Problem $4(15$ points$)$

Jad and Karim are two friends that like to train together at the same time at the gym. Their session is at $10$ am on Monday and they are the only clients at that time. However, there is only one machine of each type at their gym, and they have to share most of these machines. Jad uses the following machines in his training circuit in the following order: A$,$ B$,$ C; D$.$ Whereas Karim uses the following machines in the following order: B$,$ C$,$ A$,$ E$.$ The exercise time each need on the machine is shown in the table below.

$\backslash$table$[[,$Jad$-$Exercise Time,Karim$-$Exercise Time$],[$A$,2,3],[$B$,4,5],[$C$,2,6],[$D$,6,],[$E$,-,2]]$

a$.$ Find the optimal schedule that minimizes the makespan of the training programs for Jad and Karim.

b$.$ Draw the associated Gantt chart.