An Adaptive Quantum-inspired Differential Evolution Algorithm for 0-1 Knapsack Problem

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1 An Adapive Quanu-inspired Differenial Evoluion Algorih for 0- Knapsack Proble Ashish Ranan Hoa Deparen of Elecrical Engineering Indian Insiue of Technology Kharagpur, India Anki Pa Deparen of Maheaics Indian Insiue of Technology Kharagpur, India Absrac Differenial evoluion (DE is a populaion based evoluionary algorih widely used for solving ulidiensional global opiizaion probles over coninuous spaces. However, he design of is operaors akes i unsuiable for any real-life consrained cobinaorial opiizaion probles which operae on binary space. On he oher hand, he quanu inspired evoluionary algorih (QEA is very well suiable for handling such probles by applying several quanu copuing echniques such as Q-bi represenaion and roaion gae operaor, ec. This paper exends he concep of differenial operaors wih adapive paraeer conrol o he quanu paradig and proposes he adapive quanu-inspired differenial evoluion algorih (AQDE. The perforance of AQDE is found o be significanly superior as copared o QEA and a discree version of DE on he sandard 0- knapsack proble for all he considered es cases. Keywords- differenial evoluion; quanu inspired evoluionary algoih; 0- knapsack proble; quanu copuing I. INTRODUCTION Differenial Algorih (DE, inroduced by Sorn and Price [,2] has been shown o give significanly beer perforance in ers of efficiency and robusness on any benchark uliodal coninuous funcions han oher populaion based evoluionary algorihs. For exploraion of he search space and o inroduce diversiy, i eploys wo siple uaion and crossover operaors respecively followed by a greedy replaceen sraegy. The perforance is found o be very sensiive o he uaion and crossover paraeers chosen and he bes cobinaion of boh he paraeers changes fro one funcion o anoher. Thus, a large nuber of odificaions have been proposed o ake he selecion of conrol paraeers adapive and free fro funcion dependency [3-7]. Because of is superior perforance on coninuous opiizaion probles, several odificaions have been inroduced in he pas, so ha i operaes on binary space. Papara, Engelbrech and Franken [8] proposed an angle odulaion schee (AMDE o ap he coninuous space o binary. On siilar lines, binary differenial evoluion (binde and noralizaion DE (norde were proposed based on sigoid funcion apping and noralizaion of coninuous space respecively [9] giving beer resuls as copared o AMDE. A discree binary version of differenial evoluion (DBDE for solving 0- knapsack proble was also proposed [0]. To solve various opiizaion probles beer han he convenional evoluionary algorihs, a broad class of algorihs have been proposed by applying several conceps of quanu copuing in he pas decade. Quanu copuing uses he quanu echanical phenoena like superposiion, enangleen, inerference, de-coherence, ec o develop quanu algorihs. Many quanu algorihs have been shown o be exponenially faser and assively parallel as copared o classical algorihs [, 2]. Thus quanu inspired geneic algorihs wih inerference as crossover operaor [3], quanu inspired evoluionary algorihs (QEA [4], quanu behaved paricle swar opiizaion [5] ec has been developed for boh coninuous and binary spaces. QEA uses superposiion of binary bis known as Q-bi for represenaion of individuals and updaes he individuals depending on heir values wih respec o he global bes soluion by suiably deciding he paraeer of he roaion gae operaor. Broadly i coes under he class of esiaion of disribuion algorihs (EDA [23]. QEA has deonsraed quie significan resuls on binary opiizaion probles and soe iproveens on QEA have also been proposed [6, 7]. QEAs have been exended by differenial operaors o solve flow shop scheduling probles [8], N- queen s proble [9], for classificaion rule discovery [20] and soe benchark funcions [2]. In his paper, an adapive quanu-inspired differenial evoluion algorih (AQDE is proposed wih adapive conrol of uaion and crossover paraeers and he operaors acing direcly on he superposiion saes of he individual. The proposed AQDE ouperfors QEA and DBDE under differen condiions of populaion size and ie size of he 0- knapsack proble. The res of his paper is organized as follows: Secion II gives a brief inroducion of knapsack proble, DE, DBDE and QEA. The proposed AQDE is explained in deail in secion III. Experienal seings and he resuls obained are enioned under secion IV. Finally, secion V concludes he paper.

2 II. BACKGROUND A. 0 - Knapsack Proble: The 0- knapsack proble is a classical proble in cobinaorial opiizaion. Proble Descripion: In a given se of ies each ie has an ineger weigh w and an ineger profi p. The proble is o selec a subse fro he se of ies such ha he overall profi is axiized wihou exceeding a given weigh capaciy W. I is an NP-Hard proble and hence doesn have a polynoial ie algorih. The proble ay be aheaically odeled as follows: Maxiize: pixi ( i= Subec o he consrain: wx i i W, xi {0,} (2 i= where x i akes values of eiher or 0 represening he selecion or reecion of he i h ie. B. Differenial Evoluion : In classical DE, each eber of he populaion is represened by a real valued D-diensional vecor. A ypical ieraion of he DE algorih consiss of hree aor operaions uaion, crossover and selecion, which are carried ou for each eber of he populaion (called as arge vecor. Muaion on each arge vecor of he populaion generaes a new uan vecor uniquely associaed wih i. Then he crossover operaion generaes a new rial vecor using he uan vecor and he arge vecor iself. In selecion phase he finess of he rial vecor is copared wih he arge vecor and he vecor wih higher finess replaces he arge vecor in he populaion for he nex ieraion. The hree operaions uaion, crossover and selecion, are discussed in deail below. Muaion: The uan V i vecor on a arge vecor X i is generaed by adding a randoly seleced vecor X r fro he populaion, wih a weighed difference of wo oher randoly seleced vecors X r2, X r3 fro he populaion. V = X + F.( X X (3 i r r2 r3 where r,r2 and r3 are all disinc and differen fro i. The paraeer denoes he generaion. F is a conrol paraeer whose value is ypically chosen beween 0 and 2. Crossover: The crossover operaion generaes a rial vecor U i fro is corresponding arge vecor X i and uan vecor V i, by using he following relaion: u v, if ( rand (0, CR or( = Irand = x, i, if ( rand (0, > CR and( Irand where =,2,..D, U i = (u,i, u 2,i,.., u D,i, rand is he h evaluaion of a rando nuber generaor in [0,] fro a unifor disribuion. I rand is a randoly chosen diension index fro {,2,..,D} which ensures ha he new rail vecor is differen fro he arge vecor. CR is a conrol (4 paraeer which decides he crossover rae and is value is ypically chosen in he range of 0 o. Selecion: If he rial vecor U i has a beer finess value copared o he arge vecor, hen i replaces he arge vecor in he populaion in he nex ieraion. Oherwise, he arge vecor reains unchanged in he populaion. C. Discree Binary version of Differenial evoluion The discree binary version of differenial evoluion (DBDE [0] was an aep o develop an algorih which worked on siilar lines as DE bu on a binary D-diensional space. DBDE has is roos in DE and a discree binary version of paricle swar opiizaion (DPSO [22]. Here he individuals are iniialized as a binary sring. The uaion operaor is exacly siilar o ha of DE, bu he resulan uan vecor is no longer binary because of he difference operaor and he conrol paraeer. Therefore he discreizaion process fro a real coninuous space o a binary space is done according o he following equaion: v id,, if rand(0, sig( vid, = 0, if rand(0, > sig( vid, where rand is a rando nuber in he range [0,] seleced uniforly a rando. sig( is a sigoid liiing ransforaion funcion and v i,d is d h diensional value of he i h uaed vecor in generaion. The crossover and selecion operaions in DBDE, are sae as in DE. D. Quanu- Inspired Evoluionary Algorih Quanu-inspired Evoluionary Algorih (QEA, as is nae indicaes, is inspired fro he principles of quanu copuing, bu i is designed o run on a classical copuer. In QEA, he salles uni of inforaion is called Q-bi and is defined as [α,β] T, where α and β are coplex nubers ha specify he probabiliy apliude of he respecive Q-bi saes such ha α 2 + β 2 =. α 2 represens he probabiliy ha he Q-bi will be in sae 0 and β 2 represens he probabiliy ha he Q-bi will be in sae. The represenaion for an individual q of QEA wih -bi is given as follows: q α α... α 2 = β β2... β where α i 2 + β i 2 =, i=,2,.. Algorih Descripion: In he beginning, he populaion is iniialized wih he α and β of all bis of all individuals se o / 2. In each generaion, binary srings are generaed fro he respecive Q-bi srings by observing he Q-bi saes using he following crieria: P i, 2, if rand( < βi, = 0, oherwise where P i, is he h bi of i h individual in he populaion. Once he populaion consising of he binary srings has been generaed, he finess value of hese srings is evaluaed and he bes soluions are sored separaely in a global pool B. (5 (6 (7

3 The global bes soluion b aong all he soluions in B is deerined. Then, a quanu roaion gae U(θ is used o updae he values of he Q-bis of each individual as follows: cos( Δθi sin( Δθi U ( Δ θi = sin( Δθi cos( Δθi where Δθ i, i=,2,. is he roaion angle of each Q-bi owards eiher 0 or depending on is sign. The paraeer Δθ i is decided by coparing he value of he bi in he individual and he corresponding bi in he global bes individual as per Table I (reproduced fro [4]. Table I. Look up Table for Δθ i (f(. is he profi and b i and x i are i h bi of bes soluion b and binary soluion x Then he global pool is updaed wih finess based replaceen by beer individuals of he presen generaion and he previous global pool. The global bes individual is also updaed accordingly. A global and local igraion is invoked wih a definie frequency, in which all or soe of he individuals of he global pool are replaced by he global bes or he local bes individuals respecively. The deailed procedure of QEA [4] is provided below for beer undersanding. III. ADAPTIVE QUANTUM DIFFERENTIAL EVOLUTION This secion describes he adapive quanu-inspired differenial evoluion algorih (AQDE. A. Represenaion Insead of using [α,β] T like QEA as he represenaion of Q-bis, AQDE uses he variable θ for reasons discussed laer. Since α 2 + β 2 =, i basically represens he equaion of a uni circle and each poin on is perieer can be represened by a single variable θ wih he Caresian co-ordinaes given by cosθ and sinθ where θ is defined in [0,2π]. In AQDE, he Q-bis (θ are iniialized uniforly a rando in [0, 2π] for all he bis for all he individuals in he populaion. The binary populaion is derived as follows: P x i b i f(x f(b Δθ i 0 0 false rue 0 0 false 0.0π 0 rue 0 0 false -0.0π 0 rue 0 false 0 rue 0 (8 2, if rand(0, < sin ( θ, i = (9 0, oherwise where P,i is he h bi of he i h individual in he populaion and θ,i is he corresponding Q-bi. Procedure QEA begin 0 iniialize Q(; ake P( fro Q( by (7 evaluae P( B( P( b bes soluion aong B( while <T do + ake P( fro Q( by (7 evaluae P( updae Q( using (8 sore bes soluions aong B(- and P(in B( sore bes soluion b aong B( if (igraion condiion igrae b or b o B( globally or locally respecively endif end while end Figure. QEA pseudo code B. Muaion Operaor Muaion operaor in AQDE is siilar o ha of classical DE, bu insead of operaing on he individual direcly, i is applied on he Q-bi (θ. Since θ conains inforaion abou boh α and β, i is ore appropriae o generae he uan vecor in ers of θ. Moreover, unlike he case of classical DE, i inherenly avoids he proble of consrain violaion, i.e. he uan vecor exceeding he prescribed doain. This is because boh cosine and sine funcions are periodic wih period 2π. The represenaion of Q-bis was changed o θ keeping his in ind. The uan Q-bis θ are generaed for all he individuals in he populaion in every generaion. Muan Q-bis of he i h individual in generaion are deerined as follows: θ = θ + F.( θ θ (0 i r r2 r3 where r,r2,r3 and i are uually disinc and F is he uaion conrol paraeer which is deerined in every generaion as per he following equaion, F = rand. rand.(0. ( 2 where rand, rand 2 are rando nubers generaed fro a unifor disribuion on [0,]. The purpose of uliplying one rando nuber is o ake values for F on he inerval [0,0.]. One ore independen rando nuber is furher uliplied o probabilisically generae ore values close o zero. This is because, he qualiy of soluion is found o be highly sensiive owards radical perurbaion of he Q-bi. C. Crossover Operaor The crossover operaion operaes on he original Q-bis and he respecive uan Q-bis in he following

4 anner: θ θ, if ( rand (0, CR or( = I, ( (0, ( c, i rand = θ, i if rand > CR and Irand (2 where θ c i is he h Q-bi of ih individuals afer he crossover operaion. I rand is a nuber randoly chosen fro {,2, D} which ensures a leas one Q-bi is differen fro he original se in each individual. CR is he conrol paraeer which is deerined in every ieraion as follows: = (0.5, (3 CR G rand where G rand generaes a rando nuber fro he Gaussian disribuion wih ean 0.5 and sandard deviaion As a resul, CR lies in a 0.5 neighborhood of 0.5 wih a probabiliy of Thus he value of CR is seleced very close o 0.5 in alos all he cases, which is found o he bes value experienally. D. Selecion The populaion and he Q-bis are updaed in a greedy fashion. By observing he sae of he newly obained Q-bis (θ c i, by (9, a new se of individuals are obained which replace he corresponding individual in he populaion if heir finess values are higher. The replaceen is done using he following equaions: c c + Pi, if( f( Pi > f( Pi Pi = (4a P, oherwise and i c c θ, if( f( Pi f( Pi + > θ = θ, oherwise (4b where P c i is he ih individual by observing he Q-bis odified afer crossover (θ c i. f(p i is he finess value of he corresponding individual. Procedure AQDE for knapsack begin 0 iniialize Q(; ake P( fro Q( by (9 repair P( evaluae finess of P( while <T do + deerine F and CR by ( and (3 apply uaion on Q( using (0 obain Q ( by crossover using (2 ake P ( fro Q ( using (9 repair P ( evaluae finess of P( updae P(+ and Q(+ by (4 end while end Figure 2. AQDE pseudo code Thus AQDE is an adapive algorih, which eploys differenial operaors on he superposiion sae of Q-bis and can be applied o binary opiizaion probles direcly. The pseudo code of AQDE applied o 0- knapsack proble is given in Fig. 2. IV. EXPERIMENTAL SETTINGS AND RESULTS To es he perforance of AQDE, i was copared wih boh QEA and DBDE on he 0- knapsack proble. In all es cases, srongly correlaed ses of daa were considered. The weighs w i, respecive prices p i and he knapsack capaciy W were calculaed as follows [4]. wi = rand[,0] pi = wi + 5, i =, 2,... W = wi 2 i= (5 where rand[,0] generaes an ineger in {,2,.,0} uniforly a rando. For saisfying he consrain of he knapsack proble, he repair ehod given in [4] is applied o all he algorihs. If he consrain is violaed, he repair ehod randoly chooses an ie and reoves i fro he collecion unil he consrain is us saisfied. Afer ha i sars adding ies randoly again. When he consrain is us violaed, i reoves he las added ie and sops. Three knapsack probles wih 00, 250, and 500 ies were considered wih unsored daa obained as above. For each knapsack proble, he algorih was esed for a populaion size of 30 and 50. The axiu nuber of generaions in all cases was chosen as 000. The ean bes profis of 30 runs and he respecive sandard deviaions were abulaed (Table I.The variaion of ean bes profi wih no. of generaions were ploed (Fig Ie size Table II. Perforance coparison on 0- Knapsack proble ( ( (0.77 QEA DBDE AQDE ( ( ( ( ( ( ( ( ( ( ( ( ( ( (.27

5 Figure 3. Populaion Size= 30, Ie Size = 00 Figure 4. Populaion Size= 50, Ie Size = 00 Figure 5. Populaion Size= 30, Ie Size = 250 Figure 6. Populaion Size= 50, Ie Size = 250 Figure 7. Populaion Size= 30, Ie Size = 500 Figure 8. Populaion Size= 50, Ie Size = 500

6 Figure 3-8 show he progress of he convergence by depicing he average of bes profis over 30 runs for he previously enioned populaion sizes and ie sizes. For all he cases considered, he curve of ean bes profi for AQDE lies slighly below he curves of QEA and DBDE for he iniial 50 generaions, bu soon afer ha, i goes above he curves of QEA and DBDE, hereby showing significanly beer resuls. The plos sugges a preaure convergence of boh QEA and DBDE as copared o AQDE. V. CONCLUSION In his paper, we have proposed a novel AQDE algorih for solving he 0- Knapsack proble. The proposed algorih is a hybrid of QEA and DE along wih a novel adapive paraeer conrol ehod. The experienal resuls have proved he superior perforance of AQDE copared o QEA and DBDE. Here, he perforance of AQDE was esed only on he 0- Knapsack proble. Wih soe odificaions, he concep of he algorih ay be exended o oher discree cobinaorial opiizaion probles. REFERENCES [] K. Price and R. Sorn, Differenial Evoluion a siple and efficien adapive schee for global opiizaion over coninuous spaces, Technical Repor, Inernaional Copuer Science Insiue, Berkley, [2] R. Sorn and K. Price, Differenial Evoluion a siple and efficien Heurisic for global opiizaion over coninuous spaces, Journal Global Opiizaion, Vol., 997, pp [3] J. Teo, Exploring Dynaic Self-adapive Populaions in Differenial Evoluion, Sof Copuing - A Fusion of Foundaions, Mehodologies and Applicaions, Vol. 0 (8, 2006, pp [4] J. Bres, S. Greiner, B. Boˇskovi c,m. Mernik and V. ˇZuer, Self- Adaping Conrol Paraeers in Differenial Evoluion: A Coparaive Sudy on Nuerical Benchark Probles, IEEE Transacions on Evoluionary Copuaion, Vol. 0(6, 2006, pp [5] Z. Yang, K. Tang and X. Yao, Self-adapive Differenial Evoluion wih Neighborhood Search, In Proc. IEEE Congress on Evoluionary Copuaion, Hong Kong, 2008, pp [6] S. Das, A. Konar and U.K. Chakrabory, Two iproved differenial evoluion schees for faser global search, ACM-SIGEVO Proceedings of GECCO, Washingon D.C., 2005, pp [7] U.K. Chakrabory, Advances in Differenial Evoluion, (Ed. Springer-Verlag, Heidelberg, [8] G. Papara, A. Engelbrech and N. Franken, Binary Differenial Evoluion, In Proc. of he IEEE Congress on Evoluionary Copuaion, 2006,pp [9] A. Engelbrech and G. Papara. Binary differenial evoluion sraegies, In Proc. of he IEEE Congress on Evoluionary Copuaion, 2007,pp [0] C. Peng, L. Jian and L. Zhiing. Solving 0- knapsack Probles by a Discree Binary Version of Differenial Evoluion, In Proc. Second Inernaional Syposiu on Inelligen Inforaion Technology Applicaion, 2008, pp [] L. K. Grover, A fas quanu echanical algorih for daabase search, in Proc. 28h ACM Syp. Theory of Copuing, 996, pp [2] P. Shor, Polynoial-ie algorihs for prie facorizaion and discree logarihs on a quanu copuer. SIAM J. Copuing, 26, 997, pp [3] A. Narayanan and M. Moore, Quanu-inspired geneic algorihs, in Proc. 996 IEEE In. Conf. Evoluionary Copuaion. Piscaaway,NJ,996, pp [4] Han K-H and Ki J-H. Quanu-inspired evoluionary algorih for a class of cobinaorial opiizaion. IEEE Transacion on Evoluionary Copuaion, 6(6,2002, pp [5] J. Sun, B. Feng and W.B. Xu, Paricle swar opiizaion wih paricles having quanu behavior, In proc of he IEEE Congress on Evoluionary Copuaion, 2004, pp [6] K.-H. Han and J.-H. Ki, Quanu-inspired evoluionary algorihs wih a new erinaion crierion, H gae and wo-phase schee, IEEE Transacions on Evoluionary Copuaion, 8(2, 2004,pp [7] M. D. Plael, S. Schliebs and N. Kasabov, A versaile quanu-inspired evoluionary algorih, in Proc. IEEE Congress on Evoluionary Copuaion CEC 07, 2007, pp [8] B. Jiao, X. Gu and G. Xu, An Iproved Quanu Differenial Algorih for Sochasic Flow Shop Scheduling Proble. In Proc. IEEE Inernaional Conference on Conrol and Auoaion 2009,pp [9] A. Draa, S. Meshoul, H. Talbi and M. Baouche, A Quanu- Inspired Differenial Evoluion Algorih for Solving he N-Queens Proble. The Inernaional Arab Journal of Inforaion Technology, Vol. 7, No., 200 pp [20] H. Su, Y. Yang and L. Zhao, Classificaion rule discovery wih DE/QDE algorih, Exper Syses wih Applicaions 37 (200 pp [2] Su, H. and Yang, Y. Quanu-inspired differenial evoluion for binary opiizaion, In The 4-h inernaional conference on naural copuaion,2008,pp [22] J. Kennedy and R.C. Eberhar, A discree binary version of he paricle swar algorih, Proceedings of he 997 Conference on Syses, Man, and Cyberneics, 997,pp [23] M. D. Plael, S. Schliebs and N. Kasabov, Quanu-Inspired Evoluionary Algorih:A Muliodel EDA. IEEE Transacions On Evoluionary Copuaion, 3(6,2009, pp