Lecture8 - i th unit vector where i is the variable leaving...

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Advanced Operations Research Techniques IE316 Lecture 8 Dr. Ted Ralphs
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IE316 Lecture 8 1 Reading for This Lecture Bertsimas 3.3
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IE316 Lecture 8 2 What the Tableau Looks Like The tableau looks like this - c T B B - 1 b c T - c T B B - 1 A B - 1 b B - 1 A In more detail, this is - c T B x B ¯ c 1 ··· ¯ c n x B (1) . . . B - 1 A 1 ··· B - 1 A n x B ( m )
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IE316 Lecture 8 3 Parts of the Tableau Row zero contains the reduced costs. Column zero contains the values of the current basic variables. The upper left-hand corner entry is the opposite of the current objective function value. Each nonbasic column contains the feasible direction corresponding to increasing the given nonbasic variable. The basic columns are the columns of B - 1 B = I , i.e., they are the unit vectors. All the information needed to perform an iteration of the simplex method is readily available. If variable j is to enter the basis, perform elementary row operations to turn column j of the tableau into the
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Unformatted text preview: i th unit vector, where i is the variable leaving the basis. IE316 Lecture 8 4 Implementing the Tableau Method 1. Start with the tableau associated with a specified BFS and associated basis B . 2. Examine the reduced costs in row zero and select a pivot column with ¯ c j < if there is one. Otherwise, the current BFS is optimal . 3. Consider u = B-1 A j , the j th column of the tableau. If no component of u is positive, then the LP is unbounded . 4. Otherwise, compute the step size using the minimum ratio rule and determine the pivot row . 5. Scale the pivot row so that the pivot element becomes one. 6. Add a constant multiple of the pivot row to each other row of the tableau so that all other elements of the pivot column become zero. 7. Go to Step 2....
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Lecture8 - i th unit vector where i is the variable leaving...

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