Consider the project network given below:

The data for normal times, crash times, and crashing costs are given as follows:

Let $\backslash ($ T $\backslash )$ represent the earliest completion time of the project.

$($a$),$ Determine the maximum and the minimum value of $\backslash ($ T $\backslash ).$

$($b$)$ Set up the linear program to solve the CPM problem if the project is to be completed in $21$ days at minimum cost.

$($c$)$ Given the overhead costs as $\backslash (\backslash$$ $5\backslash )$ per day, determine the optimal duration of the project in terms of both the crashing and the overhead costs by direct enumeration method.