a$.$ Determine the sequence that will minimize the makespan.

To minimize the makespan, we need to prioritize jobs with shorter processing times first. Let's calculate the completion time for each job and sort them accordingly:

Job $2$: Machine A $(3$ minutes$)+$ Machine B $(4$ minutes$)=7$ minutes

Job $1$: Machine A $(5$ minutes$)+$ Machine B $(6$ minutes$)=11$ minutes

Job $3$: Machine A $(6$ minutes$)+$ Machine B $(9$ minutes$)=15$ minutes

Job $4$: Machine A $(7$ minutes$)+$ Machine B $(8$ minutes$)=15$ minutes

Job $5$: Machine A $(9$ minutes$)+$ Machine B $(2$ minutes$)=11$ minutes

Explanation:

So$,$ the sequence that minimizes the make span is: Job $2,$ Job $5,$ Job $1,$ Job $4,$ Job $3.$

b$.$ Determine machine A$\text{'}$s idle time by creating a load Gantt chart for the resulting sequence.

Machine A$\text{'}$s idle time $=12$ minutes $($from the completion of Job $2$ to the start of Job $1)$

Step $2$

c$.$ For the sequence determined in part a$,$ can machine A$\text{'}$s idle time be reduced by splitting some jobs in half? If so$,$ which jobs?

To reduce machine A$\text{'}$s idle time, we can consider splitting longer jobs in half and interleaving them with shorter jobs. In this case, we can split Job $3$ and Job $5,$ as they have longer processing times.

d$.$ Divide tasks c and e$,$ then create a new load Gantt chart.

Let's split Job $3$ and Job $5$:

Job $3$: Split into Job $3$a $($Machine A: $3$ minutes, Machine B: $4\mathrm{.}5$ minutes$)$ and Job $3$b $($Machine A: $3$ minutes, Machine B: $4\mathrm{.}5$ minutes$).$ Job $5$: Split into Job $5$a $($Machine A: $4\mathrm{.}5$ minutes, Machine B: $1$ minute$)$ and Job $5$b $($Machine A: $4\mathrm{.}5$ minutes, Machine B: $1$ minute$).$

Updated sequence:

Job $2$

Job $5$a

Job $3$a

Job $1$

Job $4$

Job $5$b

Job $3$b

Explanation:

Machine A$\text{'}$s idle time is now reduced to $4\mathrm{.}5$ minutes.

Answer

a$.$ The optimal sequence to minimize make$-$span is: Job $2,$ Job $1,$ Job $4,$ Job $3,$ Job $5.$

b$.$ Load Gantt chart: Machine A: $[0,3,8,15,24,33]$ Machine B: $[5,9,15,23,31]$ Idle time for Machine A: $0$ minutes.

c$.$ Machine A$\text{'}$s idle time can be reduced by splitting Job $3$ and Job $5.$

d$.$ After splitting jobs c and e$,$ the new load Gantt chart will adjust accordingly.