OTHER ALGORITHMS FOR LINEAR PROGRAMMING

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The key to the extremely widespread use of linear programming is the availability of an exceptionally efficient algorithm—the simplex method—that will routinely solve

the large-size problems that typically arise in practice. However, the simplex method is only part of the arsenal of algorithms regularly used by linear programming practitioners. We now turn to these other algorithms.

This chapter begins with three algorithms that are, in fact, variants of the simplex method. In particular, the next three sections introduce the dual simplex method (a modification par- ticularly useful for sensitivity analysis), parametric linear programming (an extension for sys- tematic sensitivity analysis), and the upper bound technique (a streamlined version of the simplex method for dealing with variables having upper bounds). We will not go into the kind of detail with these algorithms that we did with the simplex method in Chaps. 4 and 5. The goal instead will be to briefly introduce their main ideas.

Section 4.9 introduced another algorithmic approach to linear programming—a type of algorithm that moves through the interior of the feasible region. We describe this interior- point approach further in Sec. 8.4.

A supplement to this chapter on the book’s website also introduces linear goal programming. In this case, rather than having a single objective (maximize or minimize Z ) as for linear programming, the problem instead has several goals toward which we must strive simultaneously. Certain formulation techniques enable converting a linear goal programming problem back into a linear programming problem so that solution procedures based on the simplex method can still be used. The supplement describes these techniques and procedures.

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