The Ace Manufacturing Company has orders for three similar products.

Product Orders

$($units$)$

A A $1,800$

B B $700$

C C $1,400$

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$)$

$111,500$

$221,400$

$331,100$

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$

$($a$)$

Develop the linear programming formulation of this problem. $($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.$)$

Min

Correct: Your answer is correct.

s$.$t$.$ Machine $1$ Capacity

Correct: Your answer is correct. Machine $2$ Capacity

Correct: Your answer is correct. Machine $3$ Capacity

Correct: Your answer is correct. Product A Orders

Correct: Your answer is correct. Product B Orders

Correct: Your answer is correct. Product C Orders

Correct: Your answer is correct. xij $>=0$ for all i$,$ j$.$

$($b$)$

Solve the transportation model for the minimum cost production schedule for the products and machines.

Show the production schedule.

$($xA$1,$ xA$2,$ xA$3,$ xB$1,$ xB$2,$ xB$3,$ xC$1,$ xC$2,$ xC$3)=$

Determine the cost $($in dollars$)$ of the production schedule.

Total $=$ $