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:

	Blend
Bean	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 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.

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 three decimal places. For subtractive or negative numbers use a minus sign even if there is a plus sign before the blank. (Example: -300)

Max	BR	+	BD	+	CR	+	CD
s.t.
Regular blend	BR			+	CR			=
DeCaf blend			BD			+	CD	=
Regular production	BR				CR			=
DeCaf production			BD			+	CD	=
BR, BD, CR, CD ≥ 0

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.

Value of the optimal solution = $  

Answer 1

STEP 1

A) let BR be the pounds of Brazilian beans purchased to produce regular. BD be the pounds of Brazilian beans purchased to produce DeCaf , CR be the pounds of Colombian beans purchased Regular, CD be the pounds of Colombian beans purchased to produce the DeCaf.

STEP 2

They sell Regular blend for $3.60 per pound and the DeCaf blend for $4.40 per pound. their total revenue is the sum of the revenues per pound of coffee.

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

STEP 3

The current market price is $0.47 per pound for Brazilian natural ad $0.62 per pound for Colombian Mild. they pay 10% more than the market price for the beans. calculate the total cost of the beans.

Total cost of beans = 1.1(0.47BR + 0.47BD + 0.62CR + 0.62CD)

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

STEP 4

The production cost is $0.80 per pound for the regular blend and production cost is $1.05 per pound for the DeCaf blend. calculate

Total production cost = 0.8BR + 1.05BD + 0.8CR + 1.05CD

STEP 5

The total contribution to profit is total revenue minus total cost of beans minus total production cost

Total profit = (total revenue) - ( total cost of beans ) - ( total production cost )

= 2.033BR + 2.53BD + 1.868CR + 2.418CD

STEP 6

A linear program model is a mathematical model with linear objective function, a set of linear constraints, and nonnegative variables.

Max	2.033BR + 2.593BD + 1.868CR + 2.418CD
s.t.
	0.25BR - 0.75CR = 0	Regular percent
	0.6 BD - 0.4CD=0	DeCaf percent
	R + CR = 1,000	pounds of regular
	BD+CD = 500	pounds of DeCaf
	BR,BD, CR, CD >=0

STEP 7

use EXCEL to solve for the optimal solution and sensitivity report using the fallowing steps.

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 variable cells box select 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 reapers

click the check box for make unconstrained variables non-negative

6. select the Select a solving method drop-down button

select Simple x LP*

7. click solve

8. when the solver results dialog box appears

select keep solver solution

select Sensitivity in the Reports box

click OK

STEP 8

the optimal solution is BR = 750, BD = 200 , CR = 250, and CD = 300. substitute the optimal solution into the objective function to find the contribution to profit.

2.033BR + 2.583BD + 1.868CR + 2.418CD

= 2.033(750) + 2.583(200) + 1.868(250) + 2.418(300)

=$ 3,233.75