# Linear Programming : Defining Costraints and Maximizing Profit

Tom and Jerry, Inc., supplies its ice cream parlors with three flavors of ice cream: chocolate, vanilla, and banana. Due to extremely hot weather and a high demand for its products, the company has run short of its supply of ingredients: milk, sugar, and cream. Hence, they will not be able to fill all the orders received from their retail outlets, the ice cream parlors. Due to these circumstances, the company has decided to choose the amount of each flavor to produce that will maximize total profit, given the constraints on supply of the basic ingredients.

The chocolate, vanilla, and banana flavors generate, respectively, $1.00, $0.90, and $0.95 of profit per gallon sold. The company has only 200 gallons of milk, 150 pounds of sugar, and 60 gallons of cream left in its inventory. The linear programming formulation for this problem is shown below in algebraic form.

Let C = gallons of chocolate ice cream produced,

V = gallons of vanilla ice cream produced,

B = gallons of banana ice cream produced.

Maximize Profit = 1.00 C + 0.90 V + 0.95 B,

subject to

Milk: 0.45 C + 0.50 V + 0.40 B ≤ 200 gallons

Sugar: 0.50 C + 0.40 V + 0.40 B ≤ 150 pounds

Cream: 0.10 C + 0.15 V + 0.20 B ≤ 60 gallons

and

C ≥ 0, V ≥ 0, B ≥ 0.

This problem was solved using the Excel Solver. The spreadsheet (already solved) and the sensitivity report are shown below. [Note: The numbers in the sensitivity report for the milk constraint are missing on purpose, since you will be asked to fill in these numbers in part (f).]

See attached file for full problem description.

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#### Solution Preview

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Tom and Jerry, Inc., supplies its ice cream parlors with three flavors of ice cream: chocolate, vanilla, and banana. Due to extremely hot weather and a high demand for its products, the company has run short of its supply of ingredients: milk, sugar, and cream. Hence, they will not be able to fill all the orders received from their retail outlets, the ice cream parlors. Due to these circumstances, the company has decided to choose the amount of each flavor to produce that will maximize total profit, given the constraints on supply of the basic ingredients.

The chocolate, vanilla, and banana flavors generate, respectively, $1.00, $0.90, and $0.95 of profit per gallon sold. The company has only 200 gallons of milk, 150 pounds of sugar, and 60 gallons of cream left in its inventory. The linear programming formulation for this problem is shown below in algebraic form.

Let C = gallons of chocolate ice cream produced,

V = gallons of vanilla ice cream produced,

B = gallons of banana ice cream produced.

Maximize Profit = 1.00 C + 0.90 V + 0.95 B,

subject to

Milk: 0.45 C + 0.50 V + 0.40 B ≤ 200 gallons

Sugar: 0.50 C + 0.40 V + 0.40 B ≤ 150 pounds

Cream: 0.10 C + 0.15 V + 0.20 B ≤ 60 ...

#### Solution Summary

An LP problem is solved. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.