Question

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:

                                                                Blend

Bean Regular                 DeCaf

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 Columbian coffee beans that will enable the production of 1000 pounds of Romans Regular coffee and 500 pound 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 Columbian Mild that will maximize the total contribution to profit. What is the optimal solution and what is the contribution profit?

Answer 1

Concepts and reason

The Linear Programming problem with more than two variables can be simplified with simplex method. The simplex method is used to move from one corner point of the solution space to the better part of the solution until the optimum feasible solution is obtained.

Fundamentals

Instructions for Excel, to find the optimal solution of the LPP model:

1.Select the Data tab from the Ribbon.

2.Select solver from the Analysis Group.

3.When the solver parameters dialog appears:

Enter the objective function.

Select the To: Max option.

Enter the decision variables into the By Changing Variables Cells box.

Select the option “Add”.

4.When the Add Constraint dialog box appears:

Enter the constraints in the Cell Reference box.

Select =

Enter the right hand side in the Constraint box.

Click OK.

5.When the solver Parameters dialog box reappears:

Click the checkbox for Make Unconstrained Variables Non-negative.

6.Select the option “Select a solving Method” drop down button.

Select Simplex LP.

7.Click Solve.

8.When the Solver Results dialog box appears:

Select Keep Solver Solution.

Select Sensitivity in the Reports box.

Click OK.

Total Revenue is expressed as shown below:

${\rm{Total Revenue}} = 3.6BR + 4.4BD + 3.6CR + 4.4CD$

The total cost of bean will be,

$\begin{array}{c}\\{\rm{Total cost of Beans}} = 1.1\left( {0.47BR + 0.47BD + 0.62CR + 0.62CD} \right)\\\\ = 0.517BR + 0.517BD + 0.682CR + 0.682CD\\\end{array}$

The total production cost will be,

${\rm{Total production cost}} = 0.8BR + 1.05BD + 0.8CR + 1.05CD$

Then the total contribution of profit is,

$\begin{array}{c}\\{\rm{Total profit = Total Revenue}} - {\rm{Total cost of Beans}} - {\rm{Total production cost}}{\rm{.}}\\\\ = \left\{ \begin{array}{c}\\3.6BR + 4.4BD + 3.6CR + 4.4CD\\\\ - \left( {0.517BR + 0.517BD + 0.682CR + 0.682CD} \right)\\\\ - \left( {0.8BR + 1.05BD + 0.8CR + 1.05CD} \right)\\\end{array} \right\}\\\\ = 2.033BR + 2.583BD + 1.868CR + 2.418CD\\\end{array}$

The LPP model to the given problem can be written as follows:

Objective function:

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

Subject to constraints:

$\begin{array}{l}\\0.25BR - 0.75CR = 0\\\\0.6BD - 0.4CD = 0\\\\BR + CR = 1000\\\\BD + CD = 500\\\end{array}$

Non-negative restrictions:

$BR,BD,CR,CD \ge 0$

Use Excel, to find the optimal solution of the LPP model.

Excel Output:

Ans:

From the output, the optimal solution is,

$BR = 750$ , $BD = 200$ , $CR = 250$ and $CD = 300$

Then the optimum solution is,

$\begin{array}{c}\\MaxZ = 2.033BR + 2.583BD + 1.868CR + 2.418CD\\\\ = \left( {2.033 \times 750} \right) + \left( {2.583 \times 200} \right) + \left( {1.868 \times 250} \right) + \left( {2.418 \times 300} \right)\\\\ = 1524.75 + 516.6 + 467 + 725.4\\\\ = 3233.75\\\end{array}$