1. Not guaranteed to find best answer, but run in a reasonable time
Heuristic Algorithms
Not guaranteed to find best answer, but run in a reasonable time

2. Heuristic 1 Ideas? Overall: O(M*N) + O(N*N) + O(M) = O(M*N + N2)
Go from any depot to nearest pickup, p
while (packages to deliver)
drop off package at delivery[p].dropOff
p = nearest remaining pickup }
Go to nearest depot
O(M*N)
N iterations
O(N)
O(M)
Overall: O(M*N) + O(N*N) + O(M)
= O(M*N + N2)

3. Fast Enough? O(M*N + N2) Ideas? Say M = 10, N = 100
Time = C * [10 * ]
Time = C * 11000
Call m3 findpath to evaluate each possible travel time?
C  0.1 s
Time  1100 s  too big!
Ideas?
Estimate travel time as  geometric distance
Now C  10-7
Time  10-7 * = 1 ms  more than fast enough!
Call find_path on final order: 200 calls  20s  OK!

4. Heuristic 1 D A A B B C C D
Much better than using given/random order Can we improve more?

5. Heuristic 1++ Overall: O(M*N) + O(2N*N) + O(M) = O(M*N + N2)
Go from any depot to nearest pickup, p while (packages to deliver) p = nearest legal pickUp or dropOff solution += path to p } Go to nearest depot
O(M*N)
2N iterations
O(N)
O(M)
Overall: O(M*N) + O(2N*N) + O(M)
= O(M*N + N2)

6. Heuristic 1++ Better still Not much more code; slightly more CPU timeA A B B C C D
Better still Not much more code; slightly more CPU time

7. Heuristic 1 & 1++ are “Greedy”
Make the next decision “greedily”
Best decision you can make now
And never go back to change a decision
Greedy algorithms
Generally fast
But often don’t get the best solution
Smarter variants: use guess of what may happen in future to make a better decision

8. 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?
Force it to start at a different depot
Start with the 2nd smallest travel time from depot to a pick up
Have a 10% chance of taking the second smallest travel time when choosing each next delivery
...

9. Heuristic 1++ (Greedy) AgainB A A B
Optimal?

10. Heuristic 2: Multi-StartB A A B
Force route to start on 2nd best depot Better final answer, even though first hop is longer!