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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 zrow s 1 6 4 1 24 s 1row s 2 1 2 1 6 s 2row s 3 0 1 1 1 1 s 3row s 4 1 1 2 s 4row 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 zrow must be 0, the only 1 appears in the row correspond ing to this basic variable. Also, if we dont count the zrow, then the mvectors 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 righthand 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 Mmethod the twophase 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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 Spring '08
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