GMBA 402C HOMEWORK #1

DUE SEP. 15

1. Fresh Dairy Farms has two different machines for processing raw milk into low-fat milk, butter, and cheese. The amount of time required on each machine to produce each unit of resulting product and the net profits are provided in the following table:

Low-fat Milk Butter Cheese Machine 1 0.2 min/gal 0.5 min/lb 1.5 min/lb Machine 2 0.3 min/gal 0.7 min/lb 1.2 min/lb Net Profit $0.22 /gal $0.38 /lb $0.72 /lb

Assuming that 8 hours of time are available each day on each machine, as manager of the Production Department, formulate a model to determine a daily production plan that maximizes the net corporate profits and yields a minimum of 300 gallons of low-fat milk, 200 pounds of butter, and 100 pounds of cheese. (Note: Any amount of milk can be produced on either machine without having to use the other machine. The same is true for each of the other two products.) Build a mathematical model using variables, an objective function, and constraints and classify the model using the scheme presented in class. (15 points)

2. The Leather Company produces baseball gloves, footballs, and leather straps using raw

leather that is processed on a machine. For the coming week, there are 1000 square meters of leather and 40 hours of machine time available. The manager wants to determine a production plan for this week that maximizes the net corporate profits. (a) Identify the variables. (5 points) (b) Identify and assign a symbolic name to each additional data value you would have to

obtain to be able to formulate a mathematical model. (5 points) (c) Formulate a mathematical model using the variable names from part (a) and the symbolic

names for the data from part (b). Classify the model using the scheme presented in class. (5 points)

3. Each week, Florida Citrus, Inc., uses a single machine for 150 hours to distill orange and

grapefruit juice into concentrates that are then stored in two separate 1000-gallon tanks (one for orange juice and one for grapefruit juice) before being frozen. The machine can process 25 gallons of orange juice per hour but only 20 gallons of grapefruit juice. Each gallon of orange juice cost $1.50 and loses 30% in water content when distilled into a concentrate that then sells for $6.00 per gallon. Each gallon of grapefruit juice costs $2.00 and loses 25% when distilled into a concentrate that then sells for $8.00 per gallon. Formulate a model to determine a production plan to maximize the profit for the coming week using the following variables: (15 points)

OJ = the number of gallons of orange juice to use this week, GJ = the number of gallons of grapefruit juice to use this week.

4. ASA is a large steel company that produces each of five different types of iron plates at its eight factories. At a recent strategic-planning meeting, the management allocated the budgets in Table 1 to each of the eight factories for the next fiscal year. These budgets were based, in part, on the demands in Table 2 for the five iron plates, provided by the Sales Department. As vice president of production, you, Mr. Leroy Adams, have been asked to determine a production plan for each of the eight factories. (50 points)

Table 1. The Budget Assigned to Each Factory (in Thousands of Dollars) Factory 1 2 3 4 5 6 7 8 Budget 900 1050 950 1050 1000 1600 950 1050

Table 2. The Demand in Tons for Each Type of Iron Plate

Plate Type 1 2 3 4 5 Demand 450 800 500 650 180

In preparation to formulate such a plan, you asked your associate, Mr. James Aaron, to obtain production costs for each of the five iron plates at each of the eight factories. At your next meeting, Mr. Aaron brought the cost data in Table 3. He also pointed out that, as external suppliers, ASA has three subcontractors from which they can purchase precisely the amounts of different types of iron plates at the prices shown in Table 4. Mr. Aaron reminded you that contractual obligations require that if a particular subcontractor is chosen, ASA must purchase all of that suppliers five types of iron plates at the indicated prices. You can, of course, choose to purchase from none, one, two, or all three subcontractors. When you relayed this information to Mr. Charles Bentley, the CEO, he approved the use of subcontractors if it would lower the total costs and informed you that the cost for buying the iron plates from the subcontractors would not affect the budgets of any of the factories (see Table 1). As production manager, formulate a model to determine a production plan for each of the eight factories and which subcontractor(s), if any, to buy iron plates from to minimize the total costs while staying within each factory's budget and at least meeting the given demands.

The Table 3. The Cost of Producing One Ton of Each Iron Plate Type i at Factory j

(in Thousands of Dollars)

Factory 1 2 3 4 5 6 7 8 Plate Type

1 5 3 4 3 3 4 6 4 2 3 4 6 2 5 3 6 4 3 7 6 5 8 4 3 4 5 4 6 6 6 5 3 5 5 4 5 8 9 8 7 10 9 8 6

Table 4. The Cost per Ton and the Amount of Each Plate Type Available from Each Subcontractor

Subcontractor 1 Subcontractor 2 Subcontractor 3 Plate Type Costa Amountb Costa Amountb Costa Amountb

1 5 40 4 10 5 20 2 5 20 8 80 6 40 3 5 30 6 50 6 10 4 3 40 4 20 5 10 5 5 20 3 10 4 50

a Cost ($000) to purchase one ton of plate type i from a subcontractor. b For any subcontractor used, all of its available amounts of all five types of iron plates must be purchased at the indicated costs.