?Romans Food Market, located in Saratoga, New York, carries a variety of special

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 $0.48 per pound for Brazilian Natural and $0.69 per pound for Colombian Mild. The compositions of each coffee blend are as follows

Romans sells the Regular blend for $3.5 per pound and the DeCaf blend for $4.4 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 450 pounds of Romans DeCaf coffee. The production cost is $0.77 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.

If required, round your answers to four decimal places. For subtractive or negative numbers use a minus sign even if there is a plus sign before the blank. (Example: -300)

The complete linear program is

What is the optimal solution and what is the contribution to profit? If required, round your answer to the nearest whole number.

Optimal solution:

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 CD = pounds of Colombian beans purchased to produce DeCaf

Explanation / Answer

Cost of Brazilian Natural = $0.48 x 1.11 = $0.5328 per lb.
Cost of Colombian Mild = $0.69 x 1.11 = $0.7659 per lb.

Selling price for Regular Coffee = $3.5 per lb.
Selling price for Decaf Coffee = $4.4 per lb.

Production cost of Regular Coffee = $0.77 per lb.
Production cost of Decaf Coffee = $1.09 per lb.

Packaging cost for both the blends = $0.25 per lb.

Profit function, P = $3.5 (BR+CR) + $4.4 (BD+CD) - $0.5328 (BR+BD) - $0.7659 (CR+CD) - $0.77 (BR+CR) - $1.09 (BD+CD) - $0.25 (BR+BD+CR+CD)

or. P = 1.9472 BR + 2.5272 BD + 1.7141 CR + 2.2941 CD

Formulation

Maximize 1.9472 BR + 2.5272 BD + 1.7141 CR + 2.2941 CD
s.t.
1 BR + 0 BD + 1 CR + 0 CD = 900
0 BR + 1 BD + 0 CR + 1 CD = 450
0.3 BR + 0 BD - 0.7 CR + 0 CD = 0
0 BR + 0.6 BD + 0 CR - 0.4 CD = 0
BR, BD, CR, CD >= 0

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LINDO code

Max 1.9472 BR + 2.5272 BD + 1.7141 CR + 2.2941 CD
s.t.
1 BR + 0 BD + 1 CR + 0 CD = 900
0 BR + 1 BD + 0 CR + 1 CD = 450
0.3 BR + 0 BD - 0.7 CR + 0 CD = 0
0 BR + 0.6 BD + 0 CR - 0.4 CD = 0
end

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Optimal solution

BR = 630
BD = 180
CR = 270
CD = 270

Objective value at optimal solution = $2763.85

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