each activity. The process to find the LSTs is as follows:
By finding the ESTs and LSTs for each activity it makes finding the critical path easier. All you have to do is find the boxes that contain the same number. For this example the critical path is: A, F, I, M.
As the title suggests crashing involves reducing the overall completion time by completing some of the activities more quickly.
It is often a trial and error process but it should be noted:
This is an artificial activity that has a time duration of 0. It is added to a network diagram to ensure that all predecessor activities are properly accounted for.
When performing a critical path analysis and you have to construct your own network from a table you need to ensure that:
Essential Further Mathematics 4ed 2012 (Example 8 p.654)
In the above example, a network has been constructed using the information contained in the table. A dummy activity (D1) was needed because the predecessor of activities C and D is A and B. The dummy activity makes it easier for us to calculate the ESTs and LSTs correctly.
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