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 10% 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	Blend
	Regular	DeCaf
Brazilian Natural	75%	40%
Colombian Mild	25%	60%

Romans sells the Regular blend for $3.60 per pound and the DeCaf blend for $4.40 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 $0.80 per pound for the Regular blend. Because of the extra steps required to produce DeCaf, the production cost for the DeCaf blend is $1.05 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. (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.)

What is the optimal solution and what is the contribution to profit (in $)? (Round your contribution to profit to two decimal places.)

(BR, BD, CR, CD)=

Profit =

Answer 1

Selling Price for Regular Coffee = 3.6 per lb              

Selling Price for DeCaf Coffee = 4.4 per lb              

Cost of Brazilian Natural = 0.47 * 1.1 = 0.517 per lb

Cost of Columbian Mild = 0.62 * 1.1 = 0.682 per lb

Production cost for Regular coffee = 0.8 per lb              

Production cost for DeCaf coffee = 1.05 per lb              

Packing cost = 0.25 per lb              

Profit = Selling - Cost - Production cost - Packing cost

= 3.6(BR + CR) + 4.4(BD + CD) - 0.517(BR + BD) - 0.682(CR + CD) - 0.8(BR + CR) - 1.05(BD + CD) - 0.25(BR + BD + CR + CD)  

= 2.033 BR + 2.583 BD + 1.868 CR + 2.418 CD  

Max: Z = 2.033 BR + 2.583 BD + 1.868 CR + 2.418 CD
s.t.  

BR + CR = 900  

BD + CD = 400  

0.25*BR -0.75*CR = 0  

0.6*BD -0.4*CD = 0  

BR =	675
BD =	160
CR =	225
CD =	240
Profit =	2786.175