Influence of Probability of Variation Operator on the Performance of Quantum-Inspired Evolutionary Algorithm for 0/1 Knapsack Problem

Transcription

1 The Open Arfcal Inellgence Journal,, 4, Open Access Influence of Probably of Varaon Operaor on he Perforance of Quanu-Inspred Eoluonary Algorh for / Knapsack Proble Mozael H.A. Khan* Deparen of Copuer Scence and Engneerng, Eas Wes Unersy, 43 Mohakhal, Dhaka, Bangladesh Absrac: Quanu-Inspred Eoluonary Algorh (QEA has been shown o be beer perforng han classcal Genec Algorh based eoluonary echnques for cobnaoral opzaon probles lke / knapsack proble. QEA uses quanu copung-nspred represenaon of soluon called Q-b nddual conssng of Q-bs. The probably apludes of he Q-bs are changed by applcaon of Q-gae operaor, whch s classcal analogous of quanu roaon operaor. The Q-gae operaor s he only araon operaor used n QEA, whch along wh soe proble specfc heursc prodes exploaon of he properes of he bes soluons. In hs paper, we analyzed he characerscs of he QEA for / knapsack proble and showed ha a probably n he range.3 o.4 for he applcaon of he Q-gae araon operaor has he greaes lkelhood of akng a good balance beween exploraon and exploaon. Experenal resuls agree wh he analycal fndng. Keywords: / knapsack proble, enropy of he probably dsrbuon for he search space, eoluonary algorh, perforance analyss, quanu-nspred eoluonary algorh.. INTRODUCTION Eoluonary Algorhs (EAs are naure-nspred, specfcally bologcal eoluon based, sochasc search or opzaon echnques. In EAs, a populaon of nal canddae soluons s eoled no new populaons of soluons fro generaon o generaon o fnd ou beer f soluons. EAs use wo apparenly conflcng echnques of exploaon and exploraon o generae new soluons. These new soluons copee for sural and f hey are beer f han he exsng soluons, hen hey sure and replace soe lower f soluons n he populaon, oherwse hey de ou. In he exploaon echnque, characerscs of beer f soluons are presered no he new soluons wh he hope o fnd ou uch beer f soluons. On he oher hand, n he exploraon echnque, a wde soluon space s searched o fnd ou beer f soluons. In praccal applcaons, a good balance beween exploaon and exploraon s requred o fnd ou global soluons whn a reasonable e. To hae a good balance beween exploaon and exploraon, he populaon dynacs such as populaon sze, paren selecon, araon operaors, reproducon and nherance, sural copeon ehod, ec. are desgned properly. A nonradonal eoluonary copuaon echnque called Quanu-Inspred Genec Algorh was deeloped by Narayanan [, ], where he concep of quanu echancal nerference was ncluded n a odfed crossoer operaor for solng raelng salesan proble. The os noable Quanu-Inspred Eoluonary Algorh (QEA was deeloped by Han [3-9]. In [3], a probablsc represenaon and a noel populaon dynacs nspred by quanu *Address correspondence o hs auhor a he Deparen of Copuer Scence and Engneerng, Eas Wes Unersy, 43 Mohakhal, Dhaka, Bangladesh; Tel: ; Fax: ; E-al: hakhan@ewubd.edu copung were proposed. In [4], he applcably of QEA o a parallel schee, parcularly, PC cluserng, was erfed successfully. In [5], he basc srucure of QEA and s characerscs were forulaed and analyzed. I was also experenally proed ha QEA s beer han classcal GA for solng / knapsack proble. In [6], a QEA-based dsk allocaon ehod (QDM was proposed, where he aerage query response es of QDM were equal o or less han hose of dsk allocaon ehods usng GA (DAGA, and he conergence speed of QDM was es faser han ha of DAGA. In [7], QEA was appled o a decson boundary opzaon for face erfcaon. Copared wh he conenonal prncpal coponens analyss (PCA ehod, proed resuls were acheed boh n ers of he face erfcaon rae and false alar rae. In [8], soe gudelnes for seng he paraeers of QEA were presened. In [9], exenson of he basc QEA of [5] such as ernaon creron, a odfed erson of he araon operaor Q-gae called H gae, and a wo-phase schee were proposed o proe he perforance of he QEA. In he QEA of [5], Q-gae araon operaor s appled on all he Q-bs of a Q-b nddual, ha s, he araon operaor s appled wh a probably of. In hs paper, we analyzed he characerscs of he QEA for / knapsack proble and showed ha a probably n he range.3 o.4 for applcaon of he Q-gae araon operaor has he greaes lkelhood of akng a good balance beween exploraon and exploaon and proes he perforance of QEA for / knapsack proble. We experened wh four ses of daa, whch agrees wh he analycal fndng. The res of he paper s organzed as follows. In Secon, background on quanu copung syse s nroduced. The / knapsack proble and he characerscs of he es daa se are dscussed n Secon 3. In Secon 4, we dscuss he srucure of he QEA for he / knapsack proble pro / Benha Open

2 38 The Open Arfcal Inellgence Journal,, Volue 4 Mozael H.A. Khan posed n [5]. We analyze he characerscs of he QEA n Secon 5. In Secon 6, we presen an analycal odel of nfluence of probably of applcaon of he Q-gae araon operaor on he perforance of he QEA for / knapsack proble. We presen he experenal resuls and relaed dscussons n Secon 7. Fnally, Secon 8 concludes he paper.. BACKGROUND ON QUANTUM COMPUTING SYSTEM Undersandng of he QEA proposed n [5] requres a sound undersandng of he quanu copung syse. In hs secon, we nroduce quanu copung syse n bref. For ore deals, readers can see he faous exbook by Nelsen and Chuang []. The basc un of nforaon n a quanu copung syse s a quanu b (qub. A qub exss n eher sae or sae, whch correspond o he logcal alues or, respecely. These saes are called he bass saes. A qub ay also exs n lnear superposon of he bass saes = + where, and are coplex nubers represenng he probably apludes of he bass saes. Afer easureen, he qub becoes wh probably and wh probably requres ha + =., where he noralzaon condon If =± and =±, hen he probably of fndng he qub n bass sae and are equal, ha eans, he qub exss n equal superposon sae. A qub can be geoercally sualzed as a un ecor locaed n any quadran of he recangular coordnae syse as shown n Fg. (. In eery case, he noralzaon condon sasfes ha ± +± =, whch ples ha he lengh of he ecor s. The sae of he qub = + s represened by he colun ecor. The sae of a qub can be changed by a -qub quanu gae, whch s characerzed by a unary operaon U acng on he colun ecor represenng he qub. The unary operaon U s represened by a unary arx sasfyng he condon U U =UU = I, where U s he heran adon of U. There are any nonral unary arces, bu only a few of he are used n quanu copuaon. The frequenly used -qub quanu gaes are Hadaard, Paul-X (also known as NOT, Paul-Y, Paul-Z, Phase, 8, and roaon gaes. The readers can see [] for ore deals on hese -qub gaes. In [5], he roaon gae s used as araon operaor. The roaon gae s defned below. Defnon. The roaon gae s defned usng he unary arx R( = cos sn. ( sn cos If he roaon gae R( s appled on he qub = +, hen he new sae of he qub wll be cos sn cos sn = R( = sn cos = sn + cos =, = + where, = cos sn and = sn + cos. The applcaon of he roaon gae on a qub can be geoercally sualzed as shown n he polar coordnae syse of Fg. (. If s pose, hen he qub s roaed angle o he couner-clockwse drecon. If s negae, hen he qub s roaed angle o he clockwse drecon. The roaon of he qub changes he relae alues of and causng he change of probably of fndng he qub n and saes. Theore. ( If he qub s n he frs or he hrd quadran, hen a couner-clockwse roaon ncreases he probably of fndng he qub n he bass sae and a clockwse roaon ncreases he probably of fndng he qub n he bass sae. ( If he qub s n he second or he fourh quadran, hen a clockwse roaon ncreases he probably of fndng he qub n he bass sae and β β α α α ββ α ββ ψ = α + β ψ = α + β ψ = α β ψ = α β Fg. (. Geoerc sualzaon of a Q-b.

3 Influence of Probably of Varaon Operaor The Open Arfcal Inellgence Journal,, Volue 4 39 θ ( α,ββ ( α,β 3 = where, he noralzaon condon sasfes ha (. = Fg. (. Geoerc sualzaon of roaon gae. a couner-clockwse roaon ncreases he probably of fndng he qub n he bass sae. Proof. Fro Fg. (3a, where he qub s roaed counerclockwse n he frs and he hrd quadrans, we see ha > and <. Ths ples ha he probably of fndng he qub n he bass sae s ncreased. Fro Fg. (3b, where he qub s roaed clockwse n he frs and he hrd quadrans, we see ha > and <. Ths ples ha he probably of fndng he qub n he bass sae s ncreased. Ths proes he frs par of he heore. Fro Fg. (3c, where he qub s roaed clockwse n he second and he fourh quadrans, we see ha > and <. Ths ples ha he probably of fndng he qub n he bass sae s ncreased. Fro Fg. 3(d, where he qub s roaed couner-clockwse n he second and he fourh quadrans, we see ha > and <. Ths ples ha he probably of fndng he qub n he bass sae s ncreased. Ths proes he second par of he heore. An -qub quanu copung syse exss n lnear superposon of saes. For exaple, f he hree nddual qubs of a 3-qub quanu copung syse are = +, = +, 3 = 3 + 3, hen he superposon of he 3-qub quanu copung syse s ( α, β ( α,β α, β Fg. (3. Roaon of qub changes he probably of fndng he bass saes. ( α,β ( An -qub quanu copung syse represens saes sulaneously n superposon. Howeer, he easureen of he syse yelds a sngle sae. If = =L = = = =L = =±, hen he - qub quanu copung syse exss n equal superposon sae, ha eans, afer easureen, he probably of fndng any of he saes are equal. The sae of an - qub quanu copung syse can be changed by applyng -qub quanu gaes on he syse or by applyng - qub gaes on he nddual qubs. An -qub quanu gae s characerzed by a unary arx. 3. THE / KNAPSACK PROBLEM AND THE CHAR- ACTERISTICS OF THE TEST DATA SET In [5], he perforance of he QEA s esed usng / knapsack proble and we also do he sae. For hs reason, undersandngs of he / knapsack proble and he characerscs of he es daa se should be ade clear. The knapsack proble can be descrbed as selecon of a subse of es fro a gen se of es, ogeher wh her weghs and profs, such ha he oal wegh of he selecon does no exceed he gen bound, ha s, he capacy of he knapsack and he oal prof of he selecon s axzed. The knapsack proble s NP-hard. The / knapsack proble s descrbed as follows. Defnon. Gen a se of es wh wegh and prof p for =,,L, and a knapsack wh capacy C, fnd a bnary ecor x = (x x Lx such ha he oal wegh w x C and he oal prof = f = w ( x = p x s axu. If x =, hen he h e s seleced for he knapsack. The perforances of he algorhs for he / knapsack proble are norally analyzed and copared by runnng he algorhs on seeral ses of randoly generaed es probles. Snce he dffculy of such probles s grealy af- ( α, β ( α,β ( α,β ( α, β ( α,β ( α, β ( α, β ( α,β ( α, β ( α,β ( α,β ( α, β (a (b (c (d

4 4 The Open Arfcal Inellgence Journal,, Volue 4 Mozael H.A. Khan feced by he correlaon beween weghs and profs [, ], hree randoly generaed ses of daa are usually used: uncorrelaed: [ ] [ ] w = unforly rando L p = unforly rando L weakly correlaed: [ ] w = unforly rando L p = w + unforly rando [ r Lr ] > srongly correlaed: [ ] w = unforly rando L p = w + r As repored n [], hgher correlaon probles hae hgher expeced dffculy. In [5], srongly correlaed daa ses are used and he daa ses are generaed wh = and r = 5. In [], he followng wo knapsack capaces are suggesed: resrce knapsack capacy: C = aerage knapsack capacy: C =.5 w = As repored n [], n he knapsack wh resrce capacy, he opal soluon conans ery few es. An area, for whch condons are no fulflled, occupes alos he whole doan. In he knapsack wh aerage capacy, he opal soluon conans abou half of he es. Bu, hs saeen s no rue for srongly correlaed daa se and aerage knapsack capacy as eden fro Theore and Corollares and below. In [5], experens are done wh srongly correlaed daa se wh =, r = 5 and aerage knapsack capacy. We also do he sae. Ths proble requres soe n deph dscusson. Theore. In he case of srongly correlaed daa se and aerage knapsack capacy, he global soluon conans ore han es wh saller weghs, where s he nuber of es. Proof. In he case of srongly correlaed daa se and aerage knapsack capacy, he prof can be expressed as f (x = p x = w x + r x. (3 = = = Fro (3, we see ha he prof consss of wo pars he frs par s equal o he oal wegh of he seleced es and he second par s equal o r es he nuber of es seleced. Therefore, he prof can be axzed by akng he oal wegh of he seleced es equal o he knapsack capacy C ha ncreases he alue of he frs par of (3 and sulaneously selecng ore es ha ncreases he alue of he second par of (3. These nerdepen condons nuely sugges ha selecng ore es of saller weghs whn he capacy consran wll ncrease he prof. The weghs of he es are unforly dsrbued n he range [ L ]. If we sor he es n he ascng order of her weghs, hen he weghs wll be unforly dsrbued fro o n he ascng order and wll for an arhec progresson. In hs case, he oal wegh of he frs half of es wll be less han he oal wegh of he second half of es. Obously, he oal wegh of he frs half of es wll be less han he knapsack capacy C, snce C s half of he oal wegh of all he es. Therefore, f we selec ore han es fro he lower whn he capacy consrans, hen he nuber of seleced es wll be axu and he prof wll also be axu. Corollary. In he case of srongly correlaed daa se wh = and aerage knapsack capacy, he global soluon conans abou.67 es wh saller weghs, where s he nuber of es. Proof. The weghs of he es are unforly dsrbued n he range [ L]. If we sor he es n he ascng order of her weghs, hen he weghs wll be unforly dsrbued fro o n he ascng order and wll for an arhec progresson wh coon dfference d = ( ( = 9 (. Then he su of weghs of all es wll be ( + S = = 5.5 and he capacy of he knapsack wll be C = S = 5.5 =.75. (4 Now, le us assue ha he su of he weghs of he frs es axzes he oal wegh sasfyng he capacy consran S C. The su of he weghs of he frs es s S = [ + ( d ] = ( =. (5 [ ] ( Usng he equaly condon S = C and fro (4 and (5, we hae [ ] ( + 9 =.75. (6 Afer soe algebrac anpulaon of (6, we hae

5 Influence of Probably of Varaon Operaor The Open Arfcal Inellgence Journal,, Volue ( + ( =. (7 Fro (7, we hae ha ( ± ( = ( ± , hen ( If > 5 ( =. (8 wll be negae and we can no consder sgn for he par of (8, snce we wll be workng wh a large alue of. Then, we can wre ( =. (9 8 For known alues of, fro (9 we hae he alues of and as shown n Table. Table. Values of and fro ( As we hae consdered he equaly condon S = C o deerne he alue of raher han consderng he capacy consran condon S C, hen fro Table, we can conseraely ake he alue of o be.67. Thus, abou.67 es wh saller weghs wll axze he seleced wegh and wll also axze he nuber of es seleced. For (3 and subsequen dscusson, we hae ha hs selecon wll produce he global opal soluon. For experenaon, we hae generaed four ses of srongly correlaed daa se wh = and r = 5 for =, 5, 5,. Afer sorng he generaed daa ses n he ascng order of weghs, we hae copued he alue of such ha he su of he frs es axzes he oal wegh whn he capacy consran S C as shown n Table. The experenal daa of Table s reasonably conssen wh he daa fro Table. Table. Values of and fro Generaed Daa Ses Corollary. In he case of srongly correlaed daa se wh = and aerage knapsack capacy, f less han.67 es, where s he nuber of es, axze he oal seleced wegh whn he capacy consran, hen he selecon wll produce a local soluon. Proof. Fro Corollary, we see ha f less han.67 es axze he oal wegh whn he capacy consran, hen he seleced es wll be of hgher weghs and he oal nuber of es seleced wll be ery sall. Fro (3, we see ha hs selecon can no produce a global soluon and herefore, wll produce a local soluon. Corollares and can be used o deelop new heurscs for explong he search echans. 4. THE QUANTUM-INSPIRED EVOLUTIONARY ALGORITHM FOR / KNAPSACK PROBLEM In he QEA proposed n [5], wo sulaneous represenaons of he soluons ono he ndduals are used one s qub based represenaon and he oher s bnary soluon correspondng o he ecor x as defned n Defnon. The represenaon of he bnary ecor x s already clear. Here, we descrbe he qub based represenaon fro [5]. Defnon 3. A Q-b s defned as he salles un of nforaon n he QEA, whch s defned as a par of nubers, as ( such ha + =. wll produce a and wll produce a. ges he probably he Q-b ges he probably ha he Q-b The Q-b s an adapaon of he concep of qub for he QEA. Lke qub, a Q-b ay be n he sae, n he sae, or n a lnear superposon of he wo. Defnon 4. A Q-b nddual s a srng of Q-bs defned as L + such ha = for =,, L,. The adanage of represenng soluon usng Q-b nddual s ha, fro (, we see ha a sngle Q-b nddual s capable of represenng bnary soluons probablscally. The QEA anans four daa srucures as descrbed below. Populaon of Q-b ndduals: The populaon of Q- b ndduals denoed Q( = q,q {,L,q n } s a populaon of n (sze of he populaon Q-b ndduals of lengh (lengh of each nddual, s he generaon nuber, and q for =,, L, n s defned as n Defnon 4.

6 4 The Open Arfcal Inellgence Journal,, Volue 4 Mozael H.A. Khan Populaon of obsered bnary soluons: The populaon of obsered bnary soluons denoed { } s a populaon of n bnary srngs P( = x, x,l, x n of lengh each obsered fro Q (, where s he generaon nuber and x s obsered fro q for =,, L, n. The obseraon s ade by he QEA operaon ake and wll be dscussed laer on. Populaon of sored bnary soluons: The populaon of sored bnary soluons denoed B( = b,b {,L,b n } s a populaon of bes n bnary srngs of lengh each seleced fro he populaons P ( and B (. The populaon B ( sores he bes n soluons so far generaed. The bes bnary soluon: The bes bnary soluon denoed b sores he bes bnary soluon so far generaed. The srucure of he QEA for he / knapsack proble s gen below. Procedure QEA for he / Knapsack Proble begn ( nalze Q( ( ake P ( by obserng Q( ( repar P( ( ealuae P( ( sore P ( n B( whle ( < MAX_GEN do begn + ( ake P( by obserng Q( ( repar P( ( ealuae P( (x updae Q( (x sore he bes n soluons aong B( and P ( no B( (x (x (x sore he bes soluon aong B( o b f (global graon condon hen grae b o B( globally else f (local graon condon hen grae b n B( o B( locally The seps of he QEA are dscussed below sep wse Sep : In he sep of nalze Q(, alues of for =,, L, n are n- for =,, L, of all q alzed o. I eans ha one Q-b nddual, q represens he lnear superposon of all soluons wh equal probably. and possble bnary Sep : Ths sep akes bnary soluons n P ( by obserng he saes of Q (, where P( = x, x,l, x n a generaon =. One bnary soluon, { } =,, L, n s a bnary srng of lengh, whch s fored usng he followng ake ( x procedure, whch s a QEA operaon. The ake operaon probablscally deernes he sae of a Q-b o be eher or depng on he alue of he probables srng of lengh, x x for and. Thus, a bnary s fored fro he Q-b nddual q, whch represens a soluon obsered fro he Q-b nddual q. For noaonal splcy, x s wren as x n he procedure ake ( x. Procedure ake ( x begn whle ( < do begn + f rando [ L] < hen x else x Sep : Ths sep repars he bnary soluon x =,, L, n n P ( for oerflled and under-flled correcons usng he followng repar ( x procedure, whch s a QEA operaon. If he knapsack s oerflled, hen he frs whle loop of he repar ( x procedure coners soe randoly seleced s o s o reduce he oal wegh whn he capacy consran. If he knapsack s under-flled, he repar ( x procedure coners soe randoly seleced s o s o axze he oal wegh whn he capacy consran. Procedure repar ( x begn knapsack-oerflled false for

7 Influence of Probably of Varaon Operaor The Open Arfcal Inellgence Journal,, Volue 4 43 f ( w x > C = hen knapsack-oerflled rue whle (knapsack-oerflled do begn randoly selec an such ha x = x f ( w x C = hen knapsack-oerflled false whle (no knapsack-oerflled do begn randoly selec an such ha x = x x f ( w x > C = hen knapsack-oerflled rue Sep : In hs sep of ealuae P (, he fness of each of he nal bnary soluon x for =,, L, n n P ( s copued usng he prof equaon f ( x = p x. = Sep : In hs sep, he nal bnary soluons n P ( s sored n B ( as bes soluons so far generaed. Sep : Ths sep, whn he whle loop, akes he bnary ndduals n P( by obserng he Q-b ndduals n Q( as n sep. Sep : Ths sep repars he bnary ndduals n P( as n sep. Sep : Ths sep ealuaes he bnary soluons n P( as n sep. Sep x: In he updae Q( sep, Q-b ndduals n Q( are updaed by usng he updae ( q procedure, whch uses Q-gaes as araon operaor as defned below. Defnon 5. A Q-gae s defned as a araon operaor of he QEA, by whch he alues of and of a Q-b are updaed o and such ha he noralzaon condon + = s sasfed. In [5], he roaon gae R( as defned n Defnon s used as a Q-gae. Fro Theore, we see ha he roaon gae R ( as a Q-gae roaes he Q-b angle owards eher or depng on he sgn of he angle and he quadran of he Q-b. The alue of he angle s deerned as a funcon of he h b of he bes soluon b and he h b of he obsered bnary soluon x. In [5], he experenally found bes alue of as a funcon of he h b of he bes soluon b and he h b of he obsered bnary soluon x s repored as n Table 3. Table 3. Experenally Found Bes Value of Used n he Roaon Gae R( as Varaon Operaor n [5] x b f ( x f (b false rue false. rue false. rue false rue The updae ( q procedure s gen below. Procedure updae ( q begn whle ( < do begn + deerne fro Table 3. f ( q s locaed n he frs or hrd quadran hen = R( else = R( q q The roaon gae used as a Q-gae n he updae ( q procedure nduces he conergence of each Q-b o eher

8 44 The Open Arfcal Inellgence Journal,, Volue 4 Mozael H.A. Khan or. Howeer, a Q-b conerged o eher or canno escape he sae by self, he ake ( x procedure wll always obsere he conerged alue resrcng any furher exploraon of he soluon space. To preen he preaure conergence of Q-b, a odfed for of he roaon gae s proposed n [9] and called H gae as defned n Defnon 6. Defnon 6. A H gae s defned as a Q-gae exed fro he roaon gae as = H (,, where for = R( : ( f and, hen = ; ( f and, hen = ; ( oherwse = where <<<, R ( s he roaon gae, and s he roaon angle. The H gae can be sualzed as shown n Fg. (4. [9], where l H ( s he sae as he roaon gae. Whle he roaon gae akes he probably of or conerge o eher or, H gae akes conerge o or (. I should be noed ha f s oo bg, he conergence ency of a Q-b nddual ay dsappear. In [9], =. s suggesed. β θ Fg. (4. H gae based on roaon gae [9]. α β (a H gae (b consrans α Sep x: In hs sep, he bes n soluons aong B( and P ( are seleced and sored no B (. Sep x: In hs sep, f he bes soluon sored n B ( s fer han he sored bes soluon b, hen b s replaced by he bes soluon sored n B (. Sep x: In hs sep, afer a specfed perod known as global graon perod, he bes soluon b s coped o all bnary ndduals n B (. Sep x: In hs sep, afer a specfed perod known as local graon perod, he bes soluon b aong soe of he soluons n B ( s coped o he. 5. ANALYSIS OF THE QEA FOR / KNAPSACK PROBLEM In hs secon, we analyze he characerscs of he QEA operaons. Theore 3. The ake ( x procedure explores he soluon space. Proof. Fro Defnon 3, we see ha a Q-b s represened by wo probably apludes and, where s he probably ha he Q-b wll produce a and s he probably ha he Q-b wll produce a. Thus, fro Defnon 4, we see ha a Q-b nddual of lengh represens all soluons probablscally. The ake ( x procedure produces eher or depng on he alue of of he Q-b and he rando nuber generaed whn he procedure. Though, here s a bas of producng eher or depng on he alue of, bu probablscally any of or ay be generaed, whch praccally explores he search space. Theore 4. The repar ( x procedure leads he soluon owards a local opa. Proof. If he knapsack s oerflled, hen he repar ( x procedure reduces he oal wegh whn he capacy consran by reducng he nuber of s n he soluon, bu he possble axu wegh s produced. Ths axzes he frs par of (3. The nuber of s ay be sall by selecng hgher-wegh es or large by selecng lower-wegh es. As here s no dea abou he nuber of s n he repared soluon, we hae no dea wheher he second par of (3 s axzed or no. As he frs par of (3 s axzed, we can say ha he soluon goes owards local opa. If he knapsack s under-flled, hen he repar ( x procedure ncreases he oal wegh by ncreasng he nuber of s n he soluon. In hs case, boh he frs par and he second par of (3 are ncreased and he soluon goes owards local opa, f no he global opa. Theore 5. (a If f ( x f (b s false, hen he updae ( q procedure conerges he Q-b nddual q o

9 Influence of Probably of Varaon Operaor The Open Arfcal Inellgence Journal,, Volue 4 45 bnary soluon b and (b f f ( x f (b s rue, hen he q un- updae ( q procedure keeps he Q-b nddual changed. Proof. Fro he updae ( q procedure and Table 3, we see ha f f ( x f (b s false, ha s, he fness of he obsered bnary nddual x fro he Q-b nddual q s less han he fness of he correspondng sored bnary soluon b : ( If x =, b =, and he Q-b s locaed n he frs or he hrd quadran, hen he roaon angle s =., whch ncreases he probably of as eden fro Theore. Tha eans, he Q-b conerges o b =. ( If x =, b =, and he Q-b s locaed n he second or he fourh quadran, hen he roaon angle s =., whch ncreases he probably of as eden fro Theore. Tha eans, he Q-b conerges o b =. ( If x =, b =, and he Q-b s locaed n he frs or he hrd quadran, hen he roaon angle s =., whch ncreases he probably of as eden fro Theore. Tha eans, he Q-b conerges o b =. ( If x =, b =, and he Q-b s locaed n he second or he fourh quadran, hen he roaon angle s = (. =., whch ncreases he probably of as eden fro Theore. Tha eans, he Q-b conerges o b =. ( If x = b = or x = b =, and he Q-b s locaed n any of he four quadrans, hen he roaon angle s =, whch does no change he probables of and. In hs case, he Q-b s already conerged o b. The aboe dscussons reeal ha he updae ( q procedure conerges he Q-b nddual q o sored bnary soluon b, whch proes he par (a of he heore. If f ( x f ( b s rue, ha s, he fness of he obsered bnary nddual x fro he Q-b nddual q s greaer han or equal o he fness of he correspondng sored bnary soluon b, hen for any cobnaon of x and b, and for any locaon of he Q-b n any quadran, he roaon angle s =, whch does no change he probables of and. The condon f ( x f ( b happens due o probablsc exploraon of he search space by he ake ( x procedure. Ths new beer f x wll be hen sored n B( n sep x. Ths proes par (b of he heore. Theore 6. The updae ( q procedure explos he propery of he already generaed bes soluons. Proof. Fro he proof of Theore 5, we see ha f f ( x f ( b s rue, hen he Q-b nddual s no changed. Bu, f f ( x f ( b s false, hen he updae ( q procedure conerges he Q-b nddual q o he sored bnary soluon b, whch explos he already generaed bes soluon. Theore 7. The graon operaon allows he QEA o escape local opa. Proof. If he probables of all Q-bs of he Q-b nddual q for =,,L,n conerged o he correspondng bs (eher or of he sored bnary soluon b for =,,L,n, hen he ake ( x procedure wll produce he sae obsered bnary soluon x repeaedly n he subsequen generaons and he produced x s wll be exacly equal o he correspondng sored bnary soluon b s. In hs suaon, no b wll be replaced n sep x and he QEA wll be suck n local opa. In hs suaon, f he bes soluon b s coped o all b s as he global graon process, hen he updae ( q procedure wll hae scope o change he probables of soe Q-bs of soe Q-b ndduals and he ake ( x procedure wll hae scope o explore new soluons o escape he local opa. Slarly, f he probables of all Q-bs of a subse of Q-b ndduals conerged o he correspondng bs of her correspondng sored bnary soluons, hen ha subse wll suffer fro slar proble. In hs suaon, copyng he bes soluon b of he se o all soluons of he se as he local graon process wll help o escape he proble. Theore 8. If he sored bes soluon b and he sored bes soluons b for =,, L, n n B ( are equal, hen he graon process fals o escape he local opa. Proof. If all of he sored bes soluons b for =,, L, n n B ( are equal o he sored bes soluon b, hen boh local and global graon wll fal o change any of he sored bnary soluon n B (. Therefore, he QEA wll be suck a local opa. The graon condons are desgn paraeers and should be carefully deerned o ake balance beween exploraon and exploaon. 6. INFLUENCE OF PROBABILITY OF VARIATION OPERATOR ON THE PERFORMANCE OF THE QEA FOR / KNAPSACK PROBLEM Fro Theore 4, we see ha he repar ( x procedure generaes local soluons. On he oher hand, fro Theores 5 and 6, we see ha he updae ( q procedure explos he sored local soluons and conerge he Q-b ndduals o hose local soluons. Fro Theore 7, we see ha by usng graon we can escape fro beng suck no hese local

10 46 The Open Arfcal Inellgence Journal,, Volue 4 Mozael H.A. Khan soluons. Bu, fro Theore 8, we see ha he QEA ay fal o escape fro local opa. The prary reason s ha he updae ( q procedure conerges he Q-b ndduals o already generaed local soluons. Therefore, n he subsequen generaons, he ake ( x procedure fals o explore new soluons. In general, we can say ha he proposed QEA explos ore han explores. To proe he perforance of he QEA, we should nesgae soe oher eans of akng balance beween he exploaon and exploraon, such ha he QEA can reach he global soluon. In hs secon, we nesgae he nfluence of he probably of applcaon of he araon operaor Q-gae n he updae ( q procedure on he perforance of he QEA. Defnon 7. The Hang dsance H d of wo bnary srngs, x and x, s he nuber of posons where he bs of he wo srngs are no equal and s copued as H d ( x ( x x =, x = where, s he lengh of he bnary srngs and s odulo addon. Theore 9. The enropy of he probably dsrbuon for he search space explored by he QEA wh probably p of applcaon of he Q-gae araon operaor n he updae ( q procedure s! H = ( h! ( h! p ( h ( h p log ( p h h p ( h = where s he lengh of he Q-b nddual. Proof. Le x be he obsered bnary soluon fro he Q-b nddual q and b be he correspondng sored bnary soluon. Le h be he hang dsance beween x and b. In he updae ( q procedure, h Q-bs of he Q-b nddual q wll conerge o her correspondng bs n he p. Then he probably of con- b wh a probably of ergng o b s p(b = ( p ( h p h, and he nuber of all possble b s! n(b = h!( h!. Therefore, he enropy [3] of he probably dsrbuon for he search space explored by he QEA wh probably p of applcaon of he Q-gae araon operaor n he updae ( q procedure s ( H = n(b p(b log p(b = h= h= ( (! ( h! ( h! p ( h p h log ( p ( h p h The expresson of enropy of he probably dsrbuon for he search space of ( s ald for p <. < Theore. The probably of applcaon of he Q-gae araon operaor n he range.3 o.4 wll hae he greaes lkelhood of akng a good balance beween he exploraon and he exploaon. Proof. We hae copued he alue of he enropy H of he search space fro ( for =,, 3 and dfferen alues of p and abulaed n Table 4. Fro Table 4, we see ha p =. 5 prodes he hghes exploraon of he search space and s no desrable. If we choose p >. 5, hen exploraon wll be reduced bu ore Q-bs of he Q-b nddual q wll be conerged o he sored bnary soluon b and wll prode ore exploaon leadng o local opa. If we choose p <<. 5, hen exploraon wll be reduced bu ery few Q-bs of he Q-b nddual q wll be conerged o he sored bnary soluon b and wll prode ery sall exploaon. Therefore, here s he greaes lkelhood of akng a good balance beween he exploaon and exploraon for he alue of p n he range.3 o.4. Table 4. Enropy H of he Search Space for Dfferen Values of p and p EXPERIMENTAL RESULTS To erfy he cla of Theore, we experened wh four knapsack probles wh, 5, 5, and es. We consdered srongly correlaed daa se wh = and r = 5, and aerage knapsack capacy. The daa were unsored. We experened wh wo QEAs. The populaon szes of QEA and QEA were and, respecely. The global graon perod for QEA was generaon and local graon was no used. For each QEA, boh roaon gae and H gae were used as araon operaors. The probably of he applcaon of he araon operaor was aken o be.,.,.3,.4,.5,.6,.7,.8,.9, and.. As he perforance easure, we colleced he bes soluon

11 Influence of Probably of Varaon Operaor The Open Arfcal Inellgence Journal,, Volue 4 47 Table 5. Experenally Found Aerage Bes Prof Generaed whn Generaons oer 3 Runs p Ie QEA Q-gae QEA R H QEA R H QEA R H QEA R H QEA R H QEA R H QEA R H QEA R H found whn generaons and aeraged he oer 3 runs. In eery run, he rando generaor was seeded wh rando seed. The resuls are suarzed n Table 5. In eery case, he bes prof s shown n bold face. Fro Table 5, we see ha bes resuls are found for hree cases a P =.3, for one case a P =.3,. 4, for en cases a P =.4, for one case a P =. 5, and for one case a P =.6. Fro hese obseraons, we see ha P n he range.3 o.4 wll ge he greaes lkelhood of geng he bes resul, whch agrees wh he Theore. We also hae colleced he bes soluon generaed n eery generaon and aeraged he oer 3 runs. The conergence rs for QEA wh es and roaon gae as araon operaor for p =.,.4,.7,. are shown n Fg. (5. Fro Fg. (5, we see ha p =. conerges ery slowly and generaes he salles prof, whch ndcaes ha prodes ery sall exploaon and randoly explores he search space. On he oher hand, p =.7 and. preaurely conerged due o ore exploaon han exploraon and produced local soluon. p =.4 reasonably explored and exploed and generaed he bes soluon. CONCLUSION In [5], a non-radonal eoluonary algorh called Quanu-Inspred Eoluonary Algorh (QEA was proposed and showed o be beer perforng han classcal Genec Algorhs for / knapsack proble. In [9], soe proeens were proposed. In hs paper, we analyzed he characerscs of he QEA operaors and showed ha he Prof P=. P=.7 P=. P= Generaon Fg. (5. Conergence rs for QEA wh es and roaon gae araon operaor for dfferen alues of probably of applcaon of he araon operaor. QEA ay fal o produce global soluon due o oer exploaon han exploraon. In he QEA of [5], he araon operaor was appled wh probably. We analycally show ha a lower probably of applcaon of he araon operaor wll prode a good balance beween he exploraon and exploaon. A ery low probably of applcaon of he araon operaor explos less and explores he search space randoly and akes larger nuber of generaon o produce een he local soluon. A large probably of he applcaon of he araon operaor explos ore and preaurely conerged o local soluon. We show ha he probably of applcaon of he araon operaor n he range.3 o.4 wll hae he greaes lkelhood of prodng good balance

12 48 The Open Arfcal Inellgence Journal,, Volue 4 Mozael H.A. Khan beween exploraon and exploaon. The experenal resuls agree wh he cla. We analyzed he behaor of he knapsack proble wh srongly correlaed daa se and aerage knapsack capacy and showed ha he global soluon conans ore han 5% es wh lower weghs. Ths knowledge can be used o dece new heursc o proe he perforance of he QEA for / knapsack proble. REFERENCES [] A. Narayan, and M. Moore, Quanu-nspred genec algorhs, n IEEE Inernaonal Conference on Eoluonary Copuaon, 996, pp [] A. Narayan, Quanu copung for begnners, n IEEE Congress on Eoluonary Copuaon, 999, pp [3] K.-H. Han and J.-H. K, Genec quanu algorh and s applcaon o cobnaoral opzaon proble, n IEEE Congress on Eoluonary Copuaon,, pp [4] K.-H. Han, K.-H. Park, C.-H. Lee, and J.-H. K, Parallel quanu-nspred genec algorh for cobnaoral opzaon proble, n IEEE Congress on Eoluonary Copuaon,, pp [5] K.-H. Han and J.-H. K, Quanu-nspred eoluonary algorh for a class of cobnaoral opzaon, IEEE Trans. Eol. Copu., ol. 6, pp , Deceber. [6] K.-H. K, J.-Y. Hwang, K.-H. Han, J.-H. K, and K.-H. Park, A quanu-nspred eoluonary copung algorh for dsk allocaon ehod, IEICE Trans. Infor. Sys., ol. E86-D, pp , March 3. [7] J.-S. Jang, K.-H. Han, and J.-H. K, Quanu-nspred eoluonary algorh-based face erfcaon, n Genec Eoluonary Copuaon Conference, 3, pp [8] K.-H. Han and J.-H. K, On seng he paraeers of quanunspred eoluonary algorh for praccal applcaons, n IEEE Congress on Eoluonary Copuaon, 3, pp [9] K.-H. Han and J.-H. K, Quanu-nspred eoluonary algorhs wh a new ernaon creron, H gae, and wo-phase schee, IEEE Trans. Eol. Copu., ol. 8, pp , Aprl 4. [] M. A. Nelsen and I. L. Chuang, Quanu copuaon and quanu nforaon. Cabrdge Unersy Press,. [] S. Marello and P. Toh, Knapsack probles, Chcheser, UK: John Wley, 99. [] Z. Mchalewcz, Genec algorhs + daa Srucures = eoluon progras, New York: Sprnger-Verlag, 999. [3] S. J. Russel and P. Norg, Arfcal nellgence: a odern approach. Englewood Clffs, NJ: Prence-Hall, 995. Receed: Deceber 7, 9 Resed: March 6, Acceped: March 8, Mozael H.A. Khan; Lcensee Benha Open. Ths s an open access arcle lcensed under he ers of he Creae Coons Arbuon Non-Coercal Lcense (hp://creaecoons.org/- lcenses/by-nc/3./ whch pers unresrced, non-coercal use, dsrbuon and reproducon n any edu, proded he work s properly ced.