a$.$ Formulate an ILP model to determine how many units of each product to produce on each machine in order to meet demand at a minimum cost.

b$.$ Implement your model in a spreadsheet and solve it$.$

c$.$ What is the optimal solution?

$28.$ Clampett Oil purchases crude oil products from suppliers in Texas $($TX$),$ Oklahoma $($OK$),$ Pennsylvania $($PA$),$ and Alabama $($AL$),$ from which it refines four end$-$products: gasoline, kerosene, heating oil, and asphalt. Because of differences in the quality and chemical characteristics of the oil from the different suppliers, the amount of each end$-$product that can be refined from a barrel of crude oil varies depending on the source of the crude. Additionally, the amount of crude available from each source varies, as does the cost of a barrel of crude from each supplier. These values are summarized in the following table. For example, the first line of this table indicates that a barrel of crude oil from Texas can be refined into $2$ barrels of gasoline, $2\mathrm{.}8$ barrels of kerosene, $1\mathrm{.}7$ barrels of heating oil, or $2\mathrm{.}4$ barrels of asphalt. Each supplier requires a minimum purchase of at least $500$ barrels:

Raw Material Characteristics

Possible Production per Barrel

$\backslash$table$[[$Crude Oils,Barrels Available,Possible Production per Barrel,Cost per Barrel,Trucking Cost$],[$Gas$,$Kero.,Heat,Asphalt$],[$TX$,1,500,2\mathrm{.}00,2\mathrm{.}80,1\mathrm{.}70,2\mathrm{.}40,$$$22,$$$1,500$