Question

g 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 1000 pounds of Romans Regular coffee and 500 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.

Answer 1

Romans Food Market, located in Saratoga, New York, carries a variety of specialty foods from around the world. Two of the stores leading products use the Romans Food Market name: Romans Regular Coffee and Romans DeCaf Coffee. These coffees are blends of Brazilian Natural and Columbian mild coffee beans, which are purchased from a distributor from New York City. Because Romans purchases large quantities the coffee beans may be purchased om an as need basis for the price of 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 Columbian Mild The composition of each coffee blend are as follows:

                                                             

Bean Regular                 DeCaf                      Blend

Brazilian Natural               75%                      40%

Columbian 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 1000 pounds of Romans Regular coffee and 500 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.

Answer:

$\mathbf{Max \ Z = 2.033 BR + 2.583 BD + 1.868 CR + 2.418 CD}$

Explanation:

From the given information:

The total revenue can be illustrated as :

Total revenue = 3.6 BR +  4.4 BD + 3.6 CR + 4.4 CD

On the other hand; the total cost  of the beans is:

= 1.1 (0.47 BR  + 0.47 BD + 0.62 CR + 0.62 CD)

=  0.517  BR + 0.517 BD + 0.682 CR + 0.682 CD

Also; The total production cost is :

= 0.8 BR + 1.05 BD + 0.8 CR + 1.05 CD

The total profit  = Total revenue - Total Cost of Beans - Total Production Cost

The total profit  =  $\left[\begin{array}{}3.6 BR + 4.4 BD + 3.6 CR + 4.4 CD\\- (0.517 BR + 0.517 BD + 0.682 CR + 0.682 CD)\\-(0.8 BR + 1.05 BD + 0.8 CR + 1.05 CD)\end{array}\right]$

The total profit  = 2.033 BR + 2.583 BD + 1.868 CR + 2.418 CD

Therefore the  linear programming model represents the Objective function of the total profit as:

$\mathbf{Max \ Z = 2.033 BR + 2.583 BD + 1.868 CR + 2.418 CD}$