Lecture8 - Lecture 8: Simplex Methods with Artificial...

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Unformatted text preview: Lecture 8: Simplex Methods with Artificial Variables September 13, 2010. Simplex Tableau Simplex tableau for the Reddy Mikks Model: Basic z x 1 x 2 s 1 s 2 s 3 s 4 Solution z 1 -5 -4 z-row s 1 6 4 1 24 s 1-row s 2 1 2 1 6 s 2-row s 3 0 -1 1 1 1 s 3-row s 4 1 1 2 s 4-row Note: In a simplex tableau, the column corresponding to a basic variable must be a unit ( m +1)-vector. In this vector, the element in the z-row must be 0, the only 1 appears in the row correspond- ing to this basic variable. Also, if we dont count the z-row, then the m-vectors corresponding to the m basic variables can form an identity matrix (if listed in a proper order). The above points are crucial for being able to apply the optimality condition, the ratio test, and other row operations. 2 Simplex Tableau: Initial BFS Get back to the Reddy Mikks Model. Max z = 5 x 1 + 4 x 2 s.t. 6 x 1 + 4 x 2 + s 1 = 24 x 1 + 2 x 2 + s 2 = 6- x 1 + x 2 + s 3 = 1 x 2 + s 4 = 2 x 1 , x 2 , s 1 , s 2 , s 3 , s 4 In this example, all the constraints are constraints with nonnegative right-hand sides, therefore we can chose all the slack variables as the basic variables and easily find the initial BFS. However, it is NOT always the case, for example: min z = 4 x 1 + x 2 s.t. 3 x 1 + x 2 = 3 4 x 1 + 3 x 2 6 x 1 + 2 x 2 4 x 1 ,x 2 3 Artificial Starting Solution Introduce artificial variables (nonnegative) so that we can easily find an initial BSF. Get rid of these artificial variables legitimately at a later it- eration. Two methods:- the M-method- the two-phase method NOTE : Artificial variables are different to slack variables al- though they look similar. Introducing slack variables does not change the LP, however, introducing artificial variables changes the LP and therefore they have to be dropped at a later iteration....
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Lecture8 - Lecture 8: Simplex Methods with Artificial...

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