The Ace Manufacturing Company has orders for three similar products.

Product Orders

$($units$)$

A A $1,800$

B B $400$

C C $1,300$

Three machines are available for the manufacturing operations. All three machines can produce all the products at the same production rate. However, due to varying defect percentages of each product on each machine, the unit costs of the products vary depending on the machine used. Machine capacities for the next week and the unit costs are shown below.

Machine Capacity

$($units$)$

$111,700$

$221,500$

$33700$

Machine

Product $123$

A $$1\mathrm{.}00$ $$1\mathrm{.}30$ $$1\mathrm{.}10$

B $$1\mathrm{.}20$ $$1\mathrm{.}40$ $$1\mathrm{.}00$

C $$0\mathrm{.}90$ $$1\mathrm{.}20$ $$1\mathrm{.}20$

Use the transportation model to develop the minimum cost production schedule for the products and machines.

$($a$)$

Show the linear programming formulation. $($Let xA$1$ be the number of units of product A produced by machine $1,$

xij

be the number of units of product i produced by machine j$,$ etc.$)$