Swarm Intelligence Traveling Salesman Problem and Ant System

Transcription

1 Swarm Intelligence Leslie Pérez áceres Hayfa Hammami IRIIA Université Libre de ruxelles (UL) ruxelles, elgium

2 Outline 1.oncept review 2.Travelling salesman problem Problem definition xamples 3.Ant System Algorithm escription Applied to TSP 4.lass exercise 5.Practical exercise 2/24

3 oncept review Optimization problems Objective function Search space Local / global optima Searching xact vs. approximation methods onstructive vs. perturbative xploration and exploitation 3/24

4 Traveling Salesman Problem Informal definition Given a set of customer cities, a salesman from his home town needs to find a shortest tour that takes him through all customers just once and then back home. 4/24

5 Traveling Salesman Problem (TSP) Main reasons for choosing the TSP: It is a classical combinatorial optimization problem. It is NP hard. It is the problem to which the Ant System algorithm was first applied. Often used to test new algorithms and variants. 5/24

6 Traveling Salesman Problem Formal efinition The TSP can be modelled as a Graph G(N,A) where: N is the set of nodes representing the cities A is the set of arcs ach arc is assign a cost value (length) d dij is the arc cost, or the length from city i to city j 7 10 A /24

7 Traveling Salesman Problem Formal definition Find a minimum length f(π) Hamiltonian circuit of a graph G(N,A), where n is number of nodes and π is a permutation of the nodes indices. n 1 f (π)= d π(i)π(i+1) + d π(n)π (1) i=1 7/24

8 Traveling Tournament Problem First attempt to solve onstructive heuristic The nearest neighborhood heuristic is a simple greedy-type construction heuristic It starts from a randomly chosen city Greedy rule: select the closest city that is not yet visited 7 10 A Initial city: losest city: losest city: losest city: losest city: Return city A cost: 8 cost: 7 cost: 13 cost: 7 cost: 9 Total: 44 8/24

9 Traveling Tournament Problem First attempt to solve The nearest neighbour algorithm is easy to implement and executes quickly. Usually the last a few edges added are extremely large, due to the greedy" nature. In some cases it even constructs the unique worst possible tour. How to generate a tour more intelligently? Learn from the previous constructions! 9/24

10 Ant System Ant System is a basic ant behaviour based algorithm. Ants visit the cities sequentially till they obtain a tour. Transition from city i to j depends on: Heuristic desirability to visit city j when in city i, associated to a static value based on the edgecost (distance) ηij Pheromone that represents the learned desirability to visit city i when in city j associated to a dynamic value τij 10/24

11 Ant System Stochastic Solution onstruction Use memory to remember partial tours. eing at a city i choose next city j probabilistically among feasible neighbouring cities. Probabilistic choice depends on: pheromone trails τij heuristic information ηij = 1/dij Random proportional rule at node i is: pkij (t)= [τij (t)]α [ηij ]β k, if j Ν i α β [τ (t )] [η ] il il k l Νi 11/24

12 Ant System Pheromone Update Use pheromone evaporation to avoid unlimited increase of pheromone trails and allow forgetting of earlier choices Pheromone evaporation rate 0 < ρ 1 Use pheromone deposite to positive feedback, reinforcing components of good solutions etter solutions give more feedback 12/24

13 Ant System Pheromone Update xample of pheromone update m τij (t ) = (1 ρ) τ(t 1) + k Δ τ ij k=1 1 Δ τ =, if arc (i, j) is used by ant k on its tour Lk k ij Lk: Tour length of ant k m: number of ants 13/24

14 Ant System Simple pseudo code While!termination() For k = 1 To m o #m number of ants ants[k][1] SelectRandomity() For i = 2 To n o #n number of cities ants[k][i] ASecisionRule(ants, i) ndfor ants[k][n+1] ants[k][1] #to complete the tour ndfor UpdatePheromone(ants) ndwhile 14/24

15 Ant System Simple example For our example with #ants=3, α=2, β=1, ρ=0.5 and τ0=1 1 6 A Pheromone trails 6 Heuristic Information nij A A 1/1 ½ ½ 1/6 1/1 1/6 1/8 1/10 ½ 1/6 1/12 ¼ ½ 1/8 1/12 1/1 1/6 1/10 ¼ 1/1 - tij A A /24

16 Ant System Simple example For ant #1 we start from city (random), selection probabilities α k ij p (t )= [τij (t )] [ηij ] pij [τil (t )]α [ηil ]β l Ν β A [ 0, 0.264, 0.323, 0.354, 1 ] k i Select a city rand 0.80 pij ity selected A [ 0, 0.267, 0.494, 1 ] Select a city rand 0.27 pij ity selected A [ 0, 0.843, 1 ] Select a city rand 0.88 ity selected 16/24

17 Ant System Simple example First iteration we can have: Ant #1: ----A- Ant #2: A-----A Ant #3: ----A- Update the pheromone using this tours m τij (t )=[1 ρ] τ(t 1)+ Δ τijk k=1 And then iterate tij A A /24

18 Ant System xercise #1 Implement Ant System according to the provided template. ++ The following slides give a practical view of the Ant System algorithm procedures. 18/24

19 Ant System Algorithm Solution onstruction 1 Procedure onstructsolutions () 2 For k = 1 To m o #m number of ants 3 For i = 1 To n o #n number of cities 4 ant[k].visited[i] false 5 ndfor 6 ndfor 7 step 1 8 For k = 1 To m o 9 r random{1,..., n} 10 ant[k].tour [step] r 11 ant[k].visited [r] true 12 ndfor 13 While (step < n) o 14 step step For k = 1 To m o 16 ASecisionRule(k, step) 17 ndfor 18 ndwhile 19 For k = 1 To m o 20 ant[k].tour [n+1] ant[k].tour[1] 21 ant[k].tour length omputetourlength(k) 22 ndfor 23 ndprocedure 19/24

20 Ant System Algorithm ecision Rule Procedure ASecisionRule(k, i) #k ant identifier #i counter for construction step c ant[k].tour[i-1] sum_prob = 0.0 For j = 1 To n o If ant[k].visited[j] Then selection_prob[j] 0.0 lse selection_prob[j] choice_info[c][j] sum_prob sum_prob + selection_prob[j] ndif ndfor r random[0, sum_prob] j 1 p selection_prob[j] While (p < r ) o j j + 1 p p + selection_prob[j] ndwhile ant[k].tour[i] j ant[k].visited[j] true ndprocedure 20/24

21 Ant System Algorithm Pheromone Update Procedure ASPheromoneUpdate () vaporate() For k = 1 To m o epositpheromone(k) ndfor omputehoiceinformation() ndprocedure 21/24

22 Ant System Algorithm Pheromone Update Procedure vaporate For i = 1 To n o For j = i To n o pheromone[i][j] (1 ρ) pheromone[i][j] pheromone[j][i] pheromone[i][j] #pheromones are symmetric ndfor ndfor ndprocedure 22/24

23 Ant System Algorithm Pheromone Update 1 Procedure epositpheromone(k) 2 #k ant identifier 3 τ 1/ant[k].tour_length 4 For i = 1 To n o 5 j ant[k].tour[i] 6 l ant[k].tour[i+1] 7 pheromone[j][l] pheromone[j][l] + τ 8 pheromone[l][j] pheromone[j][l] 9 ndfor 10 ndprocedure 23/24

24 Ant System xercise #2 Test and analyse the behaviour of the algorithm. Modify some parameters: Number of ants α, β, ρ What effect can you appreciate? What is the reason? 24/24