From: ICAPS-03 Procdings. Copyright 2003, AAAI (www.aaai.org). All rights rsrvd. A Multi-Huristic GA for Schdul Rpair in Prcast Plant Production Wng-Tat Chan* and Tan Hng W** *Associat Profssor, Dpartmnt of Civil Enginring, National Univrsity of Singapor, 0 Knt Ridg Crscn Singapor 9260; TEL 65-68742576; cvcwt@nus.du.sg **Rsarch Scholar, Dpartmnt of Civil Enginring, National Univrsity of Singapor, 0 Knt Ridg Crscn Singapor 9260; TEL 65-68744643; g020248@nus.du.sg Abstract A multi-huristic schdul rpair modl for schdul conflict rsolution is prsntd and its application in rpairing th schduls of a prfabrication plant is dscribd in this papr. Th modl combins huristic stratgis with Gntic Algorithms to rpair schduls with rsourc constraints. Th GA dtrmins th bst squnc of rsolving schdul disturbancs using huristic ruls slctd from a library of huristics commonly usd in industry. W compar quantitativly th advantags of using this modl for schdul rpair against xisting singlhuristic schdul rpair tchniqus with a multi-critria valuation function. Rsults on th macroscopic and microscopic lvls ar prsntd to undrstand th strngths and waknsss of th modl. Ky words: application of planning and schduling; dynamic schduling. Introduction Our rsarch is basd on a ral-lif application of production planning and (r)schduling in prfabrication plants. In Singapor, th incrasd us of prfabricatd building componnts and industrializd building mthods has bn idntifid as th mans of improving both th ovrall productivity at th construction sit and th quality of th construction facility. Th dmand for diffrnt typs of prfabricatd building componnts has bn on th incras, spcially in public housing and transport infrastructur projcts. As a rsul th prfabrication plants and th gnral contractors using ths prfabricatd componnts in thir projcts form a short but conomically significant construction supply chain. Th typs of prfabricatd componnts usd in a construction projct and th rat of th projct s progrss significantly influnc th production schdul of th prfabrication plant supplying thos componnts. Mor spcifically, th plant nds to schdul th production of spcific componnts rquird by th gnral contractor and Copyright 2003, Amrican Association for Artificial Intllignc (www.aaai.org). All rights rsrvd. dlivr thm to th construction sit by th du dats dtrmind largly by th pac of th construction sit schdul. Du to this intimat rlationship, a chang in componnt spcifications, th quantitis rquird or th du dats by th contractors invitably lads to a rviw of th prfabrication plant s production schdul. Conflicts in production schduls aris whn th rviw shows that production rsourcs ar ovr-committd to mt nw dlivry du dats. At last on of th production oprations has to b rschduld and this is calld a (schdul) disturbanc. Rschduling is furthr complicatd as prfabrication plants usually supply diffrnt htrognous componnts to a numbr of construction projcts simultanously at any on tim. In updating th production schdul, plant oprators tnd to utiliz thir own prfrrd huristic, usually th on that had provn asy to apply and rasonably fficint from past xprinc. Morovr, th sam huristic is likly to b applid to rsolv all schdul disturbancs. Howvr, huristics ar known to b problm spcific and cannot guarant good solutions for all cass. W propos to lt an volutionary sarch dcid th bst huristic to apply to a particular disturbanc, as wll as th ordr of rsolving disturbancs by combining th us of huristics and gntic algorithms (GA) in a mthod w call th Multi-huristics Schdul Rpair Modl. A custom chromosom rprsntation is proposd to ncod th dcisions involving th ordr of rsolving disturbancs and th huristic bst suitd to rsolv disturbancs. Th GA volvs th chromosoms to dtrmin both th optimal rpair squnc as wll as th bst combination of huristics from a pool of slctd huristic stratgis. W invstigat th fficincy of th proposd schdul rpair modl in gnrating high-quality rpaird schduls, and compar th schduls gnratd with th us of this modl against thos gnratd by th singl-huristic approachs currntly usd in th industry. This comparison is basd on a multi-critria valuation function drivd from factors prtinnt to industry practics. 236 ICAPS 2003
2 Litratur Rviw Th widr us of prfabricatd building componnts has ld to rsarch on planning and schduling mthods in th prcast industry. Warszawski (982, 990 and 999) providd a gnral framwork of th main faturs to a proposd information for planning, cost and quality control in prfabricatd plant oprations, basd on a mathmatical prcast schduling modl dfind in trms of dcision variabls. Furthrmor, Warszawski (984) proposd a modl for short and long-rangd production planning of componnts in mak-to-ordr manufacturing systms. Dawood and Nal (993) dvlopd a computr-basd capacity basd modl using th backward schduling tchniqu to hlp managrs crat long trm capacity plan, mak bttr planning dcisions and xplor options. In th gnral application of GA for schduling optimization, Chan t al. (996) proposd a gnric GA modl suitabl for schduling and rsourc allocation problms. Th random kys concpt (Ban 994) was usd in th modl to nsur that thr was no illgal schdul. On th application for GA to th optimization of production schduling of prfabricatd componnts, Chan and Hu (998, 999 and 2002) dvlopd a flow shop squncing modl for spcializd prcast production schduling, and a hybrid gntic algorithm constraint programming (GA- CP) approach to solv comprhnsiv prcast schduling. Lu and Hwang (200) proposd th usag of GA to obtain optimal rsourc-constraind production schduls for rptitiv prfabricatd componnts. On dvlopmnt that is prtinnt to industrial practic is that of ractiv schduling from artificial intllignc rsarch. Much rsarch on schdul coordination and rpair in th manufacturing industry has bn don using this schduling concpt (Zwnbn t al. 990; Smith 994 and Sadh 994). Howvr, thr has not bn much application of such concpts in th construction industry. Similaritis btwn th production procsss in th prcast factory and th assmbly lin in th manufacturing procss opns th possibility of th transfr of rsarch findings and practical xprinc of schdul rpair btwn ths two aras. Extrnal factors causing schduling disturbancs Architct Gnral contractor Nw businss opportunity Chang in spcification Chang in quantity Chang in du dats Rjctd lmnts for quality Nw lmnts rquird Mold modification Rsourc shortag Storag/production spac shortag Mod of transport In-hous factors causing schduling disturbancs Machin maintnanc and claning Mandatory chang Potntial changs in production schdul Accommodativ chang X Y Dnots factor X has a dirct influnc on factor Y Figur : Extrnal and In-hous factors causing schdul disturbancs ICAPS 2003 237
3 Schdul Disturbancs and Huristic Stratgis Important background information on how schdul disturbancs occur and th varity of huristic stratgis usd was obtaind through intrviws with industry practitionrs during th cours of this study. 3. Schdul disturbancs Thr ar svral common causs of schdul disturbancs, ranging from quantity and dsign spcification changs to poor quality and machin brakdowns. Ths causs hav bn catgorizd as ithr in-hous factors or xtrnal factors, dpnding on whthr th caus is within th control of th factory or not. Schdul changs may or may not b rquird in rspons to ths disturbancs. For xampl, th plant oprator may choos to forgo nw ordrs and not disrupt xisting schduls but is complld to chang his schduls if this involvs contractual obligations. Figur illustrats th spcific xtrnal and in-hous factors causing schdul disturbancs, as wll as thir influncs on on anothr. 3.2 Huristic stratgis for rpairing schduls Production schduling is carrid out for a fixd planning horizon (usually 30 days ahad) according to an agrd schdul for dlivring componnts. Among th huristic ruls usd by plant oprators to rschdul disturbancs and rpair thir production schduls includ: () Right shift (RS): rsolvs conflicts by pushing th production forward in tim until th disturbanc is rsolvd (Fig. 2.); (2) Lft shift (LS): a similar stratgy that shifts an opration backwards in tim. It is particularly usful whn a hard constraint that prviously prohibitd th commncmnt of th opration is softnd or rmovd (Fig. 2.2); (3) Opportunistic insrtion (OI): maks us of idl days in th schdul to accommodat a disturbanc by braking it into smallr parts and fitting ths smallr parts into th schdul in an opportunistic first fit mannr. Th fficincy of this huristic rul largly dpnds on th initial utilization lvl of th production facilitis (Fig. 2.3); (4) Dtrministic Insrtion (DI): similar to opportunistic insrtion but th disturbancs hav priority ovr alrady schduld production and displac thm from th schdul. Th lattr ar rschduld using OI (Fig. 2.4); (5) As-soon-as-possibl (ASAP) / Backward Schduling (BS): th ASAP mthod schduls th disturbanc basd on th arlist start tim (EST); th BS mthod Bfor rsolution Start of disruption Rsolution by RS Bfor rsolution Rsolution by LS Disturbanc Rvisd arlist start dat Bfor rsolution Disturbanc Aftr rsolution Production lin End of disruption Figur 2.: Right shifting Figur 2.2: Lft shifting Initial arlist start Figur 2.3: Opportunistic insrtion Point of insrtion Figur 2.4: Dtrministic insrtion Idl stat for production lin Occupid stat for production lin Disturbanc Figur 2: Illustrations of som huristic ruls schduls th disturbancs basd on th latst start tim (LST); (6) Multipl mold approach: rsolvs th disturbanc by assigning similar componnts within th sam group of componnts to any on of svral molds capabl of producing th componnts using a OI or DI stratgy; (7) Sub-contracting: this stratgy outsourcs production to othr oprators and is usd whn th plant is alrady producing at its pak capacity or it is conomically mor bnficial to do so. Th huristic ruls mntiond abov wr solicitd from xprincd plant oprators through prsonal intrviw. Th plant oprators dpndd on prvious xprinc whn choosing ruls to rsolv disturbancs and did not 238 ICAPS 2003
sm to hav a formal quantitativ way of dciding on how bst to rpair schduls. Tim prssur oftn prvntd thm from trying altrnativ ways of rsolving disturbancs or considring th ffct of rsolving svral disturbancs togthr. Th multi-huristics schdul rpair modl could hlp addrss ths dficincis and provid altrnativ high quality rpaird schduls. 4 Multi-huristic Schdul Rpair Modl Our proposd modl supports th dtrmination of priority for conflict rsolution using huristic stratgis that ar bst suitd to incorporat th conflict-causing disturbancs into an xisting schdul. Th rpair actions ar also likly to caus furthr disturbancs which thn hav to b rsolvd. Thrfor, it is ncssary to considr not only how to rsolv th conflict but also th ordr in which th conflicts ar to b rsolvd as both hav a baring on th dsirability of th final rpaird schdul. Th proposd modl supports this important considration by sarching for th bst combination of conflict-rsolving squnc (ordr) and huristics usd (how) from many possibl combinations using GA. Gntic algorithms ar stochastic sarch mthods basd on th mchanism of slction and volution, and hav bn succssfully applid in schduling problms including that of prcast lmnt production. Dtails of a GA adaptation for our proposd modl ar dscribd as follows. 4. Constraints Production schduling rquirs allocating rsourcs ovr tim to a st of jobs whil satisfying a varity of constraints and objctivs. Hard constraints must always b satisfid for a (rpaird) schdul to b valid. Soft constraints on th othr hand, could b rlaxd whn ncssary. Basd on th rsults of th industry study, w hav catgorizd th hard constraints in our proposd modl as functional, capacity and availability constraints, whil th soft constraints ar dlivry and invntory constraints. Th rprsntation of ths constraints in mathmatical trms is ncssary for thir us in GA. Th following sction discusss th mathmatical formulation of ths constraints in trms of binary dcision variabls dfind in Tabl. Functional constraint: to maintain th production intgrity of th prfabrication plant by limiting th typs of lmnts that a spcific mold can produc. Although it is possibl for a mold to produc svral diffrnt typs of lmnts, w hav rstrictd this capability to lmnts within a mold group within which thr ar only minor variations in mold dtails. This is ncssary as convrting a mold to a diffrnt mold group is rarly don in practic du to substantial convrsion tim and costs incurrd. 5 4 = 4 m= 3 6 = m= 5 x = 0 for all t () m, x = 0 for all t (2) m, Capacity constraints: Following industry norms, ach Paramtrs x m, T M E S o, S S S R D N m L L,r T T n Dscription A binary dcision variabl, whr x m, = mans that mould m is assignd to produc lmnt on day whilst x m, =0 will man th opposit; t = 0,, 2 T, schduling priods in days; m = 0,, 2 M, mould srial numbrs; = 0,, 2 E, typs of lmnts to b producd; Initial stock of lmnt typ at th bginning of th schduling priod (t = 0); Numbr of lmnt typ in stockyard on day t; Maximum allowabl storag lvl of lmnt in stockyard; Minimum buffr storag lvl of lmnt in stockyard; Numbr of lmnt typ rquird to b dlivrd on day t; Numbr of lmnt typ dlivrd on day t; Numbr of changovrs for mould m in th schduling priod; Lad tim of lmnt typ btwn production and dlivry; Minimum lad tim rquird for lmnt typ btwn production and dlivry; Prsnt tim; Total numbr of working days, obtain by subtracting th numbr of Sundays from T. Tabl. Paramtrs for mathmatical rprsntations ICAPS 2003 239
mold is limitd to produc only on lmnt pr working day (Equation 3). Thrfor th daily maximum capacity of th prcast yard is qual to th total numbr of molds (Equation 4). W hav furthr assumd that thr is no production during Sundays and public holidays (Equation 5). E = M x (0,) for all m,t (3) m, = E m= = x m, M for all t (4) x m, = 0 for t Sundays and public holidays (5) Availability constraint: spcifis th tim rquird for ach producd lmnt to b rady for dlivry. A minimum lad tim btwn production and dlivry must b obsrvd for th componnts to attain approximatly 70% of thir 28-day strngth, which rfrs to th spcific strngth that concrt gains as it stiffns from an initial plastic stat aftr a stting tim of 28 days. Traditional curing taks up to svn days, although th local practic of controlld acclratd curing in a curing chambr rducs this lad tim to just two days. L L, r whr L, r = 48 hours (6) Dlivry constraint: spcifis th dlivry rquirmnts of th componnts to th construction sits. Du to th larg sizs of th prfabricatd lmnts and th shortag of storag spac on th construction sits, plant oprators ar usually not allowd to dlivr th lmnts any arlir than th stipulatd dat of dlivry, nor dlivr mor than what is rquird (Equation 7). Furthrmor th sum of th initial stock lvl and th total production of any lmnt bfor ach dlivry dat should b at last as many as th numbr of lmnts rquird to b dlivrd (Equation 8). D t Rt, for all, whr t t2 (7), 2 T M t= m= T S0, + x,, R, for all (8) t m t= t Invntory constraint: limits th numbr of prfabricatd componnts to b stord in th invntory. It also spcifis th lvl of buffr invntory. In shor th invntory constraint srvs to dfin th oprating rang for stock lvls of ach prfabricatd componnt. Du to spac constraints, th total numbr of producd componnts that can b kpt in a plant s stockyard is limitd. Howvr, plant oprators ar highly rsourcful in sking nw avnus for storing invntory and hav bn known to stor lmnts tmporarily on transportation trailrs. Thy also kp a minimum numbr of various componnts to srv as buffrs to unxpctd or urgnt dmand. Thrfor th cumulativ numbr of any producd componnts lss dlivrd in any priod should b lss than th maximum allowabl storag limit but mor than th minimum buffr lvl. T M T S S + x D S 0, m, t= m= t= for all (9) 4.2 Objctiv functions Local prcast plants produc prfabricatd componnts mainly on a contractual basis, apart from producing som standard lmnts for anticipatd dmand. Plant oprators hav to mt contractual du dats for dlivris whil kping an accptabl lvl of invntory in th stockyard to buffr any unanticipatd dmand. Counting th numbr of lmnts that was not dlivrd on tim and th numbr of ovr or undr-stockd lmnts in th invntory will thn rflct on th fficincy of th (rpaird) schduls. Plant oprators also try to mak full us of thir molds and minimiz th numbr of changovrs rquird. Efficint lmnt to mold assignmnt is thrfor important to fficint schduling, as that will minimiz th cost of changovrs. Hnc, th numbr of changovrs incurrd bcoms our third paramtrs for valuating (rpaird) schdul fficincy. Plant oprators tnd to minimiz th numbr of idl days during th planning horizon, as it is sn as a wast of rsourcs. Howvr, thy hav to balanc btwn th costs and ffcts of xcssiv production. Production of any particular lmnt on a prmannt basis will kp th numbr of mold changs down and improv th mold utilization rat. Howvr it will also incras th ovrstocking of th lmnt thrby affcting th production of othr componnts, which can rsult in lat dlivris for th lattr. It is clar thn that th oprators hav to sk a balanc btwn th diffrnt objctivs of mting du dats, minimizing mold changs, maintaining optimum invntory lvls and kping non-productiv working days to th minimum. Th mathmatical rprsntations of ths paramtrs ar as follows: Numbr of lmnts in xcss/inadquat invntory lvl: th invntory lvl of any lmnt is bst maintaind at an optimum rang for spatial and buffr considrations. Thrfor th total numbr of lmnts in xcss of or blow dsird invntory lvls should b minimizd T E + + Min Z = {( S S ) + ( S S ) } S t= = ( + whr S S ) = max{0,( S S )} 240 ICAPS 2003
( S S, ) + t = max{0,( S S )} (0) D-Gn H-Gn Numbr of mold changs: in ordr to produc diffrnt lmnts of th sam mold group, a mold must undrgo minor modification, thrby incurring both cost and tim. Thrfor, fficint lmnt to mold assignmnt is ndd to minimiz th total numbr of mold changovrs. Min M Z M = N m m= () Numbr of lmnts not dlivrd on du dats: failur to dlivr th stipulatd numbr of lmnts on tim would incur financial pnaltis and bring dtrimnts to th rputation of plant oprators. Thrfor th total numbr of lmnts not dlivrd on tim should also b minimizd. Min Z D = T E t = = ( R D ) (2) Numbr of ffctiv idl days: th maximum numbr of lmnts that can b producd pr day is M, and th total production capacity within a planning horizon cannot b mor than MT n. A mor accurat rflction of th numbr of idl days would thrfor b rprsntd by: T M E Min Z I = MTn xt, m, (3) M t = m= = Du to th diffrnt units of masurmnt of th 4 valuation paramtrs, it would not b maningful to add thm dirctly; hnc, thr is a nd to normaliz thm into a dimnsionlss quantity. On approach is to divid ach paramtr by a constant (.g. th man valu of a distribution) and thn sum up th numbrs into an fficincy indx. Howvr this would rsult in a biasd analysis favoring paramtrs which xhibit high variability thus rsulting in high normalizd valus, as ths tnd to dominat th fficincy indx. W hav usd 4 planning ruls and th intgr programming approach to gnrat 25 psudo-schduls at various rsourc utilization lvls. Th hard constraints wr obsrvd in th cration of ths schduls to b usd for our rpair algorithms. Ths schduls wr thn valuatd sparatly using ach of th four paramtrs, rsulting in a rang of prformanc valuations for ach of th four paramtrs. Th raw valuation valus wr mappd onto a rang btwn 0 and 0.25 by mans of linar rgrssion. Doing so mant that w assumd that ach of th 4 paramtrs was qually important. Th summation of th four paramtrs cratd a dimnsionlss objctiv function which minimizd th dominanc of any paramtr. This normalizd objctiv function gav an indication of th rlativ prformanc on ach paramtr. 842 854 563 536 23 2 4 3 3 D D2 D3 D4 D5 H H2 H4 H3 H3 Figur 3: Chromosom rprsntation D-Gn H-Gn 23 536 563 842 854 2 4 3 3 D5 D4 D3 D D2 H H2 H4 H3 H3 2 4 3 3 H H2 H4 H3 H3 Figur 4: Dcoding of chromosom Th highr th indx valu, th poorr was th prformanc ranking. Th objctiv function is thrfor dfind as: Min Z = 0.2275 + 0.0008Z S + 0.0076Z M + 0.0042Z D + 0.0264Z I (4) 4.3 GA rprsntation As shown in Fig. 3, th chromosom string is mad up of qual numbr of D-gns (disturbanc gn) and H-gn (huristic gns). Each conflict to b rsolvd is rprsntd by a pair of D and H-gns. Th D-gns ncod ral numbrs that srvs as sort kys to dtrmin priority of rsolution, whilst th H-gns ncod th ordinal valu of th huristics usd to rsolv th conflict. Th proprtis of ach disturbanc and th rsolving algorithm for ach huristic ar dfind on thir rspctiv tabu. To dcod th chromosom, th squnc of rsolving conflicts is dtrmind by sorting th disturbancs in incrasing ordr of th D-gn valus. Th corrsponding huristics dfind in th H-gns ar thn usd to incorporat th disturbancs into an xisting schdul, as illustratd in Fig. 4. In this cas, th squnc of conflict rsolution with corrsponding huristics is: D5 (H) D4 (H2) D3 (H4) D (H3) D2 (H3). Thr ar svral paramtrs that can dtrmin th prformanc of GA but thir optimal valus cannot b ascrtaind by applying fixd ruls. In fac optimal GA paramtrs ar known to b notoriously difficult to dtrmin (Myrs and Hancock 200). Ths paramtrs includ th population siz, th numbr of itrations prformd, th crossovr ra th mutation rat and th trmination critrion. ICAPS 2003 24
Disturbanc Elmnt Typ Quantity Du Dat for Dlivry Natur of Disturbanc D E Day 5 To rplac a rjctd lmnt D2 E2 2 Day 7 Dsign chang to E2 lmnt D3 E3 2 Day 9 Dsign chang to E3 lmnt D4 E3 2 Day 5 To rplac a rjctd lmnt D5 E2 2 Day 7 To rplac a rjctd lmnt Tabl 2. Charactristics of disturbancs Original production schdul Rpaird production schdul Day Day 2 3 4 5 6 7 8 9 0 2 3 4 5 6 7 8 9 0 L E E E E E E E E E N E E E E E E E E E E Production lins L2 E E N N N N N N N N Schdul Rpair E E N N N N N E3 E3 E3 L3 E3 E3 E3 E3 E3 E3 E3 E3 E3 N E3 E3 E3 E3 E3 E3 E3 E3 E3 E3 L4 N N N N N N N N N N E2 E2 N N N N N N E2 E2 Th Multi-huristic Schdul Rpair Modl dtrmins th optimal squnc (D D2 D4 D3 D5) and th bst-suitd huristic to incorporat ach disturbanc dfind in Tabl 2 into th original schdul. Each cll rprsnts an lmnt typ schduld to b producd in a spcific production lin on a particular day. For xampl, production lin L is schduld to produc lmnt typ E on th first day of th original production schdul. "N" dnots no production; thrfor production lin L4 is not schdul for any production in th original schdul. Gry clls in th rpaird production schdul rprsnt th incorporatd disturbancs. Figur 5: Rpaird schdul dtrmind by GA In our proposd modl, a two-point crossovr is usd to combin th gn valus of two chromosoms to crat a nw pair of chromosoms. Mutation oprats on a singl chromosom and producs a nw gnotyp by making a random chang to th valu of on or mor of th gns in th chromosom string. Th sttings for ths ky paramtrs ar: population siz (00), numbr of itrations (500), probability of crossovr (0.85) and mutation (0.00). Ths valus wr dtrmind by fin-tuning dfault valus ovr svral runs of th GA on a similar problm. Th PGAPack opratd on a Silicon Graphics workstation in th UNIX nvironmnt was adoptd as th GA softwar usd. It is a paralll gntic algorithm library that is intndd to provid most of th capabilitis ndd for ncoding GA applications in an intgratd, samlss and portabl mannr. 5 Exprimnts Th application of our proposd modl prsntd involvs th schdul rpair of four molds ovr a priod of two wks (0 work days). Th plant producs thr typs of lmnts, namly E, E 2 and E 3, which can b producd by any of th four molds with minimal modification. Fiv disturbancs occur during th planning priod and th charactristics of ths disturbancs ar shown in Tabl 2. Svn huristic ruls wr slctd to b includd in th huristics pool. Six of th huristics wr basd on th multipl mold approach whr mor than on mold could b usd to rsolv a conflict. Th sarch for th point of insrtion into th original schdul can b prformd in a paralll mannr across all mold schduls simultanously or for ach mold schdul in squnc. Th first 6 242 ICAPS 2003
Mthod of Rsolution Multihuristic S/BS/OI P/ASAP/OI S/ASAP/OI P/BS/OI Low utilization lvl (0.5-0.64) Total numbr of idl days (days).25.25.25.25.25 Total numbr of lat dlivris (lmnts) 0 0 0 0 2 Total numbr of ovr/undr stocking (lmnt-days) 27 25 2 9 9 Total numbr of mould changs (tims) 3 7 5 7 7 Bst Indx 0.3049 0.3337 0.353 0.3289 0.3373 Multipl-huristic yild 0% 9.45% 3.4% 7.87% 0.63% Middl utilization lvl (0.65-0.80) Total numbr of idl days 0.25 0.25 0.25 0.25 0.25 Total numbr of lat dlivris 2 4 4 3 5 Total numbr of ovr/undr stocking 7 5 6 9 8 Total numbr of mould changs 5 6 6 7 6 Bst Indx 0.294 0.3005 0.303 0.307 0.307 Multipl-huristic yild 0% 2.8% 2.45% 4.42% 4.42% High utilization lvl (>0.8) Total numbr of idl days 0 0 0 0 0 Total numbr of lat dlivris 8 9 9 9 9 Total numbr of ovr/undr stocking 32 32 28 36 3 Total numbr of mould changs 6 7 7 8 8 Bst Indx 0.3323 0.344 0.3409 0.3549 0.3509 Multipl-huristic yild 0% 3.55% 2.59% 6.80% 5.60% Tabl 3. Bst prformanc of multi-huristic approach compard to th singl huristics huristics ar dnotd as S/ASAP/OI, S/BS/OI, S/ASAP/DI, P/ASAP/OI, P/BS/OI, P/ASAP/DI. Th last huristic considrd is sub-contracting. In th naming schm mployd, th first part of th nam squnc dnots th sarch squnc (paralll or squntial), th scond part dnots th dirction of sarch (from th bginning or from th nd), and th last part dnots th mannr of insrtion (opportunistic or dtrministic fit). To tst our proposd modl, 5 schduls wr artificially constructd using a random procss to giv mold utilization rats varying from 0.6 to 0.8; this rang was chosn to rflct th utilization rats commonly sn in local practic. Th initial invntoris for E, E2 and E3 ar assumd to b 6, 2 and 6 lmnts rspctivly. For ach of ths schduls, a tst was conductd using th baslin / original schdul as a basis within which to schdul th disturbancs shown in Tabl 2. Th GA procdur was thn usd to construct modifid schduls whrin th disturbancs had bn insrtd. Th rsult of on such tst is shown in Fig. 5 as spac dos not allow showing th rsults of all th tsts. Anothr 4 sts of xprimnts wr conductd, again using th sam baslin schduls but this tim allowing GA to apply only on of four huristics (S/ASAP/OI, S/BS/OI, P/ASAP/OI and P/BS/OI). Ths 4 huristics wr chosn bcaus thy ar industry s favorits. Th prformanc of our proposd multi-huristic schdul rpair modl is compard to th singl-huristic approach at both th macro and microscopic lvl. At th macroscopic prspctiv, w compar th valuation indx valus obtaind by both approachs. Th improvmnt obtaind by th multi-huristic approach is also discussd. At th microscopic lvl, w analyz th prformancs in trms of ach of th physical paramtrs that constitut th valuation indx. 5. Macroscopic analysis Having vrifid that th indx valus satisfy th normality and corrlation tsts, 4 sparat sts of paird-sampl t- tsts wr prformd to valuat th significanc of th diffrnc btwn th man indx valus of our multihuristic modl with ach of th 4 singl-huristic approachs. Th tsts rvald rsults that wr vry ncouraging. Our multi-huristics modl has, in all th 4 sparat t-tsts, producd lowr man indx valus than ach of th 4 singl-huristic approachs with p-valus vry clos to ICAPS 2003 243
Man valu Indx Lat dlivry Mould chang Non-optimal invntory Singl huristic tstd against Altrnativ hypothsis P-valu S/BS/OI < 0 0.00 P/ASAP/OI < 0 0 S/ASAP/OI < 0 0 P/BS/OI < 0 0 S/BS/OI < 0 0.055 P/ASAP/OI < 0 0.357 S/ASAP/OI < 0 0.036 P/BS/OI < 0 0.002 S/BS/OI < 0 0.00 P/ASAP/OI < 0 0 S/ASAP/OI < 0 0 P/BS/OI < 0 0 S/BS/OI >0 0.03 P/ASAP/OI not = 0 0.48 S/ASAP/OI not = 0 0.362 P/BS/OI >0 0.023 approach prformd bst against th 4 singl huristic at 3 diffrnt lvls of utilization. Th prformanc of th multi-huristic approach vrsus that of th singl huristic appars marginal whn masurd on our valuation indx formulation. Howvr, th gains bcom mor tangibl whn translatd to ral physical masurs lik th numbr of lat dlivris or mould changs, which ar significant to th plant oprators. Th oprators would typically prfr not to incur any lat dlivris du to ithr contractual obligation or far of marring th plant s rputation. Thrfor, a yild of 5% on an indx valu of 0.3 would translat to an quivalnt (0.3*0.05/0.0042) 3.57 lmnts rduction in lat dlivris or a (0.3*0.05/0.0008) 8.75 lmnts-days rduction in xcss/inadquat invntory during th 0-day rpair priod. 5.2 Microscopic analysis Th sam sts of statistic tsts wr prformd on 3 of th 4 paramtrs that constitutd th valuation indx. Th rlativ prformanc of th multi-huristic approach is thn compard with ach of th 4 singl-huristic approachs. Th rsults of ths tsts ar summarizd in Tabl 4. Plant oprators prfr to kp both th numbr of lat dlivris and th numbr of mold changs during production to th minimum. Whil ovrstocking is also undsirabl, it can b rsolvd with rlativ as in comparison. From th tst rsults, it is obsrvd that th multi-huristic modl xclld in producing rpaird schduls with a minimal numbr of mold changs. This fficint lmnt to mold assignmnt is significant as changs in th molds disrupt th workflow of th production lins and incurrd additional changovr costs. Tabl 4. P-valus for paird sampl t-tsts tsting th diffrnc of th multi-huristics approach against th various sing huristics zro. Such p-valus allow us to conclud strongly that thr is significant statistical vidnc supporting our claim that th multi-huristics modl prformd bttr than any of th singl-huristic approachs in schdul rpair. Rcalling that th indx valu is mad up of 4 diffrnt paramtrs, this suggsts that our modl gnratd solutions that dominatd thos obtaind with th singlhuristic approachs. In trms of th yild, our proposd modl outprformd any singl huristic by up to 3.09%. Th cas whr th multi-huristic approach could only prform as wll as a singl huristic occurrd whn th molds xprincd high utilization rats. Th lack of idl days for schdul rpair in ths schduls limitd what any rpair stratgy could do. Tabl 3 illustrats th cass whr th multi-huristic Having kpt th numbr of mold changs to a minimum, th multi-huristic approach continud to prform rmarkably wll in minimizing th numbr of lat dlivris incurrd in th rpaird schduls it gnratd. Statistics rvald that th multi-huristic modl producd rpaird schduls that hav a lowr man numbr of lat dlivris than 2 of th singl-huristic approach at 5% lvl of significanc and of thm at 0% lvl of significanc. Howvr, thr was not nough to show that th numbr of lat dlivris is lowr whn compard to th P/ASAP/OI huristic. Th multi-huristic approach did not far as wll in minimizing th numbr of lmnts in xcss/inadquat invntory. In fac th multi-huristic approach producd rpaird schduls that hav significantly highr man valus of xcss/inadquat invntory compard to two of th singl huristics (S/BS/OI and P/ASAP/OI). Howvr, this man valu is not significantly diffrnt from th man valus of th two othr singl huristics. 244 ICAPS 2003
This analysis indicatd that th multi-huristic schdul rpair modl was abl to do bttr than any singl-huristic approach; th rpaird schduls achivd mor fficint mold utilization and fwr lat dlivris. Mor significantly, ths improvmnts wr attaind at only a sligh or no incras in th valu of xcssiv/inadquat invntory. 6 Conclusions W hav applid th multi-huristic schdul rpair modl on a ralistic planning and (r)schduling problm for a prfabrication plant. Th initial xprimntal rsults indicat that this multi-huristic approach is ffctiv in rsolving schdul disturbancs, dmonstrably mor so than th singl-huristic approachs currntly usd in industry. Th valuation indx usd as th objctiv function incorporats most of th paramtrs of concrn to industry practitionrs including fficint lmnt to mold assignmnt and minimal lat dlivris with littl or no compromis to th invntory lvl. It can b usd to gnrat non-dominatd schduls in conjunction with th sarch procdur of th GA. Howvr, th scop of th modl is quit limitd and is rstrictd to schdul rpair. For xampl, it dos not addrss th nd for bttr schdul coordination btwn lmnts of th supply chain, particularly btwn th construction sit and th production plant. Furthr work is in progrss to look into this aspct of prcast production schduling. Idally, this will thn allow both th plant oprator and th construction managr to ngotiat th prfrrd outcom in a co-oprativ rathr than advrsarial mannr. 7 Rfrncs Ban, J.C. 994. Gntic Algorithms and Random Kys for Squncing and Optimization. ORSA Journal on Computing, 6(2), pp. 54-60. Chan, W.T.; Chua, D.K.H.; and Kannan, G. 996. Construction Rsourc Schduling with Gntic Algorithms. Journal of Construction Enginring and Managmn 22(2), pp. 25-32. Chan, W.T., and Hu, H. 2002. Production Schduling for Prcast Plants Using a Flow Shop Squncing Modl. Journal of Computing in Civil Enginring, 6(3), pp. 65-74. Dawood, N., and Nal, R.H. 993. A Capacity Planning Modl for Prcast Concrt Building Products. Building and Environmn 28(), pp. 8-95. Lu, S.S., and Hwang, S.T. 200. A GA-Basd Modl for Maximizing Prcast Plant Production undr Rsourc Constraints. Enginring Optimization, Vol. 33, pp. 69-642. Myrs, R., and Hancock, E.R. 200. Empirical Modling of Gntic Algorithms. Evolutionary Computation, 9(4), pp. 46-493. Sadh, N. 994. Micro-Opportunistic Schduling: Th Micro-Boss Factory Schdulr. Intllignt Schduling, pp. 99-37. San Francisco: Morgan Kaufmann Publishrs Inc. Smith, S.F. 994. OPIS: A Mthodology and Architctur for Ractiv Schduling. Intllignt Schduling, pp. 29-67. San Francisco: Morgan Kaufmann Publishrs Inc. Warszawski, A. 982. Managrial Planning and Control in Prcast Industry. Journal of th Construction Division, 8(CO2), pp. 299-33. Warszawski, A. 984. Production Planning in Prfabrication Plant. Building and Environmn 9(2), pp. 39-47. Warszawski, A. 990. Industrialization and Robotics in Building. Nw York: Harpr & Row. Warszawski, A. 999. Industrializd and Automatd Building Systms. London: E & FN Spon. Zwbn, M.; Daun, B.; Davis, E.; and Dal, M. 994. Schduling and Rschduling with Itrativ Rpair. Intllignt Schduling, pp. 24-256. San Francisco: Morgan Kaufmann Publishrs Inc. Chan, W.T., and Hu, H. 998. Procss Schduling Using Gnric Algorithms for Construction Industry. Proc. Third Intrnational Confrnc on Managmn CHEP and Springr-vrlag, Shanghai, China. Chan, W.T., and Hu, H. 999. Procss Schduling of Prcast Production Using Gntic Algorithms. In Proc. Fifth Intrnational Confrnc on Application of Artificial Intllignc to Civil and Structural Enginring, Vol. C, pp. 25-33. Comp Prss, Oxford, U.K. ICAPS 2003 245