L9_10 - ESI 6314 Deterministic Methods in Operations...

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1/11 ESI 6314 Deterministic Methods in Operations Research Lecture Notes University of Florida Department of Industrial and Systems Engineering / REEF
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2/11 Matrix Description of bfs’s What is the relationship between bfs’s and the original problem data? Consider the problem: max 19 x 1 + 13 x 2 + 12 x 3 + 17 x 4 s . t . 3 x 1 + 2 x 2 + x 3 + 2 x 4 225 x 1 + x 2 + x 3 + x 4 117 4 x 1 + 3 x 2 + 3 x 3 + 4 x 4 420 x 1 , x 2 , x 3 , x 4 0 (1)
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3/11 Matrix Description of bfs’s After two iterations of the standard simplex method we obtain: z = 1782 - 2 . 5 x 2 + 1 . 5 x 4 - 3 . 5 x 5 - 8 . 5 x 6 x 1 = 54 - 0 . 5 x 2 - 0 . 5 x 4 - 0 . 5 x 5 + 0 . 5 x 6 x 3 = 63 - 0 . 5 x 2 - 0 . 5 x 4 + 0 . 5 x 5 - 1 . 5 x 6 x 7 = 15 + 0 . 5 x 2 - 0 . 5 x 4 + 0 . 5 x 5 + 2 . 5 x 6 (2) The first three equations of this bfs are equivalent to 3 x 1 + 2 x 2 + x 3 + 2 x 4 + x 5 = 225 x 1 + x 2 + x 3 + x 4 + x 6 = 117 4 x 1 + 3 x 2 + 3 x 3 + 4 x 4 + x 7 = 420 (3) Hence, the bfs (2) arises by solving (3) for x 1 , x 3 , and x 7 .
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L9_10 - ESI 6314 Deterministic Methods in Operations...

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