Question

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 11% higher than the market price the distributor pays for the beans. The current market price is $0.47 per pound for Brazilian Natural and $0.62 per pound for Colombian Mild. The compositions of each coffee blend are as follows:

Bean Regular DeCaf

Brazilian Natural 75% 35%

Colombian Mild 25% 65%

Romans sells the Regular blend for $3.2 per pound and the DeCaf blend for $4.3 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 500 pounds of Romans DeCaf coffee. The production cost is $0.89 per pound for the Regular blend. Because of the extra steps required to produce DeCaf, the production cost for the DeCaf blend is $1.09 per pound. Packaging costs for both products are $0.25 per pound. 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.

LetBR = pounds of Brazilian beans purchased to produce

RegularBD = pounds of Brazilian beans purchased to produce

DeCafCR = pounds of Colombian beans purchased to produce

RegularCD = pounds of Colombian beans purchased to produce DeCaf

Answer 1

Profit for Regular blend per pound = (3.2-[0.89+0.25]) = 2.06

Profit from Brazilian Natural in Regular blend per pound = 0.75*2.06 = 1.545

Profit from Colombian beans in Regular blend per pound = 0.25*2.06 = 0.515

Profit for DeCaf blend per pound = (4.3-[1.09+0.25]) = 2.96

Profit from Brazilian Natural in DeCaf blend per pound = 0.35*2.96 = 1.036

Profit from Colombian beans in DeCaf blend per pound = 0.65*2.96 = 1.924

Revenue from Brazilian Natural = 2.581

Profit from Brazilian Natural = 2.581 - 0.47*1.11 = 2.0593

Revenue from Colombian beans = 2.439

Profit from Colombian beans = 2.439 - 0.62*1.11 = 1.7508

Let BR = pounds of Brazilian beans purchased

RegularCD = pounds of Colombian beans purchased

Maximise 2.0593*BR + 1.7508*RegularCD

Constraints:

BR <= 0.75*900 + 0.35*500 = 850

RegularCD< = 0.25*900 + 0.65 *500 = 550

BR, RegularCD > 0