A Course on Meta-Heuristic Search Methods for ...

A Course on Meta-Heuristic Search Methods for Combinatorial Optimization ProblemsAnt colony algorithm Assignments submission
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Optimization Problems
AutOrI LAB, DIA, Roma Tre
Email: [email protected]
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
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Outline
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Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
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The behaviour model of Reynolds (1987):
Separation: Each agent tries to move away from its neighbors if they are too close.
Alignment: Each agent steers towards the average heading of its neighbors.
Cohesion: Each agent tries to go towards the average position of its neighbors.
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Each agent is attracted towards the location of the roost.
Each agent remembers where it was closer to the roost.
Each agent shares information with its neighbors (originally, all other agents) about its closest location to the roost.
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Particle swarm optimization working process
vi : velocity of particle i ρ1 & ρ2: Acceleration coefficients pbesti : best position of particle i gbest: Global best position
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update velocity:
vi (t) = vi (t−1)+ρ1×(pbesti−xi (t−1))+ρ2×(gbest−xi (t−1)) (1.1)
Update position:
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Types of bees:
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They are associated with food sources.
An employed bee carries with her information about the food source (its distance and direction from the nest). She performs waggle dance to share these informations with onlookers waiting at the hive.
Onlookers: Gets informations and exploits the food source. Scouts: Randomly explores the search space.
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Initial food sources are produced for all employed bees. REPEAT
Each employed bee goes to a food source in her memory and determines a neighbour source, then evaluates its nectar amount and dances in the hive.
Each onlooker watches the dance of employed bees and chooses one of their sources depending on the dances, and then goes to that source. After choosing a neighbour around that, she evaluates its nectar amount.
Abandoned food sources are determined and are replaced with the new food sources discovered by scouts.
The best food source found so far is registered. UNTIL (requirements are met)
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Calculation of new position:
xnew = xold + rand(−1, 1)× (xold − x ′) (2.1)
x ′( 6= xold): randomly chosen position from the population.
For scouts
x ji = z jmin + rand(0, 1)× (z jmax − z jmin) (2.2)
z jmin&z jmax : lower and upper bounds for x ji
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Based on the foraging behaviour of ants.
Ants are able to find the shortest path between their nest and the food source, despite they are almost blind.
Ants deposit a certain amount of pheromone trails while walking, and each ant probabilistically choose a path containing higher amount of pheromone.
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Ant colony algorithm
Ants are moving in a straight line connecting the food source and the nest.
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Those ants which have arrived just in front of the obstacle cannot continue to follow the pheromone trail and therefore they have to choose between turning right or left. In this situation, lets assume that half the ants chose to turn right and the other half turned left.
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Those ants which choose, by chance, the shorter path around the obstacle will more rapidly reconstitute the interrupted pheromone trail compared to those which choose the longer path. Thus, the shorter path will receive a greater amount of pheromone per time unit and in turn a larger number of ants will choose the shorter path.
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Ant colony algorithm working process: an example on VRP
Bell and McMullen (2004) tackled the VRP, using ACO algorithm in which an ant simulates a vehicle and its route is constructed by incrementally selecting customers. To select the next available customer j , an ant uses the equation 3.1.
j = argmax {
(τiu) (ηiu)β }
u /∈ Mk , q ≤ qo (3.1)
τiu: the amount of pheromone on the path between the current location i and possible locations u. ηiu: the inverse of the distance between two customer locations. β: the importance of distance in comparison to pheromone quantity in the selection algorithm (β > 0). Mk : ant’s working memory. q: a random uniform variable in [0,1]. qo : a user defined parameter.
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If q > qo : In this case, the ant selects a random variable (S) to be the next customer to visit based on the probability distribution of pij (equation 3.2), which favors short paths with high levels of pheromone:
pij =
0 otherwise (3.2)
Once the capacity constraint is satisfied, the ant returns back to the depot. This selection process is continued until all customers have been visited.
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For further improvements, pheromone trails are updated by the equation 3.3.
τij = (1− α)× τij + α× τo (3.3)
α: a parameter that controls the speed of evaporation. τo : an initial pheromone value assigned to all arcs in network graph. In this work, τo is set equal to L−1 (L is the best known route distance found for the particular problem).
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Submit everything (from report to the entire program folder (code and files containing saved results (Changes in the best objective function value during the search.)). Your program should run. Test 10 combinations of Algorithmic parameters, and 20 runs under each parameter setting. You can show graphs of only the best run. your convergence graph should not go up and down ! (It is a graph between the number of iterations/function evaluations/time and the best objective function value). Provide average results of several runs also. For the second assignment:
Use tournament selection (much easier to code). Population size (20-100), Crossover probability (0.45-0.95) and Mutation probability (0.005-0.01).
Deadline 1st assignment: January 28, 12:00 Midnight Deadline 2nd assignment: February 2, 12:00 Midnight
Given time limit is more than standard for a group of 2 students
References
Bell, J. E. and McMullen, P. R. (2004). Ant colony optimization techniques for the vehicle routing problem. Advanced Engineering Informatics, 18:41–48.
Dorigo, M. (1992). Optimization, leraning and natural algorithms. PhD thesis, Politecnico di Milano.
Karaboga, D. (2005). An idea based on honey bee swarm for numerical optimization. Technical report, Erciyes University.
Kennedy, J. and Eberhart, R. (1995). Particle swarm optimization. In In Proceedings of IEEE International Conference on Neural Networks, pages 1942–1948, NJ, USA. IEEE Press.
Reynolds, C. W. (1987). Flocks, heards and schools. ACM Computer Graphics, 21(4):25–34.
Particle swarm optimization
Artificial bee colony

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