An automobile company is considering manufacturing three types of cars $($compact$,$ midsize, and large$)$ and two types of minivans $($midsize and large$).$ The resources required and the profit contributions yielded by each type of vehicle are shown in Exhibit $1.$ At present, $6500$ tons of steel and $65,000$ hours of labor are available. If any vehicles of a given type are produced, production of that type of vehicle is economically feasible only if at least a minimal number of that type are produced. These minimal numbers are also listed in Exhibit $1.$

Develop an Integer Programming model to obtain an optimal production schedule. What will be the optimal solution if the steel availability changes to $6000$ tons?

Vehicle Type

Steel $($tons$)/$unit

Labour $($hours$)/$unit

Compact

Car

$1\mathrm{.}5$

$30$

Exhibit $1$

Mid$-$Size

Car

$3$

Large Car

Mid$-$Size

Minivan

Large

Minivan

$5$

$25$

$40$

$6$

$45$

$8$

$55$

Minimum Production $($if any$)$

$1000$

$1000$

$1000$

$200$

$200$

Profit Contribution per unit $(\text{'})$

$100000$

$125000$

$150000$

$275000$

$350000$