1. 1 Introduction to Scatter Search ENGG*6140 – Paper Review Presented by: Jason Harris & Stephen Coe M.Sc. Candidates University of Guelph

2. 2 Outline Introduction Scatter Search Template Diversification Generation Method Diversification Generation Method Improvement Method Improvement Method Reference Set Update Method Reference Set Update Method Subset Generation Method Subset Generation Method Solution Combination Method Solution Combination Method Example (0-1 Knapsack Problem) Comparison with GA

3. 3 Introduction Evolutionary method Fundamental concepts were first introduced in the 1970’s and are based on formulations from the 1960’s Original proposal in 1977 was tabled by Fred Glover, after that Scatter Search was left until 1990. Uses strategies that both diversify and intensify solutions Solutions are generated using combination strategies as opposed to probabilistic learning approaches

4. 4 Scatter Search Foundations Useful information about the solution is typically contained in a diverse collection of elite solutions Combination strategies incorporate both diversity (extrapolation) and intensification (interpolation). Multiple solution combination enhance the opportunity to exploit information contained in the union of elite solutions

5. 5 Introduction A B C 1 2 3 4 Convex Non-convex These solutions are in a raw form. In most scatter searches these solutions are subject to heuristic improvement.

6. 6 Scatter Search Notation RefSet – Reference set of Solutions b – Size of the reference set x i – The i th solution in the Reference set (x 1 is the best and x b is the worst). P – Set of solutions generated by diversification generation method PSize – Size of the population of diverse solutions s – A subset of reference solutions s size – The size of the subset of reference solutions

7. Scatter Search Template Improvement Method Diversification Generation Method Reference Set Update Method Subset Generation Method sss Improvement Method Solution Combination Method No New Reference Solutions Added P RefSet

8. 8 Diversification Generation Method The idea behind the diversification generation method is to generate a collection of diverse solutions The quality of the solutions is not of importance Generation methods are often customized to specific problems PSize is usually set to the maximum of 100 or 5*b Can be totally deterministic or partially random

9. 9 Improvement Method Must be able to handle both feasible and infeasible solutions It is possible to generate multiple instances of the same solution Generally employs local searches not unlike those previously introduced in this class (steepest descent) This is the only component that is not necessary to implement the scatter search algorithm

10. 10 Subset Combination Method Construct subsets by building subsets of Type 1, Type 2, Type 3 and Type 4 subsets For a RefSet of b there are approximately (3b-7)*b/2 subset combinations The number of subsets can be reduced by considering just one layer of subsets to reduce computational time

11. 11 Solution Combination Method Generally problem specific, because it is directly related to a solution representation Can generate more than one solution and can depend on the quality of the solutions being combined Can also generate infeasible solutions If a subset has been calculated on a previous iteration it is not necessary to do the calculation again

12. 12 Reference Update Method Objective is to generate a collection of both high quality solutions and diverse solutions The number of Solution included in the RefSet is usually less than 20 Consists of the b 1 best solutions from the preceding step (solution combination or diversification generation) Consists of the b 2 solutions that have the largest Euclidian distance from the current RefSet solutions Multiple techniques are employed to update the reference set (Static, Dynamic, 2-Tier etc…)

13. 13 Example (0-1 Knapsack) Maximize: 11x 1 +10x 2 +9x 3 +12x 4 +10x 5 +6x 6 +7x 7 +5x 8 +3x 9 +8x 10 (Co-efficients represent profit for each item) Subject to: 33x 1 +27x 2 +16x 3 +14x 4 +29x 5 +30x 6 +31x 7 +33x 8 +14x 9 +18x 10  100 (Co-efficients represent weight for each item) x i ={0,1} for i=1,…,10

14. 14 Diversification Generator Our goal is to generate a diverse population from a random seed: x=(0,0,0,0,0,0,0,0,0,0) Select h  n-1, we will choose h=5 x’ 1+hk =1-x 1+hk for k=0,1,…,n/h (Values that are not visited are equal to the original seed x) Another set of diverse solutions are generated base on the compliment of x’: x” i =1-x’ I hx'x'' 1(1,1,1,1,1,1,1,1,1,1)(0,0,0,0,0,0,0,0,0,0) 2(1,0,1,0,1,0,1,0,1,0)(0,1,0,1,0,1,0,1,0,1) 3(1,0,0,1,0,0,1,0,0,1)(0,1,1,0,1,1,0,1,1,0) 4(1,0,0,0,1,0,0,0,1,0)(0,1,1,1,0,1,1,1,0,1) 5(1,0,0,0,0,1,0,0,0,0)(0,1,1,1,1,0,1,1,1,1)

15. 15 Diversification Generator SolutionTrial SolutionObjective Value Weight Constraint 1(1,1,1,1,1,1,1,1,1,1)81245 2(1,0,1,0,1,0,1,0,1,0)40123 3(1,0,0,1,0,0,1,0,0,1)3896 4(1,0,0,0,1,0,0,0,1,0)2476 5(1,0,0,0,0,1,0,0,0,0)1763 6(0,0,0,0,0,0,0,0,0,0)00 7(0,1,0,1,0,1,0,1,0,1)41122 8(0,1,1,0,1,1,0,1,1,0)43149 10(0,1,1,1,1,0,1,1,1,1)64182 9 (0,1,1,1,0,1,1,1,0,1) 57 169

16. 16 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,1,1,1,0,1) Objective = 57 Weight = 169

17. 17 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,1,1,0,0,1) Objective = 52 Weight = 136 1→0

18. 18 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,0,1,0,0,1) Objective = 46 Weight = 106 1→0

19. 19 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,0,0,0,0,1) Objective = 39 Weight = 75 1→0

20. 20 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,1,0,0,0,0,1) Objective = 49 Weight = 104 1→0 0→1

21. 21 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,0,0,0,0,1) Objective = 39 Weight = 75 1→0 0→1

22. 22 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (1,1,1,1,0,0,0,0,0,1) Objective = 50 Weight = 108 1→0 0→1

23. 23 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,0,0,0,0,1) Objective = 39 Weight = 75 1→0 0→1

24. 24 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,0,1,0,0,1) Objective = 46 Weight = 106 1→0 0→1

25. 25 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,0,0,0,0,1) Objective = 39 Weight = 75 1→0 0→1

26. 26 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,0,0,0,1,1) Objective = 42 Weight = 89 1→0 0→1

27. 27 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,1,0,0,1,1) Objective = 48 Weight = 119 1→0 0→1

28. 28 Improvement Method x 1 = 0.333 x 2 = 0.370 x 3 = 0.563 x 4 = 0.857 x 5 = 0.345 x 6 = 0.200 x 7 = 0.226 x 8 = 0.152 x 9 = 0.214 x 10 = 0.444 (0,1,1,1,0,0,0,0,1,1) Objective = 42 Weight = 89 1→0 0→1 Most Improved Feasible Solution of #9

29. 29 Improvement Method SolutionTrial Solution Objective Value Improved Solution Objective Value 1(1,1,1,1,1,1,1,1,1,1)81(0,1,1,1,0,0,0,0,1,1)42 2(1,0,1,0,1,0,1,0,1,0)40(1,0,1,1,1,0,0,0,0,0)42 3(1,0,0,1,0,0,1,0,0,1)38(1,0,0,1,0,0,1,0,0,1)38 4(1,0,0,0,1,0,0,0,1,0)24(1,0,0,1,1,0,0,0,1,0)36 5(1,0,0,0,0,1,0,0,0,0)17(1,0,1,1,0,1,0,0,0,0)38 6(0,0,0,0,0,0,0,0,0,0)0(0,1,1,1,0,0,0,0,0,1)39 7(0,1,0,1,0,1,0,1,0,1)41(0,1,0,1,0,1,0,0,0,1)36 8(0,1,1,0,1,1,0,1,1,0)43(0,1,1,1,1,0,0,0,1,0)44 9(0,1,1,1,0,1,1,1,0,1)57(0,1,1,1,0,0,0,0,1,1)42 10(0,1,1,1,1,0,1,1,1,1)64(0,1,1,1,0,0,0,0,1,1)42

30. 30 Reference Set Update SolutionImproved SolutionObjective Value 42 3(1,0,0,1,0,0,1,0,0,1)38 4(1,0,0,1,1,0,0,0,1,0)36 5(1,0,1,1,0,1,0,0,0,0)38 6(0,1,1,1,0,0,0,0,0,1)39 7(0,1,0,1,0,1,0,0,0,1)36 44 9(0,1,1,1,0,0,0,0,1,1)42 10(0,1,1,1,0,0,0,0,1,1)42 1 (0,1,1,1,0,0,0,0,1,1) 2 (1,0,1,1,1,0,0,0,0,0) 8 (0,1,1,1,1,0,0,0,1,0)

31. CandidateSolution Distance to solution Minimum Distance Solution 1Solution 2Solution 8 5474 4(1,0,0,1,1,0,0,0,1,0)5332 5(1,0,1,1,0,1,0,0,0,0)5352 6(0,1,1,1,0,0,0,0,0,1)1431 3653 Reference Set Update In the previous slide, the elite solutions were taken as 1, 2 and 8. Diversity also needs to be incorporated into the Reference set by taking the solutions that are farthest from the best solutions. Distance: (Solution 1 and Solution 8) (0, 1, 1, 1, 0, 0, 0, 0, 1, 1) Solution 1 (0, 1, 1, 1, 0, 0, 0, 0, 1, 1) Solution 1 (0,1,1,1,1,0,0,0,1,0) Solution 8 (0,1,1,1,1,0,0,0,1,0) Solution 8 (0+0+0+0+1+0+0+0+0+1) = 2 (0+0+0+0+1+0+0+0+0+1) = 2 3 (1,0,0,1,0,0,1,0,0,1) 7 (0,1,0,1,0,1,0,0,0,1)

32. 32 Subset Generation Solution Objective Value 1 (0,1,1,1,0,0,0,0,1,1) 42 2 (1,0,1,1,1,0,0,0,0,0) 42 8 (0,1,1,1,1,0,0,0,1,0) 44 3 (1,0,0,1,0,0,1,0,0,1) 38 7 (0,1,0,1,0,1,0,0,0,1) 36 Type 1 Solutions (1,2) (1,8) (1,3) (1,7) (2,8) (2,3) (2,7) (8,3) (8,7) (3,7) Type 2 Solutions (1,2,8) (1,3,8) (1,7,8) (2,3,8) (2,7,8) (3,7,8) Type 3 Solutions (1,2,8,3) (1,7,8,2) (3,7,8,1) Type 4 Solutions (1,2,8,3,7)

33. 33 Solution Combination Sol’nx1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 x9x9 x 10 30.3220.000 0.3220.000 0.3220.000 0.322 70.0000.3050.0000.3050.0000.3050.000 0.305 80.0000.373 0.000 0.3730.000 Total0.3220.6780.3731.0000.3730.3050.3220.0000.3730.627

34. 34 Solution Combination If score (i) > 0.5 If score (i) ≤ 0.5 Sol’nx1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 x9x9 x 10 30.3220.000 0.3220.000 0.3220.000 0.322 70.0000.3050.0000.3050.0000.3050.000 0.305 80.0000.373 0.000 0.3730.000 Total0.3220.6780.3731.0000.3730.3050.3220.0000.3730.627 x’ = ( 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 )

35. 35 Scatter Search Comparison to GA RefSet size vs. Generation Size All solutions participate in combination in Scatter Search Evolution of the population is controlled by deterministic rules Local search procedures are integral to Scatter Search Scatter Searches are not generally limited to combining two “parent” solutions Initial population is not constructed in a random manner

36. Scatter Search Applications Vehicle Routing Rochat and Taillard (1995); Ochi, et al. (1998); Atan and Secomandi (1999); Rego and Leão (2000); Corberán, et al. (2002) TSP Olivia San Martin (2000) Arc Routing Greistorfer (1999) Quadratic Assignment Cung et al. (1996) Financial Product Design Consiglio and Zenios (1999) Neural Network Training Kelly, Rangaswamy and Xu (1996); Laguna and Martí (2001) Combat Forces Assessment Model Bulut (2001) Graph Drawing Laguna and Martí (1999) Linear Ordering Laguna, Martí and Campos (2001) Unconstrained Optimization Fleurent, et al. (1996); Laguna and Martí (2000) Bit Representation Rana and Whitely (1997) Multi-objective Assignment Laguna, Lourenço and Martí (2000) Optimization Simulation Glover, Kelly and Laguna (1996); Grant (1998) Tree Problems Canuto, Resende and Ribeiro (2001); Xu Chiu and Glover (2000)

37. 37References Cung, V., T. Mautor, P. Michelon, and A. Tavares (1997), “A Scatter Search Based Approach for the Quadratic Assignment Problem” Proceedings of the IEEE-ICEC'97 Conference in Indianapolis, April 13-16 Glover, F., A. Løkketangen and D. Woodruff (1999), “Scatter Search to Generate Diverse MIP Solutions” in OR Computing Tools for Modeling, Optimization and Simulation: Interfaces in Computer Science and Operations Research, M. Laguna and J.L. Gonazalez-Velarde (Eds.), Kluwer Academic Publishers, pp. 299-317 http://www-bus.colorado.edu/faculty/glover/ssdiversemip.pdfhttp://www-bus.colorado.edu/faculty/glover/ssdiversemip.pdf (Last Access: March 24th 2003) Glover, F., M. Laguna and R. Martí (2000), “Scatter Search” To appear in Theory and Applications of Evolutionary Computation: Recent Trends, A. Ghosh and S. Tsutsui (Eds.), Springer-Verlag http://leeds.colorado.edu/Faculty/Laguna/articles/ss2.pdfhttp://leeds.colorado.edu/Faculty/Laguna/articles/ss2.pdf (Last Access: March 24th 2003) Glover, F., M. Laguna and R. Martí (2000), “Fundamentals of Scatter Search and Path Relinking” Control and Cybernetics, 29 (3), pp. 653-684 http://leeds.colorado.edu/Faculty/Laguna/articles/ss3.pdfhttp://leeds.colorado.edu/Faculty/Laguna/articles/ss3.pdf (Last Access: March 24th 2003) Glover, F., M. Laguna and R. Martí (2002), “Fundamentals of Scatter Search and Path Relinking: Foundations and Advanced Designs” to appear in New Optimization Techniques in Engineering, Godfrey Onwubolu (Eds.) http://leeds.colorado.edu/faculty/glover/aassprad.pdfhttp://leeds.colorado.edu/faculty/glover/aassprad.pdf (Last Access: March 24th 2003) Laguna, M. (2002), “Scatter Search” in Handbook of Applied Optimization, P. M. Pardalos and M. G. C. Resende (Eds.), Oxford University Press, pp. 183-193 http://www-bus.colorado.edu/Faculty/Laguna/articles/ss1.pdfhttp://www-bus.colorado.edu/Faculty/Laguna/articles/ss1.pdf (Last Access: March 24th 2003) Laguna, M., and R. Martí (2003), “Scatter Search: Methodology and Implementations in C” Kluwer Academic Publishers, Boston, 312 pp.