# Presensattion

Topics: Optimization, Shortest path problem, Limit superior and limit inferior Pages: 34 (2146 words) Published: April 2, 2013
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

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.

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.

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.

Slide 4

Transportation Problem

Network Representation
1
s1

s2

1

c11

2

Sources

c23

d2

3

d3

c12

c13
c21
2

d1

c22

Destinations

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

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

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.

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.