A Hybrid Co-evolutionary Particle Swarm Optimization Algorithm for Solving Constrained Engineering Design Problems

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1 JOURNL OF COMPUERS VOL 5 NO 6 JUNE Hybrd Co-evolutonary Partcle Optmzaton lgorthm for Solvng Constraned Engneerng Desgn Problems Yongquan Zhou Shengyu Pe College of Mathematcs and Computer Scence Guangx Unversty for Natonaltes Nannng Chna Emal: yongquanzhou@6com bstract hs paper presents an effectve hybrd coevolutonary partcle swarm optmzaton algorthm for solvng constraned engneerng desgn problems whch s based on smulated annealng (S) employng the noton of co-evoluton to adapt penalty factors By employng the Sbased selecton for the best poston of partcles and swarms when updatng the velocty n co-evolutonary partcle swarm optmzaton algorthm Smulaton results based on well-known constraned engneerng desgn problems demonstrate the effectveness effcency and robustness on ntal populatons of the proposed and can reach a hgh precson Index erms Partcle swarm optmzaton; Smulated annealng; Constraned optmzaton; Co-evoluton; Penalty functon I INRODUCION Partcle swarm optmzaton (PSO) algorthm [] s an optmzaton method wdely used to solve contnuous nonlnear functons It s a stochastc optmzaton technque that was orgnally developed to smulate the movement of a flock of brds or a group of fsh searchng for food Many engneerng desgn problems can be formulated as constraned optmzaton problems So far penalty functon methods have been the most popular methods for constraned optmzaton due to ther smplcty and easy mplementaton However t s often not easy to set sutable penalty factors or to desgn adaptve mechansm Dstngush from penalty functon method; co-evolutonary partcle swarm optmzaton approach (CPSO) [] s an effectve method for solvng constraned optmzaton problems Partcle swarm optmzaton (PSO) s a novel populaton-based searchng technque proposed n 995 as an alternatve to genetc algorthm (G) [] Compared wth G PSO has some attractve characterstcs Frstly PSO has memory that s the knowledge of good solutons s retaned by all partcles whereas n G prevous knowledge of the problem s destroyed once the populaton changes Secondly PSO has constructve cooperaton between partcles that s partcles n the *Correspondng author el E-mal: yongquanzhou@6com(yongquan Zhou) swarm shares ther nformaton On the other hand smlar to G t s also shown that PSO s often easy to be premature convergence so that exploraton (searchng for promsng solutons wthn the entre regon) and explotaton (searchng for mproved solutons n subregons) should be enhanced and well balanced to acheve better performance hus we wll consder the development of more effectve CPSO-based approach In ths paper we wll propose an effectve hybrd strategy named CPSOS by ncorporatng the umpng mechansm of smulated annealng (S) [4] [5] nto CPSO to acheve results wth hgh qualty and relance he remanng contents are organzed as follows In Secton the problem statement of constraned engneerng desgn problems s provded In Secton the hybrd CPSOS strategy s proposed and explaned n detal Smulaton results can be found n Secton 4 Fnally n Secton 5 we end the paper wth some conclusons and future work II PROBLEM SEMEN OF CONSRINED ENGINEERING DESIGN Generally a constraned optmzaton problem can be descrbed as follows: Fnd x to Mnmze f (x) Subect to: g ( x) 0; = n h ( x) = 0 = p () where x = [ x x x d ] denotes the decson soluton vector n s the number of nequalty constrants and p s the number of equalty constrants In a common practce equalty constrant h ( x) = 0 can be h (x) δ h (x) δ (δ s a small tolerant amount) hus N = n + p replaced by a set of nequalty constraned and all constrants can be transformed to nequalty constrants Many engneerng desgn problems can be formulated as constraned optmzaton problems he presence of constrants may sgnfcantly affect the optmzaton 00 CDEMY PUBLISHER do:0404/cp

2 966 JOURNL OF COMPUERS VOL 5 NO 6 JUNE 00 performances of any optmzaton algorthms for unconstraned problems For most constrant-handng technques both nfeasble and feasble solutons could be generated at the search stage and constrants are dealt wth when evaluatng solutons he volaton of constrants for each soluton s consdered separately and the relatonshp between nfeasble solutons and feasble regons s exploted to gude search he dffculty les n that there s no certan metrc crteron to measure ths relatonshp Dstngush from penalty functon method; a co-evolutonary partcle swarm optmzaton approach (CPSO) s proposed for constraned optmzaton problems by applyng PSO and employng the noton of co-evoluton (Wang 006) III CPSOS HYBRID SREGY he standard PSO he PSO s frst ntroduced by Kennedy and Eberhard t s a stochastc optmzaton technque that can be lkened to the behavor of a flock of brds or the socologcal behavor of group problems ncludng neural network tranng and functon mnmzaton Several attempts have made to mprove the performance of the orgnal PSO some of whch are dscussed n ths secton Let s denote the swarm sze each ndvdual s has the followng attrbutes current poston n the search spaces X = x x x ] a current velocty [ s [ v v v s Y [ y y y s] V ] = ; each partcle has ts own best poston (pbest) = correspondng to the personal best obectve value obtaned so far at tme t he global best partcle (gbest) s denoted by ŷ Durng each teraton each partcle n the swarm s updated usng () and () ssumng that the functon f s to be mnmzed that the swarm conssts of n partcles and that r ~ U (0) r ~ U (0) are elements from two unform random sequences n the range of ( 0) then v ( t + ) = wv ( t) + cr [ y ( t) x ( )] + c ˆ r[ y ( t) x ( t)] () t For all { n} where v s the velocty of the th dmenson of the th partcles c and c are constants called acceleraton coeffcents w s called the nerta factor he new poston of a partcle s calculated usng x ( = n t + ) = x ( t) + v ( t + ) () he personal best poston of each partcle s updated usng equaton Y ( t) f Y ( t + ) = X( t + ) f f ( X ( t + ) f ( Y ( t)) f ( X ( t + ) < f ( Y ( t)) (4) and the global best poston found by any partcle durng all prevous steps ŷ s defned as yˆ ( t + ) = arg mn f ( Y ( t + )) s (5) Y B CPSO strategy he prncple of co-evoluton model n CPSO s shown n Fgure In CPSO two knds of swarms are used In partcular one knd of a sngle swarm (denoted by ) wth sze M s used adapt sutable penalty factors another knd of multple swarms (denoted by M ) each wth sze M are used n parallel to search good decson solutons Each partcle B n represents a set of penalty factors for partcles n where each partcle represents a decson soluton B B B M Swam M Fgure Graphcal llustraton for noton of co-evoluton In every generaton of co-evoluton process every wll evolve by usng PSO for a certan number of generatons G wth partcle B n as penalty factors for soluton evaluaton to get a new hen the ftness of each partcle B n wll be determned fter all partcles n are evaluated wll also evolve by usng PSO wth one generaton to get a new wth adusted penalty factors he above co-evoluton process wll be repeated untl a pre-defned stoppng crteron s satsfed (eg a maxmum number of coevoluton generatons G are reached) k k k 00 CDEMY PUBLISHER

3 JOURNL OF COMPUERS VOL 5 NO 6 JUNE In partcular the th partcle n n CPSO s evaluated by usng the followng formula: F() x = f () x + sum_ vol w + num_ vol w (6) where f ( x ) s the obectve value of the th partcle sum _ vol denotes the sum of all the amounts by whch the constrants are volated num _ vol denotes the number of constrants volaton w and w are penalty factors correspondng to the partcle B n he value of sum _ vol s calculated as follows: N sum _ vol = g ( x) g ( x) > 0 (7) = where N s the number of nequalty constrants (here t s assumed that all equalty constrants have been transformed to nequalty constrants) Each partcle n represents a set of factors ( w and w ) fter evolves for a certan number of generatons ( G ) the th partcle B n s evaluated as follows () If there s at least one feasble soluton n then partcle B s evaluated usng the followng formula and s called a vald partcle: (8) f feasble P( B ) = num _ feasble num _ feasble Where f feasble denotes the sum of obectve functon values of feasble solutons on and feasble s the number of feasble solutons n num _ () If there s no feasble soluton n (t can be consdered that the penalty s too low) then partcle B n s evaluated as follows and s called an nvald partcle sum _ vol P( B) = max( Pvald) + num_ vol (9) num _ vol where max( P vald ) denotes the maxmum ftness value of all vald partcles n sum _ vol denotes the sum of constrans volaton for all partcles n and num _ vol counts the total number of constrants volaton for all partcles n he partcle n encodes a set of penalty factors ( w and w ) whle the partcle n encodes a set of decson varables Both knds of partcles wll apply Eqs () and () to adust ther postons so as to obtan good decson soluton and sutable penalty factors C Hybrd CPSOS strategy he man defcency of PSO s easy to be premature convergence One reason s that n PSO ŷ g s used as ŷ n Eq (5) Consequently all partcles have the tendency to fly to the current best soluton that may be a local optmum or a soluton near local optmum so that all partcles wll concentrate to a small regon and the global exploraton ablty wll be weakened hat s to say f ŷ g s not the global optmum the algorthm evolved wth Eqs () and () may mss the regon contanng the global optmum or may trap n a local optmum s we known smulated annealng s a stochastc searchng algorthm wth umpng property motvated by the smlarty between the solds annealng procedure and optmzaton problems he most sgnfcant character of S s the probablstc umpng property e a worse soluton has a probablty to be accepted as the new soluton Moreover by adustng the temperature such a umpng probablty can be controlled In partcular the probablty s rather hgh when temperature s hgh and decreases as the temperature decreases; and when the temperature tends to zero the probablty approaches to zero so that only better solutons can be accepted It has been theoretcally proved that under certan condtons S s globally convergent n probablty In ths secton we try to ncorporate the mechansm of S nto CPSO to propose a hybrd optmzaton strategy named CPSOS s mentoned before n CPSO ŷ g s one element of the set of all y whch can be regarded as a set of local optma hus we attempt to modfy the selecton of to overcome premature convergence In partcular we employ the cross factor of S hat s we do use one of y (denoted by ŷ g ) as ŷ Borrowng the mechansm of S all other y can be regarded as specal solutons Worse than ŷ and we defne ftness value of each y to replace temperature t e ( Fyg Fy)/ t ŷ g as the ŷ at a certan 00 CDEMY PUBLISHER

4 968 JOURNL OF COMPUERS VOL 5 NO 6 JUNE 00 he whole procedure of CPSOS s descrbed as follows: Step : (Intalzaton): Intalze and and evaluate partcles n Duplcate ; M copes as M Let l = 0 t = 0 b = 0 temperature = e 8 decayscale = 07 l Step : Repeat untl a stoppng crteron s satsfed = G ): Step : Repeat untl a stoppng crteron s satsfed ( t = G ); Step : Evolve ( = M) usng PSOS wth penalty factors ( B ( = M) t = t + Step : ; Step : Calculate ftness of all partcles B n ( = M ) Step : Evolve usng PSO wth one generaton to get a new wth adusted penalty factors Step 4: Let l = l + and t = 0; Step : Output the best gbest of all Lastly the proposed CPSOS s a general optmzaton algorthm that can be appled to any constraned optmzaton problems In the next secton we wll apply such an approach for constraned engneerng desgn problems IV SIMULION RESULS ND NLYSIS In ths secton we wll carry out numercal smulaton based on some well-known constraned engneerng desgn problems to nvestgate the performances of the proposed CPSOS For each testng problem the parameters of the CPSOS are set as follows: M =50 G =5 G =0 M =0 c = c =0 w n PSO lnearly decrease from 09 to 04 he maxmum and mnmum postons for partcles n ( X max and X mn ) depend on the varable regon gven by the problems he maxmum and mnmum postons of partcles n are set as wmax = wmax =000 and w mn = w mn = 0 for the frst two problems and as wmax = wmax =0000 w = w = 5000 for the thrd problem and mn mn Moreover the maxmum and mnmum veloctes for partcles n both the knds of swarms are set as V = 0 ( X X ) and V ( ) mn = Vmax = max max mn Smulaton results for a welded beam desgn problem (Example ) he welded beam desgn problem s taken from Rao (996) n whch a welded desgned for mnmum cost subect to constrants on shear stress (τ ) bendng stress n the beam (θ ) bucklng load on the bar ( P c ) end deflecton of the beam (δ ) and sde constrans here are four desgn varables as shown n Fgure e hx ( ) lx ( ) tx ( ) and bx ( 4) he problem can be mathematcally formulated as follows: Mnmze f ( x) = 047x x x x (40 x ) (0) 4 subect to : g( x) = τ( x) g( x) = σ( x) g() x = x x4 0 g4( x) = 0047x xx4(40 + x) 50 0 g5() x = 05 x 0 g6() x = δ () x 05 0 g ( x) = 6000 Pc( x) 0 7 Where τ ( x ) = x ( τ ') + τ ' τ '' + ( τ '') R τ ' = 6000 xx τ '' = MR J M = 6000(4 + x ) R J = = x x + x + ( ) 4 x x + x xx[ + ( ) ] σ ( x ) = δ ( x ) = x4x 95 x x4 P c( x) = ( x ) x x 4 00 CDEMY PUBLISHER

5 JOURNL OF COMPUERS VOL 5 NO 6 JUNE Fgure welded beam desgn problem (Example ) he approaches appled to ths problem nclude geometrc programmng (Ragsdell and Phllps 976) genetc algorthm wth bnary representaton and tradtonal penalty functon (Deb 99) a G-based coevoluton model (Codllo000) a feasblty-based tournament selecton scheme nspred by the multobectve optmzaton technques (Coello and Montes 000) an effectve co-evolutonary partcle swarm optmzaton for constraned engneerng desgn problems (Qe He Lng Wang 006) and a hybrd partcle swarm optmzaton wth a feasblty-based rule for constraned optmzaton (He Q Wang L 007) In ths paper the CPSOS s run 0 tmes ndependently wth the followng varable regons: 0 x 0 x x x4 0 he best solutons obtaned by the abovementoned approaches are lsted n able and ther statstcal smulaton results are shown n able Desgn varables CPSOS able Comparson of the best soluton of Example found by dfferent methods CPSO (006) HPSO (007) Ragsdell and Phllps (976) Deb (99) Coello (000) Coello and Montes (00) x ( ) x () x () x ( ) g ( ) e g ( ) e N/ g ( ) e N/ g ( ) N/ g ( ) N/ g ( ) N/ g ( ) e N/ f ( x ) able Statstcal results of dfferent methods for Example Method Best Mean Worst Std Dev CPSOS e-005 CPSO(006) Belegundu(98) 8597 N/ N/ N/ rora(989) 46 N/ N/ N/ Coello(000) Coello and Montes(00) CDEMY PUBLISHER

6 970 JOURNL OF COMPUERS VOL 5 NO 6 JUNE 00 From Fgure we can found that compared wth the results obtaned by CPSOS t s demonstrated that CPSOS s of effectveness to avod beng trapped n local optma by ncorporatng the umpng mechansm of S nto pure CPSO B Smulaton results for a tenson/compresson strng desgn problem (Example ) Fgure Evoluton of gbest of hs problem s from rora (989) and Belegundu (98) whch needs to mnmze the weght (e f ( x )) of a tenson/compresson strng (as shown n Fgure 4) subect to constrants on mnmum deflecton shear stress surge frequency lmts on outsde dameter and on desgn varables he desgn varables are the mean col dameter Dx ( ) the wre dameter d( x ) and the Px ( ) number of actve cols he mathematcal formulaton of ths problem can be descrbed as follows: Mnmze f ( x) = ( x+ ) xx () subect to : x x g( x) = x 4x xx g ( x) = ( xx x ) 508x 4045x g( x) = 0 x x x + x g4 ( x) = 0 5 Desgn varables Fgure 4 tenson/compresson strng desgn problem (Example ) he approaches appled to ths problem nclude sx dfferent numercal optmzaton technques (Belegundu 98) a numercal optmzaton technque called constrant correcton at constant cost (rora 989) a G-based co-evoluton model (Coello 000) and a feasblty-based tournament selecton scheme (Coello and Montes 00) an effectve co-evolutonary partcle swarm optmzaton for constraned engneerng desgn problems (Qe He Lng Wang 006) and a hybrd partcle swarm optmzaton wth a feasblty-based rule for constraned optmzaton (He Q Wang L 007) In ths paper the CPSOS s run 0 tmes ndependently wth the followng varable regons: 005 x x able Comparson of the best soluton for Example by dfferent methods CPSOS CPSO (006) HPSO (007) 05 x 5 he best solutons obtaned by the above-mentoned approaches are lsted n able and ther statstcal smulaton results are shown n able 4 From able t can be seen that the best feasble soluton found by CPSOS s better than the best solutons found by other technques Belegundu (98) rora (989) Coello (000) Coello and Montes (00) x ( ) x ( ) x ( ) g ( ) e N/ g ( ) e e-05 N/ g ( ) N/ g ( ) N/ f ( x ) CDEMY PUBLISHER

7 JOURNL OF COMPUERS VOL 5 NO 6 JUNE able 4 Statstcal results of dfferent methods for Example Method Best Mean Worst Std Dev CPSOS e-006 CPSO(006) e-005 Belegundu(98) 0084 N/ N/ N/ rora(989) 0070 N/ N/ N/ Coello(000) e-005 Coello and Montes(00) e-005 C Smulaton results for a pressure vessel desgn problem ( Example ) In ths problem the obectve s to mnmze the total cost ( f ( x ) ) ncludng the cost of the materal formng and weldng cylndrcal vessel s capped at both ends by hemsphercal heads as shown n Fgure 5 here are four desgn varables: s ( x thckness of the h ( x thckness of the head) R ( x nner shell) radus) and L ( x 4 length of the cylndrcal secton of the vessel not ncludng the head) mong the four varables and are nteger multples of 0065 n that the s h avalable thckness of rolled steel plates and R and L are contnuous varables he problem can be formulated as follows (Kannan and Karmer 994): Mnmze f ( x) = 064xxx + 778xxx + 66x x + 984x x () subect to : g( x) = x + 009x 0 g( x) = x x 0 4 g( x) = π x x4 π x g ( x) = x Fgure 5 Center and end secton of pressure vessel desgn problem (Example ) he approaches appled to ths problem nclude genetc adaptve search (Deb 997) an augmented Lagrangan multpler approach (Kannan and Kramer 994) a branch and bound technque (Sandgren 988) a G-based co-evoluton model (Coello 000) and a feasblty-based tournament selecton scheme (Coello and Montes 00) an effectve co-evolutonary partcle swarm optmzaton for constraned engneerng desgn problems (Qe He Lng Wang 006) and a hybrd partcle swarm optmzaton wth a feasblty-based rule for constraned optmzaton (He Q Wang L 007) In ths paper the CPSOS s run 0 tmes ndependently wth the followng varable regons: x 99 x 99 0 x 00 0 x4 00 the best solutons obtaned by the above mentoned approaches are lsted n able 5 and ther statstcal smulaton results are shown n able 6 Desgn varables CPSOS CPSO (006) x ( ) s x ( ) s x able 5 Comparson of the best soluton for Example found by dfferent methods HPSO (007) Standgren (988) Kannan and Kramer (994) Deb (997) Coello (000) Coello and Montes (00) ( ) s ( ) e+0 x4 s g ( ) e N/ g ( ) N/ g ( ) e N/ g ( ) N/ f ( x ) CDEMY PUBLISHER

8 97 JOURNL OF COMPUERS VOL 5 NO 6 JUNE 00 able 6 Statstcal results of dfferent methods for Example Method Best Mean Worst Std Dev CPSOS e-006 CPSO(006) Sandgren(988) 8906 N/ N/ N/ Kannan and Kramer(994) N/ N/ N/ Deb(997) 6408 N/ N/ N/ Coello(000) Coello and Montes(00) REFERENCES Fgure 6 Evoluton of gbest of From Fgure 6 we also found that compared wth the results obtaned by CPSOS t s demonstrated that CPSOS s of effectveness to avod beng trapped n local optma by ncorporatng the umpng mechansm of S nto pure CPSO Based on the above smulaton results and comparsons t can be concluded that CPSOS s of superor searchng qualty and robustness for constraned engneerng desgn problems V CONCLUSIONS hs paper has ntroduced a novel constrant-handlng method-cpsos hs s the frst report to usng CPSOS for constraned engneerng desgn problems whch ncorporate the mechansm of S nto CPSO Smulaton results and comparsons based on some wellknown constraned engneerng desgn problems and comparsons wth prevously reported results demonstrate the effectveness effcency and robustness of CPSOS he future work s to nvestgate better mechansm nto CPSOS to acheve better performance Our future work s to study the parallel mplementaton of CPSOS and the applcaton of CPSOS for constraned combnatoral optmzaton problems CKNOWLEDGMEN hs work s supported by Grants from NSF of Chna and the proect Supported by Grants from Guangx Scence Foundaton [] RC Eberhart and J Kennedy new optmzer usng partcle swarm theory In Proc 9 th Int Conf Neural Informaton Processng Nov00 pp [] He Q Wang L hybrd partcle swarm optmzaton wth a feasblty-based rule for constraned optmzaton ppled Mathematcs and Computaton (): [] DE Goldberg Genetc lgorthms n Search Intellgence Machne Learnng ddson Wesley M 989 [4] L Wang Intellgent Optmzaton lgorthms wth pplcatons snghua Unversty and Sprnger Press Beng 00 [5] LI Lng-La LING Wang L-Heng n effectve hybrd PSOS strategy for optmzaton and ts applcaton to parameter estmaton ppled Mathematcs and Computaton 79(): [6] Carlos Coello Coello Efrén Mezura Montes Constrant-handng n genetc algorthms throught the use of domnance-based tournament selecton dvanced Engneerng Informatcs 006():9-0 [7] Deb K Optmal desgn of a welded beam vs genetc algorthms I Journal 99 9 (): 0-05 [8] Kannan BK Kramer SN n augmented Lagrange multpler based method for mxed nteger dscrete contnuous optmzaton and ts applcatons to mechancal desgn Journal of Mechancal Desgn 994 6(): [9] Montes EM Coello CC smple multmembered evoluton strategy to solve constraned optmzaton problems IEEE ransactons on Evolutonary computaton 005 9():-7 [0] Rao SS Engneerng Optmzaton Wley New York 996 [] Sandgren E 988 Nonlnear nteger and dscrete programmng n mechancal desgn In: Proceedngs of the SME Desgn echnology Conference Kssmne FL pp Yongquan Zhou PhD Professor Hs receved the BS degree n mathematcs from Xanyang Normal Unversty Xanyang Chna In 98 the MS degree n computer from Lanzhou Unversty Lanzhou Chna In 99 and PhD n computaton ntellgence from Xdan Unversty Xan Chna Hs current research nterests ncludng computaton ntellgence and apples Shengyu Pe receved the MS degree n computer from Guangx Unversty for Natonaltes Nannng Chna He current research nterests ncludng computaton ntellgence and apples 00 CDEMY PUBLISHER