Dual Populaon-Based Incremenal Learnng for Problem Opmzaon n Dynamc Envronmens hengxang Yang, Xn Yao In recen years here s a growng neres n he research of evoluonary algorhms for dynamc opmzaon problems snce real world problems are usually dynamc, whch presens serous challenges o radonal evoluonary algorhms. In hs paper, we nvesgae he applcaon of Populaon-Based Incremenal Learnng (PBIL) algorhms, a class of evoluonary algorhms, for problem opmzaon under dynamc envronmens. Inspred by he complemenary mechansm n naure, we propose a Dual PBIL ha operaes on wo probably vecors ha are dual o each oher wh respec o he cenral pon n he search space. Usng a dynamc problem generang echnque we generae a seres of dynamc knapsack problems from a randomly generaed saonary knapsack problem and carry ou expermenal sudy comparng he performance of nvesgaed PBILs and one radonal genec algorhm. Expermenal resuls show ha he nroducon of dualsm no PBIL mproves s adapably under dynamc envronmens, especally when he envronmen s subjec o sgnfcan changes n he sense of genoype space. Key words dynamc opmzaon, populaon-based ncremenal learnng, dualsm, evoluonary algorhms. Inroducon As a class of mea-heursc algorhms, evoluonary algorhms (EAs) make use of prncples of naural selecon and populaon genecs. Due o he robus capably of fndng soluons o dffcul problems, EAs have been wdely appled for solvng saonary opmzaon problems where he fness landscape does no change durng he course of compuaon [8]. However, he envronmens of real world opmzaon problems are usually dynamc,.e., he problem fness landscape changes over me. For example, n producon schedulng problems avalable resources may change over me. The nrnsc dynamc naure of problems beng solved presens serous challenge o radonal EAs snce hey canno adap well o he changed envronmen once converged. In recen years here s a growng neres n he research of applyng EAs for dynamc opmzaon problems snce many of he problems ha EAs are beng used o solve are known o vary over me [], []. Over he pas years, a number of researchers have developed many approaches no EAs o address hs problem. Branke [6] has grouped hem no four caegores ) ncreasng dversy afer a change [7], []; ) mananng dversy hroughou he run [9]; 3) memory-based mehods [0], [4]; and 4) mul-populaon approaches [5].. Deparmen of Compuer cence, Unversy of Leceser Unversy Road, Leceser LE 7RH, Uned Kngdom s.yang@mcs.le.ac.uk. chool of Compuer cence, Unversy of Brmngham Edgbason, Brmngham B5 TT, Uned Kngdom x.yao@cs.bham.ac.uk In hs paper we nvesgae he applcaon of Populaon-Based Incremenal Learnng (PBIL) algorhms, a class of EAs, for solvng dynamc opmzaon problems. We sudy he effec of nroducng mul-populaon approach no PBIL o address dynamc opmzaon problems. Inspred by he complemenary mechansm ha broadly exss n naure, we propose a Dual PBIL ha operaes on wo probably vecors ha are dual o each oher wh respec o he cenral pon n he search space. Based on a dynamc problem generang echnque [5] ha can generae dynamc envronmens from any bnary encoded saonary problem, we sysemacally consruc a seres of dynamc knapsack problems from a randomly generaed saonary knapsack problem and carry ou expermenal sudy o compare he performance of nvesgaed PBILs and one varan of radonal genec algorhm. In he res of hs paper, we frs deal several nvesgaed PBILs ncludng our proposed dual PBIL, nex presen he algorhmc es envronmens ha conss of a saonary knapsack problem and relevan dynamc problems, hen provde he expermenal resuls wh analyss, and fnally gve ou our conclusons wh dscussons on fuure work.. Populaon-Based Incremenal Learnng Algorhms. Populaon-Based Incremenal Learnng (PBIL) The PBIL algorhm, frs proposed by Baluja [3], s a combnaon of evoluonary opmzaon and compeve learnng. I s an absracon of he genec algorhm (GA) ha explcly manans he sascs conaned n a GA s populaon. PBIL has proved o be very successful when compared o sandard GAs and hll-clmbng algorhms on an amoun of benchmark and real-world problems [4]. PBIL
Procedure PBIL begn = 0; // nalze he probably vecor for = o L do 0 P [ ] = 0.5; repea = generaeamplesfrom bvecor( P, n evaluaeamples( B = selecbesoluonfrom( // learn he probably vecor oward bes soluon for = o L do P [ ] = ( α ) P [ ] + α B [ ]; = + ; unl ermnaed = rue; // e.g., > max end; Fg. Pseudo-code of PBIL wh one probably vecor. ams o generae a real probably vecor, whch creaes hgh qualy soluons wh hgh probably when sampled. PBIL sars from an nal probably vecor wh values of each enry se o 0.5. Ths means when samplng by hs nal vecor random soluons are creaed because he probably of generang a or 0 on each locus s equal. However, as he search progresses, he values n he probably vecor are gradually learn owards hose values ha represen hgh evaluaon soluons. The evoluon process s as follows. Durng each eraon, a se of samples (soluons) s creaed accordng o he curren probably vecor as follows. For each b poson of a soluon, assumng bnary encoded, f a random creaed real number n he range of [0.0,.0] s less han he correspondng probably value n he probably vecor, s se o (or 0 oherwse s se o 0 (or respecvely). The se of samples are evaluaed accordng o he problem-specfc fness funcon. Then he probably vecor s learn (pushed) owards he soluon(s) wh he hghes fness. The dsance he probably vecor s pushed depends on he parameer of learnng rae. Afer he probably vecor s updaed a new se of soluons s generaed by samplng from he new probably vecor and hs cycle s repeaed. As he search progresses, he enres n he probably vecor move away from her nal sengs of 0.5 owards eher 0.0 or.0. The search progress sops when some ermnaon condon s sasfed, e.g., he maxmum allowable number of eraons max s reached or he probably vecor s converged o eher 0.0 or.0 for each b poson. The pseudo-code for he PBIL suded n hs paper s shown n Fg.. Whn hs PBIL a eraon a se of n = 0 soluons are sampled from he probably vecor Procedure PPBIL begn = 0; // nalze probably vecors for = o L do 0 P [ ] = 0.5; 0 P [ ] = rand[0.0,.0]; // nalze sample szes for probably vecors 0 0 n = n = 0.5 n; repea = generaeamplesfrom bvecor( P, n = generaeamplesfrom bvecor( P, n evaluaea mples(, B = selecbesoluonfrom( B = selecbesoluonfrom( // learn probably vecors oward bes soluons for = o L do P [ ] = ( α ) P [ ] + α B [ ]; P [ ] = ( α ) P [ ] + α B[ ]; // adjus sample szes for probably vecors f f B ) > f ( B ) hen n = mn{ n +, }; ( Fg. Pseudo-code of he Parallel PBIL (PPBIL). P and only he bes soluon learn he probably vecor a 0.05. nmax f f B ) < f ( B ) hen n = max{ n, }; ( n = n n ; = + ; unl ermnaed = rue; // e.g., end; nmn > max B from he se s used o P. The learnng rae α s fxed. Parallel Populaon-Based Incremenal Learnng Usng mul-populaon nsead of one populaon has proved o be a good approach for mprovng he performance of EAs for dynamc opmzaon problems [5]. mlarly, we can nroduce mul-populaon no PBIL by usng mulple probably vecors. Each probably vecor s sampled o generae soluons ndependenly, and s learn accordng o he bes soluon(s) generaed by. For he sake of smplcy, n hs paper we nvesgae a Parallel PBIL wh wo parallel probably vecors, denoed by PPBIL. The pseudo-code for PPBIL s shown n Fg.. Whn PPBIL one probably vecor P s nalzed o be 0.5 for each probably value (n order o compare s
performance wh PBIL) and he oher P s randomly nalzed. The probably vecors P and P are sampled and updaed ndependenly. Boh P and P have equal nal sample sze, half of he oal number of samples n = 0. However, n order o gve he probably vecor ha performs beer more chance o generae samples, he sample szes are slghly adaped whn he range of [ n mn, nmax ] = [0.4 n, 0.6 n] = [48, 7] accordng o her relave performance. If one probably vecor ouperforms he oher, s sample sze s ncreased by = 0.05 n = 6 whle he oher's sample sze s decreased by ; oherwse, f he wo probably vecors e, here s no change o her sample szes. The learnng rae for boh P and P s he same as ha for he PBIL..3 Dual Populaon-Based Incremenal Learnng Dualsm or complemenary s que common n naure. For example, n bology he DNA molecule consss of wo complemenary srands ha are wsed ogeher no a duplex chan. Inspred by he complemenary mechansm n naure, n hs paper we propose a Dual PBIL, denoed by DPBIL. For he convenence of descrpon, here we frs nroduce he defnon of dual probably vecor. Gven a L probably vecor P = ( P[],, P[ L]) I = [0.0,.0] of fxed lengh L, s dual probably vecor s defned as P' = dual( P) = ( P'[],, P'[ L]) I where P' [ ] =.0 P[ ] ( =,, L ). Tha s, a probably vecor's dual probably vecor s he one ha s symmerc o wh respec o he cenral pon n he search space. Wh hs defnon, DPBIL consss of a par of probably vecors ha are dual o each oher. The pseudo-code of DPBIL s gven n Fg. 3. From Fg. 3 can be seen ha DPBIL dffers from PPBIL only n he defnon of he probably vecor P and he learnng mechansm. The oher aspecs of DPBIL, such as he samplng mechansm, he sample sze updang mechansm, and relevan parameers, are he same as hose of PPBIL. Whn DPBIL P s now defned o be he dual probably vecor of P. As he search progresses only P s learn from he bes soluon generaed snce P changes wh P auomacally. If he bes overall soluon s sampled by updaed owards B P,.e. ( B ) f ( B ) ; oherwse, from B, he bes soluon creaed by learnng away from f, hen P s P s updaed away P. The reason o B les n ha s equvalen o learnng owards B. The movaon of nroducng dual probably vecor no PBIL les n wo aspecs ncreasng dversy of samples and fghng sgnfcan envronmen changes. On he frs aspec, usually wh he progress of parallel PBILs he probably vecors wll converge owards each oher and he dversy of generaed samples s reduced. Ths doesn' happen wh dual probably vecors. On he second aspec, P P Procedure DPBIL begn = 0; // nalze probably vecors for = o L do 0 0 P [ ] = P [ ] = 0.5; // nalze sample szes for probably vecors 0 0 n = n = 0.5 n; repea = generaeamplesfrom bvecor( P, n = generaeamplesfrom bvecor( P, n evaluaea mples(, B = selecbesoluonfrom( B = selecbesoluonfrom( // learn probably vecors for = o L do f f ( B ) f ( B ) hen // learn P oward P [ ] = ( α ) P [ ] + α B [ ]; else // learn P away from Fg.3 Pseudo-code of he Dual PBIL (DPBIL). when he envronmen s subjec o sgnfcan changes he dual probably vecor s expeced o generae hgh evaluaon soluons and hence mprove PBIL's adapably. 3. Algorhm Tes Envronmens In order o compare he performance of dfferen PBILs, a saonary knapsack problem s randomly consruced as he es problem. A seres of dynamc knapsack problems s hen consruced from hs saonary knapsack problem by a dynamc problem generang echnque. 3. aonary Knapsack Problem B B P [ ] = ( α ) P [ ] + α (.0 B[ ] P [ ] =.0 P [ ]; // adjus sample szes for probably vecors f f B ) > f ( B ) hen n = mn{ n +, }; ( nmax f f B ) < f ( B ) hen n = max{ n, }; ( n = n n ; = + ; unl ermnaed = rue; // e.g., end; nmn > max The knapsack problem s a well-known NP-complee combnaoral opmzaon problem and has been well suded n EA's communy. The problem s o selec from a se of ems wh varyng weghs and profs hose ems ha
wll yeld he maxmal summed prof o fll n he knapsack whou exceedng s lmed wegh capacy. Gven a se of m ems and a knapsack, he 0-knapsack problem can be descrbed as follows max p x m ( ) = = = subjec o he wegh consran m = = w x C p x where x = ( x,, x m ), x s 0 or, w and p are he wegh and prof of em respecvely, and C s he wegh capacy of he knapsack. If x =, he h em s seleced for he knapsack. In hs paper we consruced a knapsack problem wh 00 ems usng srongly correlaed ses of daa, randomly generaed as follows w = unformly random neger[, 50] p = w + unformly C = 0.6 = = random neger[, 5] 00 w And gven a soluon x, s fness f (x) s evaluaed as follows. If he sum of he em weghs s whn he capacy of he knapsack he sum of he profs of he seleced ems s used as he fness. If he soluon selecs oo many ems such ha he summed wegh exceeds he capacy of he knapsack, he soluon s judged by how much exceeds he knapsack capacy (he less, he beer) and s fness s evaluaed o be he dfference beween he oal wegh of all ems and he wegh of seleced ems, mulpled by a small 0 consan 0 o ensure ha he soluons ha overfll he knapsack are no compeve wh hose whch do no. Togeher, he fness of a soluon x s evaluaed as follows f ( x) = 0 = 00 = 0 ( p x, = 00 = f w = 00 = = 00 = w x C w x ), oherwse 3. Consrucng Dynamc Knapsack Problems In hs paper, we consruc dynamc es envronmens from above saonary knapsack problem usng a dynamc problem generang echnque proposed n [5]. Ths echnque s characerzed by wo envronmenal dynamcs parameers he speed of change and he degree of change n he genoype space. The frs parameer s referred o as he envronmenal change perod, denoed by τ, and s defned as he number of EA generaons beween wo changes. Tha s, every τ generaons he fness landscape s changed. The second parameer s measured by he rao of ones n a L bnary emplae T ( k) {0,} (where L s he chromosome lengh) creaed for each envronmenal change perod, denoed by ρ. For each envronmenal change perod k we frs creae a bnary mask M } L ( k) {0, ncremenally as M ( k) = M ( k ) T ( k) where T (k) s randomly creaed for perod k wh ρ L ones and s a bwse exclusve-or (XOR) operaor (.e., = 0, 0 =, 0 0 = ). For he frs perod, M () s nalzed o be a zero vecor. When evaluang an ndvdual x } L {0, n he populaon, we frs perform he operaon x M (k) on. The XORed resul s hen evaluaed o oban a fness value for he ndvdual x. I can be seen ha he parameer ρ conrols he degree of envronmenal change. The bgger he value of ρ, he more sgnfcan he envronmenal change. Pung hngs ogeher, he envronmen dynamcs can be formulaed as follows f ( x, ) = f ( x M ( k)) where k = /τ s he perod ndex, = [( k ) τ, kτ ] s he generaon couner. In hs paper, we consruc dynamc knapsack problems as follows. For each run of an algorhm on each knapsack problem he fness landscape s perodcally changed every τ generaons. Based on our prelmnary expermenal resuls on he saonary knapsack problem (see econ 4.), τ s se o 0, 50, 00, 50 and 00 generaons respecvely n order o es each algorhm's capably of adapng o dynamc envronmen under dfferen degree of convergence (or searchng sage). In order o es he effec of he degree of envronmenal change on he performance of algorhms ρ s se o 0.05, 0.5, 0.5, 0.75, and.0 respecvely. These values represen dfferen envronmenal change level, from lgh shfng ( ρ = 0. 05 ), o medum varaon ( ρ = 0.5, 0. 5 ), o heavy change ( ρ = 0. 75 ), and o he exreme case ( ρ =. 0 ) of oscllang beween wo reversed fness landscapes. Toally, we sysemacally consruc a seres of 5 dynamc problems, 5 values of τ combned wh 5 values of ρ, from he saonary knapsack problem. 4. Compuer Expermenal udy 4. Desgn of Expermens Expermens were carred ou o compare he performance of nvesgae PBILs. In order o compare PBILs as a whole wh oher evoluonary algorhms, we also ncluded a sandard genec algorhm, denoed by GA, as a peer algorhm n he expermens. GA has followng ypcal confguraon generaonal, unform crossover wh a crossover probably p = 0. 6, radonal b muaon wh c a muaon probably p m = 0.00, fness proporonae selecon wh he ochasc Unversal amplng (U) scheme [] and whou els model, and a populaon sze of n = 0. For each expermen of combnng algorhm and problem (saonary or dynamc), 50 ndependen runs were execued wh he same 50 random seeds. For each run of an algorhm on each problem, 0 perods of envronmenal changes were
Table Expermenal resuls wh respec o he overall mean bes-of-generaon fness FBG of dfferen algorhms on dynamc knapsack problems. Fg.4 Expermenal resuls on he saonary knapsack problem. allowed and he bes-of-generaon fness was recorded every generaon. The overall performance of an algorhm on a problem s measured by he mean bes-of-generaon fness. I s defned as he bes-of-generaon fness averaged across he number of runs and hen averaged over he daa gaherng perod. More formally hs s F BG = G = G ( N j= N = j= F BGj where F BG s he mean bes-of-generaon fness, G s he number of generaons whch s equvalen o 0 perods of envronmenal changes (.e., G = 0 τ ), N = 50 s he oal number of runs, and F s he bes-of-generaon fness of BG j generaon of run j of an algorhm on a problem. 4. Expermenal Resuls on he aonary Knapsack Problem In order o help analyze he expermenal resuls on dynamc problems, prelmnary expermens were carred ou on he saonary knapsack problem. For each run of dfferen algorhm he maxmum allowable number of generaons was se o 00. The prelmnary expermenal resuls wh respec o bes-of-generaon fness agans generaons are shown n Fg. 4, where he daa were averaged over 50 runs. From Fg. 4, can be seen ha n general all PBILs ouperform GA. Ths resul s conssen wh oher researchers' sudy [4]. PBIL ouperforms PPBIL and DPBIL whle PPBIL performs as well as DPBIL. Ths resul shows ha on he saonary knapsack problem nroducng exra probably vecor may no be benefcal snce he exsence of exra probably vecor may slow down he learnng speed of he oher probably vecor, whchever of he wo vecors performs beer. 4.3 Expermenal Resuls on Dynamc Knapsack Problems ) Param. eng Index Parameer Algorhms eng ( τ, ρ) GA PBIL PPBIL DPBIL (0, 0.05) 46.6 43.7 49. 45. (0, 0.5) 406.7 40.3 48.9 44. 3 (0, 0.50) 40.4 43.0 4.4 4.8 4 (0, 0.75) 396.9 409. 408.4 43. 5 (0,.00) 37.5 406.8 404.7 48. 6 (50, 0.05) 44.9 453.4 450.8 450. 7 (50, 0.5) 45.8 46.4 47.3 44. 8 (50, 0.50) 397.7 43.3 46.5 48.6 9 (50, 0.75) 377. 40.7 405.4 43.8 0 (50,.00) 337.3 393. 390.8 465.8 (00, 0.05) 49.5 46. 44.3 44.8 (00, 0.5) 4.4 39. 40.7 406.7 3 (00, 0.50) 4. 38.7 39. 405.7 4 (00, 0.75) 387.8 368.8 383.4 408.6 5 (00,.00) 338.4 367.0 365.0 474.9 6 (50, 0.05) 43. 403.9 434.6 439.7 7 (50, 0.5) 44.6 385.4 40.5 40.5 8 (50, 0.50) 48.0 363.6 373.9 400.6 9 (50, 0.75) 400.7 340.5 367. 40.7 0 (50,.00) 348. 95.3 3.3 477.3 (00, 0.05) 433.6 39.9 40.6 439. (00, 0.5) 46.7 363.6 393.4 399.6 3 (00, 0.50) 40.3 333. 36.0 399.9 4 (00, 0.75) 407.4 36. 34.5 405.8 5 (00,.00) 36. 33.5 53.4 479. Fg.5 Expermenal resuls on dynamc knapsack problems. The expermenal resuls on dynamc problems are summarzed n Table and ploed n Fg. 5 where he envronmenal dynamcs parameer seng s ndexed accordng o Table. From Table and Fg. 5 several resuls
(a) (b) (c) (d) Fg.6 Expermenal resuls wh respec o bes-of-generaon fness agans generaons of nvesgaed algorhms on dynamc knapsack problems. The envronmenal dynamcs parameer τ = 0 and ρ s se o (a) 0.05, (b) 0.5, (c) 0.75, and (d).0. can be observed and are dscussed as follows. Frs, an obvous resul s ha for each value of τ DPBIL performs conssenly wh he ncreasng of he value of ρ. Wh each fxed τ, when ρ ncreases from 0.05 o 0.5, 0.50, 0.75 o.0 he performance curve of DPBIL looks lke a bg U whle he oher algorhms have a performance curve of fallng sone. Ths happens because ncreasng he value of ρ ncreases he magnude of envronmenal changes, whch degrades he performance of GA, PBIL and PPBIL perssenly. However, n DPBIL he nroducon of he dual probably vecor sops DPBIL s performance from droppng when he value of ρ reaches 0.5. Thereafer, DPBIL s performance rses wh he ncreasng of he value of ρ. For each fxed value of τ when ρ =.0 DPBIL acheves he hghes performance pon. Ths resul confrms our expecaon of nroducng he dual probably vecor no DPBIL. When he magnude of envronmenal change s large, he dual probably vecor akes effec quckly o adap he DPBIL o he changed envronmen. econd, PBIL s now beaen by boh PPBIL and DPBIL on mos suaons excep for when ρ s small. When ρ s small, he dynamc knapsack problems are closer o he saonary knapsack problem where nroducng an exra probably vecor may no work well. Ths s verfed by above prelmnary expermenal resuls on he saonary knapsack problem as shown n Fg.4. However, when ρ ncreases he nroducon of exra probably vecor becomes more and more benefcal. Ths s because exra probably vecor helps mprovng dversy n he samples. Thrd, as opposed o he saonary knapsack problem, GA now ouperforms PBILs on many dynamc knapsack problems, especally when he value of τ s large. When τ s large he algorhms are gven more me o search before envronmen changes and hence hey are more lkely o converge. Convergence deprves PBILs of he adapably o
(a) (b) (c) (d) Fg.7 Expermenal resuls wh respec o bes-of-generaon fness agans generaons of nvesgaed algorhms on dynamc knapsack problems. The envronmenal dynamcs parameer τ = 00 and ρ s se o (a) 0.05, (b) 0.5, (c) 0.75, and (d).0. changed envronmen. However, he muaon mechansm embedded n GA gves GA more dversy han PBILs and beer adapably o envronmen changes. Hence, GA ouperforms PBILs. In order o beer undersand he expermenal resuls, we gve ou he dynamc performance of esed algorhms wh respec o bes-of-generaon fness agans generaons on several dynamc problems n Fg. 6 and Fg. 7, where he daa were averaged over 50 runs. In Fg. 6 and Fg. 7 he value of τ s se o 0 and 00 respecvely, and whn boh fgures he value of ρ s se o 0.05, 0.5, 0.75 and.0 respecvely. From Fg. 6 and Fg. 7, can be seen ha generally speakng, he performance of he algorhms (excep for DPBIL) drops heaver and heaver wh he ncreasng of he value of ρ as well as he value of τ. Wh DPBIL when ρ =.0 s performance rses nsead of drops wh he growng of dynamc perods due o he effec of he dual probably vecor. Ths resuls n he bg U curve for DPBIL's overall performance (see Fg. 5). 5. Conclusons and Fuure Work In hs paper we nvesgae he applcaon of Populaon-Based Incremenal Learnng (PBIL) algorhms for solvng opmzaon problems under dynamc envronmens. We sudy he effec of nroducng exra probably vecor no PBIL o mprove s performance under dynamc envronmens. Inspred by he complemenary mechansm n naure, we propose a Dual PBIL ha operaes on a par of probably vecors ha are dual o each oher wh respec o he cenral pon n he genoype space. Usng a dynamc problem generang echnque we sysemacally consruc a se of dynamc knapsack problems from a randomly creaed saonary knapsack problem and based on hese saonary and dynamc knapsack problems we carry ou expermenal sudy comparng nvesgaed PBILs and one radonal GA. From he expermenal resuls he followng conclusons can be acheved.
Frs, on saonary problems nroducng exra probably vecor no PBIL may no be benefcal. However, under dynamc envronmens nroducng exra probably vecor no PBIL mproves s performance. econd, when he envronmen s subjec o sgnfcan changes n he sense of genoype space, nroducng he dual probably vecor no PBIL can acheve very hgh performance mprovemen. Thrd, hough he GA s beaen by PBILs on he saonary problem, he muaon scheme embedded n GA helps keepng he dversy n he populaon and hence mproves GA s performance under dynamc envronmens. Ths paper nvesgaed an neresng work of applyng PBILs, especally he dual PBIL, for dynamc opmzaon problems. There are several relevan works o be carred ou n he fuure. Frs, exendng he resuls n hs paper o oher Esmaon of Dsrbuon Algorhms (EDAs) [3], of whch PBILs are a sub-class, s an neresng work. econd, s also worhy o nroduce and develop more approaches, such as he hypermuaon echnque [7], [], from EA's communy o PBILs or EDAs for dynamc opmzaon problems. Fnally, formally analyzng he performance of nvesgaed PBILs for dynamc opmzaon problems s also an mporan fuure work. References [] T. Bäck (998). On he Behavor of Evoluonary Algorhms n Dynamc Fness Landscape. Proc. of he 998 IEEE In. Conf. on Evoluonary Compuaon, 446-45. IEEE Press. [] J. E. Baker (987). Reducng Bas and Ineffcency n he elecon Algorhms. In J. J. Grefenselle (ed.), Proc. of he nd In. Conf. on Genec Algorhms, 4-. Lawrence Erlbaum Assocaes. [3]. Baluja (994). Populaon-Based Incremenal Learnng A Mehod for Inegrang Genec earch Based Funcon Opmzaon and Compeve Learnng. Techncal Repor CMU-C-94-63, Carnege Mellon Unversy, UA. [4]. Baluja and R. Caruana (995). Removng he Genecs from he andard Genec Algorhm. Proc. of he h In. Conf. on Machne Learnng, 38-46. [5] J. Branke, T. Kaubler, C. chmd, and H. chmeck (000). A Mul-Populaon Approach o Dynamc Opmzaon Problems. In Proc. of he 4h In. Conf. on Adapve Compung n Desgn and Manufacurng. [6] J. Branke (00). Evoluonary Approaches o Dynamc Opmzaon Problems - Updaed urvey. GECCO Workshop on Evoluonary Algorhms for Dynamc Opmzaon Problems, 34-37. [7] H. G. Cobb (990). An Invesgaon no he Use of Hypermuaon as an Adapve Operaor n Genec Algorhms Havng Connuous, Tme-Dependen Nonsaonary Envronmens. Techncal Repor AIC-90-00, Naval Research Laboraory, Washngon, UA. [8] D. E. Goldberg (989). Genec Algorhms n earch, Opmzaon, and Machne Learnng. Readng, MA Addson-Wesley. [9] J. J. Grefensee (99). Genec Algorhms for Changng Envronmens. In R. Männer and B. Manderck (eds.), Proc. of he nd In. Conf. on Parallel Problem olvng from Naure, 37-44. [0] J. Lews, E. Har and G. Rche (998). A Comparson of Domnance Mechansms and mple Muaon on Non-aonary Problems. In A. E. Eben, T. Bäck, M. choenauer and H.-P. chwefel (eds.), Proc. of he 5h In. Conf. on Parallel Problem olvng from Naure, 39-48. [] N. Mor, H. Ka and Y. Nshkawa (997). Adapaon o Changng Envronmens by Means of he Memory Based Thermodynamcal Genec Algorhm. In T. Bäck (ed.), Proc. of he 7h In. Conf. on Genec Algorhms, 99-306. Morgan Kaufmann Publshers. [] R. W. Morrson and K. A. De Jong (000). Trggered Hypermuaon Revsed. Proc. of he 999 Congress on Evoluonary Compuaon, 05-03. [3] H. Mühlenben and G. Paaß (996). From Recombnaon of Genes o he Esmaon of Dsrbuons I. Bnary Parameers. In H.-M. Vog, W. Ebelng, I. Rechenberg, and H.-P. chwefel (eds.), Proc. of he 4h In. Conf. on Parallel Problem olvng from Naure, 78-87. [4] K. P. Ng and K. C. Wong (995). A New Dplod cheme and Domnance Change Mechansm for Non-aonary Funcon Opmsaon. In L. J. Eshelman (ed.), Proc. of he 6h In. Conf. on Genec Algorhms. Morgan Kaufmann Publshers. [5]. Yang (003). Non-aonary Problem Opmzaon Usng he Prmal-Dual Genec Algorhm. Proc. of he 003 Congress on Evoluonary Compuaon.