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Exploring the Traveling Salesman Problem: Heuristics and Applications in Routing Optimization

This lecture delves into the fundamental aspects of the Traveling Salesman Problem (TSP), a classic optimization challenge focusing on finding the shortest route that visits a set of nodes exactly once. Key assumptions include single client service per trip, maintaining truck capacity, and trip time limits. We explore various applications across business logistics and manufacturing. Additionally, the lecture examines heuristic methods for TSP solutions, including nearest neighbor and 2-opt techniques, alongside insights into state-of-the-art solutions and complexity challenges.

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Exploring the Traveling Salesman Problem: Heuristics and Applications in Routing Optimization

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

  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. 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. Applications • Business: delivery routes, facility layout • Manufacturing: Job scheduling, job execution order, robotic function ordering • State of art: Concordehttp://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 Xij is 1 if node j visited after node i in the tour Constraint: ensure all cities visited.

  6. Formulation

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

  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. 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. 2 1 3 4 5 6 1

  11. 2 1 3 4 5 6 5

  12. 2 1 3 4 5 6 3

  13. 2 1 3 4 5 6 2

  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. 2 1 3 4 5 6 5

  16. 2 1 3 4 5 6 3 2 1-5-1 (4) 1-3-5-1 : 8 1-5-3-1 : 8

  17. 2 1 3 4 3 5 6 3 2 3 6 1-5-3-1 (8) 1-6-5-3-1 : 20 1-5-6-3-1 : 15 1-5-3-6-1 : 21

  18. 2 1 3 4 6 5 6 2 2 3 2 6 4 1-5-6-3-1 (15) 1-2-5-6-3-1 : 25 1-5-2-6-3-1 : 22 1-5-6-2-3-1 : 21 1-5-6-3-2-1 : 25

  19. 2 1 3 4 7 7 5 6 2 5 3 4 2 4 1-5-6-2-3-1 (21) 1-4-5-6-2-3-1 : 32 1-5-4-6-2-3-1 : 29 1-5-6-4-2-3-1 : 26 1-5-6-2-4-3-1 : 31 1-5-6-2-3-4-1 : 34

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

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

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

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

  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

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