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Unformatted text preview: 52 Chapter 3 Analyzing Solutions of Linear Programs Slack and Tight Constraints
In any feasible solution to a linear program, an inequality constraint is said to be tight if it is satisfied as an equation and to be slack if it is satisfied as a strict inequality. Program 3.1 illustrates these definitions. It has a total of eight inequality constraints, namely, the capacity constraint on each of five shops and the nonnegativity constraint on each of three variables. Column E of Table 3.2 reports the amount of each capacity that is actu ally used by the optimal solution. Evidently, the capacity constraints on the Engine and Body shops are tight. The other three capacity constraints are slack. The nonnegativity con straint on L is also tight. The other two nonnegativity constraints are slack. In Table 3.2 and henceforth, an inequality constraint is shaded if the optimal solution causes it to be tight. It is the tight (shaded) constraints that will help us to relate the optimal solution to the situation that the linear program models. 3.4. THE PERTURBATION THEOREM
A linear program is a model; as such, it is an inexact representation of reality. Can we de scribe the optimal solution to a linear program in a way that can be implemented when the data are inexact? Yes. The “Perturbation Theorem” tells how.’ Perturbation Theorem. If the data of a linear program are perturbed by small
amounts, its optimal solution can change, but its tight constraints stay tight, and its slack constraints stay slack. The Perturbation Theorem confirms our intuition. To see why, let us ask ourselves, “What is actually meant by the optimal solution to a linear program?” There are two possi ble answers, in the context of the Recreational Vehicle example. • Make Standard model vehicles at the rate of 20 per week, make Fancy model vehi cles at the rate of 30 per week, and make no Luxury model vehicles. • Keep the Engine and Body shops busy making Standard and Fancy model vehicles, and make no Luxury model vehicles. If the model is exactly correct, these answers are identical to each other. The first an swer sets S = 20, F = 30, and L = 0, independent of the data. If the data are a little off, the first answer may not be feasible, and if it is feasible, it may not be optimal. It is the second answer that the shop manager can implement when the data are inexact. The Perturbation Theorem assures us that the second answer stays feasible and optimal for a range of data. Toward the end of this chapter, we’ll develop our geometric intuition as to why the Pertur bation Theorem is true. 3.5. HOW A LINEAR PROGRAM APPROXIMATES REALITY
The Recreational Vehicle example illustrates the principal ways in which linear programs approximate reality—uncertainty, aggregation, and linearization. This model’s data are uncertain because they cannot be measured precisely and because they can fluctuate in unpredictable ways. For instance, it is presumed that, in a
In an unpleasant situation that is known as degeneracy, the Perturbation Theorem needs to be qualified in a way that we will discuss later in this chapter. A proof of this theorem is provided in Chapter 18. ...
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This note was uploaded on 03/17/2010 for the course IOE 202 taught by Professor Marinaepelman during the Fall '09 term at University of Michigan-Dearborn.
- Fall '09