Linear Programming

Download PDF by Stephen P. Bradley: Applied Mathematical Programming

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By Stephen P. Bradley

Publication by way of Bradley, Stephen P., Hax, Arnoldo C., Magnanti, Thomas L.

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Most of the algorithms that we are going to consider are iterative, in the sense that they move from one decision point x1 , x2 , . . , xn to another. For these algorithms, we need: i) a starting point to initiate the procedure; ii) a termination criterion to indicate when a solution has been obtained; and iii) an improvement mechanism for moving from a point that is not a solution to a better point. Every algorithm that we develop should be analyzed with respect to these three requirements. In the previous section, we discussed most of these criteria for a sample linear-programming problem.

Next, observe that since the artificial variables x9 , x10 , and x11 are all nonnegative, they are all zero only when their sum x9 + x10 + x11 is zero. For the basic feasible solution just derived, this sum is 5. , minimizing the sum of all artificial variables). Since the artificial variables are all nonnegative, minimizing their sum means driving their sum towards zero. If the minimum sum is 0, then the artificial variables are all zero and a feasible, but not necessarily optimal, solution to the original problem has been obtained.

By demonstrating that there is no feasible solution; 2. by determining an optimal solution; or 3. by demonstrating that the objective function is unbounded over the feasible region. 3 We will say that an algorithm solves a problem if it always satisfies one of these three conditions. As we shall see, a major feature of the simplex method is that it solves any linear-programming problem. Most of the algorithms that we are going to consider are iterative, in the sense that they move from one decision point x1 , x2 , .

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