Quantitative Methods
MAT 540
Transportation, Transshipment, and
Assignment Problems
Objectives
• When you complete this lesson, you will be
able to solve:
• Transportation problems
• Transshipment problems
• Assignment problems
Overview
• Network flow problems
• Transportation models
• Transshipment models
• Assignment models
The Transportation Model
• Characteristics
• A product is transported from a number of
sources to a number of destinations at the
minimum cost
• Each source is able to supply a fixed amount of
the product, and each destination has a fixed
amount of demand for the product
The Transportation Model,
continued
Grain Elevator
Supply
Mill
Demand
1. Kansas City
150
A. Chicago
200
2. Omaha
175
B. St. Louis
100
3. Des Moines
275
C. Cincinnati
300
Total
600 tons
Total
600 tons
Grain Elevator
1. Kansas City
2. Omaha
3. Des Moines
Transport Cost from Grain Elevator to Mill ($/ton)
A. Chicago
B. St. Louis
C. Cincinnati
$ 6
$ 8
$10
7
11
11
4
5
12
The Transportation Model,
continued
• Linear programming model
minimize
subject to
Z  6 x1A  8 x1B  10 x1C  7 x2 A  11x2 B  11x2C  4 x3A  5 x3B  12 x3C
x1A  x1B  x1C  150
x2 A  x2 B  x2C  175
x3A  x3B  x3C  275
x1A  x2 A  x3A  200
x1B  x2 B  x3B  100
x1C  x2C  x3C  300
xij  0
Computer Solution of a
Transportation Problem
• Excel solution
Computer Solution of a
Transportation Problem, continued
• Excel QM solution
Computer Solution of a
Transportation Problem, continued
• Excel QM solution
Computer Solution of a
Transportation Problem, continued
• QM for Windows solution
The Transshipment Model
• Transshipment points
• Transportation may take place from
• Sources through transshipment points to
destinations
• One source to another
• One transshipment point to another
• One destination to another
• Sources to destinations
The Transshipment Model,
continued
• Nebraska, Colorado each harvest 300 tons
• Kansas City, Omaha, and Des Moines are
transshipment points
Farm
1. Nebraska
2. Colorado
3. Kansas City
$16
15
Grain Elevator
4. Omaha
10
14
5. Des Moines
12
17
The Transshipment Model,
continued
• Supply constraints for the farms
x13  x14  x15  300
x23  x24  x25  300
• Demand constraints at mills
x36  x46  x56  200
x37  x47  x57  100
x38  x48  x58  300
The Transshipment Model,
continued
• Grain shipped into Kansas City: x13  x23
• Grain shipped out of Kansas City: x36  x37  x38
• The two amounts must equal one another
x13  x23  x36  x37  x38
x13  x23  x36  x37  x38  0
• Constraints for Omaha and Des Moines
x14  x24  x46  x47  x48  0
x15  x25  x56  x57  x58  0
The Transshipment Model,
continued
• Linear programming model
minimize
Z  16 x13  10 x14  12 x15  15 x23  14 x24  17 x25  6 x36  8 x37  10 x38
 7 x46  11x47  11x48  4 x56  5 x57  12 x58
subject to
x13  x14  x15  300
x23  x24  x25  300
x36  x46  x56  200
x37  x47  x57  100
x38  x48  x58  300
x13  x23  x36  x37  x38  0
x14  x24  x46  x47  x48  0
x15  x25  x35  x35  x35  0
xij  0
The Transshipment Model,
continued
• Excel solution
The Assignment Problem
• All supply and demand values equal 1
• The supply at each source and the demand
at each destination are each limited to one
unit
The Assignment Problem,
continued
• Four teams of officials to four games
• Minimize distance traveled
• Supply and demand is one team of officials
per game
The Assignment Problem,
continued
• Linear programming model
minimize
Z  210 xAR  90 xAA  180 xAD  160 xAC  100 xBR  70 xBA  130 xBD
 200 xBC  175 xCR  105 xCA  140 xCD  170 xCC  80 xDR  65 xDA
 105 xDD  120 xDC
subject to
xAR  xAA  xAD  xAC  1
xBR  xBA  xBD  xBC  1
xCR  xCA  xCD  xCC  1
xDR  xDA  xDD  xDC  1
xAR  xBR  xCR  xDR  1
xAA  xBA  xCA  xDA  1
xAD  xBD  xCD  xDD  1
xAC  xBC  xCC  xDC  1
xij  0
The Assignment Problem,
continued
• Excel solution
The Assignment Problem,
continued
• Excel QM solution
The Assignment Problem,
continued
• QM for Windows Solution