TSP with GAs

1. TSP with GAs an example of problem analysis and optimization

2. TSP - by the categories • representation many choices • fitness easy • variation many choices • selection • initialization • augmented algorithms

3. Representations • TSP is a permutation problem • sequential (path) • adjacent • ordinal • boolean adjacency matrix • boolean precedence matrix

4. 3 4 6 1 1 2 3 4 5 6 3 5 4 6 1 2 5 2 Representations sequential (path) 3 - 4 - 6 - 2 - 5 - 1 adjacency ordinal 3,3,4,2,2,1 {1,2,3,4,5,6} {1,2,4,5,6} {1,2,5,6} {1,2,5} {1,5} {1}

5. 3 4 6 1 5 2 Representations adjacency matrix precedence matrix

6. 3 1 4 2 6 3 1 6 5 5 2 4 Crossover Variations based onsequential (path) representation PMX (partially-mapped crossover) • 3 - 4 - 6 - 2 - 5 - 1 • 1 - 2 - 3 - 4 - 5 - 6 swap segments • x - x - 3 - 4 - 5 - x • x - x - 6 - 2 - 5 - x retain where possible • x - x - 3 - 4 - 5 - 1 • 1- x - 6 - 2 - 5 - x map from swap • 6 - 2 - 3 - 4 - 5 - 1 • 1 - 4 - 6 - 2 - 5 - 3 analysis: what is inherited? parent 1 2 position? y y/n relative order? y y/n adjacency? y n

7. 3 1 4 2 6 3 1 6 5 5 2 4 Crossover Variations based onsequential (path) representation OX (order crossover) CX (cycle crossover) using complementary representations Grefenstette’s heuristic (p.202) ER (edge recombination) analysis: what is inherited? parent 1 2 position? ? ? relative order? ? ? adjacency? ? ?

8. 3 1 4 2 6 3 1 6 5 5 2 4 Single parent variations onsequential (path) representation Inversion (like 2-opt) Insertion Displacement Reciprocal exchange analysis: what is inherited?from parentposition ? relative order? ? adjacency? ?

9. 3 1 4 2 6 3 1 6 5 5 2 4 Multi-parent variations onsequential (path) representation • Herdy’s heuristic (p.201) analysis: what is inherited?from parentsposition ? relative order? ? adjacency? ?

10. 1 2 3 4 5 6 1 2 3 4 5 6 3 5 4 6 1 2 2 3 4 5 6 1 1 3 2 4 1 2 3 4 5 6 3 6 1 6 3 5 5 2 4 1 2 3 4 5 6 3 1 4 5 6 2 Crossover Variations based onadjacency representation Alternating edges edge from random parent repeat, random city if cycle 1 - 3 - 4 - 5 - 6 - 2 analysis: what is inherited? parent 1 2 position? ? ? relative order? ? ? adjacency? ? ?

11. 3 1 4 2 6 3 1 6 5 5 2 4 Variations based onordinal representation • classic crossover works without repair . . . but how good is it? analysis: what is inherited? parent 1 2 position? ? ? relative order? ? ? adjacency? ? ?

12. 3 4 1 2 3 4 5 6 1 6 0 0 0 0 0 0 1 2 1 0 0 0 1 0 5 2 3 1 1 0 1 1 1 4 1 1 0 0 1 1 5 1 0 0 0 0 0 1 1 0 0 1 0 6 Representations precedence matrix vij = 1 if i before j, i≠j

13. 3 1 4 2 1 1 2 2 3 3 4 4 5 5 6 6 1 1 6 3 0 0 0 1 0 1 1 0 1 0 1 0 6 1 2 2 1 0 0 0 1 0 0 1 1 1 0 1 5 5 4 2 1 0 0 1 0 0 1 1 1 1 1 1 3 3 0 1 1 0 0 0 0 0 1 1 1 1 4 4 1 0 0 0 0 0 0 0 0 0 1 0 5 5 0 1 1 0 0 0 0 0 0 1 0 0 6 6 Variations based onprecedence matrix representation • intersection • union analysis: what is inherited? parent 1 2 position? ? ? relative order? ? ? adjacency? ? ?

14. 3 4 1 2 3 4 5 6 1 6 0 0 1 0 0 0 1 2 0 0 0 0 1 0 5 2 3 0 0 0 1 0 0 4 0 0 0 0 0 1 5 1 0 0 0 0 0 0 1 0 0 0 0 6 Representations (directed) adjacency matrix vij = 1 if ij is directed edge

15. 1 2 3 4 5 6 1 0 1 0 0 0 0 2 0 0 1 0 0 0 3 1 2 4 0 0 0 1 0 0 3 1 2 3 4 5 6 0 0 0 0 1 0 4 1 3 6 0 0 1 0 0 0 6 1 0 0 0 0 0 1 5 2 0 0 0 0 1 0 1 0 0 0 0 0 6 5 5 2 4 0 0 0 1 0 0 3 0 0 0 0 0 1 4 1 0 0 0 0 0 5 0 1 0 0 0 0 6 Variations based onadjacency matrix representation • submatrix rearrangement • common links analysis: what is inherited? parent 1 2 position? ? ? relative order? ? ? adjacency? ? ?

16. Selection strategies • Incest-prevention - solutions can’t cross over with close relatives • goal: maintain variety in population • LIKE TABU

17. Extending EA’s with local search procedure evolutionary algorithm**(p.215) t <-- 0 initialize P(t) // population local-optimize(P(t)) while (~termination-condition) do t <-- t+1 P’(t) <-- alter(P(t-1)) // offspring local-optimize(P’(t)) P(t) <-- select (P’(t), P(t-1)) end end (I have revised this algorithm)

18. procedure evolutionary algorithm**(p.215) t <-- 0 initialize P(t) // population local-optimize(P(t)) while (~termination-condition) do t <-- t+1 P’(t) <-- alter(P(t-1)) // offspring local-optimize(P’(t)) P(t) <-- select (P’(t), P(t-1)) end end interpretation • populations are locally optimized • alterations produce ‘infeasible’ (not locally optimal) solutions that are repaired by local optimization • example: Muhlenbein, p.215 • OX (order crossover) and 2-opt local optimization

19. Edge assembly crossover (EAX) Extended EA algorithm • expensive crossover with local optimization heuristic (variation) • repeatedly used with same parents looking for a child better than both (locally optimized selection)

20. Edge assembly crossover (EAX) Extended EA algorithm local optimization • expensive crossover with local optimization heuristic (variation) 3 4 4 4 6 1 6 6 5 2 5 5 3 4 3 3 6 1 1 1 2 2 5 2

21. 3 4 4 4 4 4 4 4 6 1 6 6 6 6 6 6 5 2 5 5 5 5 3 4 5 5 3 3 3 3 6 3 3 1 1 1 1 1 2 2 2 2 5 2 1 1 2 2 Edge assembly crossover (EAX) Extended EA algorithm (p.217) • repeatedly used with same parents looking for a child better than both (locally optimized selection) local optimization

22. The champion TSP algorithm:Inver-over operator procedure inver-over initialize P while not done for each Si P S’ <-- inver-over-op(Si) if (eval(S’) ≤ eval(Si)) Si <-- S’

23. Inver-over operatorAsymmetric multi-parent operator inver-over-op(Si) S’ <-- Si c <-- random city from S’ repeat if (rand() < p) select c’ from S’, c’≠c else select Sj, Sj ≠ Si c’ <-- city next to c in S’ do 2-opt swap between c, c’ in S’ c <-- c’ until c’ follows c in Si return S’

24. Inver-over inver-over-op(Si) S’ <-- Si c <-- random city from S’ repeat if (rand() < p) select c’ from S’, c’≠c else select Sj, Sj ≠ Si c’ <-- city next to c in S’ do 2-opt between c, c’ in S’ c <-- c’ until c’ follows c in Si return S’ S’: 3-7-2-5-1-6-4 c <-- 2 p? c’<--6S’ c’<--1 Sj: 7-4-6-2-1-5-3 S’:3-7-2-6-1-5-4 S’: 3-7-2-1-5-6-4 c <--6 c <--1 2 new edges 1 inherited edge 1 new edge