lecture 5 - Math 482(Lecture 5 The simplex method II...

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Math 482 (Lecture 5): The simplex method II: tableaux and moving from bfs to bfs Today: Having defined a basic feasible solution (bfs), we explain how to move from a starting to bfs to a more optimal bfs. The book-keeping is done by tableaux and the process of pivoting . See Section 2.5. That is the basic idea of the simplex method. There are bunches of traps for the unwary, but I will try to guide you through some examples, in cases that bad things don't happen. Eventually we will eliminate all possible traps. This lecture will probably take us into the next class. * Usually to solve this system we use augmented matrices. * However, the book-keeping used in the simplex method is based on something slightly different, called tableaux x 1 x 2 x 3 x 4 x 5 1 3 2 1 0 0 3 5 1 1 1 0 4 2 5 1 0 1 Here the leftmost column keeps track of the coefficient vector "b" in the system Ax=b. By Gaussian elimination we obtain: x 1 x 2 x 3 x 4 x 5 1 3 2 1 0 0 2 2 -1 0 1 0 3 -1 3 0 0 1 But now there's a new thing. We think of row 1 as having "solved" for x 3 since it reads 1=3x 1 +2x 2 +x 3 Similarly, row 2 is "solved for x 4 " and row 3 is "solved for x 5 ". * So what we do is mark in each row what is the variable that has been solved for. Since these correspond to columns that form a basis, we call these basic variables and the other variables nonbasic variables . To keep track of this, we introduce an extra column to the left that marks the basic variables: x 1 x 2 x 3 x 4 x 5 x 3 1 3 2 1 0 0 x 4 2 2 -1 0 1 0
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x 5 3 -1 3 0 0 1 The basic operation of pivoting starts from such a table, that gives a bfs and trades a basic variable for a nonbasic variable.
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