Improved Cuckoo Search Algorithm with Novel Searching Mechanism for Solving Unconstrained Function Optimization Problem

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1 IAENG Internatonal Journal o Computer Scence, 44:, IJ_44 0 Improve Cuckoo Search Algorthm wth Novel Searchng Mechansm or Solvng Unconstrane Functon Optmzaton Problem Shu-Xa L, an Je-Sheng Wang Abstract Cuckoo search () algorthm s a new bologcal heurstc algorthm an smulates the cuckoo s seekng nest an spawnng behavor an ntrouces levy lght mechansm. In orer to mprove the convergence velocty an optmzaton accuracy o cuckoo search algorthm, by combnng the learnng-evolvng thought wth Gaussan strbuton, a new searchng mechansm that wth a learnng-evolvng searchng guer s propose. Then combnng the new searchng mechansm propose wth the Levy Flght searchng mechansm accorng to a selecton probablty, a new searchng mechansm cuckoo search () algorthm s orme. The result o unctons testng uner erent mensons proves the superorty o new propose algorthm. Inex Terms cuckoo search algorthm, uncton optmzaton, searchng mechansm, Gaussan strbuton C I. INTRODUCTION CUCKOO search () algorthm s put orwar by Yang an Deb n 009, whch smulates the cuckoo s seekng nest an spawnng behavor an ntrouces levy lght mechansm nto t, whch s able to quckly an ecently n the optmal soluton [-]. Ths algorthm s manly base on two aspects: the cuckoo's parastc reproucton mechansm an Levy lghts search prncple. In nature, cuckoos use a ranom manner or a quas-ranom manner to seek br's nest locaton. Most o cuckoos lay ther eggs n other br nests an let the host rase ther cubs nstea o them. I a host oun that the eggs are not ts owns, t wll ether throw these alen eggs away rom the nest or abanon ts nest an bul a new nest n other places. However, there Manuscrpt receve June 7, 06; revse October 5, 06. Ths work was supporte by the Project by Natonal Natural Scence Founaton o Chna (Grant No. 5767), the Program or Laonng Excellent Talents n Unversty (Grant No. LR04008), the Project by Laonng Provncal Natural Scence Founaton o Chna (Grant No ), the Program or Research Specal Founaton o Unversty o Scence an Technology o Laonng (Grant No. 05TD04) an the Openng Project o Natonal Fnancal Securty an System Equpment Engneerng Research Center (Grant No. USTLKFGJ050 an USTLKEC040). Shu-Xa L s a postgrauate stuent n the School o Electronc an Inormaton Engneerng, Unversty o Scence an Technology Laonng, Anshan, 405, PR Chna (e-mal: Je-Sheng Wang s wth the School o Electronc an Inormaton Engneerng, Unversty o Scence an Technology Laonng, Anshan, 405, PR Chna; Natonal Fnancal Securty an System Equpment Engneerng Research Center, Unversty o Scence an Technology Laonng. (phone: ; ax: ; e-mal: are some cuckoos choosng nest that the color an shape o the host s egg are smlar wth ther owns to wn the host s love, whch can reuce the possblty o ther eggs beng abanone an ncrease the reproucton rate o the cuckoos. Stues have prove that algorthm are better than other swarm ntellgence algorthms n convergence rate an optmzaton accuracy, such as ant colony optmzaton (ACO) algorthm [3], genetc algorthm (GA) [4], bat algorthm (BA) [5], artcal bee colony (ABC) algorthm [6], etc. In that ths algorthm has the characterstcs o ewer parameters, smple an easy to mplement, now t has been successully apple n a varety o engneerng optmzaton problems an has a very hgh potental research value [7-8]. Cuckoo algorthm s manly base on two aspects: the cuckoo's nest parastc reprouctve mechansm an Levy lghts search prncple. In nature, cuckoos use a ranom manner or a quas-ranom manner to seek br's nest locaton [9-0]. However, some cuckoos choose nest that the color an shape o the host s eggs are smlar wth ther owns to wn the host s love. The paper s organze as ollows. In secton, the mprove cuckoo search algorthm wth the novel searchng mechansm or solvng unconstrane uncton optmzaton problem s ntrouce. The smulaton experments an results analyss are ntrouce n etals n secton 3. The concluson llustrates the last part. II. IMPROVED CUCKOO SEARCH ALGORITHM WITH NOVEL SEARCHING MECHANISM A. Basc Cuckoo Search Algorthm In general, each cuckoo can only lays one egg, an each egg on behal o one soluton (cuckoo). The purpose s to make the new an potentally better solutons replace the not-so-goo solutons (cuckoos). In orer to stuy the cuckoo search algorthm better, the smplest metho s aopte, that s to say only one egg s n each nest. In ths case, an egg, a br's nest or a cuckoo s no erences, whch s to say each nest corresponng to a cuckoo s egg. For smplcty n escrbng the cuckoo search algorthm, Yang an Deb use the ollowng three ealze rules to construct the cuckoo algorthm [-]. () Each cuckoo only lays one egg at a tme, an ranomly choose br's nest to hatch the egg. () The best nest wll carry over to the next generaton. (3) The number o avalable host nests s xe, an the probablty o a host scovers an alen egg s Pa [0,]. Cuckoo algorthm s base on ranom walk o Levy lght (Avance onlne publcaton: February 07)

2 IAENG Internatonal Journal o Computer Scence, 44:, IJ_44 0 makng search. Levy lght s a ranom walk, whose step sze obeys Levy strbuton, an the recton o travel s subject to unorm strbuton. On the bass o these rules, upatng ormula o the cuckoo nest locaton s escrbe as ollows: where, x x s () ( t) t L SL levy( ) () t x represents the poston o the -th nest at the t -th x t generaton, represents the poston o the -th nest at the ( t ) -th generaton ; s step control volume; levy( ) s a vector obeyng Levy strbuton: ( )sn( / ) L( s, ),( s s 0) (3) s where, s0 0 represents the mnmum step, s a gamma uncton; the step o levy( ) obeys levy strbuton. In general, levy strbuton s usually expresse as ollows: L(, ) cos( ks)exp[ k ] k (4) 0 In the Eq. (4), there s no any orm o explct analyss; thereore, t s cult to obtan a ranom sample by the ormula. But, when s s0 0, the Eq. (4) can be approxmate as the ollowng equaton: L(, ) ( )sn( / ) (5) s When, the Eq. (5) s equvalent to Eq. (4). Although the Eq. (5) can escrbe ranom walk behavor o the cuckoo algorthm, but t s not conucve to the escrpton o the mathematcal language, an s more savantageous to the wrtng o the program. So, Yang Xn She an Deb oun that n the realzaton o the algorthm aoptng Mantegna algorthm can well smulate ranom walk behavor o levy lght [3]. In ths algorthm, the step length s can be represente as: u s, / (6) v where s s leap path o levy lyng ; Parameters u an v are subject to normal strbuton shown as Eq. (7): N v N (7) (0, ), (0, v ) ( )sn( / ) ( )/ [( ) / ] /, Ths algorthm can generate samples approxmate to levy strbute. In theory s 0 0, but n practce, s 0 can take a very small value, such as s0 0.. v (8) B. New Search Mechansm Cuckoo Search Algorthm () In the basc algorthm, step sze an the recton generatng by usng the levy lght search mechansm are hghly ranom. It s known rom Eq. (6) that the step sze o levy lght completely epens on o the ranom number u an v, whch makes the search have characterstcs o great ranomness an blnness. An n the search process there s lack o normaton communcaton between cuckoos. The search s easy to jump rom one regon to another regon, whch leas to a low search accuracy an slow convergence spee. In orer to make the search wth a rectvty an teleology an let the algorthm o search uner a gue, nspre by the shule rog leapng algorthm (SFLA) an varaton thought comng rom erental evoluton (DE) algorthm, n ths artcle, the worst rog's upate strategy o SFLA algorthm an varaton ea are ntrouce nto search o br's nest locatons. In SFLA algorthm, n orer to get more oo aster, poor rog s nluence by goo rog jump to the better rog. Base on SFLA algorthm, n orer to make a cuckoo hunt or better br's nest aster, let t learn rom the best cuckoo an mprove the capablty o communcatng normaton wth the best cuckoo. The ntroucton o varaton thought make br's nest has an ablty o sel-evolvng, whch can ncrease the versty o br's nest poston, that s to say t can ncrease the versty o solutons. For mprovng the search ablty o the algorthm, wth learnng an evolvng as the search wzar, the Gaussan strbuton s ae to the algorthm. Base on the thoughts above, the new search mechansm s as shown ollows. lem( ) c ( x x ) G c ( x x ) G (9) best r r where, x s the -th br's nest locaton, x best s the current best locaton, r an r are ranom number rom (, n ), r r, n s the number o br's nest populaton, x r an x are the br's nest locatons correspone to a ranom r number r an r, G an G obey the Gaussan strbuton, c s the learnng scale an c s evoluton scale. The values c an c control the learnng an evolvng ablty o cuckoos. I c an c are set xe values, t can make learnng an evoluton lack lexblty. In orer to make the learnng an evoluton has a lexblty c an c changng as the ollowng ormula: c (0.5 ) / (0) c (0.5 ) / () where, an are ranom number rom [0, ] an obey unorm strbuton Ranomness an strong leap characterstcs o Levy lght make the algorthm has stronger global searchng ablty. I ully use the search mechansm that propose n ths paper (Avance onlne publcaton: February 07)

3 IAENG Internatonal Journal o Computer Scence, 44:, IJ_44 0 an abanon Levy lght search mechansm, t wll lost the avantages o hgh searchng ablty owne by orgnal algorthm, whch wll result n the algorthm cult to jump out o local optmal soluton. I ully use the Levy lght search mechansm, t wll lea to the algorthm has a low search accuracy an slow convergence spee. Thereore, n orer to exert avantages o both search mechansm, combnng the new searchng mechansm propose n ths paper wth the Levy Flght accorng to a selecton probablty. It can be express as Eq. (3). S L levy( ) p cr lem( ) p cr () where, cr s selectveprobablty that balance the two search mechansm an t s constant, 0cr. An p obeys the unorm strbuton p [0,]. When p cr, aoptng search mechansm o Levy lght to search br's nest locaton o the next generaton; When p cr, usng the search mechansm that propose by ths artcle to search br's nest locaton. III. SIMULATION RESULTS In orer to very search perormance o the new search mechansm cuckoo search algorthm (). In ths paper, sx typcal unctons are chosen to test algorthm perormance; the sx unctons are shown as Table. Expermental parameters: total br's nest populaton s n 5, the step length controlle parameter 0.0. The etecton probablty Pa =0.5, step length control, the Selecton probablty cr 0.45, The number o teratons ter 500. For each kn o algorthm, the program runs 50 tmes nepenently. Evaluate algorthm perormance by statstcs o the best value, average value an the worst value o 50 tmes runnng, an the convergence curves o unctons. The numercal test results are shown n Table. Convergence curves o Functon - 6 s shown as Fgure. It can be seen rom sx convergence curves uner the erent mensons an numercal results o Table that the algorthm has a hgher convergence spee an optmzaton accuracy than the orgnal algorthm. Thereore, algorthm has superor search perormance. TABLE. FOUR BENCHMARK FUNCTIONS USED IN THE SIMULATION EXPERIMENTS. Name Functon Scope Sphere Rosenbrock Grewank Mchalewcz Rastrgrn Sumsquares x (x) [-00,00] (x) (00(x x ) (x ) ) [-.08,.08] x 3(x) ( x ) cos( ) 4000 [-300,300] m x 4( x) sn( x ) sn( ),( m 0) [0, ] 5(x) (x 0cos( x ) 0) 6 [-.5,.5] ( x) x [-0,0] TABLE. COMPARISON OF NUMERICAL TESTING RESULTS Functon Dm. Metho Mnmum Maxmum Average F e e e e e e e-03.04e-04 (Avance onlne publcaton: February 07)

4 IAENG Internatonal Journal o Computer Scence, 44:, IJ_44 0 (a) Functon () Functon 4 (b) Functon (e) Functon 5 (c) Functon 3 () Functon 6 Fg.. Convergence curve or sx benchmark unctons. (Avance onlne publcaton: February 07)

5 IAENG Internatonal Journal o Computer Scence, 44:, IJ_44 0 IV. CONCLUSION Ths paper proposes an mprove cuckoo search algorthm wth a novel searchng mechansm or solvng unconstrane uncton optmzaton problem. The smulaton results show that the mprove cuckoo search algorthm has better convergence velocty an optmzaton accuracy. In uture, ths metho coul be extene to eal wth the other optmzaton problems. REFERENCES [] X. S. Yang an S. Deb, Cuckoo search va L evy lghts, n Proc. Worl Congress on Nature & Bologcally Inspre Computng, Combatore, Ina, 009, pp [] X. S. Yang an S. Deb, Engneerng optmsaton by cuckoo search, Internatonal Journal o Mathematcal Moellng an Numercal Optmsaton, vol., no. 4, pp , Dec. 00. [3] Y. Ghanou, an G. Benchekh, "Archtecture Optmzaton an Tranng or the Multlayer Perceptron usng Ant System," IAENG Internatonal Journal o Computer Scence, vol. 43, no., pp. 0-6, 06. [4] E. Vallaa, an R. Ruz, A genetc algorthm or the unrelate parallel machne scheulng problem wth sequence epenent setup tmes, European Journal o Operatonal Research, vol., no. 3, pp. 6 6, Jun. 0. [5] Anpng Song, Mngbo L, Xueha Dng, We Cao, an Ke Pu, "Communty Detecton Usng Dscrete Bat Algorthm," IAENG Internatonal Journal o Computer Scence, vol. 43, no., pp37-43, 06. [6] Chun-Feng Wang, an Yong-Hong Zhang, "An Improve Artcal Bee Colony Algorthm or Solvng Optmzaton Problems," IAENG Internatonal Journal o Computer Scence, vol. 43, no.3, pp , 06. [7] X. S. Yang, Cuckoo search an rely algorthm. New York: Sprnger-Verlag, 04, pp. 6. [8] E. Valan, S. Mohanna, an S. Tavakol, Improve cuckoo search algorthm or eeorwar neural network tranng, Internatonal Journal o Artcal Intellgence & Applcatons, vol., no. 3, pp , Jul. 0. [9] A. H. Ganom, X. S. Yang, an A. H. Alav, Cuckoo search algorthm: a metaheurstc approach to solve structural optmzaton problems, Engneerng wth Computers, vol. 9, no., pp. 7 35, Apr. 03. [0] K. Chanrasekaran, an S. P. Smon, Mult-objectve scheulng problem: Hybr approach usng uzzy assste cuckoo search algorthm, Swarm an Evolutonary Computaton, vol. 5, pp. 6, Aug. 0. [] X. S. Yang an S. Deb, Cuckoo search: recent avances an applcatons, Neural Computng an Applcatons, vol. 4, no., pp , Dec. 04. [] A. R. Ylz, Cuckoo search algorthm or the selecton o optmal machnng parameters n mllng operatons, The Internatonal Journal o Avance Manuacturng Technology, vol. 64, no. -4, pp. 55 6, Jan. 03. [3] P. Cvcoglu, an E. Besok, A conceptual comparson o the Cuckoo-search, partcle swarm optmzaton, erental evoluton an artcal bee colony algorthms, Artcal Intellgence Revew, vol. 39, no. 4, pp , Apr. 03. Shu-Xa L s receve her B. Sc. egree rom Unversty o Scence an Technology Laonng n 0. She s currently a master stuent n School o Electronc an Inormaton Engneerng, Unversty o Scence an Technology Laonng, Chna. Her man research nterest s moelng methos o complex process an ntellgent optmzaton algorthms. Je-sheng Wang receve hs B. Sc. An M. Sc. egrees n control scence rom Unversty o Scence an Technology Laonng, Chna n 999 an 00, respectvely, an hs Ph. D. egree n control scence rom Dalan Unversty o Technology, Chna n 006. He s currently a proessor an Master's Supervsor n School o Electronc an Inormaton Engneerng, Unversty o Scence an Technology Laonng. Hs man research nterest s moelng o complex nustry process, ntellgent control an Computer ntegrate manuacturng. (Avance onlne publcaton: February 07)