Enhanced Quantum Evolutionary Algorithms for Difficult Knapsack Problems

Transcription

1 Enhanced Quanum Evoluonary Algorhms for Dffcul Knapsack Problems C. Pavardhan 1, Apurva Narayan 1, and A. Srvasav 1 Faculy of Engneerng, Dayalbagh Educaonal Insue, Agra cpavardhan@homal.com, apurvanarayan@gmal.com 2 Mahemasches Semnar, Chrsan-Albrechs-Unversa Zu Kel, Kel Germany asr@numerk.un-kel.de Absrac. Dffcul knapsack problems are problems ha are expressly desgned o be dffcul. In hs paper, enhanced Quanum Evoluonary Algorhms are desgned and her applcaon s presened for he soluon of he DKPs. The algorhms are general enough and can be used wh advanage n oher subse selecon problems. Keywords: Evoluonary Algorhms,Quanum,knapsack. 1 Inroducon The classcal knapsack problem s defned as follows: Gven a se of n ems, each em j havng an neger prof p j and an neger wegh w j, he problem s o choose a subse of he ems such ha her overall prof s maxmzed, whle he overall wegh does no exceed a gven capacy c. The problem may be formulaed as he followng neger programmng model where he bnary decson varables x j are used o ndcae wheher em j s ncluded n he knapsack or no. Whou loss of generaly may be assumed ha all profs and weghs are posve, ha all weghs are smaller han he capacy c, and ha he overall wegh of he ems exceeds c. The sandard knapsack problem (SKP) s NP-hard n he weak sense, meanng ha can be solved n pseudo-polynomal me hrough dynamc programmng. The SKPs are que easy o solve for he mos recen algorhms [1-5]. Varous branch-andbound algorhms for SKPs have been presened. The more recen of hese solve a core problem,.e. an SKP defned on a subse of he ems where here s a large probably of fndng an opmal soluon. The MT2 algorhm [1] s one of he mos advanced of hese algorhms. Realzng ha he core sze s dffcul o esmae n A. Ghosh, R.K. De, and S.K. Pal (Eds.): PReMI 2007, LNCS 4815, pp , Sprnger-Verlag Berln Hedelberg 2007

2 Enhanced Quanum Evoluonary Algorhms for Dffcul Knapsack Problems 253 advance, Psnger [5] proposed o use an expandng core algorhm. Marello and Toh [3] proposed a specal varan of he MT2 algorhm whch was developed o deal wh hard knapsack problems. One of he mos successful algorhms for SKP was presened by Marello, Psnger and Toh [7]. The algorhm can be seen as a combnaon of many dfferen conceps and s hence called Combo. The nsances do no need o be changed much before he algorhms experence a sgnfcan dffculy. These nsances of SKP are called he Dffcul Knapsack Problems (DKP) [6]. These are specally consruced problems ha are hard o solve usng he sandard mehods ha are employed for he SKPs. Psnger [6] has provded explc mehods for consrucng nsances of such problems. Heurscs whch employ hsory of beer soluons obaned n he search process, vz. GA, EA, have been proven o have beer convergence and qualy of soluon for some dffcul opmzaon problems. Bu, sll, problems of slow/premaure convergence reman and have o be ackled wh suable mplemenaon for he parcular problem a hand. Quanum Evoluonary Algorhms (QEA) s a recen branch of EAs. QEAs have proven o be effecve for opmzaon of funcons wh bnary parameers [8, 9]. Alhough he QEAs have been shown o be effecve for SKPs n [8], her performance on he more dffcul DKPs has no been nvesgaed. Exac branch and bound based mehods are no fas for DKPs. Ths provdes he movaon for aempng o desgn newer Enhanced QEAs (EQEAs) wh beer search capably for he soluon of he DKPs. In hs paper, such EQEAs are presened. The QEAs are dfferen from hose n [8] as hey nclude a varey of quanum operaors for execung he search process. The compuaonal performance of he EQEAs s esed on large nsances of nne dfferen varees of he DKPs.e. for problems up o he sze of 10,000. The resuls obaned are compared wh hose obaned by he greedy heursc. I s seen ha he EQEAs are able o provde much mproved resuls. The res of he paper s organzed as follows. In secon 2 we descrbe he varous ypes of DKPs repored n he leraure. A bref nroducon o he QEAs and some prelmnares regardng he QEAs are provded n secon 3. The EQEA s gven n he form of pseudo-code n secon 4. The compuaonal resuls are provded secon 5. Some conclusons are derved n secon 6. 2 Dffcul Knapsack Problems (DKPs) Several groups of randomly generaed nsances of DKPs have been consruced o reflec specal properes ha may nfluence he soluon process n [6]. In all nsances he weghs are unformly dsrbued n a gven nerval wh daa range wh R =1000. The profs are expressed as a funcon of he weghs, yeldng he specfc properes of each group. Nne such groups are descrbed below as nsances of DKPs. The frs sx are nsances wh large coeffcens for prof and he las hree are nsances of small coeffcens for profs. () Uncorrelaed daa nsances: p j and w j are chosen randomly n [1,R]. In hese nsances here s no correlaon beween he prof and wegh of an em. () Weakly correlaed nsances: weghs w j are chosen randomly n [1,R] and he profs p j n [w j - R/10, w j + R/10] such ha p j >= 1. Despe her name, weakly correlaed nsances have a very hgh correlaon beween he prof and wegh of an

3 254 C. Pavardhan, A. Narayan, and A. Srvasav em. Typcally he prof dffers from he wegh by only a few percen. () Srongly correlaed nsances: weghs w j are dsrbued n [1,R] and 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. (v) Inverse srongly correlaed nsances: profs p j are dsrbued n [1,R] and w j =p j +R/10. These nsances are lke srongly correlaed nsances, bu he fxed charge s negave. (v) Almos srongly correlaed nsances: weghs w j are dsrbued n [1,R] and he profs p j n [w j + R/10 R/500, w j + R/10 + R/500]. These are a knd of fxed-charge problems wh some nose added. Thus hey reflec he properes of boh srongly and weakly correlaed nsances. (v) Uncorrelaed nsances wh smlar weghs: weghs w j are dsrbued n [100000, ] and he profs p j n [1, 1000]. (v) Spanner nsances (v,m): These nsances are consruced such ha all ems are mulples of a que small se of ems he so-called spanner se. The spanner nsances span(v,m) are characerzed by he followng hree parameers: v s he sze of he spanner se, m s he mulpler lm, and any dsrbuon (uncorrelaed, weakly correlaed, srongly correlaed, ec.) of he ems n he spanner se may be aken. More formally, he nsances are generaed as follows: A se of v ems s generaed wh weghs n he nerval [1,R], and profs accordng o he dsrbuon. The ems (p k,w k ) n he spanner se are normalzed by dvdng he profs and weghs wh m+1. The n ems are hen consruced, by repeaedly choosng an em [p k,w k ] from he spanner se, and a mulpler a randomly generaed n he nerval [1,m]. The consruced em has prof and wegh (a. p k, a.w k ).The mulpler lm was chosen as m = 10. (v) Mulple srongly correlaed nsances msr(k 1,k 2,d): These nsances are consruced as a combnaon of wo ses of srongly correlaed nsances.boh nsances have profs p j := w j + k where k = 1, 2 s dfferen for he wo nsances. The mulple srongly correlaed nsances msr(k 1,k 2,d) are generaed as follows: The weghs of he n ems are randomly dsrbued n [1,R]. If he wegh w j s dvsble by d, hen we se he prof p j :=w j + k 1 oherwse se o p j := w j + k 2. The weghs w j n he frs group (.e. where pj = wj + k 1 ) wll all be mulples of d, so ha usng only hese weghs a mos d [c/d] of he capacy can be used. To oban a compleely flled knapsack some of he ems from he second dsrbuon need o be ncluded. (x) Prof celng nsances pcel(d): These nsances have he propery ha all profs are mulples of a gven parameer d. The weghs of he n ems are randomly dsrbued n [1,R], and he profs are se o p j = d[w j /d]. The parameer d was chosen as d=3. 3 QEA Prelmnares QEA s a populaon-based probablsc Evoluonary Algorhm ha negraes conceps from quanum compung for hgher represenaon power and robus search. Qub QEA uses qubs as he smalles un of nformaon for represenng ndvduals. Each

4 Enhanced Quanum Evoluonary Algorhms for Dffcul Knapsack Problems 255 qub s represened as q = α. Sae of he qub (q ) s gven by Ψ = α 0 + β 1, β α and β are complex numbers represenng probablsc sae of qub,.e. α 2 s probably of he sae beng 0 and β 2 s he probably of he sae beng 1, such ha α 2 + β 2 ι =1. For purposes of QEA, nohng s los by regardng α and β o be real numbers. Thus, a qub srng wh n bs represens a superposon of 2 n bnary saes and provdes an exremely compac represenaon of he enre search space. Observaon The process of generang bnary srngs from he qub srng, Q, s called Observaon. To observe he qub srng (Q), a srng conssng of 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. Table1 represens he observaon process. Table 1. Observaon of qub srng Ng Q R P Updang qub srng 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 ulzed for hs purpose so ha qubs rean her properes [8]. One such gae s roaon gae, whch updaes he qubs as α β cos( Δθ ) = sn( Δθ ) sn( Δθ ) α cos( Δθ ) β where, α ι +1 and β ι +1 denoe probables for h qub n (+1) h eraon and Δθ 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 QEA. The qub srng, Q, represens probablscally he search space. Observng a qub srng n mes yelds n dfferen soluons because of he probables nvolved. 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. The sequence of seps connues. The above deas can be easly generalzed o work wh mulple qub srngs. Genec operaors lke crossover and muaon can hen be nvoked o enhance he search power furher.

5 256 C. Pavardhan, A. Narayan, and A. Srvasav 4 Enhanced Quanum Evoluonary Algorhm The algorhm s explaned succncly n he form of a pseudo-code below. Noaon Max = number of ems n DKP, Cap = Capacy of knapsack Prof = Array of prof by selecng each em Wegh = Array of wegh by selecng each em Gsol = DKP Soluon by Greedy heursc, GProf = Prof of ems n Gsol P, ep,rp : Srngs of Quanum bs used n search NO1 = Operaor used o evolve ep owards bes soluon found so far NO2 = Operaor used o evove rp randomly NO3 = Roaon operaor used o evolve p bescep = Srng produced by operaor NO1 gvng bes resul on observaon bescrp = Quanum srng produced by NO2 gvng bes resul on observaon Bes = Bes prof found by EQEA of soluon maxsol Algorhm EQEA 1 Inalze eraon number =0, cap, prof, wegh, max 2 Sor ems n descendng order of prof/wegh 3 Fnd greedy soluon G wh prof Gcos. 4 Bes = Gcos, Maxsol = G 5 Inalze for every k If G[k] ==1 p[k] =ep[k]=rp[k]=0.8 else p[k] =ep[k]=rp[k]=0.2 6 Bes Cap = ep, Bes Cap = rp /*Inalzaon*/ 7 Observe p[k] o ge soluon wh cos cos ; f(cos > Bes){Bes = cos; Maxsol = sol;} 8 whle (ermnaon_creron!= TRUE) Do Seps Apply NO1 o p o generae he curren rp.e crp; for(=0;<num1;++) { Observe crp o oban soluon rsol wh cos rcos; f(rcos > Bes) {Crossover rsol wh Maxsol o oban sol wh cos cos; If(cos>rcos) {rsol = sol;rcos = cos} bescrp = crp;} If(rcos >Bes) {Bes = rcos ; maxsol = rsol;} 10 Repea sep 9 wh NO2 on p o oban bescep, ecos and esol; 11 Apply NO3 on p 12 For( = 0 ; < max; ++) {qmax[] = fndmax(bescep[], bescrp[],p[]); qmn[] = fndmx(bescep[], bescrp[],p[]); f(p[] > 0.5) {{f((maxsol[]==1)and (bescep[]==1)and (bescrp[]==1)) p[]=qmax[];} else {f((maxsol[]==0)and (bescep[]==0)and (bescrp[]==0)) p[]=qmn[];}}

6 Enhanced Quanum Evoluonary Algorhms for Dffcul Knapsack Problems Seled = number of p[] s wh α > 0.98 or α < = If(( > er _coun) or (seled > 0.98 * max)) ermnaon _ creron = TRUE Algorhm EQEA sars wh he greedy soluon and nalzes he qub srngs n accordance wh he greedy soluon as n sep 4. Observe operaon n sep 5 s a modfed form of observaon descrbed n secon 3. In hs, he soluon srng resulng from he observaon s checked for volaon of capacy consran and repared, f necessary, usng a greedy approach.e. seleced ems are deseleced n ncreasng order of prof/wegh ll capacy consran s sasfed. The crossover operaor s a smple wo-pon crossover wh greedy repar of consran volaons. Evolvng Qubs In EQEA, he qub s evolved sae of he qub, whch s a superposon of sae 0 and 1, s shfed o a new superposon sae. Ths change n probably magnudes α 2 and β 2 wh change n sae s ransformed no real valued parameers n he problem space by wo neghborhood operaors. Neghborhood Operaor 1 (NO1) generaes a new qub array crp from he rp. An array R s creaed wh max elemens generaed a random such ha every elemen n R s eher +1 or -1. Le ρ k be he k h elemen n R. Then θ k s gven by θ k = θ -1 k +ρ k *δ (4) where, δ s aleraon n angle and θ k s he roaed angle gven by arcan(β k /α k ). δ s randomly chosen n he range [0, θ -1 k ] f ρ k = -1 and n he range [θ -1 k, π/2] f ρ k = +1. The new probably ampludes, α k, β k are calculaed usng roaon gae as 1 α jk cosδ snδ α jk (5) = 1 βjk snδ cosδ βjk Neghborhood Operaor 2 (NO2) NO2 works jus as NO1 excep ha generaes a pon beween ep and BEST. I s prmarly ulzed for exploaon of search space. The raonale for wo neghborhood operaors s as follows. NO1 has a greaer endency for exploraon. NO2 has a greaer endency owards exploaon because, as he algorhm progresses, he values of ep converge owards BEST. Table 2 shows he frequency of use of NO1 and NO2. Table 2. Frequency of use of NO1 and NO2 Sage of search Proporon of NO1 (%) Proporon of NO2 (%) Frs one-ffh eraons Second one-ffh eraons Thrd one-ffh eraons Forh one-ffh eraons Las one-ffh eraons 10 90

7 258 C. Pavardhan, A. Narayan, and A. Srvasav The neghborhood operaors hus evolve new quanum srngs from he exsng srngs. Ths approach removes all dsadvanages of bnary represenaon of real numbers whle, a he same me, balances exploraon and exploaon n he sense ha adops he sep-sze from large nally o progressvely smaller sze. Updang Qub Srng(NO3) In he updang process, he ndvdual saes of all he qubs n p are modfed so ha probably of generang a soluon srng whch s smlar o he curren bes soluon s ncreased n each of he subsequen eraons. Amoun of change n hese probables s decded by Learnng Rae, Δθ, and s aken as 0.001π. Updang process s done as explaned n secon 2. Table 3 presens he choce of Δθ for varous condons of objecve funcon values and h elemen of p and BEST n h eraon. F(p) s he prof of he curren soluon observed from p. F(BEST) s he prof of maxsol. Table 3. Calculaon of Δθ for h eraon Fness Elemenal Values Δθ X p = BEST 0 F(BEST)>F(p) p >BEST 0.001π p < BEST 0.001π F(BEST)<F(p) p > BEST 0.001π p < BEST 0.001π F(p)=F(BEST) p > BEST 0 p < BEST 0 The updang process s llusraed n fgure 2 for k h elemen for h eraon.e. changes n sae of k h qub and correspondng change n probably ampludes. Fndmax fnds he maxmum of he hree argumens whereas fndmn fnds he mnmum of he hree argumens. β k +1 β k Δθ θ k 0 α k +1 α k 1 Fg. 2. Updang k h elemen of qub srng Q 5 Resuls of Compuaonal Expermens The compuaonal expermens have been performed on he above menoned nne ypes of DKPs for he daa range R = 1000 and he number of ems rangng from 500 o For each nsance ype a seres of H = 9,, h = 1,.,...H

8 Enhanced Quanum Evoluonary Algorhms for Dffcul Knapsack Problems 259 nsances s performed. The EQEA performed beer n mos of he cases where he problem dd no ge seled up wh a defne srucure. The able 4 shows compuaonal resuls. AGP s he average prof n soluons obaned by he greedy algorhm whereas AQP s he average prof of soluons by he EQEA algorhm. Table 4. Shows he relave values of soluons by Greedy and Quanum Evoluonary mehods DKP Problem sze = 500 Problem sze = 5000 Problem sze = 10,000 Grp. # # # EQEA EQEA EQEA AGP AQP beer AGP AQP beer AGP AQP beer The Thrd column n each shows he number of problems (ou of 9 red for varous capaces of he knapsack) n whch EQEA gves beer soluon han Greedy mehod for ha group. I s seen ha he EQEA provdes a consderable mprovemen n several nsances. 6 Conclusons An enhanced QEA s presened for Dffcul knapsack problems. EQEA shows good performance even for large nsances of he problems. EQEA can be used wh advanage n any subse slecon problems apar from he DKPs. Acknowledgmens The auhors are exremely graeful o he Revered Charman, Advsory Commee on Educaon, Dayalbagh for consan gudance and encouragemen. References [1] Marello, S., Toh, P.: A new algorhm for he 0-1 knapsack problem. Managemen Scence 34, (1988) [2] Marello, S., Toh, P.: Knapsack Problems: Algorhms and Compuer Implemenaons. J. Wley, Chcheser (1990)

9 260 C. Pavardhan, A. Narayan, and A. Srvasav [3] Marello, S., Toh, P.: Upper bounds and algorhms for hard 0-1 knapsack problems. Operaons Research 45, (1997) [4] der Hede, F.M.a.: A polynomal lnear search algorhm for he n-dmensonal knapsack problem. Journal of he ACM 31, (1984) [5] Psnger, D.: An expandng-core algorhm for he exac 0-1 knapsack problem. European Journal of Operaonal Research 87, (1995) [6] Psnger, D.: Where are he Hard Knapsack Problems, Techncal Repor, , DIKU, Unversy of Copenhagen, Denmark [7] Marello, S., Psnger, D., Toh, P.: Dynamc programmng and srong bounds for he 0-1 knapsack problem. Managemen Scence 45, (1999) [8] Han, K.H., Km, J.H.: Quanum-Inspred Evoluonary Algorhm for a Class of Combnaoral Opmzaon. IEEE Transacons on Evoluonary Compuaon 6(6), (2002) [9] Han, K.H., Km, J.H.: Quanum-Inspred Evoluonary Algorhms wh a New Termnaon Creron, Hε Gae, and Two-Phase Scheme. IEEE Transacons on Evoluonary Compuaon 8(2), (2002)