Case Scenario: Walsh’s Juice Company
Walsh’s Juice Company produces three products from unprocessed grape juice-bottled 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 1400 tons of unprocessed grape juice available whereas the vineyard in Ohio has 1700 tons and the vineyard in Pennsylvania has 1100 tons The processing capacity per month is 1200 tons of unprocessed juice at the plant in Virginia 1100 tons of juice at the plant in Michigan 1400 tons at the plant in Tennessee and 1400 tons at the plant of Indiana. The cost per ton of transporting unprocessed juice from the vineyards to the plant is as follows.
Plant
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Vineyard Virginia Michigan Tennessee Indiana
New York $850 $720 $910 $750
Pennsylvania 970 790 1050 880
Ohio 900 830 780 820
The plants are different ages, have different equipment and have different wage rates thus the cost of processing each product at each plant ($/ton) differs as follows.
Plant
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Product Virginia Michigan Tennessee Indiana
Juice $2100 $2350 $2200 $1900
Concentrate 4100 4300 3950 3900
Jelly 2600 2300 2500 2800
This month the company needs to process a total of 1200 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 include 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 logistics aspect of this problem and the production processing aspect. 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 cost. Help Walsh solve this problem by formulating a linear programming model and solve it using the computer.