Metaheuristic Scheduling of 300-mm Lots Containing Multiple Orders

IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 18, NO. 4, NOVEMBER 2005 633

Metaheuristic Scheduling of 300-mm LotsContaining Multiple Orders

Peng Qu and Scott J. Mason

Abstract—The standard unit of transfer in new semiconductorwafer fabrication facilities is the front opening unified pod(FOUP). Due to automated material handling system concerns,the number of FOUPs in a wafer fab is kept limited. Moreover, acertain number of new and larger 300-mm wafers will be placedin these FOUPs and this makes grouping orders from multiplecustomers into a job a necessity. Thereby, efficient utilization ofthe FOUP capacity while attaining good system performance is achallenge. We previously investigated optimization-based solutionapproaches for minimizing total weighted completion time andmaximizing on-time delivery performance for the single machinemultiple orders per job scheduling problem. We present twometaheuristic solution approaches for this scheduling problemunder two different typical wafer fab machine environments:single unit processing and single lot processing. Experimentalresults demonstrate that the metaheuristic approaches can findnear-optimal solutions for realistic-sized 300-mm schedulingproblems in an acceptable amount of computation time.

Index Terms—Metaheuristics, scheduling, wafer fabrication.

I. INTRODUCTION

DUE to ergonomic and safety concerns, the newest gener-ation of 300-mm semiconductor wafer fabrication facili-

ties (“wafer fabs”) are highly automated. To facilitate this nec-essary factory automation while simultaneously keeping tooldevelopment costs down, 300-mm tool suppliers and semicon-ductor industry partners collaborated to develop a number ofuniversal standards. The standard unit of lot transfer betweentools in a 300-mm fab is the front-opening unified pod (FOUP),containers that hold up to 25-wafer lots of 300-mm wafers inan inert, nitrogen atmosphere. By providing a clean atmospherefor the wafers, FOUPs help prevent potential contaminants fromcontacting the surface of the wafers.

Decreasing device line widths in concert with increased wafersize have resulted in an ever-increasing number of integratedcircuits (ICs) per wafer. More ICs per wafer, in turn, resultsin fewer wafers being needed to fill customer orders. Althoughevery order could be assigned to its own FOUP, FOUP acqui-sition, storage, and maintenance costs combined with poten-tial automated material handling system (AMHS) overloadinghave forced 300-mm semiconductor manufacturers, especiallyfoundries and fabs undertaking a large amount of developmentactivities, to group orders from different customers into the sameFOUP. Once multiple orders are grouped into the same FOUP

Manuscript received April 17, 2004; revised June 28, 2005.P. Qu is with the Integrated Production Scheduling Group, Advanced Micro

Devices, Austin, TX 78741 USA (e-mail: [email protected]).S. J. Mason is with the Department of Industrial Engineering, University of

Arkansas, Fayetteville, AR 72701 USA (e-mail: [email protected]).Digital Object Identifier 10.1109/TSM.2005.858503

(production job), the jobs must be scheduled effectively to pro-mote on-time delivery and high customer satisfaction. We referto this collection of problems as multiple orders per job (moj)scheduling problems [1].

Orders are grouped and split frequently in 300-mm waferfabs. In wafer fabs wherein a significant number of develop-ment wafers exist, development lots are processed in variousshort-loops and multiple development lots can only be groupedtogether in a few common operations. In foundries, as lots fromdifferent customer orders are typically of different producttypes, the grouping of lots from different sources may result infrequent setups and process instability. Therefore, order/wafergrouping and splitting usually occurs before and after an oper-ation in which lots of the same product type/requiring the samerecipe are processed. Although the moj scheduling problemcan have fab-wide impacts, the larger fab-level problem can bedecomposed into a local, operation-level problem. Therefore,we assume that all orders are of the same product type andinvestigate the moj scheduling problem in a single machineenvironment in this paper.

Our previous efforts focused on developing optimiza-tion-based solutions for the moj scheduling problem [1]. Theoptimization-based solution approaches for the single machinecase of the moj scheduling problem require a significant amountof computing time to produce the optimal solution for fairlysmall, 15-order problem instances. Although the heuristicspresented in [1] produce “good” solutions much faster thanthe optimization model, the heuristics’ performance is verysensitive to order release time variability.

The time required to compute a production schedule is veryimportant in practice. In wafer fabs, sequencing decisions aremade very frequently, thereby demanding efficient dispatching-and/or periodic scheduling-based support systems. Clearly,finding the true optimal schedule may cost a fab much morein terms of time and/or computational resources than wouldfinding an acceptable solution within a shorter time frameusing a heuristic. Although mathematical models have beenused to determine production plans for some semiconductormanufacturers [2], manufacturers often prefer to use simpler,experience-based rules of thumb and/or dispatching approacheslike critical ratio to sequence jobs at toolgroups (groups ofidentical machines operating in parallel) throughout the fab dueto the sheer size and complexity of the problem, as well as theuncertainties that are present in the fab.

Quite often, a heuristic may only work well under spe-cific situations. Metaheuristic-based solution approaches canbe used to improve upon any heuristic’s performance whenlocal search is implemented as a postprocessing mechanism.

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634 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 18, NO. 4, NOVEMBER 2005

Metaheuristic methods, such as genetic algorithms (GAs) andtabu search (TS), have been used in the literature to analyzeNP-hard problems efficiently. These methods can be thought ofas smart, “guided” enumeration approaches in that a probleminstance’s corresponding solution space is traversed using a setof prescribed “movement rules.”

In this paper, two genetic algorithms are proposed to analyzethe single machine, moj scheduling problem. If each order hasweight or priority , due date , and is completed at time

, then two performance measures of interest can be readilycalculated. Total weighted completion time, a surrogate measurefor minimizing cycle time while placing emphasis on higherpriority orders, is calculated as

(1)

Further, total weighted tardiness, a way of measuring priority-based on-time delivery performance, is calculated as

(2)

We present two metaheuristic solution approaches for thisscheduling problem under two different typical wafer fabmachine environments: 1) single unit processing (i.e., stepper)wherein job processing time is dependent on the number ofwafers in the job, and 2) single lot processing (i.e., wet sink)wherein job processing time is independent of the number ofwafers in the job. As stated above, we assume that all ordersare of the same product type. Therefore, no inter-productsequence-dependent setups are considered in this paper. Inan attempt to exploit various characteristics of the schedulingproblem under study, different GA result representations arestudied.

Using the same experimental design and its associatedproblem instances in [1], the two metaheuristic approaches willbe compared with known, optimal solutions, both in terms ofsolution quality and required computation time. Once viablesolution procedures are developed for the single machine mojscheduling problem, these procedures can be leveraged intosuitable solution approaches for multiple machine environ-ments, such as a group of identical tools that process 300-mmwafers in parallel, simultaneously (“parallel machines”).

II. GENETIC ALGORITHMS

The original idea of genetic algorithms is developed in-dependently by Holland [3] and DeJong [4]. Falkenauer andBouffouix [5] use a GA to schedule a job shop, using a pref-erence list-based approach to represent the chromosome. Thisapproach is further compared with other popular representationssuch as job-based, priority rule-based, and operation-based [6].The authors show that their preference list-based representationis superior to the other three representations for job shop sched-uling problems. Additional work involving the use of GAs toanalyze job shop scheduling problems can be found in researchof Ponnambala et al. [7], Cheung and Zhou [8], Cavalieri et al.[9], and Deo et al. [10].

Although GAs can be used to analyze various types of prob-lems, consider the case when a GA is to be used to analyze a

scheduling problem. First, the GAs chromosome structure mustbe defined. A chromosome is a collection of numbers and char-acters that completely describes a solution for the problem beinganalyzed. Each number or character in a chromosome is calleda gene. Some number of chromosomes is generated, either ran-domly or by some specified method such as dispatching rule orother heuristic, to create the GAs initial population. Let the sizeof the initial population be chromosomes.

The following operations describe one generation (iteration)of a GA. The fitness of each individual chromosome (i.e., can-didate schedule) is assessed by computing its objective functionvalue. For example, if the objective function for the schedulingproblem under study is to minimize total weighted tardiness(TWT), then the individual chromosome (schedule) with theminimum TWT objective value has the highest fitness. Onceeach chromosome’s fitness value has been assessed, the( ; typically, ) of the population withthe highest fitness values are selected for crossover operations,while the remaining of the population with lowerfitness values is discarded.

As population size is constant in each generation, a totalof new chromosomes must now be generated.Two individual candidate chromosomes are selected from theremaining “fit” population of size These two individualcandidate chromosomes are commonly referred to as parents.Some feasible subset of genes is swapped between the twoparents, producing two new chromosomes known as offspring.The selection of parents can be either dependent on or indepen-dent of each individual candidate’s fitness. However, selectingparents with higher fitness should increase the chances ofproducing high quality offspring. After the offspring (i.e.,new schedules) are created, their fitness values are assessed.Offspring determined to be “more fit” than their parents enterthe population, replacing their less fit parents. By propagatingoffspring iteratively, unfit chromosomes in the population willeventually be forced out. This process is commonly referred toas crossover.

After a few iterations of crossover operations, the objectivevalue or fitness of each chromosome in the population oftentends to reach some common value. When this happens, the GAssolution typically will not improve any further when simply run-ning more iterations. To mitigate this case, mutation is used tointroduce new genes into the current population to propagateoffspring with more diverse characteristics. Under mutation, acandidate chromosome is randomly selected. In most sched-uling problem GA applications, the genetic information in tworandomly selected genes is then swapped, thereby producinga new chromosome that (hopefully) still represents a feasibleschedule. Another mutation approach is to randomly generatea new value for a given gene within the selected chromosome.Often, feasibility checks must be performed after chromosomalmanipulations to ensure schedule integrity.

The performance of the crossover and mutation operationsstrongly depends on the results representation that is used. Agood representation can narrow the scope of the search while si-multaneously improving searching ability. The GA results rep-resentation developed by Deo et al.. [10] is applicable for ourresearch problem. In their results representation, a chromosome

QU AND MASON: METAHEURISTIC SCHEDULING OF 300-mm LOTS CONTAINING MULTIPLE ORDERS 635

is divided into two segments: assignment and sequence. Underthis representation, both mutation and crossover operations canbe applied to both the orders and the formed jobs in the mojscheduling problem.

Consider an example problem instance having six orders thatmust be grouped into at most three jobs. Let the initial assign-ment of orders to jobs assign orders 2 and 3 to job 1, orders 1 and5 to job 2, and orders 4 and 6 to job 3. Assuming jobs are sched-uled in nondecreasing order by their indices, this order groupingand job schedule can be represented as

(3)

In (3), represents the job scheduling segment, whiledenotes the order assignment for each job. Fur-

ther, the sequence in which the order assignments appear cor-responds to the job indices. This example problem instance canbe extended to describe a general results representation. First,define the following notation:

the set of orders;the set of jobs;number of jobs;number of orders;the index of the job in the th position in the schedule;

;the number of orders assigned to job ;

;the index of the th order within job ;

, ;the index of the job containing order ;

, ;Using this notation, the GAs results representation can be

more generally described as

(4)

As both mutation and crossover operations can be applied toboth the job schedules and the order-to-job assignments in thisresults representation, a randomly generated binary selector isrequired to control the operations. If the selector equals zero,the operations are applied to the job schedules. Otherwise, theoperations will focus on the order-to-job assignments.

Unfortunately, this results representation requires a two-di-mensional array to store all the pertinent information, one di-mension each for jobs and orders. Clearly, as problem instancesize increases, a significantly large amount of memory will berequired to perform the requisite computations. This will, inturn, greatly increase the associated GAs computation time. Asone of the primary concerns when applying metaheuristic solu-tion approaches is trying to produce good solutions in minimalamounts of computation time, the results representation in (4)is not considered any further.

As the moj scheduling problem involves two decisions, jobformation and job scheduling, the results representation in (4)cannot be simplified until one of the two decisions is known.Hence, using the assumption in [1] that the sequence of jobs is

Fig. 1. Example GA crossover operation.

Fig. 2. Example GA mutation operation.

known a priori, one can reduce the results representation in (3)to a single dimension

(5)

Compared to (3), the results representation in (5) can be storedin a one-dimensional array. This, in turn, will result in reducedmetaheuristic computation times, especially for larger probleminstances.

As stated previously, crossover and mutation are two majoroperations used in GA solution approaches. Fig. 1 illustrateshow crossover is applied to the results representation in (5). As-sume that six orders exist that must be grouped into at most threejobs. The starting position and the length of the gene units to becrossed-over are randomly generated. For the purposes of illus-tration, assume crossover operations occur at the first two geneelements. The swapping of information in the first two genesbetween candidate parents results in the formation of two newchromosomes (i.e., the offspring). Once the resulting offspringrepresent feasible schedules, they replace the two least fit chro-mosomes in the current population.

Fig. 2 depicts a mutation operation occurring in the same ex-ample described above. Under mutation, the number of genesto be mutated, along with their corresponding positions are ran-domly generated. Then, the information in these genes (i.e., thejobs that the corresponding orders are assigned to) is randomlychanged. In Fig. 2, the third order is selected to have its assign-ment mutated from job 1 to job 3.

As the genes used in both crossover and mutation operationsare randomly generated, the feasibility of the resulting offspringis not known in advance. Clearly, both crossover and mutationoperations do not always produce a feasible solution. Hence, anadditional feasibility checking routine is performed after eachoperation. However, once the GA approach described above wastested with a few problem instances, the “check feasibility” rou-tine was observed to be the computational bottleneck. In theworst case, the check feasibility routine could run indefinitelywhen no feasible solution can be found after each crossover ormutation operation. Although a counting variable can be placedwithin the check feasibility routine that forces loop terminationafter a prespecified number of infeasible solutions have beengenerated, a large amount of computation time would still haveelapsed. Hence, a modified GA (MGA) approach is developedthat can avoid this infeasibility issue.


Fig. 3. Example MGA crossover operation.

Fig. 4. Example MGA mutation operation.

III. MGA

A previous heuristic for the moj scheduling problem [1]demonstrated a direct relationship between order sequence andorder-to-job assignment. Assume that order-to-job assignmentis performed in a “black box” that can always find feasibleassignment solutions for any sequence of orders. In this case,feeding the resulting order sequence produced by crossoverand/or mutation operations to the assignment black box wouldavoid the schedule infeasibility issue. This modification wouldclearly enhance the efficiency of the original GA approach.

The chromosomal structure in (5) is no longer applicable inthe MGA approach. Without the existence of the assignmentblack box, this research is simply a single machine schedulingproblem. Hence, the chromosome structures developed in pre-vious research efforts for single machine scheduling problemsare applicable for use in this problem. Let denote the index ofthe order in position of the order sequence ( ,

). Equation (6) describes the correspondingresults representation used in this paper

(6)

Under this formation, one possible result of the example in (3)and (5) can be represented as

(7)

Fig. 3 depicts the crossover operation used with this chro-mosomal structure for a six order example. Again, the startingposition and number of genes to be crossed-over are randomlygenerated, then swapped between parents. If an order (e.g., order6) is the selected gene in parent 2, but not the selected gene inparent 1, a swap within parent 1 will first be applied to shift thisorder to the selected region. Under the results representation in(6), mutation becomes a swap within the chromosome of tworandomly selected genes (Fig. 4).

After each crossover or mutation operation, the resultingorder sequence is fed into the assignment black box. The blackbox is a typical “packing” problem with a size constraint-anypacking algorithm can be applied to it. Let denote thetotal number of wafers in job and denote the statusof order . If , order has been assigned to ajob; otherwise, . Procedure PolyAssignOrdersToJobsdescribes the polynomial time heuristic for the single machine,moj scheduling problem. The procedure has computational

complexity of , where is the number of orders to bescheduled.

Procedure PolyAssignOrdersToJobs—Singleunit processingOrder sequence is generated through thecrossover/mutation operation

, , ,, ,

Do WhileDo whileIf And ThenOrder is assigned to Job

If thenEnd if

Loop,

If Then

Loop

Procedure PolyAssignOrdersToJobs—Singlelot processingSequence orders by nonincreasingvalue

,Do whileDo WhileIf And ThenOrder is assigned to Job

End If

Loop

Loop

IV. COMPUTATIONAL STUDY

A. Tuning the GA and MGA Approaches

Two criteria are used to determine the proper number ofiterations to perform for both the GA and MGA metaheuristicapproaches: required computation time and resulting objectivefunction performance ratio. To study these issues, 20 ten-orderinstances of varying order sizes are randomly generated forboth single unit and single lot processing environments. Allinstances again have a FOUP capacity of . All ordersin these instances are ready at time zero. Letbe the total weighted completion time (TWC) value [as de-scribed in (1)] obtained by metaheuristic approach (i.e.,either GA or MGA) through iterations on problem in-stance . Further, let .


Fig. 5. Number of iterations versus performance ratio for GA and MGAapproaches.

Fig. 6. Number of iterations versus computation time for GA and MGAapproaches.

The numbers in Fig. 5 depict the average performance ratiocomputed over 20 replications.

Fig. 5 displays the improvement in performance ratio as afunction of . As illustrated in the figure, TWC results usu-ally cannot be improved significantly after iterationswhen using the MGA approach. When the GA approach is used,the performance ratio enters the steady state after it-erations. Though running 2000 iterations could potentially leadto a better TWC result than that produced by running 1000 iter-ations of the MGA approach (e.g., 2000 iterations results in anaverage performance ratio of 1 versus an average performanceratio of 1.000 72 for 1000 iterations), this small objective func-tion performance improvement comes at a computational cost(Fig. 6). Therefore, iterations will be used for theMGA approach in the computational experiments that follow.

Fig. 6 also depicts the computation times associated with theGA approach as a function of . These results confirm the dis-cussion in the previous section that the feasibility checking sub-procedure requires a nontrivial amount of computation time. Infact, performing iterations of the GA requires 3.1 son average, while iterations of the MGA approachonly requires 2.4 s of computation time, on average. Therefore,in the experiments that follow, iterations are used forthe GA approach.

B. Experimental Design for TWC and TWT ComputationalStudies

In order to compare the performance of GA and MGA, anextensive set of problem instances with different characteristicswere generated for both the TWC and TWT problem objectives

TABLE IEXPERIMENTAL DESIGN FOR COMPUTATIONAL STUDY

(Table I) as calculated by (1) and (2), respectively. Order weightsare uniformly distributed over the integer set . The pro-cessing time in the single unit processing environment is 30 timeunits per wafer. Alternately, time units per wafer in thesingle lot processing environment. Each factor combination inTable I is replicated 100 times.

C. TWC Computational Studies

Fig. 7 presents the TWC performance ratios of all heuristicsin the single unit processing environment, based on the exper-imental design in Table I. The calculation of individual per-formance ratios is similar to the method used previously forthe numerator terms, will in fact denote the optimalsolution produced by the linearized MIP in [1]. This is pos-sible because the same problem instances previously generatedfor testing the linearized MIP will be examined by each of theheuristic approaches. Therefore, the average performance ratiosprovide a true measure of each heuristic’s performance com-pared to the known optimal solution.

As shown in Fig. 7, MGA works surprisingly well, beatingthe GA approach on all the design factor combinations. The av-erage gap of the GA approach to the optimal solution 5.57%,two times larger than MGA’s average gap of 2.6%. The per-formance of all heuristics is robust to problem size increases[Fig. 8(a)].

Both heuristics work well when orders are released at timezero (i.e., the zero variability case). As order release time vari-ability increases, the performance ratios of all heuristics displaya clear upward trend [Fig. 8(b)]. However, their performance re-mains steady after release time variance reaches a certain level.In other words, all heuristics are robust to order release timevariability. Similar results are observed in the single lot pro-cessing environment (Fig. 9), as MGA still works best in mostcases. Similar to the single unit processing environment, the per-formance of all approaches is robust to increasing problem in-stance size [Fig. 10(a)].

In contrast to the single unit processing environment, orderrelease variability significantly impacts the performance of all


Fig. 7. Average TWC performance ratio results under single unit processing.

Fig. 8. (a) Performance against problem size in single lot problems. (b) Performance against release time factor in single lot problems.

heuristics. Both GA and MGA display an almost linear relation-ship to the release time scale [Fig. 10(b)]. Although an upwardtrend is still apparent for each of the metaheuristic approaches,the respective slopes are not very big. Across the fifteen-orderproblem instances, the performance ratio of each metaheuristicapproach is still below 1.05 (i.e., less than 5% above the optimalsolution, on average).

It makes sense that the heuristic approaches are more sensi-tive to order release variability under single lot processing, com-pared to single unit processing. Since the processing time of ajob in single lot processing is independent of its size, groupingmore orders into a job is always preferred. Hence, compared to

the single unit processing environment, more orders typicallyare grouped into the first job under single lot processing. Moreorders in the first job accounts for a potential starting delay ofthe first job due to the variable release times of the orders withinit. Hence, as order release variability increases, the starting timeof the first job in single lot processing may be further delayed,which eventually results in a higher objective value.

Tables II and III show the percentage of problem instancesfor which each heuristic finds the optimal solution under bothsingle unit and single lot processing environments, respectively.Clearly, MGA is superior to the GA in both single unit and singlelot processing environments.


Fig. 9. Average TWC performance ratio results under single lot processing.

Fig. 10. (a) Performance against problem size in single lot problems. (b) Performance against release time factor in single lot problems.

TABLE IIPERCENTAGE OF SINGLE UNIT PROCESSING PROBLEM INSTANCES

FOR WHICH HEURISTICS PRODUCE OPTIMAL TWC SOLUTION

TABLE IIIPERCENTAGE OF SINGLE LOT PROCESSING PROBLEM INSTANCES

FOR WHICH HEURISTICS PRODUCE OPTIMAL TWC SOLUTION


TABLE IVAVERAGE TWT PERFORMANCE RATIO RESULTS UNDER SINGLE UNIT PROCESSING WHEN R = 1

TABLE VAVERAGE TWT PERFORMANCE RATIO RESULTS UNDER SINGLE UNIT PROCESSING WHEN R = 2

D. TWT Computational Studies

Tables IV and V present the TWT [as calculated by (2)]performance ratios for all heuristic approaches in the singleunit processing environment. The two tables display the resultsfor two different order size distributions: Table IV displays theresults for (i.e., order sizes between three and eightwafers), while Table V shows the TWT performance ratios forthe case (order sizes between one and four wafers). Al-though the GA is inferior to the MGA for the TWT probleminstances, it still produces average TWT performance ratios of1.463. In both tables, MGA beats the GA approach in all factorcombinations.

Through the analysis of Tables IV and V, it seems that theratio of to the average order size has a significant impacton the performance of the MGA (Fig. 11). When , the av-erage order size is 5.5; it reduces to 2.5 when . In Table IV,the average performance ratio of MGA is 1.240. In these cases,the ratio . In other words, each job contains ap-proximately two orders in a feasible order-to-job assignment.When this ratio jumps to 4.56 for the case, the averageperformance ratio of MGA drops to 1.065 (Table IV). Table V

also demonstrates this trend. In this table, whenand when . Since these ratios

are all greater than 4.56, it follows that the average performanceratios of MGAs in both situations are 1.029 and 1.026, respec-tively, in Table V.

Two possible reasons come to mind that may account forthis MGA performance trend. First, the single machine TWTscheduling problem is NP-hard. Hence, there is no dispatchingrule, such as weighted shortest processing time (WSPT) for theTWC problem instances, which can find the optimal sequenceof orders in TWT problems. Moreover, the tardiness of an orderdoes not always increase with its completion time (e.g., whenthe order is early, tardiness is zero). When an order is groupedwith other orders into a job, its weighted completion time in-creases linearly with extra time delays by units per unit oftime delay. The order’s weighted tardiness, however, may con-tinue to be zero even as its completion time is delayed (e.g.,when the order’s due date is loose).

As one can imagine, the efficacy of making the order-to-jobassignment decision under these costs depends on the numberof orders within a job. When fewer orders are grouped into a


Fig. 11. Performance ratio against FOUP size.

TABLE VIAVERAGE TWT PERFORMANCE RATIO RESULTS UNDER SINGLE LOT PROCESSING WHEN R = 1

job, the completion time of the orders is delayed for a relativelysmall number of time units, which, in turn, results in a more ac-curate estimate of the cost of assigning orders to the job. TheMGA approach, on the other hand, is not sensitive to the initialorder sequence. Through both crossover and mutation opera-tions, MGA is equipped to cover many different possible ordersequences.

Second, the assignment black box used in the MGA ap-proach also contributes to this phenomenon. The order-to-jobassignment is itself a bin-packing problem that is known tobe NP-hard. However, the MGA’s assignment black box is aheuristic that aims to find a feasible solution based on a givenorder sequence. For a given set of orders to be scheduled, theMGA’s assignment heuristic will perform better when thereare few jobs to form, as the chance the heuristic will make awrong or poor assignment is reduced. Therefore, the solutionquality of the MGA improves as the ratio increases.No obvious evidence can be found in the figure that suggestsheuristic performance depends on problem size. All heuristicsinvestigated are robust to changes in due date tightness.

Table VI (Table VII) presents the TWT performance ratios forboth GA and MGA approaches under single lot processing forthe high (low) average order size case. Although MGA is stillthe leading heuristic overall, the GA approach is superior to theMGA approach in 6 of the 108 factor combinations. The av-

TABLE VIIAVERAGE TWT PERFORMANCE RATIO RESULTS

UNDER SINGLE LOT PROCESSING WHEN R = 2

erage performance ratios of MGA and GA are 1.145 and 1.195,respectively. Both heuristics are characterized by a clear upward(i.e., undesirable) trend as problem size increases. In contrast tothe single unit environment results, heuristic performance de-creases as order due dates loosen (become “less tight”).

E. Computational Studies on Larger Problem Instances

The previous analyses demonstrate the efficiency of meta-heuristics in small problem instances of the single machine moj


scheduling problem. Since these small problem instances are notrealistic for a real semiconductor wafer fab setting, the meta-heuristics are also tested with much larger problem instances of50 and 100 orders to investigate both their required computingtime and their performance in terms of solution quality. Tenproblem instances are generated randomly for both TWC andTWT problems. All orders are released at time zero with ordersize . Further, for all “large” probleminstances studied.

The results of 10 problem instance studies in each operatingenvironment for the 50 order case support the previous con-clusions that, in general, MGA outperforms GA for both TWCproblems and TWT problems. Under single unit processing, theaverage TWC performance ratio for each of the two approachesis 1.090 for GA, and 1.029 for MGA. Further, the average TWTperformance ratio for each of the two approaches under singleunit processing is 1.076 for GA, and 1.009 for MGA.

Both approaches perform comparably in the single lot pro-cessing environment; this is due to the tight number of jobs. Asproblem size increases to 50 orders, MGA requires less com-puting time than the GA approach. The average computing timeis 11.4 s for MGA. The GA approach, however, has an averagecomputing time that ranges from 21 to 180 s for 200 iterations.This increased computing time is a result of the previously dis-cussed feasibility checking procedure.

Compared to its fifty-order problem instance performance,MGA displays improved performance, especially in TWT prob-lems, when 100-order problem instances are analyzed. Undersingle unit processing, the average TWC performance ratio foreach of the two approaches is 1.032 for GA, and 1.006 for MGA.Further, the average TWT performance ratio for each of the twoapproaches under single unit processing is 1.035 for GA, and1.002 for MGA.

Both metaheuristic approaches consume much more com-puting time, as expected, when examining 100-order instancesas compared to 50 order instances. The average computing timesof MGA increases to 1.1 min. Clearly, this increase in com-puting time is not linear. The GA approach requires up to 10min to perform 200 iterations; this is quite unsatisfactory whenconsidering the MGA approach is capable of performing 2000iterations in one-ninth of the computing time. Therefore, it ap-pears that both MGA has acceptable computing time perfor-mance when analyzing larger, more realistically-sized probleminstances. This computing can be further reduced were the MGAto be coded in a compilable programming language such asC++, rather than in its current Visual Basic for Applicationsform.

F. Implementing MGA in a 300-mm Wafer FAB

Given its ability to produce high-quality multiple orderjob schedules for practical-sized problem instance, MGA haspromise for being implemented in a real 300-mm wafer fab. Asit requires a little over 1 min to schedule 100 FOUPs on a singlemachine, this computational performance is very attractive, as100 FOUPs (i.e., up to 2500 wafers) are more than the typicalnumber of FOUPs that might be sitting in queue at a given fabtool. MGA could be used either as a strategic-level tool running

at the beginning of each shift as a means for establishing shiftplans for the operators or as a local, operations schedulerrunning every 15 min that updates FOUP schedules for someperiod of time in the future. The triggering/initiation of anMGA schedule computation could also be event-driven whenunplanned disruptions like unscheduled tool failures occur inthe fab.

V. CONCLUSION AND FUTURE RESEARCH

In this paper, we extend our prior multiple orders per jobscheduling research [1] due to computing time concerns as-sociated with implementing scheduling approaches for thispractically motivated problem in 300-mm wafer fabs. Geneticalgorithms were applied to the single machine, single productmoj scheduling problem, and the corresponding result repre-sentations are studied thoroughly. Due to the FOUP capacityconstraint, the regular GA approach requires a feasibilitychecking procedure after each crossover/mutation operationin order to guarantee a feasible order-to-job assignment. Thisfeasibility checking procedure greatly increases the computingtime. Thereby, a modified genetic algorithm was developedto eliminate the feasibility checking procedure by using an“assignment black box” idea.

Both approaches are analyzed under the single unit pro-cessing and single lot processing environments for two differentobjective functions of interest: total weighted completion time(a measure of priority-based cycle time reduction) and totalweighted tardiness (a measure of priority-based on-time de-livery). Experiments showed that the MGA approach is superiorto the GA approach in terms of both the computing time andsolution quality for both processing environment under bothperformance measures of interest.

Future research efforts will investigate extending our meta-heuristic solution approaches to more complicated machine(e.g., parallel machines) environments. In order to promotepotential implementation of our solution approaches in 300-mmwafer fabs, it will be important to ensure that realistic problemsizes can be successfully analyzed by any developed solutionapproach in a variety of different machine configuration en-vironments. As our proposed MGA has been shown to be aviable means for analyzing the single machine moj schedulingproblem, it should be leveraged into one or more suitablesolution approaches for the multiple machine environments ina 300-mm wafer fab, such as a group of identical tools thatprocess 300-mm wafers in parallel, simultaneously (“parallelmachines”).

REFERENCES

[1] S. J. Mason, P. Qu, E. Kutanoglu, and J. W. Fowler, “The single ma-chine, multiple orders per job, total weighted completion time sched-uling problem,” IIE Trans., 2005, to be published.

[2] R. C. Leachman, J. Kang, and V. Lin, “SLIM: Short cycle time and lowinventory in manufacturing at samsung electronics,” Interfaces, vol. 32,pp. 61–77, 2002.

[3] J. Holland, Adaptation in Natural and Artificial Systems. Ann Arbor:Univ. Michigan Press, 1975.

[4] K. DeJong, “An analysis of the behavior of a class of genetic adaptivesystems,” Ph.D. dissertation, Univ. Michigan, Ann Arbor, 1975.

[5] E. Falkenauer and S. Bouffouix, “A genetic algorithm for job shop,” inProc. IEEE Int. Conf. Robotics Automation, 1991, pp. 824–829.


[6] S. G. Ponnambalam, P. Aravindan, and P. S. Rao, “Comparative evalu-ation of genetic algorithms for job-shop scheduling,” Production Plan-ning Control, vol. 12, pp. 560–574, 2001.

[7] S. G. Ponnambala, V. Ramkumar, and N. Jawahar, “A multiobjective ge-netic algorithm for job shop scheduling,” Production Planning Control,vol. 12, pp. 764–774, 2001.

[8] W. Cheung and H. Zhou, “Using genetic algorithms and heuristics forjob shop scheduling with sequence-dependent setup times,” Ann. Oper.Res., vol. 107, pp. 65–81, 2001.

[9] S. Cavalieri, F. Crisafulli, and O. Mirabella, “Genetic algorithm for job-shop scheduling in a semiconductor manufacturing system,” in IEEEProc. 25th Annu. Conf. IECON, 1999, pp. 957–961.

[10] S. Deo, R. Javadpour, and G. M. Knapp, “Multiple setup PCB assemblyplanning using genetic algorithms,” Comput. Ind. Eng., vol. 42, pp. 1–16,2002.

Peng Qu received the M.S. and Ph.D. degrees in in-dustrial engineering from the University of Arkansas,Fayetteville, in 2002 and 2004, respectively.

He is an Industrial Engineer in the IntegratedProduction Scheduling Group, Advanced MicroDevices, Austin, TX. His current research inter-ests include production planning and scheduling,operation research, and simulation applications insemiconductor manufacturing.

Dr. Qu is a Member of the Institute of IndustrialEngineers and INFORMS.

Scott J. Mason received the B.S.M.E degree and theM.S.E. degree in operations research from The Uni-versity of Texas, Austin, and the Ph.D. in industrialengineering from Arizona State University, Tempe.

He is an Associate Professor and the AssociateDepartment Head of Industrial Engineering atthe University of Arkansas, Fayetteville, as wellas the Chair of Industrial Engineering GraduateStudies. His research and teaching interests includeproduction planning and scheduling, semiconductormanufacturing, and manufacturing and transporta-

tion applications of operations research.Dr. Mason is a Member of INFORMS and a Senior Member of the Institute

for Industrial Engineers.

Documents

Metaheuristic Scheduling of 300-mm Lots Containing Multiple Orders