Convergence Improvement of Reliability-Based MultiobjectiveOptimization Using Hybrid MOPSO

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1 10 h World Congress on Srucural and Mulidisciplinary Opimizaion May 19-4, 013, Orlando, Florida, USA Convergence Improvemen of Reliabiliy-Based MuliobjeciveOpimizaion Using Hybrid MOPSO Shoichiro KAWAJI 1 and Nozomu Kogiso 1, Deparmen of Aerospace Engineering, Osaka Prefecure Universiy, Sakai, Japan 10 kawaji shoichiro@aero.osakafu-u.ac.jp, kogiso@aero.osakafu-u.ac.jp 1. Absrac This sudy proposes he improvemen mehod of compuaional efficiency for he hybrid-ype muliobjecive paricle swarm opimizaion mehod (MOPSO) ha he auhors developed for he reliabiliy-based muliobjecive opimizaion (RBMO). The hybrid MOPSO inegraes he consrain saisfacion echnique using gradien informaion of consrains wih a concep of he single-loop-single-vecor mehod (SLSV). The consrain saisfacion echnique consiss of wo funcions: moving he design candidae wih consrain violaion o he feasible region based on he sensiiviy of he violaed consrains and o he feasible boundary using bi-secion mehod. Some design candidaes, however, ake much compuaion wih zigzag ieraions unil he candidae moves o feasible region. This sudy proposes he improvemen mehod by eliminaing he zigzag ieraions based on an idea from he modified SLSV mehod or he conjugae mean value mehod proposed for resolving he convergence problem for he reliabiliy-based opimizaion. The efficiency of he proposed mehod is demonsraed hrough several design problems by invesigaing he convergence and diversiy of finding Pareo opimal soluions.. Keywords: Reliabiliy-based muliobjecive opimizaion, Hybrid muliobjecive PSO, Single loop reliabiliybased opimizaion, Sensiiviy analysis 3. Inroducion Muliobjecive opimizaion mehod is widely applied o variey fields of engineering design problems. Especially, he evoluionary opimizaion algorihms are widely conduced such as muliobjecive geneic algorihms [1] and muliobjecive paricle swarm opimizaion (MOPSO) []. On he oher hand, research on he reliabiliy-based muliobjecive opimizaion (RBMO) is sill developing [3 7]. Mos sudies seleced he muliobjecive GA, especially NSGA-II, as an opimizer. However, compuaional efficiency of he muliobjecive GA is no sufficien. This is because he muliobjecive GA originally developed for design problem wih discree design variables and does no use he sensiiviy informaion. A convenional single-objecive reliabiliy-based opimizaion [8] uses a gradien-based opimizaion mehod, because he sensiiviy informaion is easily obained from he reliabiliy analysis such as he firs-order reliabiliy mehod (FORM). In addiion, he efficien reliabiliy-based design approach was developed o ake consideraion in using he gradien-based opimizaion mehod such as single-loop-single-vecor (SLSV) mehod [9]. I is considered ha he sensiiviy informaion is also useful o he RBMO problem. Therefore, we proposed he compuaional efficien RBMO algorihms [10] by inegraing he hybrid-ype MOPSO [] and he SLSV conceps [9]. The hybrid MOPSO has a consrain handling echnique ha uses he sensiiviy of he consrains [11]. The consrain saisfacion echnique consiss of he wo seps. Firs, he design candidae in infeasible region is forced o move o he feasible region on he basis of he violaed consrain gradien informaion in each opimizaion sep. Then, he consrain-saisfied design candidae is moved o he feasible boundary using bi-secion mehod. For mos engineering design problem, Pareo soluions generally lay on he feasible boundary. Therefore, he moved design candidae on he feasible boundary has higher possibiliy o selec as a Pareo candidae. This sraegy makes he Pareo searching efficiency improved, ha was demonsraed hrough simple numerical design problem [11]. Then, he hybrid-ype MOPSO wih consrain saisfacion using sensiiviy was applied o he RBMO problems [10, 1]. For compuaional efficiency, he mehod inegraes he concep of SLSV mehod. The single-loop RBMO using he hybrid-ype MOPSO was demonsraed o have higher compuaional efficiency [10, 1]. However, many calculaeions were required unil he candidae moves o feasible region for some cases wih zigzag ieraions as shown in Fig. 1. This sudy proposes he improvemen mehod by eliminaing such zigzag ieraions based on an idea from he modified SLSV mehod [13, 14] or he conjugae mean value mehod [15] proposed for resolving he convergence problem for he reliabiliy-based opimizaion. The original mehod only uses he sensiiviy a he curren posiion. On he oher hand, he modified mehod uses sensiiviies a he previous posiions o improve he moving direcion o he feasible regions. The efficiency of he proposed mehod 1

2 d Feasible region Saisfied p. T * 1 1 β σ α µ * ( = d ) x z 1 * T * β σ α x 1 g 1 (x) = 0 Violaed p. g (x) = 0 Figure 1: Feasible domain and example of moving pahs o feasible boundary. Figure : SLSV. z * g 1 (d, z) = 0 g (d, z) = 0 Relaion beween variables in applying is demonsraed hrough several design problems by invesigaing he convergence and diversiy of finding Pareo opimal soluions. 4. Reliabiliy-Based Muliobjecive Opimizaion Consider he muliobjecive opimizaion problem subjeced o he reliabiliy consrains ha consider uncerainies of design parameers such as maerial properies and applied loads. The limi sae funcion ha defines he j-h failure mode is denoed as g j (d, z) ( j = 1,, k), where d = [d 1,, d n ] T and z = [z 1,, z l ] T are he design and random variables, respecively. The random variables are assumed o be independen from each oher and denoed he mean value µ = [µ 1,, µ l ] T and he sandard deviaion as σ = diag[σ 1,, σ l ] T. The failure probabiliy P f j is defined as he probabiliy ha he j-h limi sae funcion akes he negaive value: P f j = P(g j (d, z) 0) (1) The reliabiliy-based muliobjecive opimizaion (RBMO) problem ha minimize he muliple objecive funcions under he reliabiliy consrains ha he failure probabiliies be lower han he prescribed values is formulaed as follows: Minimize: F(d) = ( f 1 (d), f (d),, f k (d)) () subjec o: P f j = P(g j (d, z) < 0) Φ( β j ), ( j = 1,, m) d L d d U where f i (d) is he i-h objecive funcion, β j is he arge reliabiliy index for he j-h failure mode, Φ() is he sandardized normal disribuion funcion and d U and d L are he upper and lower bounds of design variables, respecively. For he single-objecive reliabiliy-based opimizaion, he single-loop-single-vecor mehod (SLSV) [9] is one of he approaches ha reduce he excessive compuaional cos of he double-loop formulaed problem ha adops he firs order reliabiliy mehod (FORM) as he reliabiliy analysis. The key idea of he SLSV is ha he reliabiliy consrain is convered ino he deerminisic consrain by using a proper approximaion of he design poin, someimes called mos probable poin (MPP). When he random vecor z has a normal disribuion N(µ, σ), he mean value µ and he design poin z j of he j-h mode has following relaionship as shown in Fig.. Noe ha he mean value vecor µ is reaed as design vecor, d in his figure: z j = µ β j σ T α j, (3) α j = g j(d, z j ) g j (d, z j ) where α j is he uni gradien vecor of he j-h limi sae funcion wih respec o random variables. This equaion indicaes ha he curren mean value µ saisfying he arge reliabiliy index β will agree wih he poin ha is β imes he sandard deviaion depar from he design poin, z j in he opposie direcion of α j. Based on he maer ha he design poin z j lies on he limi sae surface, he reliabiliy consrain is rewrien as a following deerminisic consrain: g j (d, µ β j σ T α j ) 0 (4) d 1

3 f Archive member in -h generaion Pareo se in +1s generaion A f Archive member in +1s generaion Removed from Pareo se A f Σ = Σ = Σ = local bes paricle Σ = B C a D f 1 0 Figure 3: Selecion of Pareo soluions in archive B C a D f 1 0 Figure 4: mehod Σ = Global guide selecion by sigma where he gradien vecor α j should evaluae a he design poin z j. However, he poin is no known a prior. Therefore, a convenional RBDO mehod requires a ieraed reliabiliy analysis such as FORM. On he oher hand, he SLSV mehod replaces he gradien vecor α j by he gradien vecor obained in he previous opimizaion loop. This approximaion makes he RBDO problem o a single loop algorihm. 5. Incorporaion of Hybrid MOPSO wih SLSV In his sudy, he RBMO problem is solved by using he hybrid MOPSO [11] incorporaed wih concep of he SLSV mehod. The hybrid MOPSO has consrain handling capabiliy using he gradien informaion of he consrain. Inegraion of he SLSV mehod ino he consrain handling, he hybrid MOPSO is exended o he RBMO problem [10]. 5.1 Muliobjecive Paricle Swarm Opimizaion (MOPSO) The MOPSO is a kind of a heurisic approach of he muliple-poin searching approach ha expands o he muliobjecive opimizaion from he paricle swarm opimizaion (PSO). In he PSO, he curren i-h design vecor di +1 is updaed from he previous design di by using he velociy vecor u+1 i as follows: = di + u+1 i, (5) u +1 i = wu i + C 1r 1 (d p i di ) + C r (dg i di ) d +1 i where d p i is called he personal bes ha is he bes soluion of he i-h design vecor unil he previous ieraion, dg i is called he group bes ha is he bes soluion in all of he individuals. r 1 and r are he uniform random variables in [0, 1] o adjus he design updae, w is called he inerial erm and C 1 and C are he parameers o adjus he effecs of d p i and dg i, respecively. In his sudy, w = 0.4 and C 1 = C = are adoped as he generally recommended value. For he muliobjecive opimizaion problem, he archive sraegy is adoped as a kind of eliis mehod [16]. The mehod archives he Pareo candidae se and updaes he se by comparing he curren candidaes wih he archived candidaes. As illusraed in Fig. 3, he curren candidaes A, B, and C are archived as new Pareo candidaes, bu he curren candidae D and he archived candidae a are removed because hey are inferior o new Pareo candidaes. In he MOPSO, several mehods were proposed o deermine d pi and d gi which are called a local guide and a global guide, respecively, because he noion bes is no suiable for he muliobjecive opimizaion. This sudy adops he sigma mehod [17] o deermine he local and global guides. The sigma mehod evaluaes he conribuion rae beween he objecive funcions as sigma by he following equaion: f1 f Σ = f 1 f 1. f1 + f (if k = ), Σ = k f fi k 1 f k (if k 3) (6) fk f 1 where Σ is a scholar for a bi-objecive problem, and Σ becomes a vecor for more han hree objecive funcion problem. As shown in Fig. 4, he Pareo candidae ha has he mos closes sigma value in he objecive funcion space is seleced as a global guide, d g i. This sraegy is known o have a higher convergency and also higher diversiy of Pareo soluions. i=1 f 1 3

4 g 1 (x) = 0 h (k) = γ(h (k-1) +h (k) ) h (k-1) h (k-) Feasible region h (k-1) h (k) θ A g (x) = 0 θ B h (k) h (k) h (k-) Figure 5: Image of zigzag eliminaion On he oher hand, as he number of archived Pareo candidaes are increased, he archived sraegy will ake more compuaional ime o compare and updae he Pareo candidaes. To save he compuaional effor, only one Pareo candidae in each prescribed sigma region is archived. 5. Consrain Handling Using Sensiiviy and Zigzag Eliminaion The consrain handling mehod proposed in our previous sudy [10 1] use he consrain sensiiviy informaion. When he design candidae violaes he consrains, i is moved o he feasible region o he direcion obained by using he violaed consrain gradien informaion. When he violaion is improved, he design candidae is moved o he same direcion unil he candidae saisfies he consrains. On he oher hand, when he consrain violaion is deerioraed, he moving direcion is updaed by evaluaing he violaed consrains and repeaed unil all of he consrains are saisfied. Finally, he candidaes is moved o he feasible boundary using bi-secion mehod. The mehod is found o have a sufficien Pareo opimal soluion searching capaciy for deerminisic muliobjecive opimizaion problem [11] and he RBMO problem inegraed wih he SLSV mehod [10, 1]. However, he mehod does no always worked well. When he violaed consrains are ofen swiched, he design candidae will approach he feasible region by zigzag as illusraed in Fig. 1. Such a case requires many ieraions and herefore akes much compuaional effor. We modify he mehod o eliminae he zigzag ieraions. Several approaches were proposed for he singleobjecive reliabiliy-based design opimizaion (RBDO) o improve he compuaional efficiency [13 15]. These mehods use no he curren sensiiviy bu he previous sensiiviy o deermine he searching direcion. Tha is, if he searching direcion is regarded as zigzag using he curren and previous sensiiviies, he direcion is modified using he sensiiviies. An image of eliminaion of he zigzag ieraions is shown in Fig. 5, where h indicaes he moving direcion vecor based on he sensiiviies of violaed consrains and k is he number of ieraions. Firs, he changing angles beween successive searching direcions are defined based on he Modified SLSV mehod [13] as follows: θ A = cos 1 h (k )T h (k) (7) θ B = cos 1 h (k 1)T h (k) (8) Tha is, A and B are he angle beween he curren and he second previous searching direcions, and he angle beween he curren and previous searching direcions, respecively. respecively. Then, we decide ha he zigzag occurred if θ A < θ B. In his case, he moving direcion is modified using following equaion: h (k) = γ(h (k 1) + h (k) ) (9) where γ is consan o adjus he magniude of he moving direcion. illusraed in Fig. 6(a) and he algorihm is described as follows. The flow of he consrain handling is Sep 1: Se sep size Se he sep size δ for he violaed design d i. Sep : Se moving direcion Evaluae he moving direcion vecor h using he gradien vecors of violaed consrains g j (d i, d i β j σ T α j ) as follows. g j (d i, d i β j σ T α j h = ) g j (d i, d i β j σ T α j ) (10) j J 4

5 Se iniial paricle d i, v i, d pi,d gi, α j = 0, =0 Evaluae searching direcion g j (d i, d i β j σ T α j ) h = Σ g j (d i, d i β j σ T j J α j ) Evaluae consrains Check Move paricle d i = d i + δh Consrain values improved Consrain values deeriolaed Saisfy all consrain Updae sep size δ No Zigzag? Modify searching direcion Yes h (k) = γ(h (k) +h (k-1) ) = +1 Evaluae objecives a curren design and consrains a approximaed design poin F(d i ), g j (d i, d i - β j σ T α j ) Violaed? No Yes Move o feasible fron d i = d i + δh Evaluae Σ i o collec Pareo candidae Updae Pareo archive Updae d pi and d gi Updae paricle posiion, velociy and normalized gradien vecor for consrains d +1 i = d i + v i v +1 i = wv i + C 1 r 1 (d pi -d i ) + C r (d gi -d i ) α j +1 = g j (d i, d i - β j σ T α j ) g j (d i, d i - β j σ T α j ) For each paricle Find acive consrain poin End = max? No Yes Pareo se in archive (a) Consrain saisfacion wih eliminaing zigzag ieraion. (b) Reliabiliy-based muliobjecive opimizaion. Figure 6: Compuaional flow of proposed hybrid-ype MOPSO wih consrain saisfacion. where J indicaes he violaed consrain se J (J = { j g j (d i, d i β j σ T α j ) < 0, ( j = 1,, m)}). Noe ha he reliabiliy consrain is evaluaed a he approximaed design poin d β j σ T α j, no he design variable d for using he concep of he SLSV mehod. Sep 3: Updae design The design vecor is updaed using he vecor h as follows: d i = d i + δh (11) Sep 4: Judgmen Evaluae all consrain condiions for he updaed design vecor. The nex sep is seleced according o he following condiions. In case ha all of consrains are saisfied Go o Sep 5. In case ha he violaed consrain value is deerioraed Go back o Sep and updae he moving direcion, h. In case ha he consrain value is improved Evaluae he changing angles in Eqs. (7) and (8). If θ A θ B, hen updae he sep size δ and go back o Sep 3. Oherwise, modify he moving direcion using he following equaion: h (k) = γ(h (k 1) + h (k) ) (1) and hen go back o Sep 3. Where, γ is consan for adjusing he sep size. Sep 5: Move o feasible boundary Move he feasible boundary (g j (d i, µ β σ T α j ) = 0) by applying bi-secion mehod using he previous and he curren design, where he previous design is violaed and he curren design is feasible. The proposed algorihm is summarized in Fig. 6 (b). Feaure of he proposed mehod is o updae he normalized gradien vecor of he reliabiliy consrain α +1 j a each sep as well as he design vecor and he velociy. 6. Numerical Examples Efficiency of he proposed mehod is demonsraed hrough several engineering examples. The firs example illusraes he compuaional efficiency of he zigzag eliminaion. Then, he efficiency of he improved hybrid-ype MOPSO is swmonsraed hrough a couple of engineering example problems. 5

6 10 g 1 = 0 8 g 3 = 0 Feasible domain d 6 d Previous 4 (1, 5) Feasible domain 1.5 Proposed 0 (3, 0.1) g = d 1 (8.5,1) Figure 7: Feasible domain and resuls of enclosing. 1 g 3 = 0 (9,1) d 1 g = 0 Figure 8: Effec of zigzag eliminaion process. Table 1: Comparison of he number of funcion evaluaions Toal Average Previous mehod Proposed mehod Efficiency of zigzag eliminaion Consider he following feasible region consising of convex and non-convex consrains [18]: g 1 (d) = d 1 d (13) g (d) = (d 1 + d 5) + (d 1 d 1) g 3 (d) = d1 + 8d d i 10, (i = 1, ) Table 1 compares he number of funcion evaluaions required for moving o he feasible region from he 11 saring grid poins composed of ineger coordinaes ha saisfies he side consrain, ha is, (0,0) o (10, 10). In his example, he consan γ in Eq. (9) ha adjuss he sep size is se o 10. Though he previous mehod have 5 poins ha exceeds more han 100 funcion evaluaions, he new mehod has only one poin ha he number is 10. The oal number of funcion evaluaions is reduced more han 30% from 6 o In Fig. 7, he blue dos are he final goal o he feasible region saring from he grid poins and he red brokenlines are examples of he moving rajecories from several poins. As a ypical case, rajecories saring from (9,1) are compared in Fig. 8, where he blue broken-line corresponds o he previous mehod and he red one o he new mehod. For boh mehod, he rajecories pass across a boundary of g = 0 unil saisfying all consrains. The previous mehod requires 45 funcion evaluaions, bu he new mehod requires only 60, abou one-fourh. 6. Crashworhiness RBMO problem The new mehod is applied o he crashworhiness RBMO problem [4] ha is known as a benchmark problem comsising of objecive funcions and 9 consrain condiions in erms of 9 design variables. The random variables x follows independen normal disribuion whose mean value are se as design variables d and he coefficien of variaions are lised in Table (a). The objecive and consrain funcions are also lised in Table (b) and (c), respecively, ha each funcion is formulaed as polynomials hrough he response surface approximaion. In MOPSO, he number of paricles is se as 100, he number of ieraions 00 and γ = 4. Fig. 9 compares he Pareo se under he condiions ha he arge reliabiliy indices β are se as 0, 3, and 6 beween he previous and he new mehods. Noe ha β = 0 corresponds o he deerminisic problem. The obained Pareo se for boh mehods are almos similar o he original one [4], hough small difference exiss only in he upper lef par of he 6

7 Table : Formulaion of crashworhiness RBMO problem [4]. (a) Design variables, side consrains, and cov in random variables Name Variable Lower bound Upper bound COV(= σ/µ) Thickness of B-pillar inner (mm) d Thickness of B-pillar reinforcemen (mm) d Thickness of floor side inner (mm) d Thickness of cross members (mm) d Thickness of door beam (mm) d Thickness of door bel line reinforcemen (mm) d Thickness of roof rail (mm) d Maerial yield sress for B-pillar inner (GPa) d Maerial yield sress for floor side inner (GPa) d (b) Two Objecive funcions Weigh f 1 = d d d d d d 7 Door velociy f = d 3 d d 5 d 6 (c) Nine Consrains Name Upper limi Formulaion Abdomen load (kn) g x x x 3 x 9 Rib deflecion upper (mm) g x 3 4.x 1 x x 6 x x 7 x 8 Rib deflecion middle (mm) g x x 1 x 11.0x x x 7 x 8 +.0x 8 x 9 Rib deflecion lower (mm) g x 1.9x 1 x 8 Public symphysis force (kn) g x x x 3 B-Pillar velociy (m/s) g x 1 x 1.95x x 8 VC upper (m/s) g x 1 x 0.188x 1 x x x x 3 x x 6 x 9 VC middle (m/s) g x x 1 x x 1 x x x x x x 3 x x 3 x x 5 x 6 VC lower (m/s) g x 0.163x 3 x x 7 x x x f(x) β = 0 β = 3 β = f 1 (x) f(x) β = 0 β = 3 β = f 1 (x) (a) Previous mehod (b) Proposed mehod Figure 9: Pareo se comparison for crashworhiness RBMO problem. Table 3: Compuaional performance of crashworhiness RBMO problem. Targe Previous Proposed reliabiliy β Number of evaluaions Raio Number of evaluaions Raio Pareo se. The oal number of funcion evaluaions and he average number of evaluaions per each paricle in each ieraion are compared in Table 3. Though he improvemen is no so large in his example, he new mehod slighly reduces he number of funcion evaluaions. 7

8 d i : Cross-secional area l i : Elemen lengh P 1 P P1 = P = 100 Figure 10: Ten-bar russ design problem Table 4: Formulaion of 10-bar russ design problem [4]. (a) Design and random variables Design variables d i Cross-secional area 0.1 d i 5, (i = 1,, 10) Tensile srengh N(5,.5), (i = 1,, 10) Random variables x i Compression srengh N( 5,.5), (i = 11,, 0) Applied load N(100, 10), (i = 1, ) (b) Reliabiliy consrains Tensile P(x i σ i 0) Φ( β ), (i = 1,, 10), (if σ i 0) Compression P(σ i x i 0) Φ( β ), (i = 11,, 0), (if σ i 0) bar russ design problem This example is exended from a ypical 10-bar russ design problem as shown in Fig. 10 ha is known as a singleobjecive benchmark problem [13]. The member cross-secional areas are reaed as design variables, d i, (i = 1,, 10) and he wo nodal applied loads and member allowable sresses are adoped as independen normal disribued random variables as lised in Table 4 (a). The wo objecive funcions is defined o minimize he oal volume and he ip displacemen. The reliabiliy consrains are formulaed ha he reliabiliy which he member sress does no exceed he allowable sress should be higher han he arge reliabiliy as lised in Table 4 (b). This RBMO problem consiss of he wo objecives and 0 reliabiliy consrains wih 10 design variables and random variables. Oher properies such as he member lengh, l i, (i = 1,, 10), and Young s modulus, E = 10 5, are reaed as deerminisic values. Noe ha his problem adops he deerminisic design variables, ha is, µ, he mean value of random variables in he second argumen of Eq. (4) does no correspond o design variable. The concep of SLSV is applicable even in his case, hough he design poin z j described in Fig. is no rigorously correc. In MOPSO, he number of paricles is se as 100, he number of ieraions 00 and γ = 4. Fig. 11 compares he Pareo se under he condiions ha he arge reliabiliy indices β are se as 0, 3, and 6 beween he previous and he new mehods. The new mehod achieves he diversiy of Pareo se. On he oher hand, he previous mehod does no obain he lower righ region of he Pareo se. The lower righ region of Pareo se corresponds o design wih smaller ip displacemen bu larger volume as shown in Fig. 1 (a). As his kind of design does no have acive reliabiliy consrains, he difference beween he arge reliabiliy indices does no appear. On he oher hand, he upper lef region of Pareo se is much differen from he above design. The configuraion is shown in Fig. 1 (b), where he design wih minimum volume bu has larger ip displacemen. As he reliabiliy consrains are acive around his region, he arge reliabiliy has significan effec on he Pareo soluions. Boh configuraions are no suiable from he viewpoin of mechanical engineering. The design configuraion corresponding o he cenral area in he Pareo curve is shown in Fig. 1 (c). This configuraion has a good balance beween he volume and he ip displacemen and is similar o he ypical design of he single-objecive opimizaion problem ha minimize he volume under he ip displacemen and member sress consrains. Finally, he compuaional effor is compared in Table 5. I is found ha he new mehod reduces he number of funcion evaluaions as much as abou 60% of he previous mehod in his example. 8

9 ip displacemen f(x) β = 0 β = 3 β = 6 ip displacemen f(x) β = 0 β = 3 β = f volume x (x) f volume x (x) (a) Previous mehod (b) Proposed mehod Figure 11: Pareo se comparison for crashworhiness RBMO problem. (a) Displacemen minimizaion design (b) Volume minimizaion design (c) Inermediae design Figure 1: Typical Pareo opimum configuraions of 10-bar russ. Table 5: Compuaional performance of 10-bar russ RBMO problem. Targe Previous Proposed reliabiliy β Number of evaluaions Raio Number of evaluaions Raio Conclusion This sudy proposes he improvemen mehod of he hybrid-ype MOPSO ha he auhors developed for he muliobjecive opimizaion formulaed as coninuous design variables and he RBMO [10 1]. The hybrid MOPSO inegraes he consrain saisfacion echnique using gradien informaion of consrains. The consrain saisfacion echnique consiss of wo funcions: moving he design candidae wih consrain violaion o he feasible region using sensiiviy informaion of he violaed consrains and o he feasible boundary using bi-secion mehod. Some design candidaes require huge numbers of ieraions unil he candidae moves o feasible region, when he violaed consrain condiions are shifed frequenly during he consrain saisfacion process. Because he original mehod only uses he sensiiviy a he curren posiion, In order o eliminae such zigzag ieraions, an idea from he modified SLSV mehod or he conjugae mean value mehod [13, 14] is applied. The improved mehod uses sensiiviies a he previous posiions o improve he moving direcion o he feasible regions. The efficiency of he proposed mehod is demonsraed hrough several design problems by invesigaing he convergence and diversiy of finding Pareo opimal soluions. References [1] K. Deb, A. Praap, S. Agarwal and T. Meyarivan, A Fas and Eliis Muliobjecive Geneic Algorihm:NSGA- II, IEEE Trans. Evol. Compu., 6(), , 00. [] M. Reyes-Sierra and C. A. Coello Coello, Muli-Objecive Paricle Swarm Opimizers: A Survey of he Sae-of-he-Ar, In. J. Compu. Inell. Res., (3), ,

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