p1_solutions - ENGRI 115 Engineering Applications of OR...

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ENGRI 115 Engineering Applications of OR Fall 2007 Solutions Prelim 1 1. (20 points) Consider the following acyclic directed graph in which the number c ij on the edge ( i, j ) represents the length of the edge. 3 1 1 2 2 1 1 5 2 4 5 3 1 We want to determine the longest path from node 1 to node 5. (a) (15) Write down the recursive equations for the critical path method and solve them. SOLUTION: We give the solution by converting the sign of the edge costs (another formulation of the equations involves “max” instead of “min”). f (5) = min { f (4) - 1 , f (3) - 5 , f (2) - 2 } f (4) = min { f (3) - 1 , f (2) - 3 , f (1) - 2 } f (3) = min { f (1) - 1 } f (2) = min { f (1) - 1 } f (1) = min { 0 } Thus, changing the order we solve: f (1) = min { 0 } = 0 f (2) = min { f (1) - 1 } = - 1 , [1] f (3) = min { f (1) - 1 } = - 1 , [1] f (4) = min { f (3) - 1 , f (2) - 3 , f (1) - 2 } = - 4 , [2] f (5) = min { f (4) - 1 , f (3) - 5 , f (2) - 2 } = - 6 , [3] (b) (5) What is the longest path from 1 to 5 - and what is its length? SOLUTION: The longest path is from 1 to 5 is 1 - 3 - 5, which has length 6.
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2. (30 points) (a) (10) In the graph G below, the bold edges denote a minimum spanning tree T . Indicate a test which uses out-of-tree edges to demonstrate minimality of T . 7 2 3 4 3 5 4 6 7 5 1 SOLUTION: In each step choose an outer-tree-edge e , and consider the unique cycle C that it forms with the spanning tree. If we have c e c f for all edges f C - e , then we must have a minimum spanning tree (see lecture about MSTs and Observation 6).
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