A COMPARISON OF HEURISTICS TO SOLVE A SINGLE MACHINE BATCHING PROBLEM WITH UNEQUAL READY TIMES OF THE JOBS. Oleh Sobeyko Lars Mönch

Transcription

1 Proceedngs of the 2011 Wnter Smulaton Conference S. Jan, R.R. Creasey, J. Hmmelspach, K.P. Whte, and M. Fu, eds. A COMPARISON OF HEURISTICS TO SOLVE A SINGLE MACHINE BATCHING PROBLEM WITH UNEQUAL READY TIMES OF THE JOBS ABSTRACT Oleh Sobeyko Lars Mönch Department of Mathematcs and Computer Scence Unverstätsstraße 1 Unversty of Hagen Hagen, 58097, GERMANY In ths paper, we dscuss a schedulng problem for a sngle batch processng machne that s motvated by problems found n semconductor manufacturng. The obs belong to dfferent ncompatble famles. Only obs of the same famly can be batched together. Unequal ready tmes of the obs are assumed. The performance measure of nterest s the total weghted tardness (TWT). We desgn a hybrdzed groupng genetc algorthm (HGGA) to tackle ths problem. In contrast to related work on genetc algorthms (GAs) for smlar problems, the representaton used n HGGA s based on a varable number of batches. We compare the HGGA wth a varable neghborhood search (VNS) technque wth respect to soluton qualty, computatonal effectveness, and mpact of the ntal soluton by usng randomly generated problem nstances. It turns out that the HGGA performs smlar to the VNS scheme wth respect to soluton qualty. At the same tme, HGGA s slghtly more robust wth respect to the qualty of the ntal soluton. 1 INTRODUCTION Schedulng problems for batchng machnes are mportant n semconductor manufacturng, because the processng tmes of the operatons on these machnes are large compared to other machnes n a wafer fab (Mönch et al. 2011). A batch s a set of obs that are processed at the same tme on the same machne. Recently, ths class of schedulng problems has attracted many researchers. Because of the computatonal complexty of batchng problems, heurstcs are proposed. GAs are often used as a metaheurstc for parallel batch machne schedulng problems n semconductor manufacturng. But wth the rare excepton of (Chang, Cheng and Fu 2010), the GAs are only used to assgn batches to ndvdual machnes and then to sequence them. The batch formaton decsons are made by dspatchng rule based heurstcs. In contrast to prevous papers that use GAs to tackle ths problem, the GA proposed by Chang, Cheng and Fu (2010) determnes the content of batches and s hybrdzed by local search. However, the approach has the dsadvantage that the number of batches s fxed. GGAs (cf. Falkenauer 1998) can be used qute naturally to tackle ths type of problems because each batch can be seen as a group of obs. The number of obs wthn each group s bounded from above by the maxmum batch sze,.e., the capacty of the batch processng machne. Because GGAs perform well for multple orders per ob schedulng problems (Mönch and Sobeyko 2011), we are nterested n the queston whether t s possble to apply GGAs n the present stuaton or not. Note that n contrast to the multple order per ob stuaton n ths paper the number of batches s varable. In addton, we compare the HGGA approach wth a VNS approach wth respect to soluton qualty, computatonal effectveness, and the mpact of the ntal soluton. The present paper can be seen as a frst step to tackle a larger class of batch schedulng problems n semconductor manufacturng by GGAs. The paper s organzed as follows. The researched problem s descrbed n the next secton. Then we contnue by descrng dfferent heurstc approaches, especally the HGGA. Computatonal experments /11/$ IEEE 2011

2 for comparng the dfferent heurstcs are presented n Secton 4. Fnally, we present some conclusons and plans for future work n Secton 5. 2 PROBLEM AND RELATED LITERATURE We consder a sngle batch processng machne. The maxmum batch sze s B,.e. at most B obs can be used to form a batch. When a batch s started on a machne no nterrupton s allowed. Totally, n obs have to be processed on the batch processng machne. Each ob has a due date d and a weght w that s used to model the mportance of the ob, and fnally a ready tme r. The obs belong to F ncompatble famles. Let us denote by s( ) { 1,,F} the famly of ob. All obs of the ncompatble ob famly f have the same processng tme p f. Only obs of the same ncompatble famly can be used to form a batch. Usng the α β γ notaton from schedulng theory (cf. Graham et al. 1979) the problem can be represented as follows: 1 batch, ncomp, r TWT, (1) where we denote by TWT the performance measure total weghted tardness. Ths measure s defned as ( C d ) n + : = w, where we denote by = 1 TWT - ( x,0 ) C the completon tme of ob. In addton, we use x + : = max for abbrevaton. Problem (1) s NP-hard, because the NP-hard schedulng problem 1 TWT can be reduced to t. Note that t mght be desrable to form non-full batches, because of the ready tmes and the TWT measure. A schedule conssts of a sequence of batches wth correspondng startng tmes. However, n the case where all obs are ready at tme zero, t s shown by Mehta and Uzsoy (1998) that all batches should be full except maybe for the last batch of each famly. Problems of ths type have been studed extensvely n recent years. We refer to the survey by Mathraan and Svakumar (2006). The sngle machne schedulng problem 1 batch, ncomp TT was studed by Mehta and Uzsoy (1998). Here, we denote by TT the total tardness,.e., the case where we have w = 1 for all obs. Ths research was later extended to the problem 1 batch, ncomp TWT by Perez et al. (2005). Dfferent metaheurstc approaches ncludng VNS and ant colony optmzaton are proposed for P batch,ncomp T W T by Almeder and Mönch (2011), where we denote by P dentcal parallel machnes. Two genetc algorthms are proposed for the parallel machne schedulng problem P batch, ncomp, r T W T by Mönch et al. (2005). A VNS approach and a decomposton heurstc based on mxed nteger programmng s presented by Klemmt et al. (2009) for the problem R batch,ncomp, M,r,B T W T, where we denote by R unrelated parallel machnes, by M machne dedcatons, and by B the maxmum batch sze for machne. Smulated annealng approaches are dscussed by Yugma et al. (2008) for a schedulng problem n the dffuson area of a wafer fab that ncludes batchng machnes wth ncompatble ob famles. Kurz and Mason (2008) propose teratve exchanges procedures for a schedulng problem smlar to the present problem. Recently, a memetc algorthm s proposed for P batch, ncomp, r T W T by Chang, Cheng, and Fu (2010). As dscussed n Secton 1, the resultant GA s used to form batches. However, the number of batches s predetermned for each problem nstance before runnng the GA and cannot be changed by the GA. In ths paper, we propose a GGA that s group-centrc and allows also for splttng groups,.e., batches. So far, GGAs are manly appled to n packng (Falkenauer 1996) and vehcle routng problems and only rarely for machne schedulng problems. 2012

3 3 HEURISTIC SOLUTION APPROACHES Sobeyko and Mönch In ths secton, we start by recallng a decomposton heurstc that s used for provdng ntal solutons and reference solutons. Then, we descrbe the HGGA n detal. For the sake of completeness, we brefly dscuss a VNS scheme that s also used for comparson. 3.1 Tme Wndow Decomposton Heurstc We summarze a lst schedulng approach for problem (1) based on a tme wndow decomposton (TWD) scheme because ntal solutons are determned usng ths heurstc. The followng four steps are performed: 1. When the sngle machne becomes avalable at tme t, only obs wth ready tmes smaller than t + t are consdered. We use the set J ~ f ( t) : = { r t + Δt,s( ) = f } for each famly nstead of the ~ set of all obs. We sequence all obs wthn J f ( t) wth respect to the ATC dspatchng rule n nonncreasng order. Ths rule s gven by the ndex: + w ( ( ) d - p + r - t I ( t) = exp : -, (2) p κp where we denote by κ a scalng parameter and by p the average processng tme of the remanng obs. 2. Then, we consder only the frst thresh obs from the sorted lst to form batches. The resultng set s called Ĵ f ( t). 3. Consder all batches formed by obs of the set ˆ ( t) J f. Each of these potental batches s assessed usng the BATC-II batchng rule (Mönch et al. 2005). The correspondng ndex to assess a batch b of famly f s gven by + + w ( d p t ( r t) ) b - b I b ( t) : = exp p - f κp, (3) b B where we denote by b the number of obs n b. Here, we use the ready tme of batch b, defned as ( r b) r : = max. b 4. The batch wth the largest BATC-II ndex s chosen among the dfferent famles. 5. The tme t t + Δt of the next event s calculated,.e., a machne becomes avalable or a new ob has arrved. Go to Step 1 when there are stll obs not sequenced, otherwse Stop. In ths research, we apply the BATC-II batchng rule wth an approprate value of the look-ahead parameter κ for sequencng the batches. We select the parameter from the nterval [ 0.5,5.0] wth step sze 0. 5 and choose the value that corresponds to the smallest TWT value. 3.2 HGGA Representaton and Intalzaton Scheme In order to determne a feasble schedule obs have to be assgned to batches and these batches must be sequenced. Ths partcularly mples that approprate groups of obs have to be formed. Therefore, t makes sense n a natural way to apply GGA type algorthms when dealng wth the problem of batch for- 2013

4 maton. The used representaton s a set of groups. Each group has at most B obs of the same famly. A GGA s a populaton based metaheurstc. The dfferent ndvduals from the populaton are called genomes. A GGA starts from an ntal populaton that can be generated ether randomly or a certan part of the populaton can be obtaned as a result of applyng heurstcs. In ths research, we use the TWD heurstc Genetc Operators We proceed wth descrng the man genetc operators of the GGA. The most mportant one s the crossover operator. It comnes parts of the parent genomes n order to obtan chld genomes that hopefully correspond to feasble schedules of hgher qualty. The crossover has to be desgned n such a way that chld genomes nhert as much valuable nformaton from ther parents as possble. For GGAs, the crossover operator has a specfc form as t has to deal wth groups of obs rather than wth ndvdual obs. Orgnally such a crossover operator s proposed by Falkenauer (1996). In ths paper, we modfy ths operator and take nto account the ncompatble ob famles. The crossover algorthm conssts of the followng steps: 1. Select randomly two crossng ponts, delmtng the crossng secton n each parent. 2. Replace the crossng secton of parent G 1 wth the crossng secton of parent G Elmnate all obs now occurrng twce from the batches where they are contaned n the frst parent. Thus, some batches orgnated from the frst parent have to be altered. 4. Insert the mssng obs nto the chld genome and adopt the resultng batches wth respect to the hard constrants and the TWT obectve. 5. Repeat Step 2 through Step 4 wth changed roles of the parents to obtan the second offsprng. The resultng crossover procedure s shown n Fgure 1. Ten obs are nvolved n ths example. The chld genome conssts of four batches. Fgure 1: Groupng crossover As descrbed n Brown and Sumchrast (2003), the renserton strategy appled n Step 4 of the crossover algorthm may has a great nfluence on the performance of the GGA. Here, we tested several renserton possltes and found that one of the best performng strateges s to assgn some obs to the batch only when t does not ncrease the ready tme r b of the entre batch b. In the opposte case, one ob wth 2014

5 large ready tme can cause some delay of the obs that are already n the batch. Pror to the renserton procedure we sort the obs n descendng order accordng to ther weghts w. A mutaton s used to avod premature convergence of the GA towards local optma. We use a swap type mutaton operator,.e., two batches contanng obs of the same famly are selected randomly n some genomes. In each of these batches we select randomly one ob and then we exchange the obs. Another posslty s exchangng more than two obs n the selected batches. In order to preserve the most promsng parts of the genomes, whch ncludes the batches themselves and ther sequence, an nverson genetc operator (Falkenaur 1996 and 1998) s appled. Ths technque allows a problem-specfc sequencng of the formed groups. In ths research, we apply the BATC-II batchng rule wth an approprate value of the look-ahead parameter κ for sequencng the groups. The approach for choosng κ s descrbed n Subsecton 3.1. A new sequence s accepted only f ts TWT value s mproved Local Search Scheme It s well-known that GAs often do not perform well, unless they are hybrdzed wth local search schemes. Several local search strateges are developed that mprove the qualty of the genomes n each teraton. The frst strategy s a par-wse ob exchange between batches of the same famly. The man dea of ths procedure s to exchange two obs only f an mprovement of the TWT value s guaranted. We label the batches that are obtaned from the genome by ther postons n the sequence. We select two dfferent batches b and b of the same famly f and try to consder all possble ob exchanges between them as shown n Fgure 2. Fgure 2: Par-wse exchange of obs between batches of the same famly Let n and nb be the number of obs n batches b and b, respectvely. Then the number of possble n + nb assgnments of the obs to the two batches s. Ths value can be large. We derve some propertes that ensure that swaps of obs across batches lead to TWT reductons wthout recalculatng the TWT n value completely. We denote by C, C b and C, C b the completon tme of batch b and b before and after the par-wse ob exchange, respectvely. The correspondng ready tmes of the batches are denoted r, r b and r, r b. The weghted tardness caused by the batches before and after the swap s denoted by by WT, WT b and WT, WT b. Here, we defne WT : = w T for a batch b. When the two condtons b k b k { C,r } + p max{ C, r } C max 1 f + 1 k =, (4) { C,r } + p max{ C, r } C b max b 1 b f b b + 1 = (5) hold, then t s easy to see that the entre TWT value assocated wth the genome s mproved, f the condton 2015

6 WT + WT < WT + WT (6) b b s fulflled. The completon tmes of batches startng from the batch wth ndex mn{, } have to be updated after an exchange. Shftng sngle obs between batches of the same famly s another local search strategy where obs are removed one by one from ther batches and assgned to a dfferent batch. We consder the stuaton that obs are removed from batch b and assgned to b, where <, as depcted n Fgure 3. Fgure 3 : Job shftng between batches We see that shftng one ob between the batches mproves the current value of the obectve functon f the nequalty C = max{ C 1,r } + p f max{ C, r + 1 } (7) mples that WT + WT < WT + WT (8) b s fulflled. We try to shft all obs from batch b. If the reassgnment s successful, the completon tmes of the batches have to be updated startng from batch b. Fnally, batch splttng s a local search strategy that allows for avodng stuatons when obs wth very dfferent ready tmes form the same batch. Thus, stuatons when the entre batch s notceably delayed because t contans obs wth large ready tmes may be prevented by assgnng the obs that can be processed later to a separate batch. We show the splttng of batch b nto two batches and n Fgure 4. b Fgure 4: Batch splttng Such a splttng guarantees the mprovement of the TWT value of the correspondng genome f the condton C = max{ r,max{ C 1,r } + p f } + p f < max{ C, r + 1 } (9) mples WT + WT < WT. We apply local search wthn each teraton of HGGA for each new genome drectly before calculatng ts ftness. The local search strateges are executed n the batch splttng par-wse exchange batch splt- 2016

7 tng shftng batch splttng order. Although the local search scheme ncreases the computatonal tme of the algorthm, t helps to mprove the convergence of the GGA sgnfcantly. 3.3 VNS The VNS scheme used s smlar to the scheme presented by Klemmt et al. (2009) for a slghtly more general stuaton. We consder four dfferent neghborhood structures. All the structures work on the fnal soluton representaton,.e., a sequence of batches. The frst neghborhood structure moves sngle entre batches to a randomly selected poston. The second one does the same for sequences of consecutve batches. The thrd and fourth neghborhood structure swap sngle batches or sequences of consecutve batches, respectvely. The neghborhood structures are appled n ths order. The local search approach wthn VNS s smlar to the local search approach used n HGGA to ensure comparalty of HGGA and VNS. 4 COMPUTATIONAL EXPERIMENTS Ths secton descrbes the desgn of experments used and computatonal results whch allow us to compare the performance and the computatonal potental of the heurstcs. 4.1 Desgn of Experments In order to assess the performance of the dfferent heurstcs we randomly generate problem nstances accordng to the desgn found n (Mönch et al. 2005). The desgn used s summarzed n Table 1. Table 1: Desgn of experments Factor Level Count Number of obs per famly 60, 80, Batch sze B 4, 8 2 Number of famles F 3 1 Famly processng tme Job weghts Ready tmes of obs r p f 2 wth probalty wth probalty wth probalty wth probalty wth probalty 0.1 w U ( 0,1) r ~ U ( 0,α / B p ), { 0.25,0.5,0.75} d r U 0, β / B p, { 0. 25, 0. 5, 0. 75} d ~ ( ) w ~ 1 α 3 Due date of obs β 3 Total parameter comnatons 54 The parameters α and β are used as scalng parameters to control the dstrbuton of ready tmes and due dates. Larger values of α lead to more spread out release dates. On the other hand, hgher values of β result n more loose due dates. We consder three dfferent ob famles. Jobs are assgned to batches and processed on a sngle batch processng machne wth four or eght obs as maxmum batch sze. Totally, 54 problem nstances are consdered. We mplement the HGGA as a steady-state GA wth overlappng populatons. The parameters of HGGA are presented n Table 2. These parameters are determned based on extensve expermentaton usng a tral and error strategy. Note the we use a prescrbed maxmum computng tme of two or fve mnutes, respectvely to allow for a far comparson of the dfferent algorthms

8 For TWD we use t : = p 4, where we denote by p the average processng tme of the obs of the problem nstance, and thresh = 15. All the algorthms are mplemented n the C++ programmng language. HGGA s coded usng the obect-orented framework GALb. A PC wth Intel Core GHz processor and 4 GB of man memory under the OS opensuse 11.4 Lnux s used to carry out the computatonal experments. For each problem nstance three ndependent runs wth dfferent seed values are performed and the average results over the runs are used for analyss. Table 2: Parameters of the GGA Parameter Value Populaton sze 70 Crossover probalty 0.8 Mutaton probalty 0.1 Replacement probalty 0.6 Computng tme per problem nstance 2-5 mn We are nterested n comparng the performance of HGGA, VNS, and TWD wth respect to soluton qualty, tme needed for computaton, and mpact of the qualty of the ntal soluton on the overall soluton qualty. 4.2 Results The obtaned computatonal results are presented n Table 3. Instead of comparng all problem nstances ndvdually, the nstances were grouped accordng to levels of factors. For example, the results for B = 4 are for all problem nstances wth B = 4 whle all other factors have been vared at ther dfferent levels. We present the TWT values of the two metaheurstcs relatve to the TWT values obtaned by TWD. Table 3: Computatonal results Factor Levels HGGA VNS (nt) HGGA VNS HGGA VNS TWD (nt) (no nt) (no nt) (no nt) (no nt) B = B = n = n = n = α = α = α = β = β = β = Overall Tme per problem nstance (mn) It appears that both the HGGA and the VNS scheme may have a dfferent performance dependng on the ntal solutons provded to the algorthms. Therefore, we consder two cases: one wth random ntal solutons denoted by HGGA (no nt) and VNS (no nt) and one wth ntalzng the algorthms wth rather hgh-qualty solutons obtaned from TWD. In case of HGGA, we run TWD wth three dfferent 2018

9 ( 0.0,5.0] Sobeyko and Mönch κ values to determne dfferent hgh-qualty solutons. Best results for HGGA and VNS wth and wthout approprate ntalzaton are marked as bold n Table 3. The results n Table 3 show that both the HGGA and the VNS scheme clearly outperform the TWD scheme n most cases. However, f there s a random ntalzaton of the algorthms they requre a rather large amount of tme to fnd hgh-qualty solutons. Sometmes the algorthms are outperformed by the TWD. On the other hand, f the HGGA and the VNS scheme are ntalzed wth solutons obtaned by TWD, ther performance ncreases sgnfcantly. Overall, both metaheurstcs are able to obtan about ten percent mprovement over the results found by TWD. Thus, t makes sense to compare the performance of the HGGA and the VNS scheme. Both the HGGA and the VNS scheme are able to outperform TWD wth a reasonable amount of computng tme. The performance of the algorthms can dffer dependng on the factors. Our computatonal experments show that the results are qute comparable n most of the stuatons. However, the HGGA slghtly outperforms the VNS scheme n case of a large maxmal batch sze,.e. B = 8, and a small or moderate number of obs. If the number of obs s large then VNS s preferable, snce t s not populaton-based and s therefore faster than the HGGA. It s nterestng to note that when one mnute computng tme per problem nstance s used (not shown here), the VNS scheme clearly outperforms HGGA wth respect to soluton qualty. We are also nterested n the effect of the comnatons of the level values for α and β, snce they reflect requests of the customers. The correspondng computatonal results are shown n Table 4. Table 4: Effect of ready tme and due date settng Factor Levels HGGA VNS HGGA VNS HGGA VNS TWD (nt) (nt) (no nt) (no nt) (no nt) (no nt) α = 0.25, β = α = 0.25, β = α = 0.25, β = α = 0.5, β = α = 0.5, β = α = 0.5, β = α = 0.75, β = α = 0.75, β = α = 0.75, β = Tme per problem nstance (mn) An analyss of the correspondng computatonal results shows that the HGGA performs slghtly better f there s a relatvely large range of ob ready tmes and the range of the due dates s moderate. Otherwse, the VNS seems to be able to fnd better solutons. It s nterestng to see that for wde spread ready tme and loose due dates HGGA and VNS mprove the TWD results up to 70%. Ths result s caused by the fact that the performance of ATC type heurstcs s low n ths stuaton. The overall performance analyss shows that there s lttle dfference between the HGGA and the VNS scheme. Accordng to our results, the HGGA performs smlar to the VNS scheme. 5 CONCLUSIONS AND FUTUTURE RESEARCH In ths paper, we provde a comparson of several heurstcs for solvng a sngle batch machne schedulng problem. Iteratve metaheurstcs such as GGA and VNS hybrdzed wth local search technques are able to fnd hgh-qualty solutons relatvely quckly. The performance of both metaheurstcs can be sgnf- 2019

10 cantly mproved f ntal solutons found by other heurstcs, for example TWD, are provded. Applyng local search technques wthn both the GGA and the VNS ncrease the performance of the algorthms dramatcally. Ths partcularly mples that ncorporatng problem specfc knowledge s of great mportance for these metaheurstcs. An overall comparson of the computatonal results shows that both metaheurstcs perform qute well and may compete wth each other. Although the HGGA has much potental for solvng batchng problems, the VNS algorthm seems to be able to fnd hgh-qualty solutons faster than the HGGA. Ths can be explaned by the fact that VNS operates only on one sngle soluton at each pont of tme, whle HGGA mantans a populaton of solutons. Therefore, t s advantageous to apply VNS n case hgh-qualty solutons have to be found relatvely fast. The HGGA can be used f several mnutes of computatonal tme per problem nstance are avalable. As a next logcal step the HGGA can be extended to the case of machne envronments wth parallel machnes. An approprate genome representaton has be developed to tackle ths problem, especally n case of unrelated parallel machnes wth dedcatons. It s nterestng to compare the performances of GGA and VNS type heurstcs to fnd out whether both of them are able to fnd solutons of smlar qualty and how tme-consumng the algorthms are. REFERENCES Almeder, C., and L. Mönch Metaheurstcs for schedulng obs wth ncompatble famles on parallel batchng machnes. Journal of the Operatonal Research Socety. Accepted for publcaton. Brown E., and R. Sumchrast Impact of the replacement heurstc n a groupng genetc algorthm. Computers & Operatons Research, 30: Chang, T.-C., H.-C. Cheng, and L.-C. Fu A memetc algorthm for mnmzng total weghted tardness on parallel batch machnes wth ncompatble ob famles. Computers & Operatons Research, 37: Falkenauer, E "A hybrd groupng genetc algorthm for n packng." Journal of Heurstcs, 2:5-30. Falkenauer, E Genetc Algorthms and Groupng Problems. Wley. Graham, R. L., E. L. Lawler, J. K. Lenstra, and A. H. G. Rnnooy Kan Optmzaton and approxmaton n determnstc sequencng and schedulng: a survey. Annals of Dscrete Mathematcs, 5: Klemmt, A., G. Wegert, C. Almeder, and L. Mönch A comparson of MIP-based decomposton technques and VNS approaches for batch schedulng problems. In Proceedngs of the 2009 Wnter Smulaton Conference, edted by M. D. Rossett, R. R. Hll, B. Johansson, A. Dunkn and R. G. Ingalls, Pscataway, New Jersey: Insttute of Electrcal and Electroncs Engneers, Inc. Kurz, M., and S. J. Mason Mnmzng total weghted tardness on a batch-processng machne wth ncompatble ob famles and ob ready tmes. Internatonal Journal of Producton Research, 46(1): Mönch, L., H. Balasubramanan, J. W. Fowler, and M. E. Pfund Heurstc schedulng of obs on parallel batch machnes wth ncompatble ob famles and unequal ready tmes. Computers & Operatons Research, 32: Mönch, L., J. W. Fowler, S. Dauzère-Pérès, S. J. Mason, and O. Rose A survey of problems, soluton technques, and future challenges n schedulng semconductor manufacturng operatons. Journal of Schedulng. Accepted for publcaton. Mathraan, M., and A. Svakumar A lterature revew, classfcaton and smple meta-analyss on schedulng of batch processors n semconductor. Internatonal Journal of Advanced Manufacturng. Technology 29: Mehta, S. V., and R. Uzsoy Mnmzng total tardness on a batch processng machne wth ncompatble ob famles. IIE Transactons, 30:

11 Perez, I., J. Fowler, and W. Carlyle Mnmzng total weghted tardness on a sngle batch processng machne wth ncompatble ob famles. Computers & Operatons Research, 32: Sobeyko, O., and L. Mönch Groupng genetc algorthms for solvng sngle machne multple orders per ob schedulng problems. Workng paper, Unversty of Hagen. Yugma, C., S. Dauzère-Pérès, A. Derreumaux, and O. Slle A batch optmzaton software for dffuson area schedulng n semconductor manufacturng. Advanced Semconductor Manufacturng Conferenc 2008 (ASMC 2008). AUTHOR BIOGRAPHIES OLEH SOBEYKO s a PhD student and research and teachng assstant n the Department of Mathematcs and Computer Scence at the Unversty of Hagen, Germany. He receved a master s degree n appled mathematcs from the Natonal Ivan Franko Unversty of Lvv, Ukrane. Hs current research nterests are n appled optmzaton applcatons n manufacturng, logstcs and servce operatons, and n mult-agent systems. Hs emal address s <Oleh.Sobeyko@fernun-hagen.de>. LARS MÖNCH s Professor n the Department of Mathematcs and Computer Scence at the Unversty of Hagen, Germany. He receved a master s degree n appled mathematcs and a Ph.D. n the same subect from the Unversty of Göttngen, Germany. Hs current research nterests are n smulaton-based producton control of semconductor wafer fabrcaton facltes, appled optmzaton and artfcal ntellgence applcatons n manufacturng, logstcs, and servce operatons. He s a member of GI (German Chapter of the ACM), GOR (German Operatons Research Socety), SCS, INFORMS, and IIE. Hs emal address s <Lars.Moench@fernun-hagen.de>. 2021