A Course on Meta-Heuristic Search Methods for Combinatorial
Optimization ProblemsAnt colony algorithm Assignments
submission
References
Optimization Problems
AutOrI LAB, DIA, Roma Tre
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Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Outline
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
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.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
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.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Particle swarm optimization working process
vi : velocity of particle i ρ1 & ρ2: Acceleration coefficients
pbesti : best position of particle i gbest: Global best
position
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
update velocity:
vi (t) = vi (t−1)+ρ1×(pbesti−xi (t−1))+ρ2×(gbest−xi (t−1))
(1.1)
Update position:
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Types of bees:
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
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.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
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)
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
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
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
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.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Ant colony algorithm
Ants are moving in a straight line connecting the food source and
the nest.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Ant colony algorithm
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.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Ant colony algorithm
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.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
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.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Ant colony algorithm working process: an example on VRP
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.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
Ant colony algorithm working process: an example on VRP
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).
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
References
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
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization Artificial bee colony
Ant colony algorithm Assignments submission
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.
Santosh Kumar Mandal, Ph.D research fellow Meta-Heuristics
Particle swarm optimization
Artificial bee colony
Ant colony algorithm