Operations

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...