$03.(1$ pt$)$

A production manager is planning the scheduling of $2$ products on $3$ machines. Each product can be manufactured on any of the machines. The unit production costs $($in $$)$ are summarized below:

Cost per unit $($$$)$ Product

Machine $12$

$146$

$247$

$355$

The time $($in hours$)$ required to produce a unit of each product on each of the machines is summarized below:

Time required per unit $($hrs$)$ Product

Machine $12$

$10\mathrm{.}300\mathrm{.}20$

$20\mathrm{.}250\mathrm{.}30$

$30\mathrm{.}200\mathrm{.}20$

The customer demand for Products $1$ and $2$ are $2000$ and $4000$ respectively. These demand requirements must be satis$$ed$.$ Moreover, the available machine hours for Machines $1,2,$ and, $3$ are $1125,900,$ and $1125$ respectively. De$$ne the following decision variables:

x$11$: the quantity of Product $1$ produced on Machine $1$ x$12$: the quantity of Product $1$ produced on Machine $2$ x$13$: the quantity of Product $1$ produced on Machine $3$ x$21$: the quantity of Product $2$ produced on Machine $1$ x$22$: the quantity of Product $2$ produced on Machine $2$ x$23$: the quantity of Product $2$ produced on Machine $3$

To optimize the total cost of production $($and assuming non$-$negativity for all decision variables$),$ the pro$-$ duction manager should solve:

$($a$)$ min $4$x$11+4$x$12+5$x$13+6$x$21+7$x$22+5$x$23$

subject to:

$0\mathrm{.}3$x$11+0\mathrm{.}2$x$211125$

$0\mathrm{.}25$x$12+0\mathrm{.}3$x$22900$

$0\mathrm{.}2$x$13+0\mathrm{.}2$x$231125$

x$11+$ x$12+$ x$13=2000$

x$21+$ x$22+$ x$23=4000$

$($b$)$ min $4$x$11+4$x$12+5$x$13+6$x$21+7$x$22+5$x$23$

subject to:

$0\mathrm{.}3$x$11+0\mathrm{.}2$x$121125$

$0\mathrm{.}25$x$21+0\mathrm{.}3$x$22900$

$0\mathrm{.}2$x$31+0\mathrm{.}2$x$321125$

x$11+$ x$21+$ x$31=2000$

x$12+$ x$22+$ x$32=4000$

$($c$)$ min $4$x$11+4$x$12+5$x$13+6$x$21+7$x$22+5$x$23$

subject to:

$0\mathrm{.}3$x$11+0\mathrm{.}2$x$121125$

$0\mathrm{.}25$x$21+0\mathrm{.}3$x$22900$

$0\mathrm{.}2$x$31+0\mathrm{.}2$x$321125$ x$11+$ x$21+$ x$312000$ x$12+$ x$22+$ x$324000$

$($d$)$ max $4$x$11+4$x$12+5$x$13+6$x$21+7$x$22+5$x$23$

subject to:

$0\mathrm{.}3$x$11+0\mathrm{.}2$x$211125$

$0\mathrm{.}25$x$12+0\mathrm{.}3$x$22900$

$0\mathrm{.}2$x$13+0\mathrm{.}2$x$231125$

x$11+$ x$12+$ x$13=2000$

x$21+$ x$22+$ x$23=4000$