Management Science 461

Lecture 7 – Routing (TSP)

October 28, 2008

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Facility Location Models

AssumesShipments are not combinedEach truck serves one client at a timeShortest path between facility and client

Can we relax this assumption?Combine shipmentsRespect truck capacityRespect trip time limit

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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….

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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 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 Xij is 1 if node j visited after node i in the tour

Constraint: ensure all cities visited.

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Formulation

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}1,0{

2subject to

minimize

ij

j j jiij

i j ijij

X

iXX

cx

Formulation Problem – Nothing prohibits a subtour Need one constraint for every possible

tour, which is 2N-1

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4

1

2

3

65

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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-optOthers – See Concorde

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Nearest Neighbor (NN)

Start with a random node nFind the nearest node to n not already

selectedSelect the node, travel thereRepeat until all nodes selected; reconnect to

n

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4

1

2

3

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From To Dist.

1 2 8

1 3 3

1 4 7

1 5 2

1 6 10

1

11

4

1

2

3

655

From To Dist.

5 2 4

5 3 3

5 4 6

5 6 4

12

4

1

2

3

65

3

From To Dist.

3 2 5

3 4 9

3 6 6

13

2 From To Dist.

2 4 6

2 6 74

1

2

3

65

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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.

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4

1

2

3

655

From To Dist.

1 2 8

1 3 3

1 4 7

1 5 2

1 6 10

16

24

1

2

3

65

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From To Dist.

1 2 8

1 3 3

1 4 7

1 6 10

5 2 4

5 3 3

5 4 6

5 6 4

1-5-1(4)

1-3-5-11-5-3-1

: 8: 8

17

33

4

1

2

3

65 6

1-5-3-1(8)

1-6-5-3-1 : 20: 15

From To Dist.

1 2 8

1 4 7

1 6 10

5 2 4

5 4 6

5 6 4

3 2 5

3 4 9

3 6 6

1-5-6-3-1: 211-5-3-6-1

2

3

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

From To Dist.

1 2 8

1 4 7

3 2 5

3 4 9

5 2 4

5 4 6

6 2 7

6 4 6: 251-5-6-3-2-1

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4

1

2

3

65

2

19

77

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

From To Dist.

1 4 7

2 4 6

3 4 9

5 4 6

6 4 6

: 311-5-6-2-4-3-1

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: 341-5-6-2-3-4-1

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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

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2-Opt Heuristic

k+1j+1

k

j

k+1j+1

k

jBefore

After: alternate linkspart of pathreversed

Repeat for all pairs of links until no improvement possible

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1

2

3

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2-opt example

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Another 2-opt example

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Not a tour!

Another example

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Comparison Between HeuristicsHeuristic avg. % above opt.Nearest neighbor (100 runs) 15.3Nearest insertion (100 runs) 17.6Cheapest insertion (100 runs) 13.9Farthest insertion (100 runs) 4.9Arbitrary insertion (100 runs) 3.8Clarke-Wright (100 runs) 3.62-Opt (25 runs) 2.22-Opt (50 runs) + 3-Opt 0.9NN + 2-Opt + 3-Opt 1.2AI + 3-Opt (10 runs) 1.0FI + 3-Opt (10 runs) 0.8Four composite heur. above 0.5