an order effect of neighborhood structures in...

Research ArticleAn Order Effect of Neighborhood Structures in VariableNeighborhood Search Algorithm for Minimizing the Makespanin an Identical Parallel Machine Scheduling

Ibrahim Alharkan1 Khaled Bamatraf1 Mohammed A Noman 1 Husam Kaid 1

Emad S Abouel Nasr12 and Abdulaziz M El-Tamimi1

1 Industrial Engineering Department College of Engineering King Saud University PO Box 800 Riyadh 11421 Saudi Arabia2Faculty of Engineering Mechanical Engineering Department Helwan University Cairo 11732 Egypt

Correspondence should be addressed to Mohammed A Noman mmohammed1ksuedusa

Received 11 September 2017 Revised 6 February 2018 Accepted 27 February 2018 Published 22 April 2018

Academic Editor Ton D Do

Copyright copy 2018 Ibrahim Alharkan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Variable neighborhood search (VNS) algorithm is proposed for scheduling identical parallel machine The objective is to studythe effect of adding a new neighborhood structure and changing the order of the neighborhood structures on minimizing themakespan To enhance the quality of the final solution a machine based encoding method and five neighborhood structures areused inVNS Two initial solutionmethodswhichwere used in two versions of improvedVNS (IVNS) are employed namely longestprocessing time (LPT) initial solution denoted as HIVNS and random initial solution denoted as RIVNSThe proposed versionsare compared with LPT simulated annealing (SA) genetic algorithm (GA) modified variable neighborhood search (MVNS) andimproved variable neighborhood search (IVNS) algorithms from the literature Computational results show that changing the orderof neighborhood structures and adding a new neighborhood structure can yield a better solution in terms of average makespan

1 Introduction

Identical parallel machine scheduling (IPMS) with the objec-tive of minimizing the makespan is one of the combinationaloptimization problems It is known to be NP-hard by Gareyand Johnson [1] since it does not have a polynomial timealgorithm Exact algorithms such as branch and bound [2]and cutting plane algorithms [3] solve this type of IPM andfind optimal solution for small size instances As the problemsize increases the exact algorithms are inefficient and takemuch time to get a solution

That disadvantages bring a need for heuristics and meta-heuristics that give optimal or near optimal solution withina reasonable amount of time Longest Processing Time Rule(LPT) proposed by Mokotoff [4] is the first heuristic appliedin IPMS which has a tight worst case performance of boundof 43ndash13119898 where 119898 is the number of parallel machinesLPT is based on distributing jobs on machines according tomaximum processing time and the remaining jobs go one by

one to the least loaded machine until assigning all the jobs tothe machinesThe LPT heuristic performs well for makespancriteria but the solution obtained is often local optima LaterCoffman et al [5] proposed MULTIFIT algorithm that isbased on techniques from bin-packing Blackstone Jr andPhillips [6] proposed a simple heuristic for improving LPTsequence by exchange jobs between processors to reducemakespan Lee and Massey [7] combine two heuristics LPTand MULTIFIT to form a new one The heuristic uses LPTheuristic as an initial solution for the MULTIFIT heuristicTheperformance of the combined heuristic is better than LPTand the error bound is not worse than theMULTIFIT Yue [8]proved the bound for MULTIFIT to be 1311 Lee and Massey[9] extend the MULTIFIT algorithm and show that the errorbound of implementing the algorithm is only 110 Gareyand Johnson [1] proposed that 3-phase composite heuristicconsists of constructive phase and two improvement phaseswith no preliminary sort of processing times They showedthat their proposed heuristic is quicker than LPT Ho and

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 3586731 8 pageshttpsdoiorg10115520183586731

2 Mathematical Problems in Engineering

Wong [10] introduce Two-Machine Optimal Schedulingwhich uses lexicographic searchTheir method performs bet-ter that LPTMULTIFIT andMULTIFIT extension algorithmand it takes less amount of CPU time than MULTIFIT andMULTIFIT extension algorithms

Riera et al [11] proposed two approximate algorithmsthat use LPT as an initial solution and compare themwith dynamic programming and MULTIFIT algorithmsAlgorithm 1 uses exchange between two jobs to improvethe makespan Algorithm 2 schedules a job such that thecompletion time and process time of the selected job arenear the bound Their second algorithm is compared withMULTIFIT algorithms and results showed similarity to theMULTIFIT algorithm but their algorithm reduces CPUtime with respect to MULTIFIT heuristic Cheng and Gen[12] applied memetic algorithm to minimize the maximumweighted absolute lateness on PMS and showed that it out-performs genetic algorithm and the conventional heuristicsGhomi and Ghazvini [13] proposed a pairwise interchangealgorithm and it gave near optimal solution in a short periodof timeMin andCheng [14] proposed a genetic algorithmGAusing machine code They showed that GA outperforms LPTand SA and is suitable for large scale IPMS problems Guptaand Ruiz-Torres [15] proposed a LISTFIT heuristic basedon bin-backing and list scheduling The LISTFIT generatean optimal or near optimal solution and outperforms LPTMULTIFIT and COMBINE heuristics Costa et al [16]proposed algorithm inspired by the immune systems ofvertebrate animals Lee et al [17] proposed a simulatedannealing (SA) approach for makespan minimization onIPMS It chooses LPT as an initial solution Computationalresults showed that the SAheuristic outperforms the LISTFITand pairwise interchange (PI) algorithms Moreover it isefficient for large scale problems Tang and Luo [18] proposea new ILS algorithm combining with a variable number ofcyclic exchanges Experiments show that the algorithm isefficient for 119875119898 119862max Akyol and Bayhan [19] proposeda dynamical neural network that employs parameters oftime varying penalty The simulation results showed thatthe proposed algorithm generated feasible solutions and itfound better makespan when compared to LPT Kashan andKarimi [20] presented discrete particle swarm optimization(DPSO) algorithm for makespan minimization Computa-tional results showed that hybridized DPSO (HDPSO) algo-rithmoutperforms both SA andDPSO algorithms Sevkli andUysal [21] proposed modified variable neighborhood search(MVNS) which is based on exchange and move neighbor-hood structures Computational results demonstrated thatthe proposed algorithm outperforms both GA and LPTalgorithms Min and Cheng [14] proposed a harmony search(HS) algorithmwith dynamic subpopulation (DSHS) Resultsshow that DSHS algorithm outperforms SA and HDPSOfor many instances Moreover the execution time is lessthan 1 sec for all computations Chen et al [22] proposeddiscrete harmony search (DHS) algorithm that uses discreteencoding scheme to initialize the harmony memory (HM)then the improvisation scheme for generating new harmonyis redefined for suitability for solving the combinational opti-mization problem In addition the study made hybridizing a

local search method with DHS to increase the speed of localsearch Computational results show that theDHS algorithm isvery competitive when compared with other heuristics in theliterature Jing and Jun-qing [23] proposed efficient variableneighborhood search that uses four neighborhood structuresand has two versions One version uses LPT sequence as aninitial solution The other version uses random sequence asan initial solution A computational result demonstrates thatEVNS is efficient in searching global or near global optimumM Sevkli and A Z Sevkli [24] proposed stochasticallyperturbed particle swarm optimization algorithm (SPPSO)The algorithm compared two recent PSO algorithms Itis concluded that SPPSO algorithm has produced betterresults than DPSO and PSOspv in terms of the optimalsolutions number Laha [25] proposed an improved simulatedannealing (SA) heuristic Computational results show thatthe proposed heuristic is better than that produced by thebest-known heuristic in the literature Other advantages ofit are the ease of implementation In this paper the proposedalgorithm of Jing and Jun-qing [23] in their paper ldquoefficientvariable neighborhood search for identical parallel machinesschedulingrdquo is used with some changes on it One of thechanges is changing in the order of the neighborhood struc-tures and the other change is adding another neighborhoodstructure to get five neighborhood structures in our proposedalgorithm

The remaining sections of this paper are organized asfollows In Section 2 a brief description of IPMS problemis mentioned In Section 3 the steps of proposed algorithmare described in detail and the neighborhood structuresof this proposed algorithm are explained In Section 4computational results are discussed Conclusion is made inSection 5

2 Problem Description

The identical parallel machine scheduling (IPMS) problemcan be described as follows

A set 119899 of an independent jobs 119869 = 1198691 1198692 119869119899 tobe processed on 119898 identical parallel machines 119872 = 11987211198722 119872119898with the processing time of job 119894on any identicalmachine is given by 119901119894

A job can only be processed on one machine simultane-ously and a machine cannot process more than one job ata time Priority and precedence constraints are not allowedThere is no job cancellation and a job completes its processingon a machine without interruption

The objective is to minimize the total completion timeldquothe makespanrdquo of scheduling jobs on the machines

This scheduling problem can be described by a triple 120572 |120573 | 120574 as follows

119875119898 119862max (1)

where 119875 indicates parallel machine environment119898 indicatesnumber of machines 120573 indicates no constraints in thisproblem and 119862max indicates that the objective is to minimizethe makespan

Mathematical Problems in Engineering 3

1st version

LPT initial solutionHIVNS1

Random initial solutionRIVNS1



2nd version

Figure 1 The two versions of the proposed algorithms

This problem is interesting because minimizing themakespan has the effect of balancing the load over the variousmachines which is an important goal in practice

3 Development of the Proposed(IVNS) Algorithm

31 Basic VNS Variable neighborhood search (VNS) is ametaheuristic proposed by Mladenovic and Hansen [26] toenhance the solution quality by systematic neighborhoodschanges The main VNS algorithm steps can be summarizedas follows Initialization choose the neighborhood structuresset (NK1015840) 119896 = 1 2 1198961015840max obtain an initial solution andselect a stopping condition Repeat the next steps until thestopping condition is satisfied

(1) Set 119896 larr 1(2) Repeat the following steps until 119896 = 119896max

(a) Shaking generate a point 1199091015840 at random from the119896th neighborhood of 119909 (1199091015840 isin 119873119896(119909))

(b) Local search apply some local search methodwith 1199091015840 as initial solution denote with 11990910158401015840 the soobtained local optimum

(c) Move or not if the local optimum 11990910158401015840 is betterthan the incumbent 119909 move there (119909 larr 11990910158401015840)and continue the search with 1198731 (119896 larr 1)otherwise set 119896 larr 119896 + 1 improved variableneighborhood search (IVNS) algorithm

As we mentioned earlier the proposed algorithm is anaddition of the proposed algorithm of Jing and Jun-qing [23]

The proposed algorithms have two versions and twotypes for each version as shown in Figure 1 In the firstversion a new neighborhood structure was added to thefour neighborhood structures which are proposed by Jingand Jun-qing [23] while in the second version the order ofthese neighborhood structures was changed Both versionsuse LPT [20] and random initial solutions and are referredto as ldquoHIVNSrdquo and ldquoRIVNSrdquo respectively All these versionsof the proposed algorithm use the same five neighborhoodstructures These neighborhood structures will be discussedin the following section

32 Neighborhood Structures Determining the neighbor-hood structures is critical in the VNS algorithm To enhance

the local searching abilities five different kinds of neigh-borhoods are utilized to find better solutions on a givenschedule in the proposed algorithm which are designedbased on such an idea that a given solution can be improvedby moving or swapping jobs between the problem machines(themachines with their finished time equal to themakespanof the solution) and any other nonproblem machines (themachines with their finished time less than the makespan ofthe solution)

The five neighborhood structures are illustrated as fol-lows

(1) Move move a job 119869119894 from 119872119901 to 119872119899119901 if condition(119862119872119901 minus 119862119872119899119901 gt 119901119894) is satisfied

(2) Exchange 1 exchange a job 119869119894 selected from119872119901 withanother job 119869119895 selected from119872119899119901 if (119901119894 minus 119901119895 gt 0) and(119862119872119901 minus 119862119872119899119901 gt 119901119894 minus 119901119895)

(3) Exchange 2 exchange two jobs 119869119894 and 119869119895 selectedfrom119872119901 with one job 119869119896 selected from119872119899119901 if (119901119894 +119901119895 minus 119901119896 gt 0) and (119862119872119901 minus 119862119872119899119901 gt 119901119894 + 119901119895 minus 119901119896)

(4) Exchange 3 exchange two jobs 119869119894 and 119869119895 selectedfrom119872119901 with two jobs 119869119896 and 119869119905 selected from119872119899119901if (119901119894 + 119901119895 minus (119901119896 + 119901119905) gt 0) and (119862119872119901 minus 119862119872119899119901 gt119901119894 + 119901119895 minus (119901119896 + 119901119905))

(5) Exchange 4 exchange one job 119869119894 selected from 119872119901with two jobs 119869119894 and 119869119896 selected from 119872119899119901 if (119901119894 minus(119901119895 + 119901119896) gt 0) and (119862119872119901 minus 119862119872119899119901 gt 119901119894 minus (119901119895 + 119901119896))

The orders assigned to the types proposed of the algo-rithm are as follows

(1) The order of ldquoHIVNS1rdquo and ldquoRIVNS1rdquo is ldquomoveexchange 1 exchange 2 exchange 3 and exchange 4rdquo

(2) The order of ldquoHIVNS2rdquo and ldquoRIVNS2rdquo is ldquoexchange3 exchange 1 move exchange 2 and exchange4rdquo Improved VNS (IVNS) flow chart is shown asFigure 2

33 Steps of IVNS The steps of IVNS for ldquoHIVNS1rdquo andldquoRIVNS1rdquo are shown as follows

Step 1 Generate initial schedule 119883 (generated randomly orobtain from the LPT rule) MaxIterNum = 100 119894 = 0 and119896max = 5

Step 2 Compute lower bound LB(119862max) = max[(1119898)sum119899119894=1 119901119894]max119894119901119894


No

Shake procedure get a new start

Yes

Yes

Neighborhood search

NoNo

Yes

Initialization set initial solution x0 max iteration

Output x0

point x

k = 1

Local search get a solution

k = 1

x = x

k = k + 1

if k gt kＧＲ

x isin Nk(x)

i = i + 1

then x0 = x

num (MaxIterNum) i = 0

i lt MaxIterNum

if (f(x) lt f(x0))

If (f(x) lt f(x))

Figure 2 Flow chart of the basic VNS algorithm

Step 3 Repeat until 119891(119883) (makespan of 119883) is equal to LB or119894 gtMaxIterNum

Step 31 For schedule119883 distinguish the problemmachine set(119878pm) and the nonproblem machine set (119878npm)

Step 32 For each machine119872119901 in 119878pm do

Step 321 For each machine119872119899119901 in 119878npm do

Step 3211 Set 119870 = 1 finish = false

Step 3212Repeat

Switch (119870) 119870 = 1 1198831015840 = Move(119872119901119872119899119901 119883) break119870 = 2 1198831015840 = Exchange 1(119872119901119872119899119901 119883)break

119870 = 3 1198831015840 = Exchange 2(119872119901119872119899119901 119883)beak119870 = 4 1198831015840 = Exchange 3(119872119901119872119899119901 119883)break119870 = 5 1198831015840 = Exchange 4(119872119901119872119899119901 119883)if (119891(1198831015840) lt 119891(119883)) then set119883 = 1198831015840 finish =true go to Step 3 otherwise set 119896 = 119896 + 1

until 119896 = 119896max

Step 33 If finish = false then 119894 = 119894 + 1

Step 4 Output the best solution119883 found so far

The steps of IVNS for ldquoHIVNS2rdquo and ldquoRIVNS2rdquo areshown as follows



Table 1 Number of machines and jobs

Number of machines 2 5 10 20Number of jobs 20 50 100 200 20 50 100 200 20 50 100 200 50 100 200







Step 3212

Repeat

Switch (119870) 119870 = 1 1198831015840 = Exchange 3(119872119901119872119899119901 119883)break119870 = 2 1198831015840 = Exchange 1(119872119901119872119899119901 119883)break119870 = 3 1198831015840 = Move(119872119901119872119899119901 119883) break119870 = 4 1198831015840 = Exchange 2(119872119901119872119899119901 119883)beak119870 = 5 1198831015840 = Exchange 4(119872119901119872119899119901 119883)if (119891(1198831015840) lt 119891(119883)) then set119883 = 1198831015840 finish =true go to Step 3 otherwise set 119896 = 119896 + 1

until 119896 = 119896max



4 Computational Results and Comparison

In this section the results of two versions of the proposedalgorithm were compared with LPT [1] SA [17] GA [14]MVNS [21] and IVNS [23] algorithms from the literatureThe two versions of the improved variable neighborhoodsearch algorithms ldquoHIVNS1 and RIVNS1rdquo and ldquoHIVNS2 andRIVNS2rdquo were coded in MATLAB R2012a and executed oni5 CPU 5GHz with 6GB of RAM All of them were stoppedafter getting the lower bound or running for 100 iterations forRIVNS1 and RIVNS2 The number of machines and numberof jobs are shown in Table 1

The processing time of the jobs is the same as Jingand Jun-qing [23] As a result 15 instances were conductedand each instance was conducted with 10 generations ofdifferent processing times The total is 150 instances The

094098102106

11

AlgorithmsAverageLower boundMaximum average

mak

espa

n m

eans

Aver

age o

f

LPT SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2

Figure 3 Histogram of averages of makespan means

performance of the algorithms is measured with respectto the average makespan (mean) and average CPU time(Avg time in second) The ldquomeanrdquo performance is a relativequality measure of solutions computed by 119862LB where 119862is the average makespan obtained for each instance with 10generations given by the algorithm and LB is the lower boundof the instance calculated in equation that mentioned inSection 33 Step 2 The ldquoAvg timerdquo refers to the total time ittakes for the algorithm to finish the solution Table 2 presentsthe results of the previous algorithms from the literaturewhileTable 3 presents the results from the proposed algorithms Bycomparing the results of the average means of the makespanIt is obvious that the proposed algorithms outperform all thealgorithms in Table 2 from the literature It is worth notingthat for each instance the proposed algorithms obtain noworse than algorithms inTable 2 except at 10machines and 20jobs instance in only HIVNS algorithm and that is due to thedifficulty facing the proposed algorithms to get a lower boundwhen the difference between the number ofmachines and thenumber of jobs is relatively small In addition by comparingthe two proposed algorithms we can see that the versionshave the same average means of the makespan in case ofrandom initial solution while as the 2nd version outperformsthe 1st version in case of LPT initial solution in both averagemeans of makespan and average CPU time

Figure 3 shows the averages of makespan mean themaximum averages for each algorithm and the lower boundIt can be observed that the two proposed versions algorithmshave makespan means averages which closed to the lowerbound especially in RIVNS1 and RIVNS2 that have the sameaverage (10054)

Figure 4 shows the averages of Avg time means for allalgorithms We can see that RIVNS1 and RIVNS2 have thesmallest Avg time which are 0008 and 0007 respectivelyMoreover it can be observed that the Avg time of HIVNS1and HIVNS2 is much higher than HIVNS because in thispaper Matlab is used to construct the code of HIVNS1 and


Table2Th

emakespanresults

before

thec

hang

ingordero

fthe

neighb

orho

odstructures

[23]

Instance

number

119898119899

LPT

SAGA

MVNS

RIVNS

HIV

NS

Mean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

e1

2

2010

033

1000

610

149

1000

002215

1000

000106

1000

0000

0910

000

000

062

5010

001

1000

010

738

1000

005131

1000

000226

1000

000016

1000

0000

033

100

1000

010

000

19382

1000

027948

1000

0004

4510

000

00031

1000

0000

054

200

1000

010

000

09218

1000

023145

1000

000905

1000

000059

1000

0000

035

5

2010

315

10264

13614

10356

03412

10127

00124

10045

00083

10043

00073

650

10053

10045

25502

10312

06935

10050

00283

10012

00027

1000

900022

7100

10005

10005

42271

10242

13862

10052

00534

10002

000

4210

000

00015

8200

10003

10003

144302

10168

26560

10084

01064

10001

00075

1000

000035

9

10

2010

794

10792

19328

11336

03575

10987

00131

10799

004

0210

590

00384

1050

10207

10207

41428

11169

16303

10315

00315

10078

00075

10036

000

6711

100

10110

10110

2013

0111421

14547

10181

006

4710

031

000

6110

011

000

4312

200

10007

10007

10165

10611

33761

10127

01561

1000

400117

10002

00034

1320

5010

510

10510

24803

13167

09728

11671

004

4110

312

01389

10304

01392

14100

10123

10123

42799

12270

18111

10857

00905

10087

00179

1004

600087

15200

10063

10063

146684

11353

344

9210

415

01840

10039

00163

10024

00113

Average

10148

10142

507

7910

827

1598

210

319

006

3510

095

00182

1007

100152


Table 3 The makespan results after the effect of the changing order of the neighborhood structures

Instance number 119898 119899RIVNS1 HIVNS1 RIVNS2 HIVNS2

Mean Avg time Mean Avg time Mean Avg time Mean Avg time1

2

20 10000 00005 10000 00004 10000 00008 10000 000052 50 10000 00002 10000 00002 10000 00002 10000 000023 100 10000 00004 10000 00005 10000 00004 10000 000834 200 10000 00002 10000 00004 10000 00002 10000 070305

5

20 10005 00014 10000 00025 10005 00014 10000 000166 50 10000 00005 10000 00028 10000 00008 10000 001757 100 10000 00235 10000 01104 10000 00258 10000 017058 200 10000 00002 10000 25874 10000 00002 10000 250579

10

20 10625 00016 10682 00040 10625 00015 10602 0004010 50 10004 00103 10000 00156 10000 00064 10000 0058011 100 10000 00052 10000 01122 10000 00054 10000 0071312 200 10000 00238 10000 18051 10000 00207 10000 155613

2050 10175 00158 10256 00542 10175 00153 10263 00672

14 100 10003 00288 10000 02312 10003 00270 10000 0022215 200 10000 00004 10000 10538 10000 00004 10000 07414

Average 10054 00075 10063 03987 10054 00071 10058 03951

SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2

CPU time 5078 1598 0064 0018 0015 0008 0399 0007 0395

0000

1000

2000

3000

4000

5000

6000

Tim

e (se

cond

)

Figure 4 Histogram of averages of Avg time means

HIVNS2 and the authors of [23] used C++ to constructHIVNS Thus the benefits of Avg time offered by C++ faroutweigh the simplicity of Matlab Avg time is especiallyimportant when dealing with algorithms sincemany calcula-tions involve immense optimization with complex equationsand algorithms or calculations with a large number ofiterations As the amount of data increases the computationtime (Avg time) for Matlab code increases significantlythereforeMatlab code takesmore time for those calculationsfor example in Table 3 the cases of 119898 = 2 5 10 20 119899 = 200are need to large number of iterations C++ is theway to go foralgorithms calculations because of its speed and versatility

5 Conclusion

In this paper two versions of improved variable neighbor-hood search (IVNS) algorithms are proposed for scheduling

identical parallelmachines IPMwith the objective of studyingthe effect of adding a newneighborhood structure and chang-ing the order of neighborhood structures on minimizing themakespan119862max In the proposed algorithms amachine basedencoding method and five neighborhood structures are usedto enhance the quality of the final solution Computationalresults showed that the proposed algorithms outperform allthe algorithms in the literature and obtain no worse thanalgorithms except when the number of machines and thenumber of jobs are relatively small which is due to thedifficulty facing the proposed algorithms to get a lower boundin that case In addition we concluded that the second versionoutperforms the first version in case of LPT initial solutionand therefore changing the order of neighborhood structureshas an effect on minimizing the makespan Further researchis to implement the proposed algorithms in scheduling ofunrelated parallel machines

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors extend their appreciation to the Deanship ofScientific Research at King Saud University for funding thiswork through Research Group no RG-1439-009

References

[1] M R Gary and D S Johnson Computers and IntractabilityA Guide to the Theory of NP-completeness WH Freeman andCompany New York NY USA 1979


[2] R L Graham ldquoBounds on multiprocessing timing anomaliesrdquoSIAM Journal onAppliedMathematics vol 17 pp 416ndash429 1969

[3] S L van de Velde ldquoDuality-based algorithms for schedulingunrelated parallel machinesrdquo ORSA Journal on Computing vol5 no 2 pp 192ndash205 1993

[4] E Mokotoff ldquoAn exact algorithm for the identical parallelmachine scheduling problemrdquo European Journal of OperationalResearch vol 152 no 3 pp 758ndash769 2004

[5] J Coffman M R Garey and D S Johnson ldquoAn application ofbin-packing to multiprocessor schedulingrdquo SIAM Journal onComputing vol 7 no 1 pp 1ndash17 1978

[6] J H Blackstone Jr andD T Phillips ldquoAn improved heuristic forminimizing makespan among m identical parallel processorsrdquoComputers amp Industrial Engineering vol 5 no 4 pp 279ndash2871981

[7] C-Y Lee and J D Massey ldquoMultiprocessor scheduling com-bining LPT andMULTIFITrdquoDiscrete Applied Mathematics vol20 no 3 pp 233ndash242 1988

[8] M Y Yue ldquoOn the exact upper bound for the multifit processorscheduling algorithmrdquo Annals of Operations Research vol 24no 1ndash4 pp 233ndash259 1990

[9] C-Y Lee and J D Massey ldquoMultiprocessor scheduling Anextension of the MULTIFIT algorithmrdquo Journal of Manufactur-ing Systems vol 7 no 1 pp 25ndash32 1988

[10] J C Ho and J S Wong ldquoMakespanminimization for m parallelidentical processorsrdquo Naval Research Logistics (NRL) vol 42no 6 pp 935ndash948 1995

[11] J Riera D Alcaide and J Sicilia ldquoApproximate algorithms forthe 119875 119862max problemrdquo TOP vol 4 no 2 pp 345ndash359 1996

[12] R Cheng and M Gen ldquoParallel machine scheduling problemsusing memetic algorithmsrdquo Computers amp Industrial Engineer-ing vol 33 no 3-4 pp 761ndash764 1997

[13] S F Ghomi and F J Ghazvini ldquoA pairwise interchange algo-rithm for parallel machine schedulingrdquo Production Planning ampControl vol 9 pp 685ndash689 1998

[14] L Min and W Cheng ldquoA genetic algorithm for minimizing themakespan in the case of scheduling identical parallel machinesrdquoArtificial Intelligence in Engineering vol 13 no 4 pp 399ndash4031999

[15] J N D Gupta and J Ruiz-Torres ldquoA LISTFIT heuristic for min-imizing makespan on identical parallel machinesrdquo ProductionPlanning and Control vol 12 no 1 pp 28ndash36 2001

[16] A M Costa P A Vargas F J Von Zuben and P M FrancaldquoMakespan minimization on parallel processors An immune-based approachrdquo in Proceedings of the 2002 Congress on Evolu-tionary Computation (CEC rsquo02) pp 920ndash925 IEEE May 2002

[17] W-C Lee C-C Wu and P Chen ldquoA simulated annealingapproach to makespan minimization on identical parallelmachinesrdquo The International Journal of Advanced Manufactur-ing Technology vol 31 no 3-4 pp 328ndash334 2006

[18] L Tang and J Luo ldquoA new ILS algorithm for parallel machinescheduling problemsrdquo Journal of Intelligent Manufacturing vol17 no 5 pp 609ndash619 2006

[19] D E Akyol andGM Bayhan ldquoMinimizingmakespan on iden-tical parallel machines using neural networksrdquo in Proceedings ofthe International Conference on Neural Information Processingvol 4234 of Lecture Notes in Computer Science pp 553ndash562Springer 2006

[20] A H Kashan and B Karimi ldquoA discrete particle swarmoptimization algorithm for scheduling parallelmachinesrdquoCom-puters amp Industrial Engineering vol 56 no 1 pp 216ndash223 2009

[21] M Sevkli and H Uysal ldquoA modified variable neighbor-hood search for minimizing the makespan onidentical parallelmachinesrdquo in Proceedings of the 2009 International Conferenceon Computers and Industrial Engineering (CIE rsquo09) pp 108ndash111IEEE July 2009

[22] J Chen Q-K Pan and H Li ldquoHarmony search algorithmwith dynamic subpopulations for scheduling identical parallelmachinesrdquo in Proceedings of the 2010 6th International Confer-ence on Natural Computation (ICNC rsquo10) pp 2369ndash2373 IEEEAugust 2010

[23] C Jing and L Jun-qing ldquoEfficient variable neighborhood searchfor identical parallel machines schedulingrdquo in Proceedings of theControl Conference (CCC rsquo12) pp 7228ndash7232 IEEE 2012

[24] M Sevkli and A Z Sevkli A Stochastically Perturbed ParticleSwarm Optimization for Identical Parallel Machine SchedulingProblems INTECH Open Access Publisher 2012

[25] D Laha ldquoA simulated annealing heuristic for minimizingmakespan in parallel machine schedulingrdquo in Proceedings of theInternational Conference on Swarm Evolutionary and MemeticComputing pp 198ndash205 Springer 2012

[26] N Mladenovic and P Hansen ldquoVariable neighborhood searchrdquoComputers amp Operations Research vol 24 no 11 pp 1097ndash11001997

Hindawiwwwhindawicom Volume 2018

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Mathematical Problems in Engineering

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Wong [10] introduce Two-Machine Optimal Schedulingwhich uses lexicographic searchTheir method performs bet-ter that LPTMULTIFIT andMULTIFIT extension algorithmand it takes less amount of CPU time than MULTIFIT andMULTIFIT extension algorithms

Riera et al [11] proposed two approximate algorithmsthat use LPT as an initial solution and compare themwith dynamic programming and MULTIFIT algorithmsAlgorithm 1 uses exchange between two jobs to improvethe makespan Algorithm 2 schedules a job such that thecompletion time and process time of the selected job arenear the bound Their second algorithm is compared withMULTIFIT algorithms and results showed similarity to theMULTIFIT algorithm but their algorithm reduces CPUtime with respect to MULTIFIT heuristic Cheng and Gen[12] applied memetic algorithm to minimize the maximumweighted absolute lateness on PMS and showed that it out-performs genetic algorithm and the conventional heuristicsGhomi and Ghazvini [13] proposed a pairwise interchangealgorithm and it gave near optimal solution in a short periodof timeMin andCheng [14] proposed a genetic algorithmGAusing machine code They showed that GA outperforms LPTand SA and is suitable for large scale IPMS problems Guptaand Ruiz-Torres [15] proposed a LISTFIT heuristic basedon bin-backing and list scheduling The LISTFIT generatean optimal or near optimal solution and outperforms LPTMULTIFIT and COMBINE heuristics Costa et al [16]proposed algorithm inspired by the immune systems ofvertebrate animals Lee et al [17] proposed a simulatedannealing (SA) approach for makespan minimization onIPMS It chooses LPT as an initial solution Computationalresults showed that the SAheuristic outperforms the LISTFITand pairwise interchange (PI) algorithms Moreover it isefficient for large scale problems Tang and Luo [18] proposea new ILS algorithm combining with a variable number ofcyclic exchanges Experiments show that the algorithm isefficient for 119875119898 119862max Akyol and Bayhan [19] proposeda dynamical neural network that employs parameters oftime varying penalty The simulation results showed thatthe proposed algorithm generated feasible solutions and itfound better makespan when compared to LPT Kashan andKarimi [20] presented discrete particle swarm optimization(DPSO) algorithm for makespan minimization Computa-tional results showed that hybridized DPSO (HDPSO) algo-rithmoutperforms both SA andDPSO algorithms Sevkli andUysal [21] proposed modified variable neighborhood search(MVNS) which is based on exchange and move neighbor-hood structures Computational results demonstrated thatthe proposed algorithm outperforms both GA and LPTalgorithms Min and Cheng [14] proposed a harmony search(HS) algorithmwith dynamic subpopulation (DSHS) Resultsshow that DSHS algorithm outperforms SA and HDPSOfor many instances Moreover the execution time is lessthan 1 sec for all computations Chen et al [22] proposeddiscrete harmony search (DHS) algorithm that uses discreteencoding scheme to initialize the harmony memory (HM)then the improvisation scheme for generating new harmonyis redefined for suitability for solving the combinational opti-mization problem In addition the study made hybridizing a

local search method with DHS to increase the speed of localsearch Computational results show that theDHS algorithm isvery competitive when compared with other heuristics in theliterature Jing and Jun-qing [23] proposed efficient variableneighborhood search that uses four neighborhood structuresand has two versions One version uses LPT sequence as aninitial solution The other version uses random sequence asan initial solution A computational result demonstrates thatEVNS is efficient in searching global or near global optimumM Sevkli and A Z Sevkli [24] proposed stochasticallyperturbed particle swarm optimization algorithm (SPPSO)The algorithm compared two recent PSO algorithms Itis concluded that SPPSO algorithm has produced betterresults than DPSO and PSOspv in terms of the optimalsolutions number Laha [25] proposed an improved simulatedannealing (SA) heuristic Computational results show thatthe proposed heuristic is better than that produced by thebest-known heuristic in the literature Other advantages ofit are the ease of implementation In this paper the proposedalgorithm of Jing and Jun-qing [23] in their paper ldquoefficientvariable neighborhood search for identical parallel machinesschedulingrdquo is used with some changes on it One of thechanges is changing in the order of the neighborhood struc-tures and the other change is adding another neighborhoodstructure to get five neighborhood structures in our proposedalgorithm

The remaining sections of this paper are organized asfollows In Section 2 a brief description of IPMS problemis mentioned In Section 3 the steps of proposed algorithmare described in detail and the neighborhood structuresof this proposed algorithm are explained In Section 4computational results are discussed Conclusion is made inSection 5

2 Problem Description

The identical parallel machine scheduling (IPMS) problemcan be described as follows

A set 119899 of an independent jobs 119869 = 1198691 1198692 119869119899 tobe processed on 119898 identical parallel machines 119872 = 11987211198722 119872119898with the processing time of job 119894on any identicalmachine is given by 119901119894

A job can only be processed on one machine simultane-ously and a machine cannot process more than one job ata time Priority and precedence constraints are not allowedThere is no job cancellation and a job completes its processingon a machine without interruption

The objective is to minimize the total completion timeldquothe makespanrdquo of scheduling jobs on the machines

This scheduling problem can be described by a triple 120572 |120573 | 120574 as follows

119875119898 119862max (1)

where 119875 indicates parallel machine environment119898 indicatesnumber of machines 120573 indicates no constraints in thisproblem and 119862max indicates that the objective is to minimizethe makespan


1st version





2nd version


























No


Yes

Yes

Neighborhood search

NoNo

Yes


Output x0

point x

k = 1


k = 1

x = x

k = k + 1

if k gt kＧＲ

x isin Nk(x)

i = i + 1

then x0 = x


i lt MaxIterNum

if (f(x) lt f(x0))

If (f(x) lt f(x))







Step 3212Repeat



until 119896 = 119896max














Step 3212

Repeat


until 119896 = 119896max






094098102106

11


mak

espa

n m

eans

Aver

age o

f

LPT SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2






Table2Th

emakespanresults

before

thec

hang

ingordero

fthe

neighb

orho

odstructures

[23]

Instance

number

119898119899

LPT

SAGA

MVNS

RIVNS

HIV

NS

Mean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

e1

2

2010

033

1000

610

149

1000

002215

1000

000106

1000

0000

0910

000

000

062

5010

001

1000

010

738

1000

005131

1000

000226

1000

000016

1000

0000

033

100

1000

010

000

19382

1000

027948

1000

0004

4510

000

00031

1000

0000

054

200

1000

010

000

09218

1000

023145

1000

000905

1000

000059

1000

0000

035

5

2010

315

10264

13614

10356

03412

10127

00124

10045

00083

10043

00073

650

10053

10045

25502

10312

06935

10050

00283

10012

00027

1000

900022

7100

10005

10005

42271

10242

13862

10052

00534

10002

000

4210

000

00015

8200

10003

10003

144302

10168

26560

10084

01064

10001

00075

1000

000035

9

10

2010

794

10792

19328

11336

03575

10987

00131

10799

004

0210

590

00384

1050

10207

10207

41428

11169

16303

10315

00315

10078

00075

10036

000

6711

100

10110

10110

2013

0111421

14547

10181

006

4710

031

000

6110

011

000

4312

200

10007

10007

10165

10611

33761

10127

01561

1000

400117

10002

00034

1320

5010

510

10510

24803

13167

09728

11671

004

4110

312

01389

10304

01392

14100

10123

10123

42799

12270

18111

10857

00905

10087

00179

1004

600087

15200

10063

10063

146684

11353

344

9210

415

01840

10039

00163

10024

00113

Average

10148

10142

507

7910

827

1598

210

319

006

3510

095

00182

1007

100152





2

20 10000 00005 10000 00004 10000 00008 10000 000052 50 10000 00002 10000 00002 10000 00002 10000 000023 100 10000 00004 10000 00005 10000 00004 10000 000834 200 10000 00002 10000 00004 10000 00002 10000 070305

5

20 10005 00014 10000 00025 10005 00014 10000 000166 50 10000 00005 10000 00028 10000 00008 10000 001757 100 10000 00235 10000 01104 10000 00258 10000 017058 200 10000 00002 10000 25874 10000 00002 10000 250579

10

20 10625 00016 10682 00040 10625 00015 10602 0004010 50 10004 00103 10000 00156 10000 00064 10000 0058011 100 10000 00052 10000 01122 10000 00054 10000 0071312 200 10000 00238 10000 18051 10000 00207 10000 155613

2050 10175 00158 10256 00542 10175 00153 10263 00672

14 100 10003 00288 10000 02312 10003 00270 10000 0022215 200 10000 00004 10000 10538 10000 00004 10000 07414

Average 10054 00075 10063 03987 10054 00071 10058 03951

SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2

CPU time 5078 1598 0064 0018 0015 0008 0399 0007 0395

0000

1000

2000

3000

4000

5000

6000

Tim

e (se

cond

)



5 Conclusion





Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









1st version





2nd version


























No


Yes

Yes

Neighborhood search

NoNo

Yes


Output x0

point x

k = 1


k = 1

x = x

k = k + 1

if k gt kＧＲ

x isin Nk(x)

i = i + 1

then x0 = x


i lt MaxIterNum

if (f(x) lt f(x0))

If (f(x) lt f(x))







Step 3212Repeat



until 119896 = 119896max














Step 3212

Repeat


until 119896 = 119896max






094098102106

11


mak

espa

n m

eans

Aver

age o

f

LPT SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2






Table2Th

emakespanresults

before

thec

hang

ingordero

fthe

neighb

orho

odstructures

[23]

Instance

number

119898119899

LPT

SAGA

MVNS

RIVNS

HIV

NS

Mean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

e1

2

2010

033

1000

610

149

1000

002215

1000

000106

1000

0000

0910

000

000

062

5010

001

1000

010

738

1000

005131

1000

000226

1000

000016

1000

0000

033

100

1000

010

000

19382

1000

027948

1000

0004

4510

000

00031

1000

0000

054

200

1000

010

000

09218

1000

023145

1000

000905

1000

000059

1000

0000

035

5

2010

315

10264

13614

10356

03412

10127

00124

10045

00083

10043

00073

650

10053

10045

25502

10312

06935

10050

00283

10012

00027

1000

900022

7100

10005

10005

42271

10242

13862

10052

00534

10002

000

4210

000

00015

8200

10003

10003

144302

10168

26560

10084

01064

10001

00075

1000

000035

9

10

2010

794

10792

19328

11336

03575

10987

00131

10799

004

0210

590

00384

1050

10207

10207

41428

11169

16303

10315

00315

10078

00075

10036

000

6711

100

10110

10110

2013

0111421

14547

10181

006

4710

031

000

6110

011

000

4312

200

10007

10007

10165

10611

33761

10127

01561

1000

400117

10002

00034

1320

5010

510

10510

24803

13167

09728

11671

004

4110

312

01389

10304

01392

14100

10123

10123

42799

12270

18111

10857

00905

10087

00179

1004

600087

15200

10063

10063

146684

11353

344

9210

415

01840

10039

00163

10024

00113

Average

10148

10142

507

7910

827

1598

210

319

006

3510

095

00182

1007

100152





2

20 10000 00005 10000 00004 10000 00008 10000 000052 50 10000 00002 10000 00002 10000 00002 10000 000023 100 10000 00004 10000 00005 10000 00004 10000 000834 200 10000 00002 10000 00004 10000 00002 10000 070305

5

20 10005 00014 10000 00025 10005 00014 10000 000166 50 10000 00005 10000 00028 10000 00008 10000 001757 100 10000 00235 10000 01104 10000 00258 10000 017058 200 10000 00002 10000 25874 10000 00002 10000 250579

10

20 10625 00016 10682 00040 10625 00015 10602 0004010 50 10004 00103 10000 00156 10000 00064 10000 0058011 100 10000 00052 10000 01122 10000 00054 10000 0071312 200 10000 00238 10000 18051 10000 00207 10000 155613

2050 10175 00158 10256 00542 10175 00153 10263 00672

14 100 10003 00288 10000 02312 10003 00270 10000 0022215 200 10000 00004 10000 10538 10000 00004 10000 07414

Average 10054 00075 10063 03987 10054 00071 10058 03951

SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2

CPU time 5078 1598 0064 0018 0015 0008 0399 0007 0395

0000

1000

2000

3000

4000

5000

6000

Tim

e (se

cond

)



5 Conclusion





Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









No


Yes

Yes

Neighborhood search

NoNo

Yes


Output x0

point x

k = 1


k = 1

x = x

k = k + 1

if k gt kＧＲ

x isin Nk(x)

i = i + 1

then x0 = x


i lt MaxIterNum

if (f(x) lt f(x0))

If (f(x) lt f(x))







Step 3212Repeat



until 119896 = 119896max














Step 3212

Repeat


until 119896 = 119896max






094098102106

11


mak

espa

n m

eans

Aver

age o

f

LPT SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2






Table2Th

emakespanresults

before

thec

hang

ingordero

fthe

neighb

orho

odstructures

[23]

Instance

number

119898119899

LPT

SAGA

MVNS

RIVNS

HIV

NS

Mean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

e1

2

2010

033

1000

610

149

1000

002215

1000

000106

1000

0000

0910

000

000

062

5010

001

1000

010

738

1000

005131

1000

000226

1000

000016

1000

0000

033

100

1000

010

000

19382

1000

027948

1000

0004

4510

000

00031

1000

0000

054

200

1000

010

000

09218

1000

023145

1000

000905

1000

000059

1000

0000

035

5

2010

315

10264

13614

10356

03412

10127

00124

10045

00083

10043

00073

650

10053

10045

25502

10312

06935

10050

00283

10012

00027

1000

900022

7100

10005

10005

42271

10242

13862

10052

00534

10002

000

4210

000

00015

8200

10003

10003

144302

10168

26560

10084

01064

10001

00075

1000

000035

9

10

2010

794

10792

19328

11336

03575

10987

00131

10799

004

0210

590

00384

1050

10207

10207

41428

11169

16303

10315

00315

10078

00075

10036

000

6711

100

10110

10110

2013

0111421

14547

10181

006

4710

031

000

6110

011

000

4312

200

10007

10007

10165

10611

33761

10127

01561

1000

400117

10002

00034

1320

5010

510

10510

24803

13167

09728

11671

004

4110

312

01389

10304

01392

14100

10123

10123

42799

12270

18111

10857

00905

10087

00179

1004

600087

15200

10063

10063

146684

11353

344

9210

415

01840

10039

00163

10024

00113

Average

10148

10142

507

7910

827

1598

210

319

006

3510

095

00182

1007

100152





2

20 10000 00005 10000 00004 10000 00008 10000 000052 50 10000 00002 10000 00002 10000 00002 10000 000023 100 10000 00004 10000 00005 10000 00004 10000 000834 200 10000 00002 10000 00004 10000 00002 10000 070305

5

20 10005 00014 10000 00025 10005 00014 10000 000166 50 10000 00005 10000 00028 10000 00008 10000 001757 100 10000 00235 10000 01104 10000 00258 10000 017058 200 10000 00002 10000 25874 10000 00002 10000 250579

10

20 10625 00016 10682 00040 10625 00015 10602 0004010 50 10004 00103 10000 00156 10000 00064 10000 0058011 100 10000 00052 10000 01122 10000 00054 10000 0071312 200 10000 00238 10000 18051 10000 00207 10000 155613

2050 10175 00158 10256 00542 10175 00153 10263 00672

14 100 10003 00288 10000 02312 10003 00270 10000 0022215 200 10000 00004 10000 10538 10000 00004 10000 07414

Average 10054 00075 10063 03987 10054 00071 10058 03951

SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2

CPU time 5078 1598 0064 0018 0015 0008 0399 0007 0395

0000

1000

2000

3000

4000

5000

6000

Tim

e (se

cond

)



5 Conclusion





Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018

















Step 3212

Repeat


until 119896 = 119896max






094098102106

11


mak

espa

n m

eans

Aver

age o

f

LPT SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2






Table2Th

emakespanresults

before

thec

hang

ingordero

fthe

neighb

orho

odstructures

[23]

Instance

number

119898119899

LPT

SAGA

MVNS

RIVNS

HIV

NS

Mean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

e1

2

2010

033

1000

610

149

1000

002215

1000

000106

1000

0000

0910

000

000

062

5010

001

1000

010

738

1000

005131

1000

000226

1000

000016

1000

0000

033

100

1000

010

000

19382

1000

027948

1000

0004

4510

000

00031

1000

0000

054

200

1000

010

000

09218

1000

023145

1000

000905

1000

000059

1000

0000

035

5

2010

315

10264

13614

10356

03412

10127

00124

10045

00083

10043

00073

650

10053

10045

25502

10312

06935

10050

00283

10012

00027

1000

900022

7100

10005

10005

42271

10242

13862

10052

00534

10002

000

4210

000

00015

8200

10003

10003

144302

10168

26560

10084

01064

10001

00075

1000

000035

9

10

2010

794

10792

19328

11336

03575

10987

00131

10799

004

0210

590

00384

1050

10207

10207

41428

11169

16303

10315

00315

10078

00075

10036

000

6711

100

10110

10110

2013

0111421

14547

10181

006

4710

031

000

6110

011

000

4312

200

10007

10007

10165

10611

33761

10127

01561

1000

400117

10002

00034

1320

5010

510

10510

24803

13167

09728

11671

004

4110

312

01389

10304

01392

14100

10123

10123

42799

12270

18111

10857

00905

10087

00179

1004

600087

15200

10063

10063

146684

11353

344

9210

415

01840

10039

00163

10024

00113

Average

10148

10142

507

7910

827

1598

210

319

006

3510

095

00182

1007

100152





2

20 10000 00005 10000 00004 10000 00008 10000 000052 50 10000 00002 10000 00002 10000 00002 10000 000023 100 10000 00004 10000 00005 10000 00004 10000 000834 200 10000 00002 10000 00004 10000 00002 10000 070305

5

20 10005 00014 10000 00025 10005 00014 10000 000166 50 10000 00005 10000 00028 10000 00008 10000 001757 100 10000 00235 10000 01104 10000 00258 10000 017058 200 10000 00002 10000 25874 10000 00002 10000 250579

10

20 10625 00016 10682 00040 10625 00015 10602 0004010 50 10004 00103 10000 00156 10000 00064 10000 0058011 100 10000 00052 10000 01122 10000 00054 10000 0071312 200 10000 00238 10000 18051 10000 00207 10000 155613

2050 10175 00158 10256 00542 10175 00153 10263 00672

14 100 10003 00288 10000 02312 10003 00270 10000 0022215 200 10000 00004 10000 10538 10000 00004 10000 07414

Average 10054 00075 10063 03987 10054 00071 10058 03951

SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2

CPU time 5078 1598 0064 0018 0015 0008 0399 0007 0395

0000

1000

2000

3000

4000

5000

6000

Tim

e (se

cond

)



5 Conclusion





Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









Table2Th

emakespanresults

before

thec

hang

ingordero

fthe

neighb

orho

odstructures

[23]

Instance

number

119898119899

LPT

SAGA

MVNS

RIVNS

HIV

NS

Mean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

eMean

Avgtim

e1

2

2010

033

1000

610

149

1000

002215

1000

000106

1000

0000

0910

000

000

062

5010

001

1000

010

738

1000

005131

1000

000226

1000

000016

1000

0000

033

100

1000

010

000

19382

1000

027948

1000

0004

4510

000

00031

1000

0000

054

200

1000

010

000

09218

1000

023145

1000

000905

1000

000059

1000

0000

035

5

2010

315

10264

13614

10356

03412

10127

00124

10045

00083

10043

00073

650

10053

10045

25502

10312

06935

10050

00283

10012

00027

1000

900022

7100

10005

10005

42271

10242

13862

10052

00534

10002

000

4210

000

00015

8200

10003

10003

144302

10168

26560

10084

01064

10001

00075

1000

000035

9

10

2010

794

10792

19328

11336

03575

10987

00131

10799

004

0210

590

00384

1050

10207

10207

41428

11169

16303

10315

00315

10078

00075

10036

000

6711

100

10110

10110

2013

0111421

14547

10181

006

4710

031

000

6110

011

000

4312

200

10007

10007

10165

10611

33761

10127

01561

1000

400117

10002

00034

1320

5010

510

10510

24803

13167

09728

11671

004

4110

312

01389

10304

01392

14100

10123

10123

42799

12270

18111

10857

00905

10087

00179

1004

600087

15200

10063

10063

146684

11353

344

9210

415

01840

10039

00163

10024

00113

Average

10148

10142

507

7910

827

1598

210

319

006

3510

095

00182

1007

100152





2

20 10000 00005 10000 00004 10000 00008 10000 000052 50 10000 00002 10000 00002 10000 00002 10000 000023 100 10000 00004 10000 00005 10000 00004 10000 000834 200 10000 00002 10000 00004 10000 00002 10000 070305

5

20 10005 00014 10000 00025 10005 00014 10000 000166 50 10000 00005 10000 00028 10000 00008 10000 001757 100 10000 00235 10000 01104 10000 00258 10000 017058 200 10000 00002 10000 25874 10000 00002 10000 250579

10

20 10625 00016 10682 00040 10625 00015 10602 0004010 50 10004 00103 10000 00156 10000 00064 10000 0058011 100 10000 00052 10000 01122 10000 00054 10000 0071312 200 10000 00238 10000 18051 10000 00207 10000 155613

2050 10175 00158 10256 00542 10175 00153 10263 00672

14 100 10003 00288 10000 02312 10003 00270 10000 0022215 200 10000 00004 10000 10538 10000 00004 10000 07414

Average 10054 00075 10063 03987 10054 00071 10058 03951

SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2

CPU time 5078 1598 0064 0018 0015 0008 0399 0007 0395

0000

1000

2000

3000

4000

5000

6000

Tim

e (se

cond

)



5 Conclusion





Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018












2

20 10000 00005 10000 00004 10000 00008 10000 000052 50 10000 00002 10000 00002 10000 00002 10000 000023 100 10000 00004 10000 00005 10000 00004 10000 000834 200 10000 00002 10000 00004 10000 00002 10000 070305

5

20 10005 00014 10000 00025 10005 00014 10000 000166 50 10000 00005 10000 00028 10000 00008 10000 001757 100 10000 00235 10000 01104 10000 00258 10000 017058 200 10000 00002 10000 25874 10000 00002 10000 250579

10

20 10625 00016 10682 00040 10625 00015 10602 0004010 50 10004 00103 10000 00156 10000 00064 10000 0058011 100 10000 00052 10000 01122 10000 00054 10000 0071312 200 10000 00238 10000 18051 10000 00207 10000 155613

2050 10175 00158 10256 00542 10175 00153 10263 00672

14 100 10003 00288 10000 02312 10003 00270 10000 0022215 200 10000 00004 10000 10538 10000 00004 10000 07414

Average 10054 00075 10063 03987 10054 00071 10058 03951

SA GA

MV

NS

RIV

NS

HIV

NS

RIV

NS1

HIV

NS1

RIV

NS2

HIV

NS2

CPU time 5078 1598 0064 0018 0015 0008 0399 0007 0395

0000

1000

2000

3000

4000

5000

6000

Tim

e (se

cond

)



5 Conclusion





Acknowledgments


References



































Journal of












Journal of







Volume 2018






Volume 2018









































Journal of












Journal of







Volume 2018






Volume 2018








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