exam2soln

# exam2soln - 1 IE 220 Fall 2006 Professor Snyder November 7,...

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1 IE 220 Fall 2006 Professor Snyder November 7, 2006 Name: Mid-Term Exam 2 INSTRUCTIONS: Write neatly and show your work for each question. I will give partial credit for partial answers. You may not use your notes, slides, or textbook. You may use a calculator. If you need more space, use the back of the page, or see the proctor and he will give you more paper. You may remove the staple if you wish. Please put the pages back in the correct order at the end of the exam. There are 100 total points. Good luck!

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2 1. ( 10 points ) Find a minimum spanning tree for the network shown below using the algorithm we discussed in class. Start the algorithm at node A . Indicate your spanning tree on the ﬁgure using boldface or squiggly lines. What is the cost of your spanning tree? G H A B I E D F C 22 21 30 23 29 18 30 15 19 16 5 15 25 4 9 6 11 11
3 2. ( 10 points ) Consider the following algorithm for ﬁnding the shortest path from a node O to a node T : 1. Initialize the algorithm by letting the current node be O . 2. Choose the shortest arc from the current node, provided that the arc doesn’t connect to a node already on the path. Add the arc (and the node at the other end of it) to the path. 3. If the node just added is T , stop. Otherwise, go to 2. This is a “greedy” algorithm, and we said in class that it is not guaranteed to ﬁnd an optimal solution to the shortest-path problem. Demonstrate this by creating a network with at most 4 nodes such that the algorithm does not ﬁnd the shortest path from O to T . Make sure your network includes nodes labeled O and T . Write each arc’s cost (i.e., distance) next to the arc. Indicate the optimal path, and the path found by the algorithm, where indicated below. Your network: The optimal path from O to T in your network: The path found by the algorithm in your network:

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3. ( 14 points ) Consider the following LP: ( P ) minimize Z = 3 x 1 - 2 x 2 + 5 x 3 + 8 x 4 subject to x 1 + x 2 + x 3 + x 4 8 9 x 1 - 2 x 2 - x 4 13 - 3 x 2 + 3 x 3 + 5 x 4 = 22 x 1 , x 2 0 x 3 0 x 4 unrestricted (a) ( 10 points ) Formulate the dual of this LP. (b) (
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## This homework help was uploaded on 04/07/2008 for the course IE 220 taught by Professor Storer during the Spring '07 term at Lehigh University .

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exam2soln - 1 IE 220 Fall 2006 Professor Snyder November 7,...

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