A linear programming computer package is needed.

Romans Food Market, located in Saratoga, New York, carries a variety of specialty foods from around the world. Two of the store's leading products use the Romans Food Market name: Romans Regular Coffee and Romans DeCaf Coffee. These coffees are blends of Brazilian Natural and Colombian Mild coffee beans, which are purchased from a distributor located in New York City. Because Romans purchases large quantities, the coffee beans may be purchased on an as$-$needed basis for a price $10\%$ higher than the market price the distributor pays for the beans. The current market price is $$0\mathrm{.}47$ per pound for Brazilian Natural and $$0\mathrm{.}62$ per pound for Colombian Mild. The compositions of each coffee blend are as follows.

Bean Blend

Regular DeCaf

Brazilian Natural $75\%40\%$

Colombian Mild $25\%60\%$

Romans sells the Regular blend for $$3\mathrm{.}60$ per pound and the DeCaf blend for $$4\mathrm{.}40$ per pound. Romans would like to place an order for the Brazilian and Colombian coffee beans that will enable the production of $900$ pounds of Romans Regular coffee and $400$ pounds of Romans DeCaf coffee. The production cost is $$0\mathrm{.}80$ per pound for the Regular blend. Because of the extra steps required to produce DeCaf, the production cost for the DeCaf blend is $$1\mathrm{.}05$ per pound. Packaging costs for both products are $$0\mathrm{.}25$ per pound.

$($a$)$

Formulate a linear programming model that can be used to determine the pounds of Brazilian Natural and Colombian Mild that will maximize the total contribution to profit. $($Let BR $=$ pounds of Brazilian beans purchased to produce Regular, BD $=$ pounds of Brazilian beans purchased to produce DeCaf, CR $=$ pounds of Colombian beans purchased to produce Regular, and CD $=$ pounds of Colombian beans purchased to produce DeCaf.$)$

Max:

s$.$t$.$

Regular $\%$ constraint

DeCaf $\%$ constraint

Pounds of Regular

Pounds of DeCaf

$($b$)$

What is the optimal solution?

$($BR$,$ BD$,$ CR$,$ CD$)=$

$($c$)$

What is the contribution to profit $($in $$)?$

$($Round your answer to two decimal places.$)$

$