scatter search methodology and applications

8/9/2019 Scatter Search Methodology and Applications

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Scatter Search:Scatter Search:

Methodology andMethodology and Applications Applications

Manuel LagunaManuel LagunaUniversity of ColoradoUniversity of Colorado

RafaelRafael MartMartUniversity of ValenciaUniversity of Valencia


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Scatter SearchScatter Search

MethodologyMethodology


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MetaheuristicMetaheuristic

A metaheuristic refers to a master strategy A metaheuristic refers to a master strategy

that guides and modifies other heuristicsthat guides and modifies other heuristicsto produce solutions beyond those that areto produce solutions beyond those that arenormally generated in a quest for localnormally generated in a quest for localoptimality.optimality.

A metaheuristic is a procedure that has A metaheuristic is a procedure that hasthe ability to escape local optimalitythe ability to escape local optimality


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Typical Search TrajectoryTypical Search Trajectory

0

10

20

30

40

50

60

70

80

90

100

1 4 7 10 13 16 19 22 25 28 31 34 37 40

Itera t ion

O b j e c

t i v e

F u n c

t i o n

Value

Best Value


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Metaheuristic ClassificationMetaheuristic Classification

x/y/zx/y/z ClassificationClassification

x = A (adaptive memory) or M (x = A (adaptive memory) or M ( memorylessmemoryless ))y = N (systematic neighborhood search) or S (randomy = N (systematic neighborhood search) or S (randomsampling)sampling)

z = 1 (one current solution) or P (population ofz = 1 (one current solution) or P (population ofsolutions)solutions)

Some ClassificationsSome ClassificationsTabuTabu search (A/N/1)search (A/N/1)Genetic Algorithms (M/S/P)Genetic Algorithms (M/S/P)Scatter Search (M/N/P)Scatter Search (M/N/P)


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Scatter SearchScatter Search

P

Diversification GenerationMethod

Repeat until |P | = PSize

Subset GenerationMethod

ImprovementMethod

Solution CombinationMethod

ImprovementMethod

Stop if no morenew solutions

Reference SetUpdate Method

RefSet


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Repeat until |P | = PSize

Scatter Search with RebuildingScatter Search with RebuildingP


Subset GenerationMethod

ImprovementMethod


ImprovementMethod

No more newsolutions

Reference SetUpdate Method

RefSet


ImprovementMethod

Stop if MaxIter reached


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TutorialTutorial

Unconstrained Nonlinear OptimizationUnconstrained Nonlinear Optimization

ProblemProblem

( ) ( ) ( ) ( )( ) ( )( ) ( )( )

4,,1for1010toSubject

118.19111.101901100Minimize

422

42

2

23

2234

21

2212

K=

++++++

i x

x x x x x x x x x x

i


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Diversification GenerationDiversification Generation

MethodMethod

Subrange 1 Subrange 2 Subrange 3 Subrange 4

-10 -5 0 +5 +10

Probability of selecting a subrange is proportional to a frequency count


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1111

Diverse SolutionsDiverse Solutions

x11.11-9.587.428.83-6.23

1.640.2-3.09-6.08

-1.97

x20.85-6.57-1.71-8.457.48

3.34-3.646.620.67

8.13

x39.48-8.819.284.526

-8.32-5.3-2.33-6.48

-5.63

x2-6.35-2.275.923.187.8

-8.66-7.03-3.121.48

8.02

f ( x)835546.21542078.9901878.0775470.7171450.5

546349.8114023.87469.1279099.9

54537.2

Solution12345

6789

10


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Improvement MethodImprovement MethodSolution x1 x2 x3 x2 f ( x)

1 2.5 5.4 2.59 5.67 1002.72 -0.52 -0.5 0.35 -0.14 138.53 -2.6 5.9 4.23 10 7653.74 0.49 0.53 2.47 5.89 213.75 -3.04 9.45 1.14 0.41 720.16 -1.4 2.46 0.37 -3.94 1646.7

7 -0.36 -0.31 0.8 1.14 57.18 -1.63 2.51 0.73 0.56 21.59 -0.8 0.69 -1.16 1.5 11.2

10 -2.47 5.32 -2.92 8.15 1416.7

Nelder and Mead (1965)


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InitialInitial RefSetRefSet

Solutionnumber in P x1 x2 x3 x4 f ( x)

35 -0.0444 0.0424 1.3577 1.8047 2.146 1.133 1.2739 -0.6999 0.5087 3.534 -0.0075 0.0438 1.4783 2.2693 3.549 1.1803 1.4606 -0.344 0.2669 5.2

38 1.0323 0.9719 -0.8251 0.695 5.3

Solution x1 x2 x3 x4 f ( x)

37 -3.4331 10 1.0756 0.3657 1104.130 3.8599 10 -4.0468 10 9332.445 -4.4942 10 3.0653 10 13706.183 -0.2414 -6.5307 -0.9449 -9.4168 17134.8

16 6.1626 10 0.1003 0.1103 78973.2

High-Quality Solutions

Diverse Solutions


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Subset Generation MethodSubset Generation Method

All pairs of reference solutions that include All pairs of reference solutions that include

at least one new solutionat least one new solution

The method generates (b2The method generates (b2 --b)/2 pairs fromb)/2 pairs fromthe initialthe initial RefSetRefSet


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Combination MethodCombination Method

1 2 3 4 5 6 7 8 9 10 11 12 13

1

23

4

5

6

7

8

9

10

x2 = (8,4)

x1 = (5,7)

x3 = x1 - r ( x2 - x1) x4 = x1 + r ( x2 - x1) x5 = x2 + r ( x2 - x1)

x4 = (6.5,5.5)r = 1/2

x5 = (11,1)r = 1

x3 = (9,7)r = 2/3

y

x


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Alternative Combination Method Alternative Combination Method

x3

1 2 3 4 5 6 7 8 9 10 11 12 13

1

2

3

4

5

6

7

8

9

10

x2 = (8,4)

x1 = (5,7)

x3 = x1 - r ( x2 - x1) x4 = x1 + r ( x2 - x1)

x5

= x2

+ r ( x2

- x1)

y

x

x4

x5


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Reference Set Update MethodReference Set Update Method

Updated RefSet

Worst

Best1

b

2...

Quality 1

b

2...

Worst

Best

New trialsolution

RefSet of size b


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RefSetRefSet after Updateafter Update x x11 x x22 x x33 x x44 f f (( x x))

1.13831.1383 1.29651.2965 0.83060.8306 0.7150.715 0.140.14

0.70160.7016 0.52970.5297 1.20781.2078 1.46331.4633 0.360.360.52690.5269 0.2870.287 1.26451.2645 1.60771.6077 0.590.59

1.19631.1963 1.39681.3968 0.68010.6801 0.4460.446 0.620.62

0.33260.3326 0.10310.1031 1.36321.3632 1.83111.8311 0.990.990.33680.3368 0.10990.1099 1.38181.3818 1.93891.9389 1.021.02

0.31270.3127 0.09490.0949 1.35121.3512 1.85891.8589 1.031.03

0.75920.7592 0.5230.523 1.31391.3139 1.71951.7195 1.181.180.20040.2004 0.03440.0344 1.40371.4037 1.94381.9438 1.241.24

1.38921.3892 1.93051.9305 0.12520.1252 --0.01520.0152 1.451.45


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TutorialTutorial

00 --1 Knapsack Problem1 Knapsack Problem

Maximize 10 x 1 + 14 x 2 + 9 x 3 + 8 x 4 + 7 x 5 + 5 x 6 + 9 x 7 + 3 x 8

S.t. 7 x 1 + 12 x 2 + 8 x 3 + 9 x 4 + 8 x 5 + 6 x 6 + 11 x 7 + 5 x 8 < 100

x i = { 0, 1} for i = 1, , 8


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Additional Strategies Additional Strategies

Reference SetReference Set

RebuildingRebuildingMultiMulti--tier tier

Subset GenerationSubset GenerationSubsets of size > 2Subsets of size > 2

Combination MethodCombination MethodVariable number of solutionsVariable number of solutions


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RebuildingRebuilding

RefSet Rebuilt RefSet

b1

b2

DiversificationGeneration Method Reference SetUpdate Method


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22 --TierTier RefSetRefSet

RefSet

b1

b2


ImprovementMethod

Try here first

If it fails, thentry here


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Subset GenerationSubset Generation

Subset Type 1:Subset Type 1: all 2all 2 --element subsets.element subsets.

Subset Type 2:Subset Type 2: 33 --element subsets derivedelement subsets derivedfrom the 2from the 2 --element subsets by augmentingelement subsets by augmentingeach 2each 2 --element subset to include the bestelement subset to include the bestsolution not in this subset.solution not in this subset.Subset Type 3:Subset Type 3: 44 --element subsets derivedelement subsets derivedfrom the 3from the 3 --element subsets by augmentingelement subsets by augmenting

each 3each 3 --element subset to include the bestelement subset to include the bestsolutions not in this subset.solutions not in this subset.Subset Type 4:Subset Type 4: the subsets consisting of thethe subsets consisting of the

bestbestii

elements, forelements, forii

= 5 to b.= 5 to b.


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Subsets of Size > 2Subsets of Size > 2

Type 1 Type 2Type 3

Type 4

LOLIB

Random0%

10%20%

30%

40%

50%

60%

70%

80%

90%


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Variable Number of SolutionsVariable Number of SolutionsQuality

Worst

Best1

b

2... Generate 5 solutions

Generate 3 solutions

Generate 1 solution

RefSet of size b


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Hybrid ApproachesHybrid Approaches

Use of MemoryUse of Memory

TabuTabu Search mechanisms for intensificationSearch mechanisms for intensificationand diversificationand diversification

GRASP ConstructionsGRASP ConstructionsCombination MethodsCombination Methods

GA OperatorsGA OperatorsPathPath RelinkingRelinking


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MultiobjectiveMultiobjective Scatter SearchScatter Search

This is a fruitful research areaThis is a fruitful research area

ManyMany multiobjectivemultiobjective evolutionaryevolutionary

approaches exist (approaches exist ( CoelloCoello , et al. 2002), et al. 2002)

SS can use similar techniques developedSS can use similar techniques developedfor MOEA (for MOEA ( multiobjectivemultiobjective evolutionaryevolutionaryapprochesapproches ))


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MultiobjectiveMultiobjective EA TechniquesEA Techniques

Independent SamplingIndependent Sampling

Search onSearch on f f (( x x)) == wwii f f ii(( x x))Change weights and rerunChange weights and rerun

Criterion SelectionCriterion SelectionDivide reference set intoDivide reference set into kk subsetssubsets

Admission to Admission to ii thth subset is according tosubset is according to f f ii(( x x))


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Scatter SearchScatter Search Applications and Implementations Applications and Implementations


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Scatter Scatter SearchSearch ElementsElements

Diversification Generator MethodDiversification Generator Method

Improvement MethodImprovement MethodReference SetReference Set

InitializationInitializationUpdateUpdateRebuildRebuild

Subset Generation MethodSubset Generation MethodSolution Combination MethodSolution Combination Method

Problem Independent


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RefSet DataRefSet Data StructureStructuretypedeftypedef structstruct REFSETREFSET

{{ intint b;b; / * Si ze/ * Si ze */*/doubledouble **sol ;**sol ; / * Sol ut i ons/ * Sol ut i ons */*/doubledouble ** Obj ValObj Val ;; // * Obj ect i ve val ue of sol ut i ons* Obj ect i ve val ue of sol ut i ons */*/intint *order;*order; / * Or der of sol ut i ons *// * Or der of sol ut i ons */intint ** i t eri t er ;; / * Ent er i ng i t er at i on * // * Ent er i ng i t er at i on * /intint NewSol ut i onsNewSol ut i ons ;; / * =1 i f new el ement has been added *// * =1 i f new el ement has been added */

}} REFSETREFSET ;;

bsol

ObjValorder

iterNewSolutions

RefSet nvar


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MainMain FunctionFunctioni nti nt mai n( voi d)mai n( voi d){{

SS *SS * pbpb ;; / * Poi nt er t o pr obl em dat a/ * Poi nt er t o pr obl em dat a ** //i nti nt nvarnvar = 10;= 10; / * Number of var i abl es/ * Number of var i abl es */*/i nti nt b = 10;b = 10; / * Si ze of r ef er ence set/ * Si ze of r ef er ence set ** //i nti nt PSi zePSi ze = 100;= 100; / * Si ze of P/ * Si ze of P */*/

pbpb == SSProblem_DefinitionSSProblem_Definition ( nvar , b, PSi ze( nvar , b, PSi ze ) ;) ;

/ * I nser t her e t he Scat t er Sear ch code *// * I nser t her e t he Scat t er Sear ch code */

SSFree_DataStructuresSSFree_DataStructures ( pb( pb ) ;) ;r et ur n 0;r et ur n 0;

}}


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BasicBasic DesignDesign

pbpb == SSPr obl em_Def i ni t i on( nvar , b, PSi zeSSPr obl em_Def i ni t i on( nvar , b, PSi ze ) ;) ;SSCreate_PSSCreate_P ( pb( pb ) ;) ;SSCreate_RefSetSSCreate_RefSet ( pb( pb ) ;) ;

Whi l eWhi l e (( pbpb -- >>r sr s -- >>NewSol ut i onsNewSol ut i ons ))SSUpdate_RefSetSSUpdate_RefSet ( pb( pb ) ;) ;

SSBest Sol ( pb, sol , &val ueSSBest Sol ( pb, sol , &val ue ) ;) ;SSFr ee_Dat aSt r uct ur es( pbSSFr ee_Dat aSt r uct ur es( pb ) ;) ;


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Advanced Advanced DesignDesignpbpb == SSPr obl em_Def i ni t i on( nvar , b, PSi zeSSPr obl em_Def i ni t i on( nvar , b, PSi ze ) ;) ;SSCreate_PSSCreate_P ( pb( pb ) ;) ;

SSCreate_RefSetSSCreate_RefSet ( pb( pb ) ;) ;

f or ( I t erf or ( I t er =1;=1; I t erI t er r sr s -- >>NewSol ut i onsNewSol ut i ons ))SSUpdate_RefSetSSUpdate_RefSet ( pb( pb ) ;) ;

el seel se

SSRebuild_RefSetSSRebuild_RefSet ( pb( pb ) ;) ;}}SSBest Sol ( pb, sol , &val ueSSBest Sol ( pb, sol , &val ue ) ;) ;SSFr ee_Dat aSt r uct ur es( pbSSFr ee_Dat aSt r uct ur es( pb ) ;) ;


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SSCreate_PSSCreate_Pwhi l e( cur r ent PSi zewhi l e( cur r ent PSi ze p -- >>PSi zePSi ze )){{

SSGener at e_Sol ( pb, solSSGener at e_Sol ( pb, sol ) ;) ;

obj _valobj _val = sol _val ue( sol ) ;= sol _val ue( sol ) ;SSImprove_solutionSSImprove_solution ( pb, sol , &obj _val( pb, sol , &obj _val ) ;) ;

/ * Check whet her sol i s a new sol ut i on *// * Check whet her sol i s a new sol ut i on */

j =1; equal =0; j =1; equal =0;whi l e( j p>p -- >>sol [ j ++] , pbsol [ j ++] , pb -- >>nvarnvar ) ;) ;

/ * Add i mpr oved sol ut i on t o P *// * Add i mpr oved sol ut i on t o P */i f ( ! equal ) {i f ( ! equal ) {f or ( j =1; j nvar ; jnvar ; j ++)++)

pbpb -- >p>p -- >>sol [ cur r ent PSi ze] [ jsol [ cur r ent PSi ze] [ j ] =sol [ j ] ;] =sol [ j ] ;

pbpb -- >p>p -- >>Obj Val [ cur r ent PSi zeObj Val [ cur r ent PSi ze ++] =++] = obj _valobj _val ;;}}}}


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SSCreate_RefSetSSCreate_RefSet/ * Or der sol ut i ons i n P *// * Or der sol ut i ons i n P */p_or der =p_or der = SSOrderSSOrder ( pb( pb -- >p>p -- >>Obj ValObj Val ,, pbpb -- >p>p -- >>PSi zePSi ze ,, pbpb -- >Opt ) ;>Opt ) ;b1 =b1 = pbpb -- >>r sr s -- >b / 2;>b / 2;

/ * Add t he/ * Add t he f i r s tf i r s t (( bestbest ) b1) b1 sol ut i ons i n P t o Ref Set */sol ut i ons i n P t o Ref Set */

/ * Comput e mi ni mum di st ances f r om P t o Ref Set *// * Comput e mi ni mum di st ances f r om P t o Ref Set */f or ( i =1; i p -- >>PSi ze; iPSi ze; i ++)++)

mi n_di st [ i ] =mi n_di st [ i ] = SSDist_RefSetSSDist_RefSet ( pb, b1, pb( pb, b1, pb -- >p>p -- >sol [ i ] ) ;>sol [ i ] ) ;

/ * Add b/ * Add b -- b1 di ver se sol ut i ons t o Ref Set */b1 di ver se sol ut i ons t o Ref Set */f or ( i =b1+1; i r sr s -- >b; i ++)>b; i ++) {{

a=a= SSMax_di st _i ndex( pb, mi n_di stSSMax_di st _i ndex( pb, mi n_di st ) ;) ;/ * Copy sol a f r om P t o/ * Copy sol a f r om P t o Ref setRef set */*/SSUpdate_distancesSSUpdate_distances ( pb, mi n_di st , i( pb, mi n_di st , i ) ;) ;

}}/ * Comput e t he or der i n Ref Set :/ * Comput e t he or der i n Ref Set : orderorder */*/

pbpb -- >>r sr s -- >>NewSol ut i onsNewSol ut i ons == 1;1; pbpb -- >>Cur r ent I t erCur r ent I t er = 1;= 1;


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SSUpdate_RefSetSSUpdate_RefSet

pbpb -- >>r sr s -- >>NewSol ut i onsNewSol ut i ons =0;=0;SSCombi ne_Ref Set ( pbSSCombi ne_Ref Set ( pb ) ;) ;pbpb -- >>Cur r ent I t erCur r ent I t er ++;++;

f or ( a=1; apool _si ze; a++)

{{val ue=val ue= sol _val ue( pbsol _val ue( pb -- >pool [ a] ) ;>pool [ a] ) ;SSI mpr ove_sol ut i on( pb, pbSSI mpr ove_sol ut i on( pb, pb -- >pool [ a] , &val ue) ;>pool [ a] , &val ue) ;

SSTr yAdd_Ref Set ( pb, pbSSTr yAdd_Ref Set ( pb, pb -- >pool [ a] , val ue) ;>pool [ a] , val ue) ;}}pbpb -- >>pool _si zepool _si ze =0;=0;


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Advanced Advanced DesignsDesigns

ReferenceReference SetSet UpdateUpdate

DynamicDynamic // StaticStatic22 Tier Tier / 3/ 3 Tier Tier

SubsetSubset GenerationGenerationUse ofUse of MemoryMemoryExplicitExplicit MemoryMemory

Attributive Attributive MemoryMemory

PathPath RelinkingRelinking


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A I


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An An InstanceInstance

1

2

34

1 2 3 4

3

4

12

3 4 1 2

0 9 8 32 0 11 4

5 3 0 122 3 4 0

0 12 5 34 0 2 6

8 3 0 911 4 2 0

p=(1,2,3,4)

cE(p)=12+5+3+2+6+9=37

p*=(3,4,1,2)

cE(p*)=9+8+3+11+4+12=47

SSSS ff hh LOPLOP


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SSSS for for thethe LOPLOP

Complete the generic codeComplete the generic code

Design specific methods for the problemDesign specific methods for the problem --dependent elementsdependent elements

Diversification Generator MethodDiversification Generator MethodImprovement MethodImprovement Method

Combination MethodCombination Method

Di ifi i GDi ifi i G


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Diversification Generator Diversification Generator Use ofUse of problemproblem structurestructure toto createcreate methodsmethods inin order order totoachieveachieve aa goodgood balancebalance betweenbetween qualityquality andand diversitydiversity ..

QualityQualityDeterministicDeterministic constructiveconstructive methodmethodDiversityDiversity

RandomRandom Generator Generator

SystematicSystematic GeneratorsGenerators (Glover, 1998)(Glover, 1998)GRASP constructions.GRASP constructions.

The method randomly selects from a short list of the mostThe method randomly selects from a short list of the mostattractive sectors.attractive sectors.

Use ofUse of MemoryMemoryMModifyingodifying a measure of attractiveness proposed by Becker witha measure of attractiveness proposed by Becker witha frequencya frequency --based memory measure that discourages sectorsbased memory measure that discourages sectorsfrom occupying positions that they have frequently occupiedfrom occupying positions that they have frequently occupied

Di i Q liDi it Q lit


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Diversity vs. QualityDiversity vs. Quality

Compare several diversification generatorsCreate a set of 100 solutions with each one

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

DG05 DG04 DG02 DG01 DG09 DG08 DG06 DG03 DG07 DG10

Procedure

C

d

C+d

d = Standardized DiversityC = Standardized Quality


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S l ti C bi ti M th dSol tion Combination Method


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Solution Combination MethodSolution Combination Method

The method scans (from left to right) each referenceThe method scans (from left to right) each referencepermutation.permutation.

Each reference permutation votes for its first element that is sEach reference permutation votes for its first element that is s tilltillnot included in the combined permutation (incipient element).not included in the combined permutation (incipient element).The voting determines the next element to enter the first stillThe voting determines the next element to enter the first stillunassigned position of the combined permutation.unassigned position of the combined permutation.

The vote of a given reference solution is weighted according toThe vote of a given reference solution is weighted according tothe incipient elements position.the incipient elements position.

Incipient element(3,1,4,2,5) votes for 4 Solution under construction:(1,4,3,5,2) votes for 4 (3,1,2,4,_ )(2,1,3,5,4) votes for 5

E i tExperiments ithwith LOLIBLOLIB


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ExperimentsExperiments withwith LOLIBLOLIB

49 Input-Output Economic Tables

GDGD CKCK CK10CK10 TSTS SSSS

OO ptimaptimadevdev iationiation

0.15%0.15% 0.15%0.15% 0.02%0.02% 0.04%0.04% 0.01%0.01%

NNumber ofumber ofoptimaoptima

1111 1111 2727 3333 4242

RR un timeun time(sec(sec ondsonds ))

0.010.01 0.100.10 1.061.06 0.490.49 2.352.35

Another Example Another Example


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pp

A commercial SS implementation A commercial SS implementationOptQuest Callable Library (by OptTek)As other context-independent methodsseparates the method and the evaluation.

OptimizationProcedure

Input

Output

SystemEvaluator

O QO tQ t b db d A li iA li ti


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OptQuestOptQuest basedbased Applications Applications

Solution Generator

Solution EvaluatorUser-writtenApplication

OptQuestCallable Library

System

Evaluator

FeasibilityFeasibility andand EvaluationEvaluation


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FeasibilityFeasibility andand EvaluationEvaluation

User Implementation

ConstraintMapping Complex SystemEvaluator PenaltyFunction x x*

F( x*)

G( x*)P( x*)

Returns toOptQuest

The OptQuest engine

generates a new solution

ComparisonComparison withwith GenocopGenocop


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ComparisonComparison withwith GenocopGenocopTestTest onon 2828 hardhard nonlinear nonlinear instancesinstances

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

1.0E+10

1.0E+11

1.0E+12

1.0E+13

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Evaluations

A v e r a g e o b

j e c t

i v e

f u n c

t i o n v a

l u e

( L o g a r

i t h m i c s c a

l e )

GenocopOCL

1.0E+03

1.0E+04

1.0E+05

1.0E+06

1.0E+07

1.0E+08

1.0E+09

1.0E+10

1.0E+11

1.0E+12

1.0E+13

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Evaluations

A v e r a g e o b

j e c t

i v e

f u n c

t i o n v a

l u e

( L o g a r

i t h m i c s c a

l e )

GenocopOCL

ConclusionsConclusions


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ConclusionsConclusions

The development of metaheuristics usuallyThe development of metaheuristics usually

entails a fair amount of experimentationentails a fair amount of experimentation(skill comes from practice).(skill comes from practice).Code objectives:Code objectives:

Quick StartQuick StartBenchmarkBenchmark

Advanced Designs Advanced DesignsScatter Search provides a flexibleScatter Search provides a flexible

framework to develop solving methodsframework to develop solving methods ..

CallCall forfor PapersPapers


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CallCall for for PapersPapers

European Journal of Operational ResearchEuropean Journal of Operational Research

Feature Issue onFeature Issue on

SCATTER SEARCH METHODSSCATTER SEARCH METHODSFOR OPTIMIZATIONFOR OPTIMIZATION

Deadline for submissions: June 30, 2003Deadline for submissions: June 30, 2003

http://http:// www.uv.es/~rmarti/ejor.htmlwww.uv.es/~rmarti/ejor.html

QuestionsQuestions


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QuestionsQuestions

scatter search methodology and applications

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