P–MetaheuristicsSwarm intelligence (Ch 3.6) and Conclusionsof P-heuristics (Ch 3.9)

Joen DahlbergCommunications and Transport SystemsDepartment of Science and TechnologyLinköping University

Outline

1 Swarm Intelligence– Short Introduction– Ant Colony– Particle Swarm

2 Conclusions of P-heuristics

Swarm Intelligence – Short Introduction Joen Dahlberg November 10, 2015 2 / 12

Introduction

Algorithms inspired from the collective behaviour of species such asants, bees, wasps, termite, fish, and birds are referred as swarmintelligence algorithms.

The main characteristics of swarm–intelligence–based algorithms areparticles are simple, they cooperate by an indirect communicationmedium, and do movements in the decision space.

Among the most successful swarm intelligence inspired optimizationalgorithms are ant colony optimization and particle swarmoptimization.

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 3 / 12

Ant Colony Optimization (ACO)

Traditionally, ACO have been applied to combinatorial optimizationproblems, e.g. scheduling, routing and assignment problem. It isinspired by ants scouting for food.

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 3 / 12

Ant Colony Optimization (ACO)

Traditionally, ACO have been applied to combinatorial optimizationproblems, e.g. scheduling, routing and assignment problem. It isinspired by ants scouting for food.

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 3 / 12

Ant Colony Optimization (ACO)

Traditionally, ACO have been applied to combinatorial optimizationproblems, e.g. scheduling, routing and assignment problem. It isinspired by ants scouting for food.

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 3 / 12

Ant Colony Optimization (ACO)

Traditionally, ACO have been applied to combinatorial optimizationproblems, e.g. scheduling, routing and assignment problem. It isinspired by ants scouting for food.

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 3 / 12

Ant Colony Optimization (ACO)

Traditionally, ACO have been applied to combinatorial optimizationproblems, e.g. scheduling, routing and assignment problem. It isinspired by ants scouting for food.

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 3 / 12

Ant Colony Optimization (ACO)

Traditionally, ACO have been applied to combinatorial optimizationproblems, e.g. scheduling, routing and assignment problem. It isinspired by ants scouting for food.

Movie: Ant leaving trail

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 4 / 12

Inspired parameters

The algorithm uses ants and pheromone trails. The pheromone trailleads the ants to good solutions. The pheromone trails decreases overtime (evaporation), but the ants adds new pheromones based on theirfindings (reinforcements).

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 5 / 12

Translation to heuristics

AntsThey are stochastic greedy procedures that construct solutions in aprobabilistic manner by adding solution components until a completesolution is derived. The problem is usually a decision problem whereants walk on edges between nodes.

Pheromone trailsEach edge in the network is given a probabilistic value, τij , that anant will choose that particular edge. The collection of τ–values is thepheromone trail(s).

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 5 / 12

Translation to heuristics

EvaporationIn the evaporation phase, the probabilistic values are reducedproportionally to the current value, τij := (1− ρ)τij , ∀(i, j) ∈ E,where ρ ∈]0, 1]. Intensification and diversification of solutions can beadjust based on the choice of ρ.

ReinforcementThere are three different strategies for increasing the probabilisticvalues. Either at each step of the ant’s walk, when a completesolution is found, or when all ants found complete solutions.

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 6 / 12

Standard algorithm

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 7 / 12

Example 3.18 ACO for the TSP

Swarm Intelligence – Ant Colony Joen Dahlberg November 10, 2015 7 / 12

Example 3.18 ACO for the TSP

τij is the standard pheromone.ηij is a problem-dependent pheromone, in this case distance−1 ( 1

dij).

α and β are parameters representing the relative influence of the twopheromones.∆ = 1

f(π) , where π is the obtained tour∗ of the ant.

∗ in the algorithm π is the best found tour, why? wrong?

Swarm Intelligence – Particle Swarm Joen Dahlberg November 10, 2015 8 / 12

Particle Swarm Optimization (PSO)

Originally, PSO has been successfully designed for continuousoptimization problems. It mimic the social behaviour of e.g. birdflocking and fish schooling.

Movie: Flock of Starlings

Swarm Intelligence – Particle Swarm Joen Dahlberg November 10, 2015 9 / 12

Inspired parameters

Each particle represents a (complete) solution, it’s current position,xi(t), in the decision space. The particle has a velocity, vi(t), whichdefines the moving direction and length of the move at every step ofthe particle. The particle have a defined neighbourhood, it’s socialinfluence. The velocity is changed based on the personal bestsolution, pbesti (pi), and the locally (amongst the particles in theneighbourhood) best found solution, lbest (pl). The particle may alsohave an x–fitness and/or p–fitness value(s).

The neighbourhood can range from all particles, a global influence,the gbest Method, to a subset of particles, down to zero distance, noinfluence at all.

Swarm Intelligence – Particle Swarm Joen Dahlberg November 10, 2015 9 / 12

Inspired parameters

Swarm Intelligence – Particle Swarm Joen Dahlberg November 10, 2015 10 / 12

Algorithm

Swarm Intelligence – Particle Swarm Joen Dahlberg November 10, 2015 10 / 12

Algorithm

vi(t) = vi(t− 1) + ρ1 · (pi − xi(t− 1)) + ρ2 · (pg/l − xi(t− 1))Adjustment towards individual best solution scaled with a random variableρ1 ∈ [0, 1].Adjustment towards global/local best solution scaled with a random variableρ2 ∈ [0, 1].

Swarm Intelligence – Particle Swarm Joen Dahlberg November 10, 2015 10 / 12

Algorithm

vi(t) = vi(t− 1) + ρ1 · (pi − xi(t− 1)) + ρ2 · (pg/l − xi(t− 1))Adjustment towards individual best solution scaled with a random variableρ1 ∈ [0, 1].Adjustment towards global/local best solution scaled with a random variableρ2 ∈ [0, 1].

Conclusions of P-heuristics Joen Dahlberg November 10, 2015 11 / 12

An overlook

Conclusions of P-heuristics Joen Dahlberg November 10, 2015 12 / 12

Final remark

P–metaheuristics and especially nature-inspired algorithms (e.g.,evolutionary algorithms, immune systems, particle swarm, bee colony,and ant colony) are well suited to solve complex optimizationproblems dealing with uncertainty and dynamicity. Indeed, in solvingthis type of problems, one must focus on a more efficientdiversification of the search to react promptly to changes of thelandscape or generate more robust solutions.

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