A simplified binary harmony search algorithm for large scale 0–1 knapsack problems

19 Pages·2015·1.388 MB·English

by  Xiangyong Kong, Liqun Gao, Haibin Ouyang, Steven Li

ExpertSystemswithApplicationsxxx(2015)xxx–xxx
ContentslistsavailableatScienceDirect
Expert Systems with Applications
journal homepage: www.elsevier.com/locate/eswa
A simplified binary harmony search algorithm for large scale 0–1
knapsack problems
Xiangyong Konga,⇑, Liqun Gaoa, Haibin Ouyanga, Steven Lib
aSchoolofInformationScienceandEngineering,NortheasternUniversity,Shenyang110004,China
bGraduateSchoolofBusinessandLaw,RMITUniversity,Melbourne3000,Australia
a r t i c l e i n f o a b s t r a c t
Articlehistory: As an important subset of combinatorial optimization, 0–1 knapsack problems, especially the high-
Availableonlinexxxx dimensionalones,areoftendifficulttosolve.Thisstudyaimstoprovideanewsimplifiedbinaryharmony
search(SBHS)algorithmtotacklesuchNP-hardproblemsarisingindiverseresearchfields.Thekeydif-
Keywords: ferencebetweenSBHSandotherHSmethodsisintheprocessofimprovisation.Thedifferencesamong
Harmonysearch harmoniesstoredinharmonymemoryratherthanthepitchadjustmentrate(PAR)andstepbandwidth
Simplifiedbinaryharmonysearch (bw)areemployedtoproducenewsolutionsandthiscangreatlyalleviatetheburdenofsettingthese
0–1knapsackproblems
importantfactorsmanually.Moreover,theharmonymemoryconsideringrate(HMCR)isdynamically
Largescale
adjustedintermsofthedimensionsizetoimproveconvergenceofthealgorithm.Therefore,theproposed
Ingeniousimprovisationscheme
method does not require any tedious process of proper parameter setting. To further enhance the
populationdiversity,aspecificheuristicbasedlocalsearcharoundinfeasiblesolutionsiscarriedoutto
obtainbetterqualitysolutions.Asetof10lowdimensionalknapsackproblemsaswellaslargescale
instances with up to 10,000 items are used to test the effectiveness of the proposed algorithm.
Extensive comparisons are made with the most well-known state-of-the-art HS methods including 9
continuous versions and 5 binary-coded variants. The results reveal that the proposed algorithm can
obtain better solutions in almost all cases and outperforms the other considered HS methods with
statisticalsignificance,especiallyforthelargescaleproblems.
(cid:2)2015ElsevierLtd.Allrightsreserved.
1.Introduction wherenisthenumberofitems.Eachitemipossessesaprofitvalue
p andavolumevaluev.V denotesthevolumecapacityofthe
i i max
Combinatorial optimizationis a mathematical optimization or knapsack. x represents the state of the item i and is restricted to
i
feasibility program to find an optimal object from a finite set of either 0 or 1. If the item i is put into the knapsack, x is set to 1,
i
objects.Amongthem,0–1knapsackproblemisthemostrepresen- otherwise, 0. Each item may be chosen at most once and cannot
tative subset and it involves important applications in various beplacedintheknapsackpartly.
fields, including factory location problem, production scheduling Ingeneral,the0–1knapsackproblemisaselectionprocessof
problem,assignmentproblemandreliabilityproblem.Thusithas items to fulfill a knapsack with some limits. The objective is to
attracted a great deal of attention and been extensively studied maximizethecumulativeprofitsoftheitemspackedintheknap-
inthelastfewdecades.Mathematically,the0–1knapsackproblem sack under the condition that the corresponding total volume is
initsstandardformcanbeexpressedas: less than or equal to a given volume capacity. The 0–1 knapsack
problemsareusuallynon-differentiable,discontinuous,unsmooth
Xn
Max fðxÞ¼ p (cid:2)x andhighlynonlinearNP-hardproblemswithplentyoflocaloptima
i i
i¼1 andcomplexconstraints.Sincethedecisionvariablesarerestricted
s:t: Xn v (cid:2)x 6V tobeeither0or1,thevariablespaceiscomposedofasetoffinite
i i max discrete points in which the global optimal solution is located.
i¼1 Therefore the most direct method is to exhaustively enumerate
x 2f0;1g; i¼1;2;...n
i allthesolutionsandselectafeasibleonewiththegreatestprofit
as the global optimum. However, it is not feasible in reality as
⇑ Correspondingauthor.Tel.:[phone]. the number of items becomes larger and larger. Meanwhile, the
E-mail addresses: [email protected] (X. Kong), [email protected] traditional methods, such as dynamic programming approach
(L.Gao),[email protected](H.Ouyang),[email protected](S.Li).
http://dx.doi.org/10.1016/j.eswa.2015.02.015
0957-4174/(cid:2)2015ElsevierLtd.Allrightsreserved.
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015
2 X.Kongetal./ExpertSystemswithApplicationsxxx(2015)xxx–xxx
(Brotcorne,Hanafi,&Mansi,2009)andbranchandboundapproach He,&Zhang,2014a)dynamicallyincreasedHMCRwithincreasing
(Fukunaga, 2011), suffer from the same problem as well. To generations while Kumar, Chhabra, and Kumar (2014) entailed
circumventtheaboveproblem,moreandmoreresearchersfocus exponential changes during the process of improvisation for
their attention on meta-heuristic algorithms that imitate specific HMCRtogettheglobaloptimalsolution.Thepitchadjustingrate
naturalphenomenon. (PAR) determines whether the pitch adjustment is employed on
Inthelastfewdecades,avarietyofmeta-heuristicoptimization thenewcandidateharmonyandthishassubstantialinfluenceon
algorithms have been developed, such as genetic algorithm (GA), thequalityoffinalsolution.TheoptimizationabilityofHSpartly
antcolonyoptimization(ACO),simulatedannealing(SA),particle reliesontheparametersettingofPARanditisimportanttochoose
swarm optimization (PSO), differential evolution (DE) etc. They anappropriatevalueforPAR.Unfortunately,thereisnoagreement
can randomly search in the variable space under certain rules reached on the best choice of PAR. Several studies are even con-
regardless of the characteristic of the problems to be solved. flicting with each other on the best choice of PAR, for example,
Unlike numerical methods, the objective function is not required the linear increment (Contreras et al., 2014; Xiang et al., 2014a;
to be differentiable or even continuous. Thus the meta-heuristic Yuan, Zhao, Yang, & Wang, 2014), the linear decrease (Mahdavi,
algorithms may be applied to solve all kinds of optimization Fesanghary, & Damangir, 2007; Yadav, Kumar, Panda, & Chang,
problems. 2012),exponentialincrement(Chen,Pan,&Li,2012)andexponen-
Among these meta-heuristic algorithms, the harmony search tialdecrease(Kumaretal.,2014)etc.Moreover,thepitchadjust-
(HS) algorithm (Geem, Kim, & Loganathan, 2001), a simple but ment step (bw) has a direct impact on the performance of HS as
powerful stochastic search technique inspired by the musician it controls the balance between the capabilities of exploration
attuning,isworthmentioning.HSimitatestheimprovisationpro- andexploitation.Theadjustmentofbwisverydifficultsinceitis
cesssuchasrockmusictofindaperfectpleasingharmonyfroman closely related to not only the search process, but the problem
aestheticpointofview.Itissimilartothesearchprocessoftheglo- beingresolvedaswell.bwshouldbelargeatearlierstageinfavor
baloptimuminoptimizationevaluatedbyanobjectivefunction.In of the global search throughout the entire space and smaller bw
particular, a musical harmony in HS can be viewed as a variable valuesarebeneficialtothelocalsearcharoundthepromisingarea
vectorandthebestharmonyachievedintheendisanalogousto to improve the accuracy as the search proceeds. IHS (Mahdavi
theglobaloptimalsolution. et al., 2007), PCOAHS (Yuan et al., 2014) and GDHS (Khalili,
ThecharacteristicsandadvantagesofHSwithrespecttoother Kharrat, Salahshoor, & Sefat, 2014) decreased bw exponentially
well-known meta-heuristics, such as GA, PSO and DE, have been with increasing generations. Similarly, Pan, Suganthan,
discussed in Geem et al. (2001); Hasançebi, Erdal, and Saka Tasgetiren, and Liang (2010) followed a linear decrease form of
(2009); Kulluk, Ozbakir, and Baykasoglu (2012). They can be bwinthefirsthalfofgenerationsandkeptitconstantintherest
summarized in following 4 aspects: (1) Each harmony vector in of generations. As known, bw is problem-dependent and should
harmonymemorymayparticipateinproducingnewsolutionvec- beupdateddynamicallyfordifferentproblemswithdifferentfea-
tors in HS, which enhances the flexibility and helps to generate tures. However, the above mentioned adaptive schemes ignored
higher quality solutions. While only two selected individuals are this key point. To overcome it, Das, Mukhopadhyay, Roy,
consideredastheparentvectors inGA, PSOupdatestheposition Abraham, and Panigrahi (2011) analyzed firstly the evolution of
ofeachparticlebysimplymovingtowarditspersonalbestlocation theexplorative searchbehavior ofHS and recomputedthe band-
andthefittestpositionvisitedbytheentireswarmbynow.Asfor widthbwforeachiterationproportionallytothestandarddevia-
anoffspringvector,themostrespectivemutationoperatorforDEis tion of current harmonies. Based on the above theoretical
carriedoutwiththreedistinctindividualsrandomlyselectedinthe analysis, Kattan and Abdullah (2013) employed another dynamic
wholepopulation.(2)UnlikePSOandDEwhichadjustthevariable bandwidth adjustment method for the pitch adjustment process.
vectorinonefixedrule,eachdecisionvariablevalueisdetermined Thebandwidthvalueswerealsocomputedbycalculatingthestan-
independentlyintheimprovisationofHS.(3)GA,PSOandDEpro- dard deviation of the respective HM column whereas the
ducemultiplesolutionssimultaneouslyinoneevolutioniteration, improvisation acceptance rate percentage was introduced to fine
whereasHSonlyobtainsonesinglesolutionvectordependingon tune the proportionality factor which was fixed in Das et al.
all the harmonies. (4) Most of other meta-heuristics including (2011).Meanwhile,Chenetal.(2012)computedthebwvaluepro-
GA, PSO and DE compare the offspring solutions with their portionaltothevariationofthecurrentdecisionvariabledynami-
corresponding parent individuals. However, the newly generated callytoobtainthebestvaluebasedontheevolutionofthesearch
harmonyjustcompareswiththeworst harmonyin theharmony processandthepropertiesoftheproblem.
memoryandreplacesitwhenithasworsefitness. Furthermore, various complicated parameter tuning methods
Sinceitsintroduction,HShasbeensuccessfullyappliedtosolve arealsoraisedduringthelastdecade.Panetal.(2010)restricted
variouscomplexreal-worldoptimizationproblemssuchasclassi- HCMR(PAR) in a normal distribution and updated the meanand
fication problem, structural optimization, stability analysis, standard deviation values by learning from their historic values
environmental/economic dispatch, parameter identification, net- corresponding to generated harmonies entering the HM.
work reconfiguration, unit commitment and scheduling problem Meantime, the best-to-worst ratio was introduced in Kattan and
(Manjarres et al., 2013). Many studies are also conducted to Abdullah (2013) to evaluate the quality of current HM solutions
enhance the accuracy and convergence speed of HS (Moh’d Alia and helped to dynamically adjust PAR values as the search pro-
& Mandava, 2011). Like other meta-heuristic algorithms, the gresses. Based on previous improvements on parameter setting,
optimizationcapacityofHSstronglyreliesontheparametersset- El-Abd (2013) linearly decreased PAR as proposed in Wang and
tings,includingharmonymemoryconsideringrate(HMCR),pitch Huang(2010)andexponentiallydecreasedbwaspreviouslypro-
adjusting rate (PAR) and pitch adjustment step (bw). Therefore, posed in Mahdavi et al. (2007) for getting a better performance.
manyresearchershavefocusedontheparametercontrol. Furthermore,Kumaretal.(2014)exploredfourdifferentcasesof
HMCR is the probability of each component generating from linearandexponentialchangesduringtheprocessofimprovisation
previous values stored in the harmony memory (HM) and varies forHMCRandPARtogettheglobaloptimalsolution.Anintelligent
between0and1.ToguaranteetheconvergenceofHS,HMCRfavors tuned harmony search algorithm (ITHS, Yadav et al. (2012)), in
large values and generally locates in the interval [0.9, 1]. To which there is no need to tune the parameters, was put forward
eliminate the drawbacks associated with fixed HMCR values, tomaintainaproperbalancebetweendiversificationandintensifi-
ABHS(Contreras,Amaya,&Correa,2014)andIGHS(Xiang,An,Li, cation throughout the search process by automatically selecting
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015
X.Kongetal./ExpertSystemswithApplicationsxxx(2015)xxx–xxx 3
theproperpitchadjustmentstrategybasedonitsharmonymem- defectonthesearchabilitydegradedbythismodificationwasana-
ory. To receive spur-in-time responses, Enayatifar, Yousefi, lyzed by Greblicki and Kotowski (2009) from the theoretical and
Abdullah, and Darus (2013) employed a learning automaton (LA) experimentalresults.Toovercometheaboveshortage,Wang,Xu,
to immediately tune the HS parameters regarding the harmony Mao,andFei(2010)presentedanewpitchadjustmentoperation
feedback. This learning-based adjustment mechanism solves the and then developed anovel discrete binaryHS algorithm(DBHS)
difficulties in parameter setting and enhances the local search to solve the discrete problems more effectively. However, BHS
abilitiesof the algorithm.It shouldbementioned thatGeem and and DBHS can only solve low-dimensional problems and can be
Sim(2010)focusedontheparameter-setting-free(PSF)technique hardly applied to high-dimensional problems. Moreover, Wang
toalleviatetheburdenofmanuallyfindingthebestparameterset- et al. (2013a) proposed an improved adaptive binary harmony
ting.ArehearsalbasedtechniquewasintroducedtoadjustHMCR search (ABHS) algorithm with a scalable adaptive strategy to
andPAR,buttherewassomethingunreasonableinthecalculation enhance the search ability and robustness. ABHS was evaluated
ofthesetwoparameterseventhoughtheauthorsdeclaredthatthe on the benchmark functions and 0–1 knapsack problems and
PSFtechniquecanfindgoodsolutionsrobustly. numerical results have demonstrated that it was more effective
Ascanbeseenfromabove,alargenumberofadaptivemecha- to solve the binary-coded problems. Thereafter ABHS was
nisms have been proposed to tune the parameters in HS. extended (ABHS1,Wang, Yang, Pardalos, Qian, & Fei (2013b)) to
Althoughbetterperformanceshavebeengivenanddemonstrated, findtheoptimalfuzzycontrollerparameterstoimprovethecon-
extraburdenforadditionalparametersettingsarealsointroduced trolperformanceowingtoitsoutstandingperformance.Basedon
inthesecases.Tolessentheparametersettingeffort,somespecific theresultstestedon10largescale0–1knapsackproblems,ABHS
medicationsofHSarepresented,especiallyfortheparameterbw. andABHS1demonstrateanoverwhelmingperformance,butthere
BorrowingtheconceptsofswarmintelligencefromPSO,thenew aretoomanyadditionalparameterstodetermine.Morerecently,a
harmonyinGHS(Omran&Mahdavi,2008)wasmodifiedtomimic novel global-best harmony search algorithm called DGHS (Xiang,
thebestharmonyintheHMandthustheparameterbwusedforthe An, Li, He, & Zhang, 2014b) was proposed to solve discrete 0–1
pitch adjustment was removed from the classical HS. Moreover, knapsack problems with binary coding. A best harmony based
WangandHuang(2010)replacedtheparameterbwbyupdating improvisation and two-phase repair operator were employed to
the new harmony according to the maximal and minimal values generatenewsolutionsafteragreedyinitialization.However,too
intheHM.UnlikemostHSvariants,tworatherthanonerandomly muchconsiderationonthebestharmonyintheHMmakesiteasy
selected harmonies were considered in GSHS (Castelli, Silva, tobetrappedinlocaloptima.
Manzoni,&Vanneschi,2014)fortheimprovisationphaseandthe Expectforthestatedbinarycodingmethods,manyreal-coded
newharmonywasobtainedmakinguseofalinearrecombination HSvariantsarealsoconsideredtosolvethediscreteproblemswith
operator that combines the information of two harmonies. Thus specific conversion of actual discrete decision values from real
there is no need to tune the PAR and bw parameters. Similar to variables.Amongthem,replacementofrealnumberwiththenear-
GSHS, Zou, Gao, Li, and Wu (2011) developed a different variant estintegeristhemostdirectandcommonlyusedstrategytoreach
ofHSnamedNovelGlobalHarmonySearch(NGHS).Twospecific a permissible discrete decision value. Considering this fact, Zou,
harmoniesintheHM,thatistheglobalbestharmonyandtheworst Gao,Wu,andLi(2010)developedanovelglobalharmonysearch
harmony,wereemployedtogeneratenewharmoniesandthenew algorithm (NGHS1) to solve the 0–1 knapsack problems. NGHS1
proposed variable updating technique excluded the PAR and bw was derived from the swarm intelligence of particle swarm and
parameters. More specifically, the new harmony always replaced replacedtheharmonymemoryconsiderationandpitchadjustment
the worst harmony in the HM even if it was far worse than the withanewpositionupdatingschemeandgeneticmutationstrat-
worst harmony. Furthermore, Valian, Tavakoli, and Mohanna egy. Similar to NGHS1, a social harmony search algorithmmodel
(2014)modifiedthe improvisationstep ofNGHSinspiringbythe was presented by Kaveh and Ahangaran (2012) for the cost
swarmintelligenceandmadethenewharmonyimitateonedimen- optimization of composite floor system with discrete variables.
sionofthebestharmonyintheHMasthatinvestigatedinGHS. TheroundingoperatorsarealsointegratedwithHStosolveother
Except for various adaptive parameter mechanisms and discrete problems, including epileptic seizure detection (Gandhi,
improvisation schemes, a series of powerful evolution strategies Chakraborty, Roy, & Panigrahi, 2012), size optimization
were introduced to ameliorate the optimization performance of (Askarzadeh, 2013) and steel frame optimization (Murren &
HS as well. These include low discrepancy sequences (Wang & Khandelwal, 2014). Besides the conversion approach, the
Huang,2010),chaosmaps(Alatas,2010),learnableevolutionmod- practitionershavedesignedafewspecialimprovisationoperators
els(Cobos,Estupinˆán,&Pérez,2011),mutationoperator(Pandi& toenablethesearchexecuteddirectlyinthediscretedomain.For
Panigrahi, 2011), dynamic subpopulations topology (Turky & example, Lee, Geem, Lee, and Bae (2005) presented a new pitch
Abdullah, 2014), and island model (Al-Betar, Awadallah, Khader, adjustmentwithneighboringvalueswhichcanhelpHStooptimize
&Abdalkareem,2015).Thankstoitsrelativeeaseandflexiblestruc- thestructureswithdiscrete-sizedmembers.Withtheassistanceof
ture,HShasbeenintegratedwithothermetaheuristiccomponents job-permutation-basedrepresentation,severalnovelpitchadjust-
orconceptstoenhancethesearchabilities,suchasdifferentialevo- ment rules have been employed to produce feasible solutions so
lution (Chakraborty, Roy, Das, Jain, & Abraham, 2009), particle thatHSiseffectiveforsolvingvariousschedulingproblems,such
swarm optimization (Pandi & Panigrahi, 2011) and genetic algo- as blocking permutation flow shop scheduling problem (Wang,
rithm(Zouetal.,2011).MoredetailedsummaryofresearchinHS Pan,&Tasgetiren,2010;Wang,Pan,&Tasgetiren,2011), no-wait
variantscanbefoundinMoh’dAliaandMandava(2011). flow shop scheduling problem (Gao, Pan, & Li, 2011), flexible job
As discussed above, a lot of attention has been given to the shop scheduling problem (Yuan, Xu, & Yang, 2013; Gao et al.,
research of HS for optimization problems in continuous space. 2014a;Gaoetal.,2014b),single-machineschedulingproblemwith
However,thereislittleworkconcentratingondiscreteproblems, planned maintenance (Zammori, Braglia, & Castellano, 2014).
especially the 0–1 optimization problems. A lot more attention Moreover, some quantum inspired operators were successfully
shouldbegiventothisareaandthisstudythusaimstocontribute combined with HS for 0–1 optimization problems (Layeb, 2013).
inthisarea. In addition, HS was mixed with ant colony optimization to solve
ThefirstbinarycodedHS(BHS)wasintroducedbyGeem(2005) the traveling salesman problem (Yun, Jeong, & Kim, 2013). It
to tackle the discrete water pump switching problem. BHS dis- should be mentioned that Geem (2008) defined a novel partial
cardedthepitchadjustmentoperatorfromclassicalHS.Thenthe stochasticderivativefordiscrete-valuedfunctionsanditcanhelp
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015
4 X.Kongetal./ExpertSystemswithApplicationsxxx(2015)xxx–xxx
HS to solve various discrete science and engineering problems the variable space and stored in the harmony memory. HS has
moreefficientlyandeffectively. three main procedures to improvise a new candidate harmony,
Although good results have been reported by the aforemen- that is, harmony memory consideration, pitch adjustment and
tionedHSvariantsfordiscreteproblems,theirperformanceisstill random search. The new harmony will replace the worst har-
notsatisfactoryandmanydrawbacksneedtobeimproved,espe- monyvectorinthewholeHMonlyifitisbetter(withhigherfit-
ciallyfor0–1optimizationproblems.Inotherwords,theresearch ness). Continue the improvisation process until a predefined
onHSfordiscreteproblemsisstillatitsinfancy. accuracy is achieved or certain number of improvisations has
Thispaperaimstofillthegapintheliteraturebyproposinga been accomplished.
simplified binary harmony search (SBHS) algorithm for solving Various steps consisted in HS is presented as follows (Geem
the 0–1 optimization problems. An ingenious improvisation etal.,2001):
schemewithoutanyparameterisintroducedbycombingharmony
memoryconsiderationwithpitchadjustment.SBHSalsodynami- Step1: Initializationofthealgorithmparameters.
callyadaptstheHMCRvaluesinaccordancewiththedimensions Thereare 5 parameters required to be set in classical HS
for problems with different properties. Furthermore, a two-stage forvariousproblems.Theyare:theharmonymemorysize
greedy procedure is embedded to repair the infeasible solutions (HMS);harmonymemoryconsideringrate(HMCR);pitch
emerged in the search process. A set of various large scale 0–1 adjusting rate (PAR); pitch adjustment step (bw); and
knapsack problems is selected to evaluate the effectiveness and maximalnumberofimprovisations(NI)orcertainsolution
superiority of SBHS. The experimental results indicate that SBHS accuracy.
issuperiortotheexistingHSvariantsinalmostallsituationsand Step2: Initializationoftheharmonymemory.
offersafasterconvergenceandhigheraccuracy. Initially,theharmonymemoryisfulfilledwithHMShar-
Themaincontributionsofthisstudyarethusasfollows.First,it moniesrandomlygeneratedinthevariablespace.
introduces a parameter-free improvisation scheme that depends 2 x1 x1 ... x1 fðx1Þ 3
on the difference between the best harmony and one randomly 1 2 n
6 x2 x2 ... x2 fðx2Þ 7
chosen harmony stored in the HM. More specifically, the pitch HM¼6 1 2 n 7 ð2Þ
adjustment parameters PAR and bw are excluded from the algo- 64 ... ... ... ... ... 75
rithmwithoutrequiringanyadditionalparameter.Thatistosay, xHMS xHMS ... xHMS fðxHMSÞ
1 2 n
SBHS has the least parameters comparedto the existing HSvari-
Step3: Improvisationofanewcandidateharmony.
ants. Second, it employs a simple but useful varying method to
Theimprovisationisconductedtogenerateanewcandi-
adapttheHMCRvalues,whichcaneffectivelyenhancetheconver-
date harmony xnew with three rules. The detailed proce-
genceandimprovetheoptimizationabilityofSBHStosuitvarious
dureworksasfollows:
problems with different dimensions. Third, it conducts a greedy
localsearch around these newinfeasible harmoniestoguarantee
the feasibility of the solutions and maintain population diversity fori=1tondo
simultaneously. The local search is accomplished in two stages ifUð0;1Þ6HMCR
relying on the specific heuristic derived form the 0–1 knapsack xniew¼xri;r2f1;2;...;HMSg
problems. Finally, it alleviates the burden of manually choosing %memoryconsideration
thebestparametersettingbecausethereareonlytwoparameters ifUð0;1Þ6PARthen
leftanddynamicadaptiveschemesaregiven. xnew¼xnew(cid:3)Uð0;1Þ(cid:4)bw
i i
The rest of this paper is organized as follows. In Section 2, a %pitchadjustment
basic process of how the HS works is laid out. Section 3 sum- endif
marizes four recent variants of HS proposed for solving discrete else
problems, including BHS, DBHS, NGHS1 and ABHS. In Section 4, xniew¼xi;minþUð0;1Þ(cid:4)ðxi;max(cid:5)xi;minÞ
the proposed simplified binary harmony search algorithm is %randomsearch
described particularly. Numerical experiments and comparisons endif
are conducted in Section 5 to evaluate the optimization perfor- endfor
mance of SBHS on large scale 0–1 knapsack problems. Section 6
givestheconcludingremarksanddirectionsforfurtherresearch.
2.Theharmonysearchalgorithm whereUð0;1Þareuniformrandomnumbersbetween0and1
independently.
Theresearchframeworkissetupinthissectionandthenota- Step4: Updateoftheharmonymemory.
tionandterminologiesusedthroughoutthepaperareclarifiedas TheworstharmonyintheHMwillbereplacedbythenew
well. In general, an optimization problem can be represented as improvised harmony when the candidate vector xnew is
follows(inthemaximizationsense): better than the worst harmony evaluated by the fitness
function.
Max fðxÞ;x¼ðx1;x2;...;xnÞ Step5: Terminationchecking.
s:t: gðxÞ<0;i¼1;2;...;p If NI harmonies have been produced or predefined accu-
i ð1Þ
hðxÞ¼0;j¼1;2;...;q racy is reached, terminate the algorithm and output the
j
x 6x 6x ;i¼1;2;...;n best harmony vector in the HM as the optimal solution.
i;min i i;max
Otherwise,returntoStep3andrepeattheimprovisation
wherenrepresentsthedimensionsize.pandqarethenumberof process.
inequalityconstraintsandequalityconstraints,respectively.
The harmony search algorithm is inspired from the music 3.RecentHSvariantsfor0–1optimizationproblems
improvisationprocess and a solutionvectoris analogy to a‘‘har-
mony’’ here. Similar to other population based heuristic meth- Inordertofindsatisfactorysolutionsforthe0–1optimization
ods, a set of harmony vectors are firstly generated randomly in problems,severalvariantsofHShavebeendevelopedtoimprove
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015
X.Kongetal./ExpertSystemswithApplicationsxxx(2015)xxx–xxx 5
itsperformance.Thissectionbrieflyreviewsfourrecentextensions 3.3.Novelglobalharmonysearch(NGHS1)algorithm
oftheHSalgorithm.
Unlike the above binary coded HS variants, Zou et al. (2011)
employed a novel global harmony search algorithm (NGHS1)
3.1.Binary-codingharmonysearch(BHS)algorithm
inspiredbytheswarmintelligenceofparticleswarmtosolvethe
0–1 knapsack problems. The search process is conducted in con-
Geem(2005)firstlyutilizedadiscreteversionofHStosolvea
tinuous space and the real values are replaced with the nearest
water pump switching problem. In this study, the float encoding
integers as the binary solutions. In fact, the genetic mutation
method in classical HS is replaced by the binary coded scheme
probabilityp introducedinNGHS1isthesameto1-HMCRwhich
becausecandidatevaluesforeachvariablearerestrictedto0and m
denotestheprobabilityofrandomlychoosingafeasiblevaluenot
1. The improvisation process of BHS is the same as the classical
related to the HM. Simultaneously, the position updating is a
HS except that the pitch adjustment operator is abolished. Each
fusionofmemoryconsiderationandpitchadjustmentinclassical
variable of the new candidate harmony is picked up from either
HS with PAR=1. The modified improvisation process in NGHS1
acorrespondinghistoricalvaluestoredintheHMorarandomfea-
worksasfollows:
siblevalueof0or1dependingonHMCR.
The detailed process of improvisation in BHS is carried out as
fori=1tondo
follows:
step ¼jxbest(cid:5)xworstj%calculationoftheadaptivestep
( i i i
xnew2fx1;x2;...;xHMSg; if Uð0;1Þ6HMCR xnew¼xbest(cid:3)Uð0;1Þ(cid:4)step %positionupdating
xnew i i i i ð3Þ i i i
i xniew2f0;1g; otherwise ifUð0;1Þ6pm
xnew¼x þUð0;1Þ(cid:4)ðx (cid:5)x Þ
i i;min i;max i;min
where xnew is the ith element of the new harmony candidate and %geneticmutation
i
eachvariableischoseninthesamemannerasEq.(3).Thecandidate endif
valueischosenfromtheexistingvaluescontainingincurrentHM endfor
withtheprobabilityofHMCRandfromthevariablespacerandomly
withtheprobability1-HMCR.
Furthermore,GreblickiandKotowski(2009)analyzedtheprop-
wherexbestandxworstdenotetheglobalbestharmonyandtheworst
ertiesofHSbasedontheone-dimensionalbinaryknapsackprob-
one in current HM, respectively. It is worth mentioning that the
lemandfoundtheperformanceofBHSunsatisfactory.
worstharmonyisalwaysreplacedbythenewgeneratedharmony
regardless of the quality of the new generated harmony, even if
3.2.Discretebinaryharmonysearch(DBHS)algorithm xnew isworsethanxworst.
Inthepositionupdatingprocess,thebestandworstharmonies
Tocompensatethedegradationcausedbydiscardingthepitch are the only two harmonies considered in current HM and the
adjustment rule in BHS, Wang et al. (2010) redesigned the pitch information used in improvisation is so little that NGHS1 easily
adjustment operation in the DBHS algorithm. In contrast to BHS leadstoprematureconvergenceandstagnation.
in whicheachvariableofthenew harmonymaybechosenfrom
differentharmonies,DBHSselectsonlyoneharmonyfromcurrent 3.4.Adaptivebinaryharmonysearch(ABHS)algorithm
HM to form the new harmony. That is, the individual strategy
operationcanbedefinedinEqs.(4)and(5)below. To tackle the 0–1 optimization problems more effectively,
Wang et al. (2013a) gave an improved adaptive binary harmony
xnew¼(cid:2)xti; if Uð0;1Þ6HMCR; t2f1;2;...;HMSg ð4Þ search(ABHS)algorithmfollowinganalyzingdrawbacksofHSfor
i R; otherwise binary-valued problems. In addition to the individual selection
strategy shown as Eqs. (4) and (5) in DBHS, ABHS employs a bit
(cid:2)0; if Uð0;1Þ60:5 selection strategy to implement harmony memory consideration
R¼ ð5Þ operation in which each element of the new harmony vector is
1; otherwise
independentlychosenfromtheHM.Thepitchadjustmentruleuti-
lized in ABHS is the same as that in DBHS. The authors focused
wheretisonespecificrandomintegerselectedfromtheinterval[1,
mainly on various former adaptive mechanisms of HMCR and
HMS].xt representstheithelementofthechosenharmonyinthe
i PAR and defined a scalable adaptive strategy of HMCR based on
HM.
theoverallresultsoftheparameteranalysistoenhanceitssearch
Forthe0–1optimizationproblems,thereareonlytwovalues,
abilityandrobustness.HMCRfinallyusedinABHSisadaptiveand
i.e.,0and1,tobeselectedforthevariablesandthereforethepitch
linearlyincreasingasEq.(7).
adjustmentstepmustbe 1in alldimensions. Tofacilitateimple-
mentation, DBHS defines the corresponding element value of the (cid:3) c(cid:4) blnnc lnn k
HMCR¼ 1(cid:5) þ þ (cid:4) ð7Þ
global optimal harmony vector in the HM as the adjusted value n n n K
forachosenelementinsteadoftheNOTgatetorealizethepitch
where c is a constant; k and K denote the current and maximal
adjustmentoperator.
iterations,respectively;bmcistheoperatortakingthelargestinte-
( gerlessthanm.Theaboveadaptivefactordynamicallyandlinearly
xbest; if Uð0;1Þ6PAR
xnew¼ i ð6Þ adjusts HMCR based on the dimension and current iteration
i xniew; otherwise number.
Basedontheexperimentalresultsonbenchmarkfunctionsand
wherexbestistheithcorrespondingelementvalueoftheglobalopti- 0–1 knapsack problems, ABHS outperforms other algorithms in
i
mal harmony vector xbest. The pitch adjustment utilized can termsofsearchaccuracyandconvergencespeed.However,ABHS
enhancethelocalsearchabilitytofindbettersolutionsforbinary suffers from two parameters c and NGC which are vital but hard
problems.Thenumberofnewlygeneratedcandidates(NGC)isalso tochoosetoguaranteegoodperformanceforproblemswithdiffer-
studiedtocheckitseffectontheperformanceofDBHS. entproperties.
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015
6 X.Kongetal./ExpertSystemswithApplicationsxxx(2015)xxx–xxx
4.Simplifiedbinaryharmonysearchalgorithm distinctfromxbest,theswappingprocesscanspeeduptheconver-
i
gence of the HM to the real global optimum. Thus the ingenious
Thissectionpresentsasimplifiedbinaryharmonysearchalgo- improvisationschemeproposedinSBHScanappropriatelybalance
rithm(SBHS)tocircumventthedrawbacksincurrentHSvariants theexplorationabilityandexploitationability,thatis,thecapabili-
forthe0–1optimizationproblems.InSBHS,onlytwoparameters, tiesoftheglobalsearchandthelocalsearchgivefullplaytothe
i.e.,HMSandHMCR,needtobesetandaningeniousimprovisation optimization. Compared to previous HS variants mentioned in
rule is introduced based on the difference between the best har- Section3,SBHSnotonlyutilizesmoreinformationforoneelement
mony and one randomly chosen harmony stored in the HM to fromtheHM,i.e., tworandomlyselectedvariablevaluesand the
implementpitchadjustmentwithoutanyparametersuchasPAR. best realized harmonies, but also considers the internal property
HMCR is linearly increasing with the dimension to improve the ofthoseharmonies,i.e.,thedifferenceofelementvaluesbetween
optimization ability on different problems. To guarantee the thebestharmonyandotherharmonies.Furthermore,SBHSavoids
feasibility of the solutions, a two-stage greedy procedure is the selection of both PAR and bw parameter values and the
employedtorepairtheinfeasiblesolutionvectorsemergedinthe improvisation process in Eq. (8) can be determined according to
HM.ThedetailsoftheSBHSalgorithmarepresentedbelow. not only the evolution of the search process, but also different
search spaces for different problems. In summary, SBHS is easy
4.1.Aningeniousimprovisationscheme toimplementandsuitableforthe0–1optimizationproblemswith
differentcharacteristics.
ComparedtothefloatcodingmethodusedinNGHS1,itismore
appropriateforthevariablestobecodedinbinaryschemefor0–1
4.2.Two-stagegreedyproceduretorepairtheinfeasiblesolution
optimizationproblems.Therefore,abinarycodingschemesimilar
tothatusedinBHS,DBHSandABHSisemployed.Notethatdiscard-
Sinceonlythefeasiblesolutionscandelegatethefeasibleregion
ingofthepitchadjustmentoperatorinBHSleadstoanunsatisfac-
inthedefinedvariablespaceforconstrainedproblemsandinfeasi-
toryperformanceforbinaryoptimizationproblems.Thusthepitch
ble solutions would mislead the search to be stagnated in the
adjustment operator is important and must be contained in the
infeasibleregion,wemustmakesurethatalltheharmoniescon-
improvisation process. The memory consideration in DBHS and
tainedintheHMarefeasible.Thesimplestwaytoachievethisis
ABHS are not ideal as the corresponding value is selected from
byremovingsomeitemsfromtheknapsackandsettingthevari-
eitheronespecificharmonyordifferentharmoniesintheharmony
ablevalueofcorrespondingitemfrom1to0.Intuitively,theitem
memory.DBHSmodifiesthepitchadjustmentoperatorbyreplac-
withgreaterprofitandsmallervolumehasmorepossibilitytobe
ing the chosen element with the corresponding element value of
packedintotheknapsackformaximizingthetotalprofits.Therela-
theglobaloptimalharmonyvectorincurrentHM.Theinformation
tiveprofitdensityproposedbyDantzig(1957)canbeusedasarule
keptinthebestharmonyvectorisutilizedtoomuchandmayresult
forchoosinganitemanditiscalculatedas
inprematureconvergenceandstagnation.Besides,thepitchadjust-
ment operator depends on the probability PAR which is decided u ¼p=v ð9Þ
regardless of the informationcontaining in the HM.To avoid the i i i
disadvantages associated with these binary-coded algorithms, an
wheretherelativeprofitdensityoftheithitemisdenotedbyu and
ingeniousimprovisationschemeisintroducedasbelow. the other two values p and v represent the profit and voluime,
i i
xnew¼xr1þð(cid:5)1Þ^ðxr1Þ(cid:4)jxbest(cid:5)xr2j; i¼1;2;...;n ð8Þ respectively.
i i i i i
A profit density based two-stage greedy procedure is used to
where r1 and r2are two distinctrandomintegers between 1 and repair the infeasible solutions emerged in initialization and
HMSforeachelement. improvisation process to ensure the availability of harmonies in
Eq. (8) combines the memory consideration and pitch adjust- theHM.Theusedrepairprocedureisaccomplishedintwostages.
ment to improvise a new harmony in each bit. The first term on Inthefirststage,afeasiblesolutionisachievedbytakingoutthe
therighthandsideofEq.(8)denotesthememoryconsideringpart items with lower profit density under the constraint condition.
and the second is the pitch adjusting part of improvisation. The After that, there may be some small space left in the knapsack
memoryconsideringpartisthesameasthatinBHSbutthepitch whichislargerthanthevolumesofotheritemsunpacked.Tofill
adjustmentdependsonthedifferencebetweenthebestharmony theknapsackasmuchaspossible,thesecondgreedyphasewhich
andrandomlyselectedharmonyintheHMratherthanaspecific isalmostoppositetothefirststageisapplied.Inthesecondstage,
valuePAR.Thelargeristheproportionofdifferencebetweenthe theitemswhichcanbepackedintotheknapsackindividuallyare
bestharmonyandotherharmonies,themorepossibilitythepitch found out and then added to the knapsackone by one according
adjustmenthappens.Iftheelementinthebestharmonyisdiffer- totheirprofitdensitylevelsindecreasingorderuntilthetotalvol-
ent from the corresponding value of the randomly selected har- umeofthechosenitemsexceedstheknapsackvolume.
mony, the considered value from the HM is changed, for Specifically, the profit density based two-stage greedy proce-
example,from 0 to 1, or from 1 to 0, and this change isupdated dureinSBHSconsistsofthefollowingsteps:
with a power function of the considered value. If the considered
value is 0, its power function of (cid:5)1 is 1 and otherwise, (cid:5)1. At Step1: Givenaninfeasibleharmonydenotedasx¼ðx ;x ;...;x Þ.
1 2 n
the same time, the differencemust be0 and 1. Values generated Step2: Calculatethe total volume V of the items chosen by the
t
byimprovisationarealsocontainedinthevariablespaceandthere infeasibleharmonyxandthevalueofconstraintviolation
isnoneedtomodifythemastheydonotviolatetheboundarycon- V :
c
straintofvariables.Therefore,Eq.(8)candirectlyrealizethebinary
opeFroartoarcwhoitsheonuxtr1a,nifythloegdiciaffleorepnecraeteoxri.sts,i.e.,xbestisdifferentfrom Vt¼Xn vi(cid:2)xi;Vc¼Vt(cid:5)Vmax ð10Þ
i i
xr2;xr1isswappedtoanothervalueinthedefineddomain{0,1}.If i¼1
i i
the chosen xri1 is identical to xbiest, the swapping process would Step3: GetthecorrespondingitemsequenceS1basedontherela-
enhance the diversity of the HM to effectively avoid it stopping tiveprofitdensityofeachitemcalculatedusingEq.(9)in
at very poor quality local optima. However, if the chosen xr1 is ascendingorder.
i
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015
X.Kongetal./ExpertSystemswithApplicationsxxx(2015)xxx–xxx 7
Step4: RemovetheitemsassequenceS1untilthetotalvolumeis Thus it is not necessary to choose a large value for HMS. In this
smallerthantheknapsackvolume,i.e.,V <0.Thepseudo paper,itissettobe5inallsituationsforSBHS.
c
codeisshownasbelow. DuetotheeliminationofparametersPARandbwinSBHS,the
HMCRvalueplaysthemostimportantroleforthealgorithmper-
j¼1; formanceasitcontrolsthebalancebetweenthecapabilitiesofglo-
whileV >0 bal searchand localsearch.Note thatevery componentobtained
c
ifx =1 by the memory consideration comes from the previous values
S1ðjÞ
x =0; storedintheHManditisfurtherdeterminedtobepitchadjusted
S1ðjÞ
Vc¼Vc(cid:5)vS1ðjÞ; or not. HMCR can take any value between 0 and 1. HMCR=0
meanseverycandidatevalueischosenfromtherangeofthevari-
endif
able space randomly. HMCR=1 deprives the chance to choose a
j¼jþ1;
valuefromoutsidetheHMtoimprovetheharmony.
endwhile
Generally speaking, a large HMCR favors the local search. To
enhance the exploration ability, HMCR value should be small in
order to lead the search proceeding in the whole variable space.
Step5: Calculate the remaining volume of the knapsack Vl, The best choice of HMCR value is normally in the interval
Vl¼(cid:5)Vc,andjudgewhetherthesmallestitemvolumeis [0.95,1]. Moreover, various HMCR adaptive methods appear to
largerthanitornot.Ifso,terminatetherepairingstrategy; ameliorate the performance and flexibility of the HS variants,
otherwise,turntoStep6. including the linear increment, linear decrease, nonlinear incre-
Step6: Receive the corresponding item sequence S2 based on ment, random increment etc. Most of them focus on the search
theirvolumesinascendingorder. process and base the HMCR value on the maximum and current
Step7: StoretheitemtabsinasequenceS3,thevolumeofwhich iterationnumbersregardlessoftheproblemdimension.
issmallerthantheremainingvolumeoftheknapsackVl. Givenann-dimensionalproblem,theexpectednumberofele-
Thepseudocodeisshownasbelow. ments chosen from the HM in the new candidate harmony is
n(cid:2)HMCR,whiletheexpectednumberofcomponentsreinitialized
j=1; randomlyfromthepossiblerangeofvaluesrelyingonthecomple-
whilej<¼nandVl>vS2ðjÞ mentationis n(cid:2) (1-HMCR). For alow-dimensional problem,n(cid:2) (1-
j¼jþ1; HMCR) is small. However, for a large scale problem with
endwhile nP100, the value n(cid:2) (1-HMCR) would be so significant that too
j¼j(cid:5)1; manyrandomlyselectedelementswoulddestroytheoptimization
S3=S2(1:j); abilityofthealgorithm.Inthiscase,itisnotthesearchprocessbut
theproblemdimensionwhichhas thebiggesteffecton thealgo-
rithm performance. Based on this observation, SBHS updates the
Step8: GetthecorrespondingtabsequenceS4ofthoseitemscon- HMCRvaluedynamicallyaccordingtoEq.(11)andremainsitcon-
tained in S3 based on their relative profit density in stantintheentiresearchprocess.
ascendingorder. HMCR¼1(cid:5)10=n; nP100 ð11Þ
Step9: Insert the items into the knapsack following the tab
sequence S4 until there is no space in the knapsack. The TheproposedHMCRtuningschemeiscapableofavoidingthe
pseudocodeisshownasbelow. number of variables randomly reinitialized being too large and
ensures the convergence of the algorithm. Moreover, reinitializa-
tion with a small probability can diversify the HM and thus to
whilej>0andV >0
ifx ¼0landV >v avoid premature stagnation and ill-convergence. This simplifica-
S3ðS4ðjÞÞ l S3ðS4ðjÞÞ tiononparametersettingalsomakesimplementationeasy.
x ¼1;
S3ðS4ðjÞÞ
V ¼V (cid:5)v ;
l l S3ðS4ðjÞÞ 4.4.ComputationalprocedureofSBHS
endif
j¼j(cid:5)1;
In summary, the computational procedure of the SBHS algo-
endwhile
rithmcanbeillustratedasfollows.
After the repair work through this mechanism, it is clear that the Table1
ParametersettingfortheHSvariants.
infeasibleharmoniesnolongerviolatetheconstraint.Becausethe
repairoperationsintheabovetwostagesincludingremovingand Variant Parametersetting
inserting are all carried out greedily based on the relative profit IHS HMCR=0.95;PARmax=0.99;PARmin=0.35;bwmax=0.05;
density,theknapsackcanbefilledupaccordingtotheitemprofit bwmin=0.0001
asmuchaspossible. GHS HMCR=0.99;PARmax=0.99;PARmin=0.01
SAHS HMCR=0.99;PARmax=1;PARmin=0
EHS HMCR=0.99;PAR=0.33;bw=1.17(cid:2)pffiVffiffiaffiffiffirffiffiðffiffixffiffiÞffiffi
4.3.Adaptiveparametertuningmechanism
NGHS Pm¼2=n
NDHS HMCR=0.99;PARmax=0.99;PARmin=0.01;ts=2
Itshouldbenotedthatnoextra parameters areintroducedin SGHS HMCRmax=1;HMCRmin=0.9;PARmax=1;PARmin=0;LP=100;
SBHS.Owingtotheingeniousimprovisationscheme,theparame- bwmax=1/10;bwmin=0.0005;HMCRm=0.98;PARm=0.9;
tersPARandbwinclassicalHSarenolongerneededinSBHSand ITHS PARmax=1;PARmin=0;HMCR=0.99
BHS HMCR=0.971;NGC=1
there are only two parameters including HMS and HMCR to be
DBHS NGC=20,HMCR=0.7,PAR=0.1
tuned. HMS is the number of harmonies preserved in the HM NGHS1 Pm¼2=n
andhaslittleeffectontheperformanceofthealgorithm.Ingen- ABHS PAR=0.2;C=15;NGC=20
eral, the larger the HMS is, the lower the convergence speed is. ABHS1 PARmax=0.25;PARmin=0.15;HMCRmax=0.97;HMCRmin=0.95
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015
8 X.Kongetal./ExpertSystemswithApplicationsxxx(2015)xxx–xxx
Table2
5.Experimentalresultsanddiscussions
Descriptionof10low-dimensional0–1knapsackproblems.
Problem n Optimum Parameters Alargenumberofexperimentalstudieson0–1knapsackprob-
KP1 10 295 v=(95,4,60,32,23,72,80,62,65,46),Vmax lemsareextensivelyinvestigatedand10low-dimensionaland16
=269,p=(55,10,47,5,4,50,8,61,85,87) largescaleinstancesareconsideredtotesttheoptimizationability
KP2 20 1024 v=(92,4,43,83,84,68,92,82,6,44,32,18,56,
in this section. To evaluate the effectiveness of SBHS, its perfor-
83,25,96,70,48,14,58),Vmax=878,p=(44,46, manceiscomparedwiththestate-of-the-artHSvariantsconsisting
90,72,91,40,75,35,8,54,78,40,77,15,61,17,
75,29,75,63) of 9 continuous and 5 binary variants. The continuous variants
KP3 4 35 v=(6,5,9,7),Vmax=20,p=(9,11,13,15) includes IHS (Mahdavi et al., 2007), GHS (Omran & Mahdavi,
KP4 4 23 v=(2,4,6,7),Vmax=11,p=(6,10,12,13) 2008), SGHS (Pan et al., 2010), SAHS (Wang & Huang, 2010),
KP5 15 481.0694 v=(56.358531,80.874050,47.987304,
PSFHS (Geem & Sim, 2010), NGHS (Zou et al., 2011), EHS (Das
89.596240,74.660482,85.894345,51.353496,
1.498459,36.445204,16.589862,44.569231, et al., 2011), NDHS (Chen et al., 2012) and ITHS (Yadav et al.,
0.466933,37.788018,57.118442,60.716575), 2012).ThebinaryonesareBHS(Geem,2005),DBHS(Wangetal.,
Vmax =375,p=(0.125126,19.330424, 2010), NGHS1 (Zou et al., 2010), ABHS (Wang et al., 2013a) and
58.500931,35.029145,82.284005,17.410810,
ABHS1 (Wang et al., 2013b). All the computational experiments
71.050142,30.399487,9.140294,14.731285,
are conducted in Matlab 7.7 using a PC with Intel(R) Core(TM) 2
98.852504,11.908322,0.891140,53.166295,
60.176397) Quad CPU Q9400 @ 2.66GHz, 3.50GB RAM and Windows XP
KP6 10 52 v=(30,25,20,18,17,11,5,2,1,1),Vmax=60, operatingsystem.
p=(20,18,17,15,15,10,5,3,1,1)
KP7 7 107 v=(31,10,20,19,4,3,6),Vmax=50,p=(70,20,
39,37,7,5,10) 5.1.Experimentalresultsanddiscussions
KP8 23 9767 v=(983,982,981,980,979,978,488,976,972,
486,486,972,972,485,485,969,966,483,964, Tomakethecomparisonasfairaspossible,thesettingsforthe
963,961,958,959),Vmax=10000, comparison HS variants follow the original referencesmentioned
p=(981,980,979,978,977,976,487,974,970,
in previous paragraph. HMSis set to be19 for BHS,50 for SAHS,
485,485,970,970,484,484,976,974,482,962,
961,959,958,857) 10forITHS,30forbothABHSandABHS1,and5forallothercom-
KP9 5 130 v=(15,20,17,8,31),Vmax=80,p=(33,24,36, parisonalgorithms.Otherparametersutilizedinourexperiments
37,12) aregiveninTable1.
KP10 20 1025 v=(84,83,43,4,44,6,82,92,25,83,56,18,58,
It should be noted that the pseudo-random number is used
14,48,70,96,32,68,92),Vmax=879,p=(91,72,
90,46,55,8,35,75,61,15,77,40,63,75,29,75, instead of the low-discrepancy sequences to initialize the HM in
17,78,40,44) SAHS.Theamountofrehearsaliscarriedoutforone-tenthofthe
totalgenerationsandthereisnoneedtosetotherparametersin
PSFHS.TheHMCR(PAR)valueinSGHSisdistributedwithstandard
deviation0.01(0.05)duringthewholeperiod.ABHSusesanadap-
Step1: Set parameters HMS and HMCR according to the dimen-
tive linearly increasing HMCR defined as Eq. (23) in Wang et al.
sionofaparticularproblem.
(2013b).
Step2: Initialize the HM using Bernoulli stochastic process and
For continuous HS variants, the variable space is restricted in
repairtheinfeasibleharmoniesthroughtheprofitdensity
[0,1]n and the nearest integers from the real numbers emerged
based two-stage greedy procedure as discussed in
in searching process are served as the binary variables without
Section 4.2. Total profits of each harmony are evaluated
anychangeontheoriginalrealnumbers.Binarycodingisapplied
accordingtotheitemsselected.
inotherbinaryHSvariants.Sincethemaximalvolumeoftheknap-
Step3: DeterminethemaximaliterationnumberNIandsetcur-
sackislimitedin0–1knapsackproblemsandsometimesthetotal
rentiterationk=1.
volumeoftheitemspackedintheknapsackmayexceedthecon-
Step4: Improvise a new harmony xnewðxnew;xnew;...;xnewÞ as
1 2 n straint,theviolationisunacceptableandmustbechecked.Abal-
belowandrepairitifinfeasible.
ance between the constraint and the objective function value
needtobefoundandmaintained.Themostdirectwaytohandle
fori=1tondo
the constraint is the penalty function method in which feasible
ifUð0;1Þ6 HMCR pointsarefavoredoverinfeasiblepointsandthepenaltyoninfea-
xniew¼xri1þð(cid:5)1Þ^ðxri1Þ(cid:4)jxbiest(cid:5)xir2j, siblesolutionsisassessedbasedonthedistanceawayfromthefea-
%improvisation sible region. This approach utilizes penalty functions to form a
else second function to be minimized. For the 0–1 knapsack problem
(cid:2)0; if Uð0;1Þ60:5 defined in the introduction, the corresponding penalty function
xniew¼ 1; otherwise canbeformedanddescribedasfollows.
!
%reinitialization Max FðxÞ¼Xn p (cid:2)x (cid:5)b(cid:2)max 0;Xn v (cid:2)x (cid:5)V
endif i i i i max ð12Þ
endfor i¼1 i¼1
s:t: x 2f0;1g; i¼1;2;...;n
i
wherebrepresentsthepenaltycoefficientandissetto1020 forall
experimentsinthispaper.
Step5: Replace the worst harmony in the HM with xnew, if and
onlyifxnew isbetterthantheworstharmony.k¼kþ1. 5.2.Comparisonsonlow-dimensional0–1knapsackproblems
Step6: Repeatsteps4–5intheimprovisationprocessuntilNInew
candidateharmonieshavebeengenerated. In this section, 10 low-dimensional 0–1 knapsack problems
Step7: Output the best harmony vector xbest in the HM as the takenfromZouetal.(2010)andWangetal.(2013a)areadopted
optimalsolution. to investigate the performance of our proposed algorithm. The
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015
X.Kongetal./ExpertSystemswithApplicationsxxx(2015)xxx–xxx 9
Table3
ResultsofcontinuousHSvariantsonKP1-KP10.
IHS GHS SAHS EHS NGHS NDHS SGHS ITHS PSFHS
KP1 SR 0.7 1 1 0.1 1 0.52 0.4 0.6 0.12
Best 295 295 295 295 295 295 295 295 295
Median 295 295 295 279 295 295 293 295 246
Worst 246 295 295 173 295 288 241 246 155
Mean 292.4 295 295 267.1 295 293.36 276.92 292.46 240.06
Std 9.14 0 0 30.06 0 2.36 22.3 8.4 39.88
KP2 SR 1 1 1 0.3 1 0.38 0.6 0.78 0.12
Best 1024 1024 1024 1024 1024 1024 1024 1024 1024
Median 1024 1024 1024 1013 1024 1018 1024 1024 987
Worst 1024 1024 1024 855 1024 945 990 995 847
Mean 1024 1024 1024 998.2 1024 1011.68 1017.44 1020.84 975.46
Std 0 0 0 34.36 0 16.98 10.29 7.53 43.57
KP3 SR 0.82 1 1 0.76 1 0.92 0.76 0.86 1
Best 35 35 35 35 35 35 35 35 35
Median 35 35 35 35 35 35 35 35 35
Worst 28 35 35 28 35 28 28 28 35
Mean 33.74 35 35 33.82 35 34.54 33.72 34.52 35
Std 2.72 0 0 2.45 0 1.7 2.58 1.47 0
KP4 SR 1 1 1 0.76 1 1 0.8 0.94 0.38
Best 23 23 23 23 23 23 23 23 23
Median 23 23 23 23 23 23 23 23 22
Worst 23 23 23 18 23 23 19 19 12
Mean 23 23 23 22.14 23 23 22.38 22.88 21.08
Std 0 0 0 1.65 0 0 1.4 0.59 2.52
KP5 SR 0.68 1 1 0.18 1 0.7 0.2 0.88 0.08
Best 481.07 481.07 481.07 481.07 481.07 481.07 481.07 481.07 481.07
Median 481.07 481.07 481.07 431.71 481.07 481.07 437.93 481.07 410.79
Worst 437.95 481.07 481.07 348.94 481.07 432.5 367.97 437.95 278.3
Mean 468.77 481.07 481.07 430.36 481.07 474.37 442.77 478.15 398.7
Std 19.45 0 0 32.23 0 13.69 25.17 10.35 47.15
KP6 SR 0.6 1 1 0.52 1 0.46 0.48 0.68 0.58
Best 52 52 52 52 52 52 52 52 52
Median 52 52 52 52 52 51 51 52 52
Worst 47 52 52 43 52 43 46 45 47
Mean 50.86 52 52 50.18 52 50.5 49.92 50.94 51.02
Std 1.65 0 0 2.42 0 1.95 2.29 1.92 1.46
KP7 SR 0.14 1 1 0.1 1 0.68 0.02 0.22 0.32
Best 107 107 107 107 107 107 107 107 107
Median 105 107 107 93 107 107 93 105 100
Worst 93 107 107 81 107 93 79 93 69
Mean 100.66 107 107 95.26 107 105.34 93.36 102.68 98.12
Std 5.84 0 0 9.19 0 3.67 9.65 5.3 9.76
KP8 SR 0.8 0.92 1 0.06 1 0.34 0.2 0.76 0
Best 9767 9767 9767 9767 9767 9767 9767 9767 9765
Median 9767 9767 9767 9759 9767 9765 9762 9767 9748
Worst 9762 9762 9767 9630 9767 9755 9748 9761 9611
Mean 9766.32 9766.84 9767 9754.16 9767 9764.5 9761.44 9766.1 9723
Std 1.52 0.74 0 19.35 0 2.98 4.25 1.73 51.83
KP9 SR 1 1 1 0.68 1 1 0.92 0.96 0.66
Best 130 130 130 130 130 130 130 130 130
Median 130 130 130 130 130 130 130 130 130
Worst 130 130 130 106 130 130 109 118 93
Mean 130 130 130 125.2 130 130 128.86 129.52 123.66
Std 0 0 0 7.52 0 0 4.06 2.38 10.07
KP10 SR 0.98 1 1 0.22 1 0.44 0.58 0.68 0.02
Best 1025 1025 1025 1025 1025 1025 1025 1025 1025
Median 1025 1025 1025 1005 1025 1019 1025 1025 956
Worst 1019 1025 1025 953 1025 930 987 1005 800
Mean 1024.88 1025 1025 999.52 1025 1013.46 1018.12 1021.86 946.82
Std 0.85 0 0 22.23 0 20.96 10.39 5.43 56.44
NS 10 10 10 10 10 10 10 10 9
MSR 3 9 10 0 10 2 0 0 1
dimension and parameters of these test problems are listed in measures,i.e.,‘‘SR’’,‘‘Best’’,‘‘Median’’,‘‘Worst’’,‘‘Mean’’and‘‘Std’’,
Table2. areusedtoevaluateeachHSalgorithm.Sincetheoptimumvalues
Themaximumnumberofiterationsissetto10,000forallbut ofthose10problemsareknown,itispossibletocalculatethesuc-
ABHSandDBHSforwhich500istaken.50independentrunsare cessrate(SR)among50runsinreachingtheappointedoptima.The
conducted to collect the statistical results. In this paper, 6 best, median and worst values are chosen from the total results
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015
10 X.Kongetal./ExpertSystemswithApplicationsxxx(2015)xxx–xxx
Table4
ResultsofbinaryHSvariantsonKP1-KP10.
KP1 KP2 KP3 KP4 KP5 KP6 KP7 KP8 KP9 KP10 MSR
BHS SR 0.78 0.92 0.98 1 0.96 0.9 0.56 0.82 0.98 0.94 1
Best 295 1024 35 23 481.07 52 107 9767 130 1025
Median 295 1024 35 23 481.07 52 107 9767 130 1025
Worst 293 1018 28 23 437.94 50 93 9762 118 1019
Mean 294.58 1023.52 34.86 23 479.55 51.84 104.34 9766.34 129.76 1024.64
Std 0.81 1.64 0.99 0 7.59 0.51 4.5 1.52 1.7 1.44
DBHS SR 1 1 1 1 1 1 1 1 1 1 10
Best 295 1024 35 23 481.07 52 107 9767 130 1025
Median 295 1024 35 23 481.07 52 107 9767 130 1025
Worst 295 1024 35 23 481.07 52 107 9767 130 1025
Mean 295 1024 35 23 481.07 52 107 9767 130 1025
Std 0 0 0 0 0 0 0 0 0 0
NGHS1 SR 1 1 1 1 1 0.96 1 0.94 1 1 8
Best 295 1024 35 23 481.07 52 107 9767 130 1025
Median 295 1024 35 23 481.07 52 107 9767 130 1025
Worst 295 1024 35 23 481.07 51 107 9765 130 1025
Mean 295 1024 35 23 481.07 51.96 107 9766.88 130 1025
Std 0 0 0 0 0 0.2 0 0.48 0 0
ABHS SR 1 1 1 1 1 1 1 1 1 1 10
Best 295 1024 35 23 481.07 52 107 9767 130 1025
Median 295 1024 35 23 481.07 52 107 9767 130 1025
Worst 295 1024 35 23 481.07 52 107 9767 130 1025
Mean 295 1024 35 23 481.07 52 107 9767 130 1025
Std 0 0 0 0 0 0 0 0 0 0
ABHS1 SR 0.86 0.96 1 0.98 0.98 0.84 0.48 0.82 1 1 3
Best 295 1024 35 23 481.07 52 107 9767 130 1025
Median 295 1024 35 23 481.07 52 105 9767 130 1025
Worst 293 1018 35 22 475.48 49 96 9762 130 1025
Mean 294.72 1023.76 35 22.98 480.96 51.68 105.18 9766.44 130 1025
Std 0.7 1.19 0 0.14 0.8 0.82 2.95 1.33 0 0
SBHS SR 1 1 1 1 1 1 1 1 1 1 10
Best 295 1024 35 23 481.07 52 107 9767 130 1025
Median 295 1024 35 23 481.07 52 107 9767 130 1025
Worst 295 1024 35 23 481.07 52 107 9767 130 1025
Mean 295 1024 35 23 481.07 52 107 9767 130 1025
Std 0 0 0 0 0 0 0 0 0 0
Table5
ResultsofranksumtestsforSBHSwithotherHSvariants.
SBHS IHS GHS SAHS EHS NGHS NDHS SGHS ITHS PSFHS BHS DBHS NGHS1 ABHS ABHS1
KP1 1 0 0 1 0 1 1 1 1 1 0 0 0 1
KP2 0 0 0 1 0 1 1 1 1 1 0 0 0 0
KP3 1 0 0 1 0 0 1 1 0 0 0 0 0 0
KP4 0 0 0 1 0 0 1 0 1 0 0 0 0 0
KP5 1 0 0 1 0 1 1 1 1 0 0 0 0 0
KP6 1 0 0 1 0 1 1 1 1 1 0 0 0 1
KP7 1 0 0 1 0 1 1 1 1 1 0 0 0 1
KP8 1 0 0 1 0 1 1 1 1 1 0 0 0 1
KP9 0 0 0 1 0 0 1 0 1 0 0 0 0 0
KP10 0 0 0 1 0 1 1 1 1 0 0 0 0 0
1 6 0 0 10 0 7 10 8 9 5 0 0 0 4
0 4 10 10 0 10 3 0 2 1 5 10 10 10 6
-1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
obtained in 50 independent runs and the mean values and stan- 9,767.LookingattheSRfigures,onlySAHSandNGHShave100%
darddeviationsofthemareintroducedtoestimatetherobustness successratesforallproblems,i.e.,theoptimumsolutionforeach
ofthealgorithms. instance is reached in each run. EHS, SGHS and ITHS have 100%
Theresultsobtainedbythese9continuousHSvariantsarepre- successratesfor8outof10problems.Moreover,thesuccessrates
sentedinTable3whiletheresultsobtainedbyanother6binaryHS ofthese3algorithmsareverylow,forexample,SGHSachievesthe
variants including SBHS proposed in this paper are presented in optimumforKP7onlyoncein50runs.GHShas100%successrates
Table 4. It should be noted that HMCR in SBHS is set to 0.95 for for9outof10instancesexceptKP8forwhichithasasuccessrate
thelow-dimensional0–1knapsackproblems. of92%.IHS,NDHSandPSFHShave100%successratesfornomore
As shownin Table 3, all continuousHS variantsexcept PSFHS than3problems.
achievetheoptimumforanylow-dimensional0–1knapsackprob- From the above analysis, 9 continuous HS variants can be
lems.PSFHSonlyachievesanoptimumvalueof9,765forKP8in50 roughly divided into 3 groups: the first group consisting of GHS,
independent runs. This is different from the optimum value of SAHS and NGHS, the second group consisting of IHS, NDHS and
Pleasecitethisarticleinpressas:Kong,X.,etal.Asimplifiedbinaryharmonysearchalgorithmforlargescale0–1knapsackproblems.ExpertSystemswith
Applications(2015),http://dx.doi.org/10.1016/j.eswa.2015.02.015