Using dispatching rules for job shop scheduling with due date-based objectives

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1 Usng sacng uls fo ob so sculng w u a-bas obcvs T. C. CHIANG * an L. C. FU Afflaon: D. Comu Scnc an Infomaon Engnng Naonal Tawan Unvsy Pos A: No. 1 Sc. 4 Roosvl Roa Ta Tawan 106 Emal: ccang@.og Pon: Fax: Afflaon: D. Comu Scnc an Infomaon Engnng Naonal Tawan Unvsy Pos A: No. 1 Sc. 4 Roosvl Roa Ta Tawan 106 Emal: lcn@nu.u.w Pon: Fax: * To wom cosonnc soul b ass. Absac Ts a asss ob so sculng oblm w u a-bas obcvs nclung ay a man anss an maxmum anss. T focus aoac s sacng uls. Egn sacng uls a slc fom lau an faus an sgn concs a scuss. Tn a sacng ul s oos w goal as acvng goo an balanc fomanc wn mo an on obcv s concn a sam m. Fs alzs goo sgn ncls cognz fom xsng uls. Scon noucs a u a xnson ocu o solv a oblm of ngav allowanc m. T a ob cana ucon mcansm s vlo o mak ul comuaonally ffcn. A las a comnsv smulaon suy s conuc w gn xsng uls as bncmaks. T xmnal suls vfy suoy of oos ul scally on ay a an man anss. 1

2 1. Inoucon Job so sculng as aac sacs fom acama an nusy fo sval cas u o s g oblm comlxy an accaly n al wol. In ob so sculng oblm a many macns an ask s o scul obs o b ocss on macns followng fn ous so as o omz o o sasfy concn fomanc ca. T common assumons of ob so sculng oblm nclu: 1) All obs a ay a bgnnng. 2) Pocssng ms a known an fx. Su ms a nclu n ocssng ms. 3) On macn can ocss only on ob a a m an on ob can b ocss on only on macn a a m. 4) Pocssng of a ob on a macn s call an oaon an s a cnc consan bwn oaons of a ob. I mans a succng oaon can no sa unl cng oaon s fns. 5) No mon s allow wc mans a onc an oaon sas mus connu unl ocssng s fns. 6) Macns a connuously avalabl. No macn bakown o mannanc aks lac. T ob so sculng oblm s ofn a as a squncng oblm o mn ocssng o of oaons on macns. In lau aoacs o solv ob so sculng oblms nclu xac algoms lk mamacal ogammng an banc an boun sac bas ma-uscs lk local sac an gnc algoms an sacng uls. Dsacng uls a wly acc n nusy (Alon-Day al Ggan al. 1998) bcaus of as of mlmnaon sasfacoy fomanc low comuaon qumn an flxbly o ncooa oman knowlg an xs. Tus many sacs n acama vo mslvs on sgnng sacng uls an suyng fomanc on ob so 2

3 sculng. Fo xaml Vsalann al. (1987) oos a aamz ul wll known as Aan Tanss Cos (ATC) ul o mnmz wg anss n ob so. I assgn os o obs accong o xc lay cos mmnn macn ocssng m. Anson al. (1990) vlo wo uls usng ynamc oaon u as bas on manng allowanc ms o mnmz u a-bas obcvs n ob so. T so floo ulzaon lvl was akn o aus wgs o ocssng m an u a nfomaon n ul oos by Ragu al. (1993). Fo o os an suvys of oy uls s Mos al. (1992) Km al. (1994) Cang al. (1996) Jong al. (1998) Ranan al. (1999) an Jayamonan al. (2000). In s a w lac focus on solvng ob so sculng oblms w u a-bas obcvs by sacng uls. In Scon 2 w summaz xsng uls fom lau an scb sgn ncls. Tn Scon 3 gvs ou nwly vlo sacng ul wc ams o ov goo an balanc fomanc w sc o mull obcvs smulanously. T fomanc of oos ul an xsng uls s xamn n Scon 4 an conclusons a ma n Scon A suvy on xsng sacng uls Noaons fo sacng uls ocssng m of mmnn oaon manng ocssng m of ob (nclung ) P q s s l l k a /k b oal ocssng m of ob quung m of mmnn oaon sysm m m a wc sacng cson s o b ma u a of ob slack of ob s = slack of mmnn oaon s= c( ) w c s a aam avag ocssng m of oaons a cun macn oal ocssng m of oaons a nx macn aam of ul ATC/COVERT 3

4 Q u k a quu conanng wang obs ow many ms u a of a ob as bn xn aam of oos ul swc of xng funcon aam of oos ul mull of manng ocssng m o xn u a allowanc m Obcvs C comlon m of a ob n numb of obs n so U U = 1 f C > ows U = 0 ay a = U n ; man anss = ( C ) / n ; maxmum anss = max {( C ) } / T w alay many sacng uls oos n lau. Som of m w ognally vlo fo flow m-bas obcvs bu w sown o b ffcv fo u a-bas obcvs. In abl 1 w summaz gn uls a w sown o fom wll o w commonly us as bncmaks n lau wc focus on u a-bas obcvs. Fo ac ul w ov s mamacal quaon fo calculang oy valus laons bwn oy valu an nvolv facos an som fncs. W a cck n column mans a ob w a lag oy valu wll b ocss al; ows ob w a small oy valu wll b ocss al. A ownwa/uwa aow n 4 o 12 columns mans a ul fs ob w a small/lag valu of cosonng faco. T noaons us n abl 1 a scb a bgnnng of s scon. << Ins abl 1 abou >> In abl 1 fs fou uls us only ocssng m-la nfomaon. T sos ocssng m (SPT) ul s unoubly on of mos common bncmak uls n lau an was sown o ov goo fomanc on mnmzng ay a n woks of Cang al. (1996) 4

5 an Ranan al. (1999) scally wn so ulzaon lvl s g o wn u as of obs a g. Ruls nclung sos manng ocssng m (SRPT) las oal wokloa (LTWK) an sos ocssng m ov oal wokloa (SPT/TWK) xn SPT conc o mull oaons an w also sown o fom wll on mnmzng ay a (Cang al. 1996). Followng os mnon abov n sam abl a fv uls a a on mo faco u a wn calculang oy valus. By smly usng u as als u a (EDD) ul was sown o b bs fo mnmzng maxmum anss n suy of Baman (1998) an was also sown o fom wll on mnmzng ay a (Cang al. 1996) an man anss (Jong al. 1998). T mof u a (MDD) ul wc s a combnaon of EDD an SRPT was sown o b goo a mnmzng man anss (Jong al an Km al. 2003). On conay oaon u a (ODD) ul acvs goo fomanc on mnmzng maxmum anss (Ragu al an Jayamonan al. 2000) by combnng EDD an longs manng ocssng m fs (LRPT). T mof oaon u a (MOD) ul wc combns ODD an SPT was vf o b goo fo mnmzng man anss n woks by Jong al. (1998) an Km al. (2003). Mo comlca combnaons of ocssng m-la an u a-la nfomaon la o uls suc as cos ov m (COVERT) aan anss cos (ATC) ccal ao (CR) CR+SPT an S/RPT+SPT. T COVERT ul s a oula bncmak ul wn man anss s cons an was sown o fom wll n suy of Ranan al. (1999) an Km al. (2003). In wok of Ranan al. (1999) COVERT also ov goo suls w sc o maxmum anss. T CR ul uss ao of allowanc m (m lf o u a) o manng ocssng m o mn obs oy. In suy of Safa al. (1999b) CR was scon bs ul fo a cos-bas fomanc consng anss an olng coss. 5

6 Abu-Sulman al. (2005) vlo a aamz CR ul wc was sown o oufom EDD an SPT on sval u a-bas obcvs. T CR ul was also ulz fo sculng n smconuco waf fabcaon facly wc can b sn as a comlx ob so n Ros (2002). Ruls CR+SPT an S/RPT+SPT w oos by Anson al. (1990) an a ynamc vson of MOD ul. Insa of fx allocaon of allowanc m o oaons s wo uls alloca allowanc m o oaons accong o ccal ao an ao of slack o manng ocssng m scvly. CR+SPT was sown o ov goo fomanc on mnmzng man anss (Anson al. 1990) an S/RPT+SPT was vf o b goo fo ucng man anss an ay a (Ragu al. 1993). T las fou uls n abl 1 a lavly nw uls an w scuss n cn sus by Ranan al. (1999) an Jayamonan al. (2000). In sus PT+PW+ODD can acv goo fomanc on mnmzng bo man anss an ay a. As fo PT+WINQ+SLACK was sown o b goo on mnmzng maxmum anss. In summay w can fn common caacscs fo uls a a goo a ac fomanc con. To mnmz ay a uls a follow conc of so ocssng m al usually ov goo suls n wc ocssng m may f o m qu o fns mmnn manng o all oaons. Consng mnmzng maxmum anss uls a ao slack ( of ob slf o of mmnn oaon) as mao faco ofn gv sasfacoy fomanc. T s lss slack al s acually a combnaon of al u a al an long manng ocssng m al. As fo uls a fom wll w sc o man anss y ofn combn sval ncls. Fo xaml COVERT an CR+SPT alz so ocssng m al long manng ocssng m al an al u a al n calculaons. I mls a mo facos soul b akn no consaon fo mnmzng man anss an o fomanc masus. 6

7 3. T oos sacng ul Rcnly auos (Cang al. 2004) oos a sacng ul nam nanc ccal ao (ECR) wos fau s o us gou nfomaon o oz obs. T oy valu of a ob ns on nflunc uon all comng obs caus by fs ocssng of. In a way s ul combns concs of so ocssng m al al u a al an long manng ocssng m al an was sown o fom wll on mnmzng ay a. Howv s combn ncl s only alcabl o non-ay obs n ECR. Fo ay obs ECR us a saa mcansm o calcula oy valus an los ms of combn ncl. In s a w xn ECR ul an mak combn ncl alcabl o all obs by noucng a u a xnson ocu. Two nw facos a nvolv n ocu an. T faco fs o ow many ms u a of a ob as bn xn an faco s u a s a xnson. T nal valu of fo ac ob s zo an nal valu of 0 s s al u a ( ) gvn by oblm. Ts wo facos a us only ns ul an xn u a wll no affc al u a. Bss a wo aams k an u n ul. T aam k s la o ow long u a wll xn an aam u conols w o x ay obs. In followng w wll sow al ss of oos ul fs an n gv xlanaons fo sgn s an usag of aams k an u. 3.1 Dal algom an sgn conc Dal ss S1. Du a xnson Fo ac ob wang n quu Q f + n = + 1 = S2. Influnc smaon 1 + k Fo ac ob sma nflunc uon all wang obs by fs ocssng of. 7

8 V = ( ) ( ) Q w ( a ) = ( + 1) u 1 f = 0 ( a ) = ( + 1) u (/a) 2 f a > 0 ( a ) = ( + 1) u f a <. Slc ob w smalls V o b nx ocssng ag. To xlan ul l us sa fom S2. In S2 nflunc caus by fs ocssng of ob s sma by cangs o g of ncy of all comng obs (nclung slf). In mos cass g of ncy of a ob s assss by squa of ao of manng ocssng m o allowanc m. A lag ao mls g ncy an squa s us so as o mak g of ncy as fas wn allowanc m s gng clos o manng ocssng m. Fo ob manng ocssng m wll cas by ocssng m of s mmnn oaon bu fo all o obs manng ocssng ms k uncang. As fo allowanc m s m o cosonng u a of ac ob ( ) lss ocssng m of mmnn oaon of ob ( ). T g of ncy wll cas fo ob an wll ncas fo all o obs. T long manng ocssng m of s mo s g of ncy cass; so ocssng m of mmnn oaon of s lss g of ncy of o comng obs ncass. Tus by gvng gs oy o ob w smalls V wc mans a fs ocssng of s ob wll manan lows oal g of ncy of all obs oos ul alzs concs of so ocssng m al al u a al an long manng ocssng m al. 8

9 Fo ay obs ao of manng ocssng m o allowanc m bcoms ngav u o ngav allowanc m. Ta causs a conflc w s of ul snc now g of ncy of a ay ob gs low an low as m gos by. In oos ul w solv s oblm by noucng u a xnson ocu n S1. Wn a ob bcoms ay w xn s u a by k ms s manng ocssng m w k s a ul aam (no a xn u a s only fo ul). By xnng u a ao n S2 wll b k osv an afomnon combn ncl of ul s now also alcabl o ay obs. T valu of aam k cs sang g of ncy of ay ob af u a xnson an s mac on fomanc wll b scuss n Scon 4 la. A las w xlan us of faco an scon aam u of oos ul n S2. T valu of sas fom zo an ncass by on ac m ob cvs a u a xnson. In o o x ay obs w mak g of ncy n S2 n ooon o (+1). In o wos ncy of a ob wll ncas o cas fas wn cvs mo u a xnsons. T ncy of a ob s confn bwn ( +1) u 1 an ( +1) u. If a ob w only on manng oaon s slc o b nx ocssng ag s ncy s s as ( +1) u 1. If a ob canno m s xn u a bcaus ano ob s slc as nx ocssng ag s ncy s s as ( +1) u. As mnon n mos cass ncy of a ob s n ooon o squa of ao of manng ocssng m o allowanc m. Inoucng faco n ncy funcon coul ffcvly uc maxmum anss; owv xng ay obs coul also ncas ay a. If mnmzng ay a as g moanc an maxmum anss s xng ffc can b swc off by cangng u fom on o zo. Ta s funcon of u. 3.2 Examl In followng w us an xaml o monsa ow oos ul woks. T a fou obs an fou macns n xaml. T ous of obs ocssng ms of oaons an u 9

10 as of obs a gvn n abl 2. Assum aal scul s as llusa n fgu 1. T sysm m s 30 an now macn M3 fnss oaon of Job1. T valus of ul aams k an u a s as wo an on scvly. T oos ul s nvok an calculaon s as follows: << Ins abl 2 an fgu 1 abou >> S0 Inalzaon Q = {2 3 4} 2 = = 25 3 = = 45 4 = = 40 S1. Du a xnson = no xnson of u a = = = = 120 = no xnson of u a S2-3. Influnc smaon V 2 ( ) 1 ( ) 2 ( ) V 3 1 ( ) 2 ( ) (: < 25) V4 ( ) 1 ( ) 2 ( ) S4. Fnally Job2 s slc as nx ocssng ag fo macn M Job cana ucon mcansm In oos ul valuang nflunc uon all comng obs s lful (as wll b sown n Scon 4) o slc an aoa nx ocssng ob. Howv s valuaon coul also ak long comuaon m u o O(n 2 ) m comlxy w n bng numb of ob canas. In s subscon w wll sow ow w mak oos ul comuaonally ffcn by a ob cana 10

11 11 ucon mcansm. T ucon mcansm s bas on followng om. Tom 1. Domnanc laons bwn ob canas Gvn wo obs an f followng wo conons a sasf n V < V n S2 of oos ul. In s conon w say ob omnas ob ob s omnan an ob s non-omnan. (1) < (2) ( ) + ( ) < ( ) + ( ). Poof: V = ) ( ) ( Q = ) ( ) ( ) ( Q V = ) ( ) ( Q V = ) ( ) ( ) ( Q L Y = Q ) ( Q ) ( V V = ( ) ( ) ( ) ( ) ( ) ( ) + Y. Bcaus < an (.) s noncasng as a cass w av Y 0. Accong o assumon w av ( ) + ( ) < ( ) + ( ). Tfo V V < 0 namly V < V.

12 Coollay 1. Job cana s ucon Bas on Tom 1 ob canas ncssay o b valua n oos ul can b uc o s of omnan obs. Poof: Assum a a ob s no omnan namly s omna by on ob ob. By Tom 1 V > V. Tus ob can no b on w smalls V valu an can no b nx ocssng ag. In o wos can b mov fom s of ob canas bng valua by oos ul. Af vaon of afomnon om an coollay wa w n s an ffcn ocu o fn ou omnan obs. H w so o fas omnan song aoac oos by K. Db al. (2002). Ts aoac was ognally us o so nvuals bas on omnanc laonss n a mul-obcv gnc algom an w ak s ocu o b a ool o ffcnly uc sz of ob cana s fo ou oos ul. Is m comlxy s O(n) mcally an algom s sown n abl 3 fo comlnss. 4. Exmns an suls 4.1 Bncmak oblms << Ins abl 3 abou >> To xamn fomanc of oos ul an xsng uls w us bncmak oblms fom Dmkol al. (1998) snc y a ublcly accssbl an u as a nvolv. In bncmaks all obs a avalabl a m zo. O la nfomaon s gvn n abl 4. Fo ac aam combnaon a fv oblm nsancs. Tus a 160 (42225) oblm nsancs n oal. 12

13 << Ins abl 4 abou >> Fom xmnal suls w foun a ay as of nsancs fom Dmkol al. a vy g. To xamn s nnn uls n ob sos w low ay as w also gna a aa s of 50 nsancs by ouslvs. In ou own aa s a 200 obs 20 macns an ocssng m s gna fom Unfom[1 100]. Du as of obs a s accong o TWK (Toal Wokloa) mo w gnss faco gna fom 6 o 10 unfomly. 4.2 Pfomanc of oos ul an gn xsng uls Pfomanc of oos ul an gn xsng uls n abl 1 a xamn w sc o u a-bas obcvs ay a man anss an maxmum anss. Fo xsng uls w aams nclung ODD MOD COVERT ATC an PT+PW+ODD n valus fom 1 o 10 w s. As fo oos ul g combnaons of wo aams (k an u) w s. T suls oban by akng man ov 160 ublc oblm nsancs an 50 slf-gna oblm nsancs a summaz n abl 5 an 6 scvly. Du o lmaon of sac w only ls fomanc suls of fv aams fo ac of xsng uls w aams (no a suc omsson os no affc lav anks bwn uls). In abl 5 an 6 oos ul s no by ECR-II. << Ins abl 5 abou >> << Ins abl 6 abou >> Pfomanc suls w ublc nsancs To mnmz ay a fo ublc nsancs bs fv uls a ECR-II SRPT LTWK PT+PW an SPT. T oos ul foms wll on mnmzng ay a w a w ang of aam valus. T common fau of bs fv uls s a y all ao so ocssng m al ncl. No a SRPT LTWK an SPT us s ncl only. T 13

14 avag ay as a vy g n s ublc nsancs by all uls. I was o a un conon n wc u as a g fo obs uls followng so ocssng m al conc usually fom wll. Ou obsvaon s conssn w fnng. To mnmz man anss bs fv uls a MDD ATC CR+SPT COVERT an MOD. Ts uls w o o b goo a ucng man anss an y all cons bo ocssng m an u a nfomaon smulanously. Ou oos ul anks sx an man anss s ga an MDD by lss an 1%. T bs fv uls fo mnmzng maxmum anss a PT+WINQ+SLACK ODD SLACK PT+PW+ODD an EDD. Aanly y a smla n us of slack-bas nfomaon. Ou oos ul s svn bs ul among nnn ons. Aloug maxmum anss acv by oos ul s 30% g an a of bs ul sgnfcanly oufoms os bs sx uls w sc o bo ay a an man anss. As can b sn n abl 5 no ul can omna all os n ms of all fomanc obcvs. W ov fgu 2 n o o xamn fomanc of nnn uls on concn obcvs asly. In s fgu fomanc masus of obcvs a nomalz w mnmum/maxmum valu bng nomalz o zo/on. W also sow aa ons oc o lan of nomalz ay a an nomalz man anss so a nomalz maxmum anss s as o cck. Obsvng oc aa ons w can fn a oos ul can omna many uls wn ay a an man anss a concn. No ul can omna oos ul fo s wo obcvs. On goun lan a wo aa ons of o uls clos o ogn wc sn SRPT an LTWK uls. Aloug y ov sasfacoy fomanc on ay a an man anss fomanc on maxmum anss s qu ba as can b sn n fgu. Consng maxmum anss w can classfy xsng uls no wo gous on fo low man anss an o fo low 14

15 maxmum anss. No also a mos uls n gou w low maxmum anss av vy g ay a. T snav ul n fs gou s ATC w man anss aoun 2000 an maxmum anss aoun T snav uls n scon gou a EDD an PT+WINQ+SLACK w man anss fom 2100 o 2200 an maxmum anss fom 3500 o Ts uls o no omna on ano. T oos ul ovs fomanc clos o ATC wn man anss an maxmum anss a cons a sam m. In summay oos ul can low ay a k las man anss an gv accabl maxmum anss wn comang w xsng uls. T ffculy o gna soluons a a goo on all obcvs coul b u o nau of oblm nsancs n aa s. Fu sac wll connu n fuu woks. << Ins fgu 2 abou >> Pfomanc suls w slf-gna nsancs T scon aa s nclus 50 slf-gna oblm nsancs w muc low ay a an fs aa s. I s us o xamn fomanc of uls n ffn so conons. In s aa s bs fv uls o mnmz ay a a ECR-II S/RPT+SPT ATC MDD an PT+PW+ODD. T ay a of scon bs ul s almos wc as g as ay a of oos ul. No a oos ul s ank as bs ul n bo aa ss fo s con. Exc oos ul no ul s ank n bs fv n bo aa ss. Ts obsvaon sows a avanag of oos ul on mnmzng ay a s obus n ffn so conons. Concnng man anss ECR-II CR+SPT ATC MOD an S/RPT+SPT a bs fv uls. Aloug CR+SPT ATC an MOD conssnly fom wll on s con n bo aa ss oos ul can ov b suls wn ay a an man anss a akn no consaon a sam m. 15

16 Fo las con maxmum anss bs fv uls nclu ODD SLACK CR CR+SPT an ATC. T oos ul anks as g. ODD an SLACK a common n bs fv uls n bo aa ss an agan sows sng of slack on ucng maxmum anss. T nomalz fomanc of uls on concn obcvs s llusa n fgu 3. T avanag of oos ul on ay a an man anss s cla n fgu. I can omna all uls wn only s wo obcvs a cons. Takng maxmum anss no consaon oos ul can omna wo- of xsng uls. As mnon n Scon 4.1 s aa s was gna fo nvonmns w low ay as an vous (ublc) aa s. Hnc s mo lkly o oban soluons a a goo on all obcvs. In s suaon oos ul s abl o consuc sculs w lows ay a las man anss an qu low maxmum anss. I s also nsng o obsv fomanc of xsng uls n wo aa ss. In vous aa s w obcvs conflc w on ano xsng uls can usually fn avanags on on obcv. Bu n s aa s w soluons a a goo on on obcv a also goo on os many xsng uls coul fom baly on all obcvs. << Ins fgu 3 abou >> 4.3 Effcs of aams of oos ul T a wo aams n ou oos ul u fo conollng xng funcon an k fo sang g of ncy of ay obs. Fs w obsv ffc of aam u on ay a an maxmum anss. W u bng zo wc mans w swc off xng funcon w can oban low ay a. By sng u as on o swc on xng funcon maxmum anss can b uc a xns of lag ay a. As fo aam k small valu of k s g sang g of ncy s assgn o ay ob. Wn w cas 16

17 valu of k sculs w low man anss can b gna bu also w a loss of m-u-a obs. W an aqua sng of s wo aams manags can gna sculs w s comoms bwn mull fomanc ca. 4.4 Comuaon ffcncy of oos ul In Scon 3.3 w mnon abou O(n 2 ) m comlxy of oos ul an also vlo a ob cana ucon mcansm o mov s comuaon ffcncy. To xamn ffc of ob cana ucon mcansm w gna ffy mo oblm nsancs only fo s xmn on comuaon ffcncy. Ts nsancs fall no fv gous w numb of obs bng an 500 scvly. T numb of macns s always n. Avag comuaon ms ov n nsancs n ac cagoy a sown n fgu 4. T SPT ul s akn as a fnc snc s calculaon s vy sml. Fom fgu 4 s aan a oos ob cana s ucon mcansm s vy ffcv on sonng comuaon m of ou ul. W ucon mcansm ou ul can un almos as fas as SPT ul. << Ins fgu 4 abou >> 5. Concluson In s a w scuss us of sacng uls o solv ob so sculng oblms concnng u a-bas ca. Egn xsng uls a w oulaly ao fo u a-bas obcvs n lau a fs xamn. W analyz avanags of s uls an co sgn concs ns m. Tn w oos a ul wc combns ky sgn concs nclung so ocssng m al al u a al an long manng ocssng m al n o o gna sculs w goo an balanc fomanc on mull u a-bas obcvs. Bss w also vlo a ob cana ucon mcansm 17

18 wc ams o mov comuaonal ffcncy of oos ul. Fnally a comnsv smulaon suy s conuc o xamn fomanc of oos ul an xsng uls. T xmnal suls confm avanag of oos ul on mull ca n ffn so conons scally wn ay a an man anss a mao concns. I ncas a oos ul s a omsng alnav fo ob sos n wc u a-bas obcvs a concn. T ob cana ucon mcansm s also sown bnfcal by obsvng a oos ul can un almos as fas as a vy sml ul. Hnc oos ul s no only ffcv bu also ffcn o b us n nvonmn w al-m sacng s qu. In xmnal suls fomanc of oos ul on mnmzng maxmum anss s sasfacoy bu no as goo as a on ay a an man anss. In ou fuu woks a lann sac cons: (1) W wll nanc fomanc of oos ul on maxmum anss wl kng s suoy on o wo ca. (2) In sos w s an ssnal conflc bwn ay a an maxmum anss wc mans a w can no uc bo of m a sam m w wan o mak oos ul abl o gna a numb of comomsng soluons oug unng aams of ul. (3) W wll y o nga oos ul an mul-obcv ma-uscs suc as gnc algoms so a unng aams of ul s on auomacally an nllgnly by ma-uscs. 18

19 Rfncs Abu-Sulman A. Pa D.B. an Boaman B T mof ccal ao: owas squncng w a connuous cson oman Innaonal Jounal of Poucon Rsac 43 (15) Anson E.J. an Nyna J.C Two nw uls o mnmz anss n a ob so Innaonal Jounal of Poucon Rsac 28 (3) Alon-Day K. an Sao L Ral-m sac gs al-m suls n AMD s Fab25 Poc. of IEEE/SEMI Avanc Smconuco Manufacung Confnc an Wokso Baman S Sml oy ul combnaons: an aoac o mov bo flow m an anss Innaonal Jounal of Poucon Rsac 35 (10) Baman S T mac of oy ul combnaons on lanss an anss IIE Tansacons Cang Y.-L. Suyos T. an Sullvan R. S Rankng sacng uls by aa nvlomn analyss n a ob so nvonmn IIE Tansacons Cang T.-C. an Fu L.-C Solvng FMS sculng oblm by ccal ao-bas uscs an gnc algom Poc. of IEEE Confnc on Robocs an Auomaon Db K. Paa A. Agawal S. an Myavan T A fas an ls mul-obcv gnc algom: NSGA-II IEEE Tansacons on Evoluonay Comuaon 6 (2) Dmkol E. Ma S. an Uzsoy R Bncmaks fo so sculng oblms Euoan Jounal of Oaonal Rsac Ggan A. an Ncolson G B sac alcaon a succss soy Poc. of IEEE/SEMI Avanc Smconuco Manufacung Confnc an Wokso Jayamonan M.S. an Ranan C Nw sacng uls fo so sculng: a s fowa Innaonal Jounal of Poucon Rsac 38 (3)

20 Jong K.-C. an Km Y.-D A al-m sculng mcansm fo a flxbl manufacung sysm: usng smulaon an sacng uls Innaonal Jounal of Poucon Rsac 36 (9) Km M. H. an Km Y.-D Smulaon-bas al-m sculng n a flxbl manufacung sysm Jounal of Manufacung Sysms 13 (2) Km Y.-D. Sm S.-O. Co B. an Hwang H Smlfcaon mos fo acclang smulaon-bas al-m sculng n a smconuco waf fabcaon facly IEEE Tansacons on Smconuco Manufacung 16 (2) Lo J. E. Jang W. an Kln C.M A nw ul fo mnmzng numb of ay obs n ynamc flow sos Euoan Jounal of Oaonal Rsac Mos M. an Engll S A suvy of oy uls fo FMS sculng an fomanc fo bncmak oblm Poc. of 31 s Confnc on Dcson an Conol Pval H. an Mbak N Dynamc slcon of sacng uls fo manufacung sysm sculng Innaonal Jounal of Poucon Rsac Ragu T.S. an Ranan C An ffcn ynamc sacng ul fo sculng n a ob so Innaonal Jounal of Poucon Economcs Ranan C. an Holaus O A comaav suy of sacng uls n ynamc flowsos an ob sos Euoan Jounal of Oaonal Rsac Ros O Som ssus of ccal ao sac ul n smconuco manufacung Poc. of 2002 Wn Smulaon Confnc Russll R.S. Da-El E.M. an Taylo III B.W A comaav analyss of COVERT ob squncng ul usng vaous so fomanc masus Innaonal Jounal of Poucon Rsac 25 (10)

21 Safa R. an Bunn P Wokso sculng usng accal (naccua) aa Pa 1: T fomanc of usc sculng uls n a ynamc ob so nvonmn usng a ollng m ozon aoac Innaonal Jounal of Poucon Rsac 37 (17) Safa R. an Bunn P Wokso sculng usng accal (naccua) aa Pa 2: An nvsgaon of obusnss of sculng uls n a ynamc an socasc nvonmn Innaonal Jounal of Poucon Rsac 37 (18) Vsalann A. P. J. an Moon T. E Poy uls fo ob sos w wg anss coss Managmn Scnc 33 (8)

22 Tabl 1 Collcon of bs xsng uls fo u a-bas obcvs SPT Z = P q s s l l Ts n M92 R93 K94 C96 P97 B97 J98 B98 R99 S99ab J00 L04 SRPT C96 J98 LTWK P K94 C96 J98 SPT/TWK /P K94 C96 J98 EDD R87 M92 R93 C96 B97 J98 B98 R99 S99ab J00 MDD max{ +} R87 K94 J98 K03 ODD s+ M92 C96 J00 MOD max{s+ } R87 M92 R93 K94 J98 S99ab J00 K03 SLACK s R87 K94 C96 J98 B98 COVERT (1/)(1 s + /(k b ( R87 M92 K94 P97 J98 R99 ))) + S99ab K03 ATC (1/)x(-(s/(k a l)) + ) R93 K94 J98 S99ab J00 CR ( )/ M92 P97 S99ab R02 A05 CR+SPT max{( )/ } A90 M92 R93 P97 S99ab S/RPT+SPT max{(s/) } A90 R93 PT+PW +q J00 PT+PW+ODD +q+s J00 WINQ l M92 C96 PT+WINQ+SLACK + l+s - R99 J00 T lag g oy T l sans fo fs l of fs auo an numb sans fo ya of ublcaon. If ncass nflunc of s cass Tabl 2 Daa of xaml oblm O1 O2 O3 O4 Du a Job1 M1 / 10 M4 / 5 M3 / 15 M2 / 9 80 Job2 M2 / 20 M3 / 10 M1 / 8 M4 / 7 70 Job3 M2 / 6 M3 / 20 M2 / 4 M1 / Job4 M1 / 7 M3 / 15 M4 / 9 M2 /

23 Tabl 3 Fas omnan song aoac P: s of all comng obs n quu P: uc s of ob canas P= fn-omnan-obs(p) P= {} Fo ac P an P P= P{} Fo ac P an If omnas n P= P\{} Els f omnas P= P\{} Tabl 4 Daa of bncmak oblm s on Poblm aam Valus cons Toal valus Pocssng m () Unfom [1 200] 1 Numb of obs (n) Numb of macns Job u a = Unfom[(R)/2 + (R)/2] = (1.0 T)n E[] T = cnag ay obs = R = u a ang =

24 Tabl 5 Pfomanc an anks of sacng uls w 160 ublc oblm nsancs Tay a (%) Man anss Maxmum anss SPT * SRPT * LTWK * SPT/TWK SLACK * EDD * MDD * CR+SPT * S/RPT+SPT c= * c= * ODD c= c= c= MOD COVERT c= * c= c= c= c= k b = * k b = k b = k b = k b = ATC k a = k a = k a = k a = * k a = * CR PT+PW * c= * PT+PW+ ODD c= c= c= c= WINQ PT+WINQ+SLACK * ECR-II u = 0 k = * ECR-II u = 0 k = * ECR-II u = 0 k = * ECR-II u = 0 k = * ECR-II u = 1 k = ECR-II u = 1 k = ECR-II u = 1 k = * ECR-II u = 1 k = * * fs o bs fv uls 24

25 Tabl 6 Pfomanc an anks of sacng uls w 50 slf-gna oblm nsancs Tay a (%) Man anss Maxmum anss SPT SRPT LTWK SPT/TWK SLACK * EDD MDD * CR+SPT * * S/RPT+SPT * * c= * c= ODD c= c= c= MOD COVERT c= * c= c= c= c= k b = k b = k b = k b = k b = ATC k a = k a = k a = k a = k a = * * * CR * PT+PW c= * PT+PW+ ODD c= c= c= c= WINQ PT+WINQ+SLACK ECR-II u = 0 k = * * ECR-II u = 0 k = * * ECR-II u = 0 k = * * ECR-II u = 0 k = * * ECR-II u = 1 k = * * ECR-II u = 1 k = * * ECR-II u = 1 k = * * ECR-II u = 1 k = * * * fs o bs fv uls 25

26 Fgus Fgu 1. T aal scul of xaml oblm Fgu 2. Nomalz fomanc of oos ul an o xsng uls n aa s 1. Fgu 3. Nomalz fomanc of oos ul an o xsng uls n aa s 2. Fgu 4. T ffc of ob cana ucon mcansm 26

27 M1 M Job 1 Job 2 Job 3 M3 30 Job 4 M4 15 m Fgu. 1 T aal scul of xaml oblm (ECR-II(g)/os(g) a aa ons of ECR-II/os oc o goun lan w nomalz maxmum anss as zo.) Fgu. 2 Nomalz fomanc of oos ul an o xsng uls n aa s 1 27

28 (ECR-II(g)/os(g) a aa ons of ECR-II/os oc o goun lan w nomalz maxmum anss as zo.) Fgu. 3 Nomalz fomanc of oos ul an o xsng uls n aa s 2 Scons Numb of obs SPT ECRII-wou ucon ECRII-w ucon Fgu. 4 T ffc of ob cana ucon mcansm 28