Metaheuristics: a quick overview

Transcription

1 Metaheuristics: a quick overview Marc Sevaux University of Valenciennes CNRS, UMR 8530, LAMIH / Production systems Marc.Sevaux@univ-valenciennes.fr Marc Sevaux TEW Antwerp

2 Outline Outline Neighborhood metaheurisitcs Application on a parallel machine scheduling problem Pm r j w j U j Conclusion on neighborhood search methods Population metaheuristics Application on a single machine scheduling problem 1 r j w j T j Conclusion on population metaheuristics Marc Sevaux TEW Antwerp

3 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... Notation a set of solutions, S a function to minimize, f(s) :s S IR an optimal solution, s S/f(s ) f(s) s S Marc Sevaux TEW Antwerp

4 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 the neighborhood. Find an initial solution Define a neighborhood Explore [exhaustively] the neighborhood Evaluate a neighbor. Notation a neighborhood of s, N(s) Marc Sevaux TEW Antwerp

5 Different approaches 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, Mladenović, 98] ILS : Iterated local search [Lorenço et al, 2000] Practical use in scheduling Deepest descent, Multi-start Simulated annealing, Tabu search Marc Sevaux TEW Antwerp

6 Simple descent Simple descent procedure Simple Descent(initial solution s) repeat Choose s N(s) if f(s ) <f(s) then s s end if until f(s ) f(s), s N(s) end Marc Sevaux TEW Antwerp

7 Simple descent Simple descent 01 Initial solution Objective value Final solution Solution space Marc Sevaux TEW Antwerp

8 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 if until f(s ) f(s), s N(s) end Marc Sevaux TEW Antwerp

9 Deepest descent Deepest descent Objective value Initial solution Final solution Solution space Marc Sevaux TEW Antwerp

10 Multi-start descent and deepest descent Multi-start descent and deepest descent procedure Multistart Descent iter =1 f(best) =+ repeat Choose a starting solution s 0 at random s Simple Descent(s 0 ) (ou Deepest Descent(s 0 )) if f(s) <f(best) then Best s end if iter iter +1 until iter = IterMax end Marc Sevaux TEW Antwerp

11 Multi-start descent and deepest descent Multi-start descent and deepest descent Initial solutions Objective value Best solution Final solutions Solution space Marc Sevaux TEW Antwerp

12 Tabu search Tabu search procedure Tabu Search(inital solution s) Best s repeat Choose 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 if until stopping conditions are met end Marc Sevaux TEW Antwerp

13 Tabu search Tabu search Initial solution Objective value Final solution Solution space Best solution Marc Sevaux TEW Antwerp

14 Simulated annealing Simulated annealing procedure Simulated Annealing(inital solution s) Best s T T init, choose α ]0, 1[, choose IterStage repeat for i =1to IterStage Choose s N(s) f(s ) f(s) if < 0or s s with the probability exp T end for T α T until stopping conditions are met end, Update Best Marc Sevaux TEW Antwerp

15 Simulated annealing Simulated annealing Final solution Objective value Initial solution Solution space Best solution Marc Sevaux TEW Antwerp

16 Threshold accepting Threshold accepting procedure Threshold(inital solution s) T T init, choose α ]0, 1[, choose IterStage Best s, moved true while moved do for i =1to IterStage Choose s N(s), f(s ) f(s) moved true if < 0 or <T then s s else moved false, end if Update Best end for T α T end while end Marc Sevaux TEW Antwerp

17 Threshold accepting Threshold accepting Final solution Objective value Initial solution Solution space Best solution Marc Sevaux TEW Antwerp

18 Variable neighborhood search Variable neighborhood search procedure VNS(inital solution s) Best s Choose a set of neighborhood structures N k k =1...k max repeat k 1 repeat Shaking: choose s N k (s) s Localsearch(s ) if f(s ) <f(s) then s s else k k +1end if Update Best until k = k max until stopping conditions are met end Marc Sevaux TEW Antwerp

19 Variable neighborhood search Variable neighborhood search Initial solution (s) Objective value N1 s N1 s N1 s s s s s N3 N1 N2 Solution space Marc Sevaux TEW Antwerp

20 Iterated local search Iterated local search procedure ILS(inital solution s 0 ) Best s 0 s Localsearch(s 0 ) repeat s P erturbation(s, history) s Localsearch(s ) s AcceptanceCriterion(s, s, history) Update Best until stopping conditions are met end Marc Sevaux TEW Antwerp

21 Iterated local search Iterated local search Initial solution s 0 Perturbation s Objective value s s Solution space Marc Sevaux TEW Antwerp

22 References References [Glover 89/90 ] Tabu Search Part I et II, ORSA Journal on Computing 1989:1(3), pp et 1990:2(1), pp4-32. [Kirkpatrick et al 83 ] Optimization by Simulated Annealing, Science 220, pp [Deuck et Scheuer 90 ] Threshold accepting: a general purpose optimization algorithm... Journal of Computational Physics 90, pp [Hansen et Mladenović 98] Variable Neighborhood Search: Principles and Applications, Rapport GERAD G [Lorenço et al 2000 ] Iterated Local Search, preprint. Marc Sevaux TEW Antwerp

23 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 Marc Sevaux TEW Antwerp

24 Population metaheuristics Population metaheuristics Basedonapopulationofsolutions How to combine different solutions? How to manage the population? How to evaluate a solution? Marc Sevaux TEW Antwerp

25 Different approaches Different approaches Genetic Algorithm, Holland 1975 Goldberg 1989 Memetic Algorithm, Moscatto 1989 Hybrid Genetic Algorithm Ant Colony Optimisation, Dorigo 1991 Scatter search, Laguna, Glover, Martì 2000 Application to scheduling Marc Sevaux TEW Antwerp

26 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 Marc Sevaux TEW Antwerp

27 Description of the algorithms Memetic algorithm Generate an initial population While stopping conditions are not met Select two individuals Apply crossover operator Improve offspring under probability Insert offspring under conditions Remove an individual under conditions End While Marc Sevaux TEW Antwerp

28 Description of the algorithms Hybrid genetic algorithm Generate an initial population While stopping conditions are not met Select two individuals Apply crossover operator Improve offspring under probability Mutate offspring under probability Insert offspring under conditions Remove an individual under conditions End While Marc Sevaux TEW Antwerp

29 Description of the algorithms Ant Colony Optimisation Nest Food Nest?????? Food?????? Obstacle Nest Food Nest Food Obstacle Obstacle Marc Sevaux TEW Antwerp

30 Description of the algorithms Ant Colony Optimisation Generate an initial colony While stopping conditions are not met 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 Marc Sevaux TEW Antwerp

31 Description of the algorithms Scatter search Generate an initial improved population Select a diverse subset (RefSet, sizeb) While stopping conditions are not met A RefSet While A 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 (outer loop) (inner loop) Marc Sevaux TEW Antwerp

32 Intensification vs diversification Intensification vs diversification GA Intensification selection, crossover, replacement MA selection, crossover, local search, replacement HGA selection, crossover, local search, replacement Diversification mutation mutation ACO pheromone information colony, ant memory reinitialization SS inner loop: crossover, local search diverse replacement Marc Sevaux TEW Antwerp

33 References References (web sites) Genetic Algorithms Holland 1975 Goldberg Ant Colony Optimisation Dorigo Memetic Algorithms Moscatto ~moscato/memetic_home.html Scatter search Laguna, Glover, Martì laguna/scattersearch.htm Marc Sevaux TEW Antwerp

34 References References Genetic Algorithms Holland 1975 Goldberg 1989 D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, Reading, Massachusetts, Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs, Springer Verlag, 1999 Ant Colony Optimisation Dorigo 1991 Dorigo M., V. Maniezzo and A. Colorni (1991). Positive Feedback as a Search Strategy. Technical Report No. 91-6, Politecnico di Milano, Italy Dorigo M. and G. Di Caro (1999). The Ant Colony Optimization Meta-Heuristic. In D. Corne, M. Dorigo and F. Glover, editors, New Ideas in Optimization, McGraw-Hill, Marc Sevaux TEW Antwerp

35 References References Memetic Algorithms Moscatto 1989 P. Moscatto, On Evolution, Search, Optimization, Genetic Algorithms and Martial Arts: Towards Memetic Algorithms, Caltech Concurrent Computation Program, Report. 790, Scatter search Laguna, Glover, Martì 2000 F. Glover, M. Laguna and R. Martì, Fundamentals of Scatter Search and Path Relinking Control and Cybernetics, vol. 39, no. 3 pp (2000) F. Glover, M. Laguna and R. Martì (forthcoming) Scatter Search Theory and Applications of Evolutionary Computation: Recent Trends, A. Ghosh and S. Tsutsui (Eds.), Springer-Verlag. Marc Sevaux TEW Antwerp

36 Application to a scheduling problem Application to a scheduling problem Scheduling Application Minimizing the total weighted tardiness on a single machine in the general case, 1 r j w j T j Aim of the study Point out the differences between an Hybrid Genetic Algorithm and a Scatter Search Implementation Both methods are compared on a same basis (common components, identical stoppping conditions, equivalent evaluations). Marc Sevaux TEW Antwerp

37 Application to a scheduling problem Previous work Complexity status 1 r j w j T j is NP-Hard in a strong sense 1 r j T j NP-Hard in a strong sense [Garey, Johnson 77] 1 w j T j NP-Hard in a strong sense [Lawler 77] 1 T j NP-Hard in an ordinary sense [Du, Leung 90] Recent approaches on 1 w j T j (OR-Library) [Crauwels, Potts, van Wassenhove 98] SA, TS, GA [Congram, Potts, van de Velde 02] Iterated dynasearch algorithm Other approaches on 1 r j w j T j [Akturk, Ozdemir 00] Exact method (up to 20 jobs) [Jouglet 02] Constraint-based method (up to 40 jobs) Marc Sevaux TEW Antwerp

38 Common components Common components Coding Permutation = natural representation for single machine scheduling problems, no clones allowed Fitness Build semi-active schedules according to the order of the permutation Crossover Standard LOX crossover Mutation GPI: General Pairwise Interchange Local search Based on the GPI, a best improvement method is applied Marc Sevaux TEW Antwerp

39 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 Marc Sevaux TEW Antwerp

40 What could happen What could happen Easy problems (SS) cannot generate a diverse population Size of the population (GA/SS) SS: Too large too slow GA: Too small not enough diversification Mutation rate (GA) Too low cannot escape from a local optima Too high cannot converge Local search rate (GA) Too low converges slowly Too high converges too quickly to a local optima Marc Sevaux TEW Antwerp

41 What could happen Influence of mutation rate Marc Sevaux TEW Antwerp

42 What could happen Influence of local search rate Marc Sevaux TEW Antwerp

43 What could happen Extreme rate values Marc Sevaux TEW Antwerp

44 Numerical results Numerical results Several sets of instances are used for numerical evaluation OR-LIB instances for 1 w j T j n {40, 50, 100}, 125 instances each Comparison to optima (when known) or best solutions ODD instances for 1 r j w j T j (initially for 1 r j w j U j ) n {20, 40, 60, 80, 100}, 20 instances each Comparison between GA and SS approaches Marc Sevaux TEW Antwerp

45 Numerical results Generic parameters Parameters will be arbitrarily fixed for the study, better solution can be 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 it. without improvement (for ORLIB 100 jobs: 120 sec.) Marc Sevaux TEW Antwerp

46 Numerical results OR LIB results Parameters need to be tuned more precisely Instances CPU GA it. SS GA = SS GA > SS Sol. ORLIB s ORLIB s ORLIB 100* 58.36s *CPU time limit is fixed to 120s GA is always limited by the limit of 10,000 it w/o improvement SS is always limited by the CPU time limit Marc Sevaux TEW Antwerp

47 Numerical results Another set of instances ODD A problem instance generator is used, it is based on a single day of 80 periods. 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 dates are 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,d i r i ] Weights uniformly distributed in [1, 10] Marc Sevaux TEW Antwerp

48 Numerical results Release date and due date repartition 6 Release dates Due dates 5 4 Frequency Time periods Marc Sevaux TEW Antwerp

49 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 s 0.0s 17 0 ODD s s 3.1s 17 3 ODD s s 19.1s 15 5 ODD s s 248.9s 9 11 ODD 11.16s s 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 improvement SS is always limited by the CPU time limit Marc Sevaux TEW Antwerp

50 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 Marc Sevaux TEW Antwerp