chap11_2011_2

Chap11_2011_2 - 1 • The Traveling Salesman Problem(TSP seeks a minimum total-length route visiting every points in a given set exactly once[11.23

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Unformatted text preview: 11/15/2011 1 • The Traveling Salesman Problem (TSP) seeks a minimum- total-length route visiting every points in a given set exactly once. [11.23] • A traveling salesman problem is symmetric if the distance or cost of passing from any point ? to any other point ? is the same as the distance from ? to ? . Otherwise, the problem is asymmetric. [11.24] 11.5 Traveling Salesman and Routing Models Example 11.8 NCB Circuit Board TSP Figure 11.4 shows the tiny example that we will investigate for fictional board manufacturer NCB. We seek an optimal route through the 10 hole locations indicated. Table 11.7 reports straight-line distances d i,j between hole locations i and j. Lines in Figure 11.4 show a fair quality solution with total length 92.8 inches. The best route is 11 inches shorter (see Section 12.6). 1 2 3 4 5 6 7 8 9 10 1 3.6 5.1 10.0 15.3 20.0 16.0 14.2 23.0 26.4 2 3.6 3.6 6.4 12.1 18.1 13.2 10.6 19.7 23.0 3 5.1 3.6 7.1 10.6 15.0 15.8 10.8 18.4 21.9 4 10.0 6.4 7.1 7.0 15.7 10.0 4.2 13.9 17.0 5 15.3 12.1 10.6 7.0 9.9 15.3 5.0 7.8 11.3 6 20.0 18.1 15.0 15.7 9.9 25.0 14.9 12.0 15.0 7 16.0 13.2 15.8 10.0 15.3 25.0 10.3 19.2 21.0 8 14.2 10.6 10.8 4.2 5.0 14.9 10.3 10.2 13.0 9 23.0 19.7 18.4 13.9 7.8 12.0 19.2 10.2 3.6 10 26.4 23.0 21.9 17.0 11.3 15.0 21.0 13.0 3.6 1 2 3 4 5 6 7 8 9 10 11/15/2011 2 • Most ILP models of the symmetric case employ decision variables, ? < ? ? ?,¡ ≡ ¢ 1 if the route includes a leg between i and j 0 otherwise • Total length of a route can be calculated by £ £ ? ?,¡ ¡>? ? ?,¡ ? • Constraints for symmetric TSP £ ? ¡,? + ¡¤? £ ? ?,¡ = 2 ??¥ ¦?? ? ¡>? Formulating the Symmetric TSP (11.15) (11.16) • Definition of subtours S proper subset of points/cities to be routed • Subtour elimination constraints £ £ ? ?,¡ + ?∉ ?∈ £ £ ? ?,¡ ≥ 2 ?∈ ?∉ • Number of legs between points in S and points not in S must be at least 2. Subtours (11.17) 1 2 3 4 5 6 7 8 9 10 11/15/2011 3 min ? ?,? ?>? ¡ ?,? ? s.t. ¡ ?,? + ?<? ¡ ?,? = 2¢¢??£¢¤??¢? ?>? ¡ ?,? + ?∉ ?∈ ¡ ?,? ≥ 2¢??£¢¤??¢¥, ¥ ≥ 3 ?∈ ?∉ ¡ ?,? = 0¢?£¢1¢??£¢¤??¢?, ? ¦ ? ILP Model of the Symmetric TSP [11.25] min ? ?,? ?≠? ¡ ?,? ? s.t. ¡ ?,? = 1¢??£¢¤??¢? ? ¡ ?,? = 1¢¢??£¢¤??¢? ? ¡ ?,? ≥ 1¢??£¢¤??¢¥, ¥ ≥ 2 ?∉ ?∈ ¡ ?,? = 0¢?£¢1¢??£¢¤??¢?, ? ILP Model of the Asymmetric TSP [11.26] 11/15/2011 4 ? ¡,? ≡ 1¢¢¢if¢¢kth¢point¢visited¢is¢i 0¢¢¢otherwise¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢¢ ¢ ¢ min £ £ ? ?,? ? £ ? ¡,? ¡ ¤ ¡+¥,? ? ¢¢(Total¢length) £ ?...
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This note was uploaded on 12/03/2011 for the course ESI 6316 taught by Professor Staff during the Summer '11 term at FIU.

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Chap11_2011_2 - 1 • The Traveling Salesman Problem(TSP seeks a minimum total-length route visiting every points in a given set exactly once[11.23

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