Region A Raw Milk Milk from region A is 54 cents per gallon up to 500 gallons and 58 cents for each gallon in excess of 500 gallons. There is no upper bound on the amount that can be purchased. Raw milk from Region A has 5% butterfat and when separated (at 5 cents per gallon) it yields two varities of milk, a high-fat variety with 80% butterfat and a low-fat variety with 1% butter fat.

The volume of milk is conserved in all separation processing.

Region B Raw Milk. The purchase price for milk from region B is 44 cents per gallon up to 700 gallons and 45 cents per gallon thereafter. Raw milk from Region B has 3.5% butterfat and when separated (at 6 cents per gallon) yields two varieties, a high-fat milk with 80% butterfat and a low-fat milk with 0.5% butterfat.

Production Process. After the raw milk is purchased and collected at the plant, some of it may drawn off and separated into the high-fat and low-fat varieties. All the resulting varieties are then mixed to produce two final products: "milk" and "cream", which have certain minimum butterfat requirements (listed below). .

Demand and Selling Price. The demand and selling price for the final products are described in the following table:

	Minimum required percent of butterfat	Maximum volume demanded in gallons	Selling price in dollars per gallon
Cream	50.0	300	4.50
Milk	2.0	2000	0.80

For example, all the cream produced must have at least 50% butterfat and sells at $4.50 per gallon, and as much as 300 gallons of cream produced can be sold. Larger amounts can be produced, and the excess over the maximum volume disposed of free of cost.

Problem.

• A. Using GAMS, formulate and solve a linear model for maximiming profit.
• B. Try changing the price for each gallon of region-B milk in excess of 700 gallons from 45 cents to 46, 47, and 48 cents, respectively, and re-solve the problem for each of these values. In each case, comment on how the optimal profit and the purchased amounts of raw milk changes.
• C. Suppose that the purchase price for milk from region B were 44 cents per gallon up to 700 gallons and 40 cents per gallon thereafter. Can you still formulate the profit maximization problem as a linear program? If not, why not?