Chapter 6: Transportation, Assignment, and
Transshipment Problems
A network model is one which can be represented by a set
of nodes, a set of arcs, and functions (e.g. costs, supplies,
demands, etc.) associated with the arcs and/or nodes.
Examples include transportation, assignment,
transshipment as well as shortest-route, maximal flow
problems, minimal spanning tree and PERT/CPM
problems.
All network problems can be formulated as linear
programs. However, there are many computer packages
that contain separate computer codes for these problems
which take advantage of their network structure.
If the right-hand side of the linear programming
formulations are all integers, then optimal solution of the
decision variables will also be integers.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
Transportation Problem
The transportation problem seeks to minimize the
total shipping costs of transporting goods from m
origins (each with a supply si) to n destinations (each
with a demand dj), when the unit shipping cost from
an origin, i, to a destination, j, is cij.
The network representation for a transportation
problem with two sources and three destinations is
given on the next slide.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 2
Transportation Problem
Network Representation
s1
s2
1
c11
c23
m Sources
d1
2
d2
3
d3
c12
c13
c21
2
1
c22
n Destinations
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 3
Transportation Problem
Linear Programming Formulation
Using the notation:
xij = number of units shipped from
origin i to destination j
cij = cost per unit of shipping from
origin i to destination j
si = supply or capacity in units at origin i
dj = demand in units at destination j
continued
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 4
Transportation Problem
Linear Programming Formulation (continued)
m
Min
n
 c
ij
xij
i 1 j 1
n
x
ij
 si
i 1, 2,
,m
Supply
ij
 dj
j 1, 2,
,n
Demand
j 1
m
x
i 1
xij > 0 for all i and j
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
Example: Transportation Problem
The Navy has depots in Albany, BenSalem, and
Winchester. Each of these three depots has 3,000
pounds of materials which the Navy wishes to ship to
three installations, namely, San Diego, Norfolk, and
Pensacola. These installations require 4,000, 2,500, and
2,500 pounds, respectively.
The shipping costs per pound for are shown on the next
slide. Formulate and solve a linear program to
determine the shipping arrangements that will
minimize the total shipping cost.
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Slide 6
Example: Transportation Problem (Continued)
Source
Destination
San Diego Norfolk Pensacola
Albany
BenSalem
Winchester
$12
20
30
$6
11
26
$5
9
28
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Slide 7
Transportation Problem: Network Representation
Source
Albany
3000
BenSalem
3000
1
Destination
c11
c13
c21
2
c22
1
San Diego
4000
2
Norfolk
2500
3
Pensacola
2500
c23
c31
Winchester
3000
c12
3
c33
c32
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Slide 8
Example: Transportation Problem (Continued)
Define the Decision Variables
We want to determine the pounds of material, xij ,
to be shipped by mode i to destination j. The
following table summarizes the decision variables:
San Diego Norfolk Pensacola
Albany
x11
x12
x13
BenSalem
x21
x22
x23
Winchester
x31
x32
x33
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Slide 9
Example: Transportation Problem (Continued)
Define the Objective Function
Minimize the total shipping cost.
Min: (shipping cost per pound for each mode per
destination pairing) x (number of pounds shipped
by mode per destination pairing).
Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23
+ 30x31 + 26x32 + 28x33
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Slide 10
Transportation Problem: Example #2
Define the Constraints
Source availability:
(1) x11 + x12 + x13 = 3000
(2) x21 + x22 + x23 = 3000
(3) x31 + x32 + x33 = 3000
Destination material requirements:
(4) x11 + x21 + x31 = 4000
(5) x12 + x22 + x32 = 2500
(6) x13 + x23 + x33 = 2500
Non-negativity of variables:
xij > 0, i = 1, 2, 3 and j = 1, 2, 3
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Slide 11
Example: Transportation Problem (Continued)
Computer Output
OBJECTIVE FUNCTION VALUE = 142000.000
Variable
Value
Reduced Cost
x11
1000.000
0.000
x12
2000.000
0.000
x13
0.000
1.000
x21
0.000
3.000
x22
500.000
0.000
x23
2500.000
0.000
x31
3000.000
0.000
x32
0.000
2.000
x33
0.000
6.000
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Slide 12
Transportation Problem: Example #2
Solution Summary
• San Diego will receive 1000 lbs. from Albany
and 3000 lbs. from Winchester.
• Norfolk will receive 2000 lbs. from Albany
and 500 lbs. from BenSalem.
• Pensacola will receive 2500 lbs. from BenSalem.
• The total shipping cost will be $142,000.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 13
Transportation Problem
LP Formulation Special Cases
• Total supply exceeds total demand:
No modification of LP formulation is necessary.
• Total demand exceeds total supply:
Add a dummy origin with supply equal to the
shortage amount. Assign a zero shipping cost
per unit. The amount “shipped” from the
dummy origin (in the solution) will not actually
be shipped.
Assign a zero shipping cost per unit
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or duplicated, or posted to a publicly accessible website, in whole or in part.
• Maximum route capacity from i to j:
Slide 14
Transportation Problem
LP Formulation Special Cases (continued)
• The objective is maximizing profit or revenue:
Solve as a maximization problem.
• Minimum shipping guarantee from i to j:
xij > Lij
• Maximum route capacity from i to j:
xij < Lij
• Unacceptable route:
Remove the corresponding decision variable.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 15
Assignment Problem
An assignment problem seeks to minimize the total
cost assignment of m workers to m jobs, given that the
cost of worker i performing job j is cij.
It assumes all workers are assigned and each job is
performed.
An assignment problem is a special case of a
transportation problem in which all supplies and all
demands are equal to 1; hence assignment problems
may be solved as linear programs.
The network representation of an assignment problem
with three workers and three jobs is shown on the
next slide.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 16
Assignment Problem
Network Representation
1
Agents
c11
c13
c21
2
1
c12
Tasks
c22
2
c23
c31
3
c33
c32
3
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Slide 17
Assignment Problem
Linear Programming Formulation
Using the notation:
xij =
1 if agent i is assigned to task j
0 otherwise
cij = cost of assigning agent i to task j
continued
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 18
Assignment Problem
Linear Programming Formulation (continued)
m
Min
n
 c
ij
xij
i 1 j 1
n
x
ij
1
i 1, 2,
,m
Agents
ij
1
j 1, 2,
,n
Tasks
j 1
m
x
i 1
xij > 0 for all i and j
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 19
Example: Assignment Problem
An electrical contractor pays his subcontractors a
fixed fee plus mileage for work performed. On a given
day the contractor is faced with three electrical jobs
associated with various projects. Given below are the
distances between the subcontractors and the projects.
Subcontractor
Westside
Federated
Goliath
Universal
Projects
A B C
50 36 16
28 30 18
35 32 20
25 25 14
How should the contractors be assigned so that total
mileage is minimized?
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 20
Example: Assignment Problem
Network Representation
West.
Subcontractors
50
36
16
28
Fed.
18
35
Gol.
Univ.
20
25
A
Projects
30
B
32
C
25
14
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 21
Assignment Problem: Example
Linear Programming Formulation
Min
s.t.
50x11+36x12+16x13+28x21+30x22+18x23
+35x31+32x32+20x33+25x41+25x42+14x43
x11+x12+x13 < 1
x21+x22+x23 < 1
Agents
x31+x32+x33 < 1
x41+x42+x43 < 1
x11+x21+x31+x41 = 1
x12+x22+x32+x42 = 1
Tasks
x13+x23+x33+x43 = 1
xij = 0 or 1 for all i and j
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 22
Assignment Problem: Example
The optimal assignment is:
Subcontractor Project Distance
Westside
C
16
Federated
A
28
Goliath
(unassigned)
Universal
B
25
Total Distance = 69 miles
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 23
Assignment Problem
LP Formulation Special Cases
•Number of agents exceeds the number of tasks:
Extra agents simply remain unassigned.
•Number of tasks exceeds the number of agents:
Add enough dummy agents to equalize the
number of agents and the number of tasks.
The objective function coefficients for these
new variable would be zero.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 24
Assignment Problem
LP Formulation Special Cases (continued)
•The assignment alternatives are evaluated in terms
of revenue or profit:
Solve as a maximization problem.
•An assignment is unacceptable:
Remove the corresponding decision variable.
•An agent is permitted to work t
tasks:
n
x
ij
t
i 1, 2,
,m
Agents
j 1
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 25
Transshipment Problem
Transshipment problems are transportation
problems in which a shipment may move through
intermediate nodes (transshipment nodes)before
reaching a particular destination node.
Transshipment problems can be converted to larger
transportation problems and solved by a special
transportation program.
Transshipment problems can also be solved by
general purpose linear programming codes.
The network representation for a transshipment
problem with two sources, three intermediate nodes,
and two destinations is shown on the next slide.
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 26
Transshipment Problem
Network Representation
s1
c15
Supply
s2
3
c13
1
c37
c14
Sources
c25
6
c46
c47
4
c23
2
c36
c56
c24
5
Demand
7
c57
d1
d2
Destinations
Intermediate Nodes
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 27
Transshipment Problem
Linear Programming Formulation
Using the notation:
xij = number of units shipped from node i to node j
cij = cost per unit of shipping from node i to node j
si = supply at origin node i
dj = demand at destination node j
continued
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 28
Transshipment Problem
Linear Programming Formulation (continued)
Min

cij xij
all arcs
s.t.

xij 
arcs out


Origin nodes i
xij  0
Transhipment nodes
xij  d j
Destination nodes j
arcs in
xij 

arcs out
arcs in


arcs in
xij  si
xij 
arcs out
xij > 0 for all i and j
continued
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 29
Transshipment Problem
LP Formulation Special Cases
• Total supply not equal to total demand
• Maximization objective function
• Route capacities or route minimums
• Unacceptable routes
The LP model modifications required here are
identical to those required for the special cases in
the transportation problem.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 30
Transshipment Problem Example
The Northside and Southside facilities of Zeron
Industries supply three firms (Zrox, Hewes, Rockrite)
with customized shelving for its offices. They both
order shelving from the same two manufacturers,
Arnold Manufacturers and Supershelf, Inc.
Currently weekly demands by the users are 50
for Zrox, 60 for Hewes, and 40 for Rockrite. Both
Arnold and Supershelf can supply at most 75 units to
its customers.
Additional data is shown on the next slide.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 31
Transshipment Problem Example
Because of long standing contracts based on
past orders, unit costs from the manufacturers to the
suppliers are:
Zeron N
Arnold
5
Supershelf
7
Zeron S
8
4
The costs to install the shelving at the various
locations are:
Zrox
Thomas
1
Washburn
3
Hewes Rockrite
5
8
4
4
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 32
Transshipment Problem Example
Network Representation
ZROX
75
ARNOLD
Arnold
5
Zeron
N
8
75
4
50
Hewes
HEWES
60
RockRite
40
5
8
3
7
Super
Shelf
1
Zrox
Zeron
WASH
BURN
S
4
4
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 33
Transshipment Problem: Example
Linear Programming Formulation
• Decision Variables Defined
xij = amount shipped from manufacturer i to supplier j
xjk = amount shipped from supplier j to customer k
where i = 1 (Arnold), 2 (Supershelf)
j = 3 (Zeron N), 4 (Zeron S)
k = 5 (Zrox), 6 (Hewes), 7 (Rockrite)
• Objective Function Defined
Minimize Overall Shipping Costs:
Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37
+ 3x45 + 4x46 + 4x47
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 34
Transshipment Problem: Example
Constraints Defined
Amount Out of Arnold:
Amount Out of Supershelf:
Amount Through Zeron N:
Amount Through Zeron S:
Amount Into Zrox:
Amount Into Hewes:
Amount Into Rockrite:
x13 + x14 < 75
x23 + x24 < 75
x13 + x23 - x35 - x36 - x37 = 0
x14 + x24 - x45 - x46 - x47 = 0
x35 + x45 = 50
x36 + x46 = 60
x37 + x47 = 40
Non-negativity of Variables: xij > 0, for all i and j.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 35
Transshipment Problem: Example
Computer Output
Objective Function Value =
1150.000
Variable
Value
Reduced Costs
X13
X14
X23
X24
X35
X36
X37
X45
X46
X47
75.000
0.000
0.000
75.000
50.000
25.000
0.000
0.000
35.000
40.000
0.000
2.000
4.000
0.000
0.000
0.000
3.000
3.000
0.000
0.000
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 36
Transshipment Problem: Example
Solution
ZROX
75
ARNOLD
Arnold
5
75
Zeron
N
8
75
4
50
Hewes
HEWES
60
RockRite
40
5
8
3 4
7
Super
Shelf
1
Zrox
Zeron
WASH
BURN
S
4
© 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 37
Transshipment Problem: Example
Computer Output (continued)
Constraint
1
2
3
4
5
6
7
Slack/Surplus
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Dual Values
0.000
2.000
-5.000
-6.000
-6.000
-10.000
-10.000
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 38
Transshipment Problem: Example
Computer Output (continued)
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
X13
X14
X23
X24
X35
X36
X37
X45
X46
X47
3.000
6.000
3.000
No Limit
No Limit
3.000
5.000
0.000
2.000
No Limit
5.000
8.000
7.000
4.000
1.000
5.000
8.000
3.000
4.000
4.000
7.000
No Limit
No Limit
6.000
4.000
7.000
No Limit
No Limit
6.000
7.000
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Slide 39
Transshipment Problem: Example
Computer Output (continued)
RIGHT HAND SIDE RANGES
Constraint
1
2
3
4
5
6
7
Lower Limit
75.000
75.000
-75.000
-25.000
0.000
35.000
15.000
Current Value Upper Limit
75.000
No Limit
75.000
100.000
0.000
0.000
0.000
0.000
50.000
50.000
60.000
60.000
40.000
40.000
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or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 40