W ALSH'S J UICE C OMPANY
Walsh's Juice Company produces three products from unprocessed grape juicebottled juice, frozen juice concentrate, and jelly . It purchases grape juice from three vineyards near the Great Lakes. The grapes are harvested at the vineyards and immediately converted into juice at plants at the vineyard sites and stored there in refrigerated tanks. The juice is then transported to four different plants in Virginia, Michigan, Tennessee, and Indiana, where it is processed into bottled grape juice, frozen juice concentrate, and jelly. Vineyard output typically differs each month in the harvesting season , and the plants have different processing capacities .
In a particular month the vineyard in New York has 1,400 tons of unprocessed grape juice available, whereas the vineyard in Ohio has 1,700 tons and the vineyard in Pennsylvania has 1,100 tons. The processing capacity per month is 1,200 tons of unprocessed juice at the plant in Virginia, 1,100 tons of juice at the plant in Michigan, 1,400 tons at the plant in Tennessee, and 1,400 tons at the plant in Indiana. The cost per ton of transporting unprocessed juice from the vineyards to the plant is as follows :
The plants are different ages, have different equipment, and have different wage rates; thus, the cost of processing each product at each plant ($/ton) differs, as follows:
This month the company needs to process a total of 1,200 tons of bottled juice, 900 tons of frozen concentrate, and 700 tons of jelly at the four plants combined. However, the production process for frozen concentrate results in some juice dehydration, and the process for jelly includes a cooking stage that evaporates water content. To process 1 ton of frozen concentrate requires 2 tons of unprocessed juice; 1 ton of jelly requires 1.5 tons of unprocessed juice; and 1 ton of bottled juice requires 1 ton of unprocessed juice.
Walsh's management wants to determine how many tons of grape juice to ship from each of the vineyards to each of the plants and the number of tons of each product to process at each plant. Thus, management needs a model that includes both the logistical aspects of this problem and the production processing aspects. It wants a solution that will minimize total costs, including the cost of transporting grape juice from the vineyards to the plants and the product processing costs. Help Walsh's solve this problem by formulating a linear programming model and solve it by using the computer.