$($a$)$ Show the linear programming formulation. $($Let $x_{A1}$ be the number of units of product A produced by machine $1,x_{ij}$ be the number of units of product i produced by machine $j,$ etc.$)$

$Min{x}_{A1}+1\mathrm{.}3x_{A2}+1\mathrm{.}1x_{A3}+1\mathrm{.}2x_{B1}+1\mathrm{.}4x_{B2}+x_{B3}+0\mathrm{.}9x_{C1}+1\mathrm{.}2x_{C2}+1\mathrm{.}2x_{C3}$

s$.$t$.$

Machine $1$ Capacit$)$

Machine $2$ Capacity

Machine $3$ Capacit

Product A Orde

Product B Order

Product $C$ Orders

$x_{ij}0$ for all $i,j.$

$($b$)$ Show the production schedule.

$(x_{A1},x_{A2},x_{A3},x_{B1},x_{B2},x_{B3},x_{C1},x_{C2},x_{C3})=(x)$

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

Total $=$$The Ace Manufacturing Company has orders for three similar products.

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.

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

$($a$)$ Show the linear programming formulation. $($Let $x_{A1}$ be the number of units of product A produced by machine $1,x_{ij}$ be the number of units of product $\frac{?}{?}$ produced by machine $j,$ etc.$)$

$Min{x}_{A1}+1\mathrm{.}3x_{A2}+1\mathrm{.}1x_{A3}+1\mathrm{.}2x_{B1}+1\mathrm{.}4x_{B2}+x_{B3}+0\mathrm{.}9x_{C1}+1\mathrm{.}2x_{C2}+1\mathrm{.}2x_{C3}$