11/4/11 Contents CS-­‐4752 Introduc3on to Computa3onal Intelligence •  Ant Colony System –  Marco Dorigo and Luca Maria Gambardella (1997) Ant Colony System: A Coopera3ve Learning Approach to the Traveling Salesman Problem, IEEE Transac3ons on Evolu3onary Computa3on, Vol.1, No.1, 1997. •  Ant SystemElite; Ant SystemRank November 4, 2011 –  Bullnheimer B, Hartl RF, Strauss C (1999) A new rank-­‐
based version of the ant system: a computa3onal study. Cent Eur J Oper Res Econ 7:25-­‐38. AS Pheromone update Rule The ACS Algorithm •  Pheromone Update Rule: τ ij (t +1) = (1 − ρ)⋅ τ ij (t) + Δτ ij (t)
€
• Pheromone on all edges decrease with the same percentage • Pheromone on more visited edges that are parts of be^er tours are increased more than that on other edges. 50 – 100 Ci3es Results •  ACS-­‐TSP has been shown to be superior over other methods like GA, SA, EP. For some bigger problems •  Candidate list: a preferred candidate list containing cl number of closest ci3es to city i; •  When selec3ng next city to visit aber i, the candidate list is checked first and the closest city is selected. •  When the candidate list is empty, one city is selected from the rest of the un-­‐visited ci3es using the new transi3on rule. •  cl is 15 in their experiments. 1 11/4/11 ACO + Candidate List Results TSP Discussion •  Greedy approaches (more exploita3on) seem to work be^er. –  Select closest city first –  Select the edge with the most pheromone and the shortest length 90% of the 3me –  Reward pheromone to edges that belong to the shortest tour. –  Would this approach work for other problems? Ant System -­‐ Elite •  Global pheromone update rule: τ ij (t +1) = ρτ ij (t) + Δτ ij + Δτ ∗ ij
m
where Δτ ij =
∑ Δτ
k
k =1
€and €
ij
⎧ Q
⎪
,Δτ k ij ⎨ Lk
⎪⎩ 0
If ant k travels on edge (i,j) otherwise ⎧
⎪σ Q If edge (i, j) is part of the best tour found Δτ ∗ ij = ⎨ L∗
⎪ 0 otherwise ⎩
Q is a constant, σ is the weight to the elite solu3on pheromone Ant System -­‐ Elite •  Edges belong to the elite ant solu3on have extra pheromone increases based on its distance and weighted by σ. •  The edges do not belong to the elite ant solu3on also increase, based on its distance but without the weight. •  No local pheromone decrease to increase diversity. €
Ant System -­‐ Rank τ ij (t + 1) = ρτ ij (t) + Δτ ij + Δτ ∗ij
σ −1
€
where Δτ ij = ∑ Δτ ijµ σ-­‐1 best ant tours are considered µ =1
and Δτ ijµ
€
€
⎧
Q
⎪(σ − µ)
= ⎨
Lµ
⎪
0
⎩
⎧
otherwise Q If edge (i,j) is part of the best tour found ⎪
and Δτ ∗ij = ⎨σ L∗
⎪
⎩
If the μ-­‐th best ant travels on edge (i,j) 0
Ant System -­‐ Rank •  Only increase pheromone on edges that belong to the σ-­‐1 (1 smaller than the weight for pheromone increase on the best tour edges) best ant tours. •  The amount of increase is weighted by its rank within the σ-­‐1 best ants. •  When all ants find tours with similar distances, distance-­‐based (1/L) pheromone increase is not able to dis3nguish good edges from bad ones. •  The rank-­‐based pheromone increase approach is be^er in separa3ng the quality of different ant tours. otherwise €
2 11/4/11 Example Exercise σ=4, μ: rank a
b
c
d
e
f
x
16.47
16.47
20.09
22.39
25.23
22
y
96.1
94.44
92.54
93.37
97.24
96.05
σ-­‐μ ant e a
f b d Δτ ijE (t) =
c fitness 31.57401 28.95417 31.04489 31.57401 27.6283 f(ant) 3 E c d e b a f 27.6283 2 B b,d,e,f,a,c 1 C f d a b e c 31.04489 1
2
Δτ Cij (t) =
Δτ ijB (t) =
f (C)
f (B)
3
f (E)
€
a €
b e 97 a
96 28.95417 f b 95 Δτ ij (t)
€
ant permuta3on A c a e f b d B b d e f a c C f d a b e c D d b f e a c E c d e b a f 98 tour d 94 c 93 92 15 c 17 19 d 21 23 e 25 27 f a b c d €
e f 3