1. Management Science 461 Lecture 7 – Routing (TSP) October 28, 2008

2. 2 Facility Location Models Assumes  Shipments are not combined  Each truck serves one client at a time  Shortest path between facility and client Can we relax this assumption?  Combine shipments  Respect truck capacity  Respect trip time limit

3. 3 Problem Description Given a set of nodes and a cost metric (distance matrix, network, time network, etc) Find a route of minimum total length that visits each node exactly once This is called the Travelling Salesman Problem. Sounds easy….

4. 4 Applications Business: delivery routes, facility layout Manufacturing: Job scheduling, job execution order, robotic function ordering State of art: Concorde http://www.tsp.gatech.edu/concorde.html http://www.tsp.gatech.edu/concorde.html Largest TSP solved: 49 in 1954; 532 in 1987; 7,397 in 1994; 24,978 in 2004 (took 8 years of computation time)

5. Modelling TSP Minimize total cost travelled, making sure each node is visited Have full distance matrix D for all O-D pairs Binary variable X ij is 1 if node j visited after node i in the tour Constraint: ensure all cities visited. 5

6. Formulation 6

7. Problem – Nothing prohibits a subtour Need one constraint for every possible tour, which is 2 N -1 7 4 1 2 3 6 5

8. 8 Heuristics for the TSP Trade-off between finding good solutions and time spent coding Simple heuristics in this class:  Construction Heuristics: Nearest Neighbor, Nearest Insertion, Cheapest Insertion, Farthest Insertion  Improvement Heuristics: 2-opt and k-opt  Others – See Concorde

9. 9 Nearest Neighbor (NN) Start with a random node n  Find the nearest node to n not already selected  Select the node, travel there  Repeat until all nodes selected; reconnect to n

10. 10 4 1 2 3 65 FromToDist. 128 133 147 152 1610 1

11. 11 4 1 2 3 65 5 FromToDist. 524 533 546 564

12. 12 4 1 2 3 65 3 FromToDist. 325 349 366

13. 13 2 FromToDist. 246 267 4 1 2 3 65

14. 14 Nearest Insertion (NI) Choose a starting node Choose a node to enter the path by considering minimum distance Consider where on the tour is the least- cost location for adding the node Repeat until all nodes are part of the tour.

15. 15 4 1 2 3 65 5 FromToDist. 128 133 147 152 1610

16. 16 2 4 1 2 3 65 3 FromToDist. 128 133 147 1610 524 533 546 564 1-5-1 (4) 1-3-5-1 1-5-3-1 : 8

17. 17 3 3 4 1 2 3 65 6 1-5-3-1 (8) 1-6-5-3-1: 20 : 15 FromToDist. 128 147 1610 524 546 564 325 349 366 1-5-6-3-1 : 211-5-3-6-1 2 3

18. 18 6 2 1-5-6-3-1 (15) 1-2-5-6-3-1: 25 : 221-5-2-6-3-1 : 211-5-6-2-3-1 2 4 3 FromToDist. 128 147 325 349 524 546 627 646 : 251-5-6-3-2-1 6 4 1 2 3 65 2

19. 19 7 7 2 1-5-6-2-3-1 (21) 1-4-5-6-2-3-1: 32 : 291-5-4-6-2-3-1 : 261-5-6-4-2-3-1 2 4 3 FromToDist. 147 246 349 546 646 : 311-5-6-2-4-3-1 5 : 341-5-6-2-3-4-1 4 1 2 3 65 4

20. 20 2-opt Heuristic Start with a completed tour Repeat until no improvements possible:  Repeat for all pairs of links on the tour: Consider deleting the pair and reconstructing a valid tour Keep the modification that most improves tour length

21. 21 2-Opt Heuristic k+1 j+1 k j k+1 j+1 k j Before After: alternate links part of path reversed Repeat for all pairs of links until no improvement possible

22. 22 4 1 2 3 6 5 2-opt example

23. 23 4 1 2 3 6 5 Another 2-opt example

24. 24 4 1 2 3 6 5 Not a tour! Another example

25. 25 Comparison Between Heuristics Heuristicavg. % above opt. Nearest neighbor (100 runs)15.3 Nearest insertion (100 runs)17.6 Cheapest insertion (100 runs)13.9 Farthest insertion (100 runs) 4.9 Arbitrary insertion (100 runs) 3.8 Clarke-Wright (100 runs) 3.6 2-Opt (25 runs) 2.2 2-Opt (50 runs) + 3-Opt 0.9 NN + 2-Opt + 3-Opt 1.2 AI + 3-Opt (10 runs) 1.0 FI + 3-Opt (10 runs) 0.8 Four composite heur. above 0.5