Optimization of dimension of weldment locus by method of geometric programming

Transcription

1 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng BRANKO PEJOVIĆ DRAGAN SEVIĆ * VLADAN MIĆIĆ ulty of tehnl sene Kosovs Mtov Se ulty of ehnology Zvon Repul of Sps B&H Sentf ppe UDC:... o:.97/zsmtp Zstt Mtel () - () Optmton of menson of welment lous y metho of geomet pogmmng ABSRAC hs ppe on the emple of typl loe wele ssemly me optmton of ts mensons n tems of the ost of welng. In suh n eloton the mthemtl optmton moel wth lmtton funtons hs lso een pesente n t shoul e ten nto ount n the poess of esgnng y the tehnologst n esgne. o solve the pesente polem the metho of geomet pogmmng ws popose tht hs n etl een elote n the ppe n the fom of n lgothm sutle fo the pplton. In ths wy the optmton o pmy ts ws eue to ul ts though pope funton whh s muh ese to solve. he metho hs een llustte on ptl omputtonl emple wth ffeent nume of lmtton funtons. It s shown tht n se of lowe egee of omplety the soluton n e ehe y mmng the oesponng ul funton y mens of mthemtl nlyss. In se of hghe egee of omplety t s neessy to use some of the methos of non-lne pogmmng. In ths se the soluton of the polem s smplfe ue to the mnmton of lne euton. Keywos: Alphet loe wele stutues the mthemtl moel of optmton the ost funton fetue lmttons geomet pogmmng postve polynomls ul funton.. MAHEMAICAL MODEL O OPIMIZAION Mthemtl ss of tehno-eonom optmton of the oets s the mthemt moel of optmton. Moel of optmton ong to fgue onssts of the omponents: - Stte funtons s ( =...) - Lmt funton (funton ouny ontons) g ( =...) - Cte optmton n - he oetve funton ( =...). he fst two omponents e. stte funtons o stte euton n the lmt funton oet of the mthemtl moel of the oet s efne. Rel oets mply we nge of phenomen poesses n systems s vey feuent oets of moellng n mehnl engneeng n mhnng s tehnologl poesses []. *Coesponng utho: Dgn Stevć e-ml: gnestev@gml.om Ppe eeve:... Ppe epte:... Ppe s vlle on the weste: gue - he stutue of the mthemtl moel of optmton oet It shoul e note tht the mthemtl moel s oppose to the physl etns the physl ntue of the ognls (el popety) showng the mthemtl stton. hs stt fom epesses the essentl physl geometl tehnologl eonom o ny othe fetues of the el oet []. he mthemtl moel of mhnng poess n e genelly shown shemtlly y gue. ZASIA MAERIJALA () o

2 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng gue - he mthemtl moel of the mhnng poess In ths t s neessy to nlye the nputs n outputs of "ntul" poess n ll sets of nputoutput vles y =... fo eh eompose "elementy" poess. hs s "ntul" poess tht onsttutes the oe of the mhnng poess n shoul ese the mthemtl moel of the poess [-7]. Afte nlyng the estng esttons one esses the mthemtl espton of the oet n the mthemtl lnguge s spef set of funtons n eutons. hus the mn fetues of the mthemtl moel e funton of the stte of s n funton lmttons g. Conseng the sheme n gue we n set up mthemtl moel of ny mhnng poess n pouton engneeng (wth n wthout emovng the hp n eyon) though the funtons of the poess: s ( y ) n the lmt funton: ( y ) g... ()... () he mthemtl moels () n () e essentlly physl geometl tehnologl n eonom epenent ulmet wthn the mhnng poess n to the mssle omn D esng. he system ( ) vetos y n enote set of vles nput- output vles of the poess. Veto htests of the poess sttes o ontolle se y ( y y y... y ) ese the stte n ehvo of the mhnng poess n the system ws ete s onseuene of nputs n. Input pmetes whh e numeous e ve nto ontolle ( ) n unontolle ( ) se of the poess. he veto nlues ll nputs to the poess whose vlue n e mesue numelly. he veto (... ) n e oen own nto p goup optml o ontol the se n goups tht e onstnt n the ouse of the poess. he fst goup n e hnge n the poess n oe to heve the ese stte of the poess ( optmum oetve funton ( ). n ) n Veto unontolle se (...) ontns one nput pmetes whose vlues n not e mesue s well s those tht n e mesue ut whose mpt on the neglgly smll. Veto uses vese ontons n vese hnges n the flow htests of the poess ( y ) o the oetve funton ( ). o the se of etemnst poesses mpt of unontolle ftos s not lge thee s oelton etween the htests of sttes y n nputs. o ths moel the se wll not e ontne n the eltons () n () so tht t s otne: ( y )... () s g ( y )... () o n the fom of n eplt y ( ) () Components of the stte funtons n funton lmttons s t s s s efne s the mthemtl moel whle the optmton te s the th omponent togethe wth the fst two sets the fmewo mthemtl moel of optmton. On the ss of these thee omponents thee s spef fom of the oetve funton (optmton funtons): ( y ) () unton () s mthemtl espton of the optml ontol poess the entfe optmton te. In theoy tehno-eonom optmton n e ette moe optmton te o oetve funtons ( ) ong to whh the optme poesses [89]:Cost ( ) Bul tme ( u ) Eonomy ( E ) Poutvty ( P ) Poftlty ( R ) ulty (K ). Bngng to te optmton ( tu E P R K...) (7) whh shows ey pouton effets n lso funtons g onstnts whh lmt the llowe omn D hnges of nput nto funtonl eltonshp wth set of nput n the othe mentone ftos ong to () n () s otne y the mthemtl moel of optmton etemnst poess: (... p ) (8) D D g (... D ) ehno-eonom optmton s eue to mthemtl pont of vew the efnton of ZASIA MAERIJALA () o

3 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng eteml (optml) oetve funton (8) n of oesponngof mnge se n htests of the stte of the poess tht poves the optmum s shown n gue []. gue - Dgm of sufe fetues to optme the optml nge he optml level o soluton (... ) of the oetve funton (8) s lle lol etemum n pont M (efne veto ) s lle pont of lol etemum. unton my hve sevel lol etemes. o fom the mthemtl moel of optmton ong to (8) n ton to mthemtl epessons oetve funton t s neessy to set up mthemtl epesson of ll neessy funtons esttons g... hs wy efnes n lmts of pemssle o the optml e egon o omn D. All these onstnts n e epesse n the fom of eutons n neultes ontnng mong othes n gven the se of sets of nput vlues [8] Aong epose t n e onlue tht the mthemtl moel of optmton no mtte wht the suet s the wo (poess system stutue mngement et.) must lwys ontn s n the se of mhnng poess fou s omponents: unton of the flty Lmt funton he te optmton unton of optmton o oetve funton. Bse on these omponents fomng the fnl shpe of the moel optmton of gven oet tht epesses the funton optmton ( ) n the pemtte omn D tnsfomtons of vles.. HE MEHOD O GEOMERIC PROGRAMMING hs metho n solve those optmton tss whose optmton funtons e n the fom of postve polynomls: ( ) B (9) whee: B -postve oeffents (onstnts) - eponents nom nume whh my e postve negtve o eo vlue -nepenent vles (vles) tht n only hve postve vlues. he lgothm of metho whh wll e summe n the followng llows to etemne the optml soluton (... ) wth the mn [-]. In mny ses the optmton of mhnng poesses n tehnology n tems of ost whee the optmton funton s epesse s postve polynoml (9) t s possle to effetve pplton of the metho of geomet pogmmng... he s neulty methos In evelopng the lgothm metho of geomet pogmmng sttng fom the mthemtl neulty etween geomet n thmet mens of non-negtve numes. hs neulty s the founton of the metho n two ses s follows [7]. (Z Z ) ZZ () hs elton epesses the vew tht geomety n not e gete thn thmet mens. Ineulty () fo vles s: Z Z () n t s vl tht the se Z postve n postve se stsfy the onton of nomlty () om euton () we n wte the s eutons of the metho of geomet pogmmng: () he pevous euton s otne when the neulty n eplng whee n Z >. () Ineulty () hs funmentl menng fo the metho of geomet pogmmng euse the pplton of ths neulty to the funton optmton (9) my hnge: ( ) B () n euton () wte the s of mthemtl metho ZASIA MAERIJALA () o

4 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng whee n L B B L () (7) It ws ponte out tht neulty () s vl fo ny postve vlues of se whh must stsfy the onton of nomlty (). Poeeng fom ths we n smplfy the neulty () the hoe of vlues fo to otn L = n euton (7) e. L (8) Euton ( X) smplfe the funmentl epesson (X) whh now es: B B (9) hs euton pples to the onton of nomlty: () othogonlty n one wth euton (7) e. L = n the onton of postvty: () > () Rght se of the neulty (9) s funton of the se of ( ) s n e seen e. B ( ) () n s lle ul funton of onve funton (9) euse the postve polynoml (9) s onve funton. Left neulty (9) howeve epens only on the nepenent vles ( ). he followng onluson s: t the ss of the funmentl neulty polynoml of postve (X) nnot e whteve n of vlues e the vles ( )smlle thn the ul funton () (X) n the pmy moel.e. mnmton funton own to the ul moel e. the mmton of the ul funton (). So thee s pmy efomulton (se sttng) n fnng the optmton ul ts sne t n e shown [7] tht s mmum vlue of the ul funton () eul to the mnmum vlue of the s funtons n the fom of postve polynomls e. mn ( ) m B () In these ontons must e met () () n () of the ul vle ( ). So optmton (pmy) ts n mn B () B ( ) m m () > () whh s muh ese to solve n espet of the pmy ts ().e. etemnt on of mn optmton funton... Algothm of metho At pesent thee e two possle ses s follows: - he se wthout esttons - he se of the onstnts In se tht thee e no lmtton eutons () o system () s e fne y the mnmum vlue (optmum) funton optmton (9) ove the mmum e. etemnng the optml level postve polynoml testfes to the etemnton of the mmum vlue ul funton (). It woul e the fst step of the metho. In the seon step s lulte the optml ul veto... ) system of eutons: ( () whh epesents the ontons of nomlty n othogonlty. In the th step s etemne the mmum vlue ul funton () whh s se on nown set (etemne n the seon step) s lulte fom the euton: ZASIA MAERIJALA () o 7

5 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng B m (7) he fouth step nvolves the etemnton of optml pmy veto (soluton) (... ). he funtons of the (9) foun on the ss of the vlues = n the th step. Between tht mnmes the funton = the optml ul veto whh stsfes the ontons of nomlty n othogonlty () the seon step eomes s follows: B (8) om euton (8) we otn the eue optml level whee the se of = n ( ) e nown s well s some of the seon o th step. In se you e gven the lmttons n the fom of the funton: g g g... (9) gp then the pmy ts es n mn () g g g... gp > > >... () > wheen the funton estton gt ( shpe of postve polynomls gt mn t... J( t ) p B t p ) hs the () Whee n: J () (...m ) - nees of nvul memes funton J () ( m +...m ) - nees of nvul memes funton f g J () ( m +...m ) - nees of nvul memes funton g... J (p) (m p m p = ) - nees of nvul memes funton gp Hee the funtons n gp e n the shpe of postve polynoml. Wth oesponng ul ts n whh the pmy ts s eue we n show n epess the system: ( ) m t J( t ) m B p t t t () t... p () () > () As we see fom () see n euton () e entee ll oeffents B ontnng funton n system funton lmttons gt ( t p ). Hee ontons () n () e ontons of nomlty othogonlty n postvty espetvely. he futhe ouse of the optmton lgothm gven oet s entl to the metho of geomet pogmmng lgothm wthout onstnts. Howeve the eutons (8) to etemne the optmum nume of the pmy veto n ths se s mofe to e s follows: B wheen: t t J( t ) J( ) J(t ). CALCULAION EXAMPLES t p (7) (8) he genel ts of optmng the mthemtl moel of the llustte two emples of the stutul unt s elte to the optmton of mensons of the wele ssemly loe n tems of the welng osts... A smple emple In the fst emple gue ut onssts of two elements: ems (ges) wth wel n hoe (elne whee the em s fe to g et wele wel I n II. ) Defnton of fe n vle se Aong to the epose the net poeue must e efne fst s well s unhngele vle esoluton mge. Contons of the polem e gven onstnt (unhngele) se: ns of mtels of mnuftues the fee length of the em (unts) L n the mmum foe on loe ems. 8 ZASIA MAERIJALA () o

6 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng Othe mensons of the ssemly e nepenent vles. hese mensons e: h l t (9) he vlue of these mensons shoul e so etemne n optme to heve n optml veto ( h l t ) of mnmum ost of welng. n = o = mn. ) Defnng the mthemtl fom of funton optmton he ost funton s funton of optmton n e wtten s [8-]. p () gue - Loe welments hese funtons onsst of thee mn omponents (ptl hges): p - the osts of pepton (peptoy opetons) -welng osts -the ost (pe) of mtel. Costs of pepton p efe to ll the neessy tehnologl eupment to pefom welng opetons: welng tools uly eupment fo settng ems on the tuss n poston ts tghtness n moe. hese osts wll e onsee onstnt (o not epen on the vles ( n ). Cost of welng opetons n e etemne f you now the elements of these osts: - the ost of usng welng epesse n monety mount pe unt of tme whh nlues the ost of motton n lon epyment pplnes the ost of uly eupment (epeton) use n welng the ost of humn wo (pesonl nome fom ontutons n othe) - he pty e. volume of wel (wel) pe unt tme n V - volume of welment wel I n II tht the emple gven s lulte s: V V V h l h l h l () s follows ong to fgue. On the ss of these elements n e wtten osts: V h l () he ost of mtels wll e: V V G () Whee n: - mtel pe of wel - mtel pe em V g - volume of the em s lulte s: V G t ( l ) () eplng () n () to () wll e: h l t (L l ) () Costs y eplng () n () n () we otn the ese shpe optmton funton (oetve funton): espetvely: p t (L l ) h l h l () p h l t (L l ) (7) o y (9): p L (8) the pesent vlues of the oeffents n L-nown of the gven ts (oetve optmton). ) Defnng n settng up system funton lmttons. Resttons on the powe of the she n the wel [-]. he tul she stess n the wel wll e n h vew of the omputtonl of wel gue. ( ) A l o llowle tenson she h l wll pply to: (9) ZASIA MAERIJALA () o 9

7 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng ZASIA MAERIJALA () o ) ( () Dvng euton () to e: () o s funton of the lmts eng: g () Shpe of funton g n othe funton of optmton n ths fom s wll e seen tht t s sutle fo optmton.. Resttons on the noml stess steth mtel of mnuftues []. he tul tenson wll e less thn the llowle: t A ) ( () espetvely: t () gven the lmttons of the funton eng: g t (). Resttons elte to the ptl posslty of gettng wels hs lmt s epesse s h s the em wth must e gete thn the wel pmete h. It follows tht: () Gven the lmttons of the funton eng: g g (7). Resttons on the non-negtvty vles. hs lmtton s epesse y the funton: g (8) ) A mthemtl moel of optmton Aong to epose eltons (8) () () (7) (8) fo the oseve stutul stutue the mthemtl moel of optmton wll e: L mn (9) D g g g g () he funton (9) the ost of pepton p s onstnt fo the oseve elton s not ten nto ount sne they o not ffet the mthemtl nlyss tht follows. One the mnmum funton of the the sme vlue must only the ost of the pepton wth espet to the elton (). By ntoung the (onstnt): () Relton (9) n () e smplfe: L mn g g () whee n: ; ; ; Aong to the lgothm n hpte.. fo the se tht thee e lmts the oesponng ul funton onseng to () wll e: ( ) L () fte the ts hs totl of s memes: = n thee n the n thee n the g sne thee e thee funton lmtton (t = ) eh hvng sngle meme. om the onton of nomlty () n () n othogonl foms system of fve eutons wth s unnown: (V ) (IV ) ) (III ) (II ) (I ()

8 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng Ovously I euton s onton of nomlty ( funton hs thee memes n theefoe ppes ( ) whle the othe euton (II-V) e othogonlty n oe to vle n. In ths euton (II) s etemne y the euton (III) y the eutons (IV) to n the euton (V) ong to tng nto ount the eponents. otl nume of ( = -) s eul to the nume of memes of the funton n the funton lmtton ( = ). Euton V y suttng euton IV t follows tht: () By usng the Gussn lgothm smple wy to show tht ll of the unnowns n e epesse n tems of. om the seon euton t follows tht: () s follows fom the III: (7) At the en of the euton I n IV t follows tht: (8) Rengng euton () t wll e smplfe: L () (9) susttutng n y () () (7) n (8) euton (9) eomes: L () (7) Ovously the ul funton () s epesse n moe thn whh ws the gol. Logthm funtons (7) wll e: ln() ln ln (7) L ln ln ( )ln Let the sttony pont of the veto n whh the () m = m then the sme ount s heve n mmum funtons ln() ong to the (7). So to lulte the evtve of the funton ln() the vle n eutes t to eo: ln( ) (7) Gven tht ths s omple funton fo smplfton to (7) we n ntoue shfts: (ln ln ) ln ln ln ( ) ln L L ( ) ln ln L ln ln ln ln Wth ths shft the funton (7) eomes: ln ( ) Devtve of (7) wll e: ln ( ) (7) (7) (7) Ptl evtve of funtons (7) fo wll e to (7): ln (ln ) ln (7) ln ln L ln ln( ) Susttutng etts ptl funtons (7) n (7) wll e fte the ngng ong to (7): ln ln( ) ln lnl ln (77) Euton (77) fte some mthemtl opetons n e summe s: ln It follows tht: L ln L n fnlly solvng to : (78) (79) (8) L ng nto ount () () (7) (8) n (8) t follows tht: ZASIA MAERIJALA () o

9 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng L L L (8) Aongly n optml ul veto hs omponent: ( ; ; ; ; ; ) (8) By settng the lulte optmum ul omponent vetos (8) oesponng to mmum of the ul funton of () ( ) m ( ) m ( ; ; ; ; ; (8) eeves the vlue of the mnmum funton optmton e.: mn m ( ) ( ) (8) Bse on lulte fom euton () to (7) omponents of the optml veto of the system: (I ) (II ) (III ) (IV ) (V ) (VI ) L ) om I n IV of the euton t follows tht: (8) (8) VI ong to the euton t follows tht: om euton IV wll e: (87) (88) Also fom the euton V wll e: (89) he eutons of system (II) (III) (VI) whh t pesent e not use n e use to ontol the esults otne wth espet to ll of the system euton must e stsfe. o emple the oseve welng em et whh e me of on stutul steel (.% C) lulte onstnts: he pty of the welng s CEN he pe of s mtel CEN he pe of eletoe mtel 7 CEN he ost of welng eve s Allowle stess of the se mtel tensle N Allowle stess of the se metl she N Mmum powe lo em N ee length of the em L Pepton osts p 7 9 CEN Constnt vlue to () wll e: CEN 88 CEN he omponents of the ul optmum veto to e (8) o ong to (8): L he optmum ul veto of (X) wll e: (9) ZASIA MAERIJALA () o

10 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng ( ; ; ; ; ; ) ( ; ; 77 ; ; 788 ; ) (9) he optml vlues of the ul funton to (9) wll e: ( ) L (9) Susttutng the vlues (9) to (9) shll e fnl: ( ) ( ) (9) whee the optml vlue of the ul funton to funton optmton: ( ) 9 CEN (9) On the ss of the vlue (8) (87) fom (88) (89) e etemne y the ese optmum veto : (9) Contol of the esults n e pefome ong to the eutons II III n VI of system (8) onseng tht the sme e not use. Now s the optml pmy veto ompletely etemne: (; ; ; ) (8; 77; 8; 8) (9) When n optml veto (9) n optmum s heve =mn ong to (): L Susttutng (9) nto (97) wll e: (97) CEN (98) As mght e epete gven (9). In lulton mnml eo oue euse of ounng of numes (fou gts). om the ove follows tht the optml vlues of the mensons of the wele ont oseve: 8 mm 7 mm h t l 8 mm 8 mm. One n esly show tht ll the ouny ontons () e fully met... A moe omple emple As moe omple polem let's te the sme emple of the ptue wth the ffeene tht we wll ntoue two new onstnts:. In vew of the spef onstuton esons thee s lmt of geomet mesue t (99). Resttons petnng to mnmum wth menson h mn = mn elow show tht t s not possle to hve tehnologl elton of hetng: h mn mn () o mn onstnt vlue s ntoue whh my elte fo emple to the mnmum mete of the pple welng eletoes [ ]. Aongly funton optmton (gol) wll e to (8): h l l t l t () he fst thee funtons to lmt () wll lso e s n the fst emple: g g g () Due to (99) n () the followng two new onstnts wll e: g g () Ovously wth ths we hve mntne the ele ms geomet se h l. t All esttons on the funtons hee hve only one meme n ong to hpte o ths se n ppopte ul funton s the nese nume of funtons of lmtton wll e moe omple: () L () ZASIA MAERIJALA () o

11 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng om the ontons of nomlty n othogonlty () we otn the system of eutons fo etemnng the optml ul vetos: () Appently solvng the system y () nnot t le n the fst emple euse t s otne y system of fve lne eutons wth eght unnowns. Se n pmetes n the euton () wll e n ths emple: 7 CEN 8 7 CEN L L 78 CEN 8 hee s the opte se = mm n vew of the mnmum mete of the eletoe fo yng out the welng opeton.... Anlyss of the level of omplety o the smple se when the nume of eutons () s eul to the nume of unnown vlues (ul vles) of the system s otne unmguously () soluton ( ). It s possle howeve n n the othe (moe omple) se wth the system () () tht the nume of unnown se s gete thn the nume of vlle eutons. hen we nnot tl out the optml veto wth eg to euse s multfte n t hs mny nfntely moe espetve solutons n the optml soluton whh heve optmum otne y mmng the ul funton () e. solvng the optml ts (ul ts optmton) efne system (). When ths s use n some of the nlytl methos fo emple non-lne pogmmng metho of [7]. Hee s the poeue of optmton ese euse ll onstnts e of lne shpe. Othe (moe omple) se s ssote wth the egee of omplety whh s efne y the euton: s ( ) () whee - nume of mnml postve polynomls n '-se elte to the nume of nepenent vles n the ove polynoml whose elty efnes the n of mt ult y the eponents n the polynoml (9) []. o the se tht s = t oes not solve the ts of optmng the ul funton () ut the s etemne unmguously y the system (). o s= (the fst emple) s shown y the optmum soluton s otne y mmng the ul funton (). Conseng the ove the eponent fo mtes nothe emple the system s etemne y the euton () wheen the fst euton of the system s not ten nto ount: M e (7) Rn of the mt (7) usng mtes n usng mt popetes (opetons wth mtes) wll e: RngM e (8) he egee of omplety s efne s s ( ) 8 ( ) (9) Hee s the nume of mnmng polynomls '-se elte to the nume of nepenent vles n postve polynoml (9) whose vlue efnes the n of the mt fome y the eponents n the polynoml (9). Gven tht s > the optml ul veto n e otne fom the euton system (9) sne the nume of these eutons s less thn the nume of unnown vlues. hen s otne y mmng the ul funton () usng one of the nlytl methos fo emple non-lne pogmmng metho [-].... Solvng polems y usng nonlne pogmmng Susttutng nown pmetes to funton () usng n ppopte pogm y usng met- ZASIA MAERIJALA () o

12 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng ho of nonlne pogmmng to otn the optml ul omponent vetos: ht s: ( ; ; 7 ; ; 788 ; hese vlues e otne onseng to () m. By enteng the lulte vlues of the omponents (X) the pe oesponng ul funton () wll e: ( ) m m ( ) ( ; ; ; ; ; ; ; ) 9 CEN 7 8 hus the mnmum vlue of funton optmton wll e: mn m ( ) ( ) 9 CEN Bss 9 lulte fom euton (97) omponents of the optml veto ong to euton (7): 7 ; ; ) () he esultng system of lne eutons ( ) s solve eltvely esly. om seon n th eutons of the system () tht the om hee t s 78 7 l 7 Now fom the fst euton of the system () tht follows: he fouth euton of system n e use to he the esults: hen fom th euton we hve: L om ths euton t follows tht he seon euton of system my lso e use to he the esults gven tht ses e nown n the eutons. he sme pples to the ffth euton system hus the optml se fo moe omplte se of optmton wll e: h 88 mm l 7 mm t mm mm. CONCLUSION A metho of pogmmng s splye n geometl opeton use pnplly n the pouton of vous tehnologes. It s shown tht the metho une etn umstnes s use n the fel of esgn. Spel methos effeny s heve when the ssote tehnology n onstuton esstne e shown n the emples. Mny of the funtons enountee n pte etn mthemtl tnsfomtons n e eue to postve polynomls n pple to the pesent moel. he moel pesente n the ppe though the ouse of the lgothm n e onsee s moe genel n n e pple n mny es of esgn whee t n e ten nto ount wthn tehnologes whle t s possle to pply vous tehnl n eonom te n optmton. All mounts to stutul n tehnologl solutons n the poess of estlshng the optml poet to etemne the est possle. Lmt funton n e ffeent oth n nume n shpe. Applton of geomet pogmmng s possle wth ffeent funtons of optmton n onstnts s lne n nonlne. Comple polems e pesent n ths system of lne eutons tht e eltvely esy to solve whh s n vntge ompe to othe methos (fo emple smple metho n gent). he soluton s lwys otne etly wthout optml seh e. Spel ttenton when pplyng the metho of geomet pogmmng shoul e poesse when the lmt funton ontns moe thn one meme. hen the ppopte meme of the effetveness of ul funton lso nlues moe memes. In most polems n the en hee ou moe eutons thn neessy. hs llows you to monto n ontol the esults wth espet to ll eutons of the system tht must e met. Also ontol n e eese tows eulty mn = m. ZASIA MAERIJALA () o

13 B. Peovć et l... Optmton of menson of welment lous y metho of geomet pogmmng Le ny metho of optmton n geomet pogmmng metho hs ts ws. he metho n not e pple to ses whee the optmton funton n onstnts e postve polynomls ( when t ppes n the polyno - ml mnus sgn). It shoul e note tht the tehnl ptes n suh ses e genelly e. nlly t shoul e ponte out tht moen optmton methos fo effent mplementton eue multsplny nowlege eue of ffeent fels: tehnology esgnonstuton eonoms mthemtl nlyss mthemtl pogmmng. hese e poly the mn esons why we e n tehnl pte nsuffently engge t nnot s well e ustfe.. REERENCES [] H.J.Jos E. Jo (988) Spnungsoptmeung Vefhensgestltung uh tehnologeshe Optmeng n t Spnunstehn Ve Velg ehn Beln. [] I.Hm R.-Gonles (97) Pouton Opt - mton Metho usng Dgtl Compute Ofo. [] Z.Men (988) ošov u teo ps Infomto Zge. [] B.Dženovć E.Humo (98) Polem upvl - nem složenm sstemm Infomto Zge. [] D.J.Wle (988) Glolly optonl esgn J.Wley New Yo. [] G.V.Relts (99) Rvn A. Engeneeng optmton Methos n ppltons J.Wley New Yo. [7] R.Gsov.M.Klov (99) Meto optme BGU MIns. [8] J.Petć L.Šen (98) Opeon stžvn I II Z ešenh t Nučn ng Beog. [9] A.M.Cln (99) Optmlnoe upvlene tehno - logčestm poessm Enegotomt Mosv. [] H.Opt (98) Moene poutontehn Stn un tenenen. Auflge Velg W. Ge Essen. [] M.E.Zoh (99) Sttstl Anlyss estmton n optmton of sufe fnsh n the gnng poess Development of Pouton System ylo-ns LD Lonon. [] J.Petć (99) Opeon stžvn Kng I II Svemen mnst Beog. [] J.Petć S. Zloe (98) Nelneno pogmne Nučn ng Beog. [] R.J.Duttn E.L.Peteson (987) Geometpogm - mng-heoy n pplton J.Wlley New Yo. [] N.J.Kuneov (99) Mtemtčesoe pogm - ovne Vsš šol Mosv. [] Aulč I. L. (99) Mtmtčseoe pogmovne v pmeh čh Vsš šol Mosv [7].S.Nov J.B.Asov (98) Optm poes - sov tehnolog metllov Mšnostoene-ehn Mosv-Sof. [8] S.Beeovć B.Stvć (998) eo meto - olog tošov Svemen mnst Beog. [9] V.Kolć (98) eo nme tošov R Beog. [] D.Zelenovć (98) Povon sstem Nučn ng Beog. [] M.Mlosvlevć M.Roovć (99) Čelčne onstue Gđevns ng Beog. [] C.Jovnovć (987) Zvene onstue Gđe - vnse onstue Gđevns ng Beog. [] R.Geen (98) Wel Desgn Pente -Hll New Yo [] C.Hse W.Rete (99) Lehuh es Lhto - genshwe-ssens C.Velg W.Get Essen. [] D.S.Mtovć D.Mhlovć (998) Lnen lge Gđevns ng Beog. [] S.Kuep (99) Mtemtč nl I II eh - nč ng Zge. IZVOD OPIMIZACIJA DIMENZIJA OPEREĆENOG ZAVARENOG SKLOPA MEODOM GEOMERIJSKOG PROGRAMIRANJA U u e n pmeu enog testčnog opteećenog venog slop všen optm negovh men s spet tošov vvn. P ovome postvlen e mtemtč moel optme s fnm ognčen oe p poetovnu mou uet u o tehnolog onstuto. Z ešvne postvlenog polem peložen e meto geometsog pogmn o e etlno đen u u u olu lgotm pogonog pmenu.n t nčn optmon l pmn t sveo se n uln t peo ogovuće fune o se ntno lše ešv. Meto e lustovn n enom čunsom ptčnom pmeu s lčtm oem fun ognčen. Pono e se sluč mneg stepen složenost o ešen može oć msmom ogovuće ulne fune pmenom mtemtče nle. Z sluč većeg stepen složenost neophon e pmen nee o meto nelnenog pogmn. U ovom sluču ešene polem e poenostvleno og svođen lnene enčne. Klučne eč: vene stutue po eenom eosleu mtemtč moel optme fun tošov testčn ognčen geometso pogmne potvn polnom vostu fun. Nun R pmlen:... R phven:... R e ostupn n stu: ZASIA MAERIJALA () o