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 . ID: 749285
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Linear Programming ModelsSlide2
In a recent survey of Fortune 500 firms, 85% of those responding said that they used
this chapter, we discuss some of the LP models that are most often
real applications. In this chapter’s examples, you will discover how to
an aggregate labor and production plan at a shoe company
a blending plan to transform crude oils into end
The two basic goals
of this chapter are to illustrate the wide range of real
can take advantage of LP and to increase your facility in modeling LP problems
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
that all of the models in this chapter are linear
described in the previous chapter. This means that the target cell is ultimately
sum of products of constants and
, where a constant is defined by the fact that it does not depend on changing cells.Slide4
Many companies spend enormous amounts of money to advertise their products.
to ensure that they are spending their money wisely.
, they want to
numbers of various groups of potential customers and keep their advertising costs
following example illustrates a simple
model - and
model - for
a company that purchases television ads
A typical advertising model is presented in Example 4.1Slide5
To this point, the advertising models have allowed
values in the changing cells. In reality, this is not allowed.
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.Slide6
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
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.Slide7
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.Slide8
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.
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
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.Slide9
Aggregate planning models
In this section, the production planning model discussed in Example 3.3 of the
is extended to include a situation where the number of workers available
possible production levels.
Example 4.3 is typical.
workforce level is allowed to change each
period through the hiring and firing of
models, where we determine
and production schedules for a
time horizon, are called
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.
can work up to
20 hours of overtime per month and is paid $13 per hour for overtime labor.Slide11
The rolling planning
In reality, an aggregate planning model is usually implemented via a rolling planning horizon.
To illustrate, we assume that
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.Slide12
Model with backlogging
In many situations
backlogging is allowed -
that is, customer
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.Slide13
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.Slide14
Model with backlogging
certain functions, including
, MAX, and ABS, are used to relate the objective cell to the changing cells,
model becomes not only nonlinear but
nonsmooth. Essentially, nonsmooth
can have sharp edges or discontinuities. Solver’s GRG nonlinear
nonlinearities, but it has trouble with
moral is that you should avoid the non-smooth functions in optimization models
Model with backlogging allowed continued
If you do use
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
Alternatively, you can use Frontline System’s Evolutionary Solver, which became available in Excel’s Solver in Excel 2010.Slide16
Linearizing the backlogging
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
In many situations, various inputs must be blended together to produce desired outputs.
of these situations, linear programming can find the optimal combination of
well as the mix of inputs that are used to produce the desired outputs.
problems are given in the table below.See Example
Production process models
LP is often used to determine the optimal method of operating a production process.
, 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
Example 4.5 is typical.Slide19
The majority of optimization examples described in management science textbooks
the area of operations: scheduling, blending, logistics, aggregate planning, and others.
This is probably warranted, because many of the most successful management
in the real world have been in these
, optimization and other management science methods have also been applied successfully in a number of financial areas
, and they deserve recognition.Slide20
Financial models continued
Several of these applications are
this book. In this section, we begin the discussion with two typical
LP in finance.
first involves investment strategy. The second involves pension
.This type of model is demonstrated in Example 4.6.Slide21
Payments due in the future
4.7 illustrates a common situation where fixed payments are
the future and current funds must be allocated and invested so that their returns
to make the payments.
place this in a pension fund context.Slide22
Data envelopment analysis
data envelopment analysis
(DEA) method can be used to determine whether a
, restaurant, or other business is operating efficiently.
used by inefficient organizations to benchmark efficient and best-practice organizations.The following example illustrates DEA and is based on Callen (1991).
DEA is demonstrated in Example 4.8.Slide23
In this chapter, we have presented LP spreadsheet models of many diverse situations.
There are several
keys you should use with most spreadsheet optimization
the changing cells, the cells that contain the values of the
. These cells should contain the values the decision maker has direct
control over, and they should determine all other outputs, either directly or indirectly.Slide24
Set up the spreadsheet model so that you can easily calculate what you want to
minimize (usually profit or cost). For example, in the aggregate
, a good way to compute total cost is to compute the monthly cost of
up the spreadsheet model so that the relationships between the cells in the spreadsheet and the problem constraints are readily apparent.Slide25
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.
in mind that LP models tend to fall into categories, but they are definitely
alike. For example, a problem might involve a combination of the ideas
the worker scheduling, blending, and production process examples of this chapter.Slide26
Summary of key management science termsSlide27
Summary of key Excel termsSlide28
End of Chapter 4