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Lecture09 - IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis...

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IE 505 MATHEMATICAL PROGRAMMING Bahar Yetis Kara Lecture # 9 Date: 21 October 2007 1
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2 [A|b]
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Tableau form of Simplex Consider 3 0 0 . . min x b Ax x c z t s z T 0 . . min x b Ax t s x c T
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Tableau form of Simplex 4 ... ... . . ... . ... . ... ... . . ... . ... . ... ... 0 ... ... 1 ... ... z 1 1 1 1 1 11 1 1 m mn mj m i in ij i n j n j n j b a a a b a a a b a a a c c c RHS x x x
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Tableau form of Simplex Given B, the same tableau can be partitioned as: 5 0 1 x z b A c RHS 0 1 RHS b N B c c x x z N B N B
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Tableau form of Simplex Now consider 6 0 , 0 . . min N B N B N T N B T B x x b Nx Bx x c x c z t s z b B Nx Bx B N B 1 1 ) ( b B Nx B Ix N B 1 1 N B Nx B b B x 1 1
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Tableau form of Simplex Now we have: 7 0 , 0 . . min 1 1 N B N B N T N B T B x x b B Nx B Ix x c x c z t s z N B Nx B b B x 1 1 0 ) ( 1 1 N T N N T B x c Nx B b B c z b B c x c N B c z T B N T N T B 1 1 ) (
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Tableau form of Simplex Thus Is equivalent to 8 0 1 RHS b N B -c -c x x z N B N B 0 1 RHS 1 1 1 1 b B N B I x b B c N B c c x x z B T B T B N N B N c N r
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Tableau form of Simplex m mp m B r rp r B p B T B p p T B p b u x b u x b u x b B c c u c RHS x z ... ... . ... . ... . ... ... . ... . ... . ... ... ... 1 ... ... ) ( ) ( 1 1 ) 1 ( 1 9 Consider where b B b 1 p p A B u 1 enters Pivot column Pivot row leaves
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Tableau form of Simplex m mp m B r rp r B p B T B p p T B p b u x b u x b u x b B c c u c RHS x z ... ... . ... . ... . ... ... . ... . ... . ... ... ... 1 ... ... ) ( ) ( 1 1 ) 1 ( 1 10 0 0 1 0 0 0 Convert to
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Example Consider 11 0 , 8 2 6 . . 3 min 2 1 2 1 2 1 2 1 x x x x x x t s x x 0 , 8 2 6 . . 3 min 2 1 4 2 1 3 2 1 2 1 x x x x x x x x t s x x 8 6 1 0 2 1 0 1 1 1 ] 0 0 3 1 [ b A c T
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Example Initial basis
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