Valley Fruit Products Company has contracted with apple growers in Ohio, Pennsylvania, and New York to purchase apples that the company then ships to its plants in Indiana and Georgia, where they are processed into apple juice. Each bushel of apples produces 2 gallons of apple juice. The juice is canned and bottled at the plants and shipped by rail and truck to warehouse/ distribution centers in Virginia, Kentucky, and Louisiana. The shipping cost per bushel from the farms to the plants and shipping costs per gallon from the plants to the distribution centers are summarized in the following tables:
Plant
Farm 4. Indiana 5. Georgia Supply (bushels)
1. Ohio .41 .57 24,000
2. Pennsylvania .37 .48 18,000
3. New York .51 .60 32,000
Plant Capacity 48,000 35,000
Distribution Center
Plant 6. Virginia 7. Kentucky 8. Louisiana
4. Indiana .22 .10 .20
5. Georgia .15 .16 .18
Demand (gal) 9,000 12,000 15,000
Formulate and solve a linear programming model to determine the optimal shipments from the farms to the plants and from the plants to the distribution centers in order to minimize total shipping costs.