Pdfs

Z07_TAYL4367_10_SE_ModB.QXD

B-2

Module B

1/9/09

8:18 AM

Page B-2

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

Supply

1. Kansas City

2. Omaha

3. Des Moines

150

175

275

Total

600 tons

Each mill demands the following number of tons of wheat per month.

Mill

Demand

A. Chicago

B. St. Louis

C. Cincinnati

200

100

300

Total

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:

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

x3B = 100...