Chapter 3
Linear Programming : Network Models
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• The Transportation Model
• The Assignment Model
• The Transshipment Model
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Transportation, Assignment, Transshipment,
and Shortest Path Models
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• A network model describes the relationship between various
locations/tasks/teams (represented usually via nodes) taking into
account the cost/time/various other factors (usually represented via
arcs).
• The four models we’ll be covering in this chapter are all examples of
network models.
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Transportation Model
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• The transportation model is described by a set of sources and their
supplies, a set of destinations and their demands, and the unit cost
associated with each source-destination pair.
• In a transportation model, each source requires a Resource Availability
Constraint, reflecting that we cannot exceed the supply capacity of a
source.
• Each destination requires a Minimum Acquisition Constraint in order to
make sure that the demand requirement at that destination is met.
• The overall goal is usually to minimize the cost (or time) of transporting
goods from the sources to the destinations. The decision variables are
the amount of goods shipped via each source-destination pair/route.
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General Transportation Model Rules
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• For a typical transportation problem where any source can send items to
any destination, we can use the following rules to write the model:
• 1) If total supply > total demand,
• i) The source constraints must be in the format of Total Amount Shipped < Capacity
• ii) The destination constraints can say Total Amount Received = or > Demand
• 2) If total supply = total demand,
• i) The source constraints can be written as Total Amount Shipped < or = Capacity
• ii) The destination constraints can say Total Amount Received = or > Demand
• 3) If total supply < total demand,
• i) The source constraints can be written as Total Amount Shipped < or = Capacity
• Ii) The destination constraints must say Total Amount Received = Demand
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Example 3.1 from the textbook
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• Please read pages 66-71 for an in-depth discussion (including
the solution procedure) for Example 3.1. As usual, the Excel file
on the publisher’s website (Chapter 3 Worksheets) corresponds
to the figures in the textbook. You can then interact with the
appropriate worksheets in that Excel file to solve this example
along the textbook.
• In general, here are the
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Assignment Model
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• The assignment model can simply be considered as a special version of
the transportation model where all the supplies (capacities) and
requirements (demands) are equal to one.
• In an assignment model, sources usually refer to teams/groups/workers,
and destinations refer to tasks/projects. So, a typical assignment
problem would aim to minimize the total cost of assigning a number of
teams to a number of projects with the restriction that a team can be
assigned to at most one project, and that a project needs to be
completed by one team only.
• The principles described for the transportation model in previous slides
also apply to the assignment model.
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Example 3.2 from the textbook
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• Please read pages 71-75 for an in-depth discussion (including
the solution procedure) for Example 3.2. As usual, the Excel file
on the publisher’s website (Chapter 3 Worksheets) corresponds
to the figures in the textbook. You can then interact with the
appropriate worksheets in that Excel file to solve this example
along the textbook.
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Transshipment Model
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• The Transshipment Model is a generalization of the transportation model, in
the sense that we are not restricted to just one set of sources and one set
of destinations. In addition, we have a new set (sometimes more than one
set) of locations that act as the transition between sources and
destinations. These are called transshipment, or intermediate nodes.
• A transportation model might include a set of plants as sources and a set of
retail stores as destinations. The transshipment model would add a set of
warehouses as the intermediate nodes.
• These intermediate nodes simply act neutrally in that they have neither any
supplies nor any demands. They simply receive goods from sources (or
other nodes “upstream”) and pass them to destinations (or other nodes
“downstream”).
• The source and destination constraints for the transshipment model remain
the same as they are in the transportation model. The constraints for the
intermediate nodes must always be written as Amount Received = Amount
Sent.
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Example 3.3 from the textbook
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• Please read pages 75-78 for an in-depth discussion (including
the solution procedure) for Example 3.3. As usual, the Excel file
on the publisher’s website (Chapter 3 Worksheets) corresponds
to the figures in the textbook. You can then interact with the
appropriate worksheets in that Excel file to solve this example
along the textbook.
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REST OF CHAPTER 3 IN THE TEXTBOOK
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PLEASE READ AND LEARN SECTION 3.5; HOWEVER, YOU CAN
SKIP SECTIONS 3.6, AND 3.7. WE WILL NOT BE COVERING THE
LATTER TWO SECTIONS
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ADDITIONAL EXERCISES
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• I have uploaded videos on Canvas within Module 3, where I go over
the following three problems from the textbook:
1) Exercise 3.3 on page 97 (Transportation Example)
2) Exercise 3.6 on page 98-99 (Assignment Example)
3) Exercise 3.7 on page 99 (Transshipment Example)
An Excel file with three worksheets for these examples can be found
in the same location on Canvas.
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