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: Concorde Largest TSP solved: 49 in 1954; 532 in 1987; 7,397 in 1994; 24,978 in 2004 (took 8 years of computation time)

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

Formulation 6

Problem – Nothing prohibits a subtour Need one constraint for every possible tour, which is 2 N

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

FromToDist

FromToDist

FromToDist

13 2 FromToDist

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.

FromToDist

FromToDist (4) : 8

(8) : 20 : 15 FromToDist :

(15) : 25 : : FromToDist :

(21) : 32 : : FromToDist : :

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+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

opt example

Another 2-opt example

Not a tour! Another example

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) Opt (25 runs) 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