The Value-Pack Canning Company Buys Fresh Fruits and Vegetables from Farmers

Essay

The Value-Pack Canning Company buys fresh fruits and vegetables from farmers, processes and cans them, and sells the output to various supermarket chains. The company is trying to determine the optimal mix of peas and green beans to process during the forthcoming period at its Fresno plant. Output is limited by the capacity of the plant and the firm's financial resources. The plant can process 4,000,000 pounds of peas or green beans (or any linear combination thereof) during the forthcoming period. The company buys peas and green beans for $.10 and $.20 per pound respectively. Purchases must be paid for at the time of delivery and the firm's current cash balance limits purchases to $600,000 during the forthcoming period. As part of long-term contracts with several supermarket chains, the firm is required to process at least 800,000 pounds of peas and 1,200,000 pounds of green beans during the period. The profit contribution of peas and green beans are $.015 and $.025 respectively. The firm desires to find the mix of peas and green beans to produce in order to maximize the total profit contribution of the plant during the forthcoming period. Let X₁ be the number of pounds of peas processed and X₂ be the number of pounds of green beans processed.
(a)Formulate the problem algebraically in the linear programming framework.
(b)Using graphical methods, determine the optimal mix of vegetables to process.