$($Product mix problem with fixed$/$setup cost and capacity constraints$)$

Three raw materials are used to produce $3$ products: a fuel additive, a solvent base, and a carpet cleaning fluid. The profit contributions are $$40$ per ton for the fuel additive, $$30$ per ton for the solvent base, and $$50$ per ton for the carpet cleaning fluid. Each ton of fuel additive is a blend of $0\mathrm{.}4$ tons of material $1$ and $0\mathrm{.}6$ tons of material $3.$Each ton of solvent base requires $0\mathrm{.}5$ tons of material $1,0\mathrm{.}2$ tons of material $2,$ and $0\mathrm{.}3$ tons

of material $3.$ Each ton of carpet cleaning fluid is a blend of $0\mathrm{.}6$ tons of material $1,0\mathrm{.}1$ tons of material $2,$ and $0\mathrm{.}3$ tons of material $3.$ RMC has $20$ tons of material $1,5$ tons of material $2,$ and $21$ tons of material $3.$

There is a fixed cost for production setup of the products, as well as a maximum production quantity for each of the three products.

Product Setup Cost Maximum Production

Fuel additive $$20050$ tons

Solvent base $$5025$ tons

Cleaning fluid $$40040$ tons

RMC is interested in determining the optimal production quantities for the upcoming planning period to maximize its profit.