# hw3 - exceed her knapsack’s capacity and maximizes the...

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ENGRI 115 Engineering Applications of OR Fall 2007 The Shortest Path Problem II. Homework 3 Due date: Tuesday, September 25 in class. References: Lecture notes; Lab 3; Winston 8.2. The first two problems are practice problems, DO NOT hand them in. The TAs will be glad to help you if you have any questions. USE THE ALGORITHM FOR ACYCLIC DIRECTED GRAPHS in all of the problems. 1. Winston, page 418, Problem 2. 2. Winston, page 1013, Problem 1. 3. Winston, page 418, Problem 4. In the previous homework we have seen that Dijkstra’s algorithm fails on this problem. Use the algorithm for acyclic directed graphs to solve this problem. 4. Winston, page 418, Problem 6. Formulate the problem as a shortest path problem, and then solve it. 5. Knapsack Problem A hiker would like to take several items with her on a trip so that she does not
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Unformatted text preview: exceed her knapsack’s capacity, and maximizes the “pleasure of the trip”, that is, the total utilities of the items. The volume and utility of each item is known to her, and she is allowed to take several copies of the same item. Formulate the problem as ﬁnding the longest path in an acyclic directed graph with the data given below; that is, write down what the nodes, arcs and arc costs represent in your model, and show why the length of a longest path (between certain nodes) gives the pleasure of an optimal packing. Then solve the problem. Is the optimal solution unique? The knapsack has capacity 6, the hiker can take apples, maps, extra sweatshirts and textbooks with her; the respective volumes of these items are in order 1 , 2 , 3 , 4; the utilities 5 , 10 , 20 and 35....
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