Question

Problem 9-13 (Algorithmic)

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 $0.47 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%	35%
Colombian Mild	25%	65%

Romans sells 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 coffee 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 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.

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 + sign before the blank. (Example: -300)

The complete linear program is

Max	BR	+	BD	+	CR	+	CD
s.t.
	BR			+	CR			=
			BD			+	CD	=
	BR				CR			=
			BD			+	CD	=
BR, BD, CR, CD ≥ 0

What is the contribution to profit?

Optimal solution:

BR =

BD =

CR =

CD =

If required, round your answer to two decimal places.

Value of the optimal solution = $  

Answer 1

Solution:-

We have the following decision variables,

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

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Given, the market price

Brazilian Natural = $0.47 per pound
Colombian Mild = $0.62 per pound
Therefore, the purchase price (11% higher)

Brazilian Natural = $0.5217 per pound
Colombian Mild = $0.6882 per pound
-------

Given, the selling price

Romans Regular - $3.2 per pound
Romans DeCaf - $4.3 per pound
-------

Given, the production price

Romans Regular - $0.89 per pound
Romans DeCaf - $1.09 per pound
-------

Given, the packaging price - $0.25 per pound

--------

Now, the amount of Regular produce needed = 900 pounds and, the amount of DeCaf Needed = 500 pounds

Therefore, based on the composition table,

0.75BR + 0.25CR >= 900 and,

0.35BD + 0.65CD >= 500

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Another set of constraints, are related to the fact that the purchases of BR and CR, BD and CD are not entirely independent of one another as each influences the other based on their ratios.

i.e. if 0.75 pounds of Brazilian is purchased for regular, 0.25 pounds of Colombian is purchased i.e. if 1 pound of Brazilian is purchased, 0.3333 pounds of Colombian needs to be purchased and so, if BR pounds of Brazilian is purchased, BR/3 pounds of Colombian needs to be purchased. Therefore,

CR = BR/3

or, BR = 3CR

Similarly, for the DeCaf,

if 0.35 pounds of Brazilian is purchased, 0.65 pounds of Colombian needs to be purchased. Therefore, if BD pounds of Brazilian is purchased, 13/7 pounds of Colombian needs to be purchased and so,

13BD = 7CD

------------------------------------

Based on the composition table, we know that the amount of Regular produce is

1R = 0.75BR + 0.25CR

Therefore, we can calculate the profit earned to be = Selling Cost - Production Cost - Packaging Cost - Material Cost which is

Regular_Profit = 3.2R - 0.89R - 0.25R - (0.5217BR + 0.6882CR)

replacing the value of R gives,

Regular_Profit = 1.0233BR - 0.1732CR

Similarly, based on the composition table, we know that the amount of DeCaf produce is,

1D = 0.35BR + 0.65CR

Therefore, the profit is

Decaf_Profit = 4.3D - 1.09D - 0.25D - (0.5217BD + 0.6882CD)

Decaf_Profit = 0.5143BD + 1.2358CD

Therefore, the total profit is

Profit = Regular_Profit + Decaf_Profit = 1.0233BR - 0.1732CR + 0.5143BD + 1.2358CD

We need to maximise this profit.

-------------------------------

Therefore, the Linear Programming model is,

max. 1.0233BR - 0.1732CR + 0.5143BD + 1.2358CD

s. t.

0.75BR + 0.25CR >= 900

0.35BD + 0.65CD >= 500

BR - 3CR = 0

13BD - 7CD = 0

BR, BD, CR, CD >= 0

----------------------------

We can solve this linear program to get the solution

BR = 1080
BD = 360
CR = 321.101
CD = 596.330

The value of the profit is - $1944.899

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