Q$.6$ Consider the following software planning problem. The system consists of $11$ modules $($M$1$ to M$11)$ whose duration $($in days$).$ The dependency graph and the duration for each module fin $\backslash$#days$)$ is shown below

The module implementation has been assigned to three programmers $\backslash (-\backslash$mathrm$\{$A$\},\backslash$mathrm$\{$B$\}\backslash ),$ and C as given above, and they will be carried out the tasks assigned in the given sequence.

$($a$)$ Using a PERT graph, give the schedule for the above development. Label the nodes with the module names and indicate the earliest possible start$/$finish time, latest possible start$/$finish time for each module, and determine the critical path, and hence determine the estimated duration for the project.

$($b$)$ What if programmer $\backslash ($ A $\backslash )$ decides to go on a $3-$day vacation after completing M$8.$ What impact would this have on the overall schedule with respect to the graph obtained in part $($a$)$ of this question? Explain.