Name: Romans Regular Cotfee and Romans DeCaf Coffee. These coffees are blends of Brazilian Natural and Colomblan 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: Romans selis 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 caffee 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 biend 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, 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 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: BR=. BD=. CR=. CD=. If required, round your answer to the nearest cent
The linear programming model can be formulated as follows:
Objective function:
Maximize profit = (3.2 - 0.89 - 0.25)BR + (4.3 - 1.09 - 0.25)BD
Subject to:
BR + BD = 900 (pounds of Romans Regular coffee)
CR + CD = 500 (pounds of Romans DeCaf coffee)
The amount of Brazilian Natural beans required for the Regular blend:
BR + BD = (0.47 + 0.11*0.47)(BR + BD)
The amount of Colombian Mild beans required for the Regular blend:
CR + CD = (0.62 + 0.11*0.62)(CR + CD)
The optimal solution can be obtained by solving this linear programming model using a solver such as the Simplex method or the Excel Solver.
Unfortunately, the given problem doesn't provide all the necessary information to determine the final optimal solution and the contribution to profit. The problem only provides the market prices of the coffee beans and the production costs for each blend, but does not give the proportion of Brazilian Natural and Colombian Mild beans in each blend.
To obtain the optimal solution and contribution to profit, the proportions of Brazilian Natural and Colombian Mild beans in each blend need to be known.
To formulate the linear programming model, we need to define the objective function and the constraints.
Objective function:
We want to maximize the total contribution to profit, which is the revenue minus the cost. The revenue is given by the selling price per pound multiplied by the pounds produced, while the cost is given by the production cost per pound plus the packaging cost per pound multiplied by the pounds produced.
Let's define:
x1 = pounds of Brazilian beans purchased
x2 = pounds of Colombian beans purchased
The objective function, then, becomes:
Maximize: 3.2(900) + 4.3(500) - (0.89(900) + 0.25(900) + 1.09(500) + 0.25(500))
Constraints:
1. The pounds of Brazilian beans purchased for the Regular blend should be enough to produce 900 pounds of Romans Regular coffee, and for the DeCaf blend should be enough to produce 500 pounds of Romans DeCaf coffee. This can be expressed as inequalities:
0.47x1 + 0.62x2 ≥ 900 (for Regular)
0.47x1 + 0.62x2 ≥ 500 (for DeCaf)
2. The pounds of Colombian beans purchased for the Regular blend should be enough to produce 900 pounds of Romans Regular coffee, and for the DeCaf blend should be enough to produce 500 pounds of Romans DeCaf coffee. This can be expressed as inequalities:
0.47x1 + 0.62x2 ≥ 900 (for Regular)
0.47x1 + 0.62x2 ≥ 500 (for DeCaf)
3. The pounds of Brazilian and Colombian beans cannot be negative, so:
x1 ≥ 0
x2 ≥ 0
Considering the above objective function and constraints, the complete linear programming model is as follows:
Maximize: 3.2(900) + 4.3(500) - (0.89(900) + 0.25(900) + 1.09(500) + 0.25(500))
Subject to:
0.47x1 + 0.62x2 ≥ 900 (for Regular)
0.47x1 + 0.62x2 ≥ 500 (for DeCaf)
x1 ≥ 0
x2 ≥ 0
To find the optimal solution and the contribution to profit, we need to solve this linear programming problem using an optimization algorithm or software. The optimal solution will provide the values for BR, BD, CR, and CD, while the objective function's value will give us the contribution to profit.