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class+April15

# class+April15 - Math 103 Section 11 Friday Homework for...

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Math 103, Section 11, Friday, April 15, 2011 Homework for Chapter 5 is due on Friday, April 15, 5 minutes before midnight. Please do Exercises ( on pages 193-196 ) 20,26,30,34,38,42 and 46 and upload to Sakai. Homework for Chapter 6 will be discussed in class and not submitted to Sakai. Exercises (on pages 229-232) 24,26,30,32,40,46 There will be a quiz on Chapter 6 on Tuesday, April 26. Homework for Friday, April 15. Page 229 Exercises 23,25,29 Chapter 6 The Mathematics of Touring Hamilton Paths and Hamilton Circuits and The Traveling Salesman Problem Graph model of a traveling salesman problem Sites vertices of the graph Costs weights of the edges Tour Hamilton circuit Optimal tour Hamilton circuit of least total weight A tour is a Hamilton circuit of the graph and an optimal tour is the Hamilton circuit of least total weight. Strategies for solving Traveling Salesman Problems (TSPs) We will learn: 1. Exhaustive Search. Also called the Brute-Force Method. 2. Nearest-Neighbor Algorithm 3. Repetitive Nearest-Neighbor Algorithm 4. The Cheapest-Link Algorithm Table 2 completed Starting vertex Route Route starting with A Cost of route A A,C,E,D,B,A A,C,E,D,B,A \$773 B B,C,A,E,D,B A,E,D,B,C,A \$722 C C,A,E,D,B,C A,E,D,B,C,A \$722 D D,B,C,A,E,D A,E,D,B,C,A \$722 E E,C,A,D,B,E A,D,B,E,C,A \$741

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Note that trips starting at vertices B,C, and D are the same trip . You can see this when you rewrite the trip so that the Route starts at vertex A. So the repetitive nearest neighbor algorithm gives the tour A,E,D,B,C,A with \$722 as the cheapest. Relative error of a tour (e) e=(cost of tour-cost of optimal tour)/cost of optimal tour relative error for the repetitive nearest neighbor algorithm=(722-676)/676= 0.068047=6.80% Exercises: Brute force and nearest neighbor exercises.
Solution to Exercise 29: Part a: You can use the following table to find the optimal tour: Hamilton circuit Total cost 1 A,B,C,D,A 48+32+18+22=120 2 A,B,D,C,A 48+20+18+28=114 3 A,C,B,D,A 28+32+20+22=102 optimal tour 4 A,C,D,B,A 28+18+20+48=114 5 A,D,B,C,A 22+20+32+28=102 another optimal tour 6 A,D,C,B,A 22+18+32+48=120 b.-f. The nearest-neighbor tours with costs, rewritten with starting vertex A: Starting vertex Route Route starting with A Cost of route A A,D,C,B,A

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