COST OPTIMIZATION OF SLAB MILLING OPERATION USING GENETIC ALGORITHMS

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1 COST OPTIMIZATIO OF SLAB MILLIG OPERATIO USIG GEETIC ALGORITHMS Bhavsa, S.. ad Saghvi, R.C. G H Pael College of Egieeig ad Techology, Vallah Vidyaaga , Aad, Gujaa sake976@yahoo.co.i; ajeshsaghvi@gce.ac.i Asac I ode o opimize he ojecive fucio fo machiig, oe eeds o deemie cuig paamees, such as speed, feed ad deph of cu. Hee, i is peseed he use of Geeic Algoihms (GAs) o miimize he cos fo sla millig opeaio o hoizoal millig machie. The deph of cu ad feed values ae se o sui he machie codiio ad suface fiish equiemes, while cuig velociy is opimized fo miimum cos of poducio. A coecio faco is icopoaed o GAs fo coecig he esuls of opimum cuig speed ad opimum cos of poducio. A he ed he cos opimum cuig speed ad opimum cos geeaed y GAs ae compaed wih he esuls geeaed y adiioal opimizaio mehod of usig Taylo s equaio of ool life, fo is validaio. Compaiso of esuls evolves he efficiecy of GAs fo fidig he cos opimum cuig paamees. Keywods: Geeic Algoihms, Sla Millig, Cos Opimizaio.. ITRODUCTIO Of vial iees o he maufacuig egiee ae poducio cos ad poducio aes. Alhough, i pacice a high poducio ae would poaly mea low poducio coss, i should e poied ou ha hese wo facos mus e cosideed sepaaely ad ha he maufacuig codiios givig maximum poducio ae will o e ideical o hose codiios givig miimum cos of poducio. The machiig ecoomics polem (shaw, M.C.; Lissama, A.J. ad Mai, E.J.; Chapma, P.C.) cosiss i deemiig he pocess paamees, usually he cuig speed, feed ae ad he deph of cu, i ode o opimize he ojecive fucio. Moeove, i deemiig hese paamees, special aeio is usually give o he esicios o he cosais imposed o he paicula opeaio y he machie ool, he cuig ool ad he wok piece. Fo he puposes of he discussio i is ecessay o explai ha feed is he disace moved y he ool elaive o he wok piece i he feed diecio fo each evoluio of he ool o he wok piece o each soke of he ool o wok piece. Cofusio may aise i ceai mulipoi ool opeaios, such as millig, whee he feed seig o he machie efe o he elaive speed ewee he ool axis ad he wok piece i he feed diecio (he feed speed). Thus, if he feed speed i he millig opeaio is v f ad he oaioal fequecy of he ool is, he wok piece feed duig each evoluio of he cue is give y v f /, ad he maximum cuig speed v i a millig opeaio is give y πd, whee d is he ool diamee. I ow follows ha if i is equied o doule he cuig speed i a millig opeaio while keepig he feed cosa, i would e ecessay o doule oh he oaioal fequecy of he cue ad he feed speed v f (Boohoyd, G.). A ume of machiig opimizaio appoaches have ee epoed, some of hese appoaches wee limied o sigle pass opeaios (Wag, J. e al.; Kiov, K.P. ad Hisov, H.I.). The paamees usually opimized i a sigle pass ae he cuig speed ad he feed ae. The ohe appoaches cosideed he muli pass opeaios (Some, A.I. e al.; Owuolu, G.C. ad Kumalo, T.; Amiolemhe, P.E. ad Ihadode, A.O.A.). I he muli pass opeaios, he paamees o e opimized ae he deph of cu o he ume of pass, he cuig speed ad he feed ae. Owuolu e al. shows ha Geeic Algoihms (GAs) is mos suiale fo he opimizaio of machiig paamees. Simulaed Aealig is also used o fid ou he opimum machiig paamees i muli pass opeaios (Saavaa, R. e al.). I his wok sla millig opeaio is seleced fo sigle pass opeaio. GAs wee applied o fid ou he cuig velociy, which gives miimum cos of poducio fo give feed ae ad deph of cu. A he same ime he esuls of opimizaio geeaed y GA ae compaed wih he esuls oaied y ouie aalyical mehod of opimizaio which uses Taylo s equaio of opimum ool life. 2. ROUTIE METHOD OF COST OPTIMIZATIO 2. Choice of Feed

2 I is kow ha he coec feed o use i oughig opeaios is he highes he machie ool ca wihsad i ems of ool foces ad powe cosumpio. Whe a fiishig cu is o e ake, he appopiae feed will e ha which will give a accepale suface fiish. 2.2 Choice of Cuig Speed 2.2. Taylo s Equaio fo Choice of Cuig Speed I choosig he cuig speed fo a machiig opeaio, we shall follow he appoach descied y Boohoyd, G. Hee, he ime spe y he opeao ad his machie i poducig a ach of compoes ca e sepaaed io hee iems:. The opoducive ime, give y l, whee l is he ime ake o load ad uload each compoe ad o eu he ool o he egiig of he cu. 2. The oal machiig ime, give y m, whee m is he machiig ime fo he compoe. 3. The ime ivolved i chagig wo ools, give y c, whee c is he ool chagig ime, ad is he ume of ools used. Thus, if M is he oal machie ad opeao ae (icludig oveheads), he oal machie ad opeao coss will e M + + ) ( ) ( l m c To his cos, mus e added he cos of he ools used, give y C, whee C is he cos of each ool. The aveage poducio cos C p fo each compoe ca ow e wie C p = M l + M m + M c + C ( 2 ) The fis iem i he expessio is he opoducive cos, which is cosa fo he paicula opeaio. The secod iem is he machiig cos, which is educed as he cuig speed is iceased a cosa feed. The fial iems ae he ool coss, which icease as he cuig speed iceases. To calculae he ume of ools used i poducig he ach of compoes, i is ecessay o kow he elaioship ewee he cuig speed ad ool life. The wok of Taylo showed ha a empiical elaioship exiss ewee hese vaiales, amely, v = v ( 3 ) whee v = cuig speed, = ool life, = cosa, = measued ool life fo a cuig speed v. The value of may e foud fo a paicula wok piece ad ool maeial ad a paicula feed eihe y expeime o fom pulished empiical daa. The idex depeds maily o ool maeial; fo high speed seel 0.25, fo caide 0.25 < < 0.3, ad fo ceamics 0.5< < 0.7. Figue gives he appoximae ages fo he values of v fo vaious ool ad wok maeials whe he ool life is 60s. The ool life fo a paicula siuaio is heefoe give y v = ( 4 ) v I should e oed ha, adiioally, he Taylo ool life equaio has ee applied i he fom v = C ( 5 ) The ume of ools used i machiig he ach of compoes is give y / assumig ha he ool is egaged wih he wok piece duig he eie machiig ime. Thus, m m v = = ( 6 ) v Fially, he machiig ime fo oe compoe is give y K m = ( 7 ) v m 2

3 Fig.. Appoximae values of he cuig speed v whe ool life = 60 sec (Boohoyd, G.) whee v is he cuig speed ad K is a cosa fo he paicula opeaio. I sla millig opeaio, he value of K will e give y π d l / f, whee l is he legh o e aveled y cue, d is he diamee of he cue ad f is he feed. The elaioship ewee he poducio cos ad he cuig speed ca ow e oaied y susiuio of equaios ( 6 ) ad ( 7 ) i equaio ( 2 ): K K ( ) ( ) C p = M l + M + M c + C v ( 8 ) v v To fid he cuig speed v c fo miimum cos, equaio ( 8 ) mus ow e diffeeiaed wih espec o v ad equaed o zeo. Thus M vc = v M c C ( 9 ) Tool Life Deemiaio I aalyzig pacical machiig opeaios i is coveie o employ expessio fo he opimum ool life fo miimum cos c. This expessio ca e oaied y susiuio of equaio ( 9 ) i Taylo s ool life equaio ( 4 ). Thus, C c = c + ( 0 ) M Fially he coespodig opimum cuig speeds ca e foud fom 3

4 vc = v Q ( ) c The value of Q fo sla millig ca e oaied fom geomeies show i figue 2. Thus fo sla millig, θ 2 ae Q = = + acsi ( 2 ) 2π 4 2π d whee a e is he wokig egageme ad d is he ool diamee. 3. Geeic Algoihms (GAs) Philosophically, GAs ae ased o Davi s heoy of suvival of he fies. GAs ae ased o he piciples of aual geeics ad aual selecio. The asic elemes of aual geeics epoducio, coss ove, ad muaio ae used i he geeic seach pocedue (Rao, S.S.) Iiially a populaio of pois is used fo saig he pocedue isead of a sigle desig poi. Repoducio is a pocess i which he idividuals fom iiial populaio ae seleced ased o hei fiess values elaive o ha of he populaio. I his pocess each idividual sig is assiged a poailiy of eig seleced fo copyig. Thus idividuals wih highe fiess values have geae chace of eig seleced fo maig ad suseque geeic acio. Cosequely, highly fi idividuals live ad epoduce, ad less fi idividuals die. Afe epoducio, he coss ove opeaio is implemeed i wo seps. Fis, wo idividual sigs ae seleced a adom fom he maig pool geeaed y he epoducio opeao. ex, a coss ove sie is seleced a adom alog he sig legh, ad he iay digis ae swep ewee he wo sigs followig he coss ove sie. The ew sigs oaied fom he coss ove (off-spigs) ae placed i he ew populaio ad he pocess is coiued. Fially, a muaio opeao is applied o he ew sig wih a specified muaio poailiy. A muaio is he occasioal adom aleaio of a iay digi. Thus, I muaio a 0 is chaged o, ad vice vesa, a a adom locaio. 4. Implemeaio The pocedue explaied i secio 3 fo GAs is implemeed i C++ laguage fo he followig cuig codiios o a Hoizoal Millig machie fo sla millig opeaio. Wok piece Maeial = Cas Io ( Tesile Segh = 425 M/m 2 ) Cuig Tool Maeial = High Speed Seel Cue Diamee, d = m Legh of Wok piece = 0. m Legh o e Taveled y Cue, l = 0.5 m Feed Rae, f = 0.0 mm/ev M = Rs. 4/- pe mi l = 80 sec = 60 sec (figue ) = 300 sec c v = 3.5 m/s (figue ) C = Rs. 00/- pe cuig ool =

5 Fig. 2. Sla millig opeaio Accodig o he machie oleale limi ad suface oughess equiemes, values of deph of cu ad feed ae have ee seleced fo machiig. Hee cuig speed is equied o e opimized fo he opeaio meioed aove. The Geeic Algoihms is plaed o ake iiial populaio of e diffee speed values fom he age of pm o 4095 pm. Fo his pupose a oal of 2 digi iay ume is geeaed fo each speed seleced i a geeaio, i.e. 2. Afe geig he values of cuig speeds, we eed o calculae he oal cos of poducio fo idividuals ( fx () wih he help of equaio (8). As GAs ae used o oai he maximum value, ou polem is coveed o maximizaio y suacig each cos value fom he maximum cos value ad gives diffeece of f (. ow o fid ou poailiy of idividual cuig speeds, each coss fo e speed values ( ) ( f ( ) is divided y f (. The cumulaive poailiy is deemied fo epoducio fom he iiial populaio. Two poi coss ove opeao is applied o his epoducio pool fo swappig he sigs. Afewads he sigs ae muaed wih poailiy of This complees oe geeaio. Tale shows he esul of sixh geeaio. Tale2 shows he opimum esul of each geeaio. The values of velociy i ale2 does o coside he effec of paamee Q =0. 4 (Boohoyd, G.), which mus e cosideed fo sla millig opeaio accodig o equaio (). Hece he coeced opimum velociy fom ale2 is, = = m/s 0.25 Q 0.4 ow he coeced miimum cos ca e calculaed y puig his value of cuig velociy i equaio (8), which is Rs. S. Biay Values Speed Cos Cos Po. Cum. Radom Sig o. of o. of Speed RPM fx ( diffeece Po. o. o. Sigs Rs f ( Rs Tale. GAs Resul A Sixh Geeaio S. o. Populaio afe Repoducio Sig o. fo Coss s sie of Coss 2 d sie of Coss Resul of Coss ove Resul of Muaio 5

6 ove ove ove Tale. Coiued Geeaio o. Miimum Cos Rs. Velociy m/s Geeaio o. Miimum Cos Rs. Velociy m/s Tale 2. Resuls of Twey Geeaios If we follow he ouie mehod of opimizaio wih Taylo s equaio, he fis we eed o deemie fom equaio (0). This is he opimum ool life fo cos miimizaio. ow fom equaio () c calculae v c, which is he opimum cuig velociy fo cos miimizaio. This value of cuig velociy gives he miimum poducio cos whe susiued i equaio (8). The values calculaed wih he help of Taylo s equaio ae as follows, which ca e compaed wih he esul of Geeic Algoihms. =2,600sec c v c = m / s C p = Rs. 5. Coclusio Difficulies aise whe he opimizaio pocess ivolves may paamees ha ieac i highly oliea way. I his siuaio ume of cosais ove he polem may also e quie high. These kid of polems ca e efficiely solved y heuisic mehods like Geeic Algoihms ad Simulaed Aealig. 6. Refeeces Amiolemhe, P. E. ad Ihadode, A. O. A. (2004). Applicaio of Geeic Algoihms deemiaio of he opimal machiig paamees i he covesio of a cylidical a sock io a coiuous fiished pofile. Ieaioal Joual of Machie Tools ad Maufacue, 44, Boohoyd, G. (985). Fudameals of meal machiig ad machie ools. McGaw-Hill Book Compay Ic. Chapma, P. C. (2002). Poducio egieeig echology. Taa McGaw-Hill Pulishig Compay Limied. Kiov, K. P. ad Hisov, H. I. (2002). Sysem fo cuig codiios opimizaio of machiig. Takya Uivesiesi ilimsel alasimala Degisi B. seisi Cil 3(2), Lissama, A. J. ad Mai, E. J. (972). Piciples of egieeig poducio. The Eglish Uivesiies Pess Limied, Lodo. 6

7 Owuolu, G. C. ad Kumalo, T. (2002). Muli pass uig opimizaio ased o Geeic Algoihms. Ieaioal Joual of Poducio Reseach, 39(6), Rao, S. S. (996). Egieeig opimizaio : heoy ad pacice. ew Age Ieaioal (P) Limied, ew delhi. Saavaa, R. e al. (2003). Machiig paamees opimizaio fo uig cylidical sock io a coiued fiished pofile usig Geeic Algoihms ad Simulaed Aealig. Ieaioal Joual of Advace Maufacuig Techology, 2, -9. Shaw, M. C. (999). Meal cuig piciples. CBS Pulishes ad Disiuos, ew Delhi. Somez, A. I. e al. (998). Dyamic opimizaio of muli pass millig opeaio via geomeic pogammig. Ieaioal Joual of Machie Tools ad Maufacue, 39, Wag, J. e al. (2002). Opimizaio of cuig codiios fo sigle pass uig opeaios usig a deemiisic appoach. Ieaioal Joual of Machie Tools ad Maufacue, 42,

Supplementary Information

Supplementary Information Supplemeay Ifomaio No-ivasive, asie deemiaio of he coe empeaue of a hea-geeaig solid body Dea Ahoy, Daipaya Saka, Aku Jai * Mechaical ad Aeospace Egieeig Depame Uivesiy of Texas a Aligo, Aligo, TX, USA.