Traveling Salesman Problem

Continued

Heuristic 1

• Ideas?– Go from depot to nearest delivery– Then to delivery closest to that– And so on until we are done with deliveries– Then go to closest depot

Heuristic 1

Much better than just using given/random order

Optimal?

Heuristic 1 is “Greedy”

• Makes the next decision “greedily”– Best decision you can make now– Without considering future consequences– And never go back to change a decision

• Greedy algorithms– Generally fast– But often don’t get the best solution

How Could We Get More Quality?

• By spending more CPU time• Heuristic 2: take best of many results

– “Multi-start”– E.g .try our greedy algorithm multiple times

• How to make it get a different answer?1. Force it to start at a different depot

2. Start with the 2nd smallest travel time from depot to a delivery

3. Have a 10% chance of taking the second smallest travel time when choosing each next delivery

4. ...

Heuristic 2: Multi-start

Different

Might be better travel time, even though first hop is longer

Heuristic 3

• Improve an existing solution?

Heuristic 3

• Improve an existing solution?– How?– Local perturbation

5

4

8

5

4

2

20

15

18

Travel time for this connection (edge)

Total travel time = 81

Heuristic 3: Local Perturbation

• E.g. 2-edge operation (2-opt)– Delete two connections in path

Heuristic 3: Local Perturbation

• E.g. 2-edge operation (2-opt)– Reconnect total path differently– But have to make legal!

Heuristic 3: Local Perturbation

• Must form a depot all delivery depot path– Swap endpoints of blue & yellow connections

15

4

8

5

2

20

15

18

9

Heuristic 3: Local Perturbation

• Must form a depot all delivery depot path– Swap endpoints of blue & yellow connections (edges)– Then reverse some edge directions to legalize path– This can affect edge travel times!

15

4

7

4

2

20

15

18

9

Total travel time = 94

Heuristic 3: Local Perturbation

• Calculate new total travel time– Update current best solution if it is an improvement– Otherwise go back to prior solution

15

4

7

4

2

20

15

18

9

Total travel time = 94

Heuristic 3: Local Perturbation

• Try another 2-opt– Swap end-points of pink & green connections

5

4

8

5

4

2

20

15

18

Total travel time = 81

Heuristic 3: Local Perturbation

• Try another 2-opt– Swap end-points of pink & green connections– Compute new connection travel times

5

4

8

5

4

2

2314

18

Heuristic 3: Local Perturbation

• Try another 2-opt– Swap end-points of pink & green connections– Compute new connection travel times– Legalize path (have to go to depot from last delivery)

5

4

8

5

4

2

2314

7

Heuristic 3: Local Perturbation

• Try another 2-opt– Compute new travel time: 72– Improvement: save in current best solution

5

4

8

5

4

2

2314

7

Heuristic 3

• Local perturbation or iterative improvement• N-1 edges/connections between delivery locations• How many different 2-opts?

– (N-1) choose 2 choices between delivery locations– N = 40 39 * 38 / 2 = 741 options– Somewhat more if we also change the depot edges– Reasonable number, but fast code can check them all

• If you find an improvement, can try all 2-opts again– Path has changed a 2-opt that didn’t work before now might

Iterative Improvement

• Can still get stuck in local minima• Not at the optimum (global minimum)• But your local perturbations all result in a

worse solution• Larger perturbations can explore a bigger

space• E.g 3-opts

– Delete 3 connections at a time– Then reconnect in various ways– Bigger change can explore more of space– But more combinations:

• N-1 choose 3: ~10,000 • Times 4 ways to reconnect each: ~40,000

Heuristic 4: Hill Climbing

• One way to get out of a rut– Change something!– Even if it looks like a bad idea at first

• Hill climbing