The single machine multiple orders per job scheduling problem

The single machine multiple rders per jb scheduling prblem Sctt J. Masn University f Arkansas 4207 Bell Engineering Center Fayetteville, AR 72701, USA Peng Qu Advanced Micr Devices 5204 East Ben White Bulevard Austin, TX 78741, USA Erhan Kutanglu Operatins Research and Industrial Engineering Prgram The University f Texas at Austin 1 University Statin C2200 Austin, TX 78712 Jhn W. Fwler Arizna State University P. O. Bx 875906 Tempe, AZ 85287-5906, USA Abstract The standard unit f transfer in new semicnductr wafer fabricatin facilities is the frnt pening unified pd (FOUP). Due t autmated material handling system and cst cncerns, the number f FOUPs in a wafer fab is kept limited. Large 300-mm wafers allw fr custmer rders t be filled with less than a full FOUP f wafers in these new fabs, thereby making gruping rders frm multiple custmers int a jb necessary. Efficient utilizatin f FOUP capacity while attaining gd system perfrmance is a challenge. We investigate the multiple rders per jb scheduling prblem, presenting a nnlinear mixed-integer prgram that encmpasses bth rder gruping (jb frmatin) and jb scheduling decisins. Recgnizing the tractability limitatins f this frmulatin, we relax the nnlinear cnstraints s that the prblem is slvable using standard cmmercial slvers. We examine a number f heuristic appraches in an attempt t btain high quality slutins in an acceptable amunt f cmputatin time. We study bth ptimizatin- and heuristic-based appraches in tw different machine prcessing envirnments in an attempt t minimize rder ttal weighted cmpletin time n a single machine. Experimental results demnstrate the difficulty f slving the prblem using an ptimizatin-based apprach. Hwever, heuristic appraches can find gd slutins in a reasnable amunt f cmputatin time t this practically mtivated scheduling prblem. Keywrds: Scheduling, ptimizatin, heuristics, semicnductr manufacturing

1 Intrductin Althugh the semicnductr industry has a shrter histry than many ther manufacturing industries, it has becme ne f the fastest grwing industries in the wrld. In recent years, the wrldwide annual revenues f the semicnductr market have grwn t ver US $200 Billin. The size f this market has led t increased cmpetitin and risk within the semicnductr industry. T survive in such an envirnment, a cmpany must nt nly imprve its quality and manufacturing techniques, but als try t meet custmers demands in a timely manner. If prduct delivery is cnsistently late, a cmpany will lse market share and its custmers gdwill. Therefre, minimizing the time required t cmplete rders is a high pririty. The industry has turned t smarter dispatching and scheduling practices in recent years due t their high effectiveness and lw cst (Pfund and Fwler 2002). Uzsy et al. (1992) describes the fur main phases f semicnductr manufacturing: wafer fabricatin (wafer fab), wafer prbe, assembly, and final testing. In a typical wafer fab, there are dzens f prcess flws, each with a number f prcessing steps that typically varies between 300 and 500. In current generatin wafer fabs, semicnductrs are manufactured in a clean rm n silicn wafers 200-mm in diameter, typically in grups f 25 wafers knwn as lts. In additin, mre than 100 types f wafer prcessing machines are usually cntained in a wafer fab. Varius perating characteristics such as sequence-dependent setups, reentrant flw, and batch prcessing cntribute t making the wafer fab a cmplex jb shp (Masn et al. 2002). The newest generatin f semicnductr wafer fabs prduces integrated circuits (ICs) n silicn wafers 300-mm in diameter. The area f a 300-mm wafer is 2.25 times larger than that f a current-generatin 200-mm wafer. Thus, rughly 125% mre die can be prduced per wafer. With this wafer size increase, a 300-mm wafer is much heavier than a 200-mm wafer. In 300-mm wafer fabs, a standard prductin lt f 25 wafers can weigh up 1

t 30 punds a lad that cannt physically be carried safely. Therefre, full factry autmatin is inevitable in 300-mm wafer fabs. T facilitate this necessary factry autmatin while simultaneusly keeping tl develpment csts dwn, 300-mm tl suppliers and semicnductr industry partners agreed upn a set f standards by which all tls will be designed fr 300-mm wafer prcessing. The standard unit f lt transfer between tls in a 300-mm fab is the frnt-pening unified pd (FOUP) (SEMI Standard E47.1-0303, 2003). FOUPs are cntainers that hld a 25-wafer (r 13-wafer) lt f 300-mm wafers in an inert, nitrgen atmsphere. By prviding a clean atmsphere fr the wafers, FOUPs help prevent ptential cntaminants frm cntacting the surface f the wafers. The number f integrated circuits (ICs) per wafer will cntinue t grw due t decreasing device line widths (currently 0.11 micrns). This increased number f die per wafer will lead t an even greater danger f particulate cntaminatin in 300-mm wafer fabs when cmpared t existing 200-mm fabs. Further, the cmbinatin f decreased line width and increased area per wafer require fewer wafers being needed t fill a custmer s IC rders. Each rder culd be assigned t its wn FOUP, but FOUPs are expensive, and mre imprtantly, assigning each custmer rder t its wn FOUP has the ptential t cause the autmated material handling system (AMHS) t becme verladed. In additin, the prcessing time f sme peratins is the same regardless f the number f wafers in the FOUP (lt prcessing). Thus, 300-mm semicnductr manufacturers ften have the need t grup rders frm different custmers int ne FOUP. Once multiple rders are gruped int the same jb (FOUP), these jbs must then be scheduled n the varius types f tl grups in the wafer fab s that effective jb prcessing can prmte n-time delivery f custmer rders. Since there are multiple perfrmance measures in which a cmpany may be interested (e.g. makespan, ttal weighted cmpletin time, ttal weighted tardiness), we refer t this cllectin f prblems as multiple rders per jb ( mj ) scheduling prblems. 2

Figure 1 gives an verview f mj scheduling prblems. Nte that there are tw primary decisins that must be made in mj prblems: 1) hw t grup the rders tgether, which we call jb frmatin, and 2) hw t schedule the jbs nce they are frmed (jb scheduling). In rder t btain the ptimal slutin t a mj scheduling prblem, these tw decisins shuld be made simultaneusly. Hwever, heuristics that decmpse the prblem by first deciding hw t frm jbs and then schedule the frmed jbs may be reasnable and/r necessary fr sme machine envirnments and perfrmance measures. Jb Frmatin Jb Scheduling Orders (O) Jbs (J) Schedule O 1 O 2 J 1 J 3 J 2 J 4 J 3 J 1 O 3 J 4 J 2 Figure 1. The Multiple Orders per Jb Scheduling Prblem The prcessing time f a jb depends n the type f machine upn which a jb is prcessed. We cnsider tw types f machines that lead t tw types f dependency between rder prcessing times and jb prcessing times: 3

(1) Single lt prcessing: A jb s prcessing time may nt depend n the size (i.e., number f wafers) r the cntents f the jb since the prcessing step may invlve either single lt r batch prcessing. In single lt prcessing, such as the peratin f a wet sink, the entire cassette (FOUP in 300mm factries) f wafers is prcessed simultaneusly (e.g., the cassette is submerged in a liquid slutin in the wet sink case). Therefre, ttal prcessing time is independent f the number f items in the cassette. Under pure batch prcessing, which ccurs at diffusin furnaces and burn-in vens, multiple lts are prcessed simultaneusly fr a prescribed perid f time, which is again independent f the number f lts being prcessed. In these cases, the crrespnding prduct type and the specific step f the prduct s recipe determine the prcessing time. Out f these tw types, we cnsider single lt prcessing in ur study. (2) Single item prcessing: In general, jb prcessing times can be a functin f jb frmatin decisins. Hence, a jb s prcessing time may depend n the size f the jb alng with the prduct type and the prcessing step f its recipe. Single item prcessing that we cnsider here makes a jb s prcessing time t be the sum f its rders individual wafer prcessing times, hence it becmes a functin f bth wafer prcessing times and the number f wafers gruped in the jb. As multiple rders per jb will be cmmn in future semicnductr manufacturing systems, as well as in less than trucklad trucking and military deplyment applicatins (see discussin in the Cnclusins sectin), we prpse new additins t the β field f the α β γ scheduling ntatin f Graham et al. (1979). We refer t mj (lt) mj(item) single lt prcessing as and single item prcessing as. Batch prcessing is specified by including mj ( lt), batch batch as an additinal β parameter (e.g., ). 4

In this paper, we investigate the prblem f frming and scheduling jbs that are made f multiple rders f varius characteristics fr a single machine under the bjective f minimizing ttal weighted cmpletin time. The remaining sectins f the paper are rganized as fllws: Sectin 2 reviews the existing literature that is mst relevant t ur prblem. In Sectin 3, we develp the mathematical mdel fr the prblem. As the mdel is nnlinear, we als prvide a linearized versin f the mdel by bserving that the jbs can be pre-sequenced in a specific rder withut lss in the mdel s accuracy. Due t the lng cmputatin time f the mathematical mdel, we then present an extensive set f heuristic appraches in Sectin 4. Finally, we cmpare the perfrmance f the prpsed heuristics t the ptimal slutins and t each ther in Sectin 5, fllwed by research cnclusins and directins fr future research in Sectin 6. 2 Literature Review The mst recent research effrt that is similar t ur prblem is Dbsn and Nambimadm (2001) s batch lading and scheduling prblem which als cntains tw decisins: batch lading (BLP prblem) and batch scheduling (BSP prblem). In their prblem, each jb has a certain size and belngs t a specific jb family. Only the jbs frm the same family can be batched tgether under the batch capacity cnstraints. The prcessing time f a batch nly depends n the family and nt n the batch size. Obviusly, if there is nly ne jb family, the prblem is similar t ur mj scheduling prblem under pure batch prcessing. The authrs study the prblem using mean weighted flw time as the bjective functin. Hwever, batches are nt the critical resurces in their prblem, as they have n limitatins n the number f batches (jbs) in the system. When the batch capacity cnstraints are nt tight, the prblem is cnsiderably simpler. There are many ther research effrts in batch scheduling prblems (Fwler et al. 1992 and 2000, Lee et al. 1992, Chandru et al. 1993, Uzsy 1994 and 1995, 5

Webster and Baker 1995, Kempf et al. 1998, Ghazvini and DuPnt 1998, Azizglu and Webster 2000, Cheng et al. 2001). The fundamental difference between these prblems and ur prblem is that the batch size in their prblems is expressed in terms f the number f jbs that can be simultaneusly prcessed under pure batching, rather than the single item and single lt scenaris we explre. Anther clsely related prblem deals with gruping techniques in Printed Circuit Bard (PCB) assembly. Gruping techniques have been extensively studied fr the PCB industry with the primary bjective f reducing the ttal setup time (see Maimn and Shtub 1991, Hashiba and Chang 1991, 1992, Sadiq and Landers 1993, Dilln et al. 1998, Rajkumar and Narendran 1998, Salnen et al. 2000, Magazine et al. 2002, De et al. 2002.). Mst f these appraches aim t grup bards that cntain a maximum number f cmmn cmpnents. Since the prduct type is the nly characteristic f each bard that may affect the setup times, ther characteristics, e.g., pririty f a bard, are seldm cnsidered during gruping. Althugh there are similarities between ur prblem and the setup reductin prblems in PCB industry (e.g., varius prduct types, machine capacity (FOUP capacity), and s n), we cannt take direct advantage f these techniques because it is hard t find cmmn attributes that accmmdate all the characteristics f each rder while simultaneusly strngly affecting the bjective functin. Anther prblem that is similar t the mj scheduling prblem is the lt-t-rder matching prblem (see Knutsn, et al. 1999, Fwler et al. 2000, Carlyle et al. 2001). This type f prblem als invlves tw decisins: assigning rders t the factry and assigning lts t rders. Similarly, the prblem can be frmulated as a nnlinear integer prgramming mdel. But these authrs nnlinearity is tractable, which makes the prblem simpler than ur prblem. Once rders are assigned t the factry, the assignment f lts t rders des nt affect the bjective value significantly. The bjective functin in ur prblem, n the ther hand, needs the 6

cnsideratins f bth decisins. S the techniques used in these three articles are nt directly applicable fr ur research. Thrugh the literature review, we are unaware f any previus research effrts addressing the prblem f scheduling prductin jbs that cntain multiple custmer rders. As a typical wafer fab s life cycle can last almst 20 years, 300-mm wafer fabs will cntinue t be built and perated arund the wrld fr many years t cme. Our research will help semicnductr manufacturers t utilize their FOUPs mre effectively and hpefully imprve custmer satisfactin thrugh timely prductin. 3 The 1 mj ( ) w j C Prblem j 3.1 Prblem Setting and Ntatin As discussed earlier, 300-mm manufacturers ften cannt allcate an individual FOUP fr each custmer s rder due t cst, strage, and especially the AMHS system cnsideratins. The number f FOUPs available in a 300-mm wafer fab typically remains fairly cnstant after the factry has been ramped t full prductin. The number f FOUPs required in a given fab can be determined via simulatin experiments during the fab design stage. Care must be taken t ensure a sufficient number f FOUPs are present t keep the carriers frm significantly cnstraining the fab s prductin utput (i.e., ensuring that FOUPs d nt becme the bttleneck tl in the fab). Unfrtunately, custmer rder patterns vary widely ver time. Typically, rder release techniques are emplyed t help smth r balance the inventry levels thrughut the wafer fab (see Glassey and Resende 1988, Fwler et al. 2002). When emplying these techniques in 300-mm wafer fabs, cmpanies ensure sufficient FOUP capacity is available prir t releasing a new grup f rders int the fab. With this in mind, we assume that the number f FOUPs available in ur scheduling prblem is fixed, knwn, and sufficient fr prcessing all rders. 7

We assume that all rders are simultaneusly available at time 0 fr jb frmatin, scheduling, and subsequent prcessing. (We will investigate nn-zer jb ready times and the interactin f rder release plicies with scheduling in future research effrts.) Thus, the prblem we study in this paper is the mj prblem fr a single machine with a static set f rders, where the machine is a single item prcessing r a single lt prcessing machine. Our primary gal is t minimize ttal weighted cmpletin time f rders. Custmers place their rders with semicnductr manufacturers at varius pints in time by phne, fax, e-mail, and/r the Internet. Let O = { 1.. n} dente set f all custmer rders that will be scheduled at a certain decisin making pint, where n is the number f rders. Each rder O has the fllwing parameters assciated with it: s, the size f rder in number f wafers. This can be determined by cnverting a custmer s required die quantity int wafers and taking int accunt prductin yield. The size f an rder determines the number f slts it ccupies in its FOUP cntainer. t, the prduct type f rder. We assume that nly rders f the same type can be gruped tgether in a jb (r FOUP) because each prduct type requires fllwing a recipe which specifies a ptentially unique ruting (sequence f machines t visit) and different prcessing requirements. In general, a custmer may cmbine different quantities f several types f wafers int ne rder while issuing the rder. We assume that these multiple-type rders are split int multiple rders f a single type befre releasing them t the fab, mainly due t the truble invlved in gruping different types f wafers tgether in ne jb. Hence, we study the single-type versin f the prblem withut lsing much frm the essence f the prblem. w, the pririty weight f rder. The manufacturer may assign the weight f each rder accrding t their relatinship with the requesting custmer, the size f the rder, r ther factrs. As we fcus n the minimizatin f the ttal weighted cmpletin time in this paper, the weights can als include terms related t inventry hlding csts. In general, there are n restrictins n hw many wafers a custmer can rder. Hence, it is pssible that a custmer s rder size is greater than r less than the FOUP capacity. Further, the average size f rders and their 8

variability may depend n the type f prducts being manufactured. Order sizes fr cmmdity prducts such as memry and micrprcessrs are typically much larger than the rder sizes fr Applicatin Specific Integrated Circuits (ASICs) and prducts made in fundries. Let K dente the capacity f a FOUP cntainer. This eventually limits the number f wafers cming frm different rders that are gruped tgether within a specific FOUP. One can interpret this as the maximum size f the jb assciated with the FOUP under cnsideratin. As the FOUPs are standardized, they all have the same capacity f K wafers. Typical values f K are 13 r 25 wafers. Fr an rder with s < K, there is a ptential that it will be cmbined with ther rders f the same type t better utilize the existing individual and ttal FOUP capacities. If s > K fr an rder, then we assume that s K full lts are frmed fr rder, and the remaining ( s md K ) wafers are explicitly cnsidered in the mdel s that its gruping pssibilities with ther rders are cnsidered. Hence, fr simplicity, we assume that all rders have sizes less than the FOUP capacity, i.e., s < K fr all rders O. Figure 1 describes ur prblem pictrially, where jb frmatin is cmpleted befre jb scheduling. In shrt, each jb ptentially culd cntain multiple rders. The ultimate perfrmance f the system depends n the cmpletin times f custmer rders, which are captured in the ttal weighted cmpletin time bjective. Hence, the assignment f rders t jbs (lts) affects the perfrmance f the system, thrugh the final scheduling f the frmed jbs, which cnsist f the rders that are gruped tgether in FOUPs. In ther wrds, different assignments f rders t jbs will lead t different scheduling perfrmance, as the jbs that are frmed will be different. Mrever, prper scheduling f the jbs will depend n the jbs cntents (rders). Semicnductr manufacturers usually grup the rders with the same basic prduct type (differing nly by patterns prduced in phtlithgraphy) in ne FOUP t reduce the sequence-dependent setup times between 9

different types f prduct. Therefre, a prblem with multiple prduct types can be decmpsed int a series f prblems, each fr a different prduct type. Hence we further assume that all custmer rders have the same basic prduct type in this article. 3.2 A Nnlinear Integer Prgramming Mdel We nw cnsider the special case f single prduct-type, multiple-rder jbs in a single machine envirnment. We assume that the number f FOUPs (which in turn determines the number f lts t be frmed) is given and fixed at a level dented by F. Set J dentes the set f jbs, indexed by j, i.e., j=1,, F. Let ρ be the prcessing time f a single item (wafer) r the time f a single lt, depending n the machine type. This time is assumed t be the same fr all wafers. Recall that K dentes the FOUP capacity r the maximum jb size (in terms f the number f wafers). We define the fllwing basic binary decisin variables: X = 1 when rder is assigned t jb j; = 0 therwise j Y jj' = 1 when jb j is scheduled befre jb j ; = 0 therwise Depending n these variables, we define ther decisin variables t capture the timing f the jbs: p j, the prcessing time f lt/jb j. As expected this is usually a functin f the unit wafer prcessing time and is likely t be dependent n the rders that g int the jb (see belw). δ, the cmpletin time f rder. We assume that all the rders in a jb are prcessed tgether and the jb cmpletin time determines the cmpletin times f all the rders. C j, the cmpletin time f jb j. Nte that the ttal prcessing times f jbs are defined as decisin variables t capture machine types that prduce prcessing times that depend n the cntents f the jbs. Hwever, fr ur single-prduct type, single-machine, single lt prcessing prblem, the jb prcessing time is just the wafer prcessing time: i.e., 10

p j = ρ, j J. In this setting, the jb prcessing times are just parameters f the prblem instead f being decisin variables. Fr the single-type, single-machine, single item prcessing prblem, the jb prcessing time is given belw: p j = ρ s O X j j (1) The bjective f the mdel is t minimize the ttal weighted cmpletin time, as shwn belw: TWC = w O δ (2) The cnstraints include jb frmatin cnstraints, scheduling cnstraints, and the cnstraints that link rders t jbs. We first frmulate the cnstraints that link the rder-t-jb assignment decisins X with the jb scheduling decisins. The cmpletin times f rders in jb j are the same as the jb cmpletin time: δ = C X O, j J j j (3) Als, rders and their crrespnding jbs will at least spend the jb s prcessing time in the system: C j p j O, j J (4) We further have t cnsider the FOUP capacity limitatins and make sure each rder is assigned t a jb exactly nce: s X j K O j J X = 1 O j j J (5) (6) Finally, the scheduling cnstraints shuld reslve the sequencing f the frmed jbs. We intrduce disjunctive cnstraints t handle the machine capacity limitatins and t find the prcessing rder f frmed jbs: C p C M ( 1 Yjj' ) j j', j J, j' J j' j' j (7) C j p j C MY j j', j J, j' J j' jj' (8) where M is a large number (big-m). 11

Here, we define the verall multi-rder jb scheduling prblem as a nnlinear integer prgramming mdel. The verall mdel is t minimize (2) subject t cnstraints (3-8). T cmplete the mdel, cnstraint set (1) is added fr the single item prcessing prblem, and jb prcessing times are set t cnstant p = ρ fr all rders fr j the single lt prcessing versin. The nnlinearity is in cnstraint set (3). Althugh, ne can try t slve this mdel directly using nnlinear ptimizatin techniques, ur apprach intrduces a linearized versin f the mdel that significantly simplifies the mdel and uses its prperties t develp a set f efficient heuristics. 3.3 The Linearized Mdel Fr the linearized versin f the mdel, we first bserve that we d nt have t schedule the jbs frm scratch. Withut lss f generality (and lss f the riginal ptimal slutin), we can assume that there is a given sequence f jbs. In this case, the main prblem is t find the rders that will fill the jbs that are prerdered. We can d this as the capacities f all the FOUPs are the same and the sequencing f the jbs is just re-indexing them. Hence, we assume that the sequence f the jbs is pre-defined and they are sequenced accrding t the increasing rder f their indices. That is, jb 1 is the first jb, jb 2 is the secnd jb, etc. This eliminates the explicit need fr decisin variables Y. We then linearize cnstraints (3) intrducing the fllwing cnstraints instead: j ( X ) δ C M 1 O, j J j (9) where M is a large number. Nte that these linearized cnstraints will wrk as lng as the bjective functin is a regular measure, that is, it is a nn-decreasing functin f the cmpletin times f rders. This is the case fr many bjective functins including the ttal weighted cmpletin time cnsidered in this paper. 12

subject t The cmplete revised mdel is thereby described by the fllwing linear integer prgram: O w δ min (10) j ( X ) δ C M 1 O, j J (11) j p j j C O, j J (12) s X K j J (13) j O j j J X = 1 O C j J \ { F} j+ 1 C j + p j+1 (14) (15) δ 0 O, C 0 j J, X {0,1} O, j J (16) j j T cmplete the mdel, we add (1) fr the single item prcessing prblem, and specify p j = ρ, j J fr the single lt prcessing prblem. Cnstraint set (15) is used t prperly sequence the jbs accrding t their indices. T create tight instances f the mdel fr given prblem data, M can be calculated a priri. T make sure that M is nt t lse, we calculate M in the single item prcessing case as fllws: M = ρs O (17) Under single lt prcessing, we calculate M as fllws: M = Fρ (18) Althugh this linear mdel can be slved using a cmmercial ptimizatin slver such as CPLEX, even this reduced mdel needs significant cmputing time, as it is still an integer prgram. T shw hw cmputatin times change as prblem size increases, we shw the average slutin time in secnds fr different numbers f rders and jb capacities under single item prcessing in Table 1 acrss ten randmly generated prblem instances. The instances were analyzed n a Pentium IV 2.0 GHz cmputer with 512 MB f RAM. 13

1 mj item w j C prblem. Table 1. Average CPLEX Cmputatin Times (secnds) fr the ( ) j Orders K = 13 K = 25 5 0.14 0.04 10 25.66 10.73 15 38,750.00 5,174.00 3.4 Insights frm the Linearized Mdel The fllwing are several bservatins abut ptimal r near ptimal slutins t the single machine mj scheduling prblem when ttal weighted cmpletin time is t be minimized: 1. An ptimal slutin will utilize all available jbs/foups under single item prcessing i.e., every jb will have at least ne rder in it (Tanrisever and Kutanglu, 2004). T see this, we nte that putting mre rders int a jb increases the number f rders that g int the jb and increases the cmpletin time f all f its rders, hence the incentive is t divide up the rders int all available jbs. By cntrast, under single lt prcessing, an ptimal slutin will utilize the minimum number f jbs pssible. One can even add valid inequalities t bth versins f the mdel t reflect these bservatins. 2. When the FOUP capacity is large enugh (lse), then the sequence f the rders between jbs will resemble the weighted shrtest prcessing time (WSPT) rule. In fact, if FOUPs (unrealistically) had unlimited capacities r had capacities where nne f the capacity cnstraints were binding, then the WSPT sequence f the rders wuld be preserved in an ptimal slutin. The prblem wuld reduce t finding a partitin f WSPT-sequenced rders that will g int the pre-sequenced jbs, which can be slved efficiently by a dynamic prgram (Tanrisever and Kutanglu, 2004). In the case f single lt prcessing 14

under lse capacities, all the rders are naturally gruped int ne jb. This bservatin will be a crnerstne f ur heuristic develpment in the next sectin as we define ur rder sequencing rules. 3. As we increase the number f FOUPS (jbs) available in the prblem ( F ), the bjective functin shuld imprve fr the single item prcessing case until all rders are in a jb by themselves. In this case, WSPT is ptimal. Hwever, as nted previusly, when each jb is ccupied by nly a single rder, material handling burden and csts increase. 4. We nte that there is a tendency fr the sequence f rders in an F- jb prblem instance t be preserved in the ptimal slutin when an additinal available jb (FOUP) is added, especially when the FOUP capacity K is fairly high. 5. Optimal slutins t the ( ) j 1 mj item w j C prblem are characterized by later jbs being mre fully laded than earlier jbs. T illustrate these insights, we use an 8-rder example where K = 15 and ρ = 10. Clearly, when F N, each rder can be assigned t its wn jb. It fllws that n rder is delayed unnecessarily due t being gruped with ther rders. In fact, the single item mj prblem reduces t the 1 w j C j scheduling prblem, fr which WSPT is ptimal. As the prcessing time f each rder under single item prcessing is directly prprtinal t rder size, w s can be cmputed fr each rder, and then the rders can be sequenced in descending rati rder t prduce the ptimal schedule when F N. Dente this heuristic as the weighted smallest size (WSS) rule. Fr the example instance in Table 2, WSS prduces the ptimal sequence 3-5-7-2-4-1-8-6 fr the single item mj case, with TWC = 7940. 15

Table 2. Example 8-rder, single item prcessing instance Order s w 1 5 4 2 8 8 3 3 10 4 6 5 5 4 11 6 7 3 7 3 7 8 7 3 Hwever, ur research is mtivated by the F < N case inherent in 300-mm semicnductr manufacturing. Table 3 displays the ptimal slutins t the single machine mj TWC prblem instance in Table 2 as a functin f F under single item prcessing. In Table 3, each jb is identified as a bx surrunding a subset f rders. As Table 3 shws, every time F decreases, an rder is gruped with anther rder int a jb. This gruping starts initially frm the last rder in the WSS sequence (i.e., when F = 8 becmes F = 7, rder 6 is gruped with rder 8), as the rders sequenced last under the WSS rule ften have very small weights. Clearly, gruping lwer weight rders results in crrespndingly smaller weighted cmpletin time increases. As F cntinues t decrease (e.g., F = 4 ), rders in the first part f the WSS sequence are frced int grups with ther rders t insure verall schedule feasibility. In Table 3, the WSS sequence is fllwed by all rders when 4 F 8. As s = 43, jbs are 43 71.67% 4(15) = full, n average. When F = 3, hwever, the O WSS sequence is n lnger fllwed when jb fullness ( FOUP utilizatin ) appraches 100%. 16

Table 3. Example prblem instance slutins fr single item mj case as a functin f F F 8 7 6 5 4 3 3 3 3 3 3 3 5 5 5 5 5 5 7 7 7 7 7 7 2 2 2 2 2 1 4 4 4 4 4 2 1 1 1 1 1 4 8 8 8 8 8 8 6 6 6 6 6 6 TWC 7940 8150 8400 8730 9360 11150 When F = 3, rders 3, 5, 7, and 1 are gruped tgether in jb 1. Althugh this assignment f rders t jbs des nt fllw the WSS sequence, sme insight can be gained by cnsidering the prblem instance s rder sizes. The ttal number f wafers in rders 3, 5, and 7 is 10; the five remaining rders cntain 33 wafers. As jbs tw and three can cntain at mst 30 wafers since K = 15, anther rder must be gruped with rders 3, 5, and 7 in rder t prduce a feasible slutin. Neither rder 2 nr rder 4, the next tw rders in the WSS sequence, can be gruped with rders 3, 5, and 7 t frm a feasible gruping because s 2 = 8 and s4 = 6. As s 1 = 5, rder 1 is the earliest rder in the WSS sequence that can feasibly be gruped with rders 3, 5, and 7. Frm this example, it is apparent that there is a direct relatinship between the WSS sequence f rders and the ptimal rder-t-jb assignment in the single machine case f the mj scheduling prblem under single item F prcessing (as mentined in Prperties 2-4 abve). As is incrementally decreased, the rders at the bttm f the WSS sequence begin t get gruped tgether in rder t minimize unnecessary rder delay times being experienced by the rders at the tp f the WSS sequence (as mentined in Prperty 5 abve). Nt until FOUP 17

capacity becmes quite scarce in the F = 3 case des the ptimal slutin depart frm the riginal WSS sequence. In terms f the single lt mj prblem, the same 8-rder example investigated fr the single item envirnment results in the identical ptimal slutin fr 3 F 8. This ptimal single lt envirnment slutin exactly resembles the F = 3 clumn in Table 3, as it is advantageus t pack as many rders as pssible int each jb when jb prcessing time is independent f the number f rders cntained within the jb. In this case, the resulting bjective functin TWC = 760 and FOUP 1 is 100% utilized, while FOUPs 2 and 3 are bth filled 93.3%. All subsequent FOUPs in the F > 3 cases are unused in the ptimal slutin fr the single lt mj prblem. 4 Heuristic Develpment Three decisins are typically made when develping heuristics fr the ( ) 1 mj prblem. w j C j First, an rder sequencing rule is selected fr arranging a given prblem instance s rders accrding t sme specified criteria, such as nn-increasing w s (i.e., weighted smallest size (WSS)). Five rder sequencing rules are studied in ur heuristic experimentatin t determine which rule (if any) prduces cnsistently superir TWC schedules: WSS: weighted smallest size; nn-increasing rder f sequencing fr single item prcessing. w s. This is equivalent t WSPT rder WLS: weighted largest size; nn-increasing rder f w s LW: largest weight first; nn-increasing rder f w SS: smallest size first; nn-decreasing rder f s 18

LS: largest size first; nn-increasing rder f s We again assume jbs are t be scheduled in ascending rder f their index (i.e., jb 1 is first, fllwed by jb 2, and s n). Once the initial rder sequence is determined, a jb filling directin is specified. Suppse the rders are srted by the WSS rule. Further, assume the rder sequence is... 1, 2 N. When using a last t first jb F filling apprach, the last jb ( ) is filled first with rders lcated at the bttm f the WSS sequence (i.e.,, N 1, etc.). In this way, less imprtant rders (i.e., rders with lw WSS values) are assigned t jbs that will N be prcessed later in the schedule, as jbs are prcessed accrding t increasing jb index (i.e., jb 1, jb 2,, jb 1 F). Finally, the mst imprtant rder ( ) is assigned t the first jb in the schedule. Using this apprach, later jbs are filled fuller than earlier jbs. Cnversely, when using a first t last jb filling apprach, the first jb is filled first with rders lcated at the tp f the WSS sequence. As the ptimal slutin t ( ) j 1 mj item w j C prblems is characterized by later jbs being mre fully laded than earlier jbs in rder t minimize unnecessary rder delay times, a last t first jb filling apprach is adpted. Cnversely, as jbs are typically frnt-laded in the ptimal slutin t 1 mj ( lt) w j C j prblems because jb prcessing time is independent f the number f rders in the jb, a first t last jb filling apprach is selected when develping heuristics fr these prblems. Thus, we nly cnsider the apprpriate jb filling directin each f the tw prblem types. Finally, an rder-t-jb assignment rule is identified fr designating hw the individually sequenced rders are assigned t jbs. Order-t-jb assignment rules are, in effect, hw the jbs (bins) are frmed (packed). A number f the appraches investigated in this paper are based n bin-packing heuristics frm the bin packing 19

literature (Dwsland and Dwsland, 1992). Six rder-t-jb assignment rules are studied in ur heuristic experimentatin t determine which rule (if any) prduces cnsistently superir TWC schedules: FFD1: This is a first-fit decreasing methd in which the sequenced rders are feasibly placed, ne by ne, int the jbs. The feasibility cnditin is due t the limiting size f the jb (capacity f FOUPs). Starting frm the tp r bttm f the rder list (depending n the jb filling directin), the selected rder is put int the first jb with available capacity. Once jb 1 is full then we start at the tp r bttm f list and g thrugh the rder list t feasibly place the available rders in jb 2 in rder t fill it as much as pssible and cntinue till all the rders have been placed. This methd requires at mst F passes thrugh the rder list. FFD1b: This rule is the same as FFD1, except that nw when the number f rders left t be placed int the jbs equals the number f jbs available, in terms f capacity, a single rder is placed in each f the jbs left. Fr example, if there are tw rders still left t be placed and nly tw jbs left with any capacity, then a single rder is placed in each f the tw jbs withut cnsideratin t utilizatin f the capacity f the jbs. While this apprach may nt be prmising fr the single lt prcessing envirnment, we include it fr cmpleteness f the verall heuristic develpment. FFDn: In this methd, rders als are assigned t jbs accrding t first-fit criteria, but by using nly a single pass thrugh the sequenced rder list, the rder being cnsidered is placed int the first jb with available capacity. Fr each rder cnsidered, all the jbs are cnsidered ne after the ther and the rder is placed in the first jb with available capacity. FFDnb: This rule is the same as FFDn, except that as sn as the number f remaining rders t be placed in jbs equals the number f jbs left in terms f capacity, a single rder is placed in each f the 20

remaining jbs. FFDAJS: While rders are assigned t jbs accrding t first-fit criteria, this methd tries t balance the sizes f the frmed jbs. Jb size is calculated as the sum f the sizes (i.e., the number f wafers) f the rders assigned t the jb. An verview f the apprach is as fllws: 1. If all rders are assigned t a jb, Stp. Otherwise, update the sum f the sizes f all rders yet t be placed int a jb. 2. Divide this updated size by the number f currently unfilled jbs (i.e., the jbs with n rders currently assigned t them). This gives the expected average jb size fr each remaining jb. 3. Start assigning unassigned rders in FFD rder t jbs withut vilating the FOUP capacity and withut exceeding the expected average jb size calculated in Step 2. Once the size f the current jb being filled exceeds the expected average jb size r n mre rders remain, stp filling the current jb. G t Step 1. Tw variants f FFDAJS are cnsidered in ur experiments. FFD_AJS1 requires that all available FOUPS have at least a single rder assigned t them, while FFD_AJS2 des nt require all FOUPS t be utilized during rder gruping. 5 Cmputatinal Study 5.1 Experimental Design A carefully designed set f experiments was cnducted t determine the efficacy f varius cmbinatins f five different rder sequencing rules and six rder-t-jb assignment rules n TWC slutin quality. The TWC perfrmance f the 5 6 = 30 unique heuristic decisin cmbinatins was cmpared t the ptimal slutin value 21

* TWC (as determined by slving the MIP frmulatin f the prblem) ver 10 replicatins (prblem instances) f the experimental design shwn in Table 4 fr bth 10- and 15-rder prblem instances. Using Table 4, rder sizes are drawn frm a discrete unifrm distributin between 1 and 5 (i.e., s ~ DU[1,5] ) when ν = 3 and between 2 and 8 ( s ~ DU[2,8] ) when ν = 5. Further, FOUP capacity K is evaluated at tw levels, K = 13 and K = 25. These experimental factr ranges are in line with the mtivating mj applicatin f 300-mm semicnductr manufacturing. In the experiments, rder weights are randm integers between 1 and 15 ( ~ DU[ 1, 15 ), the prcessing time per item r lt (depending n the prblem type) w ] ρ = 10, and the number f jbs (FOUPs) F = Nν + 1. Setting F this way generates rather capacity-tight and challenging prblems 12β since capacity-lse prblems are cnsiderably easier t slve as mentined befre. Table 4. Experimental design fr 10-rder prblem instances Factrs Levels Level Descriptin Prblem Type ( P ) 2 Single item prcessing, Single lt prcessing Order Size ( ) FOUP Capacity ( ) ν + 1 ν + 1 s 2 Discrete unifrm ν, ν + 2 2 K 2 12 + 1 DU, where ν { 3,5} β, where β { 1,2} 5.2 10- and 15-Order Prblems Cmparisn t Optimal Slutins Define TWC( S, A, I) as the ttal weighted cmpletin time resulting frm applying rder sequencing rule S and rder-t-jb assignment rule A n prblem instance I. Further, define perfrmance rati TWC( S, A, I) PR ( S, A, I) = TWC * where TWC*(I) is the ptimal slutin value fr instance I. Table 5 presents ( I) 22

TWC ( S, A), the average TWC ver all 40 instances f varying FOUP capacities and rder size distributins fr single item prcessing, and the 95% cnfidence interval (CI) fr the average perfrmance rati, PR ( S, A), fr all S and A cmbinatins under single item prcessing fr the 10-rder cases. Table 6 presents the crrespnding results under single lt prcessing. Similarly, Table 7 (Table 8) presents the results fr the 15-rder single item (single lt) experiments. The blded values in each f the fur tables dente the rder sequencing rule S and rder-t-jb assignment rule A cmbinatins that best minimize the average TWC under the assciated prcessing envirnment and fr the crrespnding prblem size. Table 5. ( S, A) TWC and ( S, A) PR under single item prcessing cnditins fr 10-rder cases Order-t-Jb Assignment Rule Order Seq Rule FFD1 FFD1b FFDn FFDnb FFD_AJS1 FFD_AJS2 LS 25,150 24,497 25,150 24,497 23,745 23,879 [1.768,1.999] [1.716,1.943] [1.768,1.999] [1.716,1.943] [1.661,1.870] [1.671,1.880] LW 19,619 17,667 19,619 17,643 15,566 15,695 [1.381,1.554] [1.249,1.354] [1.381,1.554] [1.247,1.352] [1.086,1.127] [1.094,1.138] SS 19,694 18,478 19,694 18,093 16,348 16,348 [1.392,1.520] [1.299,1.420] [1.392,1.520] [1.275,1.402] [1.135,1.207] [1.135,1.207] WSS 18,353 16,542 18,353 16,146 14,299 14,308 [1.289,1.454] [1.167,1.280] [1.289,1.454] [1.135,1.248] [1.011,1.026] [1.012,1.027] WLS 21,796 20,612 21,796 20,612 19,108 19,238 [1.525,1.707] [1.436,1.591] [1.525,1.707] [1.436,1.591] [1.310,1.415] [1.319,1.425] 23

Table 6. ( S, A) TWC and ( S, A) PR under single lt prcessing cnditins fr 10-rder cases Order-t-Jb Assignment Rule Order Seq Rule FFD1 FFD1b FFDn FFDnb FFD_AJS1 FFD_AJS2 LS 1,541 1,701 1,541 1,819 2,203 2,203 [1.293,1.366] [1.425,1.523] [1.293,1.366] [1.498,1.604] [1.894,2.099] [1.894,2.099] LW 1,189 1,245 1,189 1,270 1,459 1,453 [1.010,1.031] [1.056,1.084] [1.010,1.031] [1.066,1.105] [1.249,1.324] [1.245,1.322] SS 1,328 1,466 1,328 1,466 1,576 1,531 [1.104,1.165] [1.207,1.323] [1.104,1.165] [1.207,1.323] [1.325,1.443] [1.297,1.403] WSS 1,180 1,241 1,180 1,247 1,359 1,346 [1.005,1.021] [1.053,1.080] [1.005,1.021] [1.057,1.087] [1.165,1.228] [1.154,1.219] WLS 1,292 1,359 1,292 1,427 1,765 1,764 [1.071,1.128] [1.124,1.192] [1.071,1.128] [1.159,1.234] [1.489,1.632] [1.489,1.631] Table 7. ( S, A) TWC and ( S, A) PR under single item prcessing cnditins fr 15-rder cases Order-t-Jb Assignment Rule Order Seq Rule FFD1 FFD1b FFDn FFDnb FFD_AJS1 FFD_AJS2 LS 52,097 51,546 52,097 51,546 50,575 50,659 [1.689,1.859] [1.667,1.837] [1.689,1.859] [1.667,1.837] [1.630,1.798] [1.633,1.805] LW 39,045 37,711 39,045 37,405 33,862 33,926 [1.268,1.350] [1.223,1.293] [1.268,1.350] [1.209,1.282] [1.101,1.136] [1.103,1.140] SS 39,747 38,700 39,747 38,482 36,215 36,215 [1.285,1.354] [1.244,1.313] [1.285,1.354] [1.240,1.307] [1.159,1.229] [1.159,1.229] WSS 36,029 34,827 36,029 34,516 31,202 31,202 [1.171,1.231] [1.127,1.182] [1.171,1.231] [1.117,1.172] [1.011,1.039] [1.011,1.039] WLS 43,486 42,386 43,486 42,386 40,427 40,536 [1.402,1.543] [1.363,1.497] [1.402,1.543] [1.363,1.497] [1.300,1.404] [1.304,1.411] 24

Table 8. ( S, A) TWC and ( S, A) PR under single lt prcessing cnditins fr 15-rder cases Order-t-Jb Assignment Rule Order Seq Rule FFD1 FFD1b FFDn FFDnb FFD_AJS1 FFD_AJS2 LS 3,057 3,160 3,057 3,291 3,785 3,785 [1.422,1.547] [1.469,1.611] [1.422,1.547] [1.517,1.664] [1.798,1.985] [1.798,1.985] LW 2,128 2,158 2,128 2,175 2,493 2,493 [1.022,1.043] [1.035,1.062] [1.022,1.043] [1.043,1.069] [1.193,1.279] [1.193,1.278] SS 2,480 2,581 2,480 2,583 2,776 2,757 [1.153,1.225] [1.205,1.288] [1.153,1.225] [1.206,1.289] [1.316,1.403] [1.308,1.391] WSS 2,123 2,155 2,123 2,178 2,360 2,358 [1.017,1.032] [1.032,1.051] [1.017,1.032] [1.038,1.062] [1.131,1.188] [1.130,1.187] WLS 2,373 2,405 2,373 2,466 2,947 2,947 [1.126,1.181] [1.141,1.199] [1.126,1.181] [1.162,1.227] [1.403,1.544] [1.402,1.543] Fr the single item prcessing envirnment, chsing either FFD_AJS1 r FFD_AJS2 as the rder-t-jb assignment rule, in cmbinatin with rder sequencing rule WSS prduces heuristic slutins that are apprximately 2% abve the ptimal slutin, n average, fr bth the 10- and 15-rder mj prblem instances. Hwever, when prcessing jbs in the single lt envirnment, experimental results suggest that the mst apprpriate rder-t-jb assignment rule is either FFD1 r FFDn, again in cmbinatin with rder sequencing rule WSS. Hwever, arranging the rders in nn-increasing rder f w prduces results cmparable t (albeit slightly wrse than) WSS in bth the 10- and 15-rder prblem instances. In the single lt envirnment wherein jb prcessing time is independent f the number f rders cntained in the jb, heuristic perfrmance was 1%-2% abve the ptimal slutin n average fr the mj prblem instances investigated. While a Pentium IV 2.8 GHz PC with 1 GB f RAM requires less than ne secnd t find a heuristic slutin a 15-rder mj prblem instance n average, the ptimizatin-based apprach ften required mre than ne hur t prduce the ptimal slutin fr the mre difficult single item prblem instances when ν = 5, and 25

especially when = 1 β (i.e., K = 13 --small FOUP capacity). Hwever, the ptimizatin mdel easily slves mst single lt prblem instances in less than 10 secnds, irrespective f the values f ν and β. A clser examinatin f heuristic perfrmance is given in Figure 2 fr the 15-rder prblem instances. In Figure 2, the single item cases result frm using WSS as the rder sequencing rule and FFD_AJS1 as the rder-t-jb assignment rule. Similarly, the single lt instances used WSS tgether with FFD1 as the rder-t-jb assignment rule. It is clear that heuristic slutin quality imprves as FOUP capacity increases (i.e., β increases) in bth prcessing envirnments. The jbs (bins) are bigger and therefre easier t frm (pack). Further, as the expected rder size increases alng with the range f rder sizes (i.e., ν increases), it becmes increasingly difficult fr the best heuristic appraches t prduce near-ptimal slutins in the single lt envirnment wherein a premium is placed n packing FOUPs as fully as pssible. Hwever, in the single item envirnment, increasing expected rder size results in prer perfrmance when FOUP capacity is lw, but imprved perfrmance under a larger FOUP capacity (see key insights discussed in Sectin 3.4 abve). Avg Perfrmance Rati 1.070 1.060 1.050 1.040 1.030 1.020 1.010 1.000 ν = 3 ν = 5 Single Item β=1 Single Lt β=1 Single Lt β=2 Single Item β=2 Figure 2. Heuristic perfrmance acrss 15 rder prblem instances as a functin f ν and β. 26

5.3 50-Order Prblems Heuristic Cmparisn The reasn fr examining these larger, 50-rder instances is twfld. First, we wish t see if arranging rders in nn-increasing rder f w prduces results cmparable t WSS in larger single lt prcessing prblem instances, r if this result is nly valid fr small prblem instances. Secnd, and mre imprtantly, the practical mtivatin underlying mj scheduling prblems has its genesis in industries wherein mre than 10 r 15 rders must be gruped int jbs and scheduled. Therefre, we nw examine larger mj prblem instances cntaining 50 rders. The experimental design described in Table 4 is again used, but the sheer size f these larger prblem instances des nt lend itself t analysis with the ptimizatin-based slutin apprach in any practical, acceptable amunt f slutin time. Therefre, we cmpare each f the 5 6 = 30 unique heuristic decisin cmbinatins [ ] t each ther. As was the case in the 10- and 15-rder experiments, rder weights w ~ DU 1,15, ρ = 10, and F = Nυ + 1, are used. 12β Using ur prir definitin f TWC( S, A, I), we define heuristic rati TWC( S, A, I) HR ( S, A, I) =, TWC min ( I) where TWC min ( I) = mintwc( S, A, I ). Table 9 (Table 10) presents TWC ( S, A) and the 95% cnfidence S, A interval (CI) fr HR ( S, A) fr all S and A cmbinatins under single item (single lt) prcessing fr the 50-rder cases. Again, the blded values in each f table dente the rder sequencing rule S and rder-t-jb assignment rule A cmbinatins that best minimize TWC under the assciated prcessing envirnment. 27

Table 9. ( S, A) TWC and ( S, A) HR under single item prcessing cnditins fr 50-rder cases Order-t-Jb Assignment Rule Order Seq Rule FFD1 FFD1b FFDn FFDnb FFD_AJS1 FFD_AJS2 LS 516162 515540 516162 515540 513595 513650 [1.842,1.958] [1.839,1.955] [1.842,1.958] [1.839,1.955] [1.831,1.946] [1.832,1.947] LW 329915 327685 329915 327543 313755 313816 [1.203,1.247] [1.195,1.237] [1.203,1.247] [1.195,1.236] [1.143,1.173] [1.143,1.173] SS 346041 343623 346041 343553 335715 335715 [1.270,1.286] [1.261,1.275] [1.270,1.286] [1.261,1.274] [1.236,1.237] [1.236,1.237] WSS 288563 286310 288563 285870 276154 276154 [1.041,1.056] [1.033,1.046] [1.041,1.056] [1.032,1.045] [1.000,1.000] [1.000,1.000] WLS 394438 392962 394438 392839 384964 385079 [1.481,1.560] [1.475,1.553] [1.481,1.560] [1.475,1.552] [1.446,1.516] [1.447,1.516] Table 10. ( S, A) TWC and ( S, A) HR under single lt prcessing cnditins fr 50-rder cases Order-t-Jb Assignment Rule Order Seq Rule FFD1 FFD1b FFDn FFDnb FFD_AJS1 FFD_AJS2 LS 29108 29306 29108 29232 32557 32557 [1.714,1.817] [1.727,1.833] [1.714,1.817] [1.712,1.813] [1.940,2.121] [1.940,2.121] LW 17363 17398 17363 17364 19233 19233 [1.082,1.112] [1.084,1.114] [1.082,1.112] [1.082,1.110] [1.202,1.280] [1.202,1.280] SS 17044 17118 17044 16800 18034 18032 [1.261,1.264] [1.268,1.271] [1.261,1.264] [1.240,1.243] [1.336,1.375] [1.336,1.375] WSS 14683 14722 14683 14684 15844 15841 [1.002,1.005] [1.004,1.008] [1.002,1.005] [1.001,1.004] [1.076,1.116] [1.076,1.116] WLS 21509 21552 21509 21583 24349 24349 [1.270,1.348] [1.272,1.351] [1.270,1.348] [1.272,1.345] [1.450,1.599] [1.450,1.599] As was the case in the smaller mj prblem instances, chsing either FFD_AJS1 r FFD_AJS2 as the rder-t-jb assignment rule in the single item prcessing envirnment in cmbinatin with rder sequencing rule WSS prduced the best heuristic results in every ne f the 50-rder prblem instances investigated. As prblem instance size increases in the single lt envirnment, any rder-t-jb assignment rule that des nt take int accunt 28

average jb size (i.e., FFD1, FFD1b, FFDn, r FFDnb), in cmbinatin with rder sequencing rule WSS prduces the best heuristic results verall. Arranging rders in nn-increasing w rder n lnger appears t be an apprpriate methd f sequencing rders as the prblem size grws. A Pentium IV 2.8 GHz PC with 1 GB f RAM requires three secnds t analyze a 50-rder mj prblem instance, n average. 6 Cnclusins and Future Research We have utlined a very practical prblem mtivated by a real issue in semicnductr manufacturing: hw d we frm jbs frm rders f varius characteristics and hw d we schedule the jbs fr gd rder-based custmer service perfrmance. We cnsidered ttal weighted cmpletin time as ur measure f perfrmance in this paper. We presented a nn-linear mixed-integer prgram that encmpasses bth the jb frmatin and jb scheduling decisins in a single mdel. Recgnizing the difficulty f slving the prblem as mdeled, we relaxed the nnlinear cnstraints and re-frmulated the prblem as an MIP. With a cncern fr cmputatin time, we examined a number f heuristic slutin appraches, identifying prmising rules fr minimizing TWC t near ptimal levels in a practically acceptable amunt f cmputatin time. Multiple rders per jb scheduling prblems exist within an extensive research landscape. Our ultimate research gal is t develp scheduling appraches fr this challenging set f prblems under varius machine envirnments, such as, flw shps, flexible flw shps, jb shps, and cmplex jb shps (i.e., 300-mm semicnductr wafer fabs). We will investigate the cmbinatin f single item, single lt, and multiple lt (i.e., batch) prcessing that exists within practical examples f these machine envirnments, ften n parallel machine tl grups. We will lk at multiple, cmpeting bjectives such as thrughput, cycle time, n-time delivery, and fab inventry level. We expect t emply ptimizatin-based slutin appraches, such as mnlithic mdel 29