Chapter 4. Linear Programming Models. Introduction. In a recent survey of Fortune 500 firms, 85% of those responding said that they used linear programming .
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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.
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.
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.
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.
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.
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.
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.