The Valley Wine Company produces two kinds of wine—Valley Nectar and Valley Red. The wines are produced from 64 tons of grapes the company has acquired this season. A 1000-gallon batch of Nectar requires 4 tons of grapes and a batch of Red requires 8 tons. However, production is limited by the availability of only 50 cubic yards of storage space for aging and 120 hours of processing time. A batch of each type of wine requires 5 cubic yards of storage space. The processing time for a batch of Nectar is 15 hours and the processing time for a batch of Red is 8 hours. Demand for the two types of wine is limited to 7 batches, each. The profit for a batch of Nectar is $9,000 and the profit for a batch of red is $12,000. The company wants to determine the number of 1,000--gallon batches of Nectar and Red to produce in order to maximize profit.
a. Formulate a linear programming model for this problem.
1. How much is produced of each type of wine to maximize profits and what is the maximum profit?
2. How much of each type of resource is left over?
3. What is the value for an additional unit of each resource?
4. Over what ranges are the values in 3) above valid?
5. What is the profit-per-unit at which the production plan will change for each type of wine?