Question: The mill Mountain Coffee Shop blends coffee on the premises for its customers. It sells three basic blends in 1-pound bags, special, mountain dark, and mill regular. It uses four different types of coffee to produce the blends-Brazilian, mocha, Columbian, and mild. The shop has the following blend recipe requirements:

The cost of Brazilian coffee is $2.00 per pound, the cost of mocha is $2.75 per pound, the cost of Columbian is $2.90 per pound, and the cost of mild is $1.70 per pound. The shop has 110 pounds of Brazilian coffee, 70 pounds of mocha, 80 pounds of Columbian, and 150 pounds of mild coffee available per week. The shop wants to know the amount of each blend it should prepare each week to maximize profit.

a-Formulate a linear programming model for this problem.

b-Solve this model by using the computer.

Your work must be submitted on a single MS Word document. If using MS Word, please save it in .rtf or in MS Office XP or 2003. If using another word processor, please save it in .rtf. You will answer the questions in the word document and provide the required supporting evidence in the form of output from Excel or QM for Windows.

HINT: Your initial objective function should have 24 terms, 12 with positive coefficients and 12 with negative coefficients. You should then be able to combined common terms to reduce your objective function to twelve terms with positive coefficients.

You should have a total of 10 explicit coefficients . 4 of these are the resource constraints for the amount of each type of coffee you have to work with. Let X _ij represent the amount of coffee i used to make blend j, where i = B, M, C, I(mild), and j = S, D, R. Then your Brazilian coffee resource constraint would show as:

X _BS + X _BD + X _BR < 110

The other 6 are found in the mix requirements table. I will walk you through one of these six constraints.

Let X _ij represent the amount of coffee i used to make blend j, where i = B, M, C, I(mild), and j = S, D, R.

The amount of Special blend made is then equal to X _BS + X _MS + X _CS + X _IS . So if the Special blend has to contain at least 40% Columbian, then the ratio of Columbian to Special Blend must be at least 0.40. This is written as follows:

X _CS > 0.40
X _BS + X _MS + X _CS + X _IS

Of course, we have to re-write this constrain in standard form. Using algebra, we have:

X _CS > 0.40(X _BS + X _MS + X _CS + X _IS )

X _CS - 0.40(X _BS + X _MS + X _CS + X _IS ) > 0

0.60X _CS – 0.40X _BS – 0.40X _MS – 0.40X _IS > 0

The other 5 constraints are derived in pretty much the same way.

WARNING on objective function: You are asked to maximize profit, not revenue. Remember that profit is total revenue minus total cost. Therefore, your objective model will contain a revenue component and a cost component. The revenue component is based on the revenue for each blend, and your cost component is based on the cost for each product. You will then need to combine like terms to reduce your overall objective function expression so that you should only have 12 terms.

For example, the revenue component for the Special blend would be

6.5(X _BS + X _MS + X _CS + X _IS ). You will have two other similar expressions.

The cost component for Brazilian would be

\[2.00\left( {{X}_{BS}}+\text{ }{{X}_{BD}}+\text{ }{{X}_{BR}} \right)~~\] You will have three other similar expressions.

You will then combine the three revenue expressions and the four cost expressions to form the objective function.

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Solution: The downloadable solution consists of 5 pages
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