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 1,000-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 each type of wine is limited to seven batches. The profit for a batch of Nectar is $10,000, and the profit for a batch of Red is $15,000. The company wants to determine the number of 1,000-gallon batches of Nectar (x1) and Red (x2) to produce in order to maximize profit. (a) Define the decision variables, and write down the objective function of the linear programming model for this problem.

(b) Write down the constraints of the linear programming model for this problem.