Problems:
Linear Programming
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).]
For each of the following parts, answer the question as specifically and completely as is possible without solving the problem again on the Excel Solver. Note: Each part is independent (i.e., any change made to the model in one part does not apply to any other parts).
(a) What is the optimal solution and total profit?
(b) Suppose the profit per gallon of banana changes to $1.00. Will the optimal solution change, and what can be said about the effect on total profit?
(c) Suppose the profit per gallon of banana changes to 92¢. Will the optimal solution change, and what can be said about the effect on total profit?
(d) Suppose the company discovers that 3 gallons of cream have gone sour and so must be thrown out. Will the optimal solution change, and what can be said about the effect on total profit?
(e) Suppose the company has the opportunity to buy an additional 15 pounds of sugar at a total cost of $15. Should they? Explain.
(f) Fill in all the sensitivity report information for the milk constraint, given just the optimal solution for the problem. Explain how you were able to deduce each number.