Optimization of Multi-Pass Turning Operations Using Genetic Algorithms

Transcription

1 Poceeing o he 2012 Inenaional oneence on Inuial Engineeing an Opeaion Managemen Ianbul, ukey, July 3 6, 2012 Opimizaion o Muli-Pa uning Opeaion ing Geneic Algoihm Ami Ayana an Maghu Solimanpu Faculy o Engineeing mia niveiy mia, We Azebaijan Povince, Ian Abac uing opeaion ae one o he mo popula machining opeaion in he inuial manuacuing yem. hee opeaion coni o wo age, namely oughing an inihing. In he inihing age, machining paamee incluing cuing pee, ee ae an eph o cu ae o be eemine. In ouging age, in aiion o hee paamee, he numbe o ough cu pae houl be ecie. hi pape popoe a non-linea conaine mahemaical pogamming moel o eeminaion o he aoemenione paamee. An opimizaion echnique bae on geneic algoihm (GA) i popoe o imulaneouly opimize he muli-pa oughing an ingle-pa inihing paamee. An illuaive example aope om he lieaue i olve hough he popoe algoihm o how he eiciency an eecivene o he popoe meho. Keywo: Muli-pa uning, onain, Opimizaion, uing paamee, Geneic Algoihm Nomenclaue ni poucion co excep maeial co ($/piece) M uing co by acual ime in cu ($/piece) I Machine ile co ue o loaing an unloaing opeaion an ool ile moion ime ($/piece) R ool eplacemen co ($/piece) ool co ($/piece), uing pee in ough an inih machining (m/min) L, Lowe an uppe boun o cuing pee in ough machining (m/min) L, Lowe an uppe boun o cuing pee in inih machining (m/min), Fee ae in ough an inih machining (mm/ev) L, Lowe an uppe boun o ee ae in ough machining (mm/ev) L, Lowe an uppe boun o ee ae in inih machining (mm/ev), Deph o cu o each pae o ough an inih machining (mm) L, Lowe an uppe boun o eph o cu in ough machining (mm) L, Lowe an uppe boun o eph o cu in inih machining (mm) n Numbe o ough cu (an inege) Deph o maeial o be emove (mm) DL, Diamee an lengh o wok-piece (mm) k o Diec labo co + ovehea ($/min) k uing ege co ($/ege) 1560

2 m, m, m Rough, inih an acual machining ime (min), e h, h 1 2 ool exchange an ool eplacemen ime (min) onan elae o ool avel an appoach/epaue ime (min),, ool lie, expece ool lie o ough machining, expece ool lie o inih machining (min) p, L ool lie o weighe combinaion o an (min), ppe an lowe boun o oo lie (min), pqonan,,, o o ool lie equaion, F, F F k, µν, 1 P, P P λυ, S R SR uing oce uing ough an inih machining (kg), Maximum allowable cuing oce (kg), onan o cuing oce equaion, uing powe uing ough an inih machining (kw), Maximum allowable cuing powe (kw), onan elae o expeion o able cuing egion, Limi o able cuing egion conain, Noe aiu o cuing ool (mm), Maximum allowable uace oughne (mm), Q, Q hip-ool ineace ough an inih machining empeaue ( ), Q Maximum allowable chip-ool ineace empeaue ( ), k,,, 2 τϕδ onan elae o equaion o chip-ool ineace empeaue, k3, k4, k 5 onan o oughing an inihing paamee elaion k3, k4, k Inoucion o uvive in oay compeiive manuacuing envionmen, poucion o pouc wih a goo qualiy in a hoe ime an wih lowe co i eenial. he elecion o opimal cuing paamee i a vey impoan iue o evey machining poce. o enue he qualiy o machine uace an euce he machining co an ime, i i ineviable o opimally elec he machining paamee like he numbe o pae, eph o cu o each pa, ee ae an cuing pee. In manuacuing pocee, he mo commonly ue opimizaion cieia i he uni poucion co uncion, which ha been eal by many eeache. Ohe impoan cieia ae uch a uace qualiy, maeial emoval ae, an ool lie. In hi pape uni poucion co uncion i coniee a he objecive uncion o be opimize an an opimizaion echnique bae on geneic algoihm i popoe o olving he machining opimizaion poblem concening he muli-pa uning opeaion. Machining opimizaion poblem i inveigae in wo moe, ingle-pa an muli-pa. Howeve, moly muli-pa moe i peee ove inglepa moe becaue o economic eaon. hi opimizaion poblem ha been aempe by many eeache. Howeve, ealy uie wee limie o ingle-pa moe wihou conieaion o egula conain which gave a ubnomal ace o he poblem. Shin an Joo [1] evelope a mahemaical moel o muli-pa uning opeaion. hei moel i equenly benchmake an exene in ohe uie. hey povie a numeical example an popoe an appoach o olving i ha combine he Fibonacci each an ynamic pogamming. Aewa, Gupa e al. [2] olve he ame poblem in wo ep. he i ep o hei appoach i o calculae he minimum poucion co o he oughing an inihing opeaion o he vaiou eph o cu. he econ ep i o eemine he opimal combinaion o eph o cu o oughing an he inihing opeaion. hei appoach eemine a poucion co, which i ubanially le han when uing Shin an Joo meho. Albei an Peone [3] uggee a muli-objecive poibiliic pogamming moel in which he opimal oluion i obaine uing a geneic algoihm. hey compae hei eul wih he eul obaine by Gupa e al. [2] o he ame example. Alhough hei eul wee neve woe han hoe obaine by Gupa e al., in wo o he ix example hei oluion howe a vey iny violaion o he cuing oce conain. hen an ai [4] exene he muli- 1561

3 pa uning opeaion moel o Shin an Joo [1] by aing even moe conain. hey alo popoe an appoach combining he imulae annealing algoihm an a paen each echnique o olving he exene moel. Onwubolu an Kumalo [5] popoe a echnique bae on a geneic algoihm o eemine he opimal machining paamee o he exene moel o hen an ai. hey aive a an even lowe poucion co han ha obaine by hen an ai [4]. Howeve, he opimal value obaine by Onwubolu an Kumalo wa ague impacical by hen an hen [6]. hey poine ou ha Onwubolu an Kumalo i no limi he numbe o ough cu o an inege value. Fo he ame poblem, ijayakuma e al. [7] popoe an an colony opimizaion meho an claime ha hei an colony-bae appoach oun an even bee oluion wih le un han eihe he appoach o hen an ai o he GA-bae appoach o Onwubolu an Kumalo. Howeve, hei oluion wa lae poven o be invali [8]. In hi pape an opimizaion echnique bae on geneic algoihm ha been aope o opimize he muli-pa uning opeaion, ubjec o eveal conain povie by he machining moel o hen an ai [4]. Heeae, he pape i oganize a ollow. Secion 2 ecibe he mahemaical omulaion o he aempe poblem. Secion 3 biely ecibe he popoe hybi geneic algoihm. An illuaive example i olve an icue in Secion 4 o how he applicabiliy an valiiy o he omulae moel. 2. Muli-Pa uning Moel he main objecive o he peen opimizaion cuing moel i o eemine he opimal machining paamee incluing cuing pee, ee ae, eph o cu an numbe o ough cu in oe o minimize he uni poucion co an imulaneouly achieve an accepable uace inih, wihou violaing any impoe cuing conain. he mahemaical moel ue by hen an ai [4] i aope in hi wok o eemining he opimal machining paamee menione above o uncion he uni poucion co,, excep maeial co, o he muli-pa uning opeaion poblem coni o ou baic co componen [1]. 1. Machining co by acual ime in cuing opeaion, M. 2. Machine ile co ue o loaing an unloaing opeaion an ile ool moion, I. 3. ool eplacemen co, R. 4. ool co,. hen an ai [4] coniee all he ou baic co componen icue by Shin an Joo [1]. he peen wok conie hee baic co componen becaue hey eem o be aequae in moeling he muli-pa uning opeaion poblem. hee componen ae now ummaize. Fo moe eail, ee hen an ai [4] Machining co he cuing poce i ivie ino muli-pa oughing an inihing [1]. he machining co, M, i expee a: M = k (1) Machine ile co he machine ile co i ivie in a conan em ue o loaing an unloaing opeaion an a vaiable em ue o ile ool moion [1] expee a: I = k0 c + ( hl 1 + h2) ool eplacemen co he aylo ool-lie equaion i ue a given in Amaego an Bown [9]: (2) 0 =. p q (3) 1562

4 Owing o he ieen machining coniion, he wea ae o ool uually ie beween oughing an inihing. he ool lie in uch a cae can be expee a in [1]: Whee = θ + (1 θ), (4) p o o =, =. (5) p q p q he ool eplacemen ime can be wien in em o ool lie ( p ), ime equie o exchange a ool ( e ) an cuing ime [1]. I can be omulae a: e R = ko p. (6) ool co he ool co can be omulae a ollow: k = p. (7) ni poucion co ing equaion (1), (2), (6) an (7), he uni poucion co,, i eine a: = M + I + R + = ko e + ko c + ( hl 1 + h2) ko + p k p. (8) 2.2. Suace qualiy uncion he uace qualiy uncion i ecibe a [10]: R kv = (9) x1 x2 x 3 a whee x 1, x 2, x 3 an k ae he conan elevan o a peciic ool wokpiece combinaion. Since eaching an ieal uace inih i no neceaily equie in inuy, o in hi pape he uace qualiy uncion i eae a a conain an heeoe conain aiacion meho i aope in hee uing coniion conain Seveal cuing conain ae coniee uing oughing an inihing opeaion a ollow: 1. Paamee boun 2. ool-lie conain 1563

5 3. uing oce conain 4. Powe conain 5. Suace inih conain 6. Sable cuing egion conain 7. hip-ool ineace empeaue conain 8. Suace inih conain (o inihing age only) 9. Roughing an inihing paamee elaion. 10. Deph o cu equaliy conain Final cuing moel he opimizaion moel o muli-pa uning opeaion i omulae a: Subjec o F( X ) Min = (10) Roughing: L (11) L (12) L (13) L (14) ( ) µ ( ) ν F = k1 F (15) P F = P (16) 6120η Q = k ( )( )( ) Q (17) τ φ δ 2 u λ ν S (18) Finihing: L (19) L (20) L (21) L (22) ( ) µ ( ) ν F = k1 F (23) 1564

6 F P = P (24) 6120η Q = k ( )( )( ) Q (25) τ φ δ 2 u λ υ S (26) 2 SR 8R (27) Relaion: k (28) 3 k (29) 4 5 k (30) Moeove, he oveall eph o cu in ough pae plu eph o inih cu mu be equal o he oal eph o maeial o be emove: A a eul, he numbe o ough pae can be expee a: n + = (31) n = (32) I i o be noe ha numbe o cu (n) mu be an inege value. he cuing oce conain o equaion (15) an he powe conain o equaion (16) ollow he omulaion o Shin an Joo [1]. he chip-ool ineace conain o equaion (17) ollow he omulaion o Hai an Rao [11]. he able cuing egion conain o equaion (18) ollow he omulaion o Naang an Fiche [12]. 3. Geneic Algoihm Geneic algoihm ae a amily o compuaional moel ha ha been popoe by Hollan in 1975 in he ligh o Dawin heoy o evoluion. hee algoihm encoe a poenial oluion o a peciic poblem on a imple aa ucue (chomoome) an apply geneic opeao o hee ucue o a o peeve ciical inomaion. An implemenaion o a geneic algoihm begin wih a populaion o (ypically anom) chomoome. One hen evaluae hee ucue an allocae epoucive oppouniie in uch a way ha hoe chomoome which epeen a bee oluion o he age poblem ae given moe chance o epouce han hoe chomoome which ae ineio oluion. he goone o a oluion i ypically meaue hough a ine uncion [10]. 3.1 Repeenaion o oluion he opimizaion poblem moele in peviou ecion ha hee inepenen vaiable o oughing poce (,, ) an hee inepenen vaiable o inihing poce (,, ). hee 6 vaiable ae epeene epecively wih ( x1, x2, x 3) an ( x4, x5, x 6) gene in binay moe. 3.2 Scaling uncion he caling uncion conve aw ine coe eune by he ine uncion o value in a ange ha i uiable o he elecion uncion. he popoe geneic algoihm ue ank uncion o caling uncion. Rank uncion cale he aw coe bae on he ank o each iniviual, ahe han i coe. he ank o an iniviual i i 1565

7 poiion in he oe coe. he ank o he ie iniviual i 1, he nex ie i 2, an o on. Rank ine caling emove he eec o he pea o he aw coe. 3.3 oove he popoe geneic algoihm ue caee uncion o coove, which ceae a anom binay veco, hen elec he gene om he i paen wheeve he veco i 1, an he gene om he econ paen wheeve he veco i 0, an combine he gene o om he chil. 3.4 Muaion Muaion bing unexpece eaue o he chilen ha o no exi in paen. Each gene i choen o muaion wih a pobabiliy o pm which i a GA paamee. he popoe geneic algoihm ue uniom uncion o muaion, which i a wo-ep poce. Fi, he algoihm elec a acion o he veco enie o an iniviual o muaion, whee each eny ha he ame pobabiliy a he muaion ae o being muae. In he econ ep, he algoihm eplace each elece eny by a anom numbe elece uniomly om he ange o ha eny. 3.5 Selecion he elecion uncion chooe paen o he nex geneaion bae on hei cale value om he ine caling uncion. In he popoe algoihm, emaine uncion i ue which aign paen eeminiically om he inege pa o each iniviual' cale value an hen ue oulee elecion on he emaining acional pa. 3.6 Implemenaion he elemen o he popoe geneic algoihm have been implemene in he geneic algoihm oolbox o he MALAB Sowae an un on a Penium 4 P wih 3.0 GHz Inel Poceo an 2 GB o RAM. he value e o ieen paamee o he geneic algoihm ae hown in able 1. able 1 Paamee o he GA Paamee alue Populaion ize 100 Elie coun 4 oove pobabiliy (%) 25 Muaion pobabiliy (%) 1 4. Illuaive Example an Dicuion An illuaive example ha been aope om [4] o emonae he peomance o he popoe appoach. able 2 how he aa o he illuaive example. Paamee/ conain alue able 2 uing moel aa o hen an ai [4] Paamee/ Paamee/ alue alue conain conain Paamee/ conain D 50 mm L 300 mm * m/min L alue 50 m/min 500 m/min L 50 m/min 0.9 mm/ev L 0.1 mm/ev 0.9 mm/ev L 0.1 mm/ev 3.0 L L 1.0 p 5 q η 0.85 k $/min k k k k 2.5 $/ege θ mm k k min 25 min max 45 min F max 200 kg. P max 5 KW 1566

8 µ 0.75 ν 0.95 SR 10 µ m R 1.2 mm λ 2 υ -1 S 140 Q u 1000 τ 0.4 φ 0.2 δ c h 4 1 7*10 h p 0.75 min/piece e 0.75 min/piece 1.5 mm/ege hi example poblem i a well-known e cae in hi iue an i equenly benchmake by ohe eeache o evaluae he eiciency o hei popoe algoihm. able 3 how a compaion o he minimum uni poucion co achieve by he popoe geneic algoihm an ohe opimizaion meho epoe in he lieaue incluing SA/PS [4], GA [6], AO [7] an PSO [13]. Alhough AO-bae appoach o ijayakuma [7] achieve a value equal o , bu i ongly violae he cuing oce conain an he powe conain boh in he ough pa o hei oluion i invali. GA-bae echnique o Onwubolu an Kumalo [6] aive a a value o bu hei opimal value i alo ongly aece by many conain violaion [13]. he popoe GA inue a 6.26% eucion in compaion wih he PSO an 7.23% eucion in compaion wih he SA/PS appoach aope by hen an ai [4]. able 4 illuae he opimal cuing paamee obaine by he popoe GA. able 3 Opimum obaine by vaiou echnique Meho Popoe GA PSO AO GA SA/PS Opimum uing paamee Opimal value obaine by popoe GA able 4 Opimal cuing paamee obaine by popoe GA oncluion In hi pape, an opimizaion echnique bae on geneic algoihm i applie o olve he machining opimizaion poblem concening he muli-pa uning opeaion. he uni poucion co i coniee a an objecive uncion o be opimize bae on a amou mahemaical moel which i ubjec o eveal pacical conain incluing eph o cu equaliy conain. hen a well-known illuaive example ha been aope om lieaue o emonae he applicabiliy o he popoe GA. ompaing he eul obaine by he popoe GA wih ohe aiional an non-aiional echnique claiie ha he popoe GA ovecome ohe meho popoe by ohe eeache. Fo uue eeach, he popoe echnique can be moiie o muli-objecive conaine opimizaion poblem in uning an alo ohe machining poblem, uch a milling an gining opeaion. Reeence [1] Shin, Y,., an Joo, Y.S., 1992, Opimizaion o machining coniion wih pacical conain, Inenaional Jounal o Poucion Reeach, 30, [2] Gupa, R., an Baa, JL., an Lal, GK, 1995, Deeminaion o opimal ubiviion o eph o cu in mulipa uning wih conain, Inenaional Jounal o Poucion Reeach, 33, [3] Albei, N., an Peone, G., 1999, Muli-pa machining opimizaion by uing uzzy poibiliic pogamming an geneic algoihm, Poceeing o he iniuion o mechanical enginee pa B, 213,

9 [4] hen, M-., an ai, D-M., 1996, A imulae annealing appoach o opimizaion o muli-pa uning opeaion, Inenaional Jounal o Poucion Reeach, 34, [5] Onwubolu, G., an Kumalo,., 2001, Muli-pa uning opeaion opimizaion bae on geneic algoihm, Poceeing o he Iniuion o Mechanical Enginee Pa B, 215, [6] hen, M-., an hen, K-Y. 2003, Opimizaion o muli-pa uning opeaion wih geneic algoihm: a noe, Inenaional Jounal o Poucion Reeach, 41, [7]. ijayakuma, K., Pabhahaan, G., Aokan, P., an Saavanan R., 2003, Opimizaion o muli-pa uning opeaion uing an colony yem, Inenaional Jounal o Machine ool an Manuacue, 43, [8] Wang, Y-., 2007, A noe on opimizaion o muli-pa uning opeaion uing an colony yem, Inenaional Jounal o Machine ool an Manuacue, 47, [9] ARMAREGO, E. J. A., an BROWN, R. H., 1969, he Machining o Meal (Englewoo li: Penice-Hall). [10] u, F., an Balic, J., 2003, Opimizaion o uing Poce by GA Appoach, Roboic an ompue Inegae Manuacuing, ol. 19, pp [11] HAI, S. K. an RAO, S. S., 1976, Deeminaion o opimum machining coniion-eeminiic an pobabiliic appoache, Jounal o Engineeing o Inuy, anacion ASME, 98, [12] NARANG, R.., an FISHER, G.W., 1993, Developmen o a ame wok o auomae poce planning uncion an eemine machining paamee, Inenaional Jounal o Poucion Reeach, 31, [13] Siniva, J., Gii, R., an Yang, S-H., 2009, Opimizaion o muli-pa uning uing paicle wam inelligence, Inenaional Jounal o Avance Manuacuing echnology, 40,