Critical path analysis

Critical path analysis

Critical Path Analysis

Once the network diagram with single time estimates has been developed, the following computational procedure may be employed for determining the critical path/s, event slacks and activity floats.

Image result for Critical path analysis in Operations Research

Calculate the Earliest Occurrence Time (EOT) for Each Event

An event occurs when all activities leading to the event have been completed. In the network diagram shown in Figure 5.10, for example, event 4 occurs when activities (2-4) and (3-4) are completed. Obviously, activity (2-4) cannot begin unless event 2 occurs, which in turn requires the completion of the activity (1-2). Likewise, activity (3-4) cannot begin unless event 3 occurs which in turn requires the completion of the activity (1-3). Thus we find that event 4 occurs when activities (1-2), (2-4), (1-3), and (3-4) are completed. In other words, event 4 occurs when paths (1-2-4) and (1-3-4) are completed.
The EOT of an event refers to the time when the event can be completed at the earliest. Looking at event 4 we find that since the paths leading to it, viz., (1-2-4) and (1-3-4) take 15 weeks and 20 weeks, respectively, the EOT of event 4 is 20 weeks. In general terms, the EOT of an event is the duration of the longest path (from the beginning event whose EOT is set at 0) leading to that event. The EOTs of various events in our illustrative project are shown in Figure 5.7. It may be noted that in Figure 5.10 and subsequent figures an event is represented by a circle. The upper half of the circle denotes the event number, the left quarter in the lower half denotes the EOT, and the right quarter in the lower half denotes the latest occurrence time, a term described later.
The EOT of the end event obviously represents the minimum time required for completing the project. To obtain the EOT of various events we start from the beginning event and move forward towards the end event. This computational procedure is referred to as the forward pass. In this computation, we assume that each activity starts immediately on the occurrence of the event preceding it. Hence the starting and finishing time for various activities obtained from this computation are the Earliest Starting Time (EST) and the Earliest Finishing Time (EFT).

Calculate the Latest Occurrence Time for Each Event

The LOT for an event represents the latest allowable time by which the event can occur, given the time that is allowed for the completion of the project (occurrence of end event). Normally, the time allowed for the completion of the project is set equal to the EOT of the end event. (In other words, the project is supposed to be completed at the earliest possible time.) This means that for the end event the LOT and EOT are set equal. The LOT for various events is obtained by working backwards from the end event. This procedure is known as the backwards pass. The LOT for event 4 in our illustrative project, for example, is equal to the LOT for event 5, the end event, minus the duration of the activity (4-5) which connects event 4 with 5. Since the LOT for event 5 is 28 weeks and the duration of activity (4-5) is 2 weeks the LOT for event 4 is 26 weeks (28-2). This represents the latest time by which event 4 should occur to enable the project to be completed in 28 weeks. Likewise, the LOT for other events can be calculated by moving backwards.

Obtain the Critical and Slack Paths

The critical path starts with the beginning event, terminates with the end event, and is marked by events which have a zero slack. This is obviously the path on which there is no slack, no cushion. Other paths are slack paths with some cushion.
The critical path is the longest path from the beginning event to the end event. Since the end can be reached, i.e., the project completed, only when this longest path is traversed, the minimum time required for completing the project is the duration on the critical path. The duration on the critical path of our project is 28 weeks; this is the minimum time required for completing the project. (It is already indicated by the EOT of event 5, the end event.)