1 introduction to scatter search engg*6140 – paper review presented by: jason harris & stephen...

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Introduction to Scatter Introduction to Scatter SearchSearch

ENGG*6140 – Paper ReviewENGG*6140 – Paper ReviewPresented by: Jason Harris & Stephen CoePresented by: Jason Harris & Stephen Coe

M.Sc. CandidatesM.Sc. CandidatesUniversity of GuelphUniversity of Guelph

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OutlineOutline

IntroductionIntroductionScatter Search TemplateScatter Search Template Diversification Generation MethodDiversification Generation Method Improvement MethodImprovement Method Reference Set Update MethodReference Set Update Method Subset Generation MethodSubset Generation Method Solution Combination MethodSolution Combination Method

Example (0-1 Knapsack Problem)Example (0-1 Knapsack Problem)Comparison with GAComparison with GA

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IntroductionIntroduction

Evolutionary methodEvolutionary methodFundamental concepts were first introduced in Fundamental concepts were first introduced in the 1970’s and are based on formulations from the 1970’s and are based on formulations from the 1960’sthe 1960’sOriginal proposal in 1977 was tabled by Fred Original proposal in 1977 was tabled by Fred Glover, after that Scatter Search was left until Glover, after that Scatter Search was left until 1990.1990.Uses strategies that both diversify and intensify Uses strategies that both diversify and intensify solutionssolutionsSolutions are generated using combination Solutions are generated using combination strategies as opposed to probabilistic learning strategies as opposed to probabilistic learning approachesapproaches

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Scatter Search FoundationsScatter Search Foundations

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

55

IntroductionIntroduction

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.

66

Scatter Search NotationScatter Search Notation

RefSet – Reference set of RefSet – Reference set of SolutionsSolutions

bb – Size of the reference set – Size of the reference set

xxii – The i – The ithth solution in the solution in the Reference set (xReference set (x11 is the is the best and xbest and xbb is the worst). is the worst).

P – Set of solutions P – Set of solutions generated by generated by diversification generation diversification generation methodmethod

PSize – Size of the PSize – Size of the population of diverse population of diverse solutionssolutions

s – A subset of reference s – A subset of reference solutionssolutions

sssizesize – The size of the subset – The size of the subset of reference solutionsof reference solutions

Scatter Search TemplateScatter Search Template

ImprovementMethod

Diversification GenerationMethod

Reference SetUpdate Method

Subset GenerationMethods ss

ImprovementMethod

Solution CombinationMethod

No New Reference Solutions Added

P

RefSet

88

Diversification Generation MethodDiversification Generation Method

The idea behind the diversification generation The idea behind the diversification generation method is to generate a collection of diverse method is to generate a collection of diverse solutionssolutions

The quality of the solutions is not of importanceThe quality of the solutions is not of importance

Generation methods are often customized to Generation methods are often customized to specific problemsspecific problems

PSize is usually set to the maximum of 100 or PSize is usually set to the maximum of 100 or 5*5*bb

Can be totally deterministic or partially randomCan be totally deterministic or partially random

99

Improvement MethodImprovement Method

Must be able to handle both feasible and Must be able to handle both feasible and infeasible solutionsinfeasible solutions

It is possible to generate multiple instances of It is possible to generate multiple instances of the same solutionthe same solution

Generally employs local searches not unlike Generally employs local searches not unlike those previously introduced in this class those previously introduced in this class (steepest descent)(steepest descent)

This is the only component that is not necessary This is the only component that is not necessary to implement the scatter search algorithmto implement the scatter search algorithm

1010

Subset Combination MethodSubset Combination Method

Construct subsets by building subsets of Construct subsets by building subsets of Type 1, Type 2, Type 3 and Type 4 Type 1, Type 2, Type 3 and Type 4 subsetssubsets

For a RefSet of For a RefSet of bb there are approximately there are approximately (3(3bb-7)*-7)*bb/2 subset combinations/2 subset combinations

The number of subsets can be reduced by The number of subsets can be reduced by considering just one layer of subsets to considering just one layer of subsets to reduce computational timereduce computational time

1111

Solution Combination MethodSolution Combination Method

Generally problem specific, because it is Generally problem specific, because it is directly related to a solution representationdirectly related to a solution representationCan generate more than one solution and Can generate more than one solution and can depend on the quality of the solutions can depend on the quality of the solutions being combinedbeing combinedCan also generate infeasible solutionsCan also generate infeasible solutionsIf a subset has been calculated on a If a subset has been calculated on a previous iteration it is not necessary to do previous iteration it is not necessary to do the calculation againthe calculation again

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Reference Update MethodReference Update Method

Objective is to generate a collection of both high Objective is to generate a collection of both high quality solutions and diverse solutionsquality solutions and diverse solutionsThe number of Solution included in the RefSet is The number of Solution included in the RefSet is usually less than 20usually less than 20Consists of the Consists of the bb11 best solutions from the best solutions from the preceding step (solution combination or preceding step (solution combination or diversification generation)diversification generation)Consists of the Consists of the bb22 solutions that have the largest solutions that have the largest Euclidian distance from the current RefSet Euclidian distance from the current RefSet solutionssolutionsMultiple techniques are employed to update the Multiple techniques are employed to update the reference set (Static, Dynamic, 2-Tier etc…)reference set (Static, Dynamic, 2-Tier etc…)

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Example (0-1 Knapsack)Example (0-1 Knapsack)

Maximize:Maximize:

11x11x11+10x+10x22+9x+9x33+12x+12x44+10x+10x55+6x+6x66+7x+7x77+5x+5x88+3x+3x99+8x+8x1010

(Co-efficients represent profit for each item)(Co-efficients represent profit for each item)

Subject to:Subject to:

33x33x11+27x+27x22+16x+16x33+14x+14x44+29x+29x55+30x+30x66+31x+31x77+33x+33x88+14x+14x99+18x+18x1010100100

(Co-efficients represent weight for each item)(Co-efficients represent weight for each item)

xxii={0,1} for i=1,…,10={0,1} for i=1,…,10

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Diversification GeneratorDiversification GeneratorOur goal is to generate a diverse population from a random seed:Our goal is to generate a diverse population from a random seed:

x=(0,0,0,0,0,0,0,0,0,0)x=(0,0,0,0,0,0,0,0,0,0)

Select h Select h n-1, we will choose h=5 n-1, we will choose h=5

x’x’1+hk1+hk=1-x=1-x1+hk1+hk for k=0,1,…,n/h for k=0,1,…,n/h

(Values that are not visited are equal to the original seed x)(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’:Another set of diverse solutions are generated base on the compliment of x’:

x”x”ii=1-x’=1-x’II

h x' 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)

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Diversification GeneratorDiversification Generator

Solution Trial Solution Objective ValueWeight

Constraint

1 (1,1,1,1,1,1,1,1,1,1) 81 245

2 (1,0,1,0,1,0,1,0,1,0) 40 123

3 (1,0,0,1,0,0,1,0,0,1) 38 96

4 (1,0,0,0,1,0,0,0,1,0) 24 76

5 (1,0,0,0,0,1,0,0,0,0) 17 63

6 (0,0,0,0,0,0,0,0,0,0) 0 0

7 (0,1,0,1,0,1,0,1,0,1) 41 122

8 (0,1,1,0,1,1,0,1,1,0) 43 149

10 (0,1,1,1,1,0,1,1,1,1) 64 182

9 (0,1,1,1,0,1,1,1,0,1) 57 169

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Constraintin xofefficient -Co

Function Objectivein xofefficient -Co Ratio

i

i

x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,1,1,1,0,1)

Objective = 57

Weight = 169

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x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,1,1,0,0,1)

Objective = 52

Weight = 1361→0

1818


x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,0,1,0,0,1)

Objective = 46

Weight = 1061→0

1→0

1919


x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,0,0,0,0,1)

Objective = 39

Weight = 751→0

1→0

1→0

2020


x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,1,0,0,0,0,1)

Objective = 49

Weight = 1041→0

1→0

1→0

0→1

2121


x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,0,0,0,0,1)

Objective = 39

Weight = 751→0

1→0

1→0

0→1

2222


x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(1,1,1,1,0,0,0,0,0,1)

Objective = 50

Weight = 1081→0

1→0

1→0

0→1

0→1

2323


x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,0,0,0,0,1)

Objective = 39

Weight = 751→0

1→0

1→0

0→1

0→1

2424


x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,0,1,0,0,1)

Objective = 46

Weight = 1061→0

1→0

1→0

0→1

0→1

0→1

2525


x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,0,0,0,0,1)

Objective = 39

Weight = 751→0

1→0

1→0

0→1

0→1

0→1

2626


x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,0,0,0,1,1)

Objective = 42

Weight = 891→0

1→0

1→0

0→1

0→1

0→1

0→1

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x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,1,0,0,1,1)

Objective = 48

Weight = 1191→0

1→0

1→0

0→1

0→1

0→1

0→1

0→1

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x1 = 0.333

x2 = 0.370

x3 = 0.563

x4 = 0.857

x5 = 0.345

x6 = 0.200

x7 = 0.226

x8 = 0.152

x9 = 0.214

x10 = 0.444

(0,1,1,1,0,0,0,0,1,1)

Objective = 42

Weight = 891→0

1→0

1→0

0→1

0→1

0→1

0→1

Most Improved Feasible Solution of #9

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Solution Trial SolutionObjective

ValueImproved 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

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Reference Set UpdateReference Set Update

Solution Improved Solution Objective Value

42

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)

Candidate Solution

Distance to solutionMinimumDistanceSolution 1 Solution 2 Solution 8

5 4 7 4

4 (1,0,0,1,1,0,0,0,1,0) 5 3 3 2

5 (1,0,1,1,0,1,0,0,0,0) 5 3 5 2

6 (0,1,1,1,0,0,0,0,0,1) 1 4 3 1

3 6 5 3

Reference Set UpdateReference Set UpdateIn the previous slide, the elite solutions were taken as 1, 2 and 8. Diversity 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 also needs to be incorporated into the Reference set by taking the solutions that are farthest from the best solutions.that are farthest from the best solutions.

Distance: (Solution 1 and Solution 8)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)(0 ,1 ,1 ,1 ,1 ,0 ,0 ,0 ,1 ,0) Solution 8 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)

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Subset GenerationSubset GenerationSolution 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)

3333

Solution CombinationSolution Combination

Sol’n x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

3 0.322 0.000 0.000 0.322 0.000 0.000 0.322 0.000 0.000 0.322

7 0.000 0.305 0.000 0.305 0.000 0.305 0.000 0.000 0.000 0.305

8 0.000 0.373 0.373 0.373 0.373 0.000 0.000 0.000 0.373 0.000

Total 0.322 0.678 0.373 1.000 0.373 0.305 0.322 0.000 0.373 0.627

Sj

Sj

jOV

jixjOV

xScore i

)(

)()(

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Solution CombinationSolution Combination

0

1ix

If score (i) > 0.5

If score (i) ≤ 0.5

Sol’n x1 x2 x3 x4 x5 x6 x7 x8 x9 x10

3 0.322 0.000 0.000 0.322 0.000 0.000 0.322 0.000 0.000 0.322

7 0.000 0.305 0.000 0.305 0.000 0.305 0.000 0.000 0.000 0.305

8 0.000 0.373 0.373 0.373 0.373 0.000 0.000 0.000 0.373 0.000

Total 0.322 0.678 0.373 1.000 0.373 0.305 0.322 0.000 0.373 0.627

x’ = ( 0, 1, 0, 1, 0, 0, 0, 0, 0, 1 )

3535

Scatter Search Comparison to GAScatter Search Comparison to GA

RefSet size vs. Generation SizeRefSet size vs. Generation SizeAll solutions participate in combination in Scatter All solutions participate in combination in Scatter SearchSearchEvolution of the population is controlled by Evolution of the population is controlled by deterministic rulesdeterministic rulesLocal search procedures are integral to Scatter Local search procedures are integral to Scatter SearchSearchScatter Searches are not generally limited to Scatter Searches are not generally limited to combining two “parent” solutionscombining two “parent” solutionsInitial population is not constructed in a random Initial population is not constructed in a random mannermanner

Scatter Search ApplicationsScatter Search ApplicationsVehicle RoutingVehicle Routing Rochat and Taillard (1995); Ochi, et al. (1998); Atan and Rochat and Taillard (1995); Ochi, et al. (1998); Atan and

Secomandi (1999); Rego and LeSecomandi (1999); Rego and Leãão (2000); Corbero (2000); Corberán, et án, et al. (2002)al. (2002)

TSPTSP Olivia San Martin (2000)Olivia San Martin (2000)

Arc RoutingArc Routing Greistorfer (1999)Greistorfer (1999)

Quadratic AssignmentQuadratic Assignment Cung et al. (1996)Cung et al. (1996)

Financial Product DesignFinancial Product Design Consiglio and Zenios (1999)Consiglio and Zenios (1999)

Neural Network TrainingNeural Network Training Kelly, Rangaswamy and Xu (1996); Laguna and MartKelly, Rangaswamy and Xu (1996); Laguna and Martí í (2001)(2001)

Combat Forces Assessment ModelCombat Forces Assessment Model Bulut (2001)Bulut (2001)

Graph DrawingGraph Drawing Laguna and MartLaguna and Martí (1999)í (1999)

Linear OrderingLinear Ordering Laguna, MartLaguna, Martí and Campos (2001)í and Campos (2001)

Unconstrained OptimizationUnconstrained Optimization Fleurent, et al. (1996); Laguna and MartFleurent, et al. (1996); Laguna and Martí (2000)í (2000)

Bit RepresentationBit Representation Rana and Whitely (1997)Rana and Whitely (1997)

Multi-objective AssignmentMulti-objective Assignment Laguna, LourenLaguna, Lourençoço and Mart and Martí (2000)í (2000)

Optimization SimulationOptimization Simulation Glover, Kelly and Laguna (1996); Grant (1998)Glover, Kelly and Laguna (1996); Grant (1998)

Tree ProblemsTree Problems Canuto, Resende and Ribeiro (2001); Xu Chiu and Canuto, Resende and Ribeiro (2001); Xu Chiu and Glover (2000)Glover (2000)

3737

ReferencesReferencesCung, 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. 299-317http://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-Verlaghttp://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. 653-684http://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.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. 183-193http://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.

1 introduction to scatter search engg*6140 – paper review presented by: jason harris & stephen...

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