1. Mental Health and Wellness Resources
Health and Wellness ( ) or Koffler Student Services.
Good 2 Talk ( ): 24 hour helpline
Other 24/7:Gerstein Centre and Toronto Distress Centre at HELP (4357). CAMH’s Center for Addiction and Mental Health at 250 College Street. 

2. M4 Iterative Algorithms, Continued

3. Solution Representation
What About Depots?
Solution Representation
Given N pick up / drop offs 2 1 3 0 2 3 2 0 1 1
Given M courier truck depots (intersections)
Don’t have to store depots Could fill in closest depot to start/end later deliveryOrder = {0, 0, 0, 1, 3, 3, 1, 2, 2, 2}

4. Local Permutations Representation
deliveryOrder = {1, 3, 2, 1, 3, 0, 0, 2} 2 1 8 15 5 2 18 3 5 0 4 20 3 0 4 1 2
Total travel time = 81

5. Local Permutations Swap order of two deliveries?
deliveryOrder = {1, 3, 2, 1, 3, 0, 0, 2}
deliveryOrder = {1, 3, 2, 1, 3, 0, 2, 0} 2 1 8 15 14 5 2 18 23 3 5 0 4 20 3 7 0 4 1 2
Total travel time = 72

6. 2-opt
deliveryOrder = {1, 3, 2, 1, 3, 0, 0, 2} Delete two connections in path 3 0 0 1 1 2 2 3

7. 2-opt
Reconnect connections differently deliveryOrder = {1, 3, 2, 1, 3, 0, 0, 2}
deliveryOrder = {1, 1 2, 3, 3, 0, 0, 2}
General:
Pick order for 3 sub-paths
Reverse any or all sub-paths
Check if legal 3 0 0 1 1 2 2 3

8. 2-opt: Complexity 2N-1 edges/connections between delivery locations
How many different 2-opts?
(2N-1) choose 2 edges we could cut
N = 100  199 * 198 / 2 = ~20,000 options
3 sub-paths
Can order in 6 ways
23 options to reverse or not 3 sub-paths
Can reverse or recombine in 48 ways
Total: ~1,000,000 2-opts
Fast code can try 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
Could make your own perturbation algorithm
Maybe 2-opts not best for traveling courier  many illegal
3-opts?

9. Heuristics Overview

10. Really a 200-dimensional space for N = 100. Huge!
Exploring the Space
Worse
Travel Time
Local Minima
Global Minimum
Better
Solution
Really a 200-dimensional space for N = Huge!

11. Solution very poor  many more bad solutions than good in this space
Pick a Random Order?
Random Solution
Worse
Travel Time
Better
Solution
Solution very poor  many more bad solutions than good in this space

12. Multi-Start? Travel Time Solution
Best of 4 random orders
Worse
Travel Time
Better
Solution
A bit better, but still not very good
Space very big, with mostly bad solution points

13. Still not locally optimal
Greedy Algorithm?
Greedy
Worse
Travel Time
Better
Solution
Much better!
Still not locally optimal

14. Local Permutation (Iterative Improvement)?
Worse
Travel Time
Better
Solution
Search Local Space
Finds Local Minimum!

15. Combine with Multi-Start?
Worse
Travel Time
Better
Solution
Yes – Find Many Local Minima
Take Best One

16. Local Permutations: Can Get Stuck
Worse
Travel Time
Better
Solution
Can’t Get Out of Local Minimum

17. More Powerful Local Permutations
Worse
Travel Time
N=100: about 1M 2-opts, but 500M 3-opts
Better
Solution
Explore Larger Part of Space  Less Prone to Getting Stuck
But More Exploration Means More CPU Time  Need Balance

18. Hill Climbing? Travel Time Solution
Worse
Travel Time
Better
Solution
Can Get You Unstuck from Local Minima
But Watch CPU Time  Need Time to Improve Solution After It Gets Worse

19. 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
Change delivery order
Even though it increases travel time
Good idea to save best solution first
Then try local perturbation around that new solution
Maybe you find something better!
If not, eventually can go back to saved solution
Lets you explore more options

20. Simulated Annealing Annealing metals Simulated annealing Cool slowly
Atoms gradually move to low energy state
Strong metal
Simulated annealing
Optimization framework that mimics annealing
Start with poor initial solution & high temperature (T)
Perturb solution (move atoms)
Most perturbations accepted when T high
Only good ones (reduce cost/energy) accepted when T approaches 0

21. Simulated Annealing
S = InitialSolution; C = Cost (S); // E.g. travel time T = high temperature; // big number while (solution changing) { Snew = perturb (S); Cnew = Cost (S); C = Cnew - C if (Cnew < C) { S = Snew; // Update solution C = Cnew; } T = reduceTemp (T);
How to perturb?
Smaller perturbations as T drops?
|| random(0,1) < e-C/T ) {
Fast update methods will let you evaluate more perturbations
How quickly to reduce T?

22. Time Limits & Using Multiple CPUs

23. Managing the Time Limit
Can get better results with more CPU time
Multi-start:
Run algorithm again with a new starting point
Keep best
Iterative improvement
Keep looking for productive changes
But 30 s limit to return solution
For every problem
Regardless of city size & num intersections
How many starting points?
How many iterative perturbations?
Solution: check how much time has passed
End optimization just before time limit

24. Time Limit
#include <chrono> // Time utilities #define TIME_LIMIT 30 // m4: 30 second time limit int main ( ) { auto startTime = std::chrono::high_resolution_clock::now(); bool timeOut = false; while (!timeOut) { myOptimizer (); auto currentTime = std::chrono::high_resolution_clock::now(); auto wallClock = std::chrono::duration_cast<chrono::duration<double>> ( currentTime - startTime); // Keep optimizing until within 10% of time limit if (wallClock.count() > 0.9 * TIME_LIMIT) timeOut = true; } ...
Inside namespace chrono
Static member function: can call without an object
Time difference
Gives actual elapsed time no matter how many CPUs you are using
Time difference in seconds