Chapter 4
Linear Programming Models
Introduction
• In a recent survey of Fortune 500 firms, 85% of those
responding said that they used linear programming.
• In this chapter, we discuss some of the LP models that
are most often applied to real applications. In this
chapter’s examples, you will discover how to build
optimization models to
– purchase television ads
– schedule postal workers
– create an aggregate labor and production plan at a shoe
company
– create a blending plan to transform crude oils into end
products, etc.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Introduction continued
• The two basic goals of this chapter are to illustrate the wide
range of real applications that can take advantage of LP
and to increase your facility in modeling LP problems in
Excel.
• We present a few principles that will help you model a wide
variety of problems.
• The best way to learn, however, is to see many examples
and work through numerous problems.
• Remember that all of the models in this chapter are linear
models as described in the previous chapter. This means
that the target cell is ultimately a sum of products of
constants and changing cells, where a constant is defined
by the fact that it does not depend on changing cells.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Advertising models
• Many companies spend enormous amounts of
money to advertise their products. They want to
ensure that they are spending their money wisely.
• Typically, they want to reach large numbers of
various groups of potential customers and keep
their advertising costs as low as possible.
• The following example illustrates a simple model and a reasonable extension of this model - for a
company that purchases television ads.
• A typical advertising model is presented in
Example 4.1
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Using integer constraints
• To this point, the advertising models have allowed
noninteger values in the changing cells. In reality,
this is not allowed.
• To force the changing cells to have integer values,
you simply add another constraint in the Solver
dialog box.
• Be aware that Solver must do a lot more work to
solve problems with integer constraints.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Using integer constraints
continued
• Consider the following about this integer solution:
– The total cost in the target cell is now worse (larger) than
before.
– The optimal integer solution is not the rounded
noninteger solution.
– When there are integer constraints, Solver uses an
algorithm - called branch and bound - that is significantly
different from the simplex method.
• Integer-constrained models are typically much harder to solve
than models without any integer constraints.
– If the model is linear except for the integer constraints,
that is, it satisfies the proportionality and additivity
assumptions of linear models, you should still select the
Simplex LP method.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Worker scheduling models
• Many organizations must determine how to
schedule employees to provide adequate service.
• The following example illustrates how LP can be
used to schedule employees.
• A typical model is presented in Example 4.2.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Multiple solutions
• Ocassionally, you may get a different schedule that is still
optimal – a solution that uses all 23 employees and meets all
constraints. This is a case of multiple optimal solutions.
• One other comment about integer constraints concerns
Solver’s Tolerance setting.
• As Solver searches for the best integer solution, it is often
able to find “good” solutions fairly quickly, but it often has to
spend a lot of time finding slightly better solutions.
• A nonzero tolerance setting allows it to quit early. The default
tolerance setting is 0.05. This means that if Solver finds a
feasible solution that is guaranteed to have an objective value
no more than 5% from the optimal value, it will quit and report
this “good” solution.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Aggregate planning models
• In this section, the production planning model
discussed in Example 3.3 of the previous chapter is
extended to include a situation where the number of
workers available influences the possible production
levels.
• Example 4.3 is typical.
• The workforce level is allowed to change each period
through the hiring and firing of workers.
• Such models, where we determine workforce levels
and production schedules for a multiperiod time
horizon, are called aggregate planning models.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Example 4.3:
Background information
• During the next four months the SureStep Company must
meet (on time) the following demands for pairs of shoes:
3,000 in month 1; 5,000 in month 2; 2,000 in month 3; and
1,000 in month 4.
• At the beginning of month 1, 500 pairs of shoes are on
hand, and SureStep has 100 workers.
• A worker is paid $1,500 per month. Each worker can work
up to 160 hours a month before he or she receives
overtime.
• A worker can work up to 20 hours of overtime per month
and is paid $13 per hour for overtime labor.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
The rolling planning horizon
approach
• In reality, an aggregate planning model is usually
implemented via a rolling planning horizon.
• To illustrate, we assume that SureStep works with
a 4-month planning horizon.
• To implement the SureStep model in the rolling
planning horizon context, we view the “demands”
as forecasts and solve a 4-month model with these
forecasts.
• However, we implement only the month 1
production and work scheduling recommendation.
• Example 4.4 is typical.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Model with backlogging
allowed
• In many situations backlogging is allowed - that is,
customer demand can be met later than it occurs.
• We’ll modify this example to include the option of
backlogged demand.
• We assume that at the end of each month a cost of $20 is
incurred for each unit of demand that remains unsatisfied at
the end of the month.
• This is easily modeled by allowing a month’s ending
inventory to be negative. The last month, month 4, should
be nonnegative. This also ensures that all demand will
eventually be met by the end of the four-month horizon.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Model with backlogging
allowed continued
• We now need to modify the monthly cost
computations to incorporate the costs due to
shortages.
• There are actually several approaches to this
backlogging problem.
• The most “natural” is shown on the next slide.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Model with backlogging
allowed continued
• When certain functions, including IF, MIN, MAX,
and ABS, are used to relate the objective cell to
the changing cells, the resulting model becomes
not only nonlinear but nonsmooth.
– Essentially, nonsmooth functions can have sharp edges
or discontinuities. Solver’s GRG nonlinear algorithm can
handle “smooth” nonlinearities, but it has trouble with
nonsmooth functions.
– The moral is that you should avoid the non-smooth
functions in optimization models.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Model with backlogging
allowed continued
• If you do use nonsmooth functions, then you must
run Solver several times, stating from different
initial solutions.
• Alternatively, non-smooth functions can be
handled with a totally different kind of algorithm
called a genetic algorithm.
• Alternatively, you can use Frontline System’s
Evolutionary Solver, which became available in
Excel’s Solver in Excel 2010.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Linearizing the backlogging
model
• Although this nonlinear model with IF functions is
“natural”, the fact that we cannot guarantee it to
find the optimal solution is disturbing.
• We can, however, handle shortages and maintain
a linear formulation.
• This method is illustrated in Example 4.3.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Blending models
• In many situations, various inputs must be blended
together to produce desired outputs.
• In many of these situations, linear programming
can find the optimal combination of outputs as well
as the mix of inputs that are used to produce the
desired outputs.
• Some examples of blending problems are given in
the table below.
• See Example 4.4
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Production process models
• LP is often used to determine the optimal method
of operating a production process.
• In particular, many oil refineries use LP to manage
their production operations.
• The models are often characterized by the fact that
some of the products produced are inputs to the
production of other products.
• Example 4.5 is typical.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Financial models
• The majority of optimization examples described in
management science textbooks are in the area of
operations: scheduling, blending, logistics,
aggregate planning, and others.
• This is probably warranted, because many of the
most successful management science applications
in the real world have been in these areas.
• However, optimization and other management
science methods have also been applied
successfully in a number of financial areas, and
they deserve recognition.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Financial models continued
• Several of these applications are discussed
throughout this book. In this section, we begin the
discussion with two typical applications of LP in
finance.
• The first involves investment strategy. The second
involves pension fund management.
• This type of model is demonstrated in Example
4.6.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Payments due in the future
• Example 4.7 illustrates a common situation where
fixed payments are due in the future and current
funds must be allocated and invested so that their
returns are sufficient to make the payments.
• We place this in a pension fund context.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Data envelopment analysis
• The data envelopment analysis (DEA) method
can be used to determine whether a university,
hospital, restaurant, or other business is operating
efficiently.
• Specifically, DEA can be used by inefficient
organizations to benchmark efficient and bestpractice organizations.
• The following example illustrates DEA and is
based on Callen (1991).
• DEA is demonstrated in Example 4.8.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Conclusion
• In this chapter, we have presented LP spreadsheet
models of many diverse situations.
• There are several keys you should use with most
spreadsheet optimization models:
– Determine the changing cells, the cells that contain the
values of the decision variables. These cells should
contain the values the decision maker has direct control
over, and they should determine all other outputs, either
directly or indirectly.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Conclusion continued
– Set up the spreadsheet model so that you can easily
calculate what you want to maximize or minimize
(usually profit or cost). For example, in the aggregate
planning model, a good way to compute total cost is to
compute the monthly cost of operation in each row.
– Set up the spreadsheet model so that the relationships
between the cells in the spreadsheet and the problem
constraints are readily apparent.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Conclusion continued
– Make your spreadsheet readable. Use descriptive
labels, use range names, use cell comments and text
boxes for explanations, and plan your model layout
before you dive in. This might not be too important for
small, straightforward models, but it is crucial for large,
complex models. Just remember that other people are
likely to be examining your spreadsheet models.
– Keep in mind that LP models tend to fall into categories,
but they are definitely not all alike. For example, a
problem might involve a combination of the ideas
discussed in the worker scheduling, blending, and
production process examples of this chapter.
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Summary of key management
science terms
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
Summary of key Excel terms
Winston/Albright
Practical Management Science, 4e
South-Western/Cengage
Learning
© 2012
Thomson/South-Western
2007
©
End of Chapter 4