1. Linear Programming 2015 1 3.3 Implementation

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6. 6  Ex 3.5: See text example 3.5 for more iterations.

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11. Linear Programming 2015 11 Practical Performance Enhancements

12. Linear Programming 2015 12 3.4 Anticycling

13. Linear Programming 2015 13 ratio = 1/3 for 1 st and 3 rd row

14. Linear Programming 2015 14  Thm : Suppose the rows in the current simplex tableau is lexicographically positive except 0-th row and lexicographic rule is used, then (a) every row except 0-th remains lexicographically positive. (b) 0-th row strictly increases lexicographically. (c) simplex method terminates finitely.

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16. Linear Programming 2015 16  Remarks : (1) To have initial lexicographically positive rows, permute the columns (variables) so that the basic variables come first in the current tableau

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19. Linear Programming 2015 19 3.5 Finding an initial b.f.s.

20. Linear Programming 2015 20 Driving artificial variables out of the basis (in tableau form) Pivot element

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22. Linear Programming 2015 22 Remarks

23. Linear Programming 2015 23  Sometimes, we may want to retain the redundant rows when we solve the problem because we do not want to change the problem data so that we can perform sensitivity analysis later, i.e. change the data a little bit and solve the problem again. Then the artificial variables corresponding to the redundant equations should remain in the basis (we should not drop the variable). It will not leave the basis in subsequent iterations since the corresponding row has all 0 coefficients.  If we do not drive the artificial variables out of the basis and perform the simplex iterations using the current b.f.s., it may happen that the value of the basic artificial variables become positive, hence gives an infeasible solution to the original problem. To avoid this, modification on the simplex method is needed or we may use the bounded variable simplex method by setting the upper bounds of the remaining artificial variables to 0. ( lower bounds are 0 )

24. Linear Programming 2015 24 Two-phase simplex method

25. Linear Programming 2015 25 3.7 Computational efficiency of the simplex method