SINGLE MACHINE MULTIPLE ORDERS PER JOB SCHEDULING USING COLUMN GENERATION

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1 SINGLE MACHINE MULTIPLE ORDERS PER JOB SCHEDULING USING COLUMN GENERATION Jagadish Jampani, Sctt J. Masn, Vishnu Erramilli Department f Industrial Engineering, 4207 Bell Engineering Center, University f Arkansas, Fayetteville, AR 72701, USA Abstract: Key wrds: Effective prductin scheduling in 300-mm semicnductr wafer fabs is a challenging prblem. Custmers rder integrated circuits, hundreds f which can be fabricated n a single silicn wafer 300-mm in diameter. Frnt pen unified pds (FOUPs) are the basic units f wafer strage and transprt in these newest semicnductr fabs. T achieve prductin efficiencies, the silicn wafers ften must be gruped tgether in FOUPs withut regard t the riginating custmer wh placed the rder. In the resulting multiple rders per jb (MOJ) scheduling prblem, scheduling is perfrmed at the FOUP (i.e., aggregated rder) level, but custmer delivery perfrmance is measured at the individual custmer rder level. Previus research effrts have illustrated the difficulty f btaining ptimal slutins fr even small MOJ prblems. We present a clumn generatin-based heuristic apprach fr analyzing the single machine MOJ scheduling prblem under tw different bjective functins: makespan and ttal weighted cmpletin time. Experimental results demnstrate the prpsed clumn generatin apprach s ability 1) t btain gd slutins t MOJ prblems in a fairly shrt amunt f cmputatin time (n average) and 2) t utperfrm cmpeting appraches in the literature fr 50-rder single machine MOJ scheduling prblem instances. semicnductr manufacturing, clumn generatin, heuristic 1. INTRODUCTION The last decade has seen tremendus grwth in the semicnductr manufacturing industry. With the ever increasing demand fr semicnductr based prducts, the industry is trying t meet ever tightening custmer deadlines while simultaneusly trying t evlve int mre efficient prductin units. There is a healthy cmpetitin in the market with every manufacturer trying t increase their market share by meeting deadlines and imprved services t the custmers. The demand f n-time delivery f high quality prducts has mtivated the manufacturers t implement technlgies based n dispatching and scheduling t reduce the prductin time. Efficient scheduling f the available resurces is an effective methd t achieve the rganizatinal gals f n-time deliveries f prducts at a lw prductin cst. Semicnductr manufacturing can be divided int fur phases: wafer fabricatin, wafer prbe, assembly and final testing (Uzsy et al. 1992). The wafer fabricatin phase is the mst cmplex phase with a number f prcess flws, each with steps assciated with it. The equipment used in wafer fabs is sphisticated and cmplex. Unique perating characteristics f the wafer fabs like re-entrant flws, batch prcessing, sequence dependent set-ups, etc., all add t the cmplexity f the wafer fabs. Due t the abve factrs Masn et al. (2002) refer t wafer fabs as cmplex jb shps. 539

2 Single Machine Multiple Orders Per Jb Scheduling Using Clumn Generatin A part f the evlutin in semicnductr manufacturing industry is the advancement in the prcesses and machine tls required t prduce the prducts. New technlgies are being intrduced in the wafer fabricatin equipment and prductin prcesses. One f the main changes has been the adptin f 300- mm diameter wafer in place f the earlier standard f 200-mm. Due t increase in diameter f the wafers mre die can be btained per wafer. This result in increment in the number f integrated circuits prduced per wafer, and better utilizatin f the highly capital-intensive machinery. But the increase in size has als brught abut an increase in weight f the basic wafer. The 300-mm wafer cannt be manually mved between wrk statins therefre cmplete autmatin f the prductin system is required. A part f this autmatin is the usage f entities called frnt-pen unified pds (FOUPs). FOUPs prvide an inert, nitrgen atmsphere thus preventing the cntaminatin f surfaces f the wafers. Custmer rders are translated in terms f the number f wafers required t fulfil the rder. Each rder can be assigned its wn FOUP while prductin but the idea is nt financially viable due t the high cst assciated with these FOUPs. Als, if each rder wuld be assigned its wn FOUP, then there wuld be t many FOUPs in the system causing the autmated material handling system (AMHS) t be verladed. T vercme these challenges ne needs t grup these multiple rders int the FOUPs (can als be cnsidered as a jb). Once the rders have been gruped int jbs, these jbs can be scheduled t ensure efficient usage f the resurces available in the fab. The research in this paper has been mainly mtivated by this challenge f gruping different custmer rders each with different attributes int single jbs and then scheduling these frmed jbs n a single machine. The bjectives f the scheduling prblem investigated in this research are minimizing the makespan (i.e., prductin cycle time) and ttal weighted rder cmpletin time (i.e., thrughput maximizatin when differing custmer pririties exist). The single lt MOJ prcessing envirnment is studied in which the prcessing time f the jb is independent f its cntents. All the wafers in the jb are simultaneusly prcessed and therefre all the rders have the same prcessing time n the cnsidered single machine, such as is the case in a wet sink r acid bath. The 1 mj ( lt) Cmax and 1 mj ( lt) w C j scheduling prblems are NP hard, as the knwn NP hard bin packing prblem j (Karp 1972) is reducible t bth f these prblems, as clearly in bth MOJ prblem cases, we are seeking t minimize the number f FOUPs (i.e., bins) that the custmer rders are assigned t. The fllwing sectins f this paper are rganized as fllws. Sectin 2 reviews the relevant literature t ur prblem. Next, the mathematical frmulatin and the slutin methdlgy are presented and explained in Sectin 3, fllwed by a cmparisn f the slutins btained frm ur prpsed slutin methdlgy and the knwn ptimal slutins in Sectin 4. Finally, we make cncluding remarks and ffer directins fr future research in Sectin LITERATURE REVIEW Scheduling jbs cntaining multiple rders is a fairly new field f research and is practically mtivated by the recent develpments in the semicnductr manufacturing industry. Kutanglu et al. (2004) address this prblem f jb frmatin and scheduling cnsidering a special case f the main prblem with an bjective f minimizing the ttal cmpletin time f the rders. It is assumed that there is n restrictin n the number f FOUPs available in the system. An algrithm is intrduced and preliminary cmputatinal results are reprted. Results were presented fr prblems cntaining up t 24 rders and six jbs/foups. Masn et al. (2004) prpse an ptimizatin mdel t slve this multiple rders per jb (MOJ) scheduling prblem with an bjective f minimizing the ttal weighted cmpletin time f the rders. Bth single lt and single item prcessing cases are cnsidered. In the single item (single lt) prcessing case, jb prcessing time is dependent n (independent f) the wafers present in the FOUP. A nn-linear mathematical mdel is develped which is linearized by assuming that the sequence f the jbs is 540

3 Single Machine Multiple Orders Per Jb Scheduling Using Clumn Generatin predefined. They are sequenced accrding t the increasing rder f their indices. It is shwn that fr the cases abve 15 rders, the ptimizatin mdel takes impractical amunts f cmputatin time. Hence, sme heuristic methds, based n the first fit decreasing apprach fr slving the traditinal bin-packing prblems, are presented t slve larger prblem instances. It is bserved that the heuristics perfrm 1% - 2% abve the ptimal slutin n average fr the 10- and 15-rders cases. Single machine scheduling prblems and their slutin appraches are addressed in Pined (1995). Van den Akker et al. (2000) presents a frmulatin f time-indexed single machine scheduling prblems and use the clumn generatin methd t slve it. In this frmulatin, assigning the jbs t the machines is cnsidered with an bjective f minimizing the ttal cst. In the literature, clumn generatin methd has been effectively used t slve large scale prblems like cutting stck, aircraft ruting, ship scheduling, vehicle ruting, bin packing, netwrk flw, and circuit card assembly. 3. MATHEMATICAL FORMULATION AND SOLUTION METHODOLOGY We assume that at an instance f time, we have a set f rders (O) that are placed by the custmers. Depending n the custmers demand and imprtance, an rder ( O) has a size s 0 and weight w 0 respectively. It is assumed that all the sizes f the rders are less than r equal t the maximum FOUP capacity. All rders are t be placed in the FOUPs and scheduled t attain the required bjective. As a lt prcessing situatin is cnsidered, it is assumed that each FOUP takes ten units f prcessing time (ten units f prcessing time is an arbitrary value and can be changed depending n the situatin). As ready times f the rders are nt cnsidered, it is assumed that any f the rders can be selected t be placed in the FOUP at any chsen instance f time. 3.1 Slutin Methdlgy In the MOJ scheduling prblem, we have a large number f variables and integrality cnstraints n the variables, bth f which cmbine t result in impractical cmputatin time. Clumn generatin (CG) methd (Dantzig et al. (1960)) is prpsed t slve this prblem. The prblem is divided int tw interrelated mdels; the master prblem (MP) and sub prblem (SP). The MP is relaxed and restricted because nn-integer values can be taken by the decisin variables in MP and nly the clumns that have the ability t imprve the bjective f the MP are generated in the SP. During each iteratin f the CG prcedure, the MP is slved t ptimality and by using its dual infrmatin, the SP generates a new clumn. This new clumn cntains the infrmatin regarding which rder ges int which jb r FOUP. This iterative prcess stps when n imprving clumns are generated by the SP. 3.2 A Heuristic fr 1 mj ( lt) Cmax The fllwing discrete ptimizatin mdel minimizes the makespan f custmer rders. Each clumn in the basis matrix represents a pattern, which indicates a way f placing different rders in a FOUP. First, we define pertinent ntatin: Index ver the ttal number f rders O ( O ) j Index ver the ttal number f patterns r clumns J ( j J ) K Maximum FOUP capacity ρ Prcessing time f a jb; assumed t be equal t 10 time units fr all jbs and hence rders. A, Number f times an rder appears in pattern j j 541

4 Single Machine Multiple Orders Per Jb Scheduling Using Clumn Generatin D X Y j Demand f each rder Number f times the pattern j is selected (variable in MP1) = 1, if rder appears in a particular pattern, 0 therwise (variable in SP1). Nw, we describe ur 1 mj ( lt) C master prblem: max Restricted Master Prblem (MP1): Minimize ρ X j (1) j J s.t. j J A, X D O (2) j j X j 0 j J (3) The bjective (1) in MP1 selects the minimum number f patterns satisfying the demand cnstraints (2) fr all the rders. The bjective is als equivalent t selecting the minimum number f FOUPs required t fill all custmer rders. The sub prblem (SP1) is slved using the dual variable infrmatin frm MP1. Parameters π represent the dual values crrespnding t cnstraint (2), which are nnnegative in sign. New clumns fr MP1 are generated by slving the simple knapsack prblem as the sub prblem. Sub Prblem (SP1): Maximize s.t. O O π (4) Y s Y K (5) Y {0,1} O (6) 3.3 A Heuristic fr 1 mj ( lt) w C O This subsectin cntains a discrete ptimizatin mdel fr minimizing ttal weighted cmpletin time f custmer rders. In this frmulatin, an upper bund n the number f FOUPs T is calculated, which is equal t the number f FOUPs in the input feasible slutin fr the clumn generatin methd. In a feasible slutin, all rders are selected nce and placed in FOUPs while satisfying the FOUP capacity cnstraint. A FOUP is divided int K slts, where K equals the maximum FOUP capacity. Further, nly ne wafer is allwed t ccupy a slt. This avids an extra cnstraint in the frmulatin regarding the FOUP capacity. In this frmulatin, a generated clumn has infrmatin that represents the slts ccupied by wafers in each FOUP. First, we define pertinent ntatin: Index ver the ttal number f rders O ( O ) k Index ver the slt number in a FOUP ( k K ) t Index ver the upper bund n the number f FOUPs ( t T ) c Index ver the clumns r patterns C () Clumns crrespnding t rder 542

5 Single Machine Multiple Orders Per Jb Scheduling Using Clumn Generatin W, c Prduct f rder s weight (w ) and the FOUP number (t) it is placed in the clumn c A k, t, c, = 1, if rder ccupies slt k in FOUP t in clumn c, 0 therwise X, c = 1, if rder is in clumn c, 0 therwise (Variable in MP2) Y k, t, c G t, = 1, if rder ccupies slt k in FOUP t in clumn c, 0 therwise (Variable in SP2) = 1, if an rder is present in fup t, 0 therwise (Variable in SP2) O Nw, we describe ur 1 mj ( lt) w C master prblem: Restricted Master Prblem (MP2): Minimize O c C ( X c c C ( ) ρ W X (7), c ), =, c s.t. 1 O (8) O c C ( ) X, c 0 A,,, X, 1 t T, k K (9) k t c c O, c C (10) The bjective functin (7) minimizes the weighted cmpletin times f the rders. Cnstraint (8) makes sure that every rder is selected nce and cnstraint (9) limits a slt in all the FOUPs t be ccupied by nly ne wafer r remains empty. Slving MP2, dual variables crrespnding t the cnstraints are btained. The set f dual values υ are unrestricted in sign and represent the dual values f the set f cnstraints (8). The set f dual values π t,k are negative in sign and crrespnd t the set f cnstraints (9). These dual variables are used in bjective functin f the sub prblem t assign the csts f the slts in the FOUPs. The sub prblem is slved fr a number f times equal t the number f rders, generating a new clumn fr each rder. All these newly generated clumns are added t the MP2. This sub prblem can be seen as a simple assignment prblem which minimizes the cst f selecting/assigning slts in the FOUPs t the rders. Sub-Prblem (SP2): Minimize s.t. t T k K t T k K k K ((( w / s ) t) Y υ (11) Y, π t, k ), k, t, c, k, t c = s (12) Y s G = 0 (13), k, t, c t t T Y {0,1} O, k K, t T, c C (14), k, t, c The SP2 bjective functin (11) minimizes the cst f placing the rders in the FOUPs. This cst is equal t the prduct f the weight f an rder (w ) and the FOUP number (t) the rder is placed in. Cnstraint (12) cnstrains that slts equal t the size f an rder are ccupied ver all the FOUPs. As rders are nt allwed t be split acrss the FOUPs, cnstraint (13) is intrduced s that either a cmplete rder is placed in a FOUP r nthing related t that rder is placed in it. 543

6 Single Machine Multiple Orders Per Jb Scheduling Using Clumn Generatin 4. EXPERIMENTAL RESULTS In rder t evaluate ur clumn generatin-based heuristics, an experimental design was develped (Table 1). The rder sizes are generated using tw discrete unifrm functins; s ~ DU[1,5] when and s ~ DU[2,8] when. The FOUP capacity is varied ver tw levels; K = 13 and K = 25. The weights f the rders are generated using the discrete unifrm functin w ~ DU[1,15]. Fr each cmbinatin f the rder size and FOUP capacity, 10 replicatins are slved. Hence, varying the tw factrs fr a 10 rder prblem instance, 40 cmputatin results are btained. This is repeated fr the 15, 20 and 50 rder prblems. The cmputatins are carried n a cmputer with 3.4 GHz prcessr and 2 GB f RAM. The slutins btained frm the ptimizatin mdel f Masn et al. (2004) and the prpsed clumn generatin methds are cmpared fr the 10, 15 and 20-rder prblems. As the ptimizatin mdel is nt able t btain slutins fr the 50 rder prblems in reasnable amunt f time, the best f the heuristic slutins frm Masn et al. (2004) is used fr cmparisn. The perfrmance rati f the clumn generatin methd is calculated by dividing the clumn generatin slutin by the ptimal slutin r best heuristic in the 50-rder case. The best, the wrst and the average perfrmance ratis fr the cmbinatin f each f the levels f the factrs are shwn in Tables 2 and 3 fr makespan and Tables 4 and 5 fr ttal weighted cmpletin time. With an bjective f minimizing the makespan (Tables 2 and 3), it is bserved that the prpsed clumn generatin-based heuristic perfrms cmparably t the heuristics f Masn et al. (2004). In terms f minimizing weighted cmpletin time, the clumn generatin s average perfrmance rati varies between ne and 2%, with an average value f 1.6%. This perfrmance is slightly better than the best previus heuristic slutin f Masn et al. (2004). As was the case in the makespan instances, the clumn generatin apprach finds many f the knwn ptimal slutin values. Fr the 50-rder ttal weighted cmpletin time cases, the clumn generatin apprach again utperfrms Masn et al. (2004). Hwever, this imprved perfrmance can cme at a cmputatinal cst. The average cmputatin times fr all the prblem instances are presented in Tables 6 and 7. It can be bserved that the clumn generatin methd btained gd slutins in reasnable amunt f cmputatin time. Unfrtunately, the average cmputatin time fr the 50-rder prblem instances is relatively high; hwever, it varies widely depending n the prblem instance. The cmputatin time als depends n hw gd the initial clumns are. As the FOUP size increases frm 13 t 25, the length f the generated clumn increases resulting in a cnsiderable increase in cmputatin time. 5. CONCLUSIONS AND FUTURE RESEARCH In this paper, the MOJ scheduling prblems f 1) minimizing makespan and 2) minimizing ttal weighted rder cmpletin time are addressed. Mathematical mdels are develped and slved using a clumn generatin heuristic apprach. Experimentatin cnducted ver a wide range f prblem instances demnstrated the efficacy f the prpsed clumn generatin apprach. The new methd, althugh utperfrming the best heuristics t date fr large 50-rder prblem instances, can result in an unacceptable amunt f cmputatin time due t increased subprblem basis matrix dimensinality. This research can be extended t address the single item prcessing envirnment in MOJ scheduling in which jb prcessing time depends n the number f items in the jb. Future research effrts will pursue this extensin, as well as the inclusin f rder ready times and jb setup times and mre advanced machine and shp envirnments in rder t increase the viability f ur heuristic appraches fr deplyment in real wrld 300-mm wafer fabs. 544

7 Single Machine Multiple Orders Per Jb Scheduling Using Clumn Generatin REFERENCES Abara J. (1989), Applying integer linear prgramming t the fleet assignment prblem, Interfaces, 19, 4, Cattrysse, D.G., Salmn, M., Van Wassenhve, L.N., (1994), A set partitining heuristic fr the generalized assignment prblem, Eurpean Jurnal f Operatinal Research, 72, 1, Dantzig, G.B., and Wlfe, P., (1960) Decmpsitin principle fr linear prgrams, Operatins Research, 8, 1, Desrsiers, J., Dumas, Y., Slmn, M.M., Sumis, F., (1996), Time cnstrained vehicle ruting and scheduling, Handbks in Operatins Research and Management Science, Vl. 8: Netwrk Ruting Ball, M.O., Magnanti, T.L., Mnma, C.L., Nemhauser, G.L., eds., Nrth-Hlland: Amsterdam, Gilmre, P.C., Gmry, R.E., (1963), A linear prgramming apprach t the cutting stck prblem-part II, Operatins Research, 11, 6, Karp, R.M., (1972), Reducibility amng cmbinatrial prblems, In Cmplexity f Cmputer Cmputatins, R.E. Miller and J.W. Thatcher, eds., Plenum Press, New Yrk, Kutanglu, E., Tanrisever, F., Masn, S.J. (2004) Frming and scheduling jbs frm multiple rders with different attributes: a cmputatinal study f special cases, 13th Annual Industrial Engineering Research Cnference, Hustn, Texas. Masn, S.J., Fwler, J.W., Carlyle, W.M. (2002) A mdified shifting bttleneck heuristic fr minimizing the ttal weighted tardiness, Jurnal f Scheduling, 5, 3, Masn, S.J., Qu, P., Kutanglu, E., Fwler, J.W. (2004) The single machine multiple rders per jb scheduling prblem, IIE Transactins, in review. Pined. M. (1995) Scheduling: Thery, Algrithms and Systems, Jhn Wiley: Englewd Cliffs, NJ Uzsy, R., Lee, C.Y., Martin-Vega, L.A. (1992) A Review f Prductin Planning and Scheduling Mdels in the Semicnductr Industry Part I: System Characteristics, Perfrmance Evaluatin and Prductin Planning, IIE Transactins, 24, Van den Akker, J.M., Hurkens, C.A.J., Salvesbergh, M.W.P. (2000) A time-indexed frmulatin fr single-machine scheduling prblems: Clumn generatin, INFORMS Jurnal n Cmputing, 12, 2, Wilhelm, E.W. (2001) A technical review f clumn generatin in integer prgramming, Optimizatin and Engineering, 2, 2,

8 Single Machine Multiple Orders Per Jb Scheduling Using Clumn Generatin Table 1. Experimental design space fr the lt prcessing instances Factrs Levels Levels Descriptin Order Size(Z) 2 Discrete Unifrm DU[ υ ( υ + 1) / 2, υ + ( υ + 1) / 2] where υ {3,5} FOUP Capacity(K) 2 12 β + 1 where β {1,2 } Table 2. Perfrmance ratis f 10 and 15-rders 1 mj ( lt) Cmax prblem instances 10 Orders 15 Orders Best Wrst Average Best Wrst Average K = K = Average Table 3. Perfrmance ratis f 20 and 50-rders mj ( lt) max prblem instances 1 C 20 Orders 50 Orders Best Wrst Average Best Wrst Average K = 13 K = Average Table 4. Perfrmance ratis f 10 and 15-rders1 j j prblem instances 10 Orders 15 Orders Best Wrst Average Best Wrst Average K = K = Average

9 Single Machine Multiple Orders Per Jb Scheduling Using Clumn Generatin Table 5. Perfrmance ratis f 20 and 50-rders1 mj ( lt) w C j j prblem instances 20 Orders 50 Orders Best Wrst Average Best Wrst Average K = 13 K = Average Table 6. Cmputatin times f mj ( lt) max prblem instances in secnds 1 C Orders Orders Orders Orders K = K = K = K = Table 7. Cmputatin times f 1 mj ( lt) w C j j prblem instances in secnds Orders Orders Orders Orders υ = υ = υ = υ =