Consider the project network given below:

Numbers along the arcs represent: crash time $($days$),$ normal time $($days$)$ per unit cost of crashing $($$$).$ Note: jobs A and $B$ cannot be crashed.

$($a$)$ Use the direct enumeration method to determine the least cost crashing schedule that will meet the project deadline of $20$ days. What is the total crashing cost?

$($b$)$ Given that we have $$50$ available for crashing, which jobs would you crash and by how much? What is the least completion time of the project?

$($c$)$ Construct an LP model to answer part $($b$).$ You must define your variables clearly, write out the constraints and the objective function.