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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 car 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
A. Chicago
B. St. Louis
C. Cincinnati
1. Kansas City
2. Omaha
3. Des Moines
$6
7
4
$ 8
11
5
$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.
Solution of the Transportation Model
B-3
minimize Z ϭ $6x1A ϩ 8x1B ϩ 10x1C ϩ 7x2A ϩ 11x2B ϩ 11x2C ϩ 4x3A ϩ 5x3B ϩ 12x3C
subject to
x1A ϩ x1B ϩ x1C ϭ 150
x2A ϩ x2B ϩ x2C ϭ 175
x3A ϩ x3B ϩ x3C ϭ 275
x1A ϩ x2A ϩ x3A ϭ 200
x1B ϩ x2B ϩ x3B ϭ 100...