Artificial Bee Colony Algorithm with Local Search for Numerical Optimization

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1 49 JOURNAL OF SOFTWARE, VOL. 6, NO. 3, MARCH Artfcal Bee Coloy Algorthm wth Local Search for Numercal Optmzato Fe Kag ad Juje L Dala Uversty of Techology / Faculty of Ifrastructure Egeerg, Dala, Cha Emal: kagfe9@63.com, ljuje@dlut.edu.c Zheyue Ma ad Haoj L Dala Uversty of Techology / School of Hydraulc Egeerg, Dala, Cha Emal: dmzy@dlut.edu.c, lhaoj983@63.com Abstract Artfcal bee coloy (ABC) algorthm s oe of the most recetly proposed swarm tellgece algorthms for global umercal optmzato. It performs well most cases; however, there stll exst some problems t caot solve very well. Ths paper presets a ovel hybrd Hooke Jeeves ABC () algorthm wth tesfcato search based o the Hooke Jeeves patter search ad the ABC. The ma purpose s to demostrate how the stadard ABC ca be mproved by corporatg a hybrdzato strategy. The proposed algorthm s tested o a comprehesve set of 36 complex bechmark fuctos ad a slope stablty aalyss problem cludg a wde rage of dmesos. Comparsos are made wth the basc ABC ad some recet algorthms. Numercal results show that the ew algorthm s promsg terms of covergece speed, success rate ad soluto accuracy Idex Terms swarm tellgece; artfcal bee coloy algorthm; patter search; umercal optmzato; crtcal crcular slp surface I. INTRODUCTION Optmzato s a feld wth wde applcatos may areas of scece ad egeerg, where mathematcal modelg s used. Global optmzato could be a very challegg task because may objectve fuctos ad real world problems are multmodal, hghly o-lear, wth steep ad flat regos ad rregulartes []. Ucostraed global optmzato problems ca be formulated as followg model: m f ( x), x ( x, x,..., x ) () where f : R R s a real-valued objectve fucto, x R, ad s the umber of the parameters to be optmzed. The foragg behavor, learg, memorzg ad formato sharg characterstcs of bees have recetly bee oe of the most terestg research areas swarm Ths work was supported by Natoal Scece Foudato for Postdoctoral Scetsts of Cha ad the State Key Program of Natoal Natural Scece of Cha (No. 9854). Correspodg author : Fe Kag (kagfe9@63.com). tellgece. Studes o hoey bees are a creasg tred the lterature durg the last few years []. Artfcal bee coloy (ABC) algorthm was proposed by Karaboga 5 [3]. It s a optmzato algorthm based o partcular tellget behavor of hoey bee swarms. ABC has bee compared wth geetc algorthm, partcle swarm optmzato (PSO), dfferetal evoluto (DE), ad evolutoary algorthms [4], [5] o a lmted umber of test fuctos. It also has bee used for desgg IIR flters [6], for the leaf-costraed mmum spag tree problem [7] ad for structural parameter verse aalyss problems [8]. Accordg to the recet studes [9], ABC s better tha or smlar to other populato-based algorthms wth the advatage of employg fewer cotrol parameters. However, t stll has some defcecy deal wth fuctos havg arrow curvg valley, fuctos wth hgh eccetrc ellpse ad some extremely complex multmodal fuctos. I order to offset the default of ABC metoed above ad mprove ts covergece speed, a Hooke Jeeves artfcal bee coloy algorthm () wth tesfcato search s proposed for umercal optmzato. The algorthm matas the ma steps of ABC ad corporates a local search techque whch s based o Hooke Jeeves method (HJ) []. The effcecy of the ew algorthm s proved by comparso wth the basc ABC ad several other well kow populato based algorthms o extesve umercal test problems. The rest of the paper s orgazed as follows. Secto II revews the fudametals of the orgal ABC. Secto III descrbes the proposed algorthm. Secto IV presets comparatve studes o bechmark fuctos ad a slope stablty aalyss problem. Coclusos are gve Secto V. Appedx A lsts all test fuctos. II. ARTIFICIAL BEE COLOBY ALGORITHM ABC s a swarm tellget optmzato algorthm spred by hoey bee foragg [3-5]. I ABC, the coloy of the artfcal bees cotas three groups of bees: employed bees, olookers ad scouts. The frst half of the coloy cossts of the employed bees ad the secod half cludes the olookers. For every food source, there s ACADEMY PUBLISHER do:.434/jsw

2 JOURNAL OF SOFTWARE, VOL. 6, NO. 3, MARCH 49 oly oe employed bee. The employed bee of a abadoed food source becomes a scout. The posto of a food source represets a possble soluto of the optmzato problem ad the ectar amout of a food source correspods to the qualty (ftess) of the assocated soluto. At the frst step, the algorthm geerates a radomly dstrbuted tal populato cotas NS solutos. Where NS s the umber of food sources ad t s equal to the umber of employed bees. Each soluto x (,,,NS) s a - dmesoal vector. I ABC, the ftess fucto s defed as follows: f ft + f () + abs( f) f < where f s the objectve fucto value of soluto, ft s the ftess value of soluto after trasformato. A olooker bee chooses a food source depedg o the probablty value p assocated wth that food source, NS p ft ft (3) j j A caddate soluto v from the old soluto x ca be geerated as v x + φ x x (4) ( ) j j j j kj are radomly chose dexes; k has to be dfferet from ; φ j s a radom umber the rage [-, ]. After each caddate source posto s produced ad evaluated by the artfcal bee, ts performace s compared wth that of ts old oe. If the ew food source has equal or better qualty tha the old source, the old oe s replaced by the ew oe. Otherwse, the old oe s retaed. If a posto caot be mproved further through a predetermed umber lmt (lmted cycles), the that food source s assumed to be abadoed. The correspodg employed bee becomes a scout. The abadoed posto wll be replaced wth a ew food source foud by the scout. Assume that the abadoed source s x, ad the the scout dscovers a ew food source as where k {,, NS} ad j {,,, } (,)( ) xj lj + rad uj l (5) j where l j ad u j are lower ad upper bouds of varable x j. III. THE PROPOSED ALGORITHM A. Hooke-Jeeves Method Hooke ad Jeeves patter search method s a smple yet very effectve optmzato techque proposed 96 []. Today, t s stll a popular tool for varous optmzato problems, especally for determstc local search. The HJ method adopted here s modfed accordg to the source code of Johso []. The ma characterstcs of the modfed HJ method are: (І) to accelerate the procedure, drect search takes advatage of ts kowledge of the sg of ts prevous move each of the drectos; (II) a dfferet step sze for each varable s used to adaptve to the scalg problems of dfferet varables. I the HJ method a combato of exploratory move (EM) ad patter move (PM) s made teratvely to search out the optmum soluto for the problem. It starts wth a exploratory move to determe a approprate drecto of search by cosderg oe varable at a tme alog the dvdual coordate drectos the eghborhood of a base pot soluto. Followg the exploratory search, a patter move s made to accelerate the search the drecto determed the exploratory search. Exploratory searches ad patter moves are repeated utl a termato crtero s met. Assume x s the curret soluto (the base pot), f m s the curret mmum value of the objectve fucto, δ(δ, δ,, δ ) s the step szes of drectos. x s a temporary vector to store the obtaed pot after EM. The ma steps of EM are descrbed Fg.. : Italze ad x x. : x x +δ ; f (f(x ) < f m ), f m f(x ), go to step 4;else go to step 3. 3: x x -δ ; f (f(x ) < f m ), f m f(x ), go to step 4; else x x. 4: If <, set + ad go to step ; else x s the result of EM ad go to step 5. 5: If f(x )> f m, falure; else success. Fgure. Ma steps of EM operator. Gve two solutos x ad x (f(x ) < f(x )), the PM takes the step x - x from x as x x +(x -x ) (6) where x s the pot obtaed by PM. The HJ patter move s a aggressve attempt of the algorthm to explot promsg search drectos because t explots formato gaed from the search durg prevous successful teratos. The dea of PM s to vestgate whether further progress s possble the geeral drecto x x (sce, f f(x ) < f(x ), the x x s clearly a promsg drecto) []. The ma steps of modfed HJ method are show Fg.. ρ.5 s the step sze reducto factor. A auxlary step sze s a s adopted to judge whe to stop the algorthm because a dfferet step sze for each varable s used. : Choose the startg pot x, the curret step sze δ (,,, ), the step sze reducto factor ρ<, the termato parameter ε>. Italzg the terato couter k, the auxlary step sze s a.. : Perform a exploratory move wth x as the base pot ad the obtaed pot s x. If the exploratory move s successful, go to step 3; else go to step 6. 3: If (x < x ) δ - δ ; else δ δ, for,,,. 4: Perform a patter move x x +(x -x ) ad set x x. 5: Perform exploratory move wth x as the base pot ad the obtaed pot s x. 6: If f(x ) < f(x ), go to step 3, else go to step 6. 7: If s a <ε, termate; else set kk+, s a s a ρ, δ δ ρ, for,,,, ad go to step. Fgure. Ma steps of modfed HJ algorthm. ACADEMY PUBLISHER

3 49 JOURNAL OF SOFTWARE, VOL. 6, NO. 3, MARCH B. ABC wth Local Search The s proposed cosderg the local search property ad patter move operator of HJ s complemetary for ABC. I the orgal ABC, the ftess value s calculated by () to select a source for a olooker bee. To mprove the robustess of the selecto strategy, rak-based ftess trasformato s adopted as ( SP )( p ) ft SP + (7) NS where p s the posto of the soluto the whole populato after rakg, SP [.,.] s the selecto pressure ad a medum value of SP.5 ca be a good choce. The ma steps of the hybrd algorthm are summarzed as below. Every terval cycles of ABC, HJ s actvated to perform a local search usg the curret best soluto as the base pot. The step sze δ should sutable to the curret states of solutos, so a adaptve step sze s adopted. It s set as a fracto of the average of dstace betwee the selected solutos ad the best soluto acheved so far. The frst % solutos after rakg are selected to calculate the step sze as follows: m ( x, ) j x best j δ j. (8) m where δ j s the step sze of the jth dmeso, m s the umber of solutos selected to calculate the step sze, x s the th soluto after rakg, x best s the curret best soluto. At early stages, the populato wll be dverse ad ths wll result larger δ j. As the populato coverges, the dstace betwee dfferet solutos decreases ad so does the step sze of HJ search. Sometme the step ca become large aga because of the scout operator to avod premature covergece. The terato tmes of HJ s cotrolled by the parameter ε, whe s a <ε the algorthm wll retur to the ma framework of. For the sake of clarty, the ma steps of are descrbed Fg. 3. The ew algorthm frst coducts the optmzato process two phases alterately: durg the explorato phase t employs the ABC algorthm to locate regos of attracto; ad subsequetly, durg the explotato phase, employs the adaptve HJ techque to make a local explotato search ear the best soluto. If the alteratve process caot mprove the best soluto ay more, HJ s actvated aga to refe the obtaed soluto. Ths process s repeated utl the termato codto s met, e.g., the maxmum umber of fucto evaluatos s reached. If callg the HJ algorthm for couter tmes ca ot mprove the best soluto, the algorthm wll ext from the ma loop ad perform a tesfcato search by HJ algorthm utl the termato codto s met. The soluto the mddle posto after rakg s replaced by the obtaed better pot after HJ search. The worst soluto the populato ca ot be replaced because that wll affect the fucto of the scout operator. : Italze the populato of solutos x,,,ns. : Evaluate the populato, cycle, k. 3: Memorze the best soluto x best ad set x best x best 4: Repeat (Explorato phase) 5: Produce ew solutos v for the employed bees by usg (4) ad evaluate them. 6: Apply the greedy selecto process for the employed bees. 7: Rak the populato ad calculate the ftess by (7) 8: Calculate the probablty p for the solutos x by (3). 9: Produce the ew solutos v for the olookers from the solutos selected depedg o p ad evaluate them. : Apply the greedy selecto process for the olookers. : Determe the abadoed soluto for the scout, f exsts, ad replace t wth a ew radomly produced soluto x. : Memorze the best soluto x best acheved so far. (Explotato phase) 3: If ((cycle mod terval)), calculate step sze δ j of HJ accordg to (8). 4: Call modfed HJ wth x best as the base pot utl s a <ε ad the obtaed pot s x best. 5: If (f(x best ) f(x best )) Replace the soluto the mddle posto after rakg by x best ad set x best x best 6: If (f(x best )<f(x best )) set x best x best ad k, else set kk+; 7: Set cyclecycle+. (Itesfcato search) 8: If (k>couter) perform tesfcato search by modfed HJ. 9: Utl a termato codto s met. Fgure 3. Ma steps of wth tesfcato search. IV. EXPERIMENTAL RESULTS AND DISCUSSION A. Comparso wth ABC, DE ad ODE A comprehesve set of bechmark fuctos cludg 3 dfferet global optmzato problems [3] (Lsted appedx A) s adopted for comparso. The four algorthms are compared by measurg the umber of fucto evaluatos (NFE) to reach a gve accuracy. Each of the expermets ths secto s repeated 5 trals wth dfferet radom seeds. The NFE max s set as 3 the comparso. The termato crtero s the NFE has reached the maxmum value or the followg codto s satsfed, * f f < ε (9) * where f s the exact global mmum, f s the best fucto value obtaed by the algorthm ad the accuracy ε s set equal to -8 [3]. I order to compare the covergece speeds of ABC ad, we use the accelerato rate (AR) whch s defed as follows, NFEABC AR () NFE A tral s successful, f (9) ca be satsfed before NFE reaches the maxmum value. The proposed ew algorthm s compared wth the basc ABC, DE ad ODE [3] terms of umber of fucto evaluatos, success rate (SR). The commo parameters of ABC ad are set as NS5 (populato sze s 5), Lmt NS ad NFE max. The other parameters of are set as terval 3, ε -3 ad couter5. Each of the expermets ths secto s repeated 5 trals. ACADEMY PUBLISHER

4 JOURNAL OF SOFTWARE, VOL. 6, NO. 3, MARCH 493 The results of solvg 3 bechmark fuctos are gve Table І ad Table II. The best results of the NFE ad SR for each fucto are hghlghted boldface. The AR ave ad the SR ave o 3 test fuctos are show the last row of Table І ad Table II. TABLE I. COMPARISON OF DIFFERENT ALGORITHMS IN TERMS OF NUMBER OF FUNCTION EVALUATIONS. F Number of fucto evaluatos DE ODE ABC AR f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f Ave As ca be see from Table І, coverges much faster tha ABC, DE ad ODE most cases. oly coverges slower tha DE o Colvlle fucto (f ), Kowalk fucto (f )ad slower tha ABC o Step fucto (f 3 ). The overall average accelerato rate of to ABC s 6.36, whch meas s o average 536% faster tha ABC. Except to two much large values of Beal fucto (f ) ad Matyas fucto (f 5 ), the average accelerato rate of to ABC s.3, whch meas s o average 3% faster tha ABC except to two very large values. Geerally speakg, ABC coverges slower tha DE ad ODE for low dmesoal problems, but the effcecy of ABC outperforms DE ad ODE as the umber of varables creases to. The results reported Table II show that there are 8~ fuctos ca ot be solved % successfully by DE, ODE ad ABC, whereas oly 5 fuctos ca ot be solved % successfully by. O the other had, performs better tha other algorthms o fuctos, whereas DE, ODE ad ABC oly perform better tha other algorthms o 5~6 fuctos. The average success rate of s.93, whch s much hgher tha the other three algorthms. TABLE II. COMPARISON OF DIFFERENT ALGORITHMS IN TERMS OF SUCCESS RATE. F Success rate DE ODE ABC f f f 3 f 4 f 5 f 6 f 7 f 8 f 9 3 f 6 f 4.9 f f f 4 4 f 5 4 f 6 4 f f 8.76 f 9 f 3 f 3 f 3 f f 4 3 f 5 3. f f f 8 3 f 9 3 f 3 3 f 3 3 f 3 3 Ave B. The Effect of Parameter couter Seve typcal fuctos hard for ABC s selected to study the effect of parameter couter. The termato cotrol parameters ε s set equal to -6 ad NFE max. The other parameters are set the same as the prevous secto. The results are lsted Table III ad Table IV. TABLE III. THE EFFECT OF PARAMETER COUNTER TO NFE AND SR F ABC couter couter5 couter f 4 NFE/SR NFE/SR NFE/SR NFE/SR f 6 3 -/. 773/. 339/. 34/. f 3 3 -/. 644/ / /. f 9 3 -/. 9796/. 8988/. 894/. f 3 4 -/. 694/ /. 66/. f 4 -/. 4/. 4663/. 355/.9 f /. 353/ /. 693/. f 4 -/. 68/.9 57/. 346/.98 The average NFE of successful rus ad the SR are lsted Table III. It ca be see that all the fucto ACADEMY PUBLISHER

5 494 JOURNAL OF SOFTWARE, VOL. 6, NO. 3, MARCH selected are hard for ABC ad ca solve all the fuctos wth hgh SR, especally whe couter5. TABLE IV. THE EFFECT OF PARAMETER COUNTER TO ACCURACY F ABC couter couter5 couter Mea Mea Mea Mea f 4.8e- 6.e-4 4.5e-9.63e-8 (.e-) (3.34e-3) (.65e-8) (.4e-7) f e- 3.58e-9.88e-8.7e-8 (.e-) (.49e-8) (.3e-7) (8.8e-8) f e+.4e-4.e-4.9e-4 (3.5e+) (5.6e-4) (3.46e-4) (3.8e-4) f e+3 4.7e-9 4.3e-6 4.5e- (.9e+3) (3.33e-8) (.79e-5) (3.9e-9) f e-3 9.6e-.43e-7 4.6e-7 (3.e-3) (5.88e-) (.77e-7) (9.58e-7) f e e e e-4 (7.473e-5) (.83e-4) (5.563e-) (6.568e-) f e- 5.67e-.67e-.77e-9 (3.86e-) (.655) (.78e-9) (8.79e-9) F f 9 f 6 f f 3 f 35 f 8 f 33 f 34 PSO-w Mea 7.96e-5 (3.56e-5) 3.8e+ (7.69e-).58e-4 (.6e-4) 9.69e- (5.e-).8e-3 (7.4e-3) 5.8e+ (.96e+) 4.5e+ (.58e+) 3.e+ (.85e+) TABLE V. RESULTS FOR -D PROBLEMS CLPSO Mea 5.5e-9 (5.5e-9).46e+ (.7e+) 4.3e-4 (.55e-4) 4.56e-3 (4.8e-3) () () () () ABC Mea.34e-3 (.74e-3).69e- (4.7e-) 3.59e- (5.9e-) 5.6e-3 (6.57e-3).89e-5 (4.73e-5) () () 7.4e-9 (.5e-8) Mea.5e-39 (.6e-39).67e-3 (.9e-3).3e-4 (4.5e-5) 6.85e-3 (6.66e-3) 8.9e-6 (.53e-5) () ().4e- (.8e-) The average best values (mea) ad the stadard devato of 5 trals are lsted Table Ⅳ. It ca be see that the accuracy of s much hgher tha ABC o fuctos havg arrow curvg valley (Rosebrock, Colvlle) ad fuctos wth hgh eccetrc ellpse (Zakharov, Schwefel s problem.). For the fuctos Perm ad Colvlle, a small couter ca get hgher accuracy ad save NFE, whle for complex multmodal fuctos Kowalk ad Fletch- Powell, larger value of couter s eeded. Geerally a moderate value couter5 s sutable to varous fuctos. I most cases, couter ca be set as a very large value ad the tesfcato search eve ca be omtted. C. Comparso wth Partcle Swarm Optmzato I ths secto, the proposed algorthm s compared wth partcle swarm optmzato (PSO) ad comprehesve learg PSO (CLPSO) [4]. Two umodal fuctos ad sx multmodal bechmark fuctos [4] (Lsted appedx A) are adopted, cosderg CLPSO s maly proposed for multmodal fuctos. The mea values ad stadard devato of the results are preseted to compare the four algorthms. All fuctos are tested o ad 3 dmesos. The maxmum ftess evaluatos s set at 3 whe solvg the -dmesoal problems ad whe solvg the 3-dmesoal problems [4]. All expermets were ru 3 tmes. The results for s show Table Ⅴ. The best results amog the four algorthms are show bold. From the results, t ca be see that ABC performs much better tha PSO o seve fuctos except to the Sphere fucto. CLPSO performs better tha other algorthms o 5 fuctos ad performs better tha other algorthms o 4 fuctos. But the result of Rosebrock fucto obtaed by s much better tha the other three algorthms. The results for 3 s show Table Ⅵ. The results show that ABC ad perform better tha PSO ad CLPSO o the 3-D problems. They perform better tha other algorthms o 4 ad 5 fuctos respectvely. The result of Rosebrock fucto obtaed by s stll much better tha the other three algorthms. F f 9 f 6 f f 3 f 35 f 8 f 33 f 34 PSO-w Mea 9.78e-3 (.5e-9).93e+ (.5e+) 3.94e-4 (.e+) 8.3e-3 (7.6e-3).3e-4 (3.3e-4).9e+ (7.7e+).97e+ (.39e+).e+3 (.56e+) TABLE VI. RESULTS FOR 3-D PROBLEMS CLPSO Mea 4.46e-4 (.73e-4).e+ (.98e+) () 3.4e- (4.64e-) 3.45e-7 (.94e-7) 4.85e- (3.63e-) 4.36e- (.44e-).7e- (8.79e-3) ABC Mea 3.37e-59 (7.3e-59).56e- (.8e-) 3.43e-4 (3.e-5) 3.e-4 (.4e-3) () () ().e- (9.94e-3) Mea 6.74e-7 (.e-7).3e-8 (7.e-8) 3.78e-4 (4.4e-5) 3.94e-4 (.e-3) 9.47e-6 (.46e-5) () ().e- (8.7e-3) By aalyzg the results o -D ad 3-D problems, we ca coclude that PSO oly perform well o the sphere fucto, CLPSO ad ABC perform well o all the fuctos except to the Rosebrock fucto. performs the best, because t ca solve all the eght problems well ad t performs better tha other algorthms o 9 scearos whereas ABC ad CLPSO oly perform better tha other algorthms o 6~7 scearos. D. Applcato o Slope Stablty Aalyss Determato of the crtcal slp surface ad the correspodg factor of safety s the cetral ssue to slope stablty aalyss. Several methods such as Mote Carlo techque [5] ad geetc algorthm [6] has bee ACADEMY PUBLISHER

6 JOURNAL OF SOFTWARE, VOL. 6, NO. 3, MARCH 495 proposed for searchg the crtcal crcular slp surface. The objectve fucto for ths problem ca be stated as m FS ( ), S ( xyd,, ) () where F s the factor of safety, xyare, the coordates of the ceter of the crcular slp surface, d y rs bottom of the slp surface ad r s the radus. A complex slope show Fg. 4 s take as a example. The parameters of sol layers are show Table Ⅶ. The factor of safety () s calculated by the ordary method. Elevato (m) Layer 4 (, 6) (8, ) Wdth (m) Fgure 4. Cross secto of the complex slope. TABLE VII. MECHANICAL PARAMETERS OF SOIL LAYERS Iteral frcto ϕ ( ) Coheso c (kn m - ) 3 (58, ) (58, 8) (58, 7) 6 Ut weght γ (kn m -3 ) The search rage s set as x [,], y [5,], d [-4,6] The NFE s set as. The statstc results durg 3 rus of ABC ad are show Table Ⅷ. The mea best objectve fucto values of ABC ad are show Fg. 5. Fgure 5. Varato of mea best objectve fucto values wth umber of fucto evaluatos. TABLE VIII. RESULTS FOR THE SLOPE STABILITY ANALYSIS PROBLEM Method Factor of safety Posto of crtcal crcle (m) M. Max. Mea SD x y r ABC e e From Table VIII ad Fg.5 t ca be see that both ABC ad ca fd the mmum factor of safety. However, performs better tha ABC the problem because t s more robust ad has hgher accuracy. A effcet global optmzato algorthm s proposed for crtcal slp surface searchg of slope stablty aalyss. V. CONCLUSION A artfcal bee coloy algorthm wth local search s proposed for global umercal optmzato problems. HJ ca erch the search drectos of ABC ad accelerate the search process. The performace of ABC o fuctos have arrow curvg valley (for example Rosebrock fucto, Colvlle fucto) ad fuctos wth hgh eccetrc ellpse (for example Zakharov fucto, Schwefel s problem.) ca be much mproved by corporate the patter move. The proposed method has bee tested o 36 bechmark mathematcal fuctos ad a slope stablty aalyss problem. Whe compared wth the basc ABC ad several most recet algorthms, the proposed ew method has demostrated strogly compettve results. APPENDIX A LIST OF TEST FUNCTIONS. Beale fucto (f ) f ( x) (.5 x+ xx ) + (.5 x+ xx ) + (.65 x+ xx ) wth 4.5 x, x 4.5, m (f) f(3,.5).. Bra RCOS fucto (f ) 5. 5 f ( x) x x + x 6 + cosx+ 4π π 8π wth 5 x, x 5, 3 m(f) f ( π ) ( π ) ( π ),.75 /,.75 / 3, De Jog s fucto 4 (f 3 ) 4 f ( x) x wth.8.8, m (f) f(,,). x 4. Easom fucto (f 4 ) f x cos( x )cos( x )exp( ( x π ) ( x π ) ) ( ) wth x, x, m (f) f ( π, π ) Matyas fucto (f 5 ) f ( x).6( x + x ).48x x wth x, x, m (f) f(,). 6. Schaffer s fucto 6 (f 6 ) f ( x) ( x x ) (+.( x + x )) s.5 wth x, x, m (f) f(,). 7. Sx Hump Camel Back fucto (f 7 ) f x 4x.x + /3x + x x 4x + 4x ( ) wth 5 x, x 5, m (f) f(.8984,-.7656)/ (-.8984,.7656). 8. Trpod fucto (f 8 ) f x px ( )( + px ( )) + ( x+ 5 px ( ))( px ( ))) ( ) ACADEMY PUBLISHER

7 496 JOURNAL OF SOFTWARE, VOL. 6, NO. 3, MARCH + ( x + 5( p( x ))) wth x, x, m (f) f(,-5) where p(x) for x, otherwse, p(x). 9. Hartma 3 fucto (f 9 ) ( j ) 4 3 j j j f( x) c exp a ( x p ) wth x, m (f) f(.464, ,.85547) The coeffcets ca be see [3].. Hartma 6 fucto (f ) f 4 6 ( x ) c exp a ( x p ) ( j ) j j j wth x, m (f) f(.69,.5,.47687,.7533,.365,.6573) The coeffcets ca be see [3].. Colvlle fucto (f ) f ( x) ( x x) + ( x ) + ( x3 ) + 9( x3 x4) +.(( x ) + ( x4 ) ) + 9.8( x )( x4 ) wth x, m (f) f(,,,).. Kowalk fucto (f ) x( b + bx) f( x) a b + bx 3+ x4 wth 5 x 5, m(f)f(.9,.9,.3,.35) 3.75e-4 a[.957,.947,.735,.6,.844,.67,.456,. 34,.33,.35,.46]; b[4.,.,.,.5,.5,.67,.5,.,.833,.74,. 65]. 3. Perm fucto (f 3 ) f ( x k k ) ( +.5)(( x / ) ) k wth x, m (f) f(,,,). 4,5,6. Shekel s Famly (f 4, f 5, f 6 ) m f( x) j 4 ( x ) j aj + c wth x, m5, m (f 4 ) f 4 (4,4,4,4) m7, m (f 5 ) f 5 (4,4,4,4) m, m (f 6 ) f 6 (4,4,4,4) Mchalewcz fucto (f 7 ) m f( x) s( x )(s( / )) x π, m wth x π, m (f ) Rastrg fucto (f 8 ) f ( x ) + ( x cos( x )) π wth 5. x 5., m (f) f(,,,). 9. Schwefel problem. (f 9 ) f ( x) ( x ) j j wth x, m (f) f(,,,).. Ackley fucto (f ) f ( x ) exp(. x ) exp( cos( x )) e π + + wth 3 x 3, m (f) f(,,,).. Alpe fucto (f ) f ( x) x s( ). x + x wth x, m (f) f(,,,).. Axs parallel hyperellpsod (f ) f ( x) x wth 5. x 5., m (f) f(,,,). 3. Grewak fucto (f 3 ) x f( x) x cos + 4 wth 6 x 6, m (f) f(,,,). 4. Levy ad Motalvo problem (f 4 ) { π π + f( x). s (3 x ) + ( x ) [+ s (3 x )] + + ( x ) [ s ( π x)] wth 5 x 5, m (f) f(,,,). 5. Quartc fucto (f 5 ) 4 ( ) [,) + f x x radom wth.8 x.8, m (f) f(,,,). 6. Rosebrock fucto (f 6 ) + f( x) [( x x ) + ( x ) ] wth 3 x 3, m (f) f(,,,). 7. Schwefel problem. (f 7 ) f ( x) max x, { } wth x, m (f) f(,,,). 8. Schwefel problem. (f 8 ) f ( x ) x + x wth x, m (f) f(,,,). 9. Sphere fucto (f 9 ) f ( x) x wth x, m (f) f(,,,). 3. Step fucto (f 3 ) ( x ) f( x) +.5 wth x, m (f) f(.5 x <.5). 3. Sum of dfferet power fucto (f 3 ) ( ) f( x) x + wth x, m (f) f(,,,). 3. Zakharov fucto (f 3 ) ( ) ( ) 4 f( x) x +.5x +.5x wth 5 x, m (f) f(,,,). 33. Nocotuous Rastrg fucto (f 33 ) ( ) + ( cos( )) π f x y y wth 5. x 5., m (f) f(,,,) where } ACADEMY PUBLISHER

8 JOURNAL OF SOFTWARE, VOL. 6, NO. 3, MARCH 497 x x < / y roud( x) / x / 34. Schwefel s problem.6 (f 34 ) f( x) x s( x ) wth 5 x 5, m (f) f(4.9687,,4.9687) 35. Weerstrass fucto (f 35 ) f max ( x k k k ) { a cos ( π b ( x.5) + k ) } k max k k ( a cos( πb ) k ), a.5, b3, k max 3 wth.5 x.5, m (f) f(,,,) 36. Fletch-Powell Problem (f 36 ) f ( x) ( A B) A ( a sα + b cos α ) j j j j j ( s cos ) j j j + j j π x π B a x b x wth, m (f), all detals ca be foud [9]. ACKNOWLEDGMENT Ths work was partly supported by the Natoal Scece Foudato for Post-doctoral Scetsts of Cha ad the State Key Program of Natoal Natural Scece of Cha (No. 9854). REFERENCES [] A. Georgeva ad I. Jordaov, Global optmzato based o ovel heurstcs, low-dscrepacy sequeces ad geetc algorthms, Eur. J. Oper. Res., vol. 96, pp. 43 4, July 9. [] A. Baykasoglu, L. Ozbakr, ad P. Tapka, Artfcal bee coloy algorthm ad ts applcato to geeralzed assgmet problem, I Swarm Itellgece, Focus o At ad Partcle Swarm Optmzato, ITech Educato ad Publshg, Vea, Austra, pp. 3 44, 7. [3] D.Karaboga, A dea based o bee swarm for umercal optmzato, Tech. Rep. TR-6, Ercyes Uversty, Egeerg Faculty,Computer Egeerg Departmet, 5. [4] D. Karaboga ad B. Basturk. A powerful ad effcet algorthm for umercal fucto optmzato: artfcal bee coloy (ABC) algorthm, J. Global Optm., vol. 39, pp , 7. [5] D. Karaboga ad B. Basturk. O the performace of artfcal bee coloy (ABC) algorthm, Appl. Soft Comput., vol. 8, pp , 8. [6] N. Karaboga, A ew desg method based o artfcal bee coloy algorthm for dgtal IIR flters, J. Frakl I., vol. 346, pp , May 9. [7] A. Sgh, A artfcal bee coloy algorthm for the leafcostraed mmum spag tree problem, Appl. Soft Comput., vol. 9, pp , March 9. [8] F. Kag, J. L, ad Q. Xu, Structural verse aalyss by hybrd smplex artfcal bee coloy algorthms, Comput. Struct., vol. 87, pp , July 9. [9] D.Karaboga ad B. Akay, A comparatve study of artfcal bee coloy algorthm, Appl. Math. Comput., vol. 4, pp. 8 3, August 9. [] R. Hooke ad T. A. Jeeves, Drect search soluto of umercal ad statstcal problems, Joural of the ACM, vol. 8, pp. -9, March 96. [] M. G. Johso, Nolear Optmzato usg the algorthm of Hooke ad Jeeves, Avalable: [] V. Torczo, O the covergece of patter search algorthms, SIAM J. Optm., vol. 7, pp. 5, 997. [3] S. Rahamaya, H. R. Tzhoosh, ad M. M. A. Salama, Opposto- based dfferetal evoluto, IEEE Tras. Evol. Comput., vol., pp , February 8. [4] J. J. Lag, A. K. Q, ad P. N. Sugatha, Comprehesve Learg partcle swarm optmzer for global optmzato of multmodal fuctos, IEEE Tras. Evol. Comput., vol., pp. 8-95, Jue 6. [5] A. L. Huse Malkaw, W. F. Hassa, ad S. K. Sarma, A effcet search method for fdg the crtcal crcular slp surface usg the Mote Carlo techque Ca. Geotech. J., vol. 38, pp. 8 89,. [6] P. McCombe, P. Wlkso, The use of the smple geetc algorthm fdg the crtcal factor of safety slope stablty aalyss, Comput. Geotech., vol. 9, pp , December. Fe Kag receved the B. Sc. ad Ph. D. degrees both frastructure egeerg from the Dala Uversty of Techology, Dala, Cha, 3 ad 9. He s curretly workg as a postdoctor at Dala Uversty of Techology. Hs curret research terests are evolutoary algorthms, swarm tellgece ad ther applcato geotechcal egeerg. He has publshed more tha papers over the past fve years. ACADEMY PUBLISHER