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Metaheuristics Network, First Milestone Meeting, 15 -16/11/2001, IRIDIA, Brussels, Belgium

Metaheuristics Network, First Milestone Meeting, 15 -16/11/2001, IRIDIA, Brussels, Belgium. ACO and ILS for the Quadratic Assignment Problem Christian Blum IRIDIA, Brussels, Belgium. Metaheuristics Network, First Milestone Meeting, 15 -16/11/2001, IRIDIA, Brussels, Belgium.

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Metaheuristics Network, First Milestone Meeting, 15 -16/11/2001, IRIDIA, Brussels, Belgium

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  1. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO and ILS for the Quadratic Assignment Problem Christian Blum IRIDIA, Brussels, Belgium

  2. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium The Qadratic Assignment problem • Given: • n facilities, n locations • A nxn matrix D keeping the distances between locations • A nxn matrix F keeping the flows between facilities • Goal: Find a permutation  on the n facilities s.t. is minimal

  3. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Pheromone representation Pheromone matrix: facilities 11 1n locations i1 in n1 nn for every location-facility pair (i,j) there is a pheromone value ij

  4. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Ant construction phase • As long as there are free locations: • choose a free location i at random • assign a free facility j by using the following transition rule:

  5. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: The hyper-cube framework • Online delayed pheromone update rule for Ant System: • where Disadvantage: upper limits of pheromone values dependent on optimal function value

  6. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: The hyper-cube framework Normalization of the online delayed pheromone update rule:

  7. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: The hyper-cube framework Solutions can be seen as binary vectors of size n2 0 0 1 3 1 2 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 Also the pheromone matrix can be written in vector form in the same way: 11 12 13 21 22 23 11 12 13 21 22 23 31 32 33 31 32 33

  8. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: The hyper-cube framework  The pheromone matrix as a vector is moving in the n2-dimensional hyper-cube Advantages: Pheromone values are limited from above by 1 Scaling of objective function values

  9. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Pheromone update MMAS in the hyper-cube framework:max=0.99 min=0.01 where where Setting dependent on branching factor

  10. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Restart mechanism • After convergence: Restart by resetting the pheromone matrix • Usually: = • Diversification (by global frequency): • gf(i,j) = # of solutions containing assignment (i,j) • if gf(i,j) high we set ij < c • if gf(i,j) low we set ij > c c c c c

  11. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ACO for the QAP: Restart mechanism • Intensification (by global desirability): • inverse of the best value of a solution containing assignment (i,j) • if gd(i,j) high we set ij > c • if gd(i,j) low we set ij < c • Daemon actions: Local Search and short runs of Tabu Search dependent on the distance dominance value • (both LS and TS provided by the Darmstadt node)

  12. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP

  13. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Perturbation mechanism • Perturbation mechanism: • Choose a number of k assignments from the current solution • Randomly reassign the k facilities to the k locations avoiding the assignments in the current solution • Parameter:size (strength) of the perturbation k • 5 <= k <= n/2

  14. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Evolution of perturbation size • if new local minimum accepted: • k = k – 3 • if new local minimum improves best solution so far: • k = 5 (minimum for k) • if new local minimum not accepted: • k = k + 1 • if beginning of a (re)start: • k = n/2 (maximum for k)

  15. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Acceptance criterion  : current solution ‘ : perturbed solution after local search

  16. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Evolution of parameter T • if the last 3 perturbed solutions in a row got accepted: • T = T * 0.9 • if 5 solutions in a row didn‘t get accepted: • T = T * 1.1

  17. Metaheuristics Network, First Milestone Meeting, 15-16/11/2001, IRIDIA, Brussels, Belgium ILS for the QAP: Restarts and usage of local search Restarts if no improvements after a number of iterations Local Search and short runs of Tabu Search dependent on the distance dominance value of the problem instance (both LS and TS provided by the Darmstadt node)

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