Coordinating Multi-Attribute Procurement Auctions in Finite Capacity Assembly Environments

Transcription

1 Coordntng Mult-Attrute Procurement Auctons n Fnte Cpcty Assemly Envronments Jong Sun nd Normn M. Sdeh Aprl 2004 CMU-ISRI e-supply Chn Mngement Lortory Insttute for Softwre Reserch Interntonl School of Computer Scence Crnege Mellon Unversty Pttsurgh, PA , USA Eml Addresses: jongs@ndrew.cmu.edu, sdeh@cs.cmu.edu Also ppers s Computer Scence Techncl Report CMU-CS The reserch reported n ths pper hs een funded y the Ntonl Scence Foundton under ITR Grnt

2 Keywords: Supply Chn Mngement, Procurement, Reverse Aucton, Fnte Cpcty Schedulng 2

3 Astrct Reserch on reverse uctons for procurement hs trdtonlly gnored the temporl nd fnte cpcty constrnts under whch mnufcturers operte. We consder the prolem fced y mnufcturer tht procures multple ey components from numer of possle supplers through mult-ttrute reverse uctons. Bds sumtted y prospectve supplers nclude prce nd delvery dte. The mnufcturer hs to select comnton of suppler ds tht wll mxmze ts overll proft. The mnufcturer s proft s determned y the revenue generted y the products t sells, the costs of the components t purchses s well s lte delvery penltes t ncurs f t fls to delver products n tme to ts own customers. We provde forml model of ths mportnt clss of prolems, dscuss ts complexty nd ntroduce rules tht cn e used to effcently prune the resultng serch spce. We proceed to show tht our model cn e chrcterzed s pseudo-erly/trdy schedulng prolem nd use ths oservton to uld n effcent heurstc serch procedure. Computtonl results show tht our heurstc procedure typclly yelds solutons tht re only few percent from the optmum. They further ndcte tht tng nto ccount the mnufcturer s cpcty sgnfcntly mproves ts ottom lne. 3

4 1. Introducton Tody s glol economy s chrcterzed y fst chngng mret demnds, short product lfecycles nd ncresng pressures to offer hgh degrees of customzton, whle eepng costs nd led tmes to mnmum. In ths context, the compettveness of oth mnufcturng nd servce compnes wll ncresngly e ted to ther lty to dentfy promsng supply chn prtners n response to chngng mret condtons. Wth the emergence of e-usness stndrds, such s exml, SOAP, UDDI nd WSDL, the Internet wll over tme fcltte the development of more flexle supply chn mngement prctces. Tody, however such prctces re confned to reltvely smple scenros such s those found n the context of MRO (Mntennce, Repr nd Opertons) procurement. The slow dopton of dynmc supply chn prctces nd the flure of mny erly electronc mretplces cn n prt e ttruted to the one-dmensonl nture of erly solutons tht forced supplers to compete solely on the ss of prce. Reserch n the re hs lso generlly gnored ey temporl nd cpcty constrnts under whch reverse uctoneers typclly operte. For nstnce, PC mnufcturer cn only ssemle so mny PCs t once nd not ll PCs re due t the sme tme. Such consdertons cn e used to help the PC mnufcturer select mong ds from competng supplers. In ths pper, we present technques med t explotng such temporl nd cpcty constrnts to help reverse uctoneer select mong competng multttrute procurement ds tht dffer n prces nd delvery dtes. We refer to ths prolem s the Fnte Cpcty Mult-Attrute Procurement (FCMAP) prolem. It s representtve of rod rnge of prctcl reverse uctons, whether n the mnufcturng or servce ndustry. Ths rtcle provdes forml defnton of the FCMAP prolem, dscusses ts complexty nd ntroduces severl rules tht cn e used to prune ts serch spce. It lso presents rnch-nd-ound lgorthm, smulted nnelng procedure nd n effcent pseudo-erly/trdy heurstc serch procedure tht ll te dvntge of these prunng rules. Computtonl results show tht ccountng for the reverse uctoneer s fnte cpcty cn sgnfcntly mprove 4

5 ts ottom lne, confrmng the mportnt role plyed y fnte cpcty consdertons n procurement prolems. Results re lso presented tht compre the performnce of our heurstcs serch procedures oth n terms of soluton qulty nd computtonl requrements under dfferent d profle ssumptons. These results suggest tht our pseudo-erly/trdy procedure s generlly cple of genertng solutons tht re just wthn few percent of the optmum nd tht t scles ncely s prolem sze ncreses. The lnce of ths pper s orgnzed s follows. Secton 2 provdes ref revew of the lterture. In secton 3, we ntroduce forml model of the FCMAP prolem. Secton 4 dentfes three rules tht cn help the reverse uctoneer (or mnufcturer) elmnte non-compettve ds or d comntons. Secton 5 ntroduces rnch-nd-ound lgorthm tht tes dvntge of our prunng rules. Ths s followed y the presentton of two heurstc serch procedures tht lso te dvntge of our prunng rules. In prtculr, Secton 6 detls rndomzed pseudoerly/trdy heurstc tht explots property of the FCMAP prolem ntroduced n Secton 4. In Secton 7, second heurstc serch procedure s presented tht comnes Smulted Annelng (SA) serch wth cost estmtor sed on the wellnown Apprent Trdy Cost rule frst ntroduced y Vepslnen nd Morton [25]. Secton 8 lso ntroduces post-processng procedure tht cn further mprove the qulty of soluton. An extensve set of computtonl results re presented nd dscussed n Secton 9. Secton 10 provdes some concludng remrs nd dscusses future extensons of ths reserch. 2. Lterture Overvew Surprsngly lttle reserch hs een reported on coordntng procurement nd fnte cpcty producton plnnng. A notle excepton s the wor of Bsso nd Aell [3] who explore sngle-perod, sngle-mchne model tht ntegrtes producton nd rw mterl orderng decsons n mnufcturng fclty wth sngle type of rw mterl nd one or more fnshed products wth stochstc demnd. Rw mterl delvery s ssumed to e stochstc: the mnufcturer typclly receves just 5

6 frcton of wht t orders. The uthors focus on determnng the qunttes n whch products re relesed nto the system, tng nto ccount the system s cpcty nd expecttons out the frcton of ordered rw mterls lely to rrve. The ojectve s to mnmze the sum of clog costs, producton costs, orderng costs, s well s rw mterl nd fnshed goods holdng costs. Gurnn et l. [12] study smlr prolem, where mnufcturer fces stochstc demnd for sngle fnshed product tht requres two crtcl components. A computtonl study s used to demonstrte the enefts of coordntng procurement nd producton decsons over more trdtonl pproch tht reles on non-coordnted procurement nd producton polces. In contrst to these erler studes, the reserch we present n ths pper ntegrtes procurement nd fnte cpcty producton plnnng prolem n envronments where orders re plced for dfferent types of fnshed products, ech possly requrng dfferent comnton of components. An mportnt spect of solvng ths more generl prolem nvolves coordntng the procurement of the multple components requred y gven order. For nstnce, there s no pont n pyng premum for hvng one component delvered erly, f the other components cn only e cqured much lter. Another dstngushng feture of our wor s the grnulrty t whch we model demnd, wth ech order hvng ts own delvery dte nd ts own mrgnl penlty for not meetng tht dte. By dfferenttng etween dfferent orders, ther ndvdul due dtes, trdness costs nd component requrements, t ecomes possle to develop solutons tht cpture the fner trdeoffs ssocted wth ther procurement requrements. Another relevnt ody of lterture revolves round reserch tht hs looed t procurement decsons suject to uncertn demnd or supply condtons. For nstnce, Song et l. [23] hve studed n ssemly envronment wth sngle fnshed product type requrng components, ech procured from pre-dentfed suppler nd delvered suject to rndom led tme. There s one-tme demnd of rndom quntty wth nown delvery dte for the sngle fnshed product. The ssemler hs to decde how much to order of ech component nd t wht tme wthout nowng the demnd quntty for the fnshed product. The ojectve s to mnmze the sum of procurement costs, holdng costs nd clog costs. In ths reserch, 6

7 ssemly tme s ssumed to e zero, therey elmntng the need for fnte cpcty producton plnnng. Gurnn et l. [11] consder product wth sngle rndom demnd nd two components; there s n ndependent suppler for ech component nd jont suppler tht cn supply the components n prs. Components rrve n the current perod wth some prolty or n the susequent perod otherwse. Once gn, the ssemly stge tes no tme. The ey decsons re how much to purchse from ech suppler to mnmze the sum of purchsng costs, component holdng costs nd clog costs. Other relted reserch ncludes the wor of Gllen nd Wen [10] who study the producton of product whose sngle component hs stochstc procurement led tme. Producton s ssumed to te no tme (whch mounts to hvng nfnte cpcty). Yno [26] consders the ssemly of product requrng two components suject to stochstc procurement nd producton led tmes. Kumr [16], Hopp nd Spermn [13] nd Shore [22] consder n envronment wth multple components, rndom procurement led tmes nd nstntneous producton. Chu et l. [4] consder multple components wth rndom procurement led tmes nd determnstc producton led tme. In contrst to the ove, the wor we present n ths pper models the fnte cpcty of the mnufcturer/ssemler nd llows for envronments wth multple fnshed products, ech requrng possly dfferent set of components. In ddton, for ech component, the mnufcturer hs to select from numer of ds from dfferent sets of prospectve supplers wth ech d possly dfferng n prce nd delvery dte. The present pper lso ssumes determnstc condtons. A thrd lne of relevnt reserch hs een concerned wth mult-perod fnte cpcty producton plnnng models. For nstnce, Crllo et l. [5] study multperod ggregted producton plnnng prolem wth sngle-product sngle-stge mnufcturer. The mnufcturer fces sttonry rndom demnd nd hs stochstc cpcty tht vres from one perod to the next. The mnufcturer hs to select the quntty of the product to relese nto the system to mnmze the sum of holdng nd clog costs. The uthors show tht the optml fnte horzon polcy s of the order-up-to-level form. Jn nd Slver [14] consder sngle-perod vrnt of the prolem studed y Crllo et l. where the mnufcturer cn py premum 7

8 to purchse dedcted cpcty from suppler. Krmrr nd Ln [15] consder mult-perod producton plnnng prolem n whch demnd nd yeld s rndom. Zpn [27] lso consders comned nventory nd producton prolem for fclty fcng stochstc demnd. Producton s ssumed to e orgnzed n lrge, dscrete tches nd s modeled s queueng system, where producton tmes cn e productspecfc nd tch-sze-dependent. Stndrd nventory nd queueng sumodels re comned nto clsscl optmzton over tch szes nd sfety stocs wth respect to holdng cost of fnshed goods plus penlty cost of clogged orders under dfferent possle control polces (e.g., Frst-Come Frst-Served). More recent wor hs lso looed t models, where ddtonl cpcty cn e purchsed (e.g. outsourcng). For nstnce, Angelus nd Porteus [1] consder comned cpcty nd producton plnnng prolem for me-to-stoc product under stochstc demnd. Rjgophln nd Swmnthn [19] report results for comned cpcty nd producton plnnng prolem nvolvng multple fnshed products nd demnd growth. Vn Meghem [24] studes the coordnton etween sucontrctng nd producton decsons n two-stge, two-plyer stochstc gme wth uncertn demnd. The ove reserch results ll rely on dfferent comntons of ggregte demnd, cpcty nd procurement models. In contrst, the wor presented n ths pper models demnd, cpcty nd procurement requrements t more detled level, enlng for the development of solutons tht explot fner trdeoffs etween dfferent procurement nd producton optons. Ths pper lso focuses on determnstc scenros. 3. The Fnte Cpcty Mult-Attrute Procurement Prolem The Fnte Cpcty Mult-Attrute Procurement (FCMAP) prolem revolves round reverse uctoneer referred elow s the mnufcturer, though t could lso e servce provder. The mnufcturer hs to stsfy set of customer commtments or orders O, M = {1,..., m} (see Fgure 1). Ech order needs to e completed y 8

9 due dte dd, nd requres one or more components (or servces), whch the mnufcturer cn otn from numer of possle supplers. The mnufcturer hs to wt for ll the components efore t cn strt processng the order (e.g., wtng for ll the components requred to ssemle gven PC). For the se of smplcty, we ssume tht the processng requred y the mnufcturer to complete wor on customer order O hs fxed durton process one order t tme ( cpcty constrnt ). du, nd tht the mnufcturer cn only Suppler Bd (Prce, Delvery Dte) Component 11 (Delvery Dte, Lte Penlty) Order 1 Suppler Bd Suppler Bd Suppler Bd Component 12 Component 21 Component 22 Order 2 Producton Fclty Customers Fgure 1. Fnte cpcty mult-ttrute procurement prolem Formlly, for ech order O nd ech component comp j, j N ={1,,n }, the mnufcturer orgnzes reverse ucton for whch t receves set of mult-ttrute j j ds β = { B 1,..., B } from prospectve supplers. Ech d B j ncludes d prce j nj j p nd proposed delvery dte j j j dl. Below we use the notton B = ( dl, p ). j Flure y the mnufcturer to meet n order O s due dte results n penlty trd T, where T s the tme y whch delvery of the product or servce s lte, nd trd s the mrgnl penlty for mssng the delvery dte. Such penltes, whch re commonly used to model mnufcturng schedulng prolems, reflect ctul 9

10 contrctul terms, loss of customer goodwll, nterests on lost profts or comnton of the ove [18]. A soluton to the FCMAP prolem conssts of: selecton of ds: Bd_Com={Bd_Com 1,,Bd_Com m }, where Bd_Com ( M order ) s comnton of n ds - one for ech of the components requred y O, nd collecton of strt tmes: ST={st 1,,st m }, where st s the tme when the j mnufcturer s scheduled to strt processng order O, nd st dl j = 1,.., n,, snce orders cnnot e processed efore ll the components they requre hve een delvered y supplers. Gven soluton (Bd_Com, ST), the proft of the mnufcturer s the dfference etween the revenue generted y ts customer orders (once they hve een completed) nd the sum of ts procurement costs nd trdness penltes. Ths s denoted: where, prof ( Bd _ Com, ST ) = j rev p trd M M j N rev s the revenue generted y the completon of order O (.e., the mount pd y the customer), j p s the prce of component comp j n Bd_Com, nd T = Mx( 0, st + du dd ) wth st eng the strt tme of order O n ST. Note tht ecuse we ssume gven set of orders, the term rev s the sme cross M ll solutons. Accordngly, mxmzng proft n Equton (1) s equvlent to mnmzng the sum of procurement nd trdness costs: cost(bd_com,st) j = p + trd T. M j N M It s worth notng tht the ove model contrsts wth erler reserch n dynmc supply chn formton, whch hs generlly ssumed mnufcturers wth nfnte T (1) 10

11 cpcty or fxed led tmes nd gnored delvery dtes nd trdness penltes ([6], [7], [9], [20]). From complexty stndpont, t cn esly e seen tht the FCMAP prolem s strongly NP-hrd, snce the specl stuton where ll components re free nd vlle t tme zero reduces to the sngle mchne totl weghted trdness prolem, tself well nown NP-hrd prolem [8]. An exmple of n exct procedure to solve FCMAP prolems nvolves loong t ll possle procurement d comntons nd, for ech such comnton, solvng to optmlty sngle mchne weghted trdness prolem wth relese dtes (e.g., usng rnch-nd-ound lgorthm). A relese dte s dte efore whch gven order s not llowed to e processed. Gven comnton of procurement ds Bd_Com, n order O hs relese dte: j r = Mx[ dl ] j N (2) where j dl denotes the delvery dte of component comp j n Bd_Com. In other words, the component tht rrves the ltest determnes the order s relese dte. Clerly, wth the excepton of frly smll prolems, the requrements of the ove procedure re computtonlly prohtve. Below, we dentfy numer of rules tht cn e used to effcently prune the serch spce ssocted wth FCMAP prolems. 4. Prunng the Serch Spce Prunng Rule 1: Elmntng Expensve Bds wth Lte Delvery Dtes Consder n FCMAP prolem P wth n order O requrng component comp j for j j whch the mnufcturer hs receved set of ds β = { B 1,..., B } from prospectve j j j j j j supplers. Let B = ( dl, p ) nd B l = ( dll, pl ) e two ds n β such tht: j j nj dl dl nd j l j p p. j l j j Then prolem P wth β = β \ { B } dmts the optml solutons wth the exct sme proft s prolem P. j j l 11

12 The correctness of ths rule should e ovous. Its pplcton s llustrted n Fgure 2, where n order requres two components: component 1 nd component 2. The mnufcturer hs receved ds for ech component. Usng Rule 1, t cn e determned, for nstnce, tht d 14 s not compettve gven tht t s more expensve thn d 13 nd rrves lte. Smlrly, d 22 nd d 24 cn lso e pruned. Bd prce d 11 d12 d 13 d 14 d 15 Component 1 Bd prce d 21 d 22 d 23 d 24 Bd delvery tme Component 2 r erlest Bd delvery tme Totl procurement cost (d12, d21) (d13, d21) (d13, d23) (d15, d23) Order relese dte Fgure 2. From 20 d comntons to 4 non-domnted ones Prunng Rule 2: Elmntng Expensve Bds wth Unnecessrly Erly Delvery Dtes Consder n FCMAP prolem P wth n order O requrng set of components j j comp j, j N = 1,... n }. Let β = { B 1,..., B } e the set of ds receved y the { j j j j mnufcturer for ech component comp j wth B = ( dl, p ). We defne the erlest possle relese dte for order nj O. It cn e computed s: erlest r s r erlest = Mx Mn dl 1 j n 1 nj j Let j B nd j Bl e two ds for component comp j such tht: p p nd j l j dl j l dl r. j erlest 12

13 j Then prolem P wth β = β \ { B } dmts the exct sme set of optml solutons s prolem P. j j l An ntutve explnton should suffce to convnce the reder. Whle d n erler delvery dte thn d j B l hs j B, ths erler dte s not worth pyng more for: t does not dd ny schedulng flexlty to the mnufcturer snce the strt of order O remns constrned y oservton. r dl. A forml proof cn esly e ult sed on ths erlest Note tht, n generl, t s not possle to prune d component comp j my hve delvery dtes tht re fter j l j B. Ths s ecuse other ds for erlest r, whch would reduce the numer of vlle schedulng optons possly ledng to lower qulty solutons. Applcton of ths rule s lso llustrted n Fgure 2, where t results n the prunng of d 11. Ths s ecuse oth d 11 nd d 12 rrve efore the order s erlest relese dte, r erlest, nd d 11 s more expensve thn d 12. Prunng Rule 3: Elmntng Expensve Bd Comntons wth Unnecessrly Erly Delvery Dtes Consder n FCMAP prolem P whose serch spce hs lredy een pruned usng j j j j j j Rule 1. In other words, gven two ds B = ( dl, p ) nd B = ( dl, p ), l, for l l l the sme component comp j, f dl > dl, then j l j p < p. j l j 1 n Let Bd _ Com = { B,..., B } e comnton of ds for the n components requred y order O. Suppose lso tht there exst two ds B = ( dl, p ) Bd _ Com nd B = ( dl, p ) Bd _ Com, nd component comp l, l such tht dl < dl dl, then l Bd _ Com s domnted y Bd _ Com, where Bd _ Com = ( Bd _ Com \ { B }) { B }. By domnted we men tht, for every soluton to prolem P nvolvng soluton where Bd _ Com s replced y Bd _ Com. Bd _ Sel, there s etter 13

14 Gven tht Bd _ Com ncludes d for second component comp l tht gets delvered t tme dl dl > dl, replcng d B wth d l B wll not dely the strt of order O nd cn only help reduce the cost of ts components snce p > p (s ndcted erler, we ssume tht Rule 1 hs lredy een ppled to prune ds). It s strghtforwrd to uld forml proof sed on the ove oservton. Note lso tht Rule 3 ctully susumes Rule 2 Rule 2 s eser to vsulze nd lso ntroduces the noton of erlest possle relese dte, whch we use lter n ths rtcle. The three prunng rules we just dentfed cn e used to prune the set of ds to e consdered. Ths s llustrted n Fgure 2, where the comnton of the three rules rngs the numer of d comntons to e consdered from 20 to just 4 nondomnted comntons. In prtculr, the pplcton of Rule 3 helps us prune d comnton (d 12, d 23 ). Ths s ecuse ths comnton s domnted y (d 13, d 23 ), whch results n the sme relese dte ut s cheper. Another d comnton pruned usng Rule 3 s (d 15, d 21 ). It should e cler tht, for ech order, Rules 1 nd 2 cn e ppled n O( c log) tme, where s n upper-ound on the numer of ds receved for gven component nd c n upper-ound on the numer of components requred y gven order. It cn lso e shown tht, for gven order O, Rule 3 cn e ppled n O( t logt) tme, where t s the totl numer of ds receved for order O cross ll the components t requres. Ths s done s follows: j j 1. For ech component comp j, crete sorted lst λ =< B,.., B > such tht j j dl dl + 1 <. Crete n overll lst of delvery dtes for ll the ds receved for O (.e., for ll the components requred y the order) nd sort the delvery dtes n ncresng order. Let 2. For ech dte r n Λ e ths sorted lst Λ, eep only those non-domnted comntons of ds tht re comptle wth hvng r s order j 1 nj O s relese dte. Note tht such d comntons re of the form { 1 n j B,..., B } where dl r, j, nd there s 14

15 no other d j B such tht dl j < dl r. In other words, for ech component j comp j, j B s the ltest d comptle wth relese dte r (nd hence lso the chepest such d). Fndng such ds requres very lttle tme, gven the sorted d lsts λ creted n step 1. j As prenthess, t s worth notng tht the three prunng rules we just ntroduced pply to scenros wth more complex constrnts, s they only te dvntge of the relese constrnt tht requres ech order to hve ll ts components efore t cn e processed. For nstnce, ths ncludes prolems where the mnufcturer s modeled s jo shop, the cpcty of some mchnes s greter thn one nd there re sequence dependent setup tmes. It cn lso e shown tht the prunng rules cn e extended to ccommodte prolems wth nventory holdng costs, s long s orders re not llowed to e shpped efore ther due dtes ths ssumpton corresponds to hvng fnshed goods nventory nd s representtve of mny supply chn stutons. Consder the non-domnted d comntons resultng from the pplcton of our three prunng rules to n FCMAP prolem. Let the non-domnted d comntons of order * O e denoted: 1 m Bd _ Com = Bd _ Com = ( r, pc ),..., Bd _ Com = ( r, pc )}, { 1 1 where r s the relese dte of d comnton Bd _ Com m m, s defned n Equton (2), nd pc s ts totl procurement cost, defned s the sum of ts component d prces. It follows tht: Property 1: For ech order O, M, t must hold tht, f r < r, then pc > pc,, {1,..., m},. In other words, the totl procurement costs of non-domnted d comntons strctly decrese s ther relese dtes ncrese. Proof: We hve lredy shown tht, followng the pplcton of Rule 1, the ds tht remn for gven component hve prces tht strctly decrese s ther delvery dtes ncrese. Let Bd _ Com e non-domnted d comnton for order O followng the pplcton of Rules 1 through 3. Let ts relese dte r e determned y the delvery 15

16 dte of component j, nmely j r = dl. Note tht, y defnton, the relese dte of d comnton s lwys determned y one or more of ts components. Gven tht Rule 3 hs lredy een ppled, the delvery dte dl of ny component must e the ltest delvery dte mong those ds for component tht stsfy dl dl. j Consder nother non-domnted d comnton Bd _ Com for order O such tht r > r. Let l e the ndex of one of the components determnng the relese dte of d comnton Bd _ Com, nmely r = dl > r = dl. Just s for d l j comnton Bd _ Com, the fct tht Rule 3 hs een ppled mples tht the delvery dte dl of ny component n Bd _ Com must e the ltest delvery dte mong those ds for component tht stsfy dl dl. Gven tht l r = dl > r = dl, t lso follows tht, for ny component, we hve l j dl dl wth strct nequlty for t lest one component, nmely component l. Gven tht Rule 1 hs een ppled, t lso follows tht, for ny component, p p wth strct nequlty for t lest one component (component l). Hence, 1 pc = p n > 1 pc = p n. Component 1 dl = dl 1 1 Bd prce Bd Bd _ Com Component 2 dl = dl = r 2 2 Bd Bd _ Com Component 3 3 dl dl = r 3 Delvery tme Fgure 3. Illustrton of Property 1 16

17 Property 1 s llustrted n Fgure 3, where we hve two d comntons Bd _ Com nd Bd _ Com for n order O tht requres three components. In ths prtculr exmple, r s determned y the delvery dte of component 2, whle r s determned y tht of component 3. The two d comntons shre the sme delvery dtes for two out of three of the components requred y order O : components 1 nd 2. The dfference n procurement cost comes from the lower prce ssocted wth the lter delvery of component 3 n d comnton (nmely, dl = r > dl ). 3 3 Bd _ Com Note lso tht, f fter pplcton of the prunng rules there exst two nondomnted d comntons, must hold tht pc = pc. Bd _ Com nd Bd _ Com, such tht r = r, t In the followng sectons, we ntroduce rnch-nd-ound lgorthm to solve the FCMAP prolem long wth two (sgnfcntly fster) heurstc serch procedures. All three procedures te dvntge of the prunng rules we just ntroduced. One of the two heurstc procedures lso tes dvntge of Property A Brnch-nd-Bound Algorthm Followng the pplcton of the prunng rules ntroduced n the prevous secton, optml solutons to the FCMAP prolem cn e otned usng rnch-nd-ound procedure. Brnchng s done over the sequence n whch orders re processed y the mnufcturer nd over the relese dtes of non-domnted d comntons of ech order. Specfclly, the lgorthm frst pcs n order to e processed y the mnufcturer then tres ll the relese dtes (of non-domnted d comntons) vlle for ths order. Note tht, s orders re sequenced n ths fshon, some of ther vlle relese dtes ecome domnted, gven pror sequencng decsons. For nstnce, consder two orders O 1 nd O 2, wth O 2 hvng two relese dtes r 21 nd r 22 wth 21 r22 r < - followng the pplcton of Prunng Rules 1 through 3. Suppose tht, t the current node, O 1 s sequenced efore O 2 nd tht O 1 s erlest completon 17

18 dte s greter thn r 22. It follows tht relese dte r 21 s strctly domnted y relese dte r 22 t ths prtculr node. Relese dtes tht ecome domnted s result of pror ssgnments cn e pruned on the fly, therey further speedng up the serch procedure. Gven node n n the serch tree, nmely prtl sequence of orders nd selecton of relese dtes for ech of the orders lredy sequenced, t s possle to compute n lower-ound for the proft of ll complete solutons (.e., lef nodes) comptle wth ths node: LB n = ( pc + trd OSn T ) + [ trd mx(0, cdos + du dd ) + mpc ], n OSn where: OS n s the set of orders sequenced t node n; pc s the totl procurement cost ssocted wth the non-domnted relese dte (or d comnton) ssgned to order O OS n nd T s ts trdness. Note tht ech order s scheduled to strt s erly s possle, gven pror sequencng decsons nd the relese dte ssgned to t: there re no enefts to strtng lter; cd OS s the completon dte of the lst order n OS n n ; mpc s the mnmum possle procurement cost of order O ths cost s node-ndependent. If the lower ound of node n s greter thn the est fesle soluton found so fr, the node n nd ll ts descendnts re pruned. 6. An Erly/Trdy Heurstc Property 1 tells us tht, followng the pplcton of the prunng rules, the procurement costs of non-domnted d comntons strctly decrese s relese dtes ncrese. Fgure 4 plots the totl procurement cost nd trdness cost of n order for dfferent possle strt tmes. Whle trdness costs ncrese lnerly wth strt tmes tht mss the order s due dte, procurement costs vry ccordng to decresng 18

19 step-wse functon. Specfclly, the crcles n Fgure 4 represent the order s nondomnted d comntons. For nstnce, f the order strts t tme st, ts procurement cost s pc, nmely the procurement cost of the ltest non-domnted d comnton comptle wth ths strt tme ( Bd _ Com ). Its trdness cost s equl to trd mx( 0, st + du dd), where trd s ts mrgnl trdness penlty, dd ts due dte nd du ts durton (or processng tme). The resultng prolem cn e vewed s pseudo erly/trdy schedulng prolem. It ers lot of smlrty to trdtonl erly/trdy schedulng prolem (e.g., [17]) ut s lso slghtly dfferent ecuse procurement costs ssocted wth dfferent d comntons led to: 1) Step-wse erlness costs n contrst to lner erlness costs found n trdtonl erly/trdy prolem; nd 2) Potentl svngs for completng the order pst ts due dte, to the extent tht there re lte d comntons tht re so chep tht t s worth fnshng the order lte. Agn ths s dfferent from trdtonl erly/trdy prolem, where fnshng n order on tme lwys leds to the lowest cost for tht order. Cost Procurement cost Trdness cost du pc Bd _ Com pc j j Bd _ Com O r erlest st dd - du r ltest Relese tme Fgure 4. An order s trdness nd procurement costs 19

20 P (S ) trd erl du 0 trd + erl erl du ln erl 0 S O -S erl du Fgure 5. Prorty functon for Pseudo-Erly/Trdy heurstc Ow nd Morton [17] hve ntroduced n erly/trdy dsptch rule for one-mchne schedulng prolems suject to lner erlness nd trdness costs. Becuse our erlness costs re not lner, ths heurstc cn not redly e ppled. Below, we refly revew some of ts ey elements nd dscuss how we hve dpted t to produce fmly of heurstc serch procedures for the FCMAP prolem. Ow nd Morton s dsptch rule nterpoltes etween two extreme cses. The frst stuton s one where ll orders re ssumed to hve plenty of tme nd where only erlness costs need to e mnmzed. The second cse s one where ll orders re ssumed to e lte nd where only trdness needs to e mnmzed. In the former cse, t cn e shown tht n optml soluton cn e ult y sequencng orders ccordng to Weghted Longest Processng Tme dsptch rule, where ech order receves prorty: P = erl du, where P s the prorty of order O, du s ts processng tme nd erl s ts mrgnl erlness cost nmely the penlty ncurred for every unt of tme the order fnshes efore ts due dte. Conversely, n the ltter cse, when ll jos re ssumed to e trdy, t cn e shown tht n optml soluton cn e ult y sequencng orders ccordng to Weghted Shortest Processng Tme dsptch rule of the form: 20

21 P = trd du, where trd s ts mrgnl trdness penlty. Le Ow nd Morton s, our erly/trdy heurstc nterpoltes etween two extreme cses: one where ll orders hve plenty of tme nd one where ll orders re lte. The prorty ssocted wth ths ltter stuton s dfferent however from the one n Ow nd Morton s rule. Ths s ecuse lter strt tmes for orders tht re lte my stll result n reductons n procurement costs. Accordngly, the prorty ssocted wth ths ltter stuton s: where 0 erl s the erlness weght t P = ( trd erl ) du, 0 st = dd du. The resultng erly/trdy heurstc ssgns ech order prorty tht vres wth ts slc S : P ( S erl ) = du trd + + erl du erl 0 + ( S ) exp[ ] du where du s the verge processng tme of n order, s loo-hed prmeter. (X) + denotes Mx (X, 0), nd slc S t tme t s defned s: S = dd du t. The ove formul cn esly e seen to reduce to the Weghted Shortest Processng Tme dsptch rule wth mrgnl trdness cost of nd to the Weghted Longest Processng Tme dsptch rule when (3) 0 trd erl when slc S 0 S. The loohed prmeter cn ntutvely e thought of s the verge numer of orders tht would e trdy f order O s selected to e scheduled next. The vlue of sclly controls the trnston etween the two extreme scenros etween whch ths rule nterpoltes. Hgher vlues of me the trnston strt erler. Ths cn e nterpreted s eng more senstve to trdness when lrger numer of orders stnd to e lte. In the FCMAP prolem, n order O cnnot strt efore ts erlest possle relese dte r erlest (see Prunng Rule 2 t should e cler tht ths relese dte s never pruned y Rule 2). In ddton, erlness costs vry ccordng to step functon. 21

22 A mrgnl erlness cost cn however e otned through regresson, whether loclly or glolly. Specfclly, we dstngush etween the followng two pproches to computng mrgnl erlness costs for n order n the FCMAP prolem: 1) Locl Erlness Weght: At tme t, the locl mrgnl erlness cost ssocted wth n order O (see Fgure 4) cn e pproxmted s the dfference n procurement costs ssocted wth the ltest non-domnted d comntons comptle wth processng the order t respectvely tme t (nmely Bd _ Com ) nd tme t + du (nmely j Bd _ Com ): pc pc L j erl =, du 2) Glol Erlness Weght: An lterntve nvolves computng sngle glol mrgnl erlness cost for ech order. Ths cn e done usng Lest Squre Regresson: pc rd n pc rd erl G =, 2 2 rd n rd where pc s the verge procurement cost of non-domnted d comntons for the order, nd rd s ther verge relese dte. The smplest possle relese polcy for the FCMAP prolem nvolves relesng ech order O t ts erlest possle relese dte, nmely r erlest. We refer to ths polcy s n Immedte Relese Polcy. It mght sometme result n relesng some orders too erly nd hence yeld unnecessrly hgh procurement costs. An lterntve s to use n Intrnsc Relese Polcy, whch releses orders when ther erly/trdy prorty P S ) ecomes postve. P S ) cn e vewed s the mrgnl cost ncurred ( for delyng the strt of order ( O t tme t. As long s ths cost s negtve, there s no eneft to relesng the order. The tppng pont, where P ( ) = 0, s the order s ntrnsc relese dte: S erl rˆ = dd du + du ln. (4) trd + erl erl 0 22

23 Here gn, one cn use ether the locl or glol erlness weghts ssocted wth n order. Intutvely, one would expect the glol erlness weght to e more pproprte for the computton of n order s relese dte nd ts locl erlness weght to e etter suted for the computton of ts prorty t prtculr pont n tme. Ths hs generlly een confrmed n our experments. In Secton 9, we only present results where prortes re computed usng locl erlness weghts. We do however report results, where relese dtes re computed wth oth locl nd glol erlness weghts, nd we found our heurstc performs etter wth glol erlness weght. Rther thn lmtng ourselves to determnstc dpttons of Ow nd Morton s dsptch rule, we hve lso expermented wth rndomzed versons, where order relese dtes nd prortes re modfed y smll stochstc perturtons. Ths enles our procedure to me up for the wy n whch t pproxmtes procurement costs, smplng the serch spce n the vcnty of ts determnstc soluton. The resultng pseudo-erly/trdy serch heurstc opertes y loopng through the followng procedure for pre-specfed mount of tme. As t tertes, the procedure lterntes etween the mmedte nd ntrnsc relese polces dscussed erler nd successvely tres numer of dfferent vlues for the heurstc s loo-hed prmeter. The followng outlnes one terton.e. wth one prtculr relese polcy nd one prtculr vlue of the loo-hed prmeter. 1. For ech order O, M = {1,2,,m}, compute the order s relese dte. When usng the mmedte relese polcy, ths smply mounts to settng the order s relese dte erlest RD = r. When usng the ntrnsc relese polcy, the order s erlest relese dte s computed s RD = Mx r,(1 + α ) rˆ }, where α s rndomly { drwn from the unform dstruton [ dev 1, +dev 1 ] (dev 1 s prmeter tht controls how wdely the procedure smples the serch spce); 2. Dsptch the orders, nmely let t 0 = Mn RD M 1) For ll those orders O tht hve not yet een scheduled nd whose relese dtes re efore t 0, compute the order s prorty t tme t 0 s: 23

24 PR ( t0 t0 ) = (1 + β ) P ( dd du ), where P s the pseudo-erly/trdy prorty defned n (3) nd β s rndomly drwn from the unform dstruton [ dev 2, +dev 2 ] (dev 2 s prmeter tht controls how wdely the procedure smples the serch spce); 2) Let order O e the order wth the hghest prorty. Schedule t 0 ; O to strt t tme 3) If ll orders hve een scheduled, then Stop. Else, let t 1 = t 0 + du nd t 2 e the erlest relese dte mong those orders tht hve not yet een scheduled. Set t = Mx t, } nd repet Steps { 1 t 2 4) Compute the proft of the resultng soluton. If t s hgher thn the est soluton otned so fr, me ths the new est soluton. A determnstc verson of ths procedure smply mounts to settng dev 1 nd dev 2 to zero. 7. A Smulted Annelng Serch Procedure A second heurstc serch procedure for the FCMAP prolem nvolves usng Smulted Annelng (SA) to explore dfferent comntons of ds. Gven selecton of non-domnted d comntons Bd Com = { Bd _ Com,..., _ 1 Bd _ Comm} one comnton per order, the procedure computes the relese dte r of ech order O nd sequences the orders, usng the Apprent Trdness Cost (ATC) dsptch rule frst ntroduced n [25]. ATC s nown to generlly yeld hgh qulty schedules for the one-mchne totl weghted trdness prolem nd hs O(m logm) complexty. As such t s n excellent estmtor for the est soluton comptle wth gven selecton of d comntons. The followng further detls the SA procedure: Step 1 Intlzton: 1 Set n ntl temperture Temp=Temp 0, nd n ntl d selecton Bd _ Com = 1 1 { Bd _ Com,..., Bd _ Com }; 1 m 24

25 Use the ATC dsptch rule to uld schedule. Let cost 1 = cost(bd_com 1, ST 1 ), where ST = { st1,..., st m } s the set of strt tmes ssgned y ATC to orders O 1 through O m. Set Bd_Com opt =Bd_Com 1 nd cost opt =cost 1. Step 2 Serch: Perform the followng step N tmes: Select Bd_Com =neghor(bd_com 1 ) (rndomly or through some heurstc), nd compute cost = cost(bd_com, ST), where ST s the set of order strt tmes ssgned y the ATC dsptch rule; If cost 1 cost cost opt, set Bd_Com 1 =Bd_Com; Else f cost>cost 1 nd rnd() exp((cost 1 -cost)/temp), set Bd_Com 1 = Bd_Com; Else f cost<cost opt, set Bd_Com opt =Bd_Com 1 =Bd_Com. If Bd_Com opt ws not modfed n the lst N tertons, decrese the temperture Temp = Temp α. Go to Step 3. Step 3 Termnton Condton: If Bd_Com opt hs not een mproved over the pst K steps, then STOP nd return (Bd_Com opt, ST opt ) s the est soluton found y the procedure, otherwse go to Step 2. The ntl d comnton Bd_Com 1 s rndomly generted. Note lso tht the ATC dsptch rule s tself prmetrc dsptch rule wth loo-hed prmeter [25]. In our experments, we systemtclly run ATC wth vlues of the loo-hed prmeter equl to 0.5, 1.0, 1.5,, 6.0 nd pc the est of the 12 solutons we hve generted. We hve studed vrtons of ths procedure tht rely on dfferent movesets. In prtculr, we hve consdered one-d moveset vrton, where we modfy the selecton of sngle d (for gven component), nd two-d moveset vrton, where two d selectons (for two dfferent components) re modfed t once. For oth types of movesets, we hve lso expermented wth two wys of selectng moves: rndom mechnsm, where move n the moveset s rndomly selected, nd 25

26 n orgnzed mechnsm tht replces the d(s) tht reduce most the proft of the current soluton (Bd_Com 1, ST 1 ), nmely, those ds for whch p j + trd T s the gretest. The experments presented n Secton 9 use the orgnzed mechnsm, s we found t to generlly yeld hgher qulty solutons thn the rndom one. We hve not found gret dfference etween vrtons of our procedure usng one-d movesets nd two d movesets. 8. Rght-Shftng Soluton Improvement Procedure Snce the totl cost functon of ech order s the sum of lnerly ncresng trdness cost functon nd step-wse non-ncresng erlness cost functon, the totl cost functon hs multple locl mnmum ponts, s shown n Fgure 6. Menwhle, the ove pseudo-erly/trdy heurstc dsptches orders s soon s the mchne ecomes vlle. Hence, the soluton produced y the pseudo-erly/trdy heurstc cn sometmes e further mproved y rght-shftng orders to lower locl mnmum ponts wthout chngng the order processng sequence. Let <1,2,,m> denote n order processng sequence produced y the ove lgorthm, st denote the strt tme of order O, {1,2,,m}. As shown n Fgure 6, the locl mnmum ponts hve one-to-one correspondence wth the non-domnted d comntons. The mprovement method processes ech order from m to 1 s follows: 1. Rght-shft erly orders: If st < dd du nd st < st +1 du, rght-shft order O untl Mn{dd du, st +1 du },.e., let st = Mn{dd du, st +1 du }. Even f ts cost remns the sme, rght-shftng O cretes more room to rght-shft erler orders, nd therefore my llow to decrese the costs of the other orders. Go to Step Under ny of the followng stutons, stop rght-shftng: st = st +1 du ; There s no locl mnmum pont etween st nd st +1 du ; or 26

27 All the locl mnmum ponts etween st nd st +1 du hve hgher costs thn the current cost, tc, of order O. Otherwse, let j* e locl mnmum pont, the cost of whch s the lowest mong ll the ponts etween st nd st +1 du. Rght-shft order O to the locl mnmum pont j*,.e., let st =st j* du. Cost Totl cost Locl mnmum pont dd - du Strt tme: st Fgure 6. An order s totl cost 9. Computtonl Evluton A numer of experments hve een run to evlute the mpct of our prunng rules, the performnce of our heurstc serch procedures, nd the enefts of our FCMAP model over trdtonl reverse ucton models tht gnore the mnufcturer s fnte cpcty. These experments re further detled elow. Emprcl Setup Prolems were rndomly generted to cover rod rnge of condtons y vryng the dstruton of d prces nd d delvery dtes s well s the overll lod fced y the mnufcturer. Results re reported for 2 groups of prolems: 1. Prolems wth 10 orders, 5 requred components per order nd 20 suppler ds per component: These prolems were ept smll enough so 27

28 tht they could e solved to optmlty wth our rnch-nd-ound lgorthm. Two sets of prolems were desgned. Key prmeter vlues of the frst set were drwn from the followng unform dstrutons: Order processng tme: U[5,25] Order mrgnl trdness cost: U[1,10] Order due dtes: 2 dstrutons:. Medum Lod (ml) prolems: U[100,300]. Hevy Lod (hl) prolems: U[100,200] Component d delveres: 2 dstrutons:. Nrrow d delvery dstruton (nd): U[0,50]. Wde d delvery dstruton (wd): U[0,100] Component d prces: 2 dstrutons:. Nrrow d prce dstruton (np): U[5,35]. Wde d prce dstruton (wp): U[5,65] The other set of prolems hs ll the sme dstrutons except for the component d delveres dstrutons:. Nrrow d delvery dstruton (nd): U[0,150]. Wde d delvery dstruton (wd): U[0,200] The frst set of prolems represents reltvely esy prolem nstnces, nd n lmost ll cses, t holds tht dd du > ltest r, where r ltest s the ltest relese dte determned y the chepest non-domnnt d comnton. Nmely, the cost functon s qusconvex nd hs no locl mnmum pont fter dd du. We cll ths set of prolems Erly-Bd prolems. The other set of prolems, whch we cll Mxed-Bd prolems, represents reltvely hrd prolem nstnces, where, n mny stutons, dd du < ltest r,.e., the cost functon hs multple locl mnmum ponts fter dd du (see Fgure 6). A totl of 20 prolems were generted n ech ctegory (ml/hl, nd/wd, np/wp), yeldng totl of 320 prolems. 2. Prolems wth 500 orders, 5 requred components per order nd 20 suppler ds per component: Whle these prolems were too lrge to e solved wth rnch-nd-ound (even wth our prunng rules), they were used to 28

29 vldte results otned on the smller sets of prolems. Ths ncludes, determnng how our heurstc serch procedures scle up nd evlutng the enefts of our FCMAP model over trdtonl reverse ucton models nd polces tht gnore the mnufcturer s fnte cpcty these ltter eng smply referred to elow s nfnte cpcty polces. Key prmeter vlues were drwn from the followng unform dstrutons: Order processng tme: U[1,5] Order mrgnl trdness cost: U[1,10] Order due dtes:. Medum Lod (ml) prolems: U[500,1500]. Hevy Lod (hl) prolems: U[500,1000] Component d delveres: 2 dstrutons:. Nrrow d delvery dstruton (nd): U[0,800]. Wde d delvery dstruton (wd): U[0,1000] Component d prces: sme 2 dstrutons s 10-order prolems (np/wp) A totl of 20 prolems were generted n ech ctegory for totl of 160 prolems. Note tht order revenues re rrelevnt, snce the orders to e produced re fxed. In other words, ll solutons dmt the sme overll revenue nd overll proft s solely determned y the sum of trdness nd procurement costs ssocted wth gven soluton see equton (1). Accordngly, we report overll costs rther thn overll profts. Dstnce from the Optmum Tles 1 nd 2 summrze results otned on 10-order/Erly-Bd prolems. Tles 3 nd 4 summrze 10-order/Mxed-Bd prolems. These tles provde the verge dstnce from the optmum of solutons otned wth dfferent vrtons of our serch heurstcs cross 16 prolem sets (four medum lod, Erly-Bd prolem sets n Tle 1, four hevy lod, Erly-Bd prolem sets n Tle 2, four medum lod, Mxed-Bd prolem sets n Tle 3, nd four hevy lod, Mxed-Bd prolem sets n 29

30 Tle 4) wth ech prolem set ncludng totl of 20 prolems. Ths dstnce from the optmum ws computed s: [cost(soluton) cost(optml_soluton)] cost(optml_soluton). Stndrd devtons re provded etween prentheses. Optml solutons were otned usng the rnch-nd-ound procedure ntroduced n Secton 5. Results re reported for the followng technques: Infnte cpcty: Ths s technque tht reflects trdtonl reverse ucton prctces, where the mnufcturer s cpcty s gnored. Specfclly, for ech order nd ech component, the mnufcturer selects the chepest d comptle wth the order s due dte. Orders re then scheduled ccordng to the ATC dsptch rule, nmely the sme rule used n our Smulted Annelng procedure (See Secton 7). Fnte cpcty: Results re reported for numer of vrtons of our serch heurstcs: o Smulted Annelng (SA) procedure: Ths s the procedure ntroduced n Secton 7 wth Temp 0 =300, α =0.95, N=60 nd K=40. The procedure ws run fve tmes on ech prolem nd we report oth verge performnce nd est performnce over 5 runs. o Pseudo-Erly/Trdy (PET) Procedure: ths s the pseudo-erly/trdy heurstc ntroduced n Secton 6. Here gn we report results for severl vrtons of ths heurstc: G-L uses glol erlness weghts n ts relese polcy nd locl erlness weghts n ts prorty computtons, L-L uses locl erlness weghts for oth relese dte nd prorty computtons, Det s determnstc vrton of the pseudo-erly/trdy heurstc descred n Secton 6, nmely dev 1 = dev 2 = 0, Rnd s stochstc vrton of the sme heurstc wth dev 1 = 0.3 nd dev 2 = 0.3. For comprson se, the CPU tme gven to ths heurstc ws the sme CPU tme requred y n verge SA run n the sme prolem ctegory, 30

31 Rnd-RS mproves the solutons produced y Rnd usng the post-processng procedure descred n Secton 8, Hyrd s heurstc tht runs SA once, PET/L-L/Rnd-RS once, PET/G-L/Rnd-RS once nd tes the est of the resultng solutons. Tle 1. Percentge devton from the optmum 10-order/erly-d/medum-lod prolems (Stndrd devtons re provded etween prentheses) np/nd wp/nd np/wd wp/wd Inf. Cp (36.84) (26.42) (84.21) (72.95) SA Avg. Best of of 5 5 Runs Runs (24.04) (21.29) (20.14) (14.21) (38.62) (24.71) (28.34) (18.88) Fnte Cpcty PET L-L Det Rnd Rnd- RS 1.82 (333) 2.06 (4.36) 7.31 (5.25) 9.74 (9.20) 0.29 (0.64) 0.03 (0.12) 3.45 (4.12) 4.08 (5.70) 0.19 (0.57) 0.03 (0.12) 1.88 (2.42) 3.19 (4.97) G-L Det Rnd. Rnd- RS 1.89 (4.69) 2.71 (5.28) 3.01 (2.86) 4.75 (5.52) 0.03 (0.12) 0.04 (0.18) 0.48 (0.81) 1.14 (2.52) 0.00 (0.00) 0.03 (0.18) 0.46 (0.79) 1.03 (2.51) Hy (0.00) 0.00 (0.00) 0.37 (0.61) 0.80 (2.45) Tle 2. Percentge devton from the optmum 10-order/erly-d/hevy-lod prolems (Stndrd devtons re provded etween prentheses) np/nd wp/nd np/wd wp/wd Inf. Cp (34.79) (41.49) (80.67) (100.54) SA Avg. Best of of 5 5 Runs Runs (22.56) (18.84) (31.14) (26.83) (25.44) (19.36) (36.86) (23.56) Fnte Cpcty PET L-L Det Rnd Rnd- RS 7.85 (5.76) (12.11) (7.84) (8.90) 2.31 (2.71) 3.59 (6.32) 6.50 (4.03) (6.33) 2.15 (2.55) 3.51 (6.28) 4.92 (3.11) 9.47 (6.04) G-L Det Rnd Rnd- RS 8.21 (6.62) (12.34) (8.74) (8.33) 2.65 (3.91) 3.13 (4.46) 3.09 (4.04) 6.13 (5.12) 2.39 (3.67) 3.07 (4.31) 3.02 (4.09) 5.75 (4.36) Hy (1.64) 1.93 (2.92) 2.28 (2.16) 5.23 (3.63) 31

32 Tle 3. Percentge devton from the optmum 10-order/mxed-d/medum-lod prolems (Stndrd devtons re provded etween prentheses) np/nd wp/nd np/wd wp/wd Inf. Cp (78.07) (52.40) (84.32) (79.84) SA Avg. Best of of 5 5 Runs Runs (40.77) (22.09) (26.04) (16.98) (58.56) (42.25) (65.82) (59.13) Fnte Cpcty PET L-L Det Rnd Rnd- RS (7.87) (9.38) (10.17) (10.25) 8.75 (6.81) (8.85) (7.29) (10.17) 6.30 (5.02) 7.30 (6.15) 7.92 (6.82) (7.82) G-L Det Rnd Rnd- RS 3.90 (4.25) 4.72 (5.03) 5.66 (3.36) 8.46 (5.58) 0.85 (1.25) 2.04 (3.10) 2.29 (2.41) 4.30 (3.61) 0.62 (1.13) 1.94 (3.08) 1.65 (1.80) 3.44 (2.80) Hy (1.14) 1.41 (2.01) 1.62 (1.82) 3.22 (2.80) Tle 4. Percentge devton from the optmum 10-order/mxed-d/hevy-lod prolems (Stndrd devtons re provded etween prentheses) np/nd wp/nd np/wd wp/wd Inf. Cp (112.91) (74.62) (170.31) (104.02) Avg. of 5 Runs (43.21) (53.18) (127.82) (77.45) SA Best of 5 Runs (17.80) (30.19) (82.99) (71.58) Fnte Cpcty PET L-L Det Rnd Rnd- RS (6.10) (9.47) (5.77) (10.77) (5.94) (5.93) (5.29) (9.13) (6.17) (5.36) (5.31) (9.24) G-L Det Rnd Rnd- RS (8.44) (7.68) (6.52) (8.71) 4.17 (4.02) 6.32 (5.01) 5.08 (4.69) 6.47 (5.11) 4.10 (3.87) 6.06 (5.01) 4.93 (4.55) 6.41 (5.09) Hy (3.62) 5.68 (4.46) 4.41 (3.60) 6.41 (5.09) Tles 1-4 yeld numer of oservtons: Importnce of the FCMAP Model: All fnte cpcty heurstcs yeld solutons wth sgnfcntly lower costs thn the nfnte cpcty one, therey confrmng the mportnce of the FCMAP model. Tng nto ccount fnte cpcty consdertons nd tghtly coordntng the procurement of the multple components requred y ech order suject to these fnte cpcty consdertons sgnfcntly mprove the mnufcturer s ottom lne. The results re generlly most mpressve on prolem ctegores np/wd nd wp/wd, where our hyrd 32

33 heurstc respectvely reduces totl costs y t lest 70% on Erly-Bd prolems nd 80% on Mxed-Bd prolems. Dstnce from the Optmum: Our hyrd heurstcs yelds solutons tht re respectvely less thn 3.3% from the optmum on medum-lod/mxed-bd prolems nd 6.5% from the optmum on hevy-lod/mxed-bd prolems. Also, our solutons re respectvely less thn 0.8% from the optmum on medumlod/erly-bd prolems nd 5.3% from the optmum on hevy-lod/erly-bd prolems. In prtculrly, our PET-GL-Rnd-RS gets the optml solutons on ll 40 rndomly generted Erly-Bd ml/np/nd nd ml/wp/nd prolems. Effectveness of Property 1: Even determnstc versons of the PET heurstc usng G-L yelds solutons tht re respectvely wthn 4.8% from the optmum on medum-lod/erly-bd prolems nd 18.8% from the optmum on hevylod/erly-bd prolems. Even wthout rght-shftng mprovement, our stochstc verson of the PET heurstc usng G-L yelds solutons tht re respectvely wthn 1.2% from the optmum on medum-lod/erly-bd prolems nd 6.2% from the optmum on hevy-lod/erly-bd prolems. Ths strongly suggests tht Property 1 nd the wy n whch our PET heurstc pproxmtes erlness costs re rther effectve. Note lso tht ths determnstc verson of our heurstc tes only tny frcton of second on these prolems. PET heurstc versus SA heurstc: Gven the sme mount of CPU tme, the PET heurstc performs sgnfcntly etter thn the SA serch procedure. Effectveness of Rght-Shftng to Improve PET: Even for the Erly-Bd prolems, PET-Rnd-RS respectvely mproves the solutons gven y PET- Rnd y 0.47% n verge usng L-L nd 0.12% n verge usng G-L. A loo t the results on Mxed-Bd prolems confrms the effectveness of PET-Rnd-RS n the cses where dd du > r ltest, nd the mprovements re respectvely 2.00% n verge usng L-L nd 0.29% n verge usng G-L. Glol versus Locl Erlness Weghts: The G-L vrton of the PET heurstc generlly performs much etter thn L-L on oth medum nd hevy lod prolems, prtculrly on ctegory wp/wd: respectvely s much s 3.7% etter on Erly-Bd prolems nd 9.4% etter on Mxed-Bd prolems (L-L performs 33