Cultural Algorithm for Engineering Design Problems

Transcription

1 IJCSI Inernaional Journal of Compuer Science Issues, Vol. 9, Issue 6, No, November Culural Algorihm for Engineering Design Problems Xuesong Yan 1,, Wei Li 1, Wei Chen 1, Wening Luo 1, Can Zhang 1 and Hanmin Liu 1, 1 School of Compuer Science, China Universiy of Geosciences Wuhan, Hubei 40074, China Deparmen of Compuer Science, Universiy of Cenral Arkansas Conway, AR 705, USA Wuhan Insiue of Ship Building Technology Wuhan, Hubei 40050, China Absrac Many engineering opimizaion problems can be sae as funcion opimizaion wih consrained, inelligence opimizaion algorihm can solve hese problems well. Culural Algorihms are a class of compuaional models derived from observing he culural evoluion process in naure, culural algorihms in he opimizaion of he complex consrained funcions of is superior performance. Experimen resuls reveal ha he proposed algorihm can find beer soluions when compared o oher heurisic mehods and is a powerful opimizaion algorihm for engineering design problems. Keywords: Engineering Design Problems, Culural Algorihms, Consrained Opimizaion, Populaion. 1. Inroducion Evoluionary compuaion has found a wide range of applicaions in various fields of science and engineering. Among ohers, evoluionary algorihms (EA) have been proved o be powerful global opimizers. Generally, evoluionary algorihms ouperform convenional opimizaion algorihms for problems which are disconinuous, non-differenial, muli-modal, noisy and no well-defined problems, such as ar design, music composiion and experimenal designs [1]. Besides, evoluionary algorihms are also well suiable for mulicrieria problems. Many engineering opimizaion design problems can be formulaed as consrained opimizaion problems. The presence of consrains may significanly affec he opimizaion performances of any opimizaion algorihms for unconsrained problems. Wih he increase of he research and applicaions based on evoluionary compuaion echniques [], consrain handling used in evoluionary compuaion echniques has been a ho opic in boh academic and engineering fields [,4]. A general consrained opimizaion problem may be wrien as follows: max f ( x ) (1) Subec o: gi( x) = ci, i = 1,,..., n, () h( x) d, = 1,,..., m. Where x is a vecor residing in an n-dimensional space, f ( x) is a scalar valued obecive funcion, gi( x) = ci, i = 1,,..., n, and h( x) d, = 1,,..., m, are consrain funcions ha need o be saisfied. Culural Algorihms (CA) proposed by Reynolds in 1994 [5]. Culural algorihm is in-deph analysis of he superioriy of he original evoluion heory on he basis of drawing on he social (culural) evoluion heory in he social sciences and has achieved broad consensus on he research resuls, and proposed a new algorihm. Culural algorihm is used o solve complex calculaions of he new global opimizaion search algorihms, culural algorihms in he opimizaion of he complex consrained funcions of is superior performance.. Culural Algorihm Evoluionary compuaion (EC) [6,7] mehods have been successful in solving many diverse problem in search and opimizaion due o he unbiased naure of heir operaions which can sill perform well in siuaion wih lile or no domain knowledge. However, here can be considerable improvemen in heir performance when problem specific knowledge is used o bias he problem solving process in order o idenify paerns in heir performance environmen. These paerns are used o promoe more insances of desirable candidaes or o reduce he number of less desirable candidaes in he populaion. In eiher case, his can afford he sysem an opporuniy o reach he desired soluion more quickly. Copyrigh (c) 01 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

2 IJCSI Inernaional Journal of Compuer Science Issues, Vol. 9, Issue 6, No, November Adapive evoluionary compuaion akes place when an EC sysem is able o incorporae such informaion ino is represenaion and operaors in order o faciliae he pruning and promoing aciviies menioned above. Some research works have shown ha self-adapaion can ake place on several levels wihin a sysem such as he populaion level, he individual level, and he componen level. A he populaion level, aspecs of he sysem parameers ha conrol all elemens of he populaion can be modified. A he individual level, aspecs of he sysem ha conrol he acion of specific individual can be modified. If he individual is specified as s collecion of componens hen componen level adapaion is possible. This involves he adapaion of parameers ha conrol he operaion of one or more componens ha make up an individual. In human socieies, culure can be a vehicle for he sorage of informaion in a form ha is independen of he individual or individuals ha generaed and are poenially accessible o all members of he sociey. As such culure is useful in guiding he problem solving aciviies and social ineracion of individuals in he populaion. This allows self-adapive informaion as well as oher knowledge o be sored and manipulaed separaely from he individuals in he social populaion. This provides a sysemaic way of uilizing self-adapive knowledge o direc he evoluion of a social populaion. Thus, culural sysems are viewed as a dual inheriance sysem where, a each ime sep, knowledge a boh he populaion level and he level of acquired beliefs is ransmied o he nex generaion. This acquired knowledge is viewed o ac as beacons by which o guide individuals owards perceived good soluions o problems and away from less desirable ones a a given ime sep. Culural Algorihms in order o model he evoluion of culural sysems based upon principles of human social evoluion aken from he social science lieraure. resul, he belief space can be used o conrol selfadapaion a any or all of hese levels. The culural algorihm is a dual inheriance sysem wih evoluion aking place a he populaion level and a he belief level. The wo componens inerac hrough a communicaions proocol. The proocol deermines he se of accepable individuals ha are able o updae he belief space. Likewise he proocol deermines how he updaed beliefs are able o impac and influence he adapaion of he populaion componen. The Culural Algorihm is a dual inheriance sysem ha characerizes evoluion in human culure a boh he macro-evoluionary level, which akes place wihin he belief space, and a he micro-evoluionary level, which occurs a he populaion space. CA consiss of a social populaion and a belief space. Experience of individuals seleced from he populaion space by he accepance funcion is used o generae problem solving knowledge ha resides in he belief space. The belief space sores and manipulaes he knowledge acquired from he experience of individuals in he populaion space. This knowledge can conrol he evoluion of he populaion componen by means of he influence funcion. As a resul, CA can provide an explici mechanism for global knowledge and a useful framework wihin which o model self-adapaion in an EC sysem. The populaion level componen of he culural algorihm will be Evoluionary Programming (EP). The global knowledge ha has been learned by he populaion will be expressed in erms of boh normaive and siuaional knowledge as discussed earlier. A flow-char of he Culural Algorihms is shown in Fig.1. Culural Algorihms are a class of compuaional models derived from observing he culural evoluion process in naure [5, 8, 9]. In his algorihm, individuals are firs evaluaed using a performance funcion. The performance informaion represens he problem-solving experience of an individual. An accepance funcion deermines which individuals in he curren populaion are able o impac, or o be voed o conribue, o he curren beliefs. The experience of hese seleced individual is used o adus he curren group beliefs. These group beliefs are hen used o guide and influence he evoluion of he populaion a he nex sep, where parameers for self-adapaion can be deermined from he belief space. Informaion ha is sored in he belief space can perain o any of he lower levels, e.g. populaion, individual, or componen. As a Copyrigh (c) 01 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

3 IJCSI Inernaional Journal of Compuer Science Issues, Vol. 9, Issue 6, No, November are seleced o impac he belief space. The seleced individuals' experiences are generalized and applied o adus he curren beliefs in he belief space via he updae funcion. The new beliefs can hen be used o guide and influence he evoluionary process for he nex generaion. Culural algorihms as described above consis of hree componens. Firs, here is a populaion componen ha conains he social populaion o be evolved and he mechanisms for is evaluaion, reproducion, and modificaion. Second here is a belief space ha represens he bias ha has been acquired by he populaion during is problem-solving process. The hird componen is he communicaions proocol ha is used o deermine he ineracion beween he populaion and heir beliefs.. Engineering Design Algorihm based on CA.1 Individual Iniializaion The radiional mehod of geneic algorihm is randomly iniialized populaion, ha is, generae a series of random numbers in he soluion space of he quesion. Design he new algorihm, we using he orhogonal iniializaion in he iniializaion phase. For he general condiion, before seeking ou he opimal soluion he locaion of he global opimal soluion is impossible o know, for some highdimensional and muli-mode funcions o opimize, he funcion iself has a lo of poles, and he global opimum locaion of he funcion is unknown. If he iniial populaion of chromosomes can be evenly disribued in he feasible soluion space, he algorihm can evenly search in he soluion space for he global opimum. Orhogonal iniializaion is o use he orhogonal able has he dispersion and uniformiy comparable; he individual will be iniialized uniformly dispersed ino he search space, so he orhogonal design mehod can be used o generae uniformly disribued iniial populaion.. Belief Space Srucure Fig. 1 Flow-har of culural algorihms In his algorihm, firs he belief space and he populaion space are iniialized. Then, he algorihm will repea processing for each generaion unil a erminaion condiion is achieved. Individuals are evaluaed using he performance funcion. The wo levels of Culural Algorihm communicae hrough he accepance funcion and he influence funcion. The accepance funcion deermines which individuals from he curren populaion In his paper, we define he belief space as < N[ n], C[ m] >, in here N denoes he normaive knowledge, consis of he change inerval informaion of variables; and C is he belief-cells informaion consis of he consrained knowledge, m is he number of cells. The normaive knowledge N, a se of inerval informaion for each of he n parameers is defined formally as 4-uple: N = I, L, U, adusn, = 1,,..., n, where I denoes he closed inerval of variable, ha is a coninuous se Copyrigh (c) 01 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

4 IJCSI Inernaional Journal of Compuer Science Issues, Vol. 9, Issue 6, No, November of real numbers x represened as a ordered number pair: I = [ l, u] = { x l x u, x R}. l (lower bound) and u (upper bound) are iniialized by he give domain values. represens he performance score of he lower L bound l for parameer. U represens he performance score of he upper bound u for parameer. The consrained informaion C[] i =< Classi, Cn1, i Cn, i Wi, Posi, Csizei >, in here Classi denoes he saus of ih uni in belief space,as feasible or infeasible. Cn1 i and Cn i denoes he number of individual locaes in feasible region or infeasible region, he iniial value is 0. Wi denoes he weigh of ih uni, in his paper he higher he finess value of he uni, he weigh value is smaller. Posi is vecor denoesthe lefmos posiion of he corner. Csizei denoes he size of he ih uni. Updae he normaive knowledge N in belief space uses he Eq. (): + 1 xi, xi, li or f( x) < Li li = li ohers + 1 f ( x) x, i li or f( x) < Li Li = Li ohers + 1 xi, xi, ui or f( x) < Ui ui = ui ohers + 1 f ( x) x, i ui or f( x) < Ui Ui = Ui ohers () Updae he C in belief space uses he Eq. (4): unknown ifcn1i = 0andCni = 0 feasible ifcn1i > 0andCni = 0 Classi = unfeasible ifcn1i = 0andCni > 0 semi _ feasible if Cn1i > 0andCni > 0 (4). Influence Funcion In his paper, he knowledge represened in he belief space can be explicily used o influence he creaion of he offspring via an influence funcion. For normaive knowledge, he influence funcion shown as he Eq. (5): xn, + ( u l) N(0,1) if xn, < l xn+ i, = xn, ( u l) N(0,1) if xn, > u (5) For consrained knowledge, he influence funcion shown as he Eq. (6): moveto( choose( Cell[ m])) if xn, { unfeasiblecells} xn, = xn, + ( u l) N(0,1)/ m oherwise (6) m In here, is he number of cells for variable, moveto () is move funcion, choose( Cell[ m ]) is selecion funcion. 4. Simulaion Experimen In his secion, we will carry ou numerical simulaion based on some well-known consrained engineering design problems o invesigae he performances of he proposed algorihm. The seleced problems have been well sudied before as benchmarks by various approaches, which is useful o show he validiy and effeciveness of he proposed algorihm. For each esing problem, he parameers of our algorihm are se as follows: he size of populaion is 100, he number of ieraion is 1000 and he run ime is Tension/Compression Sring Problem This problem is described by Arora [10], Coello and Mones [11] and Belegundu [1]. I consiss of minimizing he weigh ( f ( x )) of a ension/compression sring subec o consrains on shear sress, surge frequency and minimum deflecion as shown in Fig.. The design variables are he mean coil diameer D( = x1 ) ; he wire diameer d( = x ) and he number of acive coils N( = x ). The problem can be saed as: Minimize: f ( x) = ( x + ) xx1 (7) Subec o: xx g1( x) = 1 0, x1 4x x1x 1 g ( x) = + 1 0, ( xx 1 x1) 5108x x1 g( x) = 1 0, xx x1+ x g4 ( x) = (8) This problem has been solved by Belegundu [1] using eigh differen mahemaical opimizaion echniques, Arora [10] also solved his problem using a numerical opimizaion echnique called consrain correcion a Copyrigh (c) 01 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

5 IJCSI Inernaional Journal of Compuer Science Issues, Vol. 9, Issue 6, No, November consan cos, Addiionally, Coello solved his problem using GA-based mehod [1] and a feasibiliy-based ournamen selecion scheme [11]. Table 1 presens he bes soluion of his problem obained using he CA and compares he CA resuls wih soluions repored by oher researchers. I is obvious from he Table 1 ha he resul obained using CA is beer han hose repored previously in he lieraure. Minimize: f ( x) = 0.64x1xx xx+.1661x1x x1x (9) Subec o: g ( x) = x x 0, 1 1 g ( x) = x x 0, 4 g( x) = πxx4 πx + 1, 96, 000 0, g ( x) = x (10) This problem has been solved before by Sandgren using a branch and bound echnique [15], by Kannan and Kramer using an augmened Lagrangian Muliplier approach [16], by Deb and Gene using Geneic Adapive Search [17], by Coello using GA-based co-evoluion model [1] and a feasibiliy-based ournamen selecion scheme [11]. The comparisons of resuls are shown in Table. The resuls obained using he CA, were beer opimized han any oher earlier soluions repored in he lieraure. Fig. Tension/compression sring problem 4. Pressure Vessel Problem A cylindrical vessel is capped a boh ends by hemispherical heads as shown in Fig.. The obecive is o minimize he oal cos, including he cos of maerial, forming and welding. There are four design variables: T s (hickness of he shell, x 1 ), T h (hickness of he head, x ), R (inner radius, x ) and L (lengh of cylindrical secion of he vessel, no including he head, x 4 ). T s and T h are ineger muliples of inch, wich are he available hickness of rolled seel plaes, and R and L are coninuous. 4. Welded Beam Problem The welded beam srucure, shown in Fig. 4, is a pracical design problem ha has been ofen used as a benchmark for esing differen opimizaion mehods. The obecive is o find he minimum fabricaing cos of he welded beam subec o consrains on shear sress ( τ ), bending sress ( σ ), buckling load ( P c ), end deflecion ( δ ), and side consrain. There are four design variables: h( = x1 ) ; l( = x ); ( = x ) and b( = x4 ). Fig. 4 Welded beam problem Fig. Pressure vessel problem Using he same noaion given by Coello [14], he problem can be saed as follows: The mahemaical formulaion of he obecive funcion f ( x ), which is he oal fabricaing cos mainly comprised of he se-up, welding labor, and maerial coss, is as follows: Minimize: f ( x) = x x x x ( x ) (1) 1 4 Copyrigh (c) 01 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

6 IJCSI Inernaional Journal of Compuer Science Issues, Vol. 9, Issue 6, No, November Subec o: g1( x) = τ ( x) , g ( x) = σ ( x) , g( x) = x1 x4 0, g4( x) = x xx4( x) 5.0 0, g5( x) = 0.15 x1 0, g6 ( x) = δ ( x) 0.5 0, g7 ( x) = 6000 Pc ( x) 0, Where: x τ x τ ττ τ R ' 6000 τ =, xx ' ' '' '' ( ) = ( ) + + ( ), 1 '' MR τ =, J x M = (14 ), x x + x R = ( ), x x + x J = σ ( x) =, xx 1 x1x ( ), δ ( x) =, xx 4 4 P( x) = ( x ) x x. c 4 (1) (14) The approaches applied o his problem include geomeric programming [18], geneic algorihm wih binary represenaion and radiional penaly funcion [19], a GAbased co-evoluion model [1] and a feasibiliy-based ournamen selecion scheme inspired by he muliobecive opimizaion echniques [11]. The comparisons of resuls are shown in Table. The resuls obained using he CA, were beer opimized han any oher earlier soluions repored in he lieraure. 4.4 Saellie Coss Problem The obecive is o find he minimum cos of he saellie design, subec o consrains on Covering bandwidh S w0, Flexible ransfer ime Δ, Resoluion 0 d s0, Eclipse facor k e,max, The oal mass of he saellie M oal, Baery cycle life N BA,max and Saellie volume V sa,0. There are six design variables: Orbial aliude h, Angle of inclinaion of he orbi i, local ime of he Descending node DNT, Transfer orbi apogee radius r a, CCD camera focal lengh f c, The side lengh of he saellie srucure b and heigh l. This problem can be saed as follows [0]: min C oal (15) Subec o: R e h+ R e g1 : Sw = {sin } sini 49Np f Re + c 1 7NP sin ( ) 50 49Np + fc g : h/ fc 15/7 ( ra r1) + ( ra + r1)cosθ arccos[ )] ( r 1) ( ra r1) ( ra r1)cos a + r θ + + g : Δ = 8μ 4rr (16) a 1 ( ra + r1) + ( ra r1)cosθ Reh+ h g4 : sin ( ) < 0.5 π Re + h g5: : Moal 500 ( ra + r1 ) g6 : π 556s μ 9 g7 : l b In here, Re = 678.1km and μ = km / s. We use our algorihm for his problem and Table 4 is he resul of our algorihm for he design variable and consrained variables. Design Variables Table 4: Experimen resul Lower Bound Upper Bound Our Algorihm h (km) DNT (hr) r a (km) f c (mm) l (mm) b (mm) S w0 (km) Δ 0 (s) d s0 0 0 k e,max M oal (kg) N BA,max (s) V sa,0 (m) C oal Copyrigh (c) 01 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

7 IJCSI Inernaional Journal of Compuer Science Issues, Vol. 9, Issue 6, No, November Conclusions This paper inroduces a new mehod-culural algorihm for solving he engineering design problem. For he empirical sudies his algorihm has proved o be efficien, and he experimens resuls shown he new mehod are effecive for engineering opimizaion design. Acknowledgmens This paper is suppored by he Provincial Naural Science Foundaion of Hubei (No. 011CDB4 and 011CDB46), he Fundamenal Research Founds for Naional Universiy, China Universiy of Geosciences (Wuhan) and Naional Civil Aerospace Pre-research Proec of China. References [1] H.-P. Schwefel, Evoluion and Opimum Seeking, Wiley, [] Qie He and Ling Wang, An effecive co-evoluionary paricle swarm opimizaion for consrained engineering design problems, Engineering Applicaions of Arificial Inelligence, 0, 007, pp [] Coello, C.A.C., Mones, E.M., Consrain-handling in geneic algorihms hrough he use of dominance-based ournamen selecion, Advanced Engineering Informaics, 16, 00, pp [4] Michalewicz, Z. (Eds.), Evoluionary Algorihms in Engineering Applicaions, Springer, Berlin, 1995, pp [5] R. Reynoids, An inroducion o culural algorihms, in Proceedings of he rd Annual Conference on Evoluionary Programming, Sebald, AX; Fogel, L.J. (Ediors), River Edge, NJ, World Scienific Publishing, 1994, pp [6] Goldbeg, D. E, Geneic Algorihms in Search, Opimizaion and Machine Learning, Reading, Mass.: Addison-Wesley, [7] Michalewicz, Z, Geneic Algorihms + Daa Srucures = Evoluion Programs, rd Ed. Berlin: Springer Verlag, [8] CHUNG C, Knowledge-based approaches o selfadapaion in culural algorihms, Ph.D. Thesis, Wayne Sae Universiy, Deroi, Michigan, USA, [9] ZHANG Yin, Culural algorihm and is applicaion in he porfolio, Maser Thesis, Harbin Universiy of Science and Technology, Harbin, China, 008. (in Chinese) [10] Arora, J.S., Inroducion o Opimum Design, New York, McGraw-Hill, [11] Coello, C.A.C., Mones, E.M., Consrain-handling in geneic algorihms hrough he use of dominance-based ournamen selecion, Advanced Engineering Informaics, 00, 16, pp [1] Belegundu, A.D., A sudy of mahemaical programming mehods for srucural opimizaion, Deparmen of Civil and Environmenal Engineering, Universiy of Iowa, Iowa Ciy, Iowa, 198. [1] Coello, C.A.C., Use of a self-adapive penaly approach for engineering opimizaion problems, Compuers in Indusry, 000, 41, pp [14] Coello, C.A.C., Theoreical and numerical consrain handling echniques used wih evoluionary algorihms: a survey of he sae of he ar, Compuer Mehods in Applied Mechanics and Engineering, 00, 191 (11/1), pp [15] Sandgren, E., Nonlinear ineger and discree programming in mechanical design, In: Proceedings of he ASME Design Technology Conference, Kissimine, FL, 1988, pp [16] Kannan, B.K., Kramer, S.N., An augmened Lagrange muliplier based mehod for mixed ineger discree coninuous opimizaion and is applicaions o mechanical design, Transacions of he ASME, Journal of Mechanical Design, 116, 1994, pp [17] Deb, K., GeneAS: a robus opimal design echnique for mechanical componen design, Evoluionary Algorihms in Engineering Applicaions, 1997, pp [18] Ragsdell, K.M., Phillips, D.T., Opimal design of a class of welded srucures using geomeric programming, ASME Journal of Engineering for Indusries, 98 (), 1976, pp [19] Deb, K., Opimal design of a welded beam via geneic algorihms, AIAA Journal, 9 (11), 1991, pp [0] Xii Wang, Dayao Li, Saellie design mehodology, Beiing: Aomic Energy Press, 005. [1] Xuesong Yan, Qinghua Wu, e.al, An Efficien Funcion Opimizaion Algorihm based on Culure Evoluion, Inernaional Journal of Compuer Science Issues, Vol. 9, No., 01, pp [] X.S. Yan, Q.H. Wu, Funcion Opimizaion Based on Culural Algorihms, Journal of Compuer and Informaion Technology, Vol., 01, pp [] X.S. Yan e, al, Orhogonal evoluionary algorihm and is applicaion in relay volume opimizaion design, Compuer Engineering and Applicaions, 47(18), 011, pp (in Chinese). [4] X.S. YAN, Q.H. WU, C.Y. HU, Q.Z. LIANG, Elecronic Circuis Opimizaion Design Based On Culural Algorihms, Inernaional Journal of Informaion Processing and Managemen, (1), 011, pp [5] X.S. Yan e.al, Designing Elecronic Circuis Using Culural Algorihms, Proceedings of Third Inernaional Workshop on Advanced Compuaional Inelligence, 010, pp [6] X.S.Yan e.al, Applicaion of Culural Algorihm in Sock Daa Modeling, Compuer Engineering & Science, Vol., No.6, 010, pp (in Chinese). Xuesong Yan associae professor received him B.E. degree in Compuer Science and Technology in 000 and M.E. degree in Compuer Applicaion from China Universiy of Geosciences in 00, received he Ph.D. degree in Compuer Sofware and Theory from Wuhan Universiy in 006. He is currenly wih School of Compuer Science, China Universiy of Geosciences, Wuhan, China and now as a visiing scholar wih Deparmen of Compuer Science, Universiy of Cenral Arkansas, Conway, USA. He research ineress include evoluionary compuaion, daa mining and compuer applicaion. Copyrigh (c) 01 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

8 IJCSI Inernaional Journal of Compuer Science Issues, Vol. 9, Issue 6, No, November Wei Li received her B.E. degree in Compuer Science and Technology in 01. She is currenly is he M.E. degree candidae wih School of Compuer Science, China Universiy of Geosciences, Wuhan, China. Her research ineress include evoluionary compuaion. Can Zhang received him B.E. degree in Compuer Science and Technology in 011. He is currenly is he M.E. degree candidae wih School of Compuer Science, China Universiy of Geosciences, Wuhan, China. Her research ineress include evoluionary compuaion. Wei Chen received him B.E. degree in Compuer Science and Technology in 01. He is currenly is he M.E. degree candidae wih School of Compuer Science, China Universiy of Geosciences, Wuhan, China. Her research ineress include evoluionary compuaion. Wening Luo received her B.E. degree in Compuer Science and Technology in 01. She is currenly is he M.E. degree candidae wih School of Compuer Science, China Universiy of Geosciences, Wuhan, China. Her research ineress include evoluionary compuaion. Hanmin Liu associae professor. He is currenly as a Ph.D candidae of School of Compuer Science, China Universiy of Geosciences, Wuhan, China. He research ineress include evoluionary compuaion and applicaions. Design variables Table 1: Comparison of he bes soluion for ension/compression sring problem Belegundu (198) Arora (1989) Coello (000) Coello (00) x ( ) x ( ) x ( ) g ( ) g ( ) e-05 g ( ) g ( ) f ( x ) CA Design variables Sandgren (1988) Table : Comparison of he bes soluion for pressure vessel problem Kannan (1994) Deb (1997) Coello (000) Coello (00) x ( ) 1 T s x ( ) T h x ( ) x ( ) g ( ) g ( ) g ( ) g ( ) f ( x ) CA Copyrigh (c) 01 Inernaional Journal of Compuer Science Issues. All Righs Reserved.

9 IJCSI Inernaional Journal of Compuer Science Issues, Vol. 9, Issue 6, No, November Design variables Table : Comparison of he bes soluion for welded beam problem Ragsdell (1976) Deb (1991) Coello (000) Coello (00) x ( ) x () x () x ( ) g ( ) g ( ) g ( ) g ( ) g ( ) g ( ) g ( ) f ( x ) CA Copyrigh (c) 01 Inernaional Journal of Compuer Science Issues. All Righs Reserved.