Generalized Disjunctive Programming as a Systematic Modeling Framework to Derive Scheduling Formulations

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1 Ths s an open access arcle publshed under an ACS AuhorChoce Lcense, whch pers copyng and redsrbuon of he arcle or any adapaons for non-coercal purposes. Arcle pubs.acs.org/iecr Generalzed Dsjuncve Prograng as a Syseac Modelng Fraework o Derve Schedulng Forulaons Pedro M. Casro*,, and Ignaco E. Grossann Undade Modelac aõ e Opzac aõ de Sseas Energe cos, Laboraoŕo Naconal de Energa e Geologa, Lsboa, Porugal Deparen of Checal Engneerng, Carnege Mellon Unversy, Psburgh Pennsylvana 5-890, Uned Saes *S Supporng Inforaon Downloaded va on Noveber 7, 08 a :6:4 (UTC). See hps://pubs.acs.org/sharnggudelnes for opons on how o legaely share publshed arcles. ABSTRACT: We propose lnear generalzed dsjuncve prograng (GDP) odels for he shor-er schedulng proble of sngle sage bach plans wh parallel uns. Three dfferen conceps of connuous-e represenaon are explorededae and general precedence, as well as ulple e grds. The lnear GDP odels are hen reforulaed usng boh bg-m and convex hull reforulaons, and he resulng xed-neger lnear prograng odels copared hrough he soluon of a se of exaple probles. We show ha wo general precedence odels fro he leraure can be derved usng a bg-m reforulaon for a se of dsjuncons and a convex hull reforulaon for anoher. The bes perforer s, however, a ulple e grd odel whch can be derved fro he convex hull reforulaon followed by sple algebrac anpulaons o elnae he dsaggregaed varables and reduce he ses of consrans, hus leadng o a ore copac and effcen forulaon.. INTRODUCTION The las weny years have seen a varey of schedulng odels by he process syses engneerng couny, where ajor developens have been he unfed fraeworks for process represenaon., The sae and resource-ask neworks provde a syseac way of converng he real plan enes (e.g., reacors, producs, ulesanpower) no vrual enes ha can be used by a aheacal odel. They have been proposed wh dscree-e opzaon odels ha lke os schedulng odels feaure varables and consrans derved fro nuon and by ral and error so as o axze copuaonal perforance. Thus, and n conras o he dervaon of he process odel, no aep has been ade o unfy he aheacal odelng par. Ths aspec becoes parcularly relevan when swchng fro a dscree o a connuous-e represenaon, whch has ore do fro a odelng pon of vew. In fac, few dfferen approaches have been proposed ha can be classfed as precedence, 4 6 sngle, 7 9 or ulple e grd based. 0 8 Soe of he laer un-specfc approaches, when appled o he ore general ulpurpose producon envronen, have rased serous concerns over he years wh respec o he generaly, suggesng ha () eher he underlyng e represenaon concep does no accoun for all possbles or () no all he requred varables or consrans are effecvely par of he odel. I s hus hghly desrable o have a syseac odelng fraework ha sarng wh a soewha sple concep for e represenaon can generae he consrans ha wll ake work. Generalzed dsjuncve prograng 9 provdes a hgh level fraework for odelng xed-neger progras sarng fro a logc represenaon n whch xed-neger logc s represened hrough dsjuncons and neger logc hrough proposons. In fac, dfferen forulaons can be derved usng for exaple bg-m 0 and convex hull reforulaons, whch have copleenary srenghs. The forer has he advanage of beng spler and does no requre he defnon of addonal connuous varables and consrans, bu ay coprose copuaonal perforance. On he oher hand, he convex hull reforulaon s gher, whch generally helps o speed up he search procedure. GDP forulaons have prarly been proposed for process nework 9, 5 probles where dsjuncons reflec he opons of choosng a un, whch enforces ass balances and s assocaed o a ceran cos, or no. Oher exaples nclude srp-packng 6 and ulsage schedulng probles feaurng a sngle un per sage and he general precedence concep. 9,6 In hs paper, we propose o exane he relavely sple case of sngle sage schedulng probles wh sngle and parallel uns focusng on connuous-e represenaons and specfcally on he edae, general precedence, and ulple e grds alernaves. Sarng wh he sngle un case and nex for he case of parallel uns, GDP odels are forulaed feaurng lnear consrans ogeher wh logcal expressons nvolvng he Boolean varables of he dsjuncons. On he bass of hese GDP odels, we derve he equvalen xed-neger lnear prograng odels usng he bg-m and convex hull reforulaons. The an goal wll be o show ha well-known odels fro he leraure can be raced back o a parcular eleenary e represenaon/reforulaon par. Ther perforance s evaluaed on a se of es probles wh up o 0 orders and 5 parallel uns.. FUNDAMENTALS We consder he followng lnear generalzed dsjuncve progra: 9,, Receved: Deceber 7, 0 Revsed: Aprl 4, 0 Acceped: Aprl 5, 0 Publshed: Aprl 6, 0 0 Aercan Checal Socey 578 dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

2 Indusral & Engneerng Chesry Research Arcle Table. Consran Represenaon of Logc Proposons and Operaors n f( x) s.. gx ( ) 0 jk, k K j D A x b k jk, jk, Ω ( ) = rue n x, x 0, {rue, false} jk, (GDP) where f(x) and g(x) are lnear funcons, and dscree choces are expressed wh Boolean varables j,k n ers of dsjuncons and logc proposons Ω(). I s assued ha he dsjuncve se K s proper n ha he nersecon over j D k of he feasble regons defned by he se of pons x: A j,k x b j,k s epy. Hence he use of he exclusve OR operaor ( ) separang he dsjuncons. Dsjuncons can be ransfored no xed-neger lnear for hrough bg-m, 0 Beauon surrogae, 7 and convex hull reforulaons. Gven he fac ha for lnear consrans, he bg-m and Beauon relaxaons are equvalen, 7 he laer s no consdered n hs work... Bg-M Reforulaon. The sples represenaon of he dsjuncons k K n xed-neger lnear for are he bg- M consrans gven n (BM). Noce ha bnary varables y j,k have a one-o-one correspondence wh Boolean varables j,k and ha he las expresson gves he ghes value for paraeers M j,k, where x L and x U are he lower and upper bounds of varable x, respecvely. I s well-known ha hs se of consrans ofen yelds poor relaxaons. A x b + M ( y ) k K, j D y jk, jk, jk, jk, k jk, j Dk jk, y = k K = {0, } jk, jk, jk, L = ax{ a x b : x x x } k K, j D (BM) where j,k s he h row of M j,k and a j,k and b j,k are he h row enres of A j,k and b j,k... Convex Hull Reforulaon. The convex hull relaxaon (CH) has he advanage of beng a leas as gh as he bg-m relaxaon, hus helpng o reduce he search effor n he branch-and-bound procedure. The drawback s ha ncreases he nuber of connuous varables and consrans of he orgnal proble, whch can ake a proble ore expensve o solve, especally n larger probles. U k x = x k K j Dk jk, A x b y k K, j D jk, jk, jk, jk, k L U jk, jk, jk, jk, jk, k x y x x y k K, j D jk, j Dk y = k K x 0, y = {0, } k K, j D jk, jk, k (CH) The crcal sep s o denfy he se of dsaggregaed varables x j,k o eploy. The su of he dsaggregaed varables over he se of dsjuncons D k needs o be equal o he orgnal varables x. Two ses of consrans are hen used o relae he dsaggregaed varables wh he bnary varables y j,k. Whle he second reflecs he orgnal consran n he dsjuncon, he hrd ensures ha he new se of dsaggregaed varables s dfferen han zero only f he correspondng bnary varable s L one. The bounds x j,k and x j,k U feaure ndces j and k o ephasze ha hey can change beween dsjuncons. Furherore, hey wll ypcally be gher han he bounds of he orgnal varables x (x L and x U )... Logc Proposons. One way o derve consrans nvolvng 0 varables s o frs consder he logc expressons of he odel and, hen, ransfor he no an equvalen equaon or nequaly wh 0 varables. 8,9 Snce he Boolean varable s assocaed o a selecon or acon n he logc expressons, a bnary varable y ha has a one-o-one correspondence can be assgned o. Then, he negaon of ( ) s gven by y. The logcal value of rue corresponds o he bnary value of and false corresponds o he bnary varable of 0. The basc operaors used n proposonal logc and he represenaon of her relaonshps are shown n Table. 8 Wh he relaons n Table, one can syseacally odel an arbrary proposonal logc expresson ha s gven n ers of he dfferen operaors, as a se of lnear equaly and nequaly consrans. One approach s o conver sep by sep he logcal expresson no s equvalen conjuncve noral for represenaon. 9,0 The conjuncve noral for s a conjuncon of clauses, Q Q... Q s (.e., conneced by AND operaors ). Hence, for he conjuncve noral for o be rue, each clause Q us be rue, ndependen of he ohers. Also, snce a clause Q s a dsjuncon of non-negaed or negaed lerals,... n (.e., conneced by OR operaors ), can be expressed as he lnear nequaly n he frs row of Table. 578 dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

3 Indusral & Engneerng Chesry Research. PROBLEM DEFINITION In hs paper, we consder shor-er schedulng probles for bach plans, sarng fro he sples sequencng proble n a sngle un and endng wh ulple uns n parallel for a sngle sage. Gven are a se I of orders characerzed by processng e p as well as release r and due es d, whch are enforced as hard consrans. When n he presence of ulple uns, he processng e becoes un dependen and p s replaced by p,. For splcy n he presenaon, we assue no changeover es. The objecve funcon consdered s akespan nzaon. n MS () 4. SINGLE UNIT SEQUENCING PROBLEM The sngle un sequencng proble s he basc buldng block of any schedulng proble. Whou he ssue of sequence dependen changeovers, every sequence wll lead o he sae akespan f release and due es are negleced. However, enforcng such consrans wll ake any sequences nfeasble and ay also lead o dle es whle wang for an order o be released. The concep of sequence s presen on he hree dfferen connuous-e odels ha have been proposed o handle such proble, eher explcly hrough he defnon of sequencng varables, or plcly, by consderng a e grd conssng of e slos and assgnng orders o such slos. We now look n deal o each of hese possbles, propose he generalzed dsjuncve prograng odels, and derve her correspondng bg-m and convex hull relaxaons. 4.. General Precedence Concep. The os wdely used concep n sequencng varables based odels s he one of general precedence. Gven any wo orders and, here are jus wo sequencng possbles, eher before or before, and boh canno occur sulaneously. Ths can be odeled wh a proper dsjuncve se feaurng wo dsjuncons separaed by he exclusve OR operaor, see Fgure, where Boolean Fgure. General precedence concep for sngle un proble. varables, ndcae f order s before. Noce ha he nuber of Boolean varables can be reduced whou loss of generaly by akng <. Defnng x as he sarng e of order, hen he condon nsde he frs dsjuncon s for he endng e of order (x + p ) o be less han he sarng e of order ; see eq. The reanng ses of consrans enforce release and due es, eqs 4, and defne he akespan as he endng e of he las order, eq 5.,, I, I, < x + p x x + p x () x r I () x + p d I (4) x + p MS I (5) 4... Bg-M Reforulaon. Accordng o (BM), we need wo ses of consrans for he bg-m relaxaon of eq, see eqs 6 and 7. Noe ha he xed-neger lnear prograng (MILP) equvalen of he negaon of Boolean varable, s y, and ha he ghes bg-m paraeers are calculaed hrough eq 8. x + p x + M, ( y ) I, I, <, (6) x + p x + M < y I, I,,, (7) M = ax( x x + p), = ax( x + p) n( x ) = d r I, I, 4... Convex Hull Reforulaon. In order o denfy he requred se of dsaggregaed varables, noe ha f we copare he dsjuncons n eq wh hose n (GDP), ndex j {,} and se K = {(, ): < }. We hus need wo dsaggregaed varables for each of he varables x appearng n he dsjuncons, see eqs 9 and 0. As for he nuber of ndces, we need hree, wh he frs relang o he orgnal varable and he las wo, o he ndexes of he Boolean varable (see Appendx for furher deals). Then, eqs and reflec he consrans nsde he dsjuncons.,,,, x = x + x I, I, <,,,, x = x + x I, I, < (8) (9) (0) x,, x,, p y I, I, <, () x,, x,, p ( y ) I, I, <, () The nex sep s o deerne he lower and upper bounds of he dsaggregaed varables. Consderng varable x ȋ,, as an exaple, s lower bound s he sae as ha of he orgnal varable x : r. However, for he upper bound, he knowledge ha order s before order leads o a gher bound whenever d p < d ; see eq. Slar nsghs can be used o ge o he bounds of he reanng varables appearng n eqs 4 6.,,,, r y x n( d p, d p p) y I, I, < ax( r, r + p ) ( y ) x,,, () ( d p) ( y ) I, I, <, (4) ax( r, r + p) y x,,, ( d p ) y I, I, <, (5),,, r ( y ) x n( d p, d p p ) ( y ), I, I, < Arcle (6) 578 dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

4 Indusral & Engneerng Chesry Research 4.. Iedae Precedence Concep. The alernave o general precedence s edae precedence, whch can be que useful o accoun for sequence dependen coss n he objecve funcon. One explcly denfes he orders edaely precedng and followng order, provded ha hs s neher he frs nor he las order n he sequence. To accuraely odel he proble, four ses of dsjuncons are requred as well as addonal logc consrans. The frs s llusraed n Fgure and saes ha a parcular order s Fgure. Iedae precedence concep for sngle un proble. eher followed by anoher order or s he las one n he sequence; see eq 7. The second se of dsjuncons saes ha order s eher preceded by anoher order or s he frs one n he sequence, eq 8. Then, here can only be one frs 9 and one las order 0, and canno be he sae. The laer consran corresponds o he logc expresson n. The condons nsde he dsjuncons are shared wh he general precedence odel, ogeher wh eqs 5. Noce however, ha f order s he frs or he las n he sequence, he condons are wren for every order. las, I x + p x x + p x frs, I x + p x x + p x frs I x + p x las I x + p x frs las ( ) I (7) (8) (9) (0) () 4... Bg-M Reforulaon. Three ses of bg-m consrans resul (noe ha f he ndces are changed n he condon for, we ge he condon for, ); see eqs 4 where he M, paraeers are also calculaed by 8. x + p x + M ( y ) I, I,,, las, x + p x + M ( y ) I, I, frs, x + p x + M ( y ) I, I, () () (4) The Boolean varables nvolved n he four ses of dsjuncons are convered o bnary varables and gve rse o eqs 5 8. Recall ha he dsjuncons of a se are uually exclusve and hence he equaly o one. y + y las = I frs, y + y = I las y = I frs y = I, (5) (6) (7) (8) As for he logc expresson n, accordng o De Morgan s Theore and he second row n Table, we ge eq 9. frs las frs las y + y y + y I (9) 4... Convex Hull Reforulaon. Fndng he correc se of dsaggregaed varables o use s parcularly rcky for he edae precedence case. We know ha varables x appear n consrans relaed o,,,, frs, frs, las, las. Thus, we need o defne varables x ȋ,, and x ȋ,,, I, I,, whch read dsaggregaed varable assocaed o order and dsjuncon (, ) or(,), respecvely. In addon we need varables x frs ȋ, and x las ȋ,, I, Ieanng dsaggregaed varable lnked o order parcpang n he dsjuncon of he frs/las order. We also know ha here are four ses of dsjuncons 7 0, so here us be four ses of consrans relang he orgnal varables wh he dsaggregaed varables. Takng he frs se of dsjuncons as an exaple, s eher before an or s he las order, so we ge eq 0. Slarly for he second se of dsjuncons, see eq. x = x + x las,,, x = x + x frs,,, (0) () For generang he reanng wo ses of consrans lnkng he dsaggregaed varables o he orgnal varables, noe ha we have o ensure ha he dsaggregaed varable of a gven order can only be dfferen han zero n s own frs-order dsjuncon or n he frs dsjuncon of an order ; see eq. Lkewse for he las order dsjuncon, see eq. x = x I I frs, x = x I I las, () () The consrans nsde dsjuncons 7 and 8, feaurng he dsaggregaed raher han he orgnal varables, are gven n eqs 4 6. Noe ha dsjuncons 9 and 0 also lead o consrans 5 and 6. x x p y I, I,,,,,, (4) las las las,, x x p y I, I, frs frs frs,, x x p y I, I, Arcle (5) (6) As for he relaon beween he dsaggregaon varables and he bnary varables, he lower and upper bounds of varables x ȋ,, and x ȋ,, are he ones already gven n eqs 6. In eq dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

5 Indusral & Engneerng Chesry Research we sply use anoher ndex o reduce he nuber of consrans. Slar bounds are used for varables x frs ȋ, and x las ȋ,, he dfference beng ha n he lower bound of eq 8 and n he upper bound of eq 9 we ensure ha enough e s gven for all orders fnsh/sar before he sar/end of order f hs s n he fac he las/frs order n he sequence. [ax( r, r + p) = + r ] y, x,, [n( d p, d p p) + ( d p ) ] y =, I, I, I,, ( = = ) (ax[ rax( r + p )] + r ) y x las las =, (7) [n( d p, d p p ) + ( d p) = ] y I, I (8) frs frs =, (ax[ r, r + p] + r ) y x (n[ d pn( d p p)] + ( d p ) ) frs y I, I las = (9) The convex hull reforulaon shares wh s bg-m counerpar eqs Te Slos Concep. Raher han relyng on sequencng varables o locae orders wh respec o each oher, one can defne a grd conssng of T = I e slos and assgn orders o slos. Then, each slo ay only be occuped by an order; see Fgure. Beng x he sarng e of slo, 4... Bg-M Reforulaon. Accordng o (BM), he bg-m reforulaon for he consrans nsde he dsjuncon s gven n eqs Noce ha bg-m paraeers for a gven are p, r and ax (d p )+p d, respecvely. Because he frss are he paraeers n he orgnal consrans, eqs 4 and 4 are no bg-m consrans. Ths s he consequence of consderng he ghes, equaon-dependen, bg-m paraeers, nsead of an overall larger value. MS = T + x+ T x p y I, T, (4) x ry I, T, (4) x ( d p) y + ax( d p ) ( y ),, I, T (44) The consrans represenng he exclusve OR n eqs 40 and 4 over he bnary varables are hen shown n eqs 45 and 46. y = T, I y = I, T (45) (46) 4... Convex Hull Reforulaon. There are hree varables appearng n he consrans assocaed o dsjuncon, n eq 40, and so we need o defne he followng dsaggregaed varables: MS ȋ, x,,, and x +,,. Noe ha he frs does no need he e ndex snce only shows up for = T. The relaon beween he dsaggregaed varables and orgnal varables s gven n eqs MS = MS I Arcle (47) x = x T I,, (48) Fgure. Te slos concep for sngle un proble. hen he dfference beween x + (MS f = T ) and x us be greaer han he processng e of order f, = rue. Noe ha he equaly s no enforced so ha, f necessary, we can wa for he release e of he followng order. In addon, here are wo exra consrans assocaed o he dsjuncon, whch ensure ha he release and due e consrans are e; see eq 40. The odel s coplee wh an exra se of dsjuncons ha reflec he fac ha every order us be assgned o one slo, eq 4. Noe ha he consrans assocaed o such dsjuncons are he ones n 40., MS = T + x+ T x p T x r x + p d (40), I (4) x = x T, T + +,, I (49) Equaons 50 5 reflec he consrans assocaed o he dsjuncon n eq 40. Noce ha he laer ses also ac as he lower and upper bounds of varables x,,, respecvely. As for he relaon of he oher wo ses of dsaggregaed varables wh bnares y,, eqs 5 and 54 resul. In he laer, we have a very gh lower bound for he dsaggregaed akespan varable, whch s equal o he su of all processng es plus he nu release e over all orders. The MILP odel s coplee wh eqs 45 and 46. MS + x x p y I, T = T +,, T,,, (50) x,, ry I, T, (5) x,, ( d p) y I, T, (5) ( r + p) y x ax( d p ) y I, T, +,, I, (5) 5785 dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

6 Indusral & Engneerng Chesry Research (n r + p ) y I I MS ax( d ) y I, T, = T, (54) I, 4... Elnang Dsaggregaed Varables n Convex Hull Reforulaon. As enoned before, he an dsadvanage of he convex hull reforulaon over s bg-m counerpar s he ncreased sze resulng fro he addonal connuous varables and consrans. Ths s ofen responsble for an ncrease n he copuaonal effor offseng he evenual benefs fro a gher forulaon. One should herefore check f s possble o elnae hese exra varables o keep he benef whou he dsadvanage. Ths s forunaely he case wh he e slos convex hull forulaon. If for a gven, one sus over consrans 50 and replaces he su of he dsaggregaed varables wh her correspondng orgnal ones, accordng o eqs 47 49, one obans eq 55. Dong he sae hng for eqs 5 and 5, we oban eqs 56 and 57, whch ogeher wh 45 and 46 conclude he uch spler and consderably ore effcen forulaon, as wll be seen laer on. MS = T + T, I + x x p y T x ry T I, x ( d p) y T I, (55) (56) (57) I should also be noed ha he consrans n eqs are sronger han n eqs Ths can clearly be seen n consrans 55 and 56 where he rgh-hand sdes are gher han n eqs 4 and SINGLE STAGE PROBLEM WITH PARALLEL UNITS If nsead of a sngle un here are ulple uns n parallel he proble becoes ore challengng snce we no longer know a pror how any orders are o be allocaed o a parcular un. Ths affecs he concep of sequence, wh consrans beween any par of orders o be enforced only f boh are assgned o he un, as well as ha of e slos, snce s possble o have an nsuffcen nuber of slos, whch wll lead o an nfeasble soluon, or here ay be slos n excess, whch should be handled appropraely n order o reduce soluon degeneracy. We wll now revs he general precedence and e slos conceps and, sarng wh GDP odels, dscuss he reforulaons requred o generae soe of he MILP odels ha have been publshed n he leraure. For he general precedence concep here are acually wo dfferen alernaves as wll be shown below. 5.. General Precedence Concep (Opon ). The frs opon s o look a sequencng fro a un perspecve; see Fgure 4. Gven any wo orders (, ) wh <, he Boolean varable,, s rue f and only f s sequenced before and boh are assgned o un. Usng he sae sequencng varables of he sngle un proble and a new se of Boolean assgnen varables,,, yelds eq 58. On he oher hand, f he wo orders are assgned o bu s before, hen s,, ha s rue, eq 59. The hrd possbly, whch s no requred for he follow up, s ha he wo orders are no boh assgned o.,,,,, I, I, < M (58),,,, I, I, < M (59) I s also possble o avod varables, and replace eqs 58 and 59 by 60, whch saes ha f he wo orders are assgned o un s eher before, or before. However, such alernave leads o probles wh sgnfcanly ore bnary varables and consrans han opons and (see secon 5.) and faled o produce a beer perforance so s no consdered.,,,,, I, I, < M (60) The dsjuncons are gven n eq 6, where he processng e of order, p, s now a varable nsead of a paraeer. The nforaon gven fro he proble daa s now he processng e of n un, sop = p, f s ndeed assgned o. Noe ha we could have used p, nsead of p n he dsjuncons n eq 6, bu hs would preven us fro reachng our argeed consran, whch s gher. Snce we are sll enforcng all orders o be assgned o a un, anoher se of dsjuncons s requred, see eq 6.,,,,, I, I, x + p x x + p x < M (6), I p = p (6) The GDP odel s copleed wh consrans MILP Reforulaon. The MILP reforulaon of he GDP odel jus descrbed gves rse o he general precedence odel proposed Meńdez e al. 4 f reduced o a sngle sage. Ineresngly, one se of dsjuncons s derved fro he bg-m reforulaon, whle he oher resuls fro he convex hull. Sarng wh he frs se of dsjuncons, eq 6, s sraghforward o derve 6 and 64, where he M paraeer s deerned fro eq 8.,,, x + p x + M ( y ) I, I, < M,,, x + p x + M ( y ) I, I, < M Arcle Fgure 4. General precedence concep for sngle sage proble (opon ). (6) (64) 5786 dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

7 Indusral & Engneerng Chesry Research As for he convex hull of he second dsjuncon, we need o frs defne a new se of dsaggregaed varables, p ȋ,, whch are relaed o he known paraeers p, and he reanng varables hrough eqs Clearly, one can add over all consrans n eq 66 for a gven and use eq 65 o derve an alernave and beer se of consrans ha avods he need for he dsaggregaed varables; see eq 68. p = p I, M (65) p = p y I, M, (66) y = I, M p = p y I, M (67) (68) The nex sep s o conver he logc n eqs 58 and 59, no MILP for, where only he rgh lef plcaons are useful; see equaons 69 and 70.,,,,,,,,,,,, y + y + y + y y y + y + y I, I,,,,, < M,,,, y y + y y I, I,,,,, < M (69) (70) Afer replacng eqs n eqs 4 and 5 and eqs 6 and 64, hs yelds he fnal consrans, eqs Noe ha and 67 sll apply. x + p y M,, x + M ( y y y ) I, I,,,, < M x + p y M x + M ( + y y y ) I, I,,,, < M x + p y d I, M x + p y MS I, M (7) (7) (7) (74) 5.. General Precedence Concep (Opon ). The second opon consders Boolean varables of ype, ha are rue f boh and are assgned o he sae un and s sequenced before ; see Fgure 5. Noce ha ore varables are now requred snce even hough he se of dsjuncons s for <, we now consder, as well as,. Hence, her acual doan s for every. For he hrd dsjuncon (, )obe rue, boh orders canno be allocaed o he sae un. Noce ha only he plcaon s rue snce due o he absence of he un ndex, here ay be an acual un handlng uns and, for whch he reverse plcaon would lead o, = false, whch s correc, bu for all oher uns he sae plcaon would lead o an nconssen oucoe, = rue.,,, I, I, < x + p x x + p x (75), (,, ) I, I, < M (76) The GDP odel for opon also ncludes eqs 5 and dsjuncon MILP Reforulaon. The MILP reforulaon of he GDP general precedence odel correspondng o opon s essenally he odel proposed by Jan and Grossann. 5 The bg-m reforulaon of he consrans n he frs and second dsjuncons n eq 75, gves rse o eq 77. Then, we need o ensure ha a sngle er of he dsjuncon s acve, eq 78. As for he converson of he logc no MILP fora, hs yelds eq 79. x + p x + M ( y ) I, I,,,,,, y + y + y = I, I, <,,,, ( ),, y + y + y,, y y y I, I, < M (77) (78) (79) We now jus need o replace eq 78 n eq 79 o reove varables y, fro he forulaon and apply he convex hull reforulaon o eq 6, as descrbed n secon 5.., o derve he fnal odel. I consss of eqs, 67, 7, 74, 80, and 8. x + p y x + M ( y ) I, I, M,,, y + y y + y I, I,,,, < M Arcle Fgure 5. General precedence concep for sngle sage proble (opon ). (80) (8) 5.. Te Slos Concep wh Mulple Grds. When relyng on he concep of e slosovng fro a sngle un o ulple uns n a sngle sage has an poran plcaon: we no longer know a pror how any orders are gong o be 5787 dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

8 Indusral & Engneerng Chesry Research allocaed o a parcular un. Therefore, here s a need o defne ulple e grds, one for each un, and wll be assued ha all grds feaure he sae nuber of slos. Noe ha wh ulple e grds, he execuon of orders s led o a sngle slo, whch resuls n a beer perforance copared o he alernave of a sngle coon grd for all uns, 0 where orders ay span across ulple slos. Only by chance wll he specfed nuber of slos be equal o he nuber of allocaed orders for every un, and hence, dle slos are hghly lkely o occur. Naurally, f he nuber of slos s oo low ( T < I / M ), he odel wll be nfeasble. Fgure 6 llusraes he concep, where Boolean varables,, = rue f order s allocaed o slo of un and varables, Fgure 6. Te slos concep wh ulple grds for sngle sage proble. denfy f he slo of un has no order assgned o. Clearly, a parcular slo of a gven un ay eher have an order assgned o or none a all, leadng o he dsjuncons n eq 8. Then, a parcular order needs o be allocaed o a slo of a ceran un, eq 8. Fnally, eq 84 reduces soluon degeneracy by keepng dle slos las and nex o each oher. 5 More specfcally, dle slos are he ones furher o rgh, or n oher words, f slo n un s hen slo + us also be of orders.,, MS = T + x, + T x, p, x, r x, + p d,, MS = + x + x = 0 T, T, x M, T, 0 x ax( d), I (8),, I (8),,, + M, T (84) 5... Convex Hull Reforulaon. Slarly o he sngle un case, applyng he convex hull reforulaon and a few algebrac anpulaons akes possble o elnae he dsaggregaed varables and reduce he nuber of consrans o coe up wh a very gh and effcen odel. The fnal oucoe s essenally he odel by Casro and Grossann. 0 Equaons gve he relaon beween he orgnal and he new se of dsaggregaed varables. Noe ha four ndexes are requred, beng wo e ndexes, snce varable x, appears n boh dsjuncons (,,) and (,, ). MS = MS + MS M I,,,,, I x = x + x M, T, + +,, +, I (85) (86) x = x + x M, T, T (87) Equaons 88 9 correspond o he consrans whn he dsjuncons n eq 8, whle eq 9 s he only bound consran needed for he follow up. MS + x x = T +,, T,, p y I M, T,, (88) x ry I M, T,,,,, (89) x ( d p ) y I M, T,,,,, (90) = T +, T,, MS + x x = 0 y M, T,,, 0 y x ax( d ) y M, T I, (9) (9) To elnae he dsaggregaed varables, one only needs o add over all consrans n eqs and replace he su of he dsaggregaed varables wh he orgnal varables accordng o eqs The reanng dsaggregaed varables are hen elnaed accordng o eq 9, leadng o eq 9, or replaced by her lower or upper bounds n eq 9, resulng n eqs 94 and 95, respecvely. MS + x x = T, + T, p y M, T I,, x ry M, T,,, I,,, I I, x ( d p ) y + ax( d) y M, T (9) (94) (95) The odel s coplee wh he consrans relang he bnary varables, eqs 96 98, where he laer resuls fro he logc expressed n eq 84. y + y = M, T,, (96) I y = I,, M T, +, y y M, T, T Arcle (97) (98) 6. COMPUTATIONAL RESULTS The perforance of he dfferen MILP odels s llusraed hrough he soluon of a few exaple probles wh up o 0 orders and 5 parallel uns. We consder 0 sngle un probles wh rando generaed daa (provded as he Supporng 5788 dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

9 Indusral & Engneerng Chesry Research Arcle Table. Copuaonal Effor (CPU s) for Sngle Un Probles a proble I opu TS_CH_C TS_CH GP_BM IP_CH GP_CH TS_BM IP-BM Ex Ex Ex Ex b Ex Ex c 600 Ex b b 600 b Ex b 65 b Ex b 067 b Ex b b 600 b a The bes perforer s n bold, and he axu resource l or ou of eory ernaon s n alc. b No feasble soluon. c Subopal soluon = 797. Table. Relave Opaly Gap fro Relaxed Model n he Inal (I), Also Known As Inegraly Gap, and Fnal (F) Ieraons of he Branch-and-Bound Search wh Respec o he Known Opal Soluon TS_CH TS_BM IP_CH GP_CH GP_BM IP_BM proble I F I F I F I F I F F Ex 0.0% 0% 87% 4% 8% 0% 40% 0% 44% 0% % Ex 0.7% 0% 88% 0% 46% 0% 46% 0% 47% 0% 4% Ex 6.% 0% 89% 0% 7% 0% 7% 0% 7% 0% 0% Ex4 4.8% 0% 9% % 6% 0% 5% 8% 8% 0% 6% Ex5.9% 0% 9% 5% 8% 0% 5% 0% 8% 0% % Ex6.9% 0% 9% 6% % 0% 9% % % 0% 9% Ex7 0.0% 0% 9% 56% 5% 0% 55% % 56% 6% 54% Ex8.0% 0% 9% 8% % 0% % 7% % % % Ex9 0.7% 0% 9% 69% 40% 0% 40% 9% 40% 5% 40% Ex0.5% 0% 94% 68% 9% 0% 40% 7% 4% 8% 40% Inforaon) for coparson of he 7 dfferen forulaons derved fro he dfferen conceps for connuous-e represenaon. Then, 0 sngle sage probles wh parallel uns correspondng o probles. 5. fro Harjunkosk and Grossann 6 and probles P7 P0 n Casro and Grossann 0 are used o evaluae he wo general precedence opons and e slos alernaves. The odels were pleened n GAMS.5 and solved usng CPLEX. wh a sngle hread up o a relave opaly olerance = 0 6, or a axu copuaonal e of 600 CPUs. The hardware conssed on a lapop wh an Inel Core Duo T900.5 GHz processor, wh 4 GB of RAM and runnng Wndows Vsa Enerprse. 6.. Sngle Un Probles. The resuls are gven n Table, where he acronys for he odels are he followng. In ers of he concep for e represenaon, TS sands for e slos, GP for general precedence, and IP for edae precedence. Wh respec o he GDP reforulaon, CH refers o convex hull and BM for bg-m. In he case of TS_CH, we consder he odels wh (secon 4..) and whou (TS_CH_C) he dsaggregaed varables (secon 4..). As seen n Table, he copac e slos convex hull odel (TS_CH_C) s he bes perforer by a leas order of agnude, he excepon beng he general precedence bg-m odel (GP_BM) n Ex, whch s roughly fve es faser. Even f we use he dsaggregaed varables, we sll ge a reasonable perforance despe he sgnfcan ncrease n proble sze. The excepon s agan Ex, for whch TS_CH s unable o prove opaly (bes possble soluon a e of ernaon = 55). Clearly, he e slos concep s he bes for he sngle un proble provded ha one reles on he very gh convex hull reforulaon snce s bg-m counerpar s very poor, falng o fnd he opal soluon for EX6 and no even fndng a feasble soluon for Ex7 0. The resuls for he wo precedence conceps are que neresng. Whle s beer o rely on general precedence wh he bg-m reforulaon (GP_BM), becoes ncreasngly ore dffcul o prove opaly as he proble sze ncreases (falures n Ex7 0; see Table ). On he oher hand, for edae precedence, he convex hull s uch beer han he bg-m reforulaon, and despe he falure o fnd a feasble soluon, he bes possble soluon fro he MILP relaxaon rapdly becoes equal o he opal soluon. Snce GP_BM excels a fndng near opal soluons very rapdly, GP_BM and IP_CH have copleenary srenghs. In oher words, one can use he upper bound fro he forer and he lower bound fro he laer o prove opaly n under a nue, for all probles solved. However, such hybrd algorh would sll be worse han TS_CH_C. The convex hull reforulaon of he general precedence concep (GP_CH) can sll fnd all opal soluons, bu s resrced o provng opaly n four cases. Furherore, he opaly gap a e of ernaon s always larger han for GP_BM, despe beng norally gher; see Table. Noe ha he bg-m reforulaons of he wo precedence odels have he exac sae relaxaon (hs explans why he I colun for IP_BM s no shown). Wh respec o he convex hull reforulaon, he odel fro general precedence s soees gher and soees looser han s edae precedence counerpar. The wors odel by far n ers of relaxaon s TS_BM, he second wors perforer of he lo. Also noe ha elnang he dsaggregaed varables when gong fro 5789 dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

10 Indusral & Engneerng Chesry Research TS_CH o TS_CH_C leads o one-hrd he nuber of oal varables (see he Supporng Inforaon) and a one order of agnude reducon n he nuber of equaons bu does no affec he relaxaon. The nuber of bnary varables s dependen on he e represenaon concep and no on he reforulaon. The relave perforance of he dfferen approaches s suarzed n Table 4 usng qualave labels. The beer Table 4. Relave Perforance of Reforulaon Mehods/ Connuous-Te Represenaon Conceps connuous-e represenaon concep reforulaon general precedence (GP) edae precedence (IP) e slos (TS) bg-m (BM) edu very low very low convex hull (CH) low low hgh perforance of he e slos and general precedence conceps n he sngle un proble, explans why was chosen o dscard he edae precedence concep when gong for he ore coplexulple parallel uns scenaro. In ers of odelng gudelnes ha can be drawn for new probles, he resuls sugges fndng a GDP odel ha n he dsjuncons feaures varables wh ndexes relaed o he doan of dsjuncve se K and no o j D k (refer o secon ). Ths wll ensure ha a gh copac MILP odel can be derved followng algebrac anpulaon of soe of he varables and consrans resulng fro he convex hull reforulaon. Noce ha n eq 40 he dsjuncons are over, whle he odel varables have eher no ndex (MS) or he one assocaed o he dsjuncve se,. In conras, n he edae precedence odel, he dsjuncons feaure varables wh ndex and, beng he forer lnked o he doan of he dsjuncve se and he laer o he doan of he dsjuncons. In he general precedence odel, he dsjuncve se does feaure ndexes and bu here are wo copleenary consrans. 6.. Sngle Sage, Parallel Uns Probles. We consder wo alernaves for he e slos approach: () TS_H s a socalled heursc approach, where he gven nuber of slos s deerned fro he sandard global search procedure 8 (keep ncreasng T whle observng a change n he value of objecve funcon hen sop and repor he resul fro he prevous eraon, whch also led o he sae value and hus defnes he Arcle nu nuber of e slos); () TS_F he full approach wh T = I, feaurng enough slos o guaranee global opal soluons n a sngle run, hus provdng a farer coparson wh he general precedence ehods. The laer are naed GP_ and GP_, respecvely, fro odelng opons (see secon 5.) and (see secon 5.). The resuls n Table 5 offer no doub abou he bes perforer, he e slos approach for he nu nuber of slos, TS_H. Noce ha he opal nuber of slos lsed n colun 6 s a os he nu nuber ha poenally leads o a feasble soluon, resulng fro evenly dsrbung he orders aong he avalable uns, I / M, plus wo (n H&G5.). More han he doublng he nuber of slos up o he axu (TS_F) degrades he perforance by a leas order of agnude (noce he ncrease n negraly gap n Table 6) bu confrs he superory of he e slos concep snce s only surpassed by he general precedence odels n HG4.. The coparson beween he wo general precedence odels s no as conclusve. We can argue ha GP_, lnked o he approach by Meńdez e al., 4 has a slgh edge snce can fnd he opal soluons n 5 vs 4 cases and norally generaes beer schedules. Furherore, sees o be ore robus o an ncrease n proble sze, snce reurns beer soluons for he four larges probles (C&G7-C&G0). On he oher hand, GP_, lnked o he approach by Jan and Grossann, 5 s able o prove opaly n wo ore cases, correspondng o he salles probles H&G. and H&G.. I generaes consderably ore bnary varables, whch s copensaed by he saller nuber of equaons (see he Supporng Inforaon). Provded ha he solver does no run ou of eory, GP_ leads o a saller opaly gap a e of ernaon (consderng he real opu and no he bes feasble soluon up o ha pon); see Table 6. The negraly gap of boh forulaons s he sae, however. 7. CONCLUSIONS Ths paper has shown ha well-known MILP forulaons fro he leraure can be derved fro eleenary e represenaon conceps and her correspondng generalzed dsjuncve prograng odels, feaurng dsjuncons o consder all operaonal possbles, and sple logc. By frs proposng a GDP odel, he burden of fndng he approprae consrans for he MILP has been sgnfcanly reduced by akng advanage of reforulaon echnques such as he bg-m and convex hull o syseacally generae such consrans. The Table 5. Resuls for Sngle Sage Probles a opu subopal soluons e slos T CPUs proble (I,M) TS_F GP_ GP_ TS_H TS_F TS_H TS_F GP_ GP_ H&G. (,) H&G. (,) H&G4. (5,5) H&G4. (5,5) H&G5. (0,5) H&G5. (0,5) C&G7 (5,5) C&G8 (5,5) C&G9 (0,5) C&G0 (0,5) a Bes perforer n boldaxu resource l, or ou of eory ernaon n alc dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

11 Indusral & Engneerng Chesry Research Arcle Table 6. Relave Opaly Gap fro Relaxed Model n he Inal (I), also Known As Inegraly Gap, and Fnal (F) Ieraons of he Branch-and-Bound Search wh Respec o Known Opal Soluon TS_H TS_F GP_ GP_ proble I F I F I F I F H&G..7% 0% 6.% 0% 44% 9% 44% 0% H&G. 6.% 0% % 0% 8.7% 8.% 8.7% 0% H&G4. 5.4% 0% 9.0% 0% % % % 4% H&G4. 5% 0% 9% 0% 0% 0% 0% 0% H&G5..% 0% 5.% 0% % % % 9% H&G5. 6.4% 0% % 0% 0.7% 0.7% 0.7% 0.7% C&G7 0.5% 0% % 0% 0% 0% 0% 6% C&G8 0.5% 0% % 0% % % % % C&G9 6.6% 0% 8.9% 0% 0% 0% 0% 0% C&G0 5.9% 0% 7.8% 0% 40% 40% 40% 40% overall odelng process has hus effecvely been dvded no wo seps, where choosng he rgh e represenaon s a leas as poran as choosng he bes reforulaon ehod. More specfcally, and whle he e slos convex hull reforulaon proved o be he bes perforer for he sngle sage probles consdered, gong for a e slos bg-m odel s uch worse han usng a general precedence bg-m forulaon. Anoher neresng resul was ha he wo slar precedence conceps (general and edae) gave oppose resuls wh respec o he os effcen reforulaon. Fnally, we have also seen ha by usng he knowledge ha s plc n he dsjuncons, gher bounds can be derved for he bg-m consrans. Fuure work wll be concerned abou ore coplex schedulng probles nvolvng ulsage and ulpurpose plans, as well as sequence dependen changeovers and dfferen sorage polces. APPENDIX In order o ake easer for less experenced readers o go hrough he paper, a sple llusrave exaple s now gven o beer undersand he consrans of he sngle un, general precedence odel dscussed n secon 4.. In parcular, we focus on how ndexes are defned for he dsaggregaed varables requred by he convex hull reforulaon. Consder hree orders, {,,}. Se of consrans gves rse o he followng ndvdual consrans, where can be seen ha each sarng e varable x s presen n four dsjuncons. Ths nuber affecs he densonaly of he dsaggregaed varables. Whle one gh be eped o pu wo ndexes n he dsaggregaed varables, e.g. x ȋ, and x ȋ,, s sply no enough snce he se K ={(, ): < } conans jus hree eleens.,, x + p x x + p x,, x + p x x + p x,, x + p x x + p x The correc choce s o consder hree ndexes for he wo ses of dsaggregaed varables,.e. x ȋ,, and x ȋ,,, where ndex 579 s lnked o he varables appearng n he dsjuncon(s) assocaed o par (, ). In hs case, s eher or, leadng o he followng sx consrans resulng fro eqs 9 and 0.,, x = x + x,, x = x + x,, x = x + x,, x = x + x,, x = x + x,, x = x + x,,,,,,,,,,,, Takng he sequence as an exaple, we would requre, =, =, = rueeanng ha he nonzero dsaggregaed varables would be x,,, x,,, x,,, x,,, x,,, and x,,. ASSOCIATED CONTENT *S Supporng Inforaon Tables wh copuaonal sascs relaed o proble sze as well as daa for sngle un exaple probles Ex Ex0. Ths aeral s avalable of charge va he Inerne a hp:// pubs.acs.org/. AUTHOR INFORMATION Correspondng Auhor *Tel.: E-al: pedro.casro@lneg.p. Noes The auhors declare no copeng fnancal neres. ACKNOWLEDGMENTS The auhors graefully acknowledge fnancal suppor fro he Luso-Aercan and Naonal Scence Foundaons, under he 0 Porugal U.S. Research Neworks Progra, and fro he NSF under gran OCI REFERENCES () Kondl, E.; Paneldes, C. C.; Sargen, R. A General Algorh for Shor-Ter Schedulng of Bach Operaons - I. MILP Forulaon. Copu. Che. Eng. 99, 7,. () Paneldes, C. C. Unfed Fraeworks for he Opal Process Plannng and Schedulng. In Proceedngs of he Second Conference on dx.do.org/0.0/e00486 Ind. Eng. Che. Res. 0, 5,

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