Solving integrated process planning and scheduling problem with constraint programming

Transcription

1 Ttle Solvng ntegated pocess plannng and schedulng poblem wth constant pogammng Autho(s) Zhang, L; Wong, TN Ctaton The 13th Asa Pacfc Industal Engneeng and Management Systems Confeence (APIEMS 2012) and 15th Asa Pacfc Regonal Meetng of the Intenatonal Foundaton fo Poducton Reseach, Phuet, Thaland, 2-5 Decembe In APIEMS 2012 Confeence Poceedngs, 2012, p Issued Date 2012 URL Rghts Ths wo s lcensed unde a Ceatve Commons Attbuton- NonCommecal-NoDevatves 4.0 Intenatonal Lcense.

2 Poceedngs of the Asa Pacfc Industal Engneeng & Management Systems Confeence 2012 V. Kachtvchyanuul, H.T. Luong, and R. Ptaaso Eds. Solvng Integated Pocess Plannng and Schedulng Poblem wth Constant Pogammng Lupng Zhang, T. N. Wong Depatment of Industal & Manufactung Systems Engneeng The Unvesty of Hong Kong Pofulam Road, Hong Kong Phone: (852) E-mal : tnwong@hu.h Abstact.Pocess plannng and schedulng ae two mpotant manufactung functons whch ae usually pefomed sequentally. Howeve, due to the uncetantes and dstubances fequently occung n the manufactung envonment, the sepaately conducted pocess plan and shopfloo schedule may lose the optmalty, becomng neffectve o even nfeasble. Reseaches have consdeed the potental of ntegated pocess plannng and schedulng (IPPS) to conduct the two manufactung plannng actvtes concuently nstead of sequentally. That s, to ntegate pocess plannng wth dynamc shopfloo schedulng to cope wth the ealtmeshopfloo status.the IPPS poblem s vey complex and t has been egaded as an NP-had poblem. Many eseaches have attempted to solve the IPPS poblem wth ntellgent appoaches such as meta-heustcs and agent-based negotaton. In ths pape, a constant pogammng-based appoach s poposed and mplemented n the IPPS poblem doman. Constant pogammng (CP) featues geat modelng capabltes to eflect complex constants of a poblem, and thee s a geat potental fo CP to be used to solve IPPS poblems. The appoach s mplemented and tested on the IBM ILOG platfom, and expemental esults show that the CP can handle the IPPS poblem effcently and effectvely. Keywods:Constant pogammng; pocess plannng; ob-shop schedulng; IPPS 1. INTRODUCTION Pocess plannng and schedulng ae two cucal functons n the manufactung envonment. They ae closely elated to each othe but usually pefomed sequentally, whch sometme endes the pocess plan and schedulng nfeasble. Futhemoe, the sequental and sepaate conducton of pocess plannng and schedulng cannot satsfy the dynamc manufactung envonment wth uncetantes and dstubances. That s why the ntegated pocess plannng and schedulng (IPPS) s poposed to combne the pocess plannng and schedulng togethe nto consdeaton, espondng to the esouce estcton and dynamc manufactung stuatons. To mpove the optmalty of the pocess plans and the schedule, some eseaches have attempted to solve the IPPS poblem wth exact optmzaton methods and advanced seach technques.in ecent yeas, meta-heustcs ae wdely consdeed as pomsng appoaches to tacle the IPPS poblems. A smla poblem, ob-shop schedulng poblem has been addessed usng smulated annealng(stenhofel, Albecht, & Wong, 1999). The IPPS poblems of a smple veson, wthout consdeaton of altenatve pocesses have also been tacled by Moad and Zalzala wth Genetc Algothms(Moad & Zalzala, 1999).In 2007, L & McMahonalso used the smulated annealng and combned the pocess plannng and schedulng togethe(l & McMahon, 2007).A patcle swam optmsaton appoach was adopted to solve the IPPS poblems late(guo, L, Mleham, & Owen, 2009).Besdes meta-heustcs, agent-based appoaches have also been poposed fo the IPPS (Wong, Leung, Ma, & Fung, 2006a; Wong, Leungy, Ma, & Fung, 2006). Meta-heustcs and agent-based appoaches have been found to be moe effectve and effcent fo lage IPPS poblems, wth the obectve to fnd nea-optmal solutons n a easonable tme. :Coespondng Autho 1525

3 The applcaton of constant pogammng (CP) to schedulng poblems has been systematcally demonstated by Baptste and et al. n the boo (Baptste, Le Pape, & Nuten, 2001).The CP has also been mplemented by othe eseaches to solve the mult-machne assgnment schedulng poblem (Sadyov & Wolsey, 2006), and obshop schedulng poblems (Bec, Feng, & Watson, 2011).Although some eseaches have mplemented the CP n both plannng and schedulng poblems(tmpe, 2002), we have not yet come acoss any publcatons on the successful applcaton of CP n IPPS. In ths pape, we pusue the exploaton of CP to tacle the IPPS poblems. A CP-based IPPS model s poposed. And IBM ILOG s adopted to pogam and solve ou IPPS model. A gaph-based epesentaton s gven n secton 2to llustate the IPPS poblem clealy, and ou CP-based model s closely elated to the gaphc epesentaton. Afte that, the CP algothm fo ou IPPS poblem s depcted n detal n secton 3.And then, the poposed appoach s tested on IBM ILOG wth benchma poblems, and expements esults ae also gven n secton 4. Fnal conclusons ae demonstated n secton GRAPH-BASED REPRESENTATION OF IPPS PROBLEM Usually, the IPPS poblem featues n obs to be pocessed on m machnes, each ob consstsof a goup of opeatons whch may nvolve altenate pocesses and altenate machnes, and pecedenceelatonshps may exst between dffeent opeatons(wong, Leung, Ma, & Fung, 2006b).Pocess plannng altenatves of the IPPS poblem can be epesented gaphcally wth the AND/OR gaph. Fo example, the AND/OR gaphs n Fgue 1 show the espectve pocess plannng nfomaton fo 2 obs wth altogethe 11 opeatons, whee each node denotes an opeaton. An o-elatonshp s declaed fo ob 2. The followng notatons ae used to epesent pmtve elements of the IPPS poblem. J : a ob wth ndex numbe ; m : a machne wth the ndex numbe ; O : an opeaton of ob, J wth ndex numbe ; T ( O, ) : a gven type of the opeaton O ;, O : opeaton O to be pocessed on machne m, ;, ( τ O, ) : thepocessng tme of, O ; OR : an o-elatonshp wth ndex numbe. It can be expessed by attachng thee odeed opeatons OR ( O, O, O ), whch ndcates an o-elatonshp 0, 0 1, 1 2, 2 exsts among opeatons O, O, O ; 0, 0 1, 1 2, 2 Seq : a sequence elatonshp wth ndex numbe. It can be expessed by attachng a pa of opeatons Seq ( O, O ), whch means O s an mmedate 1, 1 2, 2 pedecesso of C max O ; 2, 2 : themaespan of the IPPS nstance; J : the set of obs of the IPPS poblem { J } ; O : the set of opeatons of the IPPS poblem { O, } ; P : the set of opeatons wth pocessng machne allocated { O } ;, M : the set of avalable machnes{ m 0}, and m0 s set to be a specal dummy machne; R : the set of all o-elatonshps { OR ( O, O, O )} ; 1, 1 0, 0 1, 1 2, 2 SEQ : the set of all sequence elatons defned n the IPP poblem { Seq ( O, O )}. 1, 1 2, 2 Regadng the IPPS poblem dscussed n ths pape, each machne can pocess one opeaton at a tme, and on opeaton cannot be nteupted when pocessng begns. Hence, the schedulng we dscussed hee s a typcal dsunctve non-peemptve schedulng poblem. As a esult, a dsunctve gaph model D=(O,A,R) can be used to ad the epesentaton of the IPPS poblem, as shown n Fgue 2. O s a set of nodes whch epesent opeatons, and a set of canddate machnes fo pocessng each opeaton O, s allocated. A s a set of dected acs, depctng the sequence elatonshps between dffeent opeatons. If a dected ac stats fom O and ponts to O, t means 1, 1 1, 1 O s an mmedate pedecessoof O 2, 2 2, 2.Usng notatons mentoned above, the sequence elatonshp can be expessed by Seq ( O, O ). R s a set of dummy nodes, 1, 1 2, 2 ndcatng the o-elatonshp of altenatve pocesses. Fgue 2 pesents the dsunctve gaph fo the 2 obs gven n Fgue 1. As shown n ths dagam, node OR 1 ndcates that the pocess tal ( O, O ) and ( O2,10 ) ae optonal and only one tal can be chosen n a soluton. And the o-elatonshp can be wttenby OR1 ( O2,7, O, O 2,10 ), whee O s the mmedate pedecesso of node OR 2,7 1, and O and O 2,10 ae two mmedate successos of ths onode. S and F ae two othe nds of dummy nodes to ndcate the statng pont and endng pont of ob J, wthn whch S 0 s the statng pont of the whole IPPS nstance. Fo thepupose of consstency, the statng node S and temnatng node F ae egaded to be specal opeatons. To dstngush S and F fom othe nodes, ndex numbes of S and F ae set to be 0 and 9999 (o a vey lage ntege) espectvely, that s: S = O, F = O (1),0,

4 Fgue 1: AND/OR gaph of an sample IPPS nstance S0 Job 1 Job 2 S1 S2 O1,1 O2,7 O1,2 OR1 O O2,10 O1,3 O1,4 O O1,5 OR1 O1,6 O2,11 F1 F2 Fgue 2: Dsunctve gaph fo the IPPS poblem. 1527

5 And a dummy machne m0 s allocated to pocess the dummy nodes, wth pocessng tme set to be 0: τ = τ = (2) 0 ( O ) 0 ( O ) 0,0, CP ALGORITHM FOR THE IPPS PROBLEM The CP algothm conssts of seveal modules. Based on the IBM ILOG open pogammng language (OPL), thee basc modules ae essental to establsh a CP model: vaables, obectve functon, and the defnton of constants. 3.1 Vaables To begn wth, a pmtve type of vaablesnteval should be defned. In ths pape, the nteval epesents the nteval of tme dung whch an opeaton s pocessed. An nteval can be chaactezed by ts statng tme, ts endng tme, and ts sze. In ths pape, snce the IPPS s defned as a non-peemptve schedulng poblem, the sze of an nteval equals to the dffeence between ts statng tme and endng tme. Howeve, befoe a soluton s obtaned, the statng tme and endng tme of an opeaton s nteval emans unnown, whch means the allocaton of an opeaton s nteval on a schedule s not detemned. Futhemoe, an nteval should be assgned to each opeaton when esolvng the poblem. Howeve, snce altenatve machnes and altenatve pocesses exst, t s not equed to map all ntevals onto the schedule. As a esult, anothe mpotant featue of an nteval s that t can be optonal. Intevals ae needed to be assgned to each nodes appeang n the dsunctve gaph. The oveall defnton of vaables ssummazed as follows: Inteval st sze 0; Inteval p[ O, ] optonal sze ( τ O, ) O, P ; Inteval ops[ O, ] optonal O, O ; Inteval o[ OR ] optonal R R ; Sequence mch[ m ] = { p[ O 0, ] = 0} m M ; Sequence ob[ J ] = { p[ O 0, ] = 0} J J. st s a dummy nteval assgned to the stat node of the whole IPPS poblem, whch s S0 n Fgue 2. In the above defnton, the vaable f [ E] epesents a collecton of vaables f [], contanng a data membe E as ts mplementaton obect. 0 0 p[ O ] eflects the nteval needed to pocess, opeaton O on machne m,, so the sze s equal to the pocessng tme of O,, whch s τ ( O, ). ops[ O, ] s the nteval fo the opeaton O wthout. allocaton of pocessng machne. When an p[ O, ] s pesent, the ops[ O, ] s pesent wth the same statng pont and endng pont as those of p[ O, ]. o[ OR ] s a dummy nteval. It only guaantees that when ts pedecesso opeaton O s pesent, one of ts successo opeatons O and 1, 1 O 0, 0 2, 2 wll be pesent as well. Sequence s anothe pmtve vaable type to ndcate a set of odeed nteval vaables. mch[ m ] s a subset of p[ O ] ncludng those ntevals whose opeatons ae, pocessed on m. 0 0 ob[ J ] s anothe subset of p[ O, ] contanng ntevals whose opeatons belong to ob 3.2Obectve Functon J 0. Maespan s a cucal cteon fo the plannng and schedulng poblem. Hee mnmzng the maespan s egaded to be the only obectve fo the CP algothm. Accodng to the IPPS model descbed n secton 2, each ob s temnated wth a dummy node F.The maespan of the IPPS poblem can be easly obtaned by the followng functon: C = max{ statof ( ops[ O ]) : J J} (3) m,9999 whee statof ( I ) s an expesson to access the stat pont of nteval I. As a esult, n one expement, the obectve s to mnmze the maespan, whch s: mncm = mn max{ statof ( ops[ O ]) : J J} (4) 3.3The Defnton of Constants,9999 Vaables gven above must subect to all constants lsted as follows: endbefoestat( st, ops[ O ]) : J J (5),0 endbefoestat ( ops[ fstof ( Seq )], ops[ nextof ( Seq )]) (6) : Seq SEQ noovelap( mch[ m ]) : m M (7) noovelap( ob[ J ]) : J J (8) 1528

6 altenatve( ops[ O 0, ],{ ps[ O 0, ] = 0, = 0}) (9) : O O 0, 0 altenatve( o[ OR ],{ ops[ nextof ( OR )], (10) ops[ thdof ( OR )]}) : OR R Functon (5) estcts that the algothm must stat seachng fom the oveall stat pont S0 of the IPPS poblem, whee endbefoestat ( I1, I 2 ) s a constant functon to ndcate that the end of nteval I1 s less than o equal to the stat of nteval I 2. O t can be expessed by the elatonshp endof ( I1) statof ( I2 ), whee statof ( I ) and endof ( I) ae two expessons to access the stat and end of the nteval I espectvely. In functon (6), fstof ( Seq ) and nextof ( Seq ) ae two expessons to access the fst and second elements of the opeaton pas stated n Seq. Fo example, f Seq epesents the sequence elatonshp Seq ( O, O ), the 1, 1 2, 2 fstof ( Seq ) and nextof ( Seq ) wll etun the opeaton O and O espectvely. Hence, the functon (6) 1, 1 2, 2 ealzes the sequence constants between dffeent opeatons. In functons (7) and (8), noovelap( s) s a constant to pevent all ntevals ncluded n sequence s fom ovelappng. Hence functon (7) estcts that a machne can only pocess one opeaton at a tme, and functon (8) estcts that dffeent opeatons of the same ob ae not pemtted to be pocessed smultaneously. In functons (9) and (10), altenatve( I,{ I1, I2... I n }) models an exclusve altenatve nteval among { I, I... I }.If nteval I s pesent, exactly one nteval 1 2 n among { I1, I2... I n } should be pesent, wth the stat pont and end pont equal to those of I espectvely. And n functon (10), expessons nextof ( OR ) and thdof ( OR ) can etun the second and thd opeatons stated n the opeaton goup OR ( O, O, O ), whch ae O and 1, 1 O 2, 2 0, 0 1, 1 2, 2 espectvely. So functon (9) estcts that one opeaton can only be pocessed once on a specfc machne. Functon (10) estcts that only one of the altenatve pocesses can be chosen. 3.4Constant Popagaton To educe the seach space and acceleate the seachng pocess, a constants popagaton method s desgned n ou appoach. The opeatons n an IPPS poblem ae classfed nto two categoes: compulsoy and optonal opeatons. Compulsoy opeatons efe to those whch have to be pocessed n ode to accomplsh a ob. Fo example, all opeatons othe than O, O and O shown n Fgue 2,10 2 ae compulsoy opeatons. Optonal opeatons efe to those opeatons wthout whch the ob can also be accomplshed. Usually, opeatons on altenatve pocess tals ae optonal. Fo example, O, O and O n 2,10 Fgue 2 ae optonal opeatons. Hee we use T ( O, ) to ndcate the type of each opeaton. It s defned that: 0 when O, s compulsoy T ( O, ) = (11) 1 when O, s optonal Obvously, f an nteval ops( O, ) s not pesent n a soluton when T ( O, ) = 0, the soluton can be elmnated fom the seach space. Actually, the seach space can be substantally shun due to the uppe constants. Anothe stuaton occus n the goup of optonal opeatons. Tae the optonal opeatons n Fgue 2 as an example. Although O and O ae both optonal opeatons, O has to be pesent f O s pesent n the fnal soluton. So we can popagate a new constant to assocate O wth O. Hee we desgned a new data element to ecod ths assocatve elatonshp, whch can be denotedas Asso, whee a s the ndex numbe of the a assocatve elatonshp. And smla to Seq, a pa of opeatons can be attached to Assoa to ndcate the specfc assocatve opeatons. So we can wte Asso ( O, O ), whch means f O s pesent, 1, 1 O 2, 2 a 1, 1 2, 2 have to be pesent as well. In addton, a set ASSO can be defned to estoe all of such nd of elatons, whch means ASSO = { Asso ( O, O )}. a 1, 1 2, 2 Moeove, anothe stuaton may exst: what happens O n Fgue 2 s also on an altenatve pocess tal? f 2,7 That means O tuns to be optonal n that ccumstance. 2,7 So anothe assocatve elaton may exst between the onodeor1 and O : when 2,7 O 2,7 s pesent, OR1 has to be pesent as well, and vce vesa. As a esult, anothe constant needs to be defned between the o-node and ts mmedate pedecesso. As defned n secton 2, OR can be epesented by OR ( O, O, O ), n whch O 0, 0 1, 1 2, 2 0, 0 s the mmedate pedecesso of OR and t can be accessed fom the expesson fstof ( OR ). As dscussed above, the followng constants can be popagated: 1529

7 pesenceof ( st) => pesenceof ( O ) : T ( O ) = 0 (12),, pesenceof ( ops[ fstof ( Assoa )]) => (13) pesenceof ( ops[ nextof ( Asso )]) : Asso ASSO a pesenceof ( ops[ fstof ( OR )]) => pesenceof ( o[ OR ]) : OR R a (14) whee pesenceof ( I1) => pesenceof ( I2 ) means that the pesence of nteval I1 wll enfoce the pesence of nteval I 2.Functon (12) estcts that all compulsoy opeatons have to be pesent. Functon (13) ndcates that those opeatons wth an assocatve elatonshp should be pesent o absent togethe. And functon (14) assocates the o-node wth ts mmedate pedecesso. 4. EXPERIMENTS AND DISCUSSIONS The IBM ILOG CPLEX Optmzaton Studo 12.4 edton s ntoduced to pogam and handle the CP-based IPPS model. Expements ae conducted on a Dell destoppc wth CPU and 6M RAM. Fo a benchma pupose, the test bed wth 18 obs, up to 300 opeatons, and 24 test poblems desgned by Km, et al. s adopted fo expements(km, Pa, & Ko, 2003). FalLmt s a cucal paamete fo the contol of seach temnaton n IBM ILOG, whch defnes a lmt estctng the numbe of falues dung the seach. Hee the FalLmt s set to be 200,000 fo all poblems tested n ou expements. Each poblem of the test bed s executed wth 6 uns and the aveage of the 6 smulaton uns s taen. Afte completon of the expemental uns, the aveage esults geneated by CP-based appoach ae compaed wth those geneated by two evoluton-base algothms, whch ae the coopeatve co-evolutonay genetc algothm (CCGA)(Potte & De Jong, 1994), and the symbotc evolutonay algothm (SEA)(Km et al., 2003). Table 1 accounts the mean maespan of 10 poblems geneated by CCGA, SEA and CP-based appoach. The CPU tme needed fo the CP-based appoach s also gven n the table fo efeence. Poblems 1 to 6 ae elatvely smple IPPS nstances wth 6 obs ncluded. Poblems 21 to 24 aecomplex IPPS nstances wth moe than 15 obs and nea 300 opeatons ncluded. Fstly, Table 1 demonstates that CP-based appoach has bette pefomance on the IPPS, snce ts esultsae bette than those of theothe two algothms n 9 out of 10 poblems of the test bed. Secondly, t can be found that the CP-based appoach pefoms much bette than the othe two algothms when solvng lage-sze poblems. The mpoved ate exceeds 20% fo the poblem 23 and 24, whch means the poposed appoachcan be mplemented to esolve complex combnatoal optmzaton poblems. At last, table 1 eveals that the CPbased appoach s petty effcent snce the tme consumed fo computaton does not gow apdly when the poblem doman expands qucly. The gantt chat shown n Fgue 3 depcts a schedule fo the poblem 24 wth 18 obs and 300 opeatons, whch eveals a vey hgh machne utlzaton. The accuacy and feasblty of the poposed appoach can also be vefed. Fgue 4(a) and Fgue 4(b) llustate the convegence cuves of the espectve maespans of poblems 2 and 24.The x-axs and y-axs epesent the tme and maespan s value espectvely. Actually, the convegence cuves fo poblems 3 to 6 featue the same tend as shown n Fgue 4(a), whch shows that a vey good soluton can be found wthn 10 seconds. Fgue 4(b) shows anothe convegence tend fo a complex IPPS nstance. It eveals that a good soluton can also be found wth aound 10 seconds. Howeve, t also shows that the CP-based appoach possesses potentals to exploe the seach space and contnuously fnd bette solutons f the FalLmt s bg enough. The popagated constants may povde substantal contbuton to such esults, snce nvald solutons can be qucly vefed and elmnated fom the seach space, and the sze of seach space can be dopped to a geat extent at the ealy stage of the seachng pocedue. 5. CONCLUSIONS In ths pape, constant pogammng (CP) s mplemented to solve the IPPS poblem. A complex IPPS model wth altenatve machnes and pocesses s establshed based on the CP language. Vaables and constants ae clealy defned. Futhemoe, a method of constant popagaton s gven n ode to enhance the effcency of the poposed appoach. Expements on a set of benchma test bed poblems ae conducted, whch eveals the feasblty and excellent pefomance of the CP n the doman of IPPS poblems. Howeve, anothe phenomenon s also evealed n ou expements. Results of dffeent expements on the same poblem featue a hgh level of smlaty. Fnal obectve values fo the same poblem tun out to be the same n epeated expements. Moeove, the convegence tend cuves fo the same poblem n epeated expements follow a faly smla tal. So t mght be assumed that the seach pocedue of the IBM ILOG CPLEX Optmzaton Studo mght be non-andomzed. It mght pohbt the futhe mpovement of the CP s pefomance n the feld of IPPS poblems. In ths egad, the combnaton of CP 1530

8 and heustc methods to solve the IPPS poblem mght be a good eseach decton. Table 1: The compason of aveagemaespan Poblem SEA CCGA CP-based Best Impoved ate (%) CP-based CP-based CP-based CP-based SEA CP-based CP-based CP-based CP-based CP-based 20.6 CPU tme (s) Fgue 3: A gantt chat fo poblem

9 Fgue 4(a): Maespan s convegence cuve of poblem 2 Fgue 4(b): Maespan s convegence cuve of poblem 24 ACKNOWLEDGMENT The wo descbed n ths pape s fully suppoted by a gant fom the Reseach Gants Councl of Hong Kong (Poect Code HKU E). REFERENCES Baptste, P., Le Pape, C., & Nuten, W. (2001). Constant-based schedulng : applyng constant pogammng to schedulng poblems. Boston: Kluwe Academc. Bec, J. C., Feng, T. K., & Watson, J. P. (2011). Combnng Constant Pogammng and Local Seach fo Job-Shop Schedulng. Infoms Jounal on Computng, 23(1), Guo, Y. W., L, W. D., Mleham, A. R., & Owen, G. W. (2009). Optmsaton of ntegated pocess plannng and schedulng usng a patcle swam optmsaton appoach. Intenatonal Jounal of Poducton Reseach, 47(14), Km, Y. K., Pa, K., & Ko, J. (2003). A symbotc evolutonay algothm fo the ntegaton of pocess plannng and ob shop schedulng. Computes & Opeatons Reseach, 30(8), L, W. D., & McMahon, C. A. (2007). A smulated annealng-based optmzaton appoach fo ntegated pocess plannng and schedulng. Intenatonal Jounal of Compute Integated Manufactung, 20(1), Moad, N., & Zalzala, A. (1999). Genetc algothms n ntegated pocess plannng and schedulng. Jounal of Intellgent Manufactung, 10(2), Potte, M. A., & De Jong, K. A. (1994). A coopeatve coevolutouay appoach to functon optmzaton. Paallel Poblem Solvng fom Natue - Ppsn I - Intenatonal Confeence on Evolutonay Computaton, Poceedngs, 866, Sadyov, R., & Wolsey, L. A. (2006). Intege pogammng and constant pogammng n solvng a multmachne assgnment schedulng poblem wth deadlnes and elease dates. Infoms Jounal on Computng, 18(2), Stenhofel, K., Albecht, A., & Wong, C. K. (1999). Two smulated annealng-based heustcs fo the ob shop schedulng poblem. Euopean Jounal of Opeatonal Reseach, 118(3), Tmpe, C. (2002). Solvng plannng and schedulng poblems wth combned ntege and constant pogammng. O Spectum, 24(4), Wong, T. N., Leung, C. W., Ma, K. L., & Fung, R. Y. K. (2006a). An agent-based negotaton appoach to ntegate pocess plannng and schedulng. Intenatonal Jounal of Poducton Reseach, 44(7), Wong, T. N., Leung, C. W., Ma, K. L., & Fung, R. Y. K. (2006b). Dynamc shopfloo schedulng n mult-agent manufactung systems. Expet Systems wth Applcatons, 31(3), Wong, T. N., Leungy, C. W., Ma, K. L., & Fung, R. Y. K. (2006). Integated pocess plannng and schedulng/eschedulng - an agent-based appoach. Intenatonal Jounal of Poducton Reseach, 44(18-19),