The 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.

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 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.

The 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.

Part A:

1. Formulate a linear programming model algebraically and establish a production schedule for the three products.

2. Solve the linear programming model using Excel.

Part B:

1. Systematically generate the total cost as the capacity of the plants in Virginia and Michigan increases in the increments of 100 tons from 1000 to 1500.

Plant Vineyard Virginia Michigan Tennessee Indiana New York 850 720 910 750 Pennsylvania 970 790 1050 8804 Ohio 900 820