A Heustc Algothm fo the Schedulng Poblem of Pllel Mchnes wth Mold Constnts TZUNG-PEI HONG 1, PEI-CHEN SUN 2, nd SHIN-DAI LI 2 1 Deptment of Compute Scence nd Infomton Engneeng Ntonl Unvesty of Kohsung Kohsung, 811, Twn, R.O.C. Deptment of Compute Scence nd Engneeng Ntonl Sun Yt-sen Unvesty Kohsung, 80424, Twn, R.O.C. tphong@nuk.edu.tw 2 Gdute Insttute of Infomton nd Compute Educton Ntonl Kohsung Noml Unvesty Kohsung, 802, Twn, R.O.C. sun@nuknucc.edu.tw, cuso@ceml.nknu.edu.tw Abstct: - Ths ppe ddesses the schedulng poblem of pllel mchnes wth mold constnts. Ech mchne hs to lod one knd of molds to pocess specfc ob n the poducton envonment. Becuse t tkes lots of tme to chnge one mold to nothe on sme mchne fo poducng obs wth dffeent types, t wll be effcent to put ll sml obs togethe s btch poducton. Ths wy wll, howeve, esult n the totl tdness of obs ncesng due to the dffeent due dtes of obs. Ths knd of poblems s n NP-hd poblem. In ths ppe, we buld model to descbe the poblem nd pesent heustc lgothm to solve t. Key-Wods: - Schedulng, pllel mchnes, mold constnts, heustc. 1 Intoducton In ths ppe, we dscuss schedulng poblem n whch thee e n obs nd m unelted pllel mchnes wth l molds. Ech ob eques sngle opeton whch s pocessed on one mchne equpped wth one mold. A ob s the mnml unt nd cn t be septed. Ech ob hs ts type nd fxed due dte denoted by d. Ech ob s elese tme s cuent. Ech mchne only cn lod no moe thn one mold t tme nd mold s only loded on one mchne t tme. If the type of the cuent ob s dffeent wth the type of the followng obs scheduled on the sme mchne, t needs setup tme fo chngng the onlne mold to next mold fo fomng the followng ob. The setup tme s fxed, mchnes- nd sequence- ndepent, denoted by S. Unelted mchne mens tht mchne my pocess dffeent obs n dffeent speeds. Fo exmple, mchne A pocesses ob t low speed, but t my pocess ob t hgh speed. On the othe hnd, mchne B pocesses ob t hgh speed, but t pocesses ob t low speed. If ll mchnes hve the sme speed, then the envonment s dentcl to the dentcl mchnes n pllel. Due to unelted pllel mchnes, the pecessng tme of ob deps on whch mchne t woks on. 2 Relted esech The pllel mchnes poblem s dvded nto two goups by the type of mchnes: dentcl pllel mchnes nd unelted pllel mchnes. The poblem of pllel mchnes wth setup tme s n ndepent subect to dscuss. Fo dentcl pllel mchnes wth setup tme, Wng et l. [1] consdeed tht pllel mchne wth modulo constnts, n whch the setup tme s ndepent fom lst modulo but only depent on the next modulo. He pesented heustc lgothm bsed on lst schedulng nd then used NBR (net beneft of elocton) lgothm to dust the sequence of obs on ech mchne fo mnmzng totl tdness. Schutten nd Leussnk [2] sw setup s setup obs wth elese dtes, due dtes nd pocessng tmes. They developed bnch-ndbound lgothm to solve the poblem fo mnmzng the mxmum tdness of ny ob. Lee nd Pnedo [3] consdeed tht obs wee weghted nd poposed thee-phse heustc to mnmze totl weghted tdness. Fst, fctos o sttstcs wee computed. Second, sequence ws constucted by dsptchng ule. Thd, smulted nnelng method ws ppled to mpove the soluton. Fo unelted pllel mchnes wth setup tme, sevel studes dscuss sequence- o mchnedepent setup tme. ISSN: 1109-2777 642 Issue 6, Volume 7, June 2008
Chen nd Wub [4] developed heustc bsed on theshold-cceptng methods nd tbu lsts to mnmze totl tdness. Tmk et l. [5] tnsfomed the schedulng poblem to mthemtcl pogmmng poblem nd used SA method nd genetc lgothm to solve t. Besdes, Km et l. [6] consdeed tht ech ob my efe to lot composed of dffeent tems whle evey tem wthn ech ob hs n dentcl pocessng tme wth common due dte. They used SA to detemne schedulng polcy so s to mnmze totl tdness. Although schedulng of unelted pllel mchne wth setup tme hs been studed n ecent yes, the most studes hypothesze tht des o molds cn be used on ny pllel mchne n the schedulng. In ths ppe, we del wth the schedulng poblem n whch molds do not be llowed to lod on ny mchne. 3.1 Ft molds nd ft mchnes Evey mold hs nge of type whch t cn fom. The fomng nge of ech mold my be ptl coveed. A subset of molds whch cn fom the equed type of the ob s clled ft molds of the ob. Due to the equpment nd ttbutes of mchnes, mold s not llowed to equp on ny mchne. A subset of mchnes whch cn equp the mold s clled ft mchnes of the mold. Hence ech ob s estcted to specfc subset of mchnes. Fgue 1() shows the elton of obs, molds, nd mchnes. Fo exmple, ob 3 cn be fomed by mold 4, mold 5 nd mold 6. The mold 4 cn be loded on mchne 2 nd 3. The mold 5 cn be loded on mchne 2 nd 4. The mold 6 cn be loded on mchne 3 nd 4. The Fgue 1(b) shows the elton of ths exmple. 3 Poblem defnton Afte descbng the shecdulng poblem n secton 1, we pesent the poblem n mthemtcl fomuls. The symbols we defned e shown s follows. : ndex of obs ( = 1, 2,, n), h: ndex of molds (h = 1, 2,, l), k: ndex of mchnes (k = 1, 2,, m). The pmetes e ssocted wth the ob : d : the due dte of ob, c : the complete tme of ob, p : the pocessng tme of ob whch s pocessed n the stndd speed, v k : the speed tht mchne k cn pocess ob t, t s eltve to the stndd speed, p k : the pocessng tme of ob on mchne k nd p k = p /v k. We tke n exmple to expln the elton of pocessng tme of obs nd speeds of mchnes. Thee s ob wth quntty q to poduct, nd the stndd speed v = 1. So the pocessng tme of ob e p = q /v = 6/1 = 6. Now we hve thee mchnes 1, 2 nd 3. The pocessng speeds fo ob e v 1, v 2 nd v 3, espectvely. If v 1 = v 2 = v 3 = v = 1, the pocessng tmes of ob on thee mchnes wll be p 1 = p 2 = p 3 = p = 6. Now we set v 1 = 1, v 2 = 2 nd v 3 = 3, whch mens speed of mchne 1 s equl to stndd speed, speed of mchne 2 s two tmes thn stndd speed, nd speed of mchne 3 s thee tmes thn stnded speed. The pocessng tme of ob on mchne 1, mchne 2 nd mchne 3 e p 1 = p /v 1 = (q /v)/v 1 = 6/(1*1) = 6, p 2 = 6/(1*2) = 3 nd p 3 = 6/(1*3) = 2, espectvely. Fgue 1: () The elton of obs, molds, nd mchnes. (b) The elton of ths exmple. We defne the ft molds nd ft mchnes by fomul (1) nd fomul (2). 1, f ob cn be fomed by H h = mold h 0, othewse 1, f mold h cn be loded on K hk = mchne k 0, othewse Fo ny ob nd mold h, the vlue of H h s 1 o 0. Fo ny mold h nd mchne k, the vlue of K hk s 1 o 0. Obvously, these two defntons hve to stsfy constnts of fomul (3) nd fomul (4), espectvely. l h= 1 m k = 1 H h l K hk m (1) (2) (3) (4) ISSN: 1109-2777 643 Issue 6, Volume 7, June 2008
Fgue 2: An exmple of postons. 3.2 The postons of ob n the schedule To model the poblem, we wnt to know tht ll possble postons of ob n the ente fesble schedule, whch e deped on ts t molds nd ft mchnes of ts t molds. Fstly, we should know ll possble postons of ob n t mchne. Becuse the mount of obs whch my be pocessed on mchne s the mount of possble postons of ob on tht mchne, we wnt to know how mny obs cn be scheduled on ft mchne.fo exmple, mchne 1 cn lod mold 4 nd 5. The mold 4 s one of ft molds of Job 3 nd ob 7, nd the mold 5 s one of ft molds of Job 3 nd ob 6. Hence thee e thee possble obs whch my be scheduled on mchne 1. Tht mens ob 3 hs thee possble postons on mchne 1. See Fgue 2. We denote the mount of obs whch my be pocessed on mchnes k by n k. Fo ny mchne k, the n k s devted by the fomul (5). n 1 n K k 1k H 1 K n * l * m 2 k H 2,, K 3.3 The model of the poblem We defne the symbol X kuh to epesent the poston whch ob s scheduled on. Fo ny ob, mold h, mchne k, nd poston u of mchne k, the vlue of X kuh s 1 o 0. The vlues of sequence {X kuh } e fesble schedule. X kuh 1, fobsscheduledsu th poston onmchnekequppeded (6) moldh 0, othewse The model of the poblem s epesented by the complete tme c of ob n fesble schedule, s shown n the fomul (7). lk H l c cku X kuh, 1 n (7) h 1, 2,, l, k 1, 2,, m, u 1, 2,, n The symbol c hku s defned n the fomul (8). The symbol n k epesents the mount of obs whch my be pocessed on mchne k. ck,( u1) p k X kuh, f M ( k, u 1) M ( k, u) 1 n cku ck,( u1) ( p k S) X kuh, othewse 1 n (8) h 1, 2,, l, k 1, 2,, m, u 1, 2,, n Thee e some constnts on X kuh n the fesble schedule, s shown n the fomul (9) nd fomul (10). 1 n X hku 1, h 1,2,, l, k 1,2,, m, u 1,2,, n X hku { 1,0}, 1,2,, n (5) (10) 1 h l 1 k u m 1 n k Fnlly, we defne symbol Y hkt to epesent the mchne whch mold h s equpped on t tme t, s shown n fomul (11). And the fomul (12) shows the constnt of mold n the fesble schedule. Y hkt hkt 1hl 1, fmoldhslodedonmchnek ttmet (11) 0, o. w. Y { 1, 0}, k 1, 2,, m, t nytme k K k (9) (12) 4 The heustc lgothm In ths secton, we pesent heustc lgothm to solve the schedulng poblem. Tble 1 llusttes the lgothm befly. We expln detls of ech step n the followng subsectons. ISSN: 1109-2777 644 Issue 6, Volume 7, June 2008
Tble 1: Heustc lgothm STEP 1 Sot ob set J s by EDD (Elest Due Dte) ule. Fnd the ob wth elest due dte n J s, sy ob. STEP 2 Fnd ll ps (h, k) of ob, whee h s ft mold of ob nd k s ft mchne of h. STEP 3 Fo ech (h, k) 3.1 Fnd ts cmg nd sot ech of them by EDD ule. 3.2 Fnd enough ponts nd the nteupt pont of the cmg. 3.3 Clculte the totl tdness o totl elness of cmg nd put them nto vege set A s. STEP 4 4.1 Get the mnmum A * of A s. 4.2 Check the mold h * of A * s vlble. If no, delete A * fom A s nd goto step 4.1. STEP 5 5.1 Schedule cmg. 5.2 Delete ll obs of cmg fom J s. If n nteupton ws occued n step 5.1, put the nteupted obs bck to J s. 5.3 Go to step 1 untl J s s empty. 4.1 The fst step nd the second step In the fst step, the ob set J s contns ll obs not scheduled yet. The EDD ule wll lst the ob wth ncesng due dte, nd the one wth the elest due dte wll be lsted n the most font. In the second step, the ode p (h, k) s of ob epesents ll combntons of ft molds of ob nd ft mchnes of the ft molds. They e ll possble choces of ob nd e ndexed wth. We wll exmne ech p to decde whch one s bette choce. 4.2 The thd step When mchne s on the setup tme of the chngng molds, t cn not pocess ny ob. If the setup tme s longe, the cost s gete. To educe tmes of chngng molds, the obs whch cn be fomed by the sme mold should be gtheed to pocess. So, fo ech p (h, k) of ob, we fnd ts cmg (common mold goup) nd sot t by the EDD ule. The goup collects obs of J s, whch cn be pocessed on the mchne k equpped wth the mold h. Obvously, the ob s the ledng ob of ts goups. If cmg s put on mchne loded wth ft mold to pocess, the loded mold should be used to fom enough obs befoe t s unloded fo mkng the setup tme cost-effectve. So, n the step 3.2, we fnd the enough pont of ech cmg. We dd the pocessng tme of obs of cmg one by one, untl the ccumultve pocessng tme (APT) s gete thn α. The lst dded ob s clled enough pont of cmg. The α s theshold pmete set by expets ccodng to the cost-beneft nlyss, whch mens how mny obs n cmg e pocessed, so tht the setup tme of cmg wll be cost-effectve. The obs n cmg cn be pocessed by the sme mold, but the due dtes my be dffeent getly. If the dffeence of due dtes of two contnul obs n cmg s get enough, n nteupton my be consdeed. Tht s, the ltte ob wth lte due dte nd ll obs n cmg fte t cn wt fo the next chnce to be scheduled nd nothe cmg wth ele due dte of ts ledng og cn be pocessed fst. The obs n cmg fte the ltte one hve lte due dtes then t due to EDD sotng n the step 3.1. Fom the ob of enough pont to penultmte ob of cmg, f the dffeence of due dtes between ob nd ts succeeded ob s gete thn β, we let the font ob be n nteupted pont of cmg. It must be notced tht nteuptons my ncese tmes of chngng molds. In dffeent stutons, the consdeton wll be dffeent. The β s nothe pmete set by expets. The pseudocode n Tble 2 llusttes how to fnd the enough pont nd nteupted ponts. When we ty to ssgn cmg of the p (h, k) to the mchne k, one of the followng thee condtons wll occu. The mchne h hs othe scheduled cmg nd the mold of the lst scheduled cmg, sy cmg s dffeent fom the mold h of the p (h, k). The mchne h hs othe scheduled cmg nd the mold of the lst scheduled cmg, sy cmg s the sme wth the mold h of the p (h, k). Thee s no othe scheduled cmg on the mchne k. In the fst condton, we hve two stteges: nteupt cmg nd succeed cmg. Fo the nteupted sttegy, we nteupt the scheduled cmg t one of ts nteupted ponts, ISSN: 1109-2777 645 Issue 6, Volume 7, June 2008
Tble 2: Pseudocode of fndng the enough pont nd nteupted ponts nput:,, cmg, (h, k) APT = 0; fo w s n ndex fom st ob of cmg to the lst ob of cmg do APT = APT + p wk, whee k s of (h, k) ; f APT then let the ob wth ndex w be the enough ob of cmg ; bek fo; f w s on the lst ob of cmg nd the enough pont s not set yet then let the lst ob be the enough pont of cmg ; fo the ob of enough pont of cmg to the penultmte ob of cmg do f the dffeence of due dtes between the ob nd ts succeeded ob then let the ob be n nteupted pont of cmg ; chnge molds, nd let cmg succeed the ob of nteupted pont of cmg to poduce. Becuse cmg my hve sevel nteupted ponts whch e ndexed wth t, we use vege tdness s cteon. Fo evey nteupted pont p t of cmg, we clculte the vege tdness of cmg, denoted by T (p t ). We dd the vege tdness of cmg fom the ob of the nteupted pont to the lst ob nd the totl tdness of cmg fom the fst ob to the ob of ts enough pont, then dvde the sum by the totl numbe of obs nvolved n the tdness clculton to fnd the vege tdness. See the fomul (13). T E ( pt ) ep ( p t ) ep 1 ) p ) 1 ) p t t ) ep ) 1 mx(0, c p ) 1 t b1 ep ) ( c p ) t b1 d ) mx(0, c d ) ( c d b ) b d ) The ep ) s the mount of obs of cmg fom the fst ob to the ob of ts enough pont. The p t ) s the mount of obs of cmg fom the ob of ts nteupted pont to ts lst ob. The cmg ) s the mount of obs of whole cmg. If the vege tdness s zeo, the vege elness s substtuted fo the vege tdness to fndng bette nteupted pont of ths sttegy. The b b (13) (14) fomul of vege elness s sml to the vege tdness, s shown n the fomul (14). Fo the not-nteupted sttegy, we do not nteupt the scheduled cmg, but succeed t. The vege tdness stll s cteon. We clculte the vege tdness of cmg fom the fst ob to the ob of ts enough pont, whch s denoted by nt (see fomul (15)). The pmetes nd ndexes of ths symbol men cmg do not nteupt the scheduled cmg. If the vege tdness s zeo, vege elness ne s clculted to substtute fo the vege tdness s pecedng sttegy (see fomul (16)). ep ) 1 mx(0, c d ) ep ) 1 (15) ep ) 1 ( c d ) ep ) (16) 1 In the second condton, the mold of the p (h, k) s sme s n onlne mold. It s the most esonble sttegy tht cmg succeeds cmg to pocess. We lso clculte the vege tdness sct of the cmg, whch only s nfluenced by the obs n the cmg, s shown n fomul (17). If sct s zeo, the vege elness sce wll lso eplce t. The fomul of sce s fomul (18). cmg ) sct 1 mx(0, c d ) (17) cmg ) 1 cmg ) sce 1 ( c d ) (18) cmg ) 1 ISSN: 1109-2777 646 Issue 6, Volume 7, June 2008
In the thd condton, the only sttegy s one tht cmg s dectly ssgned on the mchne k of the p (h, k), beng the the fst cmg of the mchne. In elty thee s the fst setup tme, but we leve t out to smplfy poblem. The fomuls of vege tdness nd elness of ths sttegy s ust lke those of the second condton, but we gve them dffeent symbols to dscmnte, denoted by T nd E, espectvely. Once the vege tdness o elness s clculted, we put t nto the vege set A s fo compson n step 4. Tble 3 llusttes the pseudocode of clcultng. Tble 3: Computtng vege tdness nd vege elness nput: cmg, cmg, (h, k) f nothe cmg s scheduled on the mchne k then f the mold used by cmg s sme s mold h then foech nteupted obs of cmg do clculte T nd nt ; f T o nt e zeo then clculte E o ne; put E o ne nto A s ; else put T o nt nto A s ; else clculte sct ; f sct s zeo then clculte sce; put sce nto A s ; else put sct nto A s ; else clculte T ; f T s zeo then clculte E; put E nto A s ; else put T nto A s ; If the mold s vlble, we schedule cmg on mchne k * nd lod mold h * nd dopt the sttegy decded by A *. Then we delete obs of cmg fom ob set J s. If n nteupton s occued n the step 5.1, we put the nteupted obs bck to J s. The lgothm wll epet untl J s s empty. 4.4 No stvton Becuse cmg cn be nteupted, some obs my be putted bck nto J s mny tmes. Ech ob hs fxed due dte, so ny ob wll be the ledng ob of ts cmg dung fnte tme. Even thee e no othe obs whch hve sme mold wth the ob nd the complete tme of the ob s not gete then α, the lgothm stll set t beng the enough pont of the cmg contnng the only ob tself. In the step 4.2., the pocedue of checkng mold wll delete the choce wth not vlble mold. Would t hppen tht thee s no ny choce fte the deletng pocedue? The nswe s no. Assume cmg hs only one mold h. Fo p (h, k 1 ), the mold h s not vlble, becuse t s equpped on nothe ft mchne k 2 t the sme tme. Due to the step 2.1, the p (h, k 2 ) s lso one choce of cmg nd h s vlble cetnly fo ths p. 5 An exmple of lgothm Hee n exmple s used to llustte the lgothm. Thee e 11 obs, 4 unelted pllel mchnes, nd 6 molds. Tble 4 shows the dt whch nfluence the schedulng n ths exmple ncluded pocessng tme (p k ), due dte (d ), ft molds of obs (H h ), ft mchnes of molds (K hk ), nd speed of mchnes fo ech ob (v k ). The J s s soted by EDD nd ob 1 hs the elest due dte. We fnd ll ps of ob 1 nd the cmgs, As shown n the Fgue 3. 4.3 The fouth step nd the ffth step We tke the mnml vlue of the vege set A s, denoted by A *. The A * decdes whch p nd wht sttegy of the p wll be dopted. It mens tht f we dopt the sttegy of A * to schedule cmg on the mchne nd lod the mold desgnted by the p of A *, denoted by h * nd k *, the vege tdness wll be smllest. We detemne whethe mold h * s vlble o not t tht tme by checkng whethe mold h * s used on nothe ft mchne. If the mold s not vlble, we delete the A * fom A s nd fnd the mnml of A s gn. Fgue 3: All ps of ob 1 nd the cmgs. Then we gve α = 15 nd β = 15 to fnd the enough pont nd nteupt ponts. Fo cmg 1 1 of p (A, 1) 1, the pocessng tme of ob 1 on mchne A s p 1A = p 1 /v 1A = 6/1 = 6, so the complete tme of ob 1 s c1 = ISSN: 1109-2777 647 Issue 6, Volume 7, June 2008
Tble 4: Dt of n exmple 1 1 clculte the vege elness E 1 of cmg 1 to substtute T 1 1, whch s 1/4[(6 8) + (11 24) + (16 26) + (34 45)] = 25 nd put t nto vege set A s. As the sme pocedue, we clculte the vege tdness of cmg 1 2 of (A, 2) 2, vege elness of cmg 1 3 of (C, 1) 3, nd vege tdness of cmg 1 4 of (C, 2) 4, whch e T 1 2 = 1.9, E 1 3 = 4.25, nd T 1 4 = 2.25, espectvely. The vege set A s s { 25, 1.9, 4.25, 2.25} nd the mnmum A * s E 1 1 = 25, whch mens the p (A, 1) s the bette choce nd the cmg 1 1 cn be scheduled dectly on mchne A. Fo ths s the fst un of the lgothm, mold 1 should be vlble. We schedule the cmg 1 1 whch contns ob 1, ob 4, ob 5 nd ob 2 on mchne A equpped wth mold 1 nd delete those obs fom J s. In the next un, the ob wth elest due dte s ob 9 nd ts ll ps e (A, 2) 1, (A, 3) 2, nd (C, 2) 3. See Fgue 5(). 6. Fo α s 15, ob 1 s not enough, nethe s the ob 4. Untl to ob 5, the complete tme of ob 5 s c 5 = 16 >15, so ob 5 s the enough pont of cmg 1 1. The complete tme s the ccumultve pocessng tme. Besdes, the dffeence of due dtes of ob 5 nd ob 2 s 19 whch s gete thn β (=15). We set ob 5 be ob of n nteupted pont of cmg 1 1. The llustton s shown n Fgue 4. Becuse thee s no othe cmg on mchne A, the sttegy we tke s schedulng cmg 1 1 on t dectly. Then the vege tdness of the sttegy s T 1 1 = 1/4 [mx (0, 6 8) + mx (0, 11 24) + mx (0, 16 26) + mx (0, 34 45)] = 0. Snce the vlue s zeo, we Fgue 5: () All ps nd cmgs of ob 9. (b) The enough pont of cmg 1 9. Fo the cmg 1 9 of (A, 2) 1, we ty t on mchne A to fnd ts enough pont nd nteupted ponts, s shown n Fgue 5(b). Obvously, even fo the lst ob of cmg 1 9, tht s ob 8, the complete tme s smlle thn 15. In ths stuton, we let the lst ob be the enough pont nd the cmg hs no nteupted pont. Becuse thee s scheduled cmg on mchne A, we hve two stteges: nteupt o not-nteupt. The Fgue 4: The enough pont nd nteupt ponts of cmg 1 1. ISSN: 1109-2777 648 Issue 6, Volume 7, June 2008
() (b) Fgue 6: Two stteges of cmg 1 9 : () nteupt the cmg 1 1. (b) not nteupt the cmg 1 1. Fgue 7: the outcome of the lgothm fo ths exmple. vege tdness of nteupted sttegy s T 1 9 = 16.23 nd the vege tdness of not-nteupted sttegy s nt 1 9 = 36.8. Fgue 6 llusttes those two stteges. Fo the (A, 3) 2, nd (C, 2) 3, we lso clculte the vege tdness by the sme pocedue. Fnlly vege set A s s {16.23, 36.8, 13.5, 37.5, 2.25} nd the mnmum of A s s T 3 9 = 2.25. The p (C, 2) 3 s chosen, nd cmg 3 9 cn be scheduled on mchne C dectly. Snce mold 2 s vlble, the schedule of cmg 3 9 s done nd the obs of cmg 3 9 e deleted fom ob set J s. The est uns of the lgothm e sml. Fgue 7 shows the outcome of the lgothm fo ths exmple. 6 Expements In elty, ths s the schedulng poblem of steel tube poducton compny. A steel tube s fomed fom steel sheet by fomng pocess on tube fomng mchne. The compny hs 10 pllel unelted tube fomng mchnes nd 206 molds. A ob we clled n ths ppe s wok ode of steel tubes. Snce ths poblem s mppng fom elstc poblem, we compe the effcency of heustc lgothm wth tht of mnul pocessng. We smulte the elty poducton envonment by speeds of 10 unelted pllel mchnes fo obs (v k ), the dt of ft molds of obs (H h ) nd ft mchnes of molds (K hk ). And we smulte the ob set of the dy befoe the fst tced dy whch hve 332 not pocessed obs. The tced dys e dys on whch the tced obs mpot nto the ob set of schedulng system. We tke 95 obs whch e mpoted obs of fve contnul wok dys s evluted dt. We tce the evluted dt n the schedulng pocess. Becuse wok dy hs only eght wok hous, tced obs whch even e schedeuled my not be ISSN: 1109-2777 649 Issue 6, Volume 7, June 2008
Smulted ob set Fgue 8: The desgn of expements. pocessed n dy. Moeove new obs mpot nto ob set dy by dy, those obs wll nfluence the schedule of tced obs. Those obs e clled ext obs. In ths schedulng pocess of ths expement, thee e 578 ext obs fom the fst tced dy to the dy on whch ll tced obs e be pocessed. The desgn of expements s llustted n Fgue 8. Accoddng to the schedulng expot of the compny, the setup tme of chngng mld cost bout 3 hous fo once. So mold should be on ft mchne t lest 2 dys, whch mens α s 16 (hous) fo eght wok hous pe dy. Moeove, f thee s n nteupted s need, t tkes helf dy (3 hous of 8 hous pe dy) to chnge mold, nd t lest 2 dy to pocess new cmg, nd then nothe helf dy to chnge pevous mold bck. The totl needed tme s 3 dy, so the β s set by 5 dy (40 hous) fo theoetcl sfety. But n elty, thee e mny unpedctble fctos fo chngng mold, the schedulng expot usully vod unexpected chngng mold. Hnce we e suggested to set the condton of nteupton moe stctly. We tke β by 5 dys, 7 dy nd 10 dy to expement espectvely. We compe the totl tdness, the numbe of tdy obs nd the vege tdness of evluted obs scheduled by the schedulng system wth those scheduled by mnul schedulng. The esults show tht the poposed ppoch cn getly mpove the schedulng esults. 7. Conclusons nd futue wok The pupose of ths ppe s to solve elty schedulng poblem. The poblem s modeled n mthemtcl expessons. We povde heustc lgothm s soluton nd desgn n expement to compe wth the mnul. Thee e mny spects not tken nto ccount n ths soluton lke mtel supplcton. We wll contnue to ext the schedulng poblem to solve the elty poblem compehensvely n the futue. Refeences: [1]Cheng-Yo Wng, Ln Go, Dng-We Wng, Zh-Song Yn, Shu-Nng Wng. Mnmze Totl Tdness of the Pllel Mchne wth Modulo Constnt. Jounl of Systems Engneeng, Vol.14, No. 4, 1999, pp. 345-350. [2]J.M.J. Schutten, R.A.M. Leussnk. Pllel Mchne Schedulng wth Relese dtes, Due dtes nd Fmly Setup Tmes. Intentonl Jounl of Poducton Economcs, Vol. 46-47, 1996, pp. 119-125. [3]Young-Hoon Lee, Mchel Pnedo. Schedulng Jobs on Pllel Mchnes wth Sequence-depent Setup Tmes. Euopen Jounl of Opetonl Resech Vol. 100, 1997, pp. 464-474. [4]Jeng-Fung Chen, T-Hs Wub. Totl Tdness Mnmzton on Unelted Pllel Mchne Schedulng wth Auxly Equpment Constnts. Omeg, Vol. 34, 2006, pp. 81-89 [5]Hssh Tmk, Yoshshge segw, Jun Kozs, Mtuhko Ak. Applcton of Sech Methods to Schedulng Poblem n Plstcs Fomng Plnt: A Bny Repesentton Appoch. The 32nd IEEE Confeence on Decson nd Contol, 1993, pp. 45-50. ISSN: 1109-2777 650 Issue 6, Volume 7, June 2008
[6]Dong-Won Km, Kyong-Hee Km, Wooseung Jng, F. Fnk Chen. Unelted Pllel Mchne Schedulng wth Setup Tmes Usng Smulted Annelng. Robotcs nd Compute-Integted Mnufctung, Vol. 18, 2002, pp. 223-231. ISSN: 1109-2777 651 Issue 6, Volume 7, June 2008