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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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Slide1

Chapter 4

Linear Programming Models

Slide2Introduction

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.

Slide3Introduction 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.

Slide4Advertising 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

Slide5Using i

nteger

c

onstraints

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.

Slide6Using 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.

Slide7Worker 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.

Slide8Multiple 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.

Slide9Aggregate 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.

Slide10Example 4.3:

Background

i

nformation

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.

Slide11The rolling planning

h

orizon

a

pproach

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.

Slide12Model with backlogging

a

llowed

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.

Slide13Model 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.

Slide14Model with backlogging

a

llowed 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

.

Slide15Model 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.

Slide16Linearizing the backlogging

m

odel

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

.

Slide17Blending 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

Slide18Production 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.

Slide19Financial 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.

Slide20Financial 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.

Slide21Payments 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.

Slide22Data 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 best-practice organizations.The following example illustrates DEA and is based on Callen (1991).

DEA is demonstrated in Example 4.8.

Slide23Conclusion

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.

Slide24Conclusion 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.

Slide25Conclusion 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.

Slide26Summary of key management science terms

Slide27Summary of key Excel terms

Slide28End of Chapter 4

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