Changeovers. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Transcription

1 wo ew Connuous-e odels for he Schedulng of ulsage Bach Plans wh Sequence Dependen Changeovers Pedro. Casro * gnaco E. Grossann and Auguso Q. ovas Deparaeno de odelação e Sulação de Processos E Lsboa Porugal Deparen of Checal Engneerng Carnege ellon Unversy Psburgh PA 523 USA Absrac hs paper presens wo new ulple-e grd connuous-e xed neger lnear progra (LP) odels for he shor-er schedulng of ulsage ulproduc plans feaurng equpen uns wh sequence dependen changeovers. her an dfference resuls fro he explc consderaon of changeover ass as odel varables raher han plcly hrough odel consrans. he forer approach s ore versale n ers of ype of objecve funcon ha can be effcenly handled (nzaon of oal cos oal earlness and aespan) and despe generang larger aheacal probles s also a beer perforer n sngle sage probles. he laer s beer sued for ulsage probles where he forer approach has soe dffcules even n fndng feasble soluons parcularly as he nuber of sages ncreases. he perforance of boh forulaons s copared o oher xed neger lnear progra and consran prograng odels. he resuls show ha ulple e grd odels are beer sued for sngle sage probles or when nzng oal earlness ha he consran prograng odel s he bes approach for * o who correspondence should be addressed. el.: Fax: E- al: pedro.casro@ne.p

2 aespan nzaon and ha he connuous-e odel wh global precedence varables s he bes overall perforer.. nroducon Schedulng probles can be acled by a varey of opzaon approaches as well as oher soluon ehods. For nsance aheacal prograng (P) odels usually leadng o xed neger lnear prograng (LP) probles have receved consderable aenon n he leraure. he focus has ranged essenally fro specfc o general ypes of newor confguraons fro pure bach o pure connuous ype of processes fro shor-er o perodc odes of operaon and fro dscree o connuous represenaons of e. Whle soe are ore robus han ohers sall changes n he characerscs of he proble can ae soe P odels hghly neffcen even non-applcable. Consran prograng (CP) 2 orgnally developed o solve feasbly probles has also been exended o solve opzaon probles parcularly schedulng probles. CP and P approaches have copleenary srenghs 3 and soe researchers 4-8 have already aen full advanage of hs by developng hybrd ehods ha are consderable ore effcen han he sandalone approaches. os of he recen P schedulng odels are based on a connuous-e represenaon hose eployng one or ore e grds focus on general ulpurpose plans and on he developen of ncreasngly effcen odels. An poran recen advance was he nroducon by Sundaraoorhy and ar 6 of a forulaon whou bg- consrans ha proved ore effcen han oher copeng ehods. Dscree-e forulaons for schedulng of ulpurpose plans go furher bac n e sarng wh he senal paper of ondl e al. 7 ha also had he er of nroducng he sae-as newor (S) process represenaon and whch was soon followed by he resource-as newor (R) based odel of Paneldes 8. Dscree and connuous-e approaches have copleenary srenghs and a xed-e represenaon odel has recenly been presened by aravelas 9 for he sulaneously opzaon of schedulng and supply chan anageen probles. Whle recen revews 20 have appeared n he leraure ha dscuss he relave ers of he varous P and CP approaches hey rely on perforance daa ha ofen nvolve dfferen probles 2

3 and dsnc hardware and sofware ools. hs paper avods such laons by followng a ore hands-on approach where as uch as sx alernave odels are esed on he soluon of a large se of exaple probles concernng he shor-er schedulng of sngle/ulsage ulproduc bach plans feaurng sequence dependen changeovers. Also analysed s he nfluence of he objecve funcon on odel perforance. hs paper can be vewed as he hrd par of an exensve coparave sudy where he prevous wo have focused on sngle sage 2 and ulsage 22 probles feaurng eher sequence ndependen changeovers (seup es) or none a all. However hs paper goes beyond a ere coparson beween dfferen ehods snce wo of he are new. Boh are ulple e grd connuous-e odels ha are exensons o he one presened by Casro and Grossann 22 wh respec o he handlng of sequence dependen changeovers. hey are concepually dfferen on how changeover ass are handled: eher explcly whch s a ore general approach n ers of varey of objecve funcons ha can be handled effcenly or plcly n he odel consrans whch has he advanage of generang saller szed odels. Anoher poran novel aspec s he cobnaon of processng and changeover ass no a sngle se of ass whch conrbues o boh reducon n he nuber of odel varables and soluon degeneracy and hs s done for one of he new connuouse odels as well as for he dscree-e odel. he oher approaches nvolved n he coparson nclude a connuous-e odel wh global precedence sequencng varables 5 (SV) a CP odel based on OPL Sudo odellng language 23 and a hybrd LP/CP odel (sngle sage only). o ncluded s he sngle grd connuous-e odel of Casro e al. 4 sply because was shown o be a poor perforer n he prevous sudes 2-22 and due o he fac ha he nuber of even pons requred o fnd he opal soluon would ncrease even ore wh he consderaon of changeover ass whch would nevably lead o even larger LPs. Oher unfor e grd connuous-e forulaons 6 are expeced o have slar drawbacs and hus are also no consdered. he res of he paper s srucured as follows. Secon 2 defnes he schedulng proble under consderaon. Secon 3 gves a horough descrpon of he proble hghlghng soe of he concepually dfferen approaches ha can be used o odel. Concernng he handlng of 3

4 changeovers wo alernave connuous-e odellng opons wll be denfed leadng o he developen of wo new LP odels ha are presened n secons 4 and 5. he oher feaured approaches are descrbed n secon 6 whle secon 7 presens he dealed copuaonal sudes. he srenghs and laons of each approach are suarzed n secon 8 whle he conclusons are lef for secon Proble defnon n hs paper he shor-er schedulng proble of ulsage ulproduc bach plans s consdered. Gven are: a se of produc orders ha us follow a sequence of processng sages o reach he condon of fnal producs; a se of avalable equpen uns each belongng o a sngle sage wh se ncludng hose uns belongng o sage. Gven also are he duraon of he processng ass p and hose of he cleanng ass cl ; he release r and due daes d all beng enforced as hard consrans. Addonal daa conssng of he processng cos of order n achne c s requred whenever he objecve s he nzaon of he oal cos. s assued ha all orders go hrough all sages ha here s a unque sequence of sages for all orders and ha unled neredae sorage (US) s avalable beween sages. on-zero processng es are requred o allow for an order o be processed on a gven achne. Hence he se of orders ha can be processed n achne s defned by = { : p > 0}. s also assued ha any gven order s only execued once over he e horzon. hs aes possble o defne cl =0. 3. Concepual represenaon: process vs. odel enes Processes nvolvng sequence dependen changeovers are generally ore dffcul o odel han hose nvolvng sequence ndependen seup es or no seup es a all. Soe odels le hose usng global precedence sequencng varables (SV) (see secon 6.) or based on consran prograng (CP) (see secon 6.3) hardly requre any changes. n conras e-grd based approaches need o be sgnfcanly alered boh dscree- (D) and connuous-e approaches he laer rrespecve of beng sngle or ulple e grd odels. aurally odel adapably o a dfferen ype of proble s drecly relaed o he ypes of varables and consrans ha are used 4

5 whch n urn are derved fro soe concepualzaon. When developng a odel he odeler us ae a few fundaenal decsons ha wll have a ajor pac on s srucure and perforance. For hs specfc ype of schedulng proble here are wo an decsons. he frs concerns he way n whch he processng and cleanng ass are odeled. he second concerns he reaen of e. he processng and cleanng ass can be handled eher explcly as odel varables or ebedded n soe of he odel consrans. e grd-based approaches consder he processng ass explcly and bnary varables are used o denfy her sarng e pon on he grd. Concernng changeover ass dscree-e 7-8 and connuous-e R-based 24 odels defne he explcly whereas connuous-e S-based 9-2 as well as consran prograng odels 2 consder he plcly. Connuous-e odels based on sequencng varables 53 consder all ypes of ass plcly. he wo new ulple e grd connuous-e odels ha are gven n he nex couple of secons handle changeover ass n an oppose way. he frs and ore general one (n ers of varey of objecve funcons ha can handle) uses a se of bnary varables ha denfy he execuon of order n un sarng a e pon ogeher wh he changeover as o allow for order o edae follow on he sae equpen. hs groupng of processng and cleanng ass no a sngle se of varables nsead of consderng he separaely s a novel dea ha proves he perforance of he odel due o he use of fewer varables and reduces he degeneracy snce he cleanng as s perfored edaely afer he processng as has ended. oe also ha s possble o have = n cases where wll be he las order o be processed on he achne under consderaon (he duraon of he cobned as s equal o ha of jus he processng as snce cl =0) alhough s no andaory ha all he achnes end wh such ass snce any wll be non-lng achnes ha can have cl 0 whou coprosng he opal soluon. 3.. Resource as newor process represenaon he use of cobned processng and changeover ass (naed DO n he llusraons) gves rse o he Resource as ewor 2 (R) process represenaons gven n Fgure and Fgure 2 whch 5

6 are he bass of boh he ulple e grd connuous-e forulaon (see secon 4) and he dscree-e represenaon (see secon 6.2). s poran o ephasze ha he ass e ndex () s no poran for he represenaon eanng ha he Rs are vald despe he fac ha ndex has a slghly dfferen eanng n each forulaon as wll be seen laer on. hus hree ndces rean (wo order ndces and one achne ndex). he execuon of a as nvolves consupon/producon of several resources ha alhough reaed n exacly he sae way by he aheacal forulaons are spl no hree dfferen ypes o faclae odel undersandng: a) he equpen resources (he eleens of se ); b) he aeral saes whch are drecly assocaed o he order under consderaon and o he sage where he aeral s produced; c) he cleanng saes whch are lned o he order () ha equpen un s ready o handle. Due o large nuber of resources nvolved and for he sae of clary we have dvded he overall R no wo supersrucures. Whle Fgure focuses on he frs wo resource ypes and on he aeral saes odfcaons due o he processng par of he as Fgure 2 focuses on cleanng saes changes caused by he changeover par of he as. he forer R apples for a gven order (.e. ) whle he laer for a gven achne (.e. ). ( ) (- ) DO Duraon=p DO ( ) Duraon=p DO (- ) Duraon=p - DO 2_ DO 2_( ) DO 2_(- ) Duraon=p cl 2 Dur.=p cl 2 Dur.=p - cl DO DO ( ) DO (- ) Duraon=p cl Dur.=p cl Dur.=p - cl (-)... DO DO ( 2 ) DO Dur.=p cl Dur.=p 2 cl 2 Dur.=p cl DO Duraon=p DO ( 2 ) Duraon=p 2 DO Duraon=p ( 2 ) Sage Sage 2 Sage Fgure. R process represenaon for order feaurng a oal of achnes and sages (changes on he uns cleanng saes oed for splfcaon). 6

7 n Fgure he equpen uns and aeral saes (crcles) are represened whn he boundares of he correspondng sage (vercal dash lnes). For exaple sage 2 (2) ncludes all achnes belongng o se 2 whose eleens are hose of se wh nubers (f equpen nuberng s based on an ordered dsrbuon of uns aong sages) rangng fro o 2. A parcular cobned as represened as a recangle (e.g. DO 2_( ) consues boh he equpen un (e.g. ( )) where s processed and he correspondng aeral sae whch s assocaed o he sage pror o he one he as belongs o (e.g. ). Boh resources are consued when he as begns. he sae as also produces wo resources bu now hese evens can occur a dfferen pons n e. Whle he sae equpen un s produced when he as ends (e.g. ( )) hus regenerang he equpen resource (he assocaed dashed lnes have arrows n boh ends) he aeral sae (e.g. 2) can be produced earler (whenever cl 0) exacly p e uns afer he as has sared. hs s he reason why he orgn of he arrow ha denoes he producon of he aeral resource s furher o he lef. s also worh nong ha he explc consderaon of aeral saes as odel varables s only possble when usng a sngle e grd le for he dscree-e forulaon (see secon 6.2). he ulple e grd connuous-e forulaon (see secon 4) uses dfferen ses of varables and consrans. DO Duraon=p DO_2 Duraon=p 2 cl 2 DO Dur.=p cl DO 2_ Duraon=p cl 2 2 DO_2_2_ Duraon=p 2... DO 2_ Dur.=p cl DO Duraon=p cl DO_2 Duraon=p 2 cl 2 DO Duraon=p Fgure 2. R process represenaon for un showng all possble cleanng saes (changes on he orders aeral saes oed for splfcaon). For he cobned as o be execued n an equpen un us be n an approprae cleanng sae. For un he several possbles are llusraed n Fgure 2. ass wh he sae order ndex (= ) consue and produce he sae cleanng sae (besdes he equpen resource) respecvely a he sar and end of he as. However ass nvolvng dfferen orders are ore 7

8 frequen and nvolve dfferen cleanng saes. For exaple as DO_2 consues resources and 2 a s sar and produces resources and a s end. he nal cleanng sae of every equpen wll be a varable of he odel n order for he os convenen ones o be seleced. he second novel ulple e grd connuous-e forulaon (see secon 5) handles changeover es plcly so ha only he processng ass are consdered. As a consequence cleanng saes are no requred and he bnary varables ha denfy he execuon of he as nvolve only one order ndex. he R represenaon of he process feaurng hose ass s a splfed verson of he one shown n Fgure and s exacly he sae as he one used for handlng he sae ype of proble whou sequence dependen changeovers. Such supersrucure can be found n Casro and Grossann Handlng of e he several forulaons consdered n hs paper rea e dfferenly. he wo new ulple e grd connuous-e forulaons as well as he R-based dscree-e forulaon dvde he e horzon no a fxed nuber - of e nervals. he nuber of ass ha can f no he e horzon s grealy dependen on he nuber of e pons n e-grd based connuouse forulaons (also soees called even pons) and less so n dscree-e forulaons. Fewer e pons are also used by he forer ype of approach (a dozen s a praccal upper bound) whle he laer usually rely on ens or even hundreds of he. Anoher ajor dfference s ha whle he dscree-e forulaon feaures equal lengh (δ) nervals eanng ha he e correspondng o each e pon s nown a pror connuous-e forulaons rea hose es as odel varables. he new ulple e grd connuous-e forulaons use as he nae suggess several e grds o locae he ass. ore specfcally un specfc e grds are eployed. s assued ha all e grds feaure he sae nuber of even pons alhough s sraghforward o adap he o a dfferen nuber per grd. he raonale behnd hs opon s ha he use of a un dependen value for ncreases he nuber of a pror decsons o ae ha can affec he qualy of he fnal soluon and also because ha opon has been found o be an effcen one n sngle 8

9 sage 2 and ulsage 22 ulproduc schedulng probles slar o hose consdered here. everheless n soe probles ay pay off o develop a se of rules leadng o he specfcaon of a dfferen nuber of e pons for he several e grds bu hs s beyond he scope of hs paper. he selecon of he cardnaly of se nvolves he followng rade-off: oo few pons aes possble o fnd he global opu and oo any aes he proble nracable. Alhough only he fnal eraon wll be repored for each exaple proble a few LPs usually need o be solved n sequence by usng sngle ncreens n unl no furher proveens n he objecve funcon are observed. Due o he propery of every order lasng exacly one e nerval a useful lower bound on he opal value of s he followng: ax ( / ). he connuous-e grd assocaed o each equpen un s gven n Fgure 3 where he nu release dae s he lower bound on he absolue e of he frs e pon and he axu due dae as he upper bound on he absolue e of he las e pon. nerval nerval 2 nerval -2 nerval [n r ax d ] Fgure 3. Connuous-e grd eployed (one for each equpen un). n he dscree-e forulaon all he evens repor o a sngle e grd and he nervals are ofen called global e nervals. he lengh of each nerval δ s ofen chosen as he greaes coon facor beween es p and p cl snce as explaned n secon 3. here wll be evens occurrng a hese relave (o he sar of he as) es. Whenever he greaes coon facor leads o oo any e nervals eanng proble nracably a hgher value of δ s used and all he daa s rounded up o s nex neger ulple. he paral and oal duraon of he cobned ass are convered fro acual e uns o a e nerval bass by usng: τ = p /δ τ = p cl ) / δ. As for he release and due daes hey rean on he real ( e scale: r = /δ δ and d = /δ δ r d. oe ha roundng he proble daa ples he consderaon of an approxaed verson of he proble ha ay or ay no lead o he rue 9

10 opal soluon. Furherore o ensure feasbly would have been ore approprae o use = /δ δ d. he unfor dscree-e grd s gven n Fgure 4. d nerval nerval 2 nerval 3 nerval -2 nerval - δ n r ax d Fgure 4. Unfor dscree-e grd. he oher feaured connuous-e forulaon does no rely on explc e grds. nsead of allocang ass o dfferen e nervals a oally dfferen concep s exploed ha reles on sequencng varables o ensure ha every achne only handles one order a a e. As wll be seen n secon 6. he odel varables and consrans feaure no e ndex gvng an poran advanage when copared o he oher connuous-e and dscree-e forulaons: no decsons ha ay evenually coprose s soluon need o be aen before solvng he proble. ha s he odel needs only o be solved once and he resulng soluon wll always be an exac and global opal soluon (f solved o a zero opaly gap). he sae can be sad for he CP forulaon alhough hs s ore accuraely classfed as a dscree-e odel snce all he acves (.e. ass) us have neger duraons. 4. ew general ulple-e-grd connuous forulaon feaurng four-ndex varables (C4) Whle he wo new ulple-e grd connuous-e forulaons are novel n he sense ha hey can handle sequence dependen changeovers hey share any characerscs wh her sngle sage 2 and ulsage 22 predecessors. hus nsead of enerng a dealed explanaon of every aspec of he forulaons and drawng coparsons wh oher ulple e grd forulaons whch are gven n hose recen papers we wll focus osly on he aspecs relang o he novel feaure. An excepon ay occur as dscussed n secon

11 he forulaon uses he already enoned 4-ndex bnary varables o assgn he execuon of he cobned processng and changeover as o a parcular achne and also o a ceran e pon. he oher se of bnary varables used are he excess resource varables R ha denfy equpen avalably (=) a a gven even pon. he reanng varables all nonnegave are he ng varables and D whch represen he absolue e of e pons and and he ransfer e of order n sage respecvely and also he new varables C and 0 C. Whle C are also excess resource varables ha when equal o ndcae ha equpen s ready o handle order a e pon he laer arse fro he need o defne an nal sae. Fgure 5 gves an overvew of how he forulaon wors. For splcy consders only one equpen un per sage hree orders and hree sages. One poran propery of he aheacal forulaon s ha each cobned as only lass for a sngle e nerval eanng ha he requred nuber of e pons on each achne s equal o he nuber of orders assgned o ha achne plus whch for hs llusrave exaple ples 4 e pons. Alhough each as lass one e nerval does no necessarly ean ha when sarng a ends exacly a e pon. For nsance he execuon of order followed by cleanng o order 2.e. cobned as (2) whch sars a he frs even pon ends well before he e of he second even pon boh n (frs sage) and 2 (second sage). oe agan as already ephaszed when descrbng Fgure ha he begnnng of (23) n sage 2 occurs exacly afer he end n sage of he processng par of he cobned as (a 22 = 2 p 2 ) bu before s full copleon ( 22 2 p 2 cl 23 ). he changeover par of he as only affecs evens occurrng on he sae achne eanng ha he approprae cleanng e us be consdered beween he processng of dfferen orders. n he frs sage he e correspondng o he sar pon of he as us be greaer han he release dae of he order r. Accordngly n he las sage orders us be concluded before her due daes d. he ransfer e of aerals fro sage o sage us be greaer han he endng e of he order s processng par of he as a sage and us be lower han he hey can also be defned as connuous varables snce he odel consrans ensure ha R ={0}. However based on experence s beer o defne he as bnary varables.

12 order sarng e a sage. n Fgure 5 he ransfer e of order n sage (D ) can be any value [ p 2 ] (hs e span s represened as a gray-flled recangle) whereas ha of order 2 n sage (D 2 ) s equal o boh s lower and upper bounds: 2 p 2 = 22. As was enoned n secon 3 s no requred for he las as o be processed o feaure equal order ndces alhough hs wll be rue f such as s execued n he las e nerval (n Fgure 5 all equpen uns end wh (33)). he oher feaures ha are worh enonng are ha he e of he frs and las e pons respecvely n he frs and las sage do no need o ach neher he orgn (nu release dae) nor he e horzon (axu due dae) and ha order sequencng can vary fro one sage o he oher as seen for un 2 whch has -2-3 and un 3 whch has an 2--3 sequence. (2) (23) p p 2 cl (33) 2 r 2 r2 p 3 cl 23 3 r3 4 2 D D 2 D 3 (2) (23) (33) p2 cl 22 p 2 2 cl 232 p D 2 D 22 p 23 D 32 (2) (3) (33) cl 23 p 3 cl 33 p 33 n r 3 d 2 23 d d3 d ax = H Fgure 5. Possble soluon fro ulple-e grd connuous-e forulaon F (=3 =3 =3 and =4). here are oher characerscs of C4 ha are no apparen fro Fgure 5. ore specfcally he nuber of e nervals can exceed he opal nuber of ass execued on a gven achne or n oher words here can be e pons where no ass are sared. As a consequence hose e nervals can have a duraon rangng fro zero o he full (f no orders are assgned o he achne a hand) e span and can be locaed anywhere e.g. a he frs second or las e nervals. We oo full advanage of hs propery when developng he objecve funcon for oal earlness nzaon 20 whch bascally forces all duy e nervals o have zero duraon and be he las ones of he correspondng e grd. For oher objecve funcons he assgnen of orders o 2

13 e pons fro frs o las whch s llusraed n Fgure 6 s enforced by an approprae se of consrans (see eq 9) = = H 3 = Fgure 6. Possble soluon fro ulple-e grd connuous-e forulaon F (=3 =3 = and =3). Orders are assgned fro he frs o las e pons. he consrans ha copose he ulple-e grd forulaon are gven nex bu frs we wll defne soe addonal ses and paraeers ha allow o reduce he nuber of varables and hence ae he forulaon ore effcen. We sar by deernng he lowes e lb a whch order can sar o be processed n un. f he achne belongs o he frs sage hen he lower bound s he release dae of he order oherwse we have o add up o hs value he nu processng e n prevous sages (eq ). lb = r n < p () An upper bound (eq 2) can also be defned for order bu now hs wll depend on he order ha s processed nex. ub = n( d > n p p d n > p p cl p ) (2) hus he new paraeer ub feaures hree e ndces and s value can be se by eher order or. ore specfcally f order s he lng one he hghes e a whch he as can sar s deerned afer subracng fro s due dae he nu processng e on all subsequen sages plus he order processng e n un. f nsead order s he lng one a slar approach s followed n ers of. everheless addonal ers are requred o nclude p and also cl. oe also ha p s only consdered whenever. Eq 3 defnes se whch conans all orders ha can precede order n achne. 3

14 { : ub lb } = (3) Eq 4 defnes he earles e a whch un can becoe acve lb and s deerned for all e pons. n lb = n lb n( p cl ) ( ) (4) For = s equal o he nu possble sarng e of all orders whle for > we need o ae no accoun he duraon of he shores cobned as (noe ha n cl = n cl ) as any es as he nuber of exsng e nervals up o e pon. We could even be ore horough and replace he second er of eq 4 by he su of he wo salles hree salles ers for =4 and so on. p cl n ers for =3 he Fnally all hs nforaon can be cobned no he defnon of se (see eq 5) generang he doan of he bnary varables. oe ha n he las e nerval (ass sarng a =- ) only ass wh he sae order ndex can be execued snce here are no enough e nervals o process any ore ass. = : ub lb ( = )} (5) { 4.. Excess resource balance consrans he excess resource balances are ypcal ulperod aeral balance expressons n whch he excess aoun a pon s equal o he excess aoun a pon - adjused by he aouns produced/consued by all ass sarng or endng a. For he equpen resources can be sad ha he un s no beng used a e pon.e. s avalable n excess (R =) f no order sars o be processed n a oherwse here s no excess resource (R =0). he consrans are as follows. R = (6) he consrans relaed o he cleanng saes are slghly ore coplex sply because he execuon of a gven as usually nvolves producon and consupon of dfferen saes. Furherore he nal resource avalably for all equpen uns s no longer and are n fac 4

15 acual odel varables whch only appear n consrans belongng o he frs e pon (frs er on he rgh hand sde of eq 7). C C C = = 0 (7) Eq 8 ensures ha here s bu one nal equpen sae for each achne. C = 0 (8) o reduce soluon degeneracy and o prove he perforance of he odel we enforce ass o be allocaed o e pons wh as low an ndex as possble. hs s he sae as sayng ha equpen avalably ncreases fro sar o fnsh. R R (9) 4.2. ng consrans he dfference beween he absolue es of any wo e pons us be greaer han he duraon of he cobned as. )] ( [ cl p (0) Eq ensures ha he absolue e of e pon n un s greaer han s predeerned lower bound (see eq ). oe ha for achnes belongng o he frs sage we ge release dae consrans. he global lower bound as already shown n Fgure 3 s he nu release dae (eq 2). lb () r n (2) he nex consran s he equvalen upper bound consran where ub s calculaed hrough eq 2. hs s a bg- consran eanng ha s only acve when here s a as sarng a n un oherwse s relaxed o s global upper bound eq 4. ) ( H ub (3) 5

16 d H = ax (4) For ulsage probles we need o relae he absolue es of consecuve sages by eans of he ransfer e varables D. Whle eq 5a ensures ha he order ransfer e n sage - s earler han s sarng e n sage eq 6a saes ha s ransfer e n sage us be greaer han he order copleon e (jus he processng par of he as) n ha sage (processed n achne ). Boh are bg- consrans ha only becoe acve f he as sars a even pon belongng o e grd. Eqs 5a-6a can be replaced by consrans 5-6 whch usually lead o a beer perforance. hese nclude ore bnary varables nsde he bg- er whch s possble snce order can only be processed once on each achne (see eq 2). hus eq 5 ncludes all ass (fro order ) ha sar a or before ang he gher when solvng he relaxed odel and hopefully geng paron of he ass over fewer e nervals whch faclaes branchng. ) ( H D (5a) ) ( H p D (6a) ) ( H D (5) ) ( H p D (6) Whenever he objecve s aespan nzaon a new varable s requred (S) ha us be greaer han he endng e of all ass. Eq 7 s a consran ha ensures hs goal by relang he varable o he sarng e of all e pons. can be descrbed as follows. When appled o sage un and e pon he second er on he rgh-hand sde s only acve when and represens he processng e of he order sarng a e pon n un plus s nu processng e n he followng sages. s equvalen o he er used for ulsage plans whou sequence dependen changeovers 22. On he oher hand he hrd er on he RHS s only acve when dealng wh he las sage and represens he duraon of all cobned ass sarng n un a or afer e pon. s orgn resuls fro perforance ess perfored whle solvng 6

17 sngle sage plans and s a new er snce he objecve of aespan nzaon was no consdered n Casro and Grossann 2. )] ( [ )] n ( [ cl p p p S = > (7) he aespan varable can also be relaed o he ransfer e varables by a slar consran. Alhough no srcly necessary eq 8 proves he perforance of he forulaon. p D S > (8) he fnal ses of ng consrans are also boh effcen and oponal. hey ac as lower and upper bounds on he ransfer es and are concepually equvalen o eqs and 3 alhough now snce he consrans are per order and no per un he processng es are ulpled by he approprae bnary varables nsead of consderng he nu possble values (whch are plc n paraeers lb and ub see eqs -2). oe also ha no bg- ers are requred for he upper bound consrans. p r D (9) p d D > (20) 4.3. Operaonal consrans he sngle se of operaonal consrans saes ha all orders us be processed exacly once n every sage. = (2) 7

18 4.4. Objecve funcons he aheacal forulaon can handle he hree alernave objecve funcons consdered n hs paper. hese are oal cos nzaon eq 22 oal earlness nzaon eq 23 and aespan nzaon eq 24. n c (22) n d ( p ) H[ ( )] (23) n S (24) n suary he forulaon feaures consrans 6-6 and 9-2 as s buldng bloc. he objecve of oal cos nzaon also requres eq 22 ha of oal earlness nzaon eq 23 whle aespan nzaon also uses eqs 7 8 and ew ulple-e-grd connuous forulaon feaurng hree-ndex varables (C3) he second new connuous-e forulaon uses bnary varables wh jus hree ndces and as a consequence gves rse o uch saller aheacal forulaons han C4. he os sgnfcan concepual dfference coes fro he fac ha does no need o consder explc cleanng saes for he equpen uns snce soe of he ng consrans ae sure ha he approprae cleanng e s aen no consderaon. Because of hs s closer o he ulple e grd forulaon of Casro and Grossann 22 for ulsage plans whou sequence dependen changeovers. However as wll be descrbed nex he sraegy used for plcly handlng changeovers s copleely oppose o he one used for nzng oal earlness eanng ha he prevously developed for of hs objecve funcon (as gven n eq 23) s ncopable wh he new forulaon. Fgure 7 llusraes how C3 wors wh a sple sngle sage exaple snce he dfferences fro C4 occur whn he equpen uns (he ransfer of aeral beween sages s slar). Before gong no he deals wo new ses of paraeers need o be defned. Frs he axu changeover e fro order n un s calculaed hrough eq 25. hen eq 26 deernes he dfference beween he axu and acual changeover es fro order o n un. Wh a 8

19 vew o accoun for he acual process and changeover e an ndrec procedure s used. Assgnng order o un a e pon aes he lengh of nerval (f no he las) o equal a leas p ax cl whch corresponds o consder he wors case scenaro n ers of changeovers. o ge he rue changeover e o order we us subrac cl Δ f as s o be perfored a as can be seen beween orders and 4 n and also 5 and 2 n 2. s worh nong ha he dfference beween he absolue es of wo consecuve e pons ay even be greaer f for exaple he release dae of he followng produc s locaed furher ahead n e (see 2 n Fgure 7). hs echnque aes ore advanageous for wo consecuve orders o be execued n consecuve e nervals. Furherore all possble feasble soluons are covered snce he axu changeover er s no consdered for ass sarng a he las e nerval snce n such case only he processng e s used. hs can be seen for orders 4 2 and 3. As a consequence orders wll ypcally be assgned fro he las bu one o he frs e pons n an oppose anner o F. cl = ax cl ax (25) cl = cl cl Δ ax (26) 2 ax cl p p 4 Δ cl 4 r d 2 r4 ax cl d4 5 2 Δ p cl p r d = r2 d n r 3 23 r3 3 p 33 d3 ax d 33 = H Fgure 7. Possble soluon fro ulple-e grd connuous-e forulaon F2 (=5 =3 = and =3). Orders are assgned fro he las bu one o he frs e pons. 9

20 he consrans of he hree-ndex bnary varables connuous e forulaon are gven nex. When copared o Casro and Grossann 22 only hose ha were odfed o accoun for sequence dependen changeovers are explaned ogeher wh new ses of consrans. R = (27) (28) R R Equaon 28 s analogous o eq 9 and leads o proved perforance. he dea s once agan o reduce he nuber of degenerae soluons now by forcng all orders o be assgned fro he las o he frs e nerval. As a consequence un avalably wll decrease fro sar o fnsh wh he excepon of he las e pon whch corresponds o he end of he e horzon where all equpen uns becoe avalable. ) ( ax cl cl p Δ (29) Equaon 29 s he roo of C3 and has already been descrbed whle explanng Fgure 7. he handlng of sequence dependen changeovers aes order aggregaon no longer possble so he doan of eq 29 now feaures hree ndces: and. he frs er on he RHS can be replaced by p whch has he advanage of ang he consran easer o undersand bu has he dsadvanage of ang less gh and hence less effcen. lb (30) p p d hb = > n (3) ) ( H hb (32) Paraeer hb represens he hghes e a whch order can sar o be processed on un and s used n eq 32 o defne upper bounds for he absolue e of he several e pons. oe ha eq 32 ranslaes no he due dae consran whenever. ) ( H D (33) 20

21 (34) ) ( H p D cl cl = n n (35) )] ( [ )] n ( [ n cl p p p S = > (36) (37) p D S > p r D (38) p d D > (39) = (40) Equaon 36 s equvalen o eq 7 bu whle we can use he acual changeover e n he las er on he RHS of he laer for he forer snce changeovers are odelled plcly we are led o use he nu cleanng e as calculaed by eq 35. n c (4) he aheacal forulaon can handle as he objecve funcon he nzaon of eher he oal cos or he aespan eqs 4 and 24 respecvely. s core consss of consrans and plus eq 4 for oal cos and eqs and 37 for aespan. 6. Oher approaches he wo new connuous-e forulaons wll be copared o oher four concepually dfferen approaches. hese nclude a connuous-e forulaon based on global precedence varables (SV) nsead of relyng on an explc e grd; a unfor e grd dscree-e forulaon (D) a consran prograng (CP) odel and for oal cos nzaon of sngle sage plans a hybrd 2

22 LP/CP approach. n hs secon we hghlgh her an feaures and presen he changes requred o effcenly acle he probles under consderaon. 6.. Connuous-e forulaon wh global precedence sequencng varables (SV) he oher feaured connuous-e forulaon also gves rse o an LP and s essenally he one of Harjunos and Grossann 5 whou he operaonal desgn varables. Bnary varables y are eployed o assgn order o un and bnary sequencng varables x are used o denfy he global precedence of order over n sage. Based on hese he endng es f of any wo orders can be relaed hrough he bg- consrans gven n eqs hese now feaure an exra er (he hrd on he RHS) o accoun for he sequence dependen changeover es whch he odel by Harjunos and Grossann 5 dd no consder. Oher proveens concernng he effcen handlng of oher objecve funcons ha are relevan o hs wor can be found n Casro and Grossann 22. f f p cl x H ( 3 x y y ) > (42) f f p cl ( x ) H (2 x y y ) > (43) 6.2. R-based dscree-e forulaon (D) he dscree-e forulaon used s based on he orgnal wor of Paneldes 8 despe he fac ha he dfferen ypes of resources are no aggregaed. reles on he Resource as ewor process represenaon le C4 and C3 and hus has a few slar feaures wh s connuouse counerpars. ore so wh C4 snce also consders cobned ass and he sae ype of bnary varables. he dscree-e grd however reles on a sngle and unfor e grd ang easer o consder alernave aeral saes (hrough varables S ) and also avodng he use of ng varables. Due o he fac ha he e correspondng o each e pon s nown a pror we can deerne he lowes and hghes pons a whch each as can sar. Despe referrng o e pons nsead of acual e values eqs 44 and 45 are slar n concep o eqs and 2. n eqs 44 noe ha n r / δ 22

23 represens he e of he frs e pon n ers of nuber of e nervals (see also Fgure 4). Snce he hrd er on he RHS calculaes he nu possble duraon n prevous sages (also n nuber of e nervals) he frs and hrd er n eq 44 n fac deerne he nuber of e nervals beween he lowes possble sarng pon of he as and he frs e pon. f here are no nervals n beween hen he as can sar a he frs e pon and hs s he reason why we add a (second er n he RHS). he doan of he odel bnary varables s hen calculaed hrough eq 46. r r lb = < n ) / n ( τ δ (44) / n ) n / n / n( r d d ub = > > δ τ τ τ δ τ τ δ (45) ub lb = } : { (46) he dscree-e forulaon can be wren n a very copac for. uses 5 ses of varables and consrans (eq and eq 8) plus he objecve funcon: eq 5 for oal cos nzaon and eq 52 for oal earlness nzaon. he odel consrans are gven below and wh he excepon of he excess resource balances for he cleanng saes (varables C ) whch are new are slar o hose of Casro and Grossann 22. he reader s dreced o hs reference for a dealed explanaon of he odel consrans and also for he echnque used for aespan nzaon whch nvolves solvng he proble several es for dfferen cardnales of whle nzng oal earlness. R R = = ) ( τ τ (47) S S = τ τ (48) C C C = = ) ( 0 τ τ (49) = (50) c n (5) 23

24 n Z = d n r [ ] ( τ ) δ (52) 6.3. Consran prograng forulaon (CP) he consran prograng (CP) forulaon used s bascally he one presened by Harjunos and Grossann 5 whch s based n LOG s OPL Sudo odellng language 23. However he ssue of sequence and achne dependen changeovers s dffcul o pleen n OPL Sudo code and could only be handled afer advce fro LOG suppor saff. For hs reason we fnd relevan o nclude he full odel. he reader s also dreced o aravelas and Grossann 4 for a bref descrpon of OPL Sudo global consrans and specal consrucs specfcally developed for schedulng applcaons. he wo an coponens of schedulng odels n OPL Sudo are acves and resources. he acves correspond o he processng ass and are referred o a gven order (belongng o he enueraed ype Orders) and o a gven sage (belongng o Sages rangng fro o sages). Snce an approprae changeover e us pass beween consecuve acves we defne a ranson ype ha wll access he ranson arx gven he approprae eleen n hs case see eq 53. he ranson arx s gven by paraeer chgover[unsordersorders] (equvalen o cl ) and s assocaed o he approprae equpen un belongng o he range ype Uns. he uns are defned as unary resources snce hey can only be used by one acvy a a e (eq 54) and are also he eleens of he group of achnes he alernave resources (eq 55). oe ha n eq 54 he ranson arx s referenced wh only one () of s hree ndces. Also declarng ha he several uns are alernave fro an acvy sandpon s absoluely val o ensure an effcen CP odel. Furher relevan declaraons nvolve he bnary assgnen varable y[] and he boundares of he e horzon whch are relaed o he nu release and he axu due daes (eqs 57-58). Acvy DO[ n Orders n Sages] ransonype ; (53) UnaryResource un[ n Uns] (chgover[]); (54) AlernaveResources achnes(un); (55) var n y[ordersuns] n 0..; (56) 24

25 scheduleorgn=n( n Orders) r[]; (57) schedulehorzon = ax( n Orders) d[]; (58) he odel consrans are gven nex. Eq 59 saes ha order can only sar o be processed on he frs sage afer s release dae. he execuon of order n he las sage us also end before s due dae eq 60. Each acvy needs o be perfored n one equpen belongng o he group of alernave resources eq 6. he duraon of acves belongng o sage s hen n effec bounded by he nu and axu processng es of he order n ha sage (eq 62). Eq 63 saes ha f un s seleced o process order n sage hen he duraon of he acvy us equal he achng processng e. Also he correspondng assgnen varable us equal. f un does no belong o sage or canno process order hen canno be seleced o perfor he acvy eq 64. Fnally eq 65 saes ha order can only be processed n sage afer gong hrough he prevous sage. DO[].sar r[] (59) DO[sages].end d[] (60) DO[] requres achnes (6) n p[ ] DO[ ].duraon ax p[ ] (62) acvyhasselecedresource(do[ ] achnes un[ ]) DO[ ].duraon = p[ ]& y[ ] = (63) no acvyhasselecedresource(do[ ] achnes un[ ]) (64) DO[] precedes DO[] (65) hree alernave objecve funcons eqs for oal cos oal earlness and aespan nzaon are respecvely gven by: n y [ ] c[ ] (66) n ( d [ ] DO[ sages].end) (67) n Z = ax DO[ sages].end (68) 25

26 6.4. Hybrd forulaon (LP/CP) he hybrd odel of Jan and Grossann 6 ogeher wh he napsac consrans of aravelas and Grossann 7 o prove he neger cus s also an effcen opon for sngle sage probles where he objecve s oal cos nzaon. uses a splfed verson of odel SV one where only he assgnen varables are consdered o deerne opal assgnens of orders o achnes. Snce no sequencng varables are used soe assgnens ay be nfeasble soehng ha s checed hrough he soluon of a CP feasbly proble for each equpen un. For each nfeasble un neger cus are added o avod geng he sae assgnens on he nex soluon of he LP. Several eraons are usually requred unl all achnes are proved feasble eanng ha he opal soluon has been found. Alhough he sae decoposon sraegy can be used wh oher objecve funcons for sngle sage probles he ehod s lely o worsen as he CP s requred o solve an opzaon raher han a feasbly proble. For ulsage probles and for oal cos nzaon Harjunos and Grossann 5 also red a hybrd LP/CP ehod and found ha unle n sngle sage probles vald cus are raher wea and ha a large nuber of eraons can be expeced before he opal soluon s found. Alhough he auhors devsed sronger heursc cus hey soees cu off he rue opal soluon. n vew of he above he use of he hybrd LP/CP approach was no consdered n cases oher han oal cos nzaon for sngle sage probles. 7. Copuaonal Resuls n hs secon he perforance of he sx dfferen approaches s llusraed hrough he soluon of 39 exaple probles. hese are denfed by a nuber and wo addonal characers where he las denfes he objecve funcon beng consdered e.g. C for oal cos E for oal earlness and for aespan nzaon. os of he daa has been aen fro he exaple probles gven n Casro and Grossann 2-22 alhough he changeover es where generaed randoly (up o a axu of 60% of he uns average processng e). Snce hese changeover es ae up a lo of space n ables we have oped no nclude he daa n he paper and gve nsead as supporng nforaon (os challengng exaples only: P5-P6 P-P3). 26

27 For solvng he LPs resulng fro he connuous and dscree-e odels (C4 C3 SV D) we have used coercal solver GAS/CPLEX 9. wh a relave olerance of E-6 and all probles were solved o opaly unless oherwse saed. he consran prograng (CP) and hybrd (LP/CP) odels where pleened and solved n LOG s OPL Sudo Concernng hardware a copuer conssng of a Penu GHz processor wh GB of RA and runnng Wndows XP Professonal was used. he resuls have been grouped by ype of proble under consderaon sngle or ulsage and hen by objecve funcon. An overvew of he copuaonal effor s gven n ables whle ore dealed copuaonal sascs for soe of he probles are lef for ables 2 and 6. he dscusson of he resuls s gven n secons 7. and 7.2 by ype of proble. 7.. Sngle Sage Probles he sngle sage probles under consderaon range fro 2 orders n 3 equpen uns o 20 orders n 5 uns. Alhough a greaer nuber of probles could be consdered he large aoun of copuaonal resources used suggess hs se corresponds o a represenave se oal cos nzaon As can be seen fro able he wo new ulple e grd connuous forulaons are he bes perforers for oal cos nzaon by a sgnfcan argn n relaon o all bu he hybrd LP/CP odel. We were surprsed by he fac ha C4 was ore effcen han C3 parcularly n P5C. n able 2 one can see ha C4 exceeds he nuber of bnary varables eployed n C3 by a facor of 0 has a slghly lower bu slar negraly gap and s solved faser by alos wo orders of agnude. odel SV requred even fewer bnary varables and alhough can always fnd he global opal soluon faled o prove opaly n hree cases (P3C P5C and P6C) eher because he axu resource l was acheved or because he solver ran ou of eory. he CP odel exhbed a beer perforance han SV bu faled o fnd he opal soluon for P5C. A he boo of he ls coes he dscree-e forulaon (D) whch due o he large nuber of e pons ha are requred o handle he exac proble daa could only solve approxae versons of he probles. For nsance P2C needs 9 e pons for δ=2 whch resuls n a good 27

28 approxaon of he proble daa. For δ=5 he cobned processng es are soewha overesaed bu despe hs he opal soluon can sll be found n soe cases (P3C P5C) oal earlness nzaon For oal earlness nzaon C4 connues o be he bes perforer even hough fals o fnd he opal soluon for P6E see able 3. For hs proble we have o rely on D whch s also a very good perforer. n parcular all exaple probles excep P5E (δ=5) could be solved wh he exac proble daa whch led o a axu of 47 e pons n PE and a very large proble sze wh 38 e pons and a oal of bnary varables sngle varables and 4003 consrans for P6E. he oher wo forulaons ha can handle hs objecve funcon have a sgnfcan decrease n perforance when copared o oal cos nzaon. SV generaes search rees ha explode n sze raher rapdly and hence lead o he solver runnng ou of eory (P3E- P6E). CP perfors slghly beer snce whle also falng o fnd he opal soluons for P3E P5E and P6E fnds beer soluons for he forer exaples. Furherore solves PE-P2E sgnfcanly faser. Proble P3E s he os neresng proble of he lo snce he CP forulaon ernaed wh an opal soluon (56) ha s n fac subopal. hs fac allowed us o denfy he os sgnfcan laon of he CP forulaon whch s also a laon of odel SV. Before gong no he dealed explanaon le us provde soe relevan proble daa. he opal soluon of P3E feaures an opal sequence n. he processng es are gven by p 3 =3 p 4 =23 p =83 p 0 =73 he correspondng changeover es by cl 34 =2 cl 4 =8 cl 0 =2 he release daes by r 3 =40 r 0 =0 r =50 r 4 =60 and he due daes by d 3 =30 d 4 =200 d =300 d 0 =370. Also requred s he daa eleen cl 3 =39 and he orders opal delvery daes: and 370 respecvely. he opal schedule for un s gven n Fgure 8 ogeher wh he bes soluon ha can be obaned by odels CP and SV. he dfference beween he wo schedules s nal n he opal soluon (shown above) he sarng e of order 3 s delayed up o an absolue e of 62 allowng o end 2 e uns laer where 2 s he exac dfference beween he opal oal earlness values (559 vs. 56). So why canno order 3 sar earler n he soluon of odels F3 and F5? he reason les n oher odel consrans ha relae 28

29 o s global successors and preven hs fro happenng. Based on SV (for CP he explanaon s he sae alhough such consran s plc) eq 42 when appled o he orders and equpen un n queson explcly saes ha f f 3 22 eanng ha he dfference beween he endng es of orders and 3 us be greaer han 22 (he processng e of order plus he changeover e fro 3 o ). hus we canno ae full advanage of he fac ha here s one order 4 ha can f beween 3 and. Alhough s unlely ha such cobnaon of processng daa occurs n a real ndusral envronen hs exaple clearly hghlghs one of he srenghs of e grd-based odels when copared o approaches based on explc or plc sequencng of ass cl 3 = 39 Fgure 8. Par of he opal schedule for exaple P3E. Opal soluon (above) and subopal soluon (below) fro connuous-e LP wh global precedence sequencng varables and CP odels aespan nzaon he objecve of aespan nzaon s he os dffcul for he ulple e grd connuous forulaons. However C4 and C3 can always fnd good soluons o he proble see able 4. Due o s sgnfcanly larger sze he LPs resulng fro C4 end o orgnae faser growng branch and bound search rees eanng ha he solver runs ou of eory faser for he wo os dffcul probles (P5-P6). everheless and despe he larger sze C4 sees agan o be slghly beer despe C3 beng alos hree orders of agnude faser for P4. he oher connuous-e forulaon SV has a slar perforance when copared o he ulple e grd forulaons and also suffers fro he sae proble of generang fas-growng rees. he CP odel s he bes overall perforer for aespan nzaon snce s he fases for P-P3 and can also fnd he bes soluon for P5 for whch he opal soluon s sll 29

30 unnown. oe however ha s perforance for P6 s raher wea snce he bes soluon found afer ore han 5 h of copuaonal e 237 s sll far fro he bes nown soluon of 64. s also clear fro able 4 ha as he nuber of orders and equpen uns ncreases so does he copuaonal effor. he D odel has he er of fndng he bes soluon for P6 and dong so whle solvng an approxaed verson of he proble (δ=2). s far o say based on he values of δ ha can be used ha he perforance of D for aespan nzaon les beween ha observed for oal earlness and oal cos nzaon. he fac ha s beer han for oal cos nzaon s soewha surprsng snce aespan nzaon usually nvolves several eraons 2226 before he opal soluon s found. n hs respec s worhwhle enonng ha he observed D pea perforance for P4 s sply because he predced nu nuber of e nervals ensures feasbly and hence only one eraon s requred n he search for he opal soluon. hs unusual behavor (for exaple P requres a oal of 53 eraons) s sply because he bolenec for P4 les wh he release dae of a parcular order. n oher words he opal aespan s equal o he earles possble endng e of ha order ulsage Probles he seven ulsage probles under consderaon range fro 8 orders n 6 uns and 2 sages (6 baches) o 5 orders n 4 uns and 2 sages (30 baches) and o 8 orders n 8 uns and 4 sages (32 baches) oal Cos nzaon he resuls gven n able 5 show ha he CP and SV odels are he bes perforers for oal cos nzaon. Whle CP was always able o prove opaly SV faled o do so for P9C (he solver ran ou of eory afer ore han 2000 CPUs) bu anaged o solve all oher probles n less han 7 CPUs. he connuous-e forulaons were able o fnd he global opal soluons for all probles excep P2C. For ha proble boh C4 and C3 could no fnd even a feasble soluon up o he axu resource l of and CPUs respecvely. hs behavor was no oally unexpeced snce he ulple e grd odel fro whch hey orgnae exhbed 30

31 he sae dffcules 22 when solvng a proble also nvolvng 4 sages. Fnally he dscree-e forulaon s by far he wors perforer and he gap s ore sgnfcan han for sngle sage probles. Only relavely coarse (δ=5) e grds could be consdered and even ha value generaed very dffcul aheacal probles for P9C (no feasble soluon) and P3C (unable o prove opaly afer CPUs). s worh nong ha we now for sure ha P9C s feasble for (δ=5) snce we were able o fnd a feasble soluon for P9E for he sae δ value (see able 7) oal Earlness nzaon Le for sngle sage probles he new C4 forulaon s he bes perforer for oal earlness nzaon (see able 7). All probles can be solved o opaly n less han one hour and was able o fnd he bes soluon for P3E. However snce opaly was proved for 9 e pons only we do no now for sure f hs s n fac he global opal soluon. he oher approaches were unable o confr hs fndng: boh SV and CP can only ge o nferor soluons and D can only solve an approxaed verson of he proble. We can assue ha he bes soluon found for PE s he global opal soluon snce C4 obaned he sae resul boh for 6 (values repored n able 7) and 7 (opaly proved n 5900 CPUs) e pons. Concernng he dscree-e forulaon hs objecve enables us o use fner e grds (δ=2 for P7E and P8E) han for he wo ohers bu s he wors forulaon of he group aespan nzaon he resuls of able 8 show ha he CP odel eerges as he bes approach for aespan nzaon snce all probles can be solved o opaly n approxaely one hour. Wh he excepon of P3 he SV odel s also successful a fndng he opal soluons bu fals o prove opaly for P8 P and P3. Concernng he novel ulple e grd forulaons C3 s a leas as copeve as SV and beer han C4 whch reurned slghly worse soluons for P and P3. 3

32 8. Overvew of an Feaures of Alernave Forulaons n order o ar he end of an exensve coparson beween several concepually dfferen approaches of whch hs paper s he hrd par we fnd convenen o suarze her an characerscs and sugges a ranng. able 9 provdes he os relevan conclusons. 8.. e Grd-based Connuous-e Forulaons he research 222 has shown us ha he use of a sngle e grd o solve sngle/ulsage probles no nvolvng shared resources such as ules or anpower s clearly a bad opon. he resuls were based on he R forulaon of Casro e al 4 whch nvolves he need o specfy boh he nuber of e pons and he nuber of e nervals ha any as can span. Alhough recen developens 6 have brough a forulaon ha does no need he laer specfcaon o acheve a good perforance n ulpurpose probles we beleve ha he an drawbac of such sngle e grd forulaons les wh he large value of ha s requred o fnd global opal soluons o he proble. hs proble can becoe ore severe when sequence dependen changeover ass are nvolved snce he end of he processng as and s subsequen cleanng as wll generally occur a dfferen e pons. aurally he use of a sngle e grd s ore advanageous o odel he ransfer of aeral beween consecuve sages so can sll be useful n sall probles nvolvng shared resources and no feaurng sequence dependen changeovers. n vew of he above he developen of ulple e grd forulaons was he nex logcal sep. her ably o f any as be jus processng or cobned processng and changeover no a sngle e nerval was a sgnfcan acheveen snce allowed o consder he nu possble nuber of e pons per grd.e. one e nerval per order allocaed o he equpen un n queson. Usng a saller nuber of e pons n any gven e grd sees o be he ey ssue snce he developed ulple e grd forulaon acually requres ore e pons n oal ( ) han hose used by sngle e grd forulaons. he use of ulple e grds also allowed for a very effcen way of dealng wh he objecve of oal earlness nzaon anly because s he only connuous-e aheacal prograng approach o he bes of our nowledge for whch he soluon of he relaxed proble (LP) can dffer fro zero correspondng 32