An Adaptive Learning Particle Swarm Optimizer for Function Optimization

Transcription

1 An Adaptve Learnng Partcle Swarm Optmzer for Functon Optmzaton Changhe L and Shengxang Yang Abstract Tradtonal partcle swarm optmzaton (PSO) suffers from the premature convergence problem, whch usually results n PSO beng trapped n local optma. Ths paper presents an adaptve learnng PSO (ALPSO) based on a varant PSO learnng strategy. In ALPSO, the learnng mechansm of each partcle s separated nto three parts: ts own hstorcal best poston, the closest neghbor and the global best one. By usng ths ndvdual level adaptve technque, a partcle can well gude ts behavor of exploraton and explotaton. A set of 21 test functons were used ncludng un-rotated, rotated and composton functons to test the performance of ALPSO. From the comparson results over several varant PSO algorthms, ALPSO shows an outstandng performance on most test functons, especally the fast convergence characterstc. I. INTRODUCTION Partcle Swarm Optmzaton (PSO) was frst ntroduced by Kennedy and Eberhart n [1], [2]. PSO s motvated from the socal behavor of organsms, such as brd flockng and fsh schoolng. In PSO, a swarm of partcles fly through the search space. Each partcle follows the prevous best poston found by ts neghbor partcles and the prevous best poston found by tself. In the past decade, PSO has been actvely studed and appled for many academc and real world problems wth promsng results due to ts property of fast convergence [8]. Ever snce PSO was frst ntroduced, several major versons of the PSO algorthms have been developed [8]. Each partcle s represented by a poston and a velocty, whch are updated as follows: V d = ωv d + η 1r 1 (pbest d Xd )+η 2r 2 (gbest d X d ) (1) X d = X d + V d, (2) where X d and X d represent the current and prevous poston of d th dmenson of partcle respectvely, V and V are the current and prevous velocty of partcle respectvely, pbest and gbest are the best poston found by partcle so far and the best poston found by the whole swarm so far respectvely, ω (0, 1) s an nerta weght, whch determnes how much the prevous velocty s preserved, η 1 and η 2 are the acceleraton constants, and r 1 and r 2 are random numbers generated n the nterval [0.0, 1.0]. There are two man models of the PSO algorthms, called gbest (global best) and lbest (local best), whch dffer n the way of defnng the neghborhood of each partcle. In The authors are wth the Department of Computer Scence, Unversty of Lecester, Unversty Road, Lecester LE1 7RH, Unted Kngdom (emal: {cl160, s.yang}@mcs.le.ac.uk) Ths work was supported by the Engneerng and Physcal Scences Research Councl (EPSRC) of the Unted Kngdom under Grant EP/E060722/1. the gbest model, the neghborhood of a partcle conssts of the partcles n the whole swarm, whch share nformaton between each other. On the contrary, n the lbest model, the neghborhood of a partcle s defned by several fxed partcles. The two models gve dfferent optmzaton performances on dfferent problems. Kennedy and Eberhart [3] and Pol et al. [8] ponted out that the gbest model has a faster convergence speed wth a hgher chance of gettng stuck n local optma than lbest. On the contrary, the lbest model s less vulnerable to the attracton of local optma but wth a slower convergence speed than the gbest model. In order to mprove PSO s performance, we present an adaptve learnng PSO (ALPSO) that utlzes a new learnng strategy. In ALPSO, each partcle can adjust ts search strategy accordng to the selecton ratos of four learnng operators n dfferent surroundng envronments. The selecton rato of each operator s calculated n the same way as n [4]. For the global best partcle, we ntroduce a learnng method that can subtract the promsng nformaton from all mproved partcles. The rest of ths paper s organzed as follows. Secton II descrbes the adaptve learnng PSO. The expermental study s present n secton III and fnally conclusons are gven n secton IV. II. ADAPTIVE LEARNING PARTICLE SWARM OPTIMIZER Although there are many mproved versons of PSO, how to balance the performance of the gbest and lbest models s stll an mportant ssue, especally for mult-modal problems. In the gbest model, all partcles socal behavor s strctly constraned by learnng nformaton from the global best partcle. Hence, partcles are easly attracted by gbest and quckly converge on that regon even t s not the global optmum and gbest does not mprove. In the lbest model, attracton by the gbest s not too much but the slow convergence speed s unbearable. In the orgn PSO, each partcle learns from ts pbest and the gbest smultaneously, whch mght cause the above problems. Hence, we can separate the cognton component and the socal component to ncrease dversty, but the proper moment for a partcle to learn from gbest or pbest s very hard to know. The followng sectons wll gve an adaptve method to enable a partcle to automatcally learn from the global or local nformaton from dfferent partcles. A. Learnng Strategy n ALPSO In ALPSO, the nformaton learnt by each partcle comes from four sources: the gbest, ts own pbest, thepbest of /09/$25.00 c 2009 IEEE 381

2 the closest partcle, and a random poston around tself. The learnng equatons are as follows: b : V d a : V d d : V d = ωv d + η r d (pbest d X d ) (3) = ωv d + η r d (pbest d nearest X d ) (4) c : X d = Xd + V avg d N(0, 1) (5) = ωv d + η r d (gbestd X d ) (6) where pbest nearest s the pbest of the closest partcle to partcle, V avg s the average velocty of all partcles, and N(0, 1) s a random number from the normal dstrbuton wth mean 0 and varance 1. Learnng from the nearest neghbor enables a partcle to explore the regon of local optma around tself. Partcles that are near a local optmum wll get closer and closer to that regon because the pbest s replaced only when a better poston s found. Gradually, they wll generate a local cluster around that local optmum. Partcles n one local cluster are not nfluenced by those far away (other local clusters) even they have very good ftness. Ths strategy can help swarm fnd more local optma rather than one optmum as the orgnal PSO does, especally for mult-modal problems. Once partcles converge on a local optmum or there s a more promsng regon nearby wthout partcles coverng t, partcles should have a probablty to jump to that promsng regon. Hence, learnng from a random poston around tself s needed. In ALPSO, each partcle has four dfferent choces to adjust ts behavor. The four choces enable each partcle to move to a promsng poston wth a hgher probablty than the orgnal PSO. Here, whch choce s the most sutable depends on the around envronment where a partcle s. However, we can not know what the around envronment looks lke. Each partcle should detect the shape of the envronment where t s by tself. Hence, we use the method proposed n [4], whch enables a partcle to choose the most sutable operator automatcally. The method s descrbed n the followng secton. B. The Adaptve Learnng Mechansm Borrowed the dea of probablty matchng[12], we ntroduce an adaptve framework usng the aforementoned four learnng operators, each of whch s assgned a selecton rato. The selecton rato of each operator s equally ntalzed to 1/4 and s adaptvely updated accordng to ts relatve performance. For each partcle, one of the four learnng operators s selected accordng to ther selecton ratos and ts offsprng ftness s evaluated. The operator that results n hgher ftness values of offsprng wll have ts selecton rato ncreased. The operator that results n lower ftness values of offsprng wll have ts selecton rato decreased. Gradually, the most sutable operator wll be chosen automatcally and control the leanng behavor of each partcle n dfferent envronments. Wthout lose of generalty, we dscuss the mnmzaton optmzaton problems n ths paper. Based on our prevous work n [4], we extend the adaptve framework at the populaton level nto the ndvdual level n ths paper. The selecton ratos are updated every U f generatons, where U f s called the updatng frequency. Durng the updatng perod for each partcle, the progress value and the reward value of operator are calculated as follows. The progress value prog (t) of operator at generaton t s defned as: M prog (t) = f(p j (t)) mn (f(p j (t)),f(c j (t))), (7) j=1 where p j (t) and c j (t) denote a partcle and ts chld produced by operator at generaton t and M s the selecton tmes of operator wth the partcle. The reward value reward (t) of operator at generaton t s defned as follows: prog reward (t) = exp( (t) M (1 α)) s P N j=1 progj(t)α + +c p (t) 1 (8) where s s the counter that records the number of chldren that are ftter than ther parent partcles by applyng operator, p (t) s the selecton rato of operator at generaton t, α s a random weght between 0.0 and 1.0, N s the number of operators, and c s a penalty factor for operator, whch s defned as follows: { 0.9, f s =0and p c = (t) =max N j=1 (p j(t)) 1, otherwse Wth the above defntons, the selecton rato of operator s updated every U f generaton accordng to the followng equaton: reward (t) p (t +1)= N j=1 reward (1 N γ)+γ, (10) j(t) where γ s the mnmum selecton rato for each operator, whch s set 0.01 for all the experments n ths paper. C. Informaton Learnng for gbest In the orgnal PSO, the gbest s updated only when partcles fnd a better poston than the current gbest. Once t s updated, the nformaton of all dmensons of the gbest s replaced wth that of the better poston. Ths updatng mechansm has a dsadvantage that promsng nformaton of some dmensons of one partcle can not be kept due to bad nformaton n other dmensons that cause ts low ftness. Ths problem s called Two step forward, one step back n [13]. If a partcle gets better, nformaton of some dmenson probably becomes more promsng. Other partcles should learn some useful nformaton from the mproved one even the partcle s ftness s very low. In ALPSO, the gbest learns the useful nformaton from those dmensons of a partcle that s mproved. Once promsng nformaton s extract from those mproved dmensons of that partcle, the nformaton of correspondng dmensons of the gbest s updated. The updatng happens only when partcles are mproved, whch s as shown n Algorthm 1. (9) IEEE Congress on Evolutonary Computaton (CEC 2009)

3 Algorthm 1 GbestUpdate(partcle p) 1: for each dmenson d of gbest do 2: X t gbest := X gbest, X t gbest [d] :=X p [d] 3: f t gbest s better than gbest then 4: X gbest [d] :=X t gbest [d] 5: end f 6: end for Algorthm 2 The ALPSO Algorthm 1: Generate the ntal partcles by randomly generatng the poston and velocty for each partcle 2: Set the generaton counter t := 0 3: whle the stop crteron s not satsfed do 4: for each partcle do 5: Select one learnng operator accordng to ts selecton rato to update partcle 6: f the updated partcle s better than ts pbest then 7: Update pbest 8: Perform GbestU pdate() for gbest 9: end f 10: f the updated partcle s better than gbest then 11: Update gbest 12: end f 13: f t%u f == 0 then 14: Update the selecton rato for each learnng operator accordng to Eq. (10) 15: else 16: Calculate the accumulatve reward value of each operator 17: end f 18: end for 19: t := t +1 20: end whle We can not apply ths strategy to all partcles because the learnng method s tme comsumng. Hence, we choose the gbest as the learner. The framework of the ALPSO algorthm s gven n Algorthm 2. III. EXPERIMENTAL STUDY A. Test Functons In order to test the performance of ALPSO, we choose three unmodal functons and 18 multmodal functons, whch are wdely used as the test functons n the lterature [5], [11], [14]. The detals of these test functons are gven n Table I. Functon f 16 s a composton functon proposed by Jang et al. [5], whch s composed of ten benchmark functons: two rotated and shfted f 1, f 2, f 3, f 4,andf 5. Functons f 18 to f 21 are rotated functons, where the rotaton matrx M for each functon s obtaned usng the method n [9]. B. Expermental Settng Experments were conducted to compare fve PSO algorthms on the 21 test problems. The algorthms are lsted as follows: Standard PSO; CPSO-H k [13]; FIPS [7]; CLPSO [6]; ALPSO For the standard PSO, the acceleraton constants η 1 and η 2 are both set to be and the nerta weght ω = Equatons 1 and 2 are used for the velocty and poston update n the standard PSO. CPSO-H k [13] s a cooperatve PSO model combned wth standard PSO, the same value of k =6n [13] s used. The fully nformed PSO (FIPS) [7] wth a U-rng topology that acheved the hghest success rate s used. Comprehensve learnng PSO (CLPSO) [6] uses all other partcles hstorcal best nformaton to update a partcle s velocty. CLSPO s desgned for solvng multmodal problems, and t presents a good performance n [6] compared wth eght other PSO algorthms. To acheve better performance of ALPSO, we use partcular settngs by experence for each problem due to dfferent complexty of dfferent problems. In ALPSO, parameters are the same as standard PSO and the updatng frequency s present n Table II. Each problem wth 10 dmensons was ndependently run 30 tmes. The ntal populaton s the same for all algorthms on each test problem and the populaton sze s gven n Table II. The maxmal number of ftness evaluatons s set to for all algorthms on each test problem. The code of the fve algorthms s avalable onlne at the followng webste: C. Expermental Results and Dscussons 1) Expermental Results: Table III presents the results of mean and varance values over 30 runs for the fve algorthms on all test problems. The best results of each problem are shown n bold except functon f 6, on whch all fve algorthms obtaned the global optmum 0. Two-taled T- test wth 58 degrees of freedom at a 0.05 level of sgnfcance was conducted between ALPSO and the best results obtaned by one of the other four algorthms and the results are also shown n Table III, where *** means the result of two algorthms s the same. The performance dfference s sgnfcant f the absolute value of the T-test result s greater than Fgs. 1 and 2 descrbe the convergence speed of the fve PSOs on all test problems. Form Table III, ALPSO shows an outstandng performance on functons f 1, f 8, f 12,andf 13 over the other four algorthms. Especally for functons f 2, f 3, f 6,andf 9, ALPSO obtaned the global optmum over all 30 runs. Comparng ALPSO wth CLPSO, though the performance of ALPSO s worse than that of CLPSO on functons f 4, f 16, f 20,andf 21, t s much better than that of CLPSO on functons f 4, f 12, f 13,andf 17, and s smlar to or the same as that of CLPSO on the other functons. For unmodal functons, ALSPO shows a fast convergence speed to the global optma. For multmodal functons, ALPSO and CLPSO present a much better performance than 2009 IEEE Congress on Evolutonary Computaton (CEC 2009) 383

4 TABLE I THE TEST FUNCTIONS, WHERE n AND f mn ARE THE NUMBER OF DIMENSIONS AND THE MINIMUM VALUE OF A FUNCTION RESPECTIVELY AND S R n Test Functon n S f mn f 1 (x) = n =1 x2 10 [ 100, 100] 0 f 2 (x) = n =1 (x2 10 cos(2πx ) + 10) 10 [-5.12, 5.12] 0 f 3 (x) = n ( kmax [a k cos(2πb k (x +0.5))]) n kmax [a k cos(πb k )], 10 [-0.5,0.5] 0 =1 k=0 k=0 a =0.5,b=3,k max =20 f 4 (x) = 1 n 4000 =1 (x 100) 2 n x 100 =1cos( )+1 10 [-600, 600] 0 1 f 5 (x) = 20 exp( 0.2 n n =1 x2 ) exp( 1 n n =1 cos(2πx )) e 10 [-32, 32] 0 f 6 (x) = n =1 ( x +0.5 ) 2 10 [-100,100] 0 f 7 (x) = n =1 ẋ4 + U(0, 1) 10 [-1.28, 1.28] 0 f 8 (x) = n =1 100(x2 +1 x ) 2 +(x 1) 2 ) 10 [-30, 30] 0 f 9 (x) = n =1 x sn ( x ) 10 [-500, 500] f 10 (x) = n + n =1 x sn ( x ) 10 [-500, 500] 0 f 11 (x) = n =1 x + n =1 x 10 [-10, 10] 0 f 12 (x) = n =1 ( j=1 x j) 2 10 [-100, 100] 0 f 13 (x) =max n =1 x 10 [-100, 100] 0 f 14 (x) = π 30 {10 sn2 (πy 1 )+ n 1 =1 (y 1) 2 [ sn 2 (πy +1 )]+ 10 [-50, 50] 0 (y n 1) 2 } + n =1 u(x, 5, 100, 4),y =1+(x +1)/4 f 15 (x) =0.1{10 sn 2 (3πx 1 )+ n 1 =1 (x 1) 2 [1 + sn 2 (3πx +1 )] 10 [-50, 50] 0 +(x n 1) 2 [1 + sn 2 (2πx n )]} + n =1 u(x, 5, 100, 4) f 16 (x) =Composton functon 5 (CF5) n [5] 10 [-5, 5] 0 f 17 (x) = n =1 100(y2 +1 y ) 2 +(y 1) 2 ), y = M x 10 [-100, 100] 0 f 18 (x) = 1 n 4000 =1 (y 100) 2 n y 100 =1cos( )+1, y = M x 10 [-600, 600] 0 1 n f 19 (x) = 20 exp( 0.2 n =1 y2 ) exp( 1 n n =1 cos(2πy )) e, 10 [-32, 32] 0 y = M x f 20 (x) = n =1 (y2 10 cos(2πy ) + 10), y = M x 10 [-5, 5] 0 f 21 (x) = n ( kmax [a k cos(2πb k (y +0.5))]) n kmax [a k cos(πb k )], 10 [-0.5,0.5] 0 =1 k=0 a =0.5,b=3,k max =20, y = M x k=0 TABLE II POPULATION SIZE AND UPDATING FREQUENCY, WHERE THE UPDATING FREQUENCY IN SHOWN IN THE BRACKET f 1 f 2 f 3 f 4 f 5 f 6 f 7 5(5) 10(5) 10(5) 20(5) 10(5) 10(5) 20(5) f 8 f 9 f 10 f 11 f 12 f 13 f 14 10(1) 40(5) 40(5) 10(5) 5(5) 5(5) 10(5) f 15 f 16 f 17 f 18 f 19 f 20 f 21 10(5) 40(10) 10(5) 20(5) 30(10) 20(5) 20(15) the other three algorthms. For example, on Schwefel s functons f 9 and f 10, all the other three algorthms are trapped nto local optmum that are far away from the glbal optmum. However, ALPSO and CLPSO both successfully avod fallng nto the deep local optmum. For rotated functons, the two algorthms also show a leadng performance compared wth the other three algorthms. Among the other three algorthms, CPSO-H 6 presents a comparatvely better performance on most problems, and t obtans the best results on problem f 7, whch s a lttle better than the results got by ALPSO. The FIPS wth the U-rng model gves relatvely better results compared wth the standard PSO. The standard PSO falls nto local optma on almost all multmodal problems. From Fgs. 1 and 2, one nterestng observaton s that ALPSO presents the fastest convergence speed on all test problems. The results obvously show that the learnng strategy for gbest s effcent to solve the Two step forward, one IEEE Congress on Evolutonary Computaton (CEC 2009)

5 TABLE III COMPARISON RESULTS OF MEANS AND VARIANCES Functon f 1 f 2 f 3 f 4 f 5 f 6 f 7 ALPSO 9.42e e (±4.97e-162) (±0) (±0) (±0.0298) (±2.55e-15) (±0) (± ) CLSPO 3.81e e (±2.09e-154) (±0) (±0) (± ) (±6.49e-16) (±0) (± ) FIPS 5.16e (±2.83e-052) (±5.54) (±0) (±0.0884) (±3.28) (±0) (± ) CPSO-H e e e (±6.74e-103) (±0.344) (±2.17e-15) (±0.0166) (±4.27e-15) (±0) (± ) PSO 2.79e (±1.53e-026) (±10.8) (±0.88) (±0.0505) (±1.24) (±0) (± ) T-test -1 *** *** *** Functon f 8 f 9 f 10 f 11 f 12 f 13 f 14 ALPSO e e e e e-32 (±0.617 ) (±9.25e-13) (±2.76e-20) (±3.6e-50) (±2.08e-50) (±1.73e-67) (±5.57e-48) CLSPO e e e e-32 (±1.68 ) (±9.25e-13) (±2.76e-20) (±1.42e-51) (±1.14e-05) (±0.748) (±5.57e-48) FIPS e e e-14 5e-26 (±6.92 ) (±162) (±162) (±4.25e-17) (±1.9e+03) (±1.38e-13) (±1.82e-25) CPSO-H e e e e e-32 (±6.87 ) (±189) (±189) (±7.54e-45) (±1.9e-13) (±2.44e-18) (±5.57e-48) PSO 9.32e e e e (±2.74e+04) (±305) (±305) (±3.05) (±6.07e+03) (±2.55e-44) (±6.36) T-test *** *** *** Functon f 15 f 16 f 17 f 18 f 19 f 20 f 21 ALPSO 1.35e e (±0) (±169) (±6.31) (±0.0458) (±7.19e-015) (±2.98) (±0.735) CLSPO 1.35e e e-006 (±0) (±36.1) (±3.35) (±0.0111) (±4e-013) (±0.675) (±1.68e-05) FIPS e (±47.5) (±179) (±7.28e+07) (±0.1) (± ) (±4.24) (± ) CPSO-H e (±0) (±398) (±176) (±0.0676) (±0.497) (±3.91) (±1.78) PSO e (±118) (±195) (±4.94e+5) (±0.0648) (±0.211) (±5.37) (±1.68) T-test *** step back problem. It does help the gbest learn promsng nformaton from those mproved partcles. Fg 3 presents the selecton rato of each operator for some test problems. From the results, we can see that the selecton rato of each operator s qute dfferent from problem to problem. Even for a partcular test problem, the selecton rato of the best operator changes n dfferent evolvng stages.it can be seen from the results of F 2, F 3, F 5 and F 6 n Fg. 3, most partcles quckly learn from the best partcle when the run starts, however, the selecton rato of learnng from partcles prvate best poston surpasses the selecton rato of learnng from the best partcle when the number of generatons reaches 500. The results valdate our predcton that dfferent learnng strateges are needed n dfferent evolvng stages. 2) Dscussons: From the above results on the 21 test problems, we can conclude that ALPSO performs much better than the other three algorthms on all unmodal test problems due to the learnng strategy for gbest. Though ALPSO does not perform the best on all multmodal test problems, t presents compettve results compared wth the other three mproved PSO algorthms. We can also conclude that the adaptve learnng mechansm enables partcles to have more chances to move to a more promsng regon, especally for those partcles beng trapped nto local optma n multmodal problems. IV. CONCLUSIONS Ths paper presents an adaptve learnng PSO whch uses an adaptve framework on the ndvdual level to adapt four leanng strateges for each partcle n the swarm. A new 2009 IEEE Congress on Evolutonary Computaton (CEC 2009) 385

6 Fg. 1. Evoluton process of the average best ftness of PSO, CLPSO, CPSO-H 6, FIPS, and ALPSO on functons f 1 to f IEEE Congress on Evolutonary Computaton (CEC 2009)

7 Fg. 2. Evoluton process of the average best ftness of PSO, CLPSO, CPSO-H 6, FIPS, and ALPSO on functons f 16 to f 21. learnng mechansm for gbest s ntroduced by extractng useful nformaton from those mproved partcles on all dmensons. The four learnng strateges gve each partcle more chances to search a larger space. A partcle s not smply nfluenced by ts own prevous best poston and the global best one, the nearest neghbor also can help t search n a local regon. The balance of local search and the global search can be solved usng the adaptve technque, whch enables each partcle make ts own choce accordng to the envronment around. The performance of ALPSO s tested on 21 test problems n comparson wth other three mproved PSOs and the standard PSO. The results show that ALPSO gves the best performance on all test unmodal problems and also presents the outstandng performance on most multmodal problems. Although ALPSO s not the best one for solvng all test multmodal problems, the adaptve framework can help partcles decde ther own step drecton. Especally for real problems, we can not know the dstrbuton of soluton space. However, we can desgn dfferent strateges to deal wth dfferent stuatons, and let partcles choose the most sutable strategy by themselves accordng to the adaptve technque. REFERENCES [4] C. L, S. Yang, and I. A. Korejo. An adaptve mutaton operator for partcle swarm optmzaton. Proc. of the 2008 UK Workshop on Computatonal Intellgence, pp , [5] J.J. Lang, PN. Suganthan, and K. Deb. Novel composton test functons for numercal global optmzaton. Swarm Intellgence Symposum, pp , [6] J. J. Lang, A. K. Qn, P. N. Suganthan, and S. Baskar. Comprehensve learnng partcle swarm optmzer for global optmzaton of multmodal functons. IEEE Trans. on Evol. Comput., 10(3): , [7] R. Mendes, J. Kennedy, and J. Neves, The fully nformed partcle swarm: smple, maybe better.ieee Trans. on Evol. Comput., 8(3): , [8] R. Pol, J. Kennedy, and T. Blackwell. Partcle swarm optmzaton: An overvew. Swarm Intellgence, 1(1): 33-58, [9] R. Salomon. Reevaluatng genetc algorthm performance under coordnate rotaton of benchmark functons: A survey of some theoretcal and practcal aspects of genetc algorthms. BoSystems, 39(3): , [10] Sh, Y. and R. C. Eberhart. A modfed partcle swarm optmzer. Proc. of the IEEE Int. Conf. on Evol. Comput., pp , [11] P. N. Suganthan, N. Hansen, J. J. Lang, K. Deb, Y.-P. Chen, A. Auger, and S. Twar. Problem defntons and evaluaton crtera for the CEC 2005 specal sesson on real-parameter optmzaton. Techncal Report, Nanyang Technologcal Unversty, Sngapore, [12] D. Therens. An adaptve pursut strategy for allocatng operator probabltes.proceedngs of the 2005 conference on Genetc and evolutonary computaton, pp , New York, NY, USA, [13] F. van den Bergh, A. P. Engelbrecht. A Cooperatve approach to partcle swarm optmzaton. IEEE Trans. on Evol. Comput., 8(2): , [14] X. Yao, Y. Lu and G. Ln. Evolutonary programmng made faster. IEEE Trans. on Evol. Comput., 3(2): , [1] Eberhart, R. C. and J. Kennedy. A new optmzer usng partcle swarm theory. Proc. of the 6th Int. Symp. on Mcro Machne and Human Scence, pp , [2] Kennedy, J. and R. C. Eberhart. Partcle Swarm Optmzaton. Proc. of the 1995 IEEE Int. Conf. on Neural Networks, pp , [3] Kennedy, J. and R. C. Eberhart. Swarm Intellgence. Morgan Kaufmann Publshers.(2001) IEEE Congress on Evolutonary Computaton (CEC 2009) 387

8 Fg. 3. The process of selecton rato of each operator for dfferent problems IEEE Congress on Evolutonary Computaton (CEC 2009)