Module

B

Transportation and Assignment Solution Methods

B-1

B-2

Module B

Transportation and Assignment Solution Methods

Solution of the Transportation Model
The following example was used in Chapter 6 of the text to demonstrate the formulation of the transportation model. Wheat is harvested in the Midwest and stored in grain elevators in three different cities—Kansas City, Omaha, and Des Moines. These grain elevators supply three flour mills, located in Chicago, St. Louis, and Cincinnati. Grain is shipped to the mills in railroad cars, each of which is capable of holding one ton of wheat. Each grain elevator is able to supply the following number of tons (i.e., railroad cars) of wheat to the mills on a monthly basis:
Grain Elevator 1. Kansas City 2. Omaha 3. Des Moines Total Supply 150 175 275 600 tons

Each mill demands the following number of tons of wheat per month.
Mill A. Chicago B. St. Louis C. Cincinnati Total Demand 200 100 300 600 tons

The cost of transporting one ton of wheat from each grain elevator (source) to each mill (destination) differs according to the distance and rail system. These costs are shown in the following table. For example, the cost of shipping one ton of wheat from the grain elevator at Omaha to the mill at Chicago is $7.
Mill Grain Elevator 1. Kansas City 2. Omaha 3. Des Moines A. Chicago $6 7 4 B. St. Louis $ 8 11 5 C. Cincinnati $10 11 12

The problem is to determine how many tons of wheat to transport from each grain elevator to each mill on a monthly basis in order to minimize the total cost of transportation. The linear programming model for this problem is formulated in the equations that follow: minimize Z = $6x1A + 8x1B + 10x1C + 7x2A + 11x2B + 11x2C + 4x3A + 5x3B + 12x3C subject to x1A + x1B x2A + x2B x3A + x3B x1A + x2A x1B + x2B x1C + x2C + + + + + + x1C = 150 x2C = 175 x3C = 275 x3A = 200