s2008.prelim2 - PRELIM 2 Closed book exam. Justify all...

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PRELIM 2 ORIE 321/521 March 13, 2008 Closed book exam. Justify all work. 1. In the context of the Bellman-Ford shortest path algorithm, suppose G = ( V, E ) is a digraph with vertices V = { 1 , · · · , n } and arc cost data c i,j , ( i, j ) E , not necessarily nonnegative. (a) (10 points) Explain why any directed path from node 1 to node n with more than n - 2 inter- mediate nodes must contain a cycle. (b) (10 points) Show that if G contains no cycles of negative total length then there is a shortest path from node 1 to node n with n - 2 or fewer intermediate nodes. 2. The following allocation model was considered in the homework: Certain sensitive instruments may fail during the course of a planned scienti±c expedition, with the probability that the i th instrument will fail j times during the expedition given by p i,j . Each time the i th instrument fails, it must be replaced by a copy shipped to the expedition at cost c i . Also given for each instrument is its weight w i , and we consider the problem of determining the composition of a
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This note was uploaded on 05/10/2010 for the course OR&IE 3310 taught by Professor Trotter during the Spring '10 term at Cornell University (Engineering School).

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