BBest Ink Printing Co$.$ received an order to print a minimum of $50,000$ tickets for a concert. They have three printing machines available to meet the order they received. The set$-$up cost of these machines and the unit cost$/$ticket printed using each machine along with their maximum production are provided in the table below:

Machine

Set$-$up cost

Cost per unit

Maximum Production

A

$$7000$

$$18$

$30,000$

B

$$4000$

$$21$

$25,000$

C

$$5400$

$$24$

$30,000$

a$.$ Formulate a linear programming model and create a spreadsheet model to find two machines that could be used to print the required number of tickets.

b$.$ Find the optimal solution in part a$.$

c$.$ Find the total cost.

d$.$ Create an alternative LP model that will improve the outcome $($be realistic in the assumptions and changes$)$