Research Article A Novel Hybrid Self-Adaptive Bat Algorithm

Transcription

1 e Scentfc World Journal, Artcle I , 12 pages Research Artcle A Novel Hybrd Self-Adaptve Bat Algorthm Iztok Fster Jr., 1 Smon Fong, 2 Janez Brest, 1 and Iztok Fster 1 1 Faculty of Electrcal Engneerng and Computer Scence, Unversty of Marbor, Smetanova 17, 2000 Marbor, Slovena 2 epartment of Computer and Informaton Scence, Unversty of Macau, Avenue Padre Tomas Perera, Tapa, Macau Correspondence should be addressed to Iztok Fster Jr.; ztok.fster2@un-mb.s Receved 17 November 2013; Accepted 9 March 2014; Publshed 9 Aprl 2014 Academc Edtors: J. Shu and F. Yu Copyrght 2014 Iztok Fster Jr. et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Nature-nspred algorthms attract many researchers worldwde for solvng the hardest optmzaton problems. One of the newest members of ths extensve famly s the bat algorthm. To date, many varants of ths algorthm have emerged for solvng contnuous as well as combnatoral problems. One of the more promsng varants, a self-adaptve bat algorthm, has recently been proposed that enables a self-adaptaton of ts control parameters. In ths paper, we have hybrdzed ths algorthm usng dfferent E strateges and appled these as a local search heurstcs for mprovng the current best soluton drectng the swarm of a soluton towards the better regons wthn a search space. The results of exhaustve experments were promsng and have encouraged us to nvest more efforts nto developng n ths drecton. 1. Introducton Optmzaton has become more and more mportant, especally durng these tmes of recesson. It has been establshed throughout practcally all spheres of human actvtes, for example, fnances, ndustres, sport, pharmacy, and so forth. In each of these spheres, a goal of the optmzaton s to fnd the optmal nput parameters accordng to known outcomes and a known model. Ths model represents a problem to be solved and transforms nput parameters to output outcomes. Essentally, the optmal nput parameters must be properly evaluated n order to determne the qualty of a soluton. Indeed, an objectve functon s used that mathematcally descrbes ths qualty. In the praxs, the value of the objectve functon can be ether mnmzed or maxmzed. For example, when buyng a car ether the mnmum cost mn(f(x)) or maxmum comfort max(f(x)) s nterestng objectve for a potental buyer, where f(x) denotes the objectve functon. Usng the equaton mn(f(x)) = max( f(x)), the maxmzaton problem can be transformed nto a mnmzaton one, and vce versa. The functon expressed n ths way s also named as ftness functon n evolutonary computaton communty. In place of the objectve functon, the mnmzaton of the ftness functon s assumed n ths paper. Most of the problems arsng n practce today are NPhard. Ths means that the tme complexty of an exhaustve search algorthm runnng on a dgtal computer whch checks all solutons wthn a search space ncreases exponentally by ncreasng the nstance sze as determned by the number of nput parameters. In some stuatons, when the number of nput parameters ncreases to a certan lmt, t can be expected that a user never obtans the results from the optmzaton. As a result, algorthms solvng the NPproblemsapproxmatelyhavearsennthepast.Although these algorthms do not fnd the exact optmal soluton, n general, ther solutons are good enough n practce. For nstance, the well-known algorthms for approxmately solvng the hardest problems today are () artfcal bee colony (ABC) [1], () bat algorthm (BA) [2], () cuckoo search (CS) [3], (v) dfferental evoluton (E) [4], (v) frefly algorthm (FA) [5, 6], (v) partcle swarm optmzaton (PSO) [7], (v) many more [8].

2 2 The Scentfc World Journal The developers of the above-mentoned algorthms were typcally nspred by nature n a development phase. Namely, each nature-nspred algorthm mmcs the natural behavors of specfc bologcal, chemcal, or physcal systems n order to solve partcular problems usng a dgtal computer. Most of the nspraton from nature emanates from bology. The greatest lastng mpresson on the developers has been left by arwn s evoluton [9]. arwn observed that nature s not statc and therefore the ftter ndvduals have more chances of survvng wthn the changng envronments. Holland n [10] connected ths adaptve behavor of the natural systems to artfcal systems that smulate ther behavor by solvng the partcular problems (also optmzaton problems) on the dgtal computers. Optmzaton algorthms are controlled by algorthm parameters that can be changed determnstcally, adaptvely, and self-adaptvely [11]. etermnstc parameters are altered by usng some determnstc rules (e.g., Rechenberg s 1/5 success rule [12]). In contrast, adaptvely controlled parameters are subject to feedback from the search process that serves as an nput to the mechansm used whch determnes the drecton and magntude of the change [11]. Fnally, the self-adaptvely controlled parameters are encoded nto a representaton of the soluton and undergo the actons of varaton operators [13]. Thspaperfocusesontheadaptatonandhybrdzaton of a swarm ntellgence algorthm. Swarm ntellgence (SI) belongs to an artfcal ntellgence dscplne (AI) whch frst became popular over the last decade and stll s [14]. It s nspred by the collectve behavor of socal swarms of ants, termtes,bees,andworms,flockofbrds,andschoolsoffsh [15]. Although these swarms consst of relatvely unsophstcated ndvduals, they exhbt coordnated behavor that drects the swarms towards ther desred goals. Ths usually results n the self-organzng behavor of the whole system, and collectve ntellgence or swarm ntellgence s n essence the self-organzaton of such multagent systems, based on smple nteracton rules. The bat algorthm (BA) s one of the younger members of ths famly whch was developed by Yang [16]. It s nspred by the mcrobats that use an echolocaton for orentaton andpreyseekng.theorgnalbatalgorthmemploystwo strategy parameters: the pulse rate and the loudness. The former regulates an mprovement of the best soluton, whle the latter nfluences an acceptance of the best soluton. Both mentoned parameters are fxed durng the executon of the orgnal bat algorthm. In the self-adaptve bat algorthm (SABA) developed by Fster et al. [17] these parameters are self-adaptve. The am of ths self-adaptaton s twofold. On the one hand, t s very dffcult to guess the vald value of the parameter at the begnnng of the algorthm. On the other hand, ths value depends on the phase n whch the search process s. Ths means that the parameter settng at the begnnng of the search process can be changed when ths process reaches maturty. In ths paper, a further step forward has been taken. Hybrdzaton wth local search heurstcs has now been appled n order to further mprove the results of the SABA algorthm. oman-specfc knowledge can be ncorporated usng the local search. Although the local search s as yet an ngredent of the orgnal bat algorthm t can be replaced by dfferent E strateges [18]. Indeed, the further mprovement of the current best soluton s expected whch would drect the swarm ntellgence search towards the more promsng areas of the search space. As a result, the hybrd self-adaptve bat algorthm (HSABA) that was appled to a benchmark sute consstng of ten well-known functons from the lterature was developed. The results of HSABA obtaned durng extensve experments showed that the HSABA mproved the results of both the orgnal BA and the SABA. Moreover, the results were also comparable wth the other well-known algorthms, lke frefly (FA) [5], dfferental evoluton (E) [19], and artfcal bee colony (ABC) [20]. The structure of ths paper s as follows. Secton 2 presents an evoluton of the bat algorthms from the orgnal BA va the self-adaptve SABA to the hybrd self-adaptve HSABA algorthm. Secton 3 descrbes the expermental work, whle Secton 4 llustrates the obtaned results n detal. The paper concludes by summarzng the performed work and outlnes drectons for the further development. 2. Evoluton of Bat Algorthms The results of experments regardng the orgnal bat algorthm showed that ths algorthm s effcent especally when optmzng the problems of lower dmensons. In lne wth ths, Eben and Smth n [11] asserted that 2-dmensonal functons are not sutable for solvng wth the populatonbased stochastc algorthms (lke evolutonary algorthms and swarmntellgence),becausethesecanbesolvedoptmally usng tradtonal methods. On the other hand, these knds of algorthms could play a role as general problem solvers, becausetheysharethesameperformanceswhenaveraged over all the dscrete problems. Ths fact s the essence of the so-called No-Free Lunch (NFL) theorem [21]. In order for ths theorem to preval, there are almost two typcal mechansms for mprovng the performance of the populaton-based algorthms as follows: () self-adaptaton of control parameters, () hybrdzaton. The former enables the control parameters to be changed durng the search process n order to better sut the exploraton and explotaton components of ths search process [22], whle the latter ncorporates the problem-specfc knowledge wthn t. In the rest of ths paper, frstly the orgnal BA algorthm s presented n detal, followed by descrbng the applcaton of a self-adaptve mechansm wthn the orgnal BA algorthm whch leads to the emergence of a self-adaptve BA algorthm SABA. Fnally, a hybrdzaton of the SABA algorthm s broadly dscussed whch obtans a hybrdzed SABA algorthm named the HSABA The Orgnal Bat Algorthm. The orgnal BA s a populaton-based algorthm, where each partcle wthn the

3 The Scentfc World Journal 3 Input: Bat populaton x =(x 1,...,x ) T for,...,np,max FE. Output: The best soluton x best and ts correspondng value f mn = mn (f (x)). (1) nt bat(); (2) eval = evaluate the new populaton; (3) f mn =fnd best soluton(x best ); {ntalzaton} (4) whle termnaton condton not meet do (5) for to Np do (6) y = generate new soluton(x ); (7) f rand(0, 1) > r then (8) y =mprove the best soluton(x best ) (9) end f {local search} (10) f f new =evaluate new soluton(y); (11) eval = eval +1; (12) f f new f and N(0, 1) < A then (13) x = y; f =f new ; (14) end f {smulated annealng} (15) f mn =fnd the best soluton(x best ); (16) end for (17) end whle Algorthm 1: Pseudocode of the orgnal bat algorthm. bat populaton represents the canddate soluton. The canddate solutons are represented as vectors x =(x 1,...,x ) T for = 1 Np wth real-valued elements x j,whereeach element s drawn from nterval x j [x lb x ub ].Thus,x lb and x ub denote the correspondng lower and upper bounds, and Np determnes a populaton sze. Ths algorthm conssts of the followng man components: () an ntalzaton, () a varaton operaton, () a local search, (v) an evaluaton of a soluton, (v) a replacement. In the ntalzaton, the algorthm parameters are ntalzed, then, the ntal populaton s generated randomly, next, ths populaton s evaluated, and fnally, the best soluton n ths ntal populaton s determned. The varaton operator movesthevrtualbatsnthesearchspaceaccordngto the physcal rules of bat echolocaton. In the local search, the current best soluton s mproved by the random walk drect explotaton heurstcs (RWE) [23]. The qualty of a soluton s determned durng the evaluaton of soluton. The replacement replaces the current soluton wth the newly generated soluton regardng the some probablty. Ths component s smlar to the smulated annealng [24], where the new soluton s accepted by the acceptance probablty functon whch smulates the physcal rules of annealng. The pseudocode of ths algorthm s presented n Algorthm 1. In Algorthm1 the partcular component of the BA algorthm s denoted ether by functon name when t comprses one lne or by a comment between two curly brackets when the components are wrtten wthn a structured statement and/or t comprses more lnes. In lne wth ths, the ntalzaton comprses lnes 1 3 n Algorthm 1, the varaton operaton lne 6 (functon generate new soluton), the local search lnes 7 9, the evaluaton of the soluton (functon evaluate new soluton n lne 10), and the replacement lnes In addton, the current best soluton s determned n each generaton (functon fnd best soluton n lne 15). The varaton operaton whch s mplemented n functon generate new soluton movesthevrtualbatstowards the best current bat s poston accordng to the followng equatons: Q (t) =Q mn +(Q max Q mn )N(0, 1), k (t+1) x (t+1) = k t +(xt best)q(t), = x (t) + k (t+1), where N(0, 1) sarandomnumberdrawnfromagaussan dstrbuton wth zero mean and a standard devaton of one. A RWE heurstcs [23] mplemented n the functon mprove the best soluton modfes the current best soluton accordng to the followng equaton: x (t) = best +εa (t) N (0, 1), (2) where N(0, 1) denotes the random number drawn from a Gaussan dstrbuton wth zero mean and a standard devaton of one, ε beng the scalng factor and A (t) the loudness. A local search s launched wth the probablty of pulse rate r. As already stated, the probablty of acceptng the new best soluton n the component save the best soluton condtonaly depends on loudness A. Actually, the orgnal BA algorthm s controlled by two algorthm parameters: the pulse rate r and the loudness A. Typcally, the rate of (1)

4 4 The Scentfc World Journal pulse emsson r ncreases and the loudness A decreases when the populaton draws nearer to the local optmum. Both characterstcs mtate natural bats, where the rate of pulseemssonncreasesandtheloudnessdecreaseswhen abatfndsaprey.mathematcally,thesecharacterstcsare captured usng the followng equatons: A (t+1) =αa (t), r (t) =r (0) [1 exp ( γε)], (3) where α and γ are constants. Actually, the α parameter controls the convergence rate of the bat algorthm and therefore plays a smlar role as the coolng factor n the smulated annealng algorthm. In summary, the orgnal BA algorthm s based on a PSO algorthm [7] whch s hybrdzed wth RWE and smulated annealng heurstcs. The former represents the local search that drects the bat search process towards mprovng the best soluton, whle the latter takes care of the populaton dversty. In other words, the local search can be connected wth explotaton, whle smulated annealng uses the exploraton component of the bat search process. The explotaton s controlled by the parameter r and exploraton by the parameter A.Asaresult,theBAalgorthmsabletoexplctly control the exploraton and explotaton components wthn ts search process The Self-Adaptve Bat Algorthm. Almost two advantages canbeexpectedwhentheself-adaptatonofcontrolparameters s appled to a populaton-based algorthm [11]: () control parameters need not be set before the algorthm s run, ()thecontrolparametersareadapteddurngtherunto the ftness landscape defned by the postons of the canddate solutons wthn the search space and ther correspondng ftness values [25]. Normally, the self-adaptaton s realzed by encodng the control parameters nto representaton of canddate solutons and lettng them undergo an operaton of the varaton operators. In ths way, the self-adaptaton of the BA control parameters (the loudness and the pulse rate) s consdered. Ths means that the exstng representaton of the canddate solutons consstng of problem varables x (t) = (x (t) 1,...,x(t) ) s wdened by the control parameters A(t+1) and r (t+1) to x (t) =(x (t) 1,...,x(t),A(t),r (t) ) T, for Np, (4) where Np denotes the populaton sze. The control parameters are modfed accordng to the followng equatons: A (t+1) ={ A(t) lb + rand 0 (A (t) ub A(t) lb ) f rand 1 <τ 1, A (t) otherwse, r (t+1) ={ r(t) lb r (t) + rand 2 (r (t) ub r(t) lb ) f rand 3 <τ 2, otherwse. (5) (6) Note that the parameters τ 0 and τ 1 denote the learnng rates that were set, as τ 0 =τ 1 = 0.1, whle rand for 4 desgnate the randomly generated value from nterval [0, 1]. Theself-adaptngpartoftheSABAalgorthmsperformed by the generate the new soluton functon (lne 6 n Algorthm 1). Note that the control parameters are modfed n ths functon accordng to the learnng rates τ 0 and τ 1. In the case τ 0 = τ 1 = 0.1, each10thcanddatesoluton s modfed on average. The modfed parameters nfluence theapplcatonofthelocalsearchandtheprobabltyof the replacement. The replacement functon preserves the problem varables as well as the control parameters. Ths self-adaptng functon was nspred by Brest et al. [4] who proposed the self-adaptve verson of E, better known as je. Ths self-adaptve algorthm mproves the results of the orgnal E sgnfcantly by contnuous optmzaton The Hybrd Self-Adaptve Bat Algorthm. The populatonbased algorthms, lke evolutonary algorthms and swarm ntellgence, can be seen as some knd of general problem solvers, because they are applcable for all classes of optmzaton problems. Thus, ther results confrm the so-called No- Free Lunch (NFL) theorem by Wolpert and Macready [21]. Accordng to ths theorem any two optmzaton algorthms are equvalent when ther performances are averaged across all possble problems. In order to crcumvent the NFL theorem by solvng a specfc problem, doman-specfc knowledge must be ncorporated wthn the algorthm for solvng t. The domanspecfc knowledge can be ncorporated by the bat algorthm wthn each of ts components, that s, an ntalzaton, a varaton operator, a local search, an evaluaton functon, and areplacement. Although the SABA algorthm sgnfcantly outperformed the results of the orgnal BA algorthm, t suffers from a lack of ncorporated doman-specfc knowledge of the problem to be solved. Ths paper focuses on hybrdzng thesabausnganovellocalsearchheurstcsthatbetter explots the self-adaptaton mechansm of ths algorthm. The standard rand/1/bn E strategy and three other E strateges focusng on the mprovement of the current best soluton were used for ths purpose, where the modfcaton ofthelocalsearchnahybrdself-adaptvebaalgorthm (HSABA) s llustrated n Algorthm 2. The local search s launched accordng to a threshold determned by the self-adapted pulse rate r (lne 1). Ths parameter s modfed by each 10th vrtual bat, on average. The local search s an mplementaton of the operators crossover and mutaton borrowed from E [19]. Frstly, the four vrtual bats are selected randomly from the bat populaton (lne 2) and the random poston s chosen wthn the vrtual bat (lne 3). Then, the approprate E strategy modfes the tral soluton (lnes 4 9) as follows: E Strategy, y n ={ x (t) n f rand (0, 1) CR n=, otherwse, where the E Strategy s launched accordng to the probablty of crossover CR. The term n=ensures that almost (7)

5 The Scentfc World Journal 5 Input: Populaton x =(x 1,...,x,A,r) T for,...,np. Output: Tral soluton y =(y 1,...,y,A,r) T. (1) f rand(0, 1) > r then (2) r,...,4 = rand(0, 1) Np + 1 r 1 =r 2 =r 3 =r 4 ; (3) n=rand(1, ); (4) for to do (5) f ((rand (0, 1) < CR) (n ==)) then (6) y n = E Strategy(n,,r 1,r 2,r 3,r 4 ); (7) end f (8) n=(n+1)%( + 1); (9) end for {E Strateges} (10) end f Algorthm 2: Modfcaton n hybrd self-adaptve BA algorthm (HSABA). Table 1: Used E strateges n the HSABA algorthm. E/rand/1/bn y j =x r1,j +F (x r2,j x r3,j ) E/randToBest/1/bn y j =x,j +F (best j x,j ) F (x r1,j x r2,j ) E/best/2/bn y j = best j +F (x r1,j +x r2,j x r3,j x r4,j ) E/best/1/bn y j = best j +F (x r1,j x r2,j ) one modfcaton s performed on the tral soluton. The E Strategy functon s presented n Algorthm 3. Although more dfferent E strateges exst today, we have focused manly on those that nclude the current best soluton n the modfcaton of the tral soluton. These strateges are presented n Table 1.Note that the sutable E strategy s selected usng the global parameter strategy. The strateges llustrated n the table that use the best soluton n the modfcaton operatons, that s, rand- ToBest/1/bn, best/2/bn, and best/1/bn, typcally drect the vrtual bats towards the current best soluton. Thus, t s expected that the new best soluton s found when the vrtualbatsmoveacrossthesearchspacedrectedbythebest soluton. The rand/1/bn represents one of the most popular E strateges today that ntroduces an addtonal randomness nto a search process. Obvously, t was used also n ths study. 3. Experments The goal of our expermental work was to show that the HSABA outperforms the results of the orgnal BA as well as the self-adaptve SABA algorthms, on the one hand, and that these results were comparable wth the results of other well-known algorthms, lke frefly (FA) [5], dfferental evoluton (E) [19], and artfcal bee colony (ABC) [20]. All the mentoned algorthms were appled to a functon optmzaton that belongs to a class of combnatoral optmzaton problems. Functon optmzaton s formally defned as follows. Let us assume a ftness functon f(x), wherex = (x 1,...,x ) denotes a vector of desgn varables from a decson space x j S. The values of the desgn varables are drawn from the nterval x j [lb j,ub j ],wherelb j R and ub j R are ther correspondng lower and upper bounds, respectvely. Then, the task of functon optmzaton s to fnd the mnmum of ths objectve functon. In the remander of ths paper, the benchmark functon sute s presented, then the expermental setup s descrbed, andfnally,theconfguratonofthepersonalcomputer(pc) on whch the tests were conducted s clarfed n detal Benchmark Sute. The benchmark sute was composed of tenwell-knownfunctonsselectedfromvarouspublcatons. The reader s nvted to check deeper detals about test functons n the state-of-the-art revews [26 28]. The defntons of the benchmark functons are summarzed n Table 2 whch conssts of three columns denotng the functon tag f, the functon name,andthefunctondefnton. Each functon from the table s tagged wth ts sequence number from f 1 to f 10. Typcally, the problem becomes heaver to solve when the dmensonalty of the benchmark functons s ncreased. Therefore, benchmark functons of more dmensons needed to be optmzed durng the expermental work. The propertes of the benchmark functons can be seen n Table 3 whch conssts of fve columns: the functon tag f, the value of the optmal soluton f = f(x ),theoptmal soluton x,thefunctoncharacterstcs, anddoman. One of the more mportant characterstcs of the functons s the number of local and global optma. Accordng to ths characterstc the functons are dvded nto ether unmodal or multmodal. The former type of functons has only one global optmum, whle the latter s able to have more local and global optma throughout the whole search space. Parameter doman lmts the values of parameters to the nterval betweentherlowerandupperbounds.asamatteroffact, these bounds determne the sze of the search space. In order to make the problems heaver to solve, the parameter domans were more wdely selected n ths paper than those prescrbed n the standard publcatons Expermental Setup. The characterstcs of the HSABA were observed durng ths expermental study. Then, the best parameter settng was determned. Fnally, the results of the HSABA were compared wth the results of the orgnal BA and self-adaptve SABA algorthms. The results of the other well-known algorthms, lke FA, E, and ABC, were also added to ths comparatve study. Note that the best parameter settngs for each partcular algorthm were used n the tests, except BA algorthms, where the same setup was employed n order to make the comparson as far as possble. All the parameter settngs were found after extensve testng. urng the tests, the BA parameters were set as follows: the loudness A 0 = 0.5, thepulserater 0 = 0.5, mnmum frequency Q max = 0.0, and maxmum frequency Q max = 2.0. Thesamentalvaluesforr 0 and A 0 were also appled by SABA and HSABA, whle the frequency was captured from the same nterval Q [0.0, 2.0] as by the orgnal bat algorthm. Thus, the values for loudness are drawn from the nterval A (t) [0.9, 1.0] andthepulseratefrom the nterval r (t) [0.001, 0.1]. The addtonal parameters

6 6 The Scentfc World Journal Input: nth poston n the tral soluton, ndexes of vrtual bats, r 1,r 2,r 3,r 4. Global: Populaton x =(x 1,...,x,A,r) T for,...,np,strategy. Output: Modfed value of the tral soluton z. (1) swtch (STRATEGY) (2) case E/rand/1/bn: (3) z=x r1,n +F (x r2,n x r3,n ); (4) case E/randToBest/1/bn: (5) z=x,n +F (best n x,n ) F (x r1,n x r2,n ); (6) case E/best/2/bn: (7) z=best n +F (x r1,n +x r2,n x r3,n x r4,n ); (8) case E/best/1/bn: (9) z=best n +F (x r1,n x r2,n ); (10) end swtch (11) return z; Algorthm 3: E Strategy functon. Table 2: efntons of benchmark functons. f Functon name efnton f 1 Grewangk s functon f (x) = f 2 Rastrgn s functon f(x) =n 10+ n 1 f 3 Rosenbrock s functon f(x) = n 1 f 4 Ackley s functon f(x) = n cos ( x )+ n n x (x 2 10cos(2πx )) 100 (x +1 x 2 )2 +(x 1) 2 (20 + e 20 e (x2 +1 +x2 ) e 0.5(cos(2πx +1)+cos(2πx )) ) f 5 Schwefel s functon f(x)= x sn ( x ) f 6 e Jong s sphere functon f(x) = f 7 Easom s functon f(x) = ( 1) ( cos 2 (x )) exp [ (x π) 2 ] f 8 Mchalewcz s functon f(x) = f 9 Xn-She Yang s functon f(x) =( f 10 Zakharov s functon f(x) = Table 3: Propertes of benchmark functons. x 2 +( 1 2 x sn(x )[sn ( x2 π )] x ) exp [ sn (x 2 )] f f x Characterstcs oman f (0,0,...,0) Hghly multmodal [ 600, 600] f (0,0,...,0) Hghly multmodal [ 15, 15] f (1,1,...,1) Several local optma [ 15, 15] f (0,0,...,0) Hghly multmodal [ , ] f (0,0,...,0) Hghly multmodal [ 500, 500] f (0,0,...,0) Unmodal, convex [ 600, 600] f (π,π,...,π) Several local optma [ 2π, 2π] f ( , ) 1 Several local optma [0, π] f (0,0,...,0) Several local optma [ 2π, 2π] f (0,0,...,0) Unmodal [ 5, 10] 1 Vald for 2-dmensonal parameter space. x ) 2 +( 1 2 x ) 4

7 The Scentfc World Journal 7 controllng behavor of E strateges were set as F = 0.01 and CR = 1.0. Both parameters had a great nfluence on the results of the HSABA algorthm, whle ths settng ntated the best performance by the self-adaptve SABA framework hybrdzed wth E strateges. FA ran wth the followng set of parameters: α = 0.1, β = 0.2,andγ = 0.9, whle E was confgured as follows: the amplfcaton factor of the dfference vector F = 0.5 and the crossovercontrolparametercr= 0.9. In the ABC algorthm, the onlooker bees represented 50% of the whole colony, whle the other 50% of the colony was reserved for the employed bees. On the other hand, the scout bee was generated when ts value had not mproved n 100 generatons. In other words, the parameter lmts was set to the value 100. Eachalgorthmnthetestswasrunwthapopulatonsze of 100. Each algorthm was launched 25 tmes. The obtaned results of these algorthms were aggregated accordng to ther best, the worst, the mean, the standard devaton, and the medan values reached durng 25 runs PC Confguraton. All runs were made on HP Compaq wth the followng confguratons. (1) Processor: Intel Core (3.8) GHz. (2) RAM: 4 GB R3. (3) Operatng system: Lnux Mnt 12. Addtonally, all the tested algorthms were mplemented wthn the Eclpse Indgo CT framework. 4. The Results Our expermental work conssted of conductng the four experments, n whch the followng characterstcs of the HSABA were tested: () an nfluence of usng the dfferent E strateges, () an nfluence of the number of evaluatons, () an nfluence of the dmensonalty of problems, (v) a comparatve study. In the frst test, searchng was performed for the most approprate E strategy ncorporated wthn the HSABA algorthm. Ths strategy was then used n all the other experments that followed. The am of the second experment was to show how the results of the HSABA converge accordng to the number of ftness functon evaluatons. ependence of the results on the dmensonalty of the functons to be optmzed s presented n the thrd experment. Fnally, the results of the HSABA were compared wth the orgnal BA and self-adaptve SABA algorthms as well as the other wellknown algorthms, lke FA, E, and ABC. In the remander of ths paper, the results of the experments are presented n detal Influences of Usng the fferent E Strateges. In ths experment, the qualtes of four dfferent E strateges that were mplemented wthn the HSABA algorthm were observed. The outcome of ths experment had an mpact on the further tests, because all the tests that followed were performed wth the most promsng E strategy found durng the experment. In lne wth ths, each of the E strateges, that s, rand/1/bn, randtobest/1/bn, best/2/bn, and best/1/bn, was tested by optmzng the benchmark functon sute. The results of functon optmzaton were averaged after 25 runs and are llustrated n Fgure 1. Ths fgure s dvded nto sx dagrams, two for each observed dmenson of the functon. Each dagram represents the progress of the optmzaton n percent on the x-axs, where 100% denotes the correspondng number of ftness functon evaluatons (e.g., 10,000 for 10-dmensonal functons), whle the value of the ftness functon s placed alongsde the y-axs. In summary, the best/2/bn strategy acheved the best result when compared wth the other three strateges, on average. Interestngly, the rand/1/bn strategy was the most successful when optmzng the functon f 4 of dmenson =10. Ths behavor was n accordance wth the characterstcs of ths functon, because hghly multmodal functons demand more randomness durng the exploraton of the search space Influence of the Number of Evaluatons. Along-termlack of mprovement regardng the best result durng the run was one of the most relable ndcators of the stagnaton. If the ftness value dd not mprove over a number of generatons, ths probably means that the search process got stuck wthn a local optmum. In order to detect ths undesrable stuaton durng the run of HSABA, the ftness values were tracked at three dfferent phases of the search process, that s, at 1/25, at 1/5, and at the fnal ftness evaluaton. Thus, HSABA runs wth the best/2/bn strategy and parameters as presented n Secton 3.2.TheresultsofthstestarecollatednTable 4. ItcanbeseenfromTable 4 that HSABA successfully progressed towards the global optmum accordng to all benchmark functons, except f 7 whch fell nto a local optmum very quckly and dd not fnd any way of further mprovng the result Influence of the mensonalty of the Problem. The am of ths experment s to dscover how the qualty of the results depends on the dmenson of the problem, n other words, the dmensonalty of the functons to be optmzed. In lne wth ths, three dfferent dmensons of the benchmark functons =10, =30,and=50were taken nto account. The results of the tests accordng to fve measures are presented n Table 5. In ths experment, t was expected that the functons of the hgher dmensons would be harder to optmze and therefore the obtaned results would be worse. The results n Table 5 show that our assumpton held for the average ftness values n general, except for functon f 9,whereoptmzng ths functon of dmenson =10returned the worst ftness value as by functons of dmensons =30and =50.

8 8 The Scentfc World Journal Values of ftness functon Values of ftness functon Calculaton progress (%) Calculaton progress (%) rand/1/bn randtobest/1/bn best/2/bn best/1/bn rand/1/bn randtobest/1/bn best/2/bn best/1/bn (a) f 4 of dmenson =10 (b) f 8 of dmenson = Values of ftness functon Values of ftness functon Calculaton progress (%) Calculaton progress (%) rand/1/bn randtobest/1/bn best/2/bn best/1/bn rand/1/bn randtobest/1/bn best/2/bn best/1/bn (c) f 3 of dmenson =30 (d) f 6 of dmenson = Values of ftness functon Values of ftness functon Calculaton progress (%) Calculaton progress (%) rand/1/bn randtobest/1/bn best/2/bn best/1/bn rand/1/bn randtobest/1/bn best/2/bn best/1/bn (e) f 2 of dmenson =50 (f) f 10 of dmenson =50 Fgure 1: Impact of E strateges on the results of optmzaton.

9 The Scentfc World Journal 9 Table 4: etaled results ( =10). Evals. Meas. f 1 f 2 f 3 f 4 f 5 Best 1.62E E E E E Worst 1.03E E E E E E + 02 Mean 4.05E E E E E Stev 2.27E E E E E Mean 3.81E E E E E Best 7.90E E E E E 003 Worst 2.18E E E E E E + 03 Mean 2.91E E E E E Stev 1.52E E E E E Mean 5.36E E E E E Best 4.87E E E E E 004 Worst 1.19E E E E E E + 04 Mean 9.35E E E E E Stev 1.88E E E E E 002 Mean 2.41E E E E E Evals. Meas. f 6 f 7 f 8 f 9 f 10 Best 7.07E E E E E 02 Worst 3.28E E E E E E + 02 Mean 6.98E E E E E + 01 Stev 2.62E E E E E + 00 Mean 9.46E E E E E + 01 Best 8.13E E E E E 05 Worst 4.49E E E E E E + 03 Mean 4.75E E E E E 01 Stev 2.18E E E E E 02 Mean 9.10E E E E E 01 Best 1.01E E E E E 06 Worst 6.85E E E E E E + 04 Mean 8.92E E E E E 02 Stev 8.47E E E E E 02 Mean 1.76E E E E E Comparatve Study. The true value of each algorthm s shown for only when we compared t wth the other algorthms solvng the same problems. Therefore, the HSABA was compared wth algorthms such as BA, SABA, FA, E, and ABC. All the algorthms were solved usng the same benchmark functons as proposed n Secton 3.1 and used the parameter settng as explaned n Secton 3.2. The results of ths comparatve study obtaned by optmzng the functons of dmenson = 30 are llustrated n Table 6.The best resultsnthesetablesarepresentedasbold. ItcanbeseenfromTable 6 that the HSABA outperformed the results of the other algorthms sxfold (.e., by f 1, f 2, f 3, f 5, f 9,andf 10 ), E twce (.e., by f 4 and f 7 ), whle the other algorthms once (.e., SABA by f 6 and ABC by f 8 ). BA and FA dd not outperform any other algorthm n the test. In order to evaluate the qualty of the results statstcally, Fredman tests [29, 30] wereconductedtocomparethe average ranks of the compared algorthms. Thus, a null hypothess s placed that states the followng: two algorthms are equvalent and therefore ther ranks should be equal. When the null hypothess s rejected, the Bonferron-unn test [31] s performed. In ths test, the crtcal dfference s calculated between the average ranks of those two algorthms. If the statstcal dfference s hgher than the crtcal dfference, the algorthms are sgnfcantly dfferent. Three Fredman tests (Fgure 2) were performed regardng data obtaned by optmzng ten functons of three dfferent dmensons accordng to fve measures. As a result, each algorthm durng the tests (also the classfer) was compared wth regard to the 10 functons 5measures; ths means 50 dfferent varables. The tests were conducted at the sgnfcance level The results of the Fredman nonparametrc test can be seen n Fgure 1 that s dvded nto three dagrams. Each dagram shows the ranks and confdence ntervals (crtcal dfferences) for the algorthms under consderaton wth regard to the dmensons of the functons. Note that the sgnfcant dfference between two algorthms s observed f ther confdence ntervals denoted by thckened lnes n Fgure 1 do not overlap.

10 10 The Scentfc World Journal Table 5: Results accordng to dmensons. Meas. f 1 f 2 f 3 f 4 f 5 Best 4.87E E E E E 04 Worst 1.19E E E E E Mean 9.35E E E E E + 00 Stev 1.88E E E E E 02 Mean 2.41E E E E E + 00 Best 6.68E E E E E 03 Worst 5.58E E E E E Mean 7.71E E E E E + 02 Stev 2.85E E E E E + 01 Medan 1.26E E E E E + 02 Best 7.16E E E E E 02 Worst 8.81E E E E E Mean 1.70E E E E E + 02 Stev 8.47E E E E E + 02 Mean 2.29E E E E E + 03 Evals. Meas. f 6 f 7 f 8 f 9 f 10 Best 1.01E E E E E 06 Worst 6.85E E E E E Mean 8.92E E E E E 02 Stev 8.47E E E E E 02 Mean 1.76E E E E E 01 Best 1.96E E E E E 04 Worst 2.17E E E E E Mean 2.63E E E E E + 01 Stev 1.29E E E E E + 00 Mean 4.82E E E E E + 01 Best 4.72E E E E E 01 Worst 9.88E E E E E Mean 5.59E E E E E + 02 Stev 1.24E E E E E + 02 Mean 1.97E E E E E + 02 ABC E FA HSABA SABA BA Average rank ( =10) Average rank ( =30) Average rank ( =50) Fgure 2: Results of the Fredman nonparametrc test on the specfc large-scale graph nstances. The frst two dagrams n Fgure 1 show that the HSABA sgnfcantly outperforms the results of the other algorthms by solvng the benchmark functons wth dmensons = 10 and = 30. Although the other algorthms are not sgnfcantly dfferent between each other the algorthms E, SABA, and ABC are substantally better than the BA and FA. The thrd dagram shows that the HSABA remans sgnfcantlybetterthantheabc,fa,andba,whlethe crtcal dfference between E and SABA was reduced and therefore becomes nsgnfcant. On the other hand, ths reducton caused the dfference between E and the other algorthms n the test, except SABA, to become sgnfcant.

11 The Scentfc World Journal 11 Table 6: Obtaned results of algorthms (=30). F Meas. BA SABA HSABA FA E ABC f 1 Mean 1.16E E E E E E + 00 Stev 1.15E E E E E E 01 f 2 Mean 9.28E E E E E E + 01 Stev 8.90E E E E E E + 01 f 3 Mean 2.84E E E E E E + 02 Stev 2.95E E E E E E + 02 f 4 Mean 2.00E E E E E E + 00 Stev 2.00E E E E E E + 00 f 5 Mean 9.45E E E E E E + 03 Stev 9.52E E E E E E + 02 f 6 Mean 5.87E E E E E E + 02 Stev 6.53E E E E E E + 02 f 7 Mean 0.00E E E E E E 136 Stev 0.00E E E E E E 136 f 8 Mean 8.62E E E E E E + 01 Stev 8.39E E E E E E 01 f 9 Mean 1.57E E E E E E 11 Stev 1.03E E E E E E 12 f 10 Mean 2.76E E E E E E + 02 Stev 2.82E E E E E E Concluson The orgnal BA algorthm ganed better results by optmzng the lower-dmensonal problems. When ths algorthm was tackled wth the harder problems of hgher dmensons the results became poorer. In order to mprove performance on hgh dmensonal problems, almost two mechansms are proposed n the publcatons: frstly, self-adaptaton of control parameters and secondly hybrdzaton of the orgnal algorthm wth problem-specfc knowledge n the form of local search. In ths paper, both mechansms were ncorporated wthn the orgnal BA algorthm n order to obtan the hybrd selfadaptve HSABA. Thus, the self-adaptve mechansm was borrowedfromtheself-adaptveje,whlefourdfferente strateges were used as a local search. Fnally, the evoluton of the orgnal BA va self-adaptve SABA and fnally to hybrd HSABA s presented n detal. On the one hand, the expermental work focuses on dscoverng the characterstcs of the HSABA, whle on the other hand, t focuses on the comparatve study, n whch the HSABA was compared wth ts predecessors BA and SABA as well as wth other well-known algorthms, lke FA, E, and ABC. The results of ths extensve work showed that the HSABA sgnfcantly outperformed the results of all the other algorthms n the tests. Therefore, the assumpton taken from publcatons that self-adaptaton and hybrdzaton are approprate mechansms for mprovng the results of populaton-based algorthms was confrmed. In the future work, ths started work wll be contnued wth experments on the large-scale global optmzaton problems. Conflct of Interests All authors declare that there s no conflct of nterests regardng the publcaton of ths paper. References [1]. Karaboga and B. Basturk, On the performance of artfcal bee colony (ABC) algorthm, Appled Soft Computng Journal, vol.8,no.1,pp ,2008. [2] I.J.Fster,.Fster,andX.S.Yang, Ahybrdbatalgorthm, Electrotechncal Revew,vol.80,no.1-2,pp.1 7,2013. [3]I.F.Jr,.Fster,andI.Fster, Acomprehensverevewof cuckoo search: varants and hybrds, Internatonal Journal of Mathematcal Modellng and Numercal Optmsaton,vol.4,no. 4, pp , [4]J.Brest,S.Grener,B.Boškovć, M. Mernk, and V. Žumer, Self-adaptng control parameters n dfferental evoluton: a comparatve study on numercal benchmark problems, IEEE Transactons on Evolutonary Computaton, vol.10,no.6,pp , [5] I.Fster,I.J.Fster,X.S.Yang,andJ.Brest, Acomprehensve revew of frefly algorthms, Swarm and Evolutonary Computaton,vol.13,pp.34 46,2013. [6] I.Fster,X.S.Yang,.Fster,andI.FsterJr, Freflyalgorthm:a bref revew of the expandng lterature, n Cuckoo Search and Frefly Algorthm, pp ,Sprnger,NewYork,NY,USA, [7] J. Kennedy and R. Eberhart, Partcle swarm optmzaton, n Proceedngs of the IEEE Internatonal Conference on Neural Networks, vol. 4, pp , IEEE, ecember [8]I.J.Fster,X.S.Yang,I.Fster,J.Brest,and.Fster, A bref revew of nature-nspred algorthms for optmzaton, Electrotechncal Revew,vol.80,no.3,2013.

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