Matthieu Basseur Julien Lemesre Clarisse Dhaenens El-Ghazali Talbi

1. Cooperation between branch and bound and evolutionary approaches to solve a BiObjective Flow-shop Problem Matthieu Basseur Julien Lemesre Clarisse Dhaenens El-Ghazali Talbi Laboratoire d’Informatique Fondamentale de Lille Université des Sciences et Technologies de Lille {basseur, lemesre, dhaenens, talbi}@lifl.fr

2. Outlines • BiObjective Flow-shop Scheduling Problem (BOFSP) • Optimization methods (OMs) - Exact Method/ Metaheuristics for BOFSP resolution • Cooperative methods • Experimentations • Conclusions and perspectives

3. BiObjective Flow-shop Problem Optimisation

4. M1 M2 M3 Cmax Flow-shop Scheduling Problem • N jobs to schedule on M machines. • Permutation Flow Shop. • Objectives to minimize : • Cmax: Total completion time (Makespan). • T: Total tardiness. • Type of Scheduling Problem: F/perm, di/(Cmax,T).

5. Multiobjective optimization (I) Min F(x) = (f1(x), f2(x), …, fn(x)) n  2 (PMO) st x  C Decision variables x = (x1, x2, …, xk) Objective space: Y = F(C) Decision space F C

6. Multiobjective optimization (II) Non-dominated solution (eligible, efficient, non inferior, Pareto optimal) • Dominance • y dominates z if and only if i[1, …, n], yi  zi and i[1, …, n], yi < zi • Non-dominated solution • A solution x is non dominated if a solution which dominates x does not exist Dominated feasible solution z y Pareto set • Goal: Find a good quality and well diversified set of Pareto solutions Example with (f1,f2) minimization

7. Optimization methods (OMs) - Exact Method/ Metaheuristics for BOFSP resolution

8. Exact approach • Two-phases method (TPM) [Ulungu & Teghem 95] • Phase 1: Search of the supported solutions - criteria aggregation. • Phase 2: MultiObjective approach – Search space restriction • Application to BOFSP [Lemesre et. al 03] • Branch and bound approach. (bounds [Lageweg et. al][Lemesre et. al 03] ) • Improvement of the initial method • Parallel version Feasible solution Convex hull Supported Pareto solutions Non-supported Pareto solution C1 C2

9. Adaptive Genetic Algorithm (I) • AGA [Basseur et. al 02] • Selection of the most effective genetic operators • Adaptive mutation selection • Adaptive combined diversification Create initial population Elitist selection into the population Set new PMi Start Crossover Mutation n Computation of PO* and the population … Mutation selection Mutation 1 End of GA

10. Rk=3 Rk=1 with AGA (II): Adaptive mutation Mutation operators: - Insertion - Exchange - Random - 2-opt T Cmax

11. Adaptive Memetic Algorithm (I) • AMA: AGA with Population-based Local search (PLS) on a population (inherit from Pareto solutions). Create initial population Elitist selection into the population Set new PMi Start Crossover Generation of PLS Mutation n Mutation selection Computation of PO* and the population Compute new P value Mutation 1

12. AMA (II): PLS • Local search based on a set of individuals. • Initial solutions of the search: Offspring creation from Pareto solutions.

13. Cooperative Methods for BOFSP

14. Cooperatives methods • Low-Level/High-Level • Low-Level: Functional composition of a single OM. • High-Level: Different OMs are self-contained. • Relay/Co-evolutionary • Relay: Pipeline fashion. • Co-evolutionary: Parallel cooperative agents. 4 Classes : [Talbi02] LR LC HR HC

15. k> α AGA PLS k< α Cooperation meta/meta:AGMA • Threshold α, limit of PO* progression. • k=number of modifications of PO* since the n last generations. • If k< α -> PLS • PO* update • Restart GA after PLS HR

16. Cooperation meta/meta:AGMA Create initial population Elitist selection into the population Start Set new PMi Crossover Generation of PLS Progress>k Mutation n Mutation selection Progress<k Computation of PO* and the population Compute new P value Mutation 1

17. Cooperation Meta/Exact 1 • CME1: Exact approach. • Run the whole TPM using solutions obtained by heuristics as initial values. • Can prove the optimality of Pareto set obtained with the metaheuristic. • Cooperation LR

18. Cooperation Meta/Exact 2 (I) • CME2: Heuristic approach. • Idea: Reducing the search space size (limit the number of solutions studied by TPM). • Goal: Control the combinatorial explosion of the studied solutions. • Large neighbourhood approach(VLSN): explore only the search space around solutions found by the metaheuristic. • Cooperation HR

19. … 1(0) 2(1) 3(1) 12(1) … … … 2(0) 1(2) 2(1) 4(1) 3(1) 12(1) 12(2) 1(2) 3(1) 12(2) R: distance … … … … … … 3 1 2 1 4 3 4 2 5 4 5 4 … … … … Cooperation Meta/Exact 2 (II) Initial schedule:

20. Cooperation Meta/Exact 3 (I) • CME3 Idea: realize exact resolutions on partitions of Pareto solutions. • Cooperation HR Point 2 Point 1

21. Point 2 Point 2 Point 2 Point 1 Point 1 Point 1 Cooperation Meta/Exact 3 (II) • Use TPM to found the bests solutions between points 1 & 2. • Iterate with the set of Pareto solutions founded by the metaheuristics. • Update Pareto solutions then move Points 1 & 2.

22. 1 2 3 4 5 6 7 8 9 1 2 3 -9 -8 4 5 7 6 -7 -4 -5 -6 -5 -6 6 7 5 6 -5 -6 7 6 5 6 5 Cooperation Meta/Exact 3 (III) Initial solution:

23. Experimentations

24. Experimentations • Taillard benmarks extended to the BiOjective case. (20..200jobs/5..20machines). http://www.lifl.fr/~basseur • Test with 10 runs per instance (excepting TPM). • Comparison of the time needed to find the optimal Pareto set. • Performance metrics to compare Pareto sets to evaluate Meta/Exact cooperations.

25. Instances Original method With improvements With parallelization ta_20_5_01 30 s 17 s No need ta_20_5_02 15’ 14’ No need ta_20_10_01 One week 2days 1day ta_20_10_02 One week 2days 1day ta_20_20_01 Unsolved Unsolved Few weeks Results – exact Pareto sets TPM

26. Instances Tmin Tmax Average Std Dev ta_20_5_01 2’43s 40h57’51s 7h49’51s 9h32’45s ta_20_5_02 8h57’56s Not found X X ta_20_10_01 Not found Not found X X ta_20_10_02 Not found Not found X X ta_20_20_01 Not found Not found X X ta_50_5_01 Not found Not found X X Results – exact Pareto sets AGA

27. Instances Tmin Tmax Average Std Dev ta_20_5_01 2” 42” 15” 14” ta_20_5_02 8” 4’25” 1’14” 1’24” ta_20_10_01 2’27” 23’50”  9’46” 6’02” ta_20_10_02 10’20” 64’36” 30’35” 18’47” ta_20_20_01 3’49” 67’24” 25’10” 20’40” ta_50_5_01 26’03” 385’30” 155’03” 120’18” Results – exact Pareto sets AMA

28. Instances Tmin Tmax Average Std Dev ta_20_5_01 10” (2”) 34” (42”) 20” (15”) 6” (14”) ta_20_5_02 40” (8”) 1’41” (4’25”) 1’01” (1’14”) 18” (1’24”) ta_20_10_01 5’02” (2’27”) 14’27”  (23’50”) 9’08” (9’46”) 2’59” (6’02”) ta_20_10_02 4’45” (10’20”) 36’25” (64’36”) 18’29” (30’25”) 10’04” (18’47”) ta_20_20_01 7’59” (3’49”) 21’03” (67’24”) 12’26” (25’10”) 4’48” (20’40”) ta_50_5_01 38’09” (26’03”) 365’16” (385’30”) 139’38” (155’03”) 106’25” (120’18”) Results – exact Pareto sets AGMA (AMA)

29. Results - AGMA Evolution Example: problem 50jobsx20machines

30. Results – CM1,2,3 • CME1: Speed up the exact resolution. • CME2: To expensive computational time to be effective. • CME3: Give good results – Need performance indicators to evaluate the progress realized.

31. Performance metrics • Contribution: Compare 2 Pareto fronts – Proportion of solutions given by each metaheuristic to build PO*. C=4 W1=4 - N1=1 W2=0 - N2=1 Cont(O,X)=0,7 Cont(X,O)=0,3

32. Zref Performance metrics • S metric [Zitzler99]: Compute dominance area enclosed by PO* and a reference point.

33. Problem Cmin Cmax Average Std Dev ta_50_10_01 0.54 0.63 0.594 0.026 ta_50_20_01 0.51 0.55 0.525 0.015 ta_100_10_01 0.96 1.00 0.986 0.015 ta_100_20_01 0.73 0.96 0.876 0.062 ta_200_100_01 1.00 1.00 1.000 0.000 Results C(CME3/AGMA)

34. Problem Smin Smax Average Std Dev ta_50_10_01 0.02% 0.46% 0.185% 0.122% ta_50_20_01 0.01% 0.27% 0.093% 0.095% ta_100_10_01 0.75% 2.10% 1.119% 0.387% ta_100_20_01 0.28% 1.92% 0.970% 0.412% ta_200_100_01 8.35% 15.57% 13.094% 1.974% Results S(CME3/AGMA)

35. AGMA Pareto Set CME3 Pareto set Results Problem 100jobs*10machines

36. AGMA Pareto Set CME3 Pareto set Results Problem 200jobs*10machines

37. Conclusions and perspectives • Conclusions • Many cooperation schemes. • Interest of Meta/Meta and Meta/Exact approaches. • Best Pareto front upgraded. • Perspectives • Test other cooperation schemes. • Hybrid Meta/Exact approaches. • Parallel cooperation (LC, HC).