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p1_sp2011 - PRELIM 1 ORIE 3310/5310 Feb 17 2011 Closed book...

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PRELIM 1 ORIE 3310/5310 Feb. 17, 2011 Closed book exam. Justify all work. 1. Consider again the following LP decomposition model from the homework assignment. The first two constraints are the linking constraints ; the remaining constraints define the two subproblems, SUB1 (variables x 1 , x 2 , x 3 ) and SUB2 (variables y 1 , y 2 ). ORIGINAL PROBLEM max 15 x 1 + 7 x 2 + 15 x 3 + 20 y 1 + 12 y 2 s.t. x 1 + x 2 + x 3 + y 1 + y 2 5 3 x 1 + 2 x 2 + 4 x 3 + 5 y 1 + 2 y 2 16 4 x 1 + 4 x 2 + 5 x 3 20 2 x 1 + x 2 4 y 1 + 1 / 2 y 2 3 1 / 2 y 1 + 1 / 2 y 2 2 x 1 , x 2 , x 3 , y 1 , y 2 0 The following tableau is encountered for the restricted master program at an iteration of the decom- position procedure. The λ variables are from SUB1 and the μ variables from SUB2; s 1 , s 2 are slack variables associated with the two linking constraints. The initial basis for the restricted master con- sisted of s 1 , s 2 , the SUB1 variable λ 0 corresponding to extreme point ( x 1 , x 2 , x 3 ) = (0 , 0 , 0) , and the SUB2 variable μ 0 corresponding to extreme point ( y 1 , y 2 ) = (0 , 0) . basis λ 0 λ 1 λ 2 μ 0 μ 1 s 1 s 2 r.h.s. ( - z ) - 4 0 0 0 0 - 10 - 1 - 70 λ 1 4 / 3 1 0 0 0 1 / 3 - 1 / 6 1 / 3 λ 2 - 1 / 3 0 1 0 0 - 1 / 3 1 / 6 2 / 3 μ 1 - 1 / 3 0 0 0 1 5 / 12 - 1 / 12 5 / 12 μ 0 1 / 3 0 0 1 0 - 5 / 12 1 / 12 7 / 12 (a) (5 points) Determine by inspection the inverse of the basis matrix (i.e.,
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