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  • Topic: Optimization, Shortest path problem, Limit superior and limit inferior
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Distribution and Network Models






Transportation Problem
• Network Representation
• General LP Formulation
Transshipment Problem
• Network Representation
• General LP Formulation
Shortest route method
• Network Representation
• General LP Formulation

© 2008 Thomson South-Western. All Rights Reserved

Slide 1

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.
Transportation, assignment, transshipment,
shortest-route, and maximal flow problems of this
chapter as well as the minimal spanning tree and
PERT/CPM problems (in others chapter) are all
examples of network problems.

© 2008 Thomson South-Western. All Rights Reserved

Slide 2

Transportation, Assignment, and
Transshipment Problems






Each of the five models of this chapter can be
formulated as linear programs and solved by
general purpose linear programming codes.
For each of the five models, if the right-hand side
of the linear programming formulations are all
integers, the optimal solution will be in terms of
integer values for the decision variables.
However, there are many computer packages
(including The Management Scientist) that contain
separate computer codes for these models which
take advantage of their network structure.

© 2008 Thomson South-Western. All Rights Reserved

Slide 3

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.

© 2008 Thomson South-Western. All Rights Reserved

Slide 4

Transportation Problem


Network Representation
1
s1

s2

1

c11

2

Sources

c23

d2

3

d3

c12

c13
c21
2

d1

c22

Destinations

© 2008 Thomson South-Western. All Rights Reserved

Slide 5

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

© 2008 Thomson South-Western. All Rights Reserved

Slide 6

Transportation Problem


Linear Programming Formulation (continued)

Min

m

n

 c x
i 1 j 1
n

x
j 1
m

x
i 1

ij ij

ij

 si

i 1, 2,

,m

Supply

ij

 dj

j 1, 2,

,n

Demand

xij > 0 for all i and j

© 2008 Thomson South-Western. All Rights Reserved

Slide 7

Transportation Problem


LP Formulation Special Cases
• 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.
© 2008 Thomson South-Western. All Rights Reserved

Slide 8

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.

© 2008 Thomson South-Western. All Rights Reserved

Slide 9...
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