S1_dualsimplex - Operations Research Models and Methods...

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Operations Research Models and Methods Paul A. Jensen and Jonathan F. Bard LP Methods.S1 Dual Simplex Algorithm In the tableau implementation of the primal simplex algorithm, the right-hand-side column is always nonnegative so the basic solution is feasible at every iteration. For purposes of this section, we will say that the basis for the tableau is primal feasible if all elements of the right-hand side are nonnegative . Alternatively, when some of the elements are negative, we say that the basis is primal infeasible . Up to this point we have always been concerned with primal feasible bases. For the primal simplex algorithm, some elements in row 0 will be negative until the final iteration when the optimality conditions are satisfied. In the event that all elements of row 0 are nonnegative, we say that the associated basis is dual feasible . Alternatively, if some of the elements of row 0 are negative, we have a dual infeasible basis. As described, the primal simplex method works with primal feasible, but dual infeasible (nonoptimal) bases. At the final (optimal) solution, the basis is both primal and dual feasible. Throughout the process we maintain primal feasibility and drive toward dual feasibility. In this section, a variant of the primal approach, known as the dual simplex method, is considered that works in just the opposite fashion. Until the final iteration, each basis examined is primal infeasible (some negative values on the right-hand side) and dual feasible (all elements in row 0 are nonnegative). At the final (optimal) iteration the solution will be both primal and dual feasible. Throughout the process we maintain dual feasibility and drive toward primal feasibility. For a given problem, both the primal and dual simplex algorithms will terminate at the same solution but arrive there from different directions. The dual simplex algorithm is most suited for problems for which an initial dual feasible solution is easily available. It is particularly useful for reoptimizing a problem after a constraint has been added or some parameters have been changed so that the previously optimal basis is no longer feasible. We will have much more to say about duality and the relationship between primal and dual solutions in Chapter 5; however, in this section, we are principally concerned with the mechanics of implementing the dual simplex method in the tableau format. We will see that the dual simplex algorithm is very similar to the primal simplex algorithm. Algorithm
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S1_dualsimplex - Operations Research Models and Methods...

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