Iterated Local Search (ILS) for the Quadratic Assignment Problem (QAP)

Tim Daniëlse en Vincent Delvigne

QAP

• facilities (items)• locations• distancematrix A (between locations)• flowmatrix B (between facilities)• Assign facilities to locations, minimizing

transportation needed.

QAP (2)

• is the distance between locations and • is the flow between facilties and • is the set of all permutations of • gives the location of item in solution

Complexity

• NP-hard• Largest non-trivial instance solved to

optimality: • Thus use heuristics:– Simulated annealing– Tabu search– Memetic algorithms– Ant algorithms– Scatter search

Iterated Local Search

Procedure Iterated Local SearchGenerateInitialSolution LocalSearch()

repeat Perturbation() LocalSearch(’) AcceptanceCriterion()

Until termination condition metEndProcedure

Iterated Local Search (2)

• Generate Initial Solution– No known, well performing construction algorithm– Randomized assignment

Iterated Local Search (3)

• Local Search– 2-opt– First-improvement pivot

– Don’t look bits

Iterated Local Search (4)

• Perturbation– random k-opt– best value of k not known a priori– Adaptive, between and

Iterated Local Search (5)

• Acceptance Criterion– standard criterion: accept only improvements– varies among the algorithm-variants

QAP instance classes

• QAPLIB benchmark library• 4 instance classes:– randomly generated (class i)– Manhattan distance matrix (class ii)– real-life instances (class iii)– random, resembling real-life (class iv)

QAP instance classes (2)

• Functions to differentiate amongst classes:- Flow dominance

- Distance dominance

- Sparsity

where is number of “0” entries in A or B

Analysis of search space

• Fitness-Distance correlation analysis

• 5000 LS runs• 1000 ILS runs with iterations.

Analysis of search space (2)

Analysis of search space (3)

Distance-Fitness Correlation

Distance-Fitness Correlation (2)

Stagnation Detection• Empirical run-time

distribution (RTD)• RTD develops below

exponential distribution (stagnation)– perform restart

Algorithm Variations

• Soft Restarts– Use of history– Random new solution.

• Random Walk (RW)– Accept answer regardless of improvement– Combination with default “Better” might improve

even more

Algorithm Variations (2)

• Large Step Markov Chains (LSMC)– Accepts worse solution with certain probability.– Similar to simulated annealing – Uses temperature parameter

Algorithm Variations (3)

• Population Based Extensions• Replace-Worst– Start with solutions each with standard ILS– Every iterations a copy of the best replaces the

worst

Algorithm Variations (4)

• Evolution strategies (ES)– Population of solutions– Each iteration new solutions are generated– Uses distance to determine membership of

population– Parameter

Algorithm Comparison

• Robust Tabu search (RoTS)– Best for class and

• Max-Min Ant System (MMAS)– Best for class and

Algorithm Comparison (2)

Algorithm Comparison (3)

Evolutionary Variant

• Variant of the Evolutionary Strategy.

• Optimized Local Search• Different parameter settings

Evolutionary Variant (3)

Conclusions

• Fitness-Distance Correlation analysis• ILS runtime analysis• Acceptance Criteria analysis• ES-MN best performing algorithm.

References

• Stutzle, Thomas (2006): Iterated Local Search for the Quadratic Assignment Problem. European Journal of Operational Research 174 (3), 1519-1539.