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Unformatted text preview: 7 Finding an initial feasible tableau To begin the simplex method for solving a linear programming problem in standard equality form, we need to find an initial feasible tableau. Some times, this is easy. For example, the first linear programming problem we studied was the problem maximize x 1 + x 2 subject to x 1 ≤ 2 x 1 + 2 x 2 ≤ 4 x 1 , x 2 ≥ . We transformed this problem to standard equality form by introducing slack variables: maximize x 1 + x 2 = z subject to x 1 + x 3 = 2 x 1 + 2 x 2 + x 4 = 4 x 1 , x 2 , x 3 , x 4 ≥ . Our initial system of equations is then z x 1 x 2 = x 1 + x 3 = 2 x 1 + 2 x 2 + x 4 = 4 , and, as we observed, this system is already a feasible tableau, corresponding to the basis [3 , 4]. More generally, given any linear programming problem in standard inequality form, maximize c T x subject to Ax ≤ b x ≥ , if we transform to standard equality form by introducing slack variables, then the original system of constraints constitute a tableau for the trans formed problem, where the basic variables are exactly the slack variables. Furthermore, this tableau is feasible, provided b ≥ 0. In general, however, it may not so easy to find an initial feasible tableau, or even to decide whether a system of constraints has a feasible solution. We next introduce a technique to resolve this difficulty. We illustrate with an example. 40 Consider the linear programming problem (7.1) maximize 2 x 1 + x 2 subject to x 1 + x 2 = 3 x 1 + x 2 x 3 = 1 x 1 , x 2 , x 3 ≥ . For the moment, we ignore the objective function, and simply try to decide whether or not the problem has a feasible solution, and, if so, to find one. With this in mind, we introduce two new artificial variables that measure the error in each of the equality constraints: ( * ) x 4 = 3 ( x 1 + x 2 ) x 5 = 1 ( x 1 + x 2 x 3 ) . We will try to force these errors to zero by solving the linear programming problem (7.2) max n x 4 x 5 : ( * ) holds and x 1 , . . . , x 5 ≥ o . Central to the success of this approach is the following property relationship between the linear programming problems (7.1) and (7.2): (7.1) is feasible ⇔ (7.2) has an optimal solution with value zero....
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This note was uploaded on 10/31/2010 for the course ORIE 5300 at Cornell.
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