Balanced Quantum-Inspired Evolutionary Algorithm for Multiple Knapsack Problem

Transcription

1 I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11 Publshed Onlne Ocober 2014 n MECS (hp:// DOI: /jsa Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem C. Pavardhan, Sulabh Bansal* Faculy of Engneerng, Dayalbagh Educaonal Insue, Dayalbagh, Agra Anand Srvasav Chrsan-Albrechs-Unversä zu Kel, Insu für Informak, Chrsan-Albrechs-Plaz 4, Kel, Germany Absrac 0/1 Mulple Knapsack Problem, a generalzaon of more popular 0/1 Knapsack Problem, s NP-hard and consdered harder han smple Knapsack Problem. 0/1 Mulple Knapsack Problem has many applcaons n dscplnes relaed o compuer scence and operaons research. Quanum Inspred Evoluonary Algorhms (QIEAs), a subclass of Evoluonary algorhms, are consdered effecve o solve dffcul problems parcularly NP-hard combnaoral opmzaon problems. A hybrd QIEA s presened for mulple knapsack problem whch ncorporaes several feaures for beer balance beween exploraon and exploaon. The proposed QIEA, dubbed QIEA-M, provdes sgnfcanly mproved performance over smple QIEA from boh he perspecves vz., he qualy of soluons and compuaonal effor requred o reach he bes soluon. QIEA-M s also able o provde he soluons ha are beer han hose obaned usng a well known heursc alone. Index Terms Hybrd Evoluonary Algorhm, Quanum Inspred Evoluonary Algorhm, Combnaoral Opmzaon, Mulple Knapsack Problem I. INTRODUCTION 0-1 Mulple Knapsack Problem (M) s a generalzaon of he sandard 0-1 knapsack problem () where mulple knapsacks are consdered o be flled nsead of one. The M p roblem s srongly NPcomplee and no FPTAS s possble for M [1]. Evoluonary A lgorhms (EAs) refer o a class of populaon based search echnque used o oban good soluons for hard opmzaon problems n general. Indvduals n populaon map o soluons of he problem. The new generaons are evolved wh an objecve o mprove qualy of soluons. Evoluon of a new populaon nvolves applcaon of varous operaors on members of exsng populaon. Quanum-Inspred Evoluonary Algorhms (QIEAs) s subclass of EAs where he represenaon of ndvduals and operaors nvolved n generaon of new ndvduals are boh desgned based on he concep of Quanum Compung. Varous forms of QIEAs have been used o solve a varey of dffcul problems for example [2,3,4,5,6,7,8,9,10,11]. QIEAs have been observed as a powerful ool because of her beer represenaon power [12,13], EDA syle of funconng [14,15], flexbly necessary for he ncluson of feaures approprae for a gven problem owards delverng beer search performance [15], nheren qualy of sarng wh exploraon and gradually shfng owards he exploaon [13]. QIEA n self only provdes a very broad framework. QIEA, jus as oher EAs, suffers from several lmaons. Small qub roaons lead o slow convergence whle large qub roaons may cause he algorhm o mss a good soluon compleely. Incluson of feaures promong faser convergence may cause he algorhm o ge suck n local opma. Slow convergence lms he problem szes ha can be ackled usng QIEAs. Implemenaon of QIEAs, herefore, s more an ar. Any aemp o solve a dffcul problem has o use hs framework judcously and nclude feaures sued o he parcular problem n order o ge he desred performance. The objecve n any aemped mplemenaon of QIEA s o balance exploraon and exploaon hus achevng convergence o opmal or near opmal soluons whou requrng prohbvely large compuaon even for larger problem szes. Hybrdzng he populaon based mea-heursc search echnque wh heurscs algorhms avalable for parcular problem s an approach ha s popular snce las few decades o solve dffcul opmzaon problems [16]. The man movaon behnd he hybrdzaon of dfferen algorhms s o explo he complemenary characer of dfferen opmzaon sraeges. Hybrd mea-heursc algorhms ry o esablsh a balance beween exploraon and exploaon of he search space. The populaon based search approaches are good a exploraon of he search space and denfyng areas wh hgh qualy soluons whle hey are no so effecve n exploaon of hese hgh qualy areas. On he oher hand he srengh of local search s he capably of quckly fndng beer soluons n he vcny of sarng soluons. Generally he heurscs avalable for opmzaon problems are based on some knd of local search on an nal soluon. Thus n such algorhms, whch hybrdze a mea-heursc wh a heursc, he mea-heursc approach can gude he global search and problem doman specfc heursc can help searchng locally around he good soluons found from global perspecve. Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

2 2 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem Table 1. Modfcaons appled on QIEA s for n he leraure Modfcaons reference e.g. Type Maxmum Sze n= ems coun m= knapsacks coun Orgnal QIEA (QIEA-o) [17,18] Modfed nalzaon of Qub ndvduals n QIEA [18] [19] [20] [21] Modfcaon n Termnaon Crera [18] Modfcaon n aracor, gae, ec whle Updang he qub [18] [4] [3] [11] [22] D Q MoK n=10,000 n=200 n=750,m=4 Modfyng Repar funcon based on doman knowledge [19] Replacng local and/or global mgraon of QIEA-o wh oher sraegy Incorporaon of genec operaor muaon [23] [24] [3] [11] D Q n=10,000 n=200 Changng number or lengh of q b ndvduals [25] MoK n=750,m=4 Renalzaon of Qubs [23] Incluson of doman knowledge n he search process. [26] [25] [3] MoK MoK D n=750, m=2 n=750, m=4 n=10,000 QIEA-o wh parallel mplemenaon [27] QIEA-o mplemenaon on GPU [28] n=250 *= 0/1 Knapsack Problem, MoK = Mul-objecve Knapsack Problem, Q = Quadrac Knapsack Problem, D= Dffcul 0/1 Knapsack problems Thus, QIEAs wh varous modfcaons have been appled o dfferen varans of popular combnaoral problem called Knapsack Problem (). Table 1 lss he modfcaons appled o QIEAs wh he example exsng n leraure. In hs paper an aemp s made o desgn a hybrd QIEA balanced n s power o explo and explore he search space n order o solve nsances of M. The populaon based mea-heursc QIEA s hybrdsed wh an exsng heursc for M known as MTHM [29] and some addonal feaures of populaon based search are judcously ncorporaed n order o solve randomly generaed nsances of M. Ths s frs aemp o solve M usng such a mea-heursc echnque. The res of he paper s organzed as follows. A bref descrpon of M wh a survey of approaches exsng n leraure for M s presened n secon II. A bref concepual descrpon of a ypcal QIEA s gven n secon III. The MTHM heursc used for hybrdzng QIEA s dscussed n secon IV. In secon V he QIEA framework used here and he proposed QIEA-M s explaned n deal. Compuaonal performance of QIEA-M s presened n secon VI. Conclusons are presened n secon VII. II. MULTIPLE KNAPSACK PROBLEM (M) Gven a se of n ems wh her profs p j and weghs w j, { }, and m knapsacks wh capaces c, { }, he M s o selec a subse of ems o fll gven m knapsacks such ha he oal prof s maxmzed and sum of weghs n each knapsack doesn exceed he capacy c. m n maxmze : j p j x j (1) subjec o : n j w j x j c { m} (2) m x j j { n} (3) (4) x j { } { m} j { n} where x j = 1 f em j s assgned o knapsack, x j = 0 oherwse and coeffcens p j, w j and c are posve negers. In order o avod any rval case, he followng assumpons are made 1. Every em has a chance o be placed a leas n larges knapsack: max j N w j max m c (5) 2. The smalles knapsack can be flled a leas by he smalles em: mn m c mn j N w j (6) 3. There s no knapsack whch can be flled wh all ems of N: n j w j c { m}. (7) The subse sum varan of M havng p j w j j { n} s known as mulple subse sum problem (MSSP). M has many applcaons n felds relaed o compuer scence and operaons research. An applcaon Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

3 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem 3 s seen when schedulng jobs on processors where some machnes unavalable for a fxed duraon or some hgh proprey jobs are pre-assgned o processors [30]. A real world applcaon of M s he problem of cargo loadng where some conaners need o be chosen from a se of n conaners o be loaded n m vessels wh dfferen loadng capaces for he shpmen of he conaners [31]. Anoher real world problem for MSSP s menoned n [32] from a company producng objecs of marble. The company receves m marble slabs, of unform sze, from a quarry. Each produc, ha company produces, requres a pece from marble slab havng a specfed lengh. Ou of he ls of producs o be prepared, some producs need o be seleced and cu from he slabs so ha he oal amoun of wased marble s mnmzed. Ths problem s modelled as MSSP wh slabs correspondng o knapsacks and he lenghs correspondng o he weghs. Several aemps o solve he M exs e.g. [33,29,34] ec. Hung & Fsk [33] presened a deph frs branch and bound algorhm where he upper bounds were derved by Lagrangan relaxaon. Marello & Toh [29] proposed a dfferen branch and bound algorhm where consran m x j s omed a each decson node and he branchng em was chosen as an em whch had been packed n k>1 knapsacks of he relaxed problem. In a laer work Marello & Toh [34] proposed a bound and bound algorhm where a each node of he branchng ree boh he upper bound and lower bound are derved. Some approxmaon algorhms also exs for M. Kellerer [35] presened he PTAS for M wh dencal capaces. Chekur & Khanna [1] generalzed and presened he PTAS for M. He also dscussed M as a specal case of he generalzed assgnmen problem (GAP). GA P s APX-hard and only a 2-approxmaon exss. I s also shown ha no FPTAS s possble for M. A PTAS conanng he wo seps, guessng he ems as frs and packng hem as second, s presened subsequenly n [1]. The EPTAS s desgned based on he LP relaxaon of he M by Jansen [36,37]. Jansen [37] presened a faser verson of he algorhm desgned n [36]. These approxmaon schemes clam o provde soluon havng approxmaon rao of 1/ϵ. The algorhm s shown o ake polynomal me wh respec o sze of he problem bu s exponenal wh respec o he 1/ϵ. III. QUANTUM INSPIRED EVOLUTIONARY ALGORITHM (QIEA) The QIEAs nroduced n [17] are populaon-based sochasc evoluonary algorhms. They use he qub, a vecor, o represen he probablsc sae of ndvdual. Each qub s represened as q * + are complex numbers so ha s he probably of sae beng 1 and s he probably of sae beng 0 such ha. For he purpose of QIEAs, and are assumed o be real. Thus, a qub srng wh n bs represens a superposon of 2 n bnary saes and provdes an exremely compac represenaon of enre space. The process of generang bnary srngs from he qub srng, Q, s known as observaon. To observe he qub srng Q, a srng conssng of he same number of random numbers beween 0 and 1 (R) s generaed. The elemen P s se o 0 f R s less han square of Q and 1 oherwse. In each of he eraons, several soluon srngs are generaed from Q by observaon as gven above and her fness values are compued. The soluon wh bes fness s denfed. The updang process moves he elemens of Q owards he bes soluon slghly such ha here s a hgher probably of generaon of soluon srngs, whch are smlar o bes soluon, n subsequen eraons. A quanum gae s ulsed for hs purpose [17]. Procedure QIEA-orgnal 1 2 nalze Q() 3 make P() by observng he saes of Q() 4 repar P() 5 evaluae P() 6 sore he bes soluons among P() no B() 7 whle ( <MAX_GEN) 8 { 9 10 make P() by observng he saes of Q(-1) 11 repar P() 12 evaluae P() 13 updae Q() 14 sore he bes soluons among B(-1) and P() no B() 15 f (mgraon-perod) 16 {mgrae b or o B() globally or locally, respecvely.} 17 } Fg. 1. Quanum Inspred Evoluonary Algorhm One such gae, used by he QIEAs presened n hs work, s he Roaon Gae, whch updaes he qubs as follows: [ ] [cos sn sn cos ] [ ] (8) where, and denoe probables for h qub n ( + 1) h eraon and s equvalen o he sep sze n ypcal erave algorhms n he sense ha defnes he rae of movemen owards he currenly perceved opmum. The above descrpon oulnes he basc elemens of QIEA Observng a qub srng n mes yelds n dfferen soluons because of he probables nvolved. The fness of hese s compued and he qub srng Q s updaed owards hgher probably of producng srngs smlar o he one wh hghes fness. Ths sequence of seps connues; hese deas can be easly generalsed o work wh mulple qub srngs. Pseudo-code for he QIEA orgnally proposed by Han & Km [17] s presened n fg 1. Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

4 4 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem IV. MTHM HEURISTIC FOR M Marello & Toh [29] gave a heursc algorhm, called MTHM, for solvng he M. Ths algorhm consss of hree phases as descrbed n he followng. The deals are avalable n [38]. 1. Inal feasble soluon s obaned n he frs phase of MTHM by applyng he Greedy algorhm o he frs knapsack; a se of remanng ems s obaned, hen he same procedure s appled for he second knapsack; hs s connued ll he m h knapsack. 2. The nal soluon s mproved durng he second phase by swappng every pars of ems assgned o dfferen knapsacks and nser a new em such ha he oal prof s ncreased. 3. Each seleced em s red o be replaced by one or more remanng ems durng he las f possble so ha he oal prof sum s ncreased. The MTHM heursc has he advanage ha some ems can be exchanged from a knapsack o anoher or excluded from he soluon se so ha oal prof ncreases, whch can lead o an effcen and fas soluon when he soluon gven by he frs phase s good. Moreover hey can be appled o any feasble soluon effecvely. The man drawback of MTHM heursc s ha consders only he exchanges beween a pars of ems and no he combnaons of ems. Varous phases of hs heursc can be nroduced n QIEA o mprove he qualy of soluons found. In secon V use of hese o mprove he performance of QIEA-M s examned n deal. The frs phase of hs heursc s used o repar he nfeasble soluons generaed by collapse operaon of QIEA. The second and hrd phases are appled as local search echnque on he soluons mproved from global perspecve durng he eraons of QIEA. V. QIEA-M In hs work a modfed QIEA framework s used as he sarng pon. In a QIEA he updae operaor s used o gradually modfy he qub ndvduals usng an aracor such ha can generae soluons more smlar o he aracor. QIEA as descrbed by Han & Km [17] manans a populaon of local bes ndvduals correspondng o he populaon of qub ndvduals besdes he global bes ndvdual generaed so far. Thus, he qub ndvduals evolve accordng o dfferen local bes soluons bu same global bes soluon. The updae operaor s appled eher locally or globally, a local level he correspondng local bes ndvdual s used as an aracor and a global level he global bes ndvdual s used as an aracor. The global and/or local bes soluons are replaced f found worse han he new soluons generaed usng a qub ndvdual. These new soluons are used as aracors subsequenly. Here nsead of generang sngle soluon, a qub ndvdual generaes mulple ndvduals every me before comparng hose wh sored bes ndvduals generaed so far such ha only he bes ou of all hese mulple soluons generaed s acually compared wh local or global bes ndvduals avalable. Ths helps n exploaon of he areas of soluon space represened by he parcular qub ndvduals more nensvely before roang hem owards an aracor. The qub ndvdual s roaed owards he local bes soluon more ofen han he global bes soluon. Such an arrangemen esablshes he balanced capably o explore and explo smulaneously whch s a foremos requremen for any mea-heursc mplemenaon. The above descrpon provdes a broad framework of QIEA wh scope for enhancemens for rapd soluon of a specfc problem. Ths frame work s used as he sarng pon here whch has been enhanced wh several feaures. The modfed algorhm desgned for M s named as QIEA-M. The mprovemens brough abou n QIEA-M are as follows. ()The ems are sored n order of decreasng prof by wegh rao o mprove he doman knowledge n he followng a. Inalzng he Qub Indvduals so ha hey generae beer soluons. b. Modfyng he repar funcon o mprove qualy of soluons. () Improvng he local bes soluons usng local search ()Muaon of soluons when hey appear o be suck n a local opmum (v) Re-nalzaon of Qub ndvduals (v) Local exploaon before global exploraon A. Sorng of ems n npu The ems havng a greaer prof by wegh rao are consdered o have hgher probably of her ncluson n he opmal soluon. Thus he ems n npu are sored n he decreasng order of her prof by wegh rao. Ths sorng s used o nalze qub ndvduals so ha hey can generae beer soluons and also o mprove repar procedure so ha provdes beer soluons. Inalzng he Qub Indvduals o depc he beer esmaons of dsrbuon models: In order o nalze he qub ndvduals, he ems are dvded n o 3 classes based on where hey le n order of preference; he frs class conans ems havng hgh preference for selecon n a knapsack, ems n second par have nermedae preference and hrd conan ems havng low preference. Hence, qubs for ems lyng n frs class (hrd class) are assgned values closer o 1 (0) so ha hey have hgh (low) probably of collapsng o value 1. Iems lyng n he second class requre more processng for convergence o eher 0 or 1, hence nermedae values beween 0 and 1 are assgned o hem. As a resul, QIEA-M sars explong he area or regon n soluon space havng hgher probably of havng soluons closer o opmal. Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

5 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem 5 Fg. 2. Inalzaon of qubs. Qubs for ems shown are sored n decreasng prof by wegh rao from lef o rgh Procedure ReparM (x) 1 le { }; 2 for from 1 o m{ 3 whle ( ) { 4 { } 5 6 }//*whle*// 7 }//* for *// 8 for j from 1 o n { 9 f (k = { } ) {x kj 1; ;} 10 } //*for j*// Fg. 3. Pseudo-code for ReparM Modfyng he repar funcon o mprove qualy of soluons: QIEA uses a smple repar funcon afer observes he qubs hrough he make procedure o make he observed soluon feasble. In QIEA-M, he repar funcon s modfed o mprove he qualy of soluons whle makng hem feasble based on he phase 1 of MTHM heursc presened n secon IV. Ths mproves he speed of convergence. As explaned earler he ems are sored n order of her preference o nclude hem no a knapsack. So, n each repar sep, ems closes o he end are removed and ems closes o he begnnng are added as necessary. The knapsacks are assumed o be sored n order of her ncreasng capacy and ha s he order n whch hey are consdered when ems are added no knapsacks. The pseudo-code s gven n Fg. 3. B. Improvng he local bes soluons The local bes soluons are furher mproved n wo sages based on he phases 1 and 2 of MTHM heursc [29] descrbed n secon IV. In frs sage res o exchange every par of ems assgned o dfferen knapsacks along wh nserng a new em so ha oal prof s ncreased. Secondly, every seleced em (sarng from las n he sored order) s red o be replaced by one of he remanng ems so ha he oal prof sum s ncreased. The pseudo-code of hese procedures s gven n Fg. 4. and Fg. 5. C. Muaon of soluons appearng o be suck n local opmum. EAs suffer from endency of geng suck n local opma. All he modfcaons descrbed above help he algorhm o explo he search space around he greedy soluons ncreasng he speed of convergence bu hey also ncrease he endency o ge suck n local opma. To comba hs problem, f a new soluon generaed s seen o be close o global bes soluon found so far s muaed. Durng muaon, afer 2-3 bs n he soluon vecor are randomly seleced and changed o 0, he paral soluon s mproved usng ImroveSage1. To check closeness of wo soluons, Hammng dsance beween hem s calculaed. Such an operaor mproves dversy whou ncreasng he compuaonal effor. I helps o explore he soluon space around a curren soluon such ha local opmal n vcny s no mssed. Ths mproves he chances of fndng opmal n case s n vcny of he convergng soluon. Procedure ImproveSage1(x) 1 le { }; 2 for each par and j n n 3 f ( { } ){ 4 f { } { 5 le k = ( { } ) 6 f ( ){ } 7 else { } 8 }//* f*// 9 }//* f*// 10 }//*for par and j*// Fg. 4. Pseudo-code for ImproveSage1 Procedure ImproveSage2(x) 1 le { }; 2 for { }{ 3 f ( { } { 4 for { } { } { 5 f(r u w w j P j p j p ; else P j ; 6 }//* for j*/ 7 le P k max j { n}p j ; 8 f( P k { x u x uk R u R u w w k } 9 }//*If u *// 10 }//*for *// Fg. 5. Pseudo-code for ImproveSage2 D. Re-nalzaon of Qub ndvduals. I may happen even afer applyng muaon as explaned n secon 5.3 ha all he soluons generaed from a qub ndvdual are sll same afer a sequence of generaons. I clearly ndcaes ha such a qub ndvdual has converged and no furher new soluons can be generaed usng hem. Thus, each qub n ndvduals whch generae same soluon for more ha 3 mes ou of 5 s rese as explaned n secon 5.1. I Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

6 6 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem ncreases he dversy of soluons explored hrough he qub ndvduals whou ncreasng he compuaonal effor. E. Local exploaon before global exploraon. QIEA has he propery ha updaes he qubs over he me perod such ha hey represen Esmaon of Dsrbuon models. Thus, basc seps of QIEA are execued for some eraons o updae some of he qub ndvduals nalzed as descrbed n secon V.A.1. The seps lsed n he followng are performed on half of he qub ndvduals for small number of mes (emprcally se as 15 for hs work) before sarng QIEA -M on he enre populaon. Make ReparM ImproveSage1 ImproveSage2 Updae The execuon of hese seps of QIEA wll also explo nensvely he area represened by he curren qub ndvduals before sarng he QIEA-M whch s balanced wh respec o s power of exploaon and exploraon. The resulng qub ndvduals hus favour he small soluon subspace close o beer soluons. A soluon o M mus specfy wheher he em s ncluded or no and f s ncluded hen he ndex of knapsack has been pu no. Hence an opmal soluon does no requre only he selecon of correc ems bu also ha hey are packed no he correc knapsacks. In hs work, wo componens (n he form of wo bnary srngs) are used o represen a soluon ndvdual where, frs of lengh n conans 0 or 1 for each em conveyng abou s selecon saus, and second of lengh (n* ) conans he ndex n bnary of knapsack n whch s packed. The negers n m are represened by a b srng of lengh. A qub ndvdual s hus represened by wo componens of lenghs n and n* correspondngly. The complee pseudo-code for QIEA-M s presened n Fg. 6. In he pseudo-code for M; refers o he curren eraon, wo componens of populaon of qub ndvduals afer h eraon are represened usng Q1() and Q2(), P1() and P2() represen he populaon of ndvdual soluons, B1() and B2() s he se of bes soluons correspondng o each ndvdual, c s he capacy of he h knapsack. Indvduals represened by Q1() and Q2() are referred o as q j (composed of q j and q j ), ndvduals n P1() and P2() are referred o as p j (composed of p j and p j ), smlarly ndvduals n B1() and B2() are referred o as b j (composed of b j and b j ) for each j n ; b refers o global bes soluon havng componens vz., b1 and b2. In he followng paragraph a bref descrpon s gven for procedures called n QIEA-M bu have no been descrbed ye. InalzeGreedy (q j ): where q j s j h qub ndvdual n a populaon Q1(). The procedure nalzes he qub ndvdual q j as explaned n secon V.A. Make P1() and P2() from Q1() and Q2(): The procedure collapses he qub ndvduals n Q1() (or Q2()) observng soluon ndvduals n P1() (or P2()). HamDsance( p j s, b1): Reurns hammng dsance beween wo bnary srngs. Muae( p j s ): Muaes he soluon p j s as descrbed n secon V.D. Updae q j based on b j : Roaes he qubs q j bs n b j and q j owards bs n b j owards as explaned earler and defned n [17] usng he roaon angle as Evaluae P(s): Assgn ndvduals n P(s) o her prof values. The maxmum number of eraons n algorhm s conrolled usng a global consan, MaxIeraons. VI. RESULTS AND DISCUSSION The expermens are done on Inel Xeon Processor E5645 ( 12M Cache, 2.40 GHz, 5.86 GT/s Inel QPI ). The machne uses Red Ha Lnux Enerprse 6. The soluons converged for mos of he problem nsances consdered here whn 10 eraons hence maxieraons s se o 10. Emprcally, and are se o 5 and populaon sze s se o 10. The expermens are performed o observe he effec of he modfcaons presened n secons V.A hrough V.E on basc QIEA framework dscussed n he begnnng of secon V. The performance of QIEA and QIEA-M s observed on randomly generaed nsances havng elemens 1000, 5000 and wh number of knapsacks as 2, 5 and 10. The problem nsances are randomly generaed, usng he generaor of nsances avalable a he Psnger s home page vz. hp:// where weghs w j are dsrbued n [1, R] and profs p j are calculaed as p j = w j + R/10. Such nsances correspond o a real-lfe suaon where he reurn s proporonal o he nvesmen plus some fxed charge for each projec. Two dfferen classes of capaces are consdered for he knapsacks n randomly generaed nsances vz., smlar and dssmlar. In nsances wh smlar capaces he frs m-1 capaces c m- are dsrbued n [ n w j j m n w j j m] m- whle nsances wh dssmlar capaces have hese c dsrbued n - * ( n j w j - k c k )+ for m- (10) The las capacy c m n boh classes s chosen as (9) c m n j w j m c (11) Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

7 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem 7 Tables 2 and 3 presen he performance of QIEA and QIEA-M for he nsances havng smlar capaces. Tables 4 and 5 presen he performance of QIEA and QIEA-M for he nsances havng dssmlar capaces. Insances are randomly generaed wh number of elemens rangng from 1000 o and number of knapsacks rangng from 2 o 100. The nsances havng 100 knapsacks of dssmlar capaces could no be generaed. Procedure QIEA-M 1 SorGreedy he Inpu; 2 ; b 0; 3 InalzeGreedy ( q j ) for each j { n}; 4 Inalze q j o for each j { n log m}; 5 Make P1() and P2() from Q1() and Q2(); 6 ReparM (P1()); 7 copy P1() o B1() and P2() o B2(); 8 for each j { n } { 9 Make p j from q j ; 10 ReparM (P1()); 11 ImproveSage1(P1()); 12 ImproveSage2(P1()); 13 f ( p j s beer han b j ) b j p j ; 14 } 15 whle ( <MaxIeraons) { 16 ;cn j 17 for r from 0 o do 18 for s from 0 o do 19 Make P(s) from Q(); 20 ReparM (P(s)); 21 Evaluae P(s); 22 for each j { s n} f p j beer han p j ) p j p s j ; else cn j cn j ; 23 } //*for s*// 24 for each j { n} { 25 f (HamDsance( s p j,b)<2) {Muae( p s s j );ImproveSage1( p j ;} 26 }//*for j*// 27 f(cn j ){ 28 InalzeGreedy ( q j ) for each j { n}; 29 Inalze q j o for each j { n log m}; 30 }//*f cn j *// 31 for j { n } ImproveSage2( p j ); 32 for each j { n} f ( p j s beer han b j ) b j p j ; 33 for each j { n} f (b j s beer han b b b j ; 34 for each j { n} Updae q j based on b j ; 35 } //*for r*// 36 for each j { n} Updae q j based on b; 37 } // *whle*// Fg. 6. Pseudo-code for QIEA-M The ables show a comparson of prof values obaned usng he specfc forma of QIEA wh values obaned usng he heursc menoned n secon IV. The bes, average, wors, sandard devaon n prof values obaned whn 30 ndependen runs of he algorhm have been repored along wh he mnmum and average compuaonal effor requred n erms of number of funcon evaluaons (FES) as o reach he bes (MnFES and AvgFES respecvely) and RDH relave dsance of bes soluon from heursc. If prof for he nsances obaned usng exac algorhm s P h and bes prof obaned s P b he RDH s calculaed as follows. R H ((P b -P h ) P h ) (12) The convergence of prof values acheved usng QIEA and QIEA-M s suded and compared. Fg 7 shows how he bes prof value obaned hrough QIEA and QIEA-M converges. The eraon number and he prof acheved have been ploed on x-axs and y-axs respecvely. QIEA-M execues he basc seps of QIEA for 15 mes durng nalzaon of half of he qub ndvduals as explaned n secon V.E. Resuls for hese addonal seps are ploed usng eraon numbers from I0 o I15 on x-axs, where values exs only for QIEA-M. I s clear from fg 7 ha Boh algorhms sar wh smlar profs. QIEA-M forgoes ahead of QIEA durng nalzaon seps. However, QIEA-M fnds sgnfcanly beer soluons wh n a few eraons of he man loop of he algorhm. Even afer 50 eraons of he man loop, QIEA s sll far away from he value acheved usng QIEA -M n a few eraons Table 2. Performance of QIEA on nsances wh smlar capaces n m Heursc (MTHM) Bes Average Wors Sddev MnFES AvgFES RDH Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

8 8 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem Table 3. Performance of QIEA-M on nsances wh Smlar Capaces n m Heursc (MTHM) Bes Average Wors Sddev MnFES AvgFES RDH Table 4. Performance of QIEA on nsances wh dssmlar capaces n m Heursc (MTHM) Bes Average Wors Sddev MnFES AvgFES RDH Table 5. Performance of QIEA-M on nsances wh dssmlar capaces n m Heursc (MTHM) Bes Average Wors Sddev MnFES AvgFES RDH Fg. 7. Comparng convergence of bes value acheved usng QIEA and QIEA-M (an nsance havng 0, m=100) Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

9 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem 9 Insances wh Smlar Capaces compuaonal effor. RDH of he soluons obaned usng QIEA and QIEA-M wh respec o sze of problem nsances havng dfferen classes of capaces consdered s shown n fg 8. Fg 9 plos he Average FES requred usng QIEA and QIEA-M wh respec o sze of problem nsances havng dfferen classes of capaces consdered. Followng pons are observed from he resuls: QIEA-M shows consderable mprovemen boh n qualy of soluons obaned and compuaonal effor requred o reach he bes as compared o QIEA. The soluons obaned from QIEA-M are much beer han he heursc used o mprove soluons locally for mos of he nsances esed. Ths mprovemen s beer for nsances whch requred o fll more number of knapsacks. The hybrdsed QIEA-M s able o provde much beer soluons whn very less number of FES as compared o QIEA used as he base. The graph showng qualy of soluons wh respec o heursc soluon (.e. RDH) for QIEA-M s smlar n shape as of QIEA bu wh a shf from area havng soluon worse han heursc o he area havng beer soluon han heursc. Insances wh Dssmlar Capaces Fg. 8. Comparson of QIEA and QIEA-M on he bass of RDH Insances wh Smlar Capaces VII. CONCLUSIONS The QIEA s mproved by embeddng whn he local mprovemen based on a known effecve heursc for M wh an objecve o mprove exploaon. Apar from some echnques vz. muaon of soluons appearng o be close o local opma, and renalzng qub ndvduals found ncapable o generae new soluons are also nduced n order o mprove he power o explore he search space. Ths way he proposed QIEA, wh balanced power of exploaon and exploraon of search space, provde sgnfcanly beer soluons wh respec o boh he QIEA used as base and he heursc used for proposed hybrdzaon. The proposed QIEA provde beer resuls usng much reduced compuaonal effor han basc QIEA. ACKNOWLEDGMENTS Auhors are graeful o Deparmen of Scence and Technology, Inda (DST) and Deusche Forschungs - gemenschaf, Germany (DFG) for he suppor under projec No. INT/FRG/DFG/P-38 led Algorhm Engneerng of Quanum Evoluonary Algorhms for Hard Opmzaon Problems". Insances wh dssmlar capaces Fg. 9. Comparson of QIEA and QIEA-M on he bass of FES requred reachng her bes soluon. Fgures 8 and 9 presen a comparson of QIEA and QIEA-M on he bass of qualy of soluons and REFERENCES [1] C. Chekur and S. Khanna, "A PTAS for he mulple knapsack problem," SIAM Journal on Compung, vol. 35, pp , Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

10 10 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem [2] S.Y. Yang, M. Wang, and L.C. Jao, "A novel quanum evoluonary algorhm and s applcaon," n Proc CEC, 2004, pp [3] C. Pavardhan, Apurva Narayan, and A. Srvasav, "Enhanced Quanum Evoluonary Algorhms for Dffcul Knapsack Problems," n PReMI'07 Proceedngs of he 2nd nernaonal conference on Paern recognon and machne nellgence, 2007, pp [4] M. D. Plael, S. Schlebs, and N. Kasabov, "A versale quanum-nspred evoluonary algorhm," n Proc. CEC, 2007, pp [5] G. S. Salesh Babu, D. Bhagwan Das, and C. Pavardhan, "Real-Parameer quanum evoluonary algorhm for economc load dspach," IET Gener. Transm. Dsrb., vol. 2, no. 1, pp , [6] A. Man and C. Pavardhan, "A Hybrd quanum evoluonary algorhm for solvng engneerng opmzaon problems," Inernaonal Journal of Hybrd Inellgen Sysems, vol. 7, pp , [7] Lng Wang and Lng-po L, "An effecve hybrd quanumnspred evoluonary algorhm for parameer esmaon of chaoc sysems," Exper Sysems wh Applcaons, vol. 37, no. 2, pp , [8] Shuyuan Yang, Mn Wang, and Lcheng Jao, "Quanumnspred mmune clone algorhm and mulscale Bandele based mage represenaon," Paern Recognon Leers, vol. 31, no. 13, pp , [9] Jng Xao, YuPng Yan, Jun Zhang, and Yong Tang, "A quanum-nspred genec algorhm for k-means cluserng," Exper Sysems wh Applcaons, vol. 37, no. 7, pp , [10] P. Arpaa, D. Maso, and C. Manna, "A Quanum-nspred Evoluonary Algorhm wh a compeve varaon operaor for Mulple-Faul Dagnoss," Appled Sof Compung, vol. 11, no. 8, pp , [11] C Pavardhan, P Prakash, and A Srvasav, "A novel quanum-nspred evoluonary algorhm for he quadrac knapsack problem," In. J. Mahemacs n Operaonal Research, vol. 4, no. 2, pp , [12] K. Han and J. Km, "On seng he parameers of quanum-nspred evoluonary algorhm for praccal applcaon.," n Proc. CEC, 2003, pp [13] K-H Han, "On he Analyss of he Quanum-nspred Evoluonary Algorhm wh a Sngle Indvdual," n IEEE Congress on Evoluonary Compuaon, Vancouver, Canada, [14] M. D. Plael, Sefan Schlebs, and Nkola Kasabov, "Quanum-Inspred Evoluonary Algorhm: A Mulmodel EDA," IEEE Transacons on Evoluonary Compuaon, vol. 13, no. 6, pp , [15] Gexang Zhang, "Quanum-nspred evoluonary algorhms: a survey and emprcal sudy," Journal of Heurscs, vol. 17, no. 3, pp , [16] C. Blum, J. Puchnger, G. R. Radl, and A. Rol, "Hybrd meaheurscs n combnaoral opmzaon: A survey," Appled Sof Compung, vol. 11, pp , [17] Kuk-Hyun Han and Jong-Hwan Km, "Quanum-Inspred Evoluonary Algorhm for a Class of Combnaoral Opmzaon," IEEE Transacons on Evoluonary Compuaon, vol. 6, no. 6, pp , December [18] K. Han and J. Km, "Quanum-nspred evoluonary algorhms wh a new ermnaon creron, h-epslon gae, and wo-phase scheme.," IEEE Trans. Evol. Compu., vol. 8, no. 2, pp , [19] Zhongyu Zhao, Xyuan Peng, Yu Peng, and Enzhe Yu, "An Effecve Repar Procedure based on Quanumnspred Evoluonary Algorhm for 0/1 Knapsack Problems," n Proceedngs of he 5h WSEAS In. Conf. on Insrumenaon, Measuremen, Crcus and Sysems, Hangzhou, 2006, pp [20] G.X. Zhang, M. Gheorghe, and C.Z. Wu, "A quanumnspred evoluonary algorhm based on p sysems for knapsack problem," Fund. Inf., vol. 87, no. 1, pp , [21] M. -H. Tayaran-N and M. -R. Akbarzadeh-T, "A Snusod Sze Rng Srucure Quanum Evoluonary Algorhm," n IEEE Conference on Cybernecs and Inellgen Sysems, 2008, pp [22] Zhyong L, Güner Rudolph, and Kenl L, "Convergence performance comparson of quanum-nspred mulobjecve evoluonary algorhms," Compuers and Mahemacs wh Applcaons, vol. 57, pp , [23] Parvaz Mahdab, Saeed Jall, and Mahd Abad, "A mulsar quanum-nspred evoluonary algorhm for solvng combnaoral opmzaon problems," n (GECCO '08) Proceedngs of he 10h annual conference on Genec and evoluonary compuaon, 2008, pp [24] Takahro Imabeppu, Shgeru Nakayama, and Saosh Ono, "A sudy on a quanum-nspred evoluonary algorhm based on par swap," Arf Lfe Robocs, vol. 12, pp , [25] T-C Lu and G-R Yu, "An adapve populaon mulobjecve quanum-nspred evoluonary algorhm for mul-objecve 0/1 knapsack problems," Informaon Scences, vol. 243, pp , [26] Y. Km, J. H. Km, and K. H. Han, "Quanum-nspred mulobjecve evoluonary algorhm for mulobjecve 0/1 knapsack problems," n Proc. CEC, 2006, pp [27] K. Han, K. Park, C. Lee, and J. Km, "Parallel quanumnspred genec algorhm for combnaoral opmzaon prblem," n Proc. CEC, vol. 2, 2001, pp [28] R. Nowonak and J. Kucharsk, "GPU-based unng of quanum-nspred genec algorhm for a combnaoral opmzaon problem," BULLETIN OF THE POLISH ACADEMY OF SCIENCES: TECHNICAL SCIENCES, vol. 60, no. 2, pp , [29] S. Marello and P. Toh, "Soluon of he zero-one mulple knapsack problem," European Journal of Operaonal Research, vol. 4, no. 4, pp , [30] F. Dedrch and K. Jansen, "Improved approxmaon algorhms for schedulng wh fxed jobs," n Proceedngs of he 20h ACM -SIAM Symposum on Dscree Algorhms (SODA2009), 2009, pp [31] S. Elon and N. Chrsofdes, "The loadng problem," Managemen Scence, vol. 17, pp , [32] Hans Kellerer, Ulrch Pferschy, and Davd Psnger, Knapsack Problems. Berln.Hedelberg: Sprnger-Verlag, [33] M. S. Hung and J. C. Fsk, "An algorhm for he 0-1 mulple knapsack problems," Naval Reserach Logscs Quarerly, vol. 24, pp , [34] S. Marello and P. Toh, "A bound and bound algorhm for he zero-one," Dscree Appled Mahemacs, vol. 3, pp , [35] H. Kellerer, "A polynomal me approxmaon scheme for he mulple knapsack problem," n Proceedngs of he 2nd Workshop on Approxmaon Algorhms for Combnaoral Opmzaon Problems, APPROX(1999), LNCS 1671, 1999, pp [36] K. Jansen, "Parameerzed approxmaon scheme for he mulple knapsack problem," Sam Journal on Compung, vol. 39, no. 4, pp , Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11

11 Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem 11 [37] K. Jansen, "A fas approxmaon scheme for he mulple knapsack problem," n 38h Conference on Curren Trends n Theory and Pracce of Compuer Scence, LNCS 7147, 2012, pp [38] S. Marello and P. Toh, Knapsack Problems: Algorhms and Compuer Implemenaons. Chcheser, UK: Wley, How o ce hs paper: C. Pavardhan, Sulabh Bansal, Anand Srvasav,"Balanced Quanum-Inspred Evoluonary Algorhm for Mulple Knapsack Problem", Inernaonal Journal of Inellgen Sysems and Applcaons(IJISA), vol.6, no.11, pp.1-11, DOI: /jsa Auhors Profles C. Pavardhan s workng n he Dayalbagh Educaonal Insue, Agra as Professor n Elecrcal Engneerng and Coordnaor of he PG program n Faculy of Engneerng, DEI. He obaned hs BSc (Engg.) degree from Dayalbagh n 1987, MTech from IISc, Bangalore n Compuer Scence n 1989 and hs PhD from DEI n He has publshed more han 250 papers n journals and proceedngs of conferences and has won 20 Bes Paper Awards. He has also publshed one book and has been an Edor of hree Conference proceedngs. He has been he Invesgaor/Co-Invesgaor of several funded R&D projecs. He has been a consulan on Advanced Algorhms o varous EDA ndusres. Hs curren research neress are quanum and sof compung and mage processng. Sulabh Bansal s a Research Assocae and Ph.D. scholar wh he Deparmen of Elecrcal Engneerng, Dayalbagh Educaonal Insue, Agra. He obaned BE (Hons.)(Compuer Scence and Engneerng) n 2000 and MTech n He has publshed several research papers n varous naonal and nernaonal conferences and journals. He has rch 12 years experence dversfed n sofware ndusry as developer and eam leader and n academcs as an educaor and researcher. He has successfully managed several sofware developmen projecs n he ndusry and guded several B.Tech. and M.Tech. level projecs. Hs curren research neress are evoluonary algorhms, opmzaon problems and parallel programmng. Anand Srvasav s a Professor n he Deparmen of Compuer Scence, Chrsan-Albrechs-Unversa zu Kel, Germany. He obaned hs docorae from he Unversy of Munser n Thereafer, he worked as an Asssan Professor n he Unversy of Bonn, he New York Unversy, Yale Unversy, he Free Unversy, Berln and Humbold Unversy of Berln. He has held senor posons a several nsuons, and has been drecor of he Compuaonal Scence Cener a Chrsan- Albrechs-Unversa zu Kel snce He was a speaker of he DFG Research Tranng Group Effcen Algorhms and Mulscale Mehods from 2005, and has been he PI of he German Research Foundaon s Fuure Ocean Cluser a Chrsan-Albrechs-Unversa snce He s also he PI for he Maerals for Lfe Cluser of Excellence proposed n 2010 a Chrsan-Albrechs-Unversa, Kel. Copyrgh 2014 MECS I.J. Inellgen Sysems and Applcaons, 2014, 11, 1-11