Amsterdam, The Netherlands

July 06-10, 2013

Ant Colony Optimization

and Swarm Intelligence

Migration Study on a

Pareto-based Island

Model for MOACOs

Antonio M. Mora, P. García-Sánchez,

J.J. Merelo, P.A. Castillo

Depto. de Arquitectura y Tecnología de Computadores

UNIVERSIDAD DE GRANADA

• Ant Colony Optimization

• Multi-objective Optimization

• Island Model

• Pareto-based Island Model – Model Factors

– Topologies

• MOACS Algorithm

• Experiments and Results – Attainment Surfaces

– Metrics and Indicators

– Statistics

• Conclusions and Future Work

• Inspired in the behaviour of natural ants when searching for food.

• They cooperate to get the fastest paths between the nest and the source of food.

• They use a chemical substance named pheromone.

In ACO algorithms

• There are a set of agents called (artificial) ants. – All of them move in a graph following and depositing (artificial) pheromone.

– They cooperate to find a solution (usually every ant yields a complete solution).

• There are some formulae applied in the run: – state transition rule decides the next step for each ant

– pheromone updating contribution and evaporation

– evaluation function assigns the cost to every solution

Ant Colony Optimization

• Optimize several (independent) objectives at a time.

• There is a set of optimal solutions. Those which are the best considering all the objectives than the rest are named non-dominated solutions.

• The ideal set of non-dominated solutions is called the Pareto Set (PS).

• Its graphical representation is the Pareto Front (PF).

Multi-Objective Optimization

Example of PF for two objective functions.

• If you search in Google (images):

Island Model

• Classical distribution model in EAs: – Divide or replicate the population in

different subpopulations (islands)

– They evolve independently

– After some generations, some individuals are chosen in every island

– They migrate to a neighbour island, following a specific neighbourhood topology

– They replace other individuals in the receiver islands.

• It Improves the algorithm performance, not just the running time

Island Model

Island model. Ring neighbourhood topology.

Image by Pablo García-Sánchez

• Just a few works in distributing MOACOs.

• No island model in this type of algorithms, just island model for ACOs.

• Not take advantage of the multi-colony division both: – From the algorithmic point of view: very important in Multi-objective

optimization.

– From the computational cost point of view: due to parallelization.

• Previous works by us in this line comparing multi-colony, sub-colony and island models for different MOACOs.

• Island model with a fixed topology and migration rate.

MOACOs

• The colony is divided into several subcolonies or islands.

• Every island searches in a different area of the space of solutions (PF).

• A l parameter is used for splitting the space in this way.

Model Factors

• Migration policy the best ants (solutions in fact) are migrated. In this case, the best regarding the prioritary objective in every island.

• Migrants influence every migrant contributes to the pheromone matrix of the receiver island, in order to guide the search to its own area.

• Replacement policy the migrant is included in the island’s PS, removing those dominated solutions (by itself) in that set.

• Migration rate some different values (number of iterations) are tested.

Pareto-based Island Model

• Migrants move to the closest neighbour island in the direction of the prioritary objective in the colony (sense of the PF).

• Tries to cover the gaps between colonies.

Pareto-based Island Model

Pareto-based Island Model

• Migrants move to the two closest neighbour islands in both directions.

• Aims for a better spread of solutions in the inter-colony gaps.

Pareto-based Island Model

• Every migrant moves from one island to the rest.

• Aims for a better spread of solutions along the whole PF.

>>>>>>>>>>>>>>>>>>>

This is a simplified example showing just the migrations from one island to the rest.

>>>>>>>>>>>>>>>>>>>

• Proposed by Barán et al. in 2003 for solving the VRPTW.

• Proved to be competitive in solving the bicriteria TSP (our test problem) in previous works.

• It considers one colony, one pheromone matrix and two heuristic functions (one per objective).

• The state transition rule considers l parameter for weighting the importance of the heuristic and memoristic (pheromone) information.

• Initially l took a different value per ant in the colony, in order to explore the whole space of solutions (PF).

• It applies a pheromone reinitialization mechanism, in order to avoid being stagnated.

MOACS

• 16 islands, everyone in a different processor of a cluster.

• Bicriteria TSP problem: kroAB100, kroAB150 and kroAB200.

• Implemented with MPI (Message Passing Interface).

• 20 runs per experiment.

• q0 and r take values for promoting the exploration more than usual.

• Global PS of all the executions for computing the metrics.

Experiments and Results

• Unidirectional

Experiments and Results

• Bidirectional

Experiments and Results

• Broadcast

Experiments and Results

• Best of every

topology

Experiments and Results

• Hypervolume: it calculates the volume, in the objective space, covered by a set of non-

dominated solutions (PS). A higher value means a better result.

Experiments and Results

• Spread: it measures the extent of spread of a PS. It considers the Euclidean distance between

consecutive solutions on average and extreme distances. A value 0 means an ideal spread.

Experiments and Results

• Epsilon: it is a measure of the smallest distance it would be necessary to translate every

solution in a PS so that it dominates the optimal PF of the problem. Smaller values are better.

Experiments and Results

• Cardinality: number of non-dominated solutions in the obtained PS.

Experiments and Results

• Time: the worst time among all the processors in one execution has been chosen as the

running time of that execution.

Experiments and Results

• A novel island model for MOACOs has been tested with different topologies and migration rates.

• Bidirectional approach yields a very good balance between quality and spread in the set of non-dominated solutions (PS).

• It has a flaw concerning the computational time. If it is relevant for the user, then unidirectional approach would be a better option.

• Regarding the migration rate, high values perform better.

Future Work

• Study some other parameters in the island model.

• Implement and test in other MOACO approaches.

• Compare with state-of-the-art distributed algorithms (EAs, island models)

Conclusions

Questions?

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