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Notes 5 EnterVar

# Notes 5 EnterVar - File EnterVar Deciding What Variable...

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File: EnterVar.doc Deciding What Variable Enters the Basis ( x B ) The analysis of which variable should enter the basis ( x B ) from the non-basic set ( x N ) has been avoided until now. We can select any variable from x N as the entering variable and find an adjacent basic feasible solution. However, the most promising variable to enter the basis would be one that yields the maximum increase in the value of the objective function relative to the current solution. Recall that we are developing the methodology to solve linear programming problems. So consider the linear objective function z c x i i n i T T T = = = + = 1 c x c x c x B B N N that we add to the system of constraints Ax Bx Nx b x 0 B N B = + = , . Given a basis, x B , we can select the most promising variable from x N to enter the basis by choosing the variable which yields the maximum change in z. However, obtaining these increases in z requires a considerable computation. For the time being, we will select the variable that yields the “best” incremental change in z; that is, has the “best” slope. The best is in quotes here since we may be maximizing in one case and minimizing in another. Obviously, the select should change with the sense of the optimization. For a maximization problem, the “best” select would be the variable with the largest slope. For a minimization problem, the “best” select would be the variable with the most negative slope. How can we determine what the contribution to the objective function would be for bringing a non-basic variable into the basis? This question is answered, as one might deduce by now, by considering the general solution relative to the current basis. The objective function value associated with a the current basic feasible solution is given by z T T = + c x c x B B N N .

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