Chapter 5
Network Models
Introduction
• Many important optimization models have a natural
graphical network representation. In this chapter, we
discuss some specific examples of network models.
There are several reasons for distinguishing network
models from other LP models:
– The network structure of these models allows them to be
represented graphically in a way that is intuitive to users.
This graphical representation can then be used as an
aid in the spreadsheet model development. In fact, for a
book at this level, the best argument for singling out
network problems for special consideration is the fact
that they can be represented graphically.
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Introduction continued
– Many companies have real problems, often extremely
large, that can be represented as network models. In
fact, many of the best management science success
stories have involved large network models.
– Specialized solution techniques have been developed
specifically for network models. Although we do not
discuss the details of these solution techniques - and
they are not implemented in Excel’s Solver - they are
important in real-world applications because they allow
companies to solve huge problems that could not be
solved by the usual LP algorithms.
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Transportation models
• In many situations, a company produces products
at locations called origins and ships these
products to customer locations called
destinations.
– Typically, each origin has a limited amount that it can
ship, and each customer destination must receive a
required quantity of the product.
• Spreadsheet optimization models can be used to
determine the minimum-cost shipping plan for
satisfying customer demands.
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Transportation models
continued
• For now, we assume that the only possible shipments
are those directly from an origin to a destination.
– That is, no shipments between origins or between
destinations are possible.
• This problem - generally called the transportation
problem - has been studied extensively in
management science.
– In fact, it was one of the first management science
models developed, more than half a century ago.
• Example 5.1 is a typical example of a small
transportation problem.
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A typical transportation
problem
• A typical transportation problem requires three sets
of numbers: capacities (or supplies), demands (or
requirements), and unit shipping (and possibly
production) costs.
• The capacities indicate the most each plant can
supply in a given amount of time under current
operating conditions. In some cases it might be
possible to increase the “base” capacities, by using
overtime, for example. In such cases the model
could be modified to determine the amounts of
additional capacity to use (and pay for).
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A typical transportation
problem continued
• The customer demands are typically estimated
from some type of forecasting model. The
forecasts are often based on historical customer
demand data.
• The unit shipping costs come from a
transportation cost analysis - what does it really
cost to send a single automobile from any plant to
any region?
– The unit “shipping” cost can also include the unit
production cost at each plant.
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Representing transportation
as a network model
• A network diagram of this model is shown here.
This diagram is typical of network models.
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Representing transportation
as a network model continued
• A node, indicated by a circle, generally represents
a geographical location.
• An arc, indicated by an arrow, generally
represents a route for getting a product from one
node to another.
• The decision variables are usually called flows.
They represent the amounts shipped on the
various arcs.
• Upper limits are called arc capacities, and they
can also be shown on the model.
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An alternative model
• The transportation model already shown is very natural.
• From its graphical representation one can see that all arcs
go from left to right, that is, from plants to regions.
• Therefore, the rectangular range of shipments allows us to
calculate shipments out of plants as row sums and
shipments into regions as column sums.
• In anticipation of later models in this chapter, however,
where the graphical network can be more complex, we
present an alternative model of the transportation problem.
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An alternative model
• First, it is useful to introduce some additional network
terminology.
• An arc pointed into a node is called an inflow. An arrow
pointed out of a node is called an outflow.
• General networks can have both inflows and outflows for
any given node.
• Typical network models have one changing cell per arc.
• It is useful to model network problems by listing all of the
arcs and their corresponding flows in one long list.
Constraints are placed in a separate section.
• For each node in the network, there is a flow balance
constraint.
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Transportation models:
Modeling issues
• Some modeling issues to note include:
1. How the demand constraints are expressed ( “>=” or
“<=” or “=”) depends on the context of the problem.
2. If all supplies and demands are integers it is not
necessary to add explicit integer constraints. This
allows us to use the “fast” simplex method.
3. Shipping costs are often nonlinear due to quantity
discounts.
4. There is a streamlined version of the simplex method
designed for transportation problems, called the
transportation simplex method.
5. LeBlanc and Galbreth (2007a, 2007b) discuss a large
network model they developed for a client. They
recommend writing a macro in VBA to sum the
appropriate flows in and out of nodes.
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Extending the transportation
model
• Extending the basic Grand Prix transportation
model is fairly easy, even when the cost structure
is considerably more complex.
• We illustrate one such extension in example 5.2.
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Assignment models
• Assignment models are used to assign, on a
one-to-one basis, members of one set to members
of another set in a least-cost (or least-time)
manner.
• The prototype assignment model is the assignment
of machines to jobs.
– For example, suppose there are four jobs and five
machines. Every pairing of a machine and a job has a
given job completion time. The problem is to assign the
machines to the jobs so that the total time to complete
all jobs is minimized.
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Assignment models continued
– To see how this is a network problem, recall the
transportation problem of sending goods from suppliers to
customers.
– Now think of the machines as the suppliers, the jobs as the
customers, and the job completion times as the unit shipping
costs.
– The capacity of any machine represents the most jobs it can
handle. The “demand” of any job is the number of times it
must be done, usually 1. Finally, there is an arc from every
machine to every job it can handle, and the allowable flows
on these arcs are all 0 or 1 - a particular machine is either
paired with a particular job (a flow of 1) or it isn’t (a flow of 0).
– Therefore, this assignment problem can be modeled exactly
like the transportation problem.
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Assignment models continued
• An example of this model appears below.
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Assignment models continued
• The optimal solution above indicates, by the 1s
and 0s in the changing cells, which machines are
assigned to which jobs.
• Specifically, machine 2 is assigned to job 4,
machine 3 is assigned to job 3, machine 4 is
assigned to jobs 1 and 2, and machines 1 and 5
are not assigned to any jobs. With this optimal
assignment, it takes 14 time units to complete all
jobs.
• The following example is a somewhat different and
less obvious type of assignment problem.
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Other logistics models
• The objective of many real-world network models is to ship
goods from one set of locations to another set of locations
at minimum cost, subject to various constraints. There are
many variations of these models.
• The general logistics problem is similar to the transportation
problem except for two possible differences.
– First, arc capacities are often imposed on some or all of the
arcs. These become simple upper bound constraints in the
model.
– Second and more significantly, inflows and outflows can be
associated with any node. Nodes are generally categorized
as origins, destinations, and transshipment points.
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Other logistics models
continued
– An origin is a location that starts with a certain supply
(or possibly a capacity for supplying). A destination is
the opposite; it requires a certain amount to end up
there. A transshipment point is a location where goods
simply pass through.
• The best way to think of these categories is in
terms of net inflow and net outflow.
– The net inflow for any node is defined as total inflow
minus total outflow for that node.
– The net outflow is the negative of this, total outflow
minus total inflow.
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Other logistics models
continued
• Using the above convention, an origin is a node
with positive net outflow, a destination is a node
with positive net inflow, and a transshipment
point is a node with net outflow (and net inflow)
equal to 0.
• It is important to realize that inflows are sometimes
allowed to origins, but their net outflows must be
positive.
• Similarly, outflows from destinations are
sometimes allowed, but their net inflows must be
positive.
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Other logistics models
continued
• There are typically two types of constraints in
logistics models (other than nonnegativity of flows).
– The first type represents the arc capacity constraints,
which are simple upper bounds on the arc flows.
– The second type represents the flow balance
constraints, one for each node.
• For an origin, this constraint is typically of the form Net Outflow =
Original Supply or possibly Net Outflow < Capacity.
• For a destination, it is typically of the form Net Inflow > Demand
or possibly Net Inflow = Demand.
• For a transshipment point, it is of the form Net Inflow = 0 (which
is equivalent to Net Outflow = 0).
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Other logistics models
continued
• It is easy to visualize these constraints in a
graphical representation of the network by
examining the flows of the arrows leading into and
out of the various nodes.
• We illustrate a typical logistics model in the
example 5.4.
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Variations of the model
• There are many variations of the RedBrand shipping
problem that can be handled by a network formulation. We
consider two possible variations.
• First, suppose RedBrand ships two products along the
given network. We assume that the unit shipping costs are
the same for either product, but the arc capacity (300)
represents the maximum flow of both products that can flow
on any arc.
• In this sense the two products are competing for arc
capacity. Each plant has a separate production capacity for
each product and each customer has a separate demand
for each product.
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Shortest path models
• In many applications, the objective is to find the
shortest path between two points in a network.
• Sometimes this problem occurs in a geographical
context where, for example, the objective is to find
the shortest path on interstate freeways from
Seattle to Miami.
• There are also problems that do not look like
shortest path problems but can be modeled in the
same way. We look at one possibility where the
objective is to find an optimal schedule for
replacing equipment.
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Shortest path models
continued
• The typical shortest path problem is a special case
of the network flow problem from the previous
section.
• To see why this is the case, suppose that you want
to find the shortest path between node 1 and node
N in a network.
• To find this shortest path, you create a network
flow model where the supply for node 1 is 1, and
the demand for node N is 1. All other nodes are
transshipment nodes.
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Shortest path models
continued
• If an arc joins two nodes in the network, the
“shipping cost” is equal to the length of the arc.
• The “flow” through each arc in the network (in the
optimal solution) is either 1 or 0, depending on
whether the shortest path includes the arc.
• No arc capacities are required in the model.
• The value of the objective is then equal to the sum
of the distances of the arcs involved in the path.
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Geographical shortest path
models
• Example 5.5 illustrates the shortest path model in
the context of a geographic network.
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Equipment replacement
models
• Although shortest path problems often involve
traveling through a network, this is not always the
case.
• For example, when should you trade your car in for
a new car? As a car gets older, the maintenance
cost per year increases, and it might become
worthwhile to buy a new car.
• If your goal is to minimize the average annual cost
of owning a car (ignoring the time value of money),
then it is possible to set up a shortest path
representation of this problem.
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Network models in the airline
industry
• We conclude this chapter with two network models
that apply to the airline industry. (The airline
industry is famous for using management science
in a variety of ways to help manage operations and
save on costs.)
• Neither of these problems looks like a network at
first glance, but some creative thinking reveals
underlying network structures.
– The first problem turns out to be an assignment model;
the second is similar to the RedBrand logistics model.
Note that these two examples are considerably more
difficult than any covered so far in this chapter.
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Summary of key management
science terms
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Summary of key Excel terms
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End of Chapter 5