metaheuristics: a quick overview

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�Metaheuristics: a quick overview

Marc Sevaux

University of Valenciennes

CNRS, UMR 8530, LAMIH / Production systems

[email protected]

Marc Sevaux TEW – Antwerp 2003 1

Outline

Outline

• Neighborhood metaheurisitcs

• Application on a parallel machine scheduling problemPm|rj |

∑wjUj

• Conclusion on neighborhood search methods

• Population metaheuristics

• Application on a single machine scheduling problem1|rj |

∑wjTj

• Conclusion on population metaheuristics


Motivations for neighborhood metaheuristics

Motivations for neighborhood metaheuristics

• NP-hard problems

• Obtain rapidly solutions

• Short time of implementation

• No/few knowledge of the problem

• ... it’s easier...

Notationa set of solutions, S

a function to minimize, f(s) : s ∈ S → IR

an optimal solution, s� ∈ S / f(s�) ≤ f(s) ∀s ∈ S


Neighborhood and initial solution

Neighborhood and initial solution

First, an initial solution is needed. Then, to explore a solution space,we need to know how to move from one solution to another. It’s theneighborhood.

• Find an initial solution

• Define a neighborhood

• Explore [exhaustively] the neighborhood

• Evaluate a neighbor.

Notationa neighborhood of s, N(s)


Different approaches


Descent methods

• Simple descent, deepest descent

• Multi-start

Metaheuristic methods

• TS : Tabu search [Glover, 89 et 90]

• SA : Simulated annealing [Kirckpatrick, 83]

• TA : Threshold accepting [Deuck, Scheuer, 90]

• VNS : Variable neighborhood [Hansen, Mladenovic, 98]

• ILS : Iterated local search [Lorenco et al, 2000]

Practical use in scheduling

• Deepest descent, Multi-start

• Simulated annealing, Tabu search


Simple descent

Simple descent

procedure Simple Descent(initial solution s)repeat

Choose s′ ∈ N(s)if f(s′) < f(s) then

s ← s′

end ifuntil f(s′) ≥ f(s), ∀s′ ∈ N(s)

end


Simple descent

Simple descent

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Initial solution

Final solution

Solution space

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Deepest descent

Deepest descent

procedure Deepest Descent(initial solution s)repeat

Choose s′ ∈ N(s) / f(s′) ≤ f(s′′) ∀s′′ ∈ N(s)if f(s′) < f(s) then

s ← s′

end ifuntil f(s′) ≥ f(s), ∀s′ ∈ N(s)

end


Deepest descent

Deepest descent

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Initial solution

Final solution

Solution space

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Multi-start descent and deepest descent


procedure Multistart Descentiter = 1f(Best) = +∞repeat

Choose a starting solution s0 at randoms ← Simple Descent(s0) (ou Deepest Descent(s0))if f(s) < f(Best) then

Best ← s

end ifiter ← iter + 1

until iter = IterMax

end




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Initial solutions

Final solutions

Solution space

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Best solution


Tabu search

Tabu search

procedure Tabu Search(inital solution s)Best ← s

repeatChoose s′ ∈ N(s) / f(s′) ≤ f(s′′) ∀s′′ ∈ N(s)if s′ is not tabu or f(s′) < f(Best) then

s ← s′

Update tabu list, update Best

end ifuntil stopping conditions are met

end


Tabu search

Tabu search

Solution space

Obj

ectiv

e va

lue

Final solution

Best solution

Initial solution


Simulated annealing

Simulated annealing

procedure Simulated Annealing(inital solution s)Best ← s

T ← Tinit, choose α ∈]0, 1[, choose IterStage

repeatfor i = 1 to IterStage

Choose s′ ∈ N(s)∆ ← f(s′) − f(s)

s ← s′

if ∆ < 0 or

with the probability exp −∆T

, Update Best

end forT ← α × T

until stopping conditions are metend


Simulated annealing

Simulated annealing

Solution space

Obj

ectiv

e va

lue

Best solution

Initial solution

Final solution


Threshold accepting

Threshold accepting

procedure Threshold(inital solution s)T ← Tinit, choose α ∈]0, 1[, choose IterStage

Best ← s, moved ← truewhile moved do

for i = 1 to IterStage

Choose s′ ∈ N(s), ∆ ← f(s′) − f(s)moved ← trueif ∆ < 0 or ∆ < T then s ← s′

else moved ← false, end ifUpdate Best

end forT ← α × T

end whileend


Threshold accepting

Threshold accepting

Solution space

Obj

ectiv

e va

lue

Best solution

Initial solution

Final solution


Variable neighborhood search


procedure VNS(inital solution s)Best ← s

Choose a set of neighborhood structures Nk k = 1 . . . kmax

repeatk ← 1repeat

Shaking: choose s′ ∈ Nk(s)s′′ ← Localsearch(s′)if f(s′′) < f(s) then s ← s′′

else k ← k + 1 end ifUpdate Best

until k = kmax





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Initial solution (s)

s’s’’

N1s’’

N1s’’

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Solution space

N2N3

s’’

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Iterated local search


procedure ILS(inital solution s0)Best ← s0

s ← Localsearch(s0)repeat

s′ ← Perturbation(s, history)s′′ ← Localsearch(s′)s ← AcceptanceCriterion(s, s′′, history)Update Best




Iterated local searchInitial solution s

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s’

s’’

Perturbation

Solution space

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lue


References

References

[Glover 89/90 ] Tabu Search Part I et II, ORSA Journal onComputing 1989:1(3), pp190-206 et 1990:2(1), pp4-32.

[Kirkpatrick et al 83 ] Optimization by Simulated Annealing,Science 220, pp671-680

[Deuck et Scheuer 90 ] Threshold accepting: a general purposeoptimization algorithm... Journal of Computational Physics 90,pp161-175.

[Hansen et Mladenovic 98 ] Variable Neighborhood Search:Principles and Applications, Rapport GERAD G-28-20.

[Lorenco et al 2000 ] Iterated Local Search, preprint.http://www.intellektik.informatik.tu-darmstadt.de/tom/


Conclusion on neighborhood search methods

Conclusion on neighborhood search methods

These metatheuristics

• are easy to understand and control

• give solutions rapidly

• need a short time for implementation

• and few knowledge of the problem

But to be really efficient

• tricks have to be added

• we need to know the problem very well

• try many of them

• and why not combine them


Population metaheuristics

Population metaheuristics

• Based on a population of solutions

• How to combine different solutions?

• How to manage the population?

• How to evaluate a solution?




• Genetic Algorithm, Holland 1975 – Goldberg 1989

• Memetic Algorithm, Moscatto 1989

• Hybrid Genetic Algorithm

• Ant Colony Optimisation, Dorigo 1991

• Scatter search, Laguna, Glover, Martı 2000

• Application to scheduling


Description of the algorithms

Genetic algorithm

Generate an initial population

While stopping conditions are not met

Select two individuals

Apply crossover operator

Mutate offspring under probability

Insert offspring under conditions

Remove an individual under conditions

End While



Memetic algorithm





Improve offspring under probability



End While



Hybrid genetic algorithm





Improve offspring under probability

Mutate offspring under probability



End While



Ant Colony Optimisation

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Ant Colony Optimisation

Generate an initial colony


For each ant of the colony

initialize ant memory

While Curent State �= Target state

Apply ant decision under pheromone information

Move to next state

Deposit pheromone information on the transition

End While

End For

Evaporate pheromone information

End While



Scatter search

Generate an initial improved population

Select a diverse subset (RefSet, size b)

While stopping conditions are not met (outer loop)

A ← RefSet

While A �= ∅ (inner loop)

Combine solutions (B ← RefSet × A)

Improve solutions

Update RefSet (keep b best from RefSet ∪ B)

A ← B − RefSet

End While

Remove the worst b/2 solutions from RefSet

Add b/2 diverse solutions to RefSet

End While


Intensification vs diversification

Intensification vs diversification

Intensification Diversification

GA selection, crossover, replace-ment

mutation

MA selection, crossover, localsearch, replacement

HGA selection, crossover, localsearch, replacement

mutation

ACO pheromone information colony, ant memory re-initialization

SS inner loop: crossover, localsearch

diverse replacement


References

References (web sites)

Genetic Algorithms Holland 1975 – Goldberg 1989http://www-illigal.ge.uiuc.edu/index.php3

Ant Colony Optimisation Dorigo 1991http://iridia.ulb.ac.be/~mdorigo/ACO/ACO.html

Memetic Algorithms Moscatto 1989http://www.densis.fee.unicamp.br/

~moscato/memetic_home.html

Scatter search Laguna, Glover, Martı 2000http://www-bus.colorado.edu/faculty/

laguna/scattersearch.htm


References

References

Genetic Algorithms Holland 1975 – Goldberg 1989D. E. Goldberg, Genetic Algorithms in Search, Optimization,and Machine Learning, Addison-Wesley, Reading, Massachusetts,1989.Z. Michalewicz, Genetic Algorithms + Data Structures =Evolution Programs, Springer Verlag, 1999

Ant Colony Optimisation Dorigo 1991Dorigo M., V. Maniezzo and A. Colorni (1991). PositiveFeedback as a Search Strategy. Technical Report No. 91-016,Politecnico di Milano, ItalyDorigo M. and G. Di Caro (1999). The Ant Colony OptimizationMeta-Heuristic. In D. Corne, M. Dorigo and F. Glover, editors,New Ideas in Optimization, McGraw-Hill, 11-32.


References

References

Memetic Algorithms Moscatto 1989P. Moscatto, On Evolution, Search, Optimization, GeneticAlgorithms and Martial Arts: Towards Memetic Algorithms,Caltech Concurrent Computation Program, Report. 790, 1989.

Scatter search Laguna, Glover, Martı 2000F. Glover, M. Laguna and R. Martı, Fundamentals of ScatterSearch and Path Relinking Control and Cybernetics, vol. 39, no.3 pp. 653-684 (2000)F. Glover, M . Laguna and R. Martı (forthcoming) ScatterSearch Theory and Applications of Evolutionary Computation:Recent Trends, A. Ghosh and S. Tsutsui (Eds.), Springer-Verlag.


Application to a scheduling problem


Scheduling ApplicationMinimizing the total weighted tardiness on a single machine inthe general case, 1|rj |

∑wjTj

Aim of the studyPoint out the differences between an Hybrid Genetic Algorithmand a Scatter Search

ImplementationBoth methods are compared on a same basis (commoncomponents, identical stoppping conditions, equivalentevaluations).



Previous work

Complexity status 1|rj |∑

wjTj is NP-Hard in a strong sense

1|rj |∑

Tj NP-Hard in a strong sense [Garey, Johnson 77]

1| |∑wjTj NP-Hard in a strong sense [Lawler 77]

1| |∑Tj NP-Hard in an ordinary sense [Du, Leung 90]

Recent approaches on 1| |∑wjTj (OR-Library)[Crauwels, Potts, van Wassenhove 98] SA, TS, GA[Congram, Potts, van de Velde 02] Iterated dynasearch algorithm

Other approaches on 1|rj |∑

wjTj

[Akturk, Ozdemir 00] Exact method (up to 20 jobs)[Jouglet 02] Constraint-based method (up to 40 jobs)


Common components

Common components

CodingPermutation = natural representation for single machinescheduling problems, no clones allowed

FitnessBuild semi-active schedules according to the order of thepermutation

CrossoverStandard LOX crossover

MutationGPI: General Pairwise Interchange

Local searchBased on the GPI, a best improvement method is applied


Common components

Parameters (choose or tune)

Chosen according to the model

• crossover operator

• mutation operator

• local search method

• distance (for diverse replacement method)

Tuned according to the problem

• mutation rate

• local search rate

• population/RefSet size

• stopping conditions


What could happen

What could happen

• Easy problems (SS)cannot generate a diverse population

• Size of the population (GA/SS)SS: Too large → too slowGA: Too small → not enough diversification

• Mutation rate (GA)Too low → cannot escape from a local optimaToo high → cannot converge

• Local search rate (GA)Too low → converges slowlyToo high → converges too quickly to a local optima


What could happen

Influence of mutation rate


What could happen

Influence of local search rate


What could happen

Extreme rate values


Numerical results

Numerical results

Several sets of instances are used for numerical evaluation

OR-LIB instances for 1| |∑wjTj

n ∈ {40, 50, 100}, 125 instances eachComparison to optima (when known) or best solutions

ODD instances for 1|rj |∑

wjTj (initially for 1|rj |∑

wjUj)n ∈ {20, 40, 60, 80, 100}, 20 instances eachComparison between GA and SS approaches


Numerical results

Generic parameters

Parameters will be arbitrarily fixed for the study, better solution canbe found by tunning the parameters correctly.

Population size (SS) NPop = 10

RefSet size b = 5

Population size (GA) NPop = n/2

Mutation rate mr = 0.1

Local search rate lsr = 0.1

Stopping conditions 1200 sec. or 10000 it. without improvement(for ORLIB 100 jobs: 120 sec.)


Numerical results

OR LIB results

Parameters need to be tuned more precisely

Instances CPU GA it. SS GA = SS GA > SS Sol.

ORLIB 40 3.89s 4836 119 6 125

ORLIB 50 8.16s 2349 0 125 124

ORLIB 100* 58.36s 2 42 62 54

*CPU time limit is fixed to 120s

GA is always limited by the limit of 10,000 it w/o improvementSS is always limited by the CPU time limit


Numerical results

Another set of instances ODD

A problem instance generator is used, it is based on a single day of 80periods. Twenty instances are generated for each value of n

(n ∈ {20, 40, 60, 80, 100}).Orders have a high probability of arriving in the morning. Due datesare in the afternoon with high probability.

Release dates distributed according to a Gamma law Γ(0.2, 4)(Mean = 20, St. dev. = 10)

Due dates distributed according to the same Gamma law (Mean =20, St. dev. = 10) but starting at the end of the horizon

Processing times uniformly distributed in [1, di − ri]

Weights uniformly distributed in [1, 10]


Numerical results

Release date and due date repartition

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70 80

Fre

quen

cy

Time periods

Release datesDue dates


Numerical results

ODD results

Stopping conditions:- CPU time limit = 1200s and 10,000 it. w/o improvement

Inst. CPU GA it. SS GA best SS best GA = SS GA > SS

ODD 20* 13.34s 8610 0.4s 0.0s 17 0

ODD 40 5.60s 1253 0.6s 3.1s 17 3

ODD 60 21.21s 291 4.6s 19.1s 15 5

ODD 80 62.68s 99 9.2s 248.9s 9 11

ODD 100 111.16s 43 33.6s 289.0s 9 10

*3 instances cannot be solved by SS, cannot generate 10 diverse solutions

GA is always limited by the limit of 10,000 it w/o improvementSS is always limited by the CPU time limit


Conclusion

Conclusion

GA and SS are both very powerful with similar kind of efforts

Hybrid Genetic Algorithm

• “easy” to implement

• lot of parameters to be tuned

• different class of instances → different parameters

• need automatic parameter configuration procedures

Scatter Search Algorithm

• “somewhat harder” to implement

• few parameters to be tuned

• need to improve the local search procedure


metaheuristics: a quick overview

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