he Ace Manufacturing Company has orders for three similar products:

Product Order $($Units$)$

A $2000$

B $500$

C $1200$

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 as follows:

Machine Capacity $($Units$)$

$11500$

$21500$

$31000$

Product

Machine A B C

$1$ $$1\mathrm{.}05$ $$1\mathrm{.}20$ $$0\mathrm{.}90$

$2$ $$1\mathrm{.}30$ $$1\mathrm{.}40$ $$1\mathrm{.}20$

$3$ $$1\mathrm{.}10$ $$1\mathrm{.}00$ $$1\mathrm{.}20$

Use the transportation model to develop the minimum cost production schedule for the products and machines. Show the linear programming formulation. If the constant is $"1"$ it must be entered in the box. If your answer is zero enter $"0".$

The linear programming formulation and optimal solution are shown.

Let xij $=$ Units of product j on machine i$.$

Min fill in the blank $1$x$1$A $+$ fill in the blank $2$x$1$B $+$ fill in the blank $3$x$1$C $+$ fill in the blank $4$x$2$A $+$ fill in the blank $5$x$2$B $+$ fill in the blank $6$x$2$C $+$ fill in the blank $7$x$3$A $+$ fill in the blank $8$x$3$B $+$ fill in the blank $9$x$3$C

s$.$t$.$

fill in the blank $10$x$1$A $+$ fill in the blank $11$x$1$B $+$ fill in the blank $12$x$1$C $<=$ fill in the blank $13$

fill in the blank $14$x$2$A $+$ fill in the blank $15$x$2$B $+$ fill in the blank $16$x$2$C $<=$ fill in the blank $17$

fill in the blank $18$x$3$A $+$ fill in the blank $19$x$3$B $+$ fill in the blank $20$x$3$C $<=$ fill in the blank $21$

fill in the blank $22$x$1$A $+$ fill in the blank $23$x$2$A $++$ fill in the blank $24$x$3$A $=$ fill in the blank $25$

fill in the blank $26$x$1$B $+$ fill in the blank $27$x$2$B $+$ fill in the blank $28$x$3$B $=$ fill in the blank $29$

fill in the blank $30$x$1$C $+$ fill in the blank $31$x$2$C $+$ fill in the blank $32$x$3$C $=$ fill in the blank $33$

xij $>=0$ for all i$,$ j

Optimal Solution Units Cost

$1-$A fill in the blank $34$ $ fill in the blank $35$

$1-$B fill in the blank $36$ $ fill in the blank $37$

$1-$C fill in the blank $38$ $ fill in the blank $39$

$2-$A fill in the blank $40$ $ fill in the blank $41$

$2-$B fill in the blank $42$ $ fill in the blank $43$

$2-$C fill in the blank $44$ $ fill in the blank $45$

$3-$A fill in the blank $46$ $ fill in the blank $47$

$3-$B fill in the blank $48$ $ fill in the blank $49$

$3-$C fill in the blank $50$ $ fill in the blank $51$

Total $ fill in the blank $52$

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