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s2008.prelim1

# s2008.prelim1 - PRELIM 1 Closed book exam Justify all work...

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PRELIM 1 ORIE 321/521 February 26, 2008 Closed book exam. Justify all work. 1. Consider the following linear programming problem, to which the decomposition procedure is applied using two subproblems (SUB1, SUB2). The first two constraints are the linking constraints ; the next two constraints, along with x 1 , x 2 , x 3 0 , define SUB1; SUB2 is defined by the final two constraints and y 1 , y 2 0 . 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 μ 0 μ 1 s 1 s 2 r.h.s.

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s2008.prelim1 - PRELIM 1 Closed book exam Justify all work...

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