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

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Lecture 8: Simplex Methods with Artificial Variables September 13, 2010.

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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 0 0 0 0 0 z -row s 1 0 6 4 1 0 0 0 24 s 1 -row s 2 0 1 2 0 1 0 0 6 s 2 -row s 3 0 -1 1 0 0 1 0 1 s 3 -row s 4 0 0 1 0 0 0 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 don’t 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 0 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 0 3

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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. 4
Artificial Variables 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 0 min z = 4 x 1 + x 2 s.t. 3 x 1 + x 2 = 3 4 x 1 + 3 x 2 - x 3 = 6 x 1 + 2 x 2 + x 4 = 4 x 1 , x 2 0

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