The critical path of the project is the longest sequence of tasks that must be completed on time for the project to finish on schedule. It determines the shortest possible duration of the project and shows the tasks that have no slack or float. Any delay in the critical path tasks will affect the project completion date. To find the critical path, we need to calculate the earliest start (ES), earliest finish (EF), latest start (LS), latest finish (LF), and slack (S) for each task. We can use the table and the network diagram given in the question to do this. The formula for calculating the slack is S = LF - EF or S = LS - ES. A task with zero slack is on the critical path. The calculations are as follows:
Task | Duration | ES | EF | LS | LF | S A | 10 | 0 | 10 | 0 | 10 | 0 B | 15 | 10 | 25 | 10 | 25 | 0 C | 10 | 10 | 20 | 15 | 25 | 5 D | 20 | 25 | 45 | 25 | 45 | 0 E | 10 | 25 | 35 | 35 | 45 | 10 F | 15 | 20 | 35 | 30 | 45 | 10 G | 5 | 45 | 50 | 45 | 50 | 0 H | 10 | 35 | 45 | 40 | 50 | 5 I | 20 | 35 | 55 | 30 | 50 | -5 J | 5 | 50 | 55 | 50 | 55 | 0
Based on the table, we can see that the tasks with zero slack are A, B, D, G, and J. Therefore, the critical path is A-B-D-G-J, which has a total duration of 55 days. The other options are not the correct critical paths, as they either include tasks with positive slack (such as C, E, F, and H) or have a longer duration than 55 days (such as A-C-F-I-J).
