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J Sched (2008) 11: 29–47DOI 10.1007/s10951-007-0049-1
A multi-criteria approach for scheduling semiconductor waferfabrication facilities
Michele E. Pfund · Hari Balasubramanian ·John W. Fowler · Scott J. Mason · Oliver Rose
Published online: 27 November 2007© Springer Science+Business Media, LLC 2007
Abstract In this research, we model a semiconductor waferfabrication process as a complex job shop, and adapt a Mod-ified Shifting Bottleneck Heuristic (MSBH) to facilitate themulti-criteria optimization of makespan, cycle time, andtotal weighted tardiness using a desirability function. Thedesirability function is implemented at two different lev-els of the MSBH: the subproblem solution procedure level(SSP level) and the machine criticality measure level (MCMlevel). In addition, we suggest two different methods ofchoosing the critical toolgroup at the MCM level: (1) theLocal MCM approach, which chooses the critical toolgroupbased on local desirability values from the SSP level and(2) the Global MCM approach, which chooses the criticaltoolgroup based on its impact on the desirability of the en-tire disjunctive graph. Results demonstrate the desirability-based approaches’ ability to simultaneously minimize allthree objectives.
M.E. PfundSupply Chain Management Department, Arizona State University,Tempe, AZ 85287, USA
H. BalasubramanianDepartment of Health Sciences Research, Mayo Clinic,Rochester, MN 55905, USA
J.W. Fowler (�)Department of Industrial Engineering, Arizona State University,PO Box 875906 Tempe, AZ 85287-5906, USAe-mail: [email protected]
S.J. MasonDepartment of Industrial Engineering, University of Arkansas,4207 Bell Engineering Center, Fayetteville, AR 72701, USA
O. RoseInstitute for Applied Computer Science, Technical University ofDresden, 01062 Dresden, Germany
Keywords Multicriteria · Shifting bottleneck · Complexjob shop
1 Introduction
The Factory Operations portion of the 2005 InternationalTechnology Roadmap for Semiconductors (SIA 2005) in-dicates that there is increasing pressure on semiconductormanufacturers to maximize throughput, reduce cycle timesand improve on-time delivery (OTD) of products to cus-tomers. This section of the ITRS also contains a list of po-tential solutions to the cost per function (e.g., transistor) andcycle time requirements. The potential solutions are classi-fied into planning decision support tools at the strategic leveland tools for running the factory at the tactical or executionlevel. The ITRS identifies real-time scheduling as one of theexecution-level potential solutions. In this paper, we discussthe development of a new approach for scheduling semi-conductor wafer fabrication facilities (“wafer fabs”) that at-tempts to optimize an aggregation function that combinesthroughput (equivalent to makespan (Cmax)), average cycletime (CT, which is similar to the sum of completion times),and OTD (minimize total weighted tardiness (TWT)).
In a typical wafer fab, there often are dozens of processflows. Process flows are routes that the wafer lots have tofollow in the factory. Each process flow contains 200–500processing steps and more than 100 machines. These ma-chines are expensive, ranging in price from $50,000 to over$14 million per tool. Frequently, groups of identical ma-chines process lots in parallel, thereby forming a toolgroup.The economic necessity to reduce capital spending dictatesthat such expensive machines be shared by all lots requiringthe particular processing operation(s) provided by the ma-chine, even though they may be at different stages of theirmanufacturing process flow. In fact, a given part may visit
30 J Sched (2008) 11: 29–47
a toolgroup many times as part of its process flow; this iscalled re-entrant flow. This results in a manufacturing en-vironment that is different in several ways from both tra-ditional flow shops as well as job shops. The main con-sequence of this re-entrant flow is that wafers at differentstages in their manufacturing cycle have to compete witheach other for the same machines. The manner in which thiscompetition is resolved has a clear impact on wafer fab ob-jectives.
Furthermore, the nature and duration of the various oper-ations in a wafer fab process flow differ significantly. Someoperations require 30 minutes or less to process a lot of 25wafers, while others may require over twelve hours. Manyof these long operations involve batch processing of lots andit is not uncommon for one-third of all wafer fab operationsto involve batch processing. Batch processing machines tendto off-load multiple lots (1 to 6) onto tools that are capableof processing only one lot at a time. This leads to the forma-tion of long queues in front of these serial (non-batch) toolsand ultimately a non-linear flow of products in the factory.The probabilistic occurrence of unplanned tool failures re-sults in a great deal of variability inherent in the time a lotspends in process in the wafer fab. This variability preventsaccurate prediction of production cycle times, resulting inlonger lead-time commitments. There are some wafer fabmachines, such as ion implanters, that require significantsequence-dependent setups. If not scheduled appropriately,these tools can become wafer fab bottlenecks. In order tounderstand the scheduling approaches currently being usedin the semiconductor industry, a survey instrument was cre-ated and sent to each of the Semiconductor Research Corpo-ration and International Sematech member companies. Thesurvey was designed to ask specific questions regarding thetypes of scheduling methodologies currently implemented,the limitations of these methodologies, and the needs for fu-ture generation scheduling systems. In total, 16 respondentsfrom 14 companies participated in the survey, representingwafer fabs from Europe, Asia, and North America.
Survey results indicate that many dispatching systems arein place within wafer fabs, most of which have been in-stalled for more than five years (Fowler and Pfund 2001).These systems are considered “satisfactory” in that benefitsare being received, but the majority of survey respondentsbelieve that more benefits are possible. Specifically, respon-dents indicate that better scheduling/dispatching rules, testenvironments, and reporting tools are needed. The surveyasked for the top three objectives used in wafer fabs today.The top three responses were cycle time, factory throughput,and on-time delivery. Maximizing factory throughput is sim-ilar to minimizing makespan, while TWT can be thought ofas a surrogate measure for OTD. Compared to the results ofthe 1994 Sematech survey, cycle time and OTD have gainedsignificant importance in wafer fabs (Neacy et al. 1994). The
primary goal of this research effort is to develop a solutionapproach that provides good performance for makespan, av-erage cycle time, and TWT for semiconductor wafer fabs.We use the well known Shifting Bottleneck approach as thebasis for our approach.
2 Related work
2.1 The shifting bottleneck heuristic
The disjunctive graph formulation and the Shifting Bottle-neck procedure to solve the job shop scheduling problemto minimize the makespan (Jm||Cmax in the notation α|β|γof Graham et al. 1979) was first proposed by Adams et al.(1988). Since then research has been focused on the algo-rithmic improvement of the procedure (papers that focus onthis include Dauzère-Pérès and Lasserre 1993, Balas et al.1995, and Balas and Vazacopoulos 1998) and also on ex-tending the Shifting Bottleneck procedure to more compli-cated objectives and augmenting the job shop environmentwith features that commonly occur in practice. Ovacik andUzsoy (1992) use an adapted Shifting Bottleneck procedurefor the scheduling of semiconductor testing operations. Theyincluded sequence dependent setups and used the maximumlateness as an objective (Jm|rj , sjk|Lmax). Ivens and Lam-brecht (1996) and Schutten (1998) discuss the extension ofthe disjunctive graph formulation to accommodate practicalfeatures such as due-dates, release dates, setup times, trans-portation times, parallel machines, and beginning inventory.
Pinedo and Singer (1999) develop a disjunctive graph for-mulation and used the Shifting Bottleneck procedure to min-imize total weighted tardiness (Jm|rj |∑wjTj ). Wafer fabsare modeled as complex job shops by Mason et al. (2002),who extend the classical job shop work of Pinedo and Singer(1999) to develop a disjunctive graph formulation and modi-fied Shifting Bottleneck heuristic (MSBH) for the wafer fabscheduling problem which we represent as FJc|rj , sjk,p-batch, recrc|∑wjTj . Mason et al. (2002) account for tool-groups consisting of multiple identical machines in a givenwork center which perform the same function (which is whythe environment is a flexible job shop represented as FJc),batch processing tools (p-batch), different arrival times ofjob (rj ), sequence-dependent setups (sjk), and recirculatingproduct flow (recrc); all of these are key features that char-acterize manufacturing in wafer fabs.
2.2 Multicriteria scheduling
Multicriteria scheduling research arose from the need to ad-dress real world scheduling problems, which seldom havea single objective function. A schedule that is good for oneobjective function may in fact be quite poor for another. De-cision makers must carefully evaluate the trade-offs involved
J Sched (2008) 11: 29–47 31
in considering several different criteria in practical schedul-ing applications.
Multicriteria problems can be considered in many waysand therefore it is important to point out the nature of op-timization being performed. Consider, for example, a bicri-teria scheduling problem with objectives of minimizing γ1
and γ2. If X is a feasible schedule, then let X(γ1) and X(γ2)
be the γ1 and γ2 objective function values of X. While ourdiscussion is on bicriteria problems, the ideas are easily ex-tended to problems with more criteria.
Sometimes the decision-maker has a priori informationregarding the nature of optimization to be performed. Forinstance, criteria γ1 may be of primary importance and γ2
of secondary interest. In other words, a solution best in γ2
may be desired among all solutions that are best in γ1. Thisis called hierarchical or lexicographic optimization. In othercases, the decision-maker may have a composite linear func-tion of the form Fl(γ1, γ2) = αγ1 + (1 − α)γ2, 0 ≤ α ≤ 1in mind that needs to be minimized. The weighted sumtranslates multiple objectives into a single objective valuefor a proposed schedule. In this way, alternative schedulescan be compared easily using only a single objective or fit-ness value. Some difficulty can be experienced in setting theweighting factors in practice, primarily due to the dimen-sionality of the objective function criteria in practice. Caremust be taken to ensure the scale of γ1 (e.g.,
∑wjCj ) does
not dominate γ2 (e.g., Cmax). Recently, Kacem et al. (2002)introduced a homogenization approach for dealing with thispotential scale difference. Cochran et al. (2003) discuss sev-eral schemes for weighting the objectives.
A more complex problem is to generate the set of Paretooptimal or non-dominated points for the decision-maker tochoose from. In a bicriteria context, a schedule X is calledPareto optimal or non-dominated if there exists no other fea-sible schedule X′′ such that X′′(γ1) ≤ X(γ1) and X′′(γ2) ≤X(γ2) where at least one of the inequalities is strict. The de-cision maker can now choose from this set of non-dominatedsolutions, the schedule that is most preferred. This approachis called the a posteriori approach and is generally the mostdifficult.
For comprehensive surveys on multicriteria schedulingwe refer the reader to Foote et al. (1988), Nagar et al. (1995),T’kindt and Billaut (2006) and for a more recent study toHoogeveen (2005). Of other papers, we refer here only tothe papers that involve the job shop environment. Very fewresearchers have dealt with multi-criteria scheduling in jobshops. The complexity of the problem has a major role toplay in this. Esquivel et al. (2002) and Kacem et al. (2002)investigate the generation of Pareto optimal schedules inclassical and flexible job shops. Iima et al. (1999) and Itohet al. (1993) consider the minimization a function that in-volves all the objectives under study in classical job shops.Balas et al. (1998) address the job shop scheduling problem
with deadlines (J |d̃| ∈ (Cmax, Tmax) in the terminology ofT’kindt and Billaut 2006). This can be viewed as a bicrite-ria problem involving minimax objectives. The ∈-constraintapproach involves generating the set of Pareto optimal solu-tions by solving a series of subproblems in which one cri-terion is optimized while not exceeding a certain prefixedvalue of the other criterion. Recently, Balas et al. (2005) con-sider the same problem but with the inclusion of sequencedependent setups (J |d̃, sjk| ∈ (Cmax, Tmax)).
In this paper, we consider multicriteria scheduling prob-lem in which we combine makespan (Cmax), average cycletime (
∑Cj ), and TWT into a single aggregation function.
Our aggregation function, however, is different from the lin-ear combination of objectives described earlier. We use, in-stead, the desirability function to aggregate the objectives.We assume the decision maker has a prefixed goal/targetvalue and an upper/worst-case value for every criterion. Wealso assume the decision-maker, just as in the linear com-bination case, is aware of the priorities on the objectives.The complex job shop environment has only been studiedfor TWT (Mason et al. 2002). Here we present a solutionmethodology for the multicriteria optimization of makespan,average cycle time, and TWT in a complex job shop.
Before we proceed with describing our approach, we notethat minimizing any of the three criteria considered indi-vidually in a complex job shop is strongly NP-hard (via re-duction from the classical job-shop case), and therefore themulticriteria problem involving the three criteria is stronglyNP-hard as well. Our approach, therefore, focuses on thedevelopment of a heuristic approach that is computation-ally feasible for the complex job shop environment. We alsonote that while approaches other than the Shifting Bottle-neck heuristic (enumerative approaches such as branch andbound algorithms, and meta heuristics) have been used forclassical job shops, these approaches are difficult to apply,given the additional complicating features of the complexjob shop (Mason et al. 2005). We therefore choose to buildon the MSBH of Mason et al. (2002) to extend it for themulticriteria problem.
3 Aggregation using the desirability function
The desirability function approach in optimizing multiplecriteria of interest was originally suggested by Derringer andSuich (1980). The approach transforms each objective intoa value between 0 and 1. Thus, each criterion is convertedinto an individual desirability function δi that varies overthe range zero to one. If is outside the user’s defined accept-able range, then δi = 0. However, if yi meets the goal, thenδi = 1.
In our research, all three objectives (makespan, cycletime, and TWT) are to be minimized. Let U be the maxi-mum allowable value for the response and let G be the goal
32 J Sched (2008) 11: 29–47
Fig. 1 The desirability function for measure i (Myers and Mont-gomery 1995)
value for yi . When minimizing the response yi, δi = 1 ifyi < G. Further, if G ≤ yi ≤ U ,
δi =(
U − yi
U − G
)zi
. (1)
Otherwise, if , then δi = 0. In (1), zi is a real number knownas the weight on the desirability function. When zi = 1 foreach objective i, the desirability function is linear (Fig. 1).Choosing zi > 1 places more emphasis on being close to thegoal value, while setting 0 < zi < 1 decreases importanceon proximity to the goal value.
Once the individual desirabilities have been calculated,the combined desirability D, which is to be maximized, iscomputed as the geometric mean of the individual desirabil-ities:
D = (δ1δ2 · · · δm)1/m. (2)
In (2), m is the total number of responses. For our research,m = 3, as we represent the desirabilities of makespan, cycletime and TWT as δcmax, δct, and δtwt. Each desirability valueδi in our experiments will have goal value Gi and maximum(upper) limit Ui , ∀i ∈ {C max,CT,TWT}.
It is important to link the properties of a solution that isoptimal with respect to a desirability function, and its con-nection to the classical multicriteria idea of a Pareto optimalor non-dominated solution. From the definition and discus-sion of the desirability function, it is clear that the optimalsolution depends upon the goals (G values) and upper tar-gets (U values) assumed for each criterion. We have listedbelow two specific cases below with regard to this. Whilethese cases highlight situations where optimality with re-spect to the desirability function and Pareto optimality maynot always match with each other, they also show that thesedifferences are more due to the framing of the problem withrespect to upper and lower limits than any inherent issues inthe desirability approach.
1. The way the desirability function is structured, any so-lutions that are over the pre-specified upper limit (for
a minimization problem) for any of the criteria are as-signed a value of 0. If the upper limit is fairly strict(aggressive), this could exclude non-dominated solutionsthat are at the ends of the efficient frontier. But in prac-tice this simply means that these solutions not acceptablebecause they perform poorly for at least one of the crite-ria. Thus, a non-dominated solution may not be optimalfrom a desirability point of view as certain constraints onindividual criteria have to be met.
2. Consider now the set of feasible solutions X with non-zero desirability values. Thus, for each solution in X, thecriteria values will be strictly less than the upper limits.Let x∗ be a dominated solution in X. Let NDx∗ be theset of non-dominated solutions that dominate x∗. Also,let x∗(j) represent the value of criterion j for solutionx∗ and Gj represent the goal or target value for criterionj . Then, for a given set of desirability weights, x∗ is anoptimal solution with respect to the desirability functionif: There exists no criterion j and no x∗∗ in NDx∗ suchthat x∗∗(j) <= Gj <= x∗(j).
This situation arises because the desirability functiondoes not distinguish between solutions that are below Gj
for criterion j : it assigns them a value of 1 even thoughthere are differences in criteria values. This reflects theidea that a “satisfaction level” for that criterion has beenmet. We note that while x∗ is optimal for the desirabil-ity function (despite being dominated), there exists a so-lution(s) in NDx which is (are) non-dominated and also(alternately) optimal. Also, as a special case if we wereto set Gj to 0 (or to really low value), x∗ would no longerbe optimal for the desirability function. This leads to ourdecision for choice of Gj values in our methodology(see Sect. 4.1).
4 Methodology
We first provide a brief overview of the Shifting Bottle-neck heuristic. Using the disjunctive graph representation,the SB procedure of Adams et al. (1988) decomposes theJm||Cmax problem into multiple instances of the 1|rj |Lmax
problem (‘subproblems’). The subproblems are solved ac-cording to some specified ‘subproblem solution procedure’(SSP) (a heuristic or an exact procedure depending on thecomputational requirements), and then evaluated in termsof a specified performance or ‘machine criticality’ mea-sure. A computational study of machine criticality mea-sures and sub-problem solution procedures is provided inHoltsclaw and Uzsoy (1996). The ‘most critical’ machine isthen scheduled at each iteration of the procedure.
The MSBH procedure of Mason et al. (2002) builds onthe SB procedure. It decomposes the complex job shopscheduling problem into individual toolgroup scheduling
J Sched (2008) 11: 29–47 33
instances (each toolgroup scheduling instance is a sub-problem). The toolgroups (or the set of identical parallelmachines) are then scheduled using a sub-problem solutionprocedure (SSP), and are then evaluated based on a machinecriticality measure (MCM). The most critical toolgroup isthen identified and scheduled in that iteration. Since our goalis to develop an approach that takes into account multiplecriteria, we use the desirability function aggregation of thethree objectives in our sub-problem solution procedure aswell as our machine criticality measure. We now describethe details of our approach.
The MSBH of Mason et al. (2002) is used for analyzingthe FJc|rj , sjk , p-batch, recrc|∑wjTj problem. We nowpresent the steps of the MSBH:
1. Let M denote the set of all m toolgroups. Initially theset M0 the set of toolgroups that have been sequenced orscheduled is empty.
2. Form and solve the subproblems for each toolgroup i ∈M\Mo (SSP level)
3. Identify the critical or bottleneck toolgroup k ∈ M\Mo
(MCM level)4. Sequence tool group k using the subproblem solution
from Step 2. Set M0 ∪ {k}.5. Re-optimize the schedule for each toolgroup m ∈ M0
considering the newly added disjunctive arcs for tool-group k.
6. If M = M0 stop. Otherwise, go to step 2.
At the SSP level (Step 2), each toolgroup is scheduledusing some SSP, and an objective function evaluation is ob-tained each time a proposed toolgroup schedule is insertedinto the underlying disjunctive graph. At the MCM level(Step 3), each toolgroup’s objective function is then usedto determine the most critical toolgroup in the current it-eration of the MSBH. At both the SSP and MCM levels,we propose to use the desirability function of Derringer andSuich (1980), which seeks to optimize a single aggregationfunction that combines several different objective functions.We combine makespan, cycle time (average flow time), andTWT into a single desirability function, which is then opti-mized.
4.1 SSP level
We use the Apparent Tardiness Cost with Setups and ReadyTimes (ATCSR) heuristic of Gadkari (2003) to schedule jobson each toolgroup at the SSP level. The ATCSR is a com-posite dispatching rule. It combines four different priorityrules—Weighted Shortest Processing Time (WSPT), LeastSlack, Shortest Setup, and Ready Time—into a single func-tion. Except for the WSPT rule, all other rules are raised toan exponent so that they appropriately discount the index
value for a job. In the ATCSR heuristic, an index Ij (t, l) iscalculated for every unscheduled job at time t as follows:
Ij (t, l) = wj
pj
exp
(−max(dj − pj − max(rj , t),0)
k1p̄
)
× exp
(
− slj
k2s̄
)
exp
(
−max(rj − t,0)
k3p̄
)
, (3)
where wj is job j ’s weight or priority, and rj is the readytime of job j with dj and rj set by finding the critical pathof the disjunctive graph. We do not use the dk
i,j variables ofPinedo and Singer (1999) as they require increased compu-tations and need to be calculated at each subproblem. This isespecially true for the large models we consider in our exper-iments. Further, slj is the setup time incurred when changingfrom job l to job j, p̄ is the average processing time of allremaining jobs, and s̄ is the average setup time. The threescaling parameters k1, k2, and k3 determine the relative im-portance of the exponential terms in relation to each otherand to the WSPT term. The heuristic works as follows: wefirst identify the machine j that is available to process jobsthe earliest (t denotes the time at which the machine is avail-able). Next, we calculate the Ij (t, l) value for all unsched-uled jobs and schedule, at time t on machine j , the job thathas the highest Ij (t, l) value. The time on the machine isupdated and the procedure is repeated.
ATCSR has traditionally been used only for the TWT ob-jective (Lee and Pinedo 1997 and Gadkari et al. 2007). How-ever, some of its components may be beneficial to the otherobjectives such as makespan and total completion time con-sidered in this research. In Table 1 a check signifies a posi-tive contribution by the ATCSR function’s priority rule com-ponent to the corresponding objective function criterion.
Varying the scaling parameters and therefore varying therelative importance of the different terms in ATCSR couldlead to good multicriteria schedules. As mentioned beforethe multicriteria complex job shop problem we consider inthis paper is strongly NP-hard. Even at the subproblem so-lution procedure level, multicriteria problems are generallystrongly NP-hard. It is therefore difficult to make a com-ment on how close the solutions generated by our techniqueare to being Pareto optimal or non-dominated. However, itcan be easily shown that the solutions that will be pickedat the SSP level using the desirability function will be thenon-dominated amongst the schedules that are generated by
Table 1 Components of the ATCSR rule and the criteria they impact
WSPT Least slack Shortest setup Ready time
Makespan � �CT � � �TWT � � � �
34 J Sched (2008) 11: 29–47
varying the scaling parameters of ATCSR. Using ATCSR inthis manner for multicriteria purposes is based on Balasub-ramanian et al. (2006) that empirically explore the perfor-mance of a composite dispatching rule similar to the ATCSRfor single machine bicriteria scheduling.
At the SSP level, the scaling parameters are varied us-ing a grid search approach in order to generate a wide rangeof schedules for subsequent consideration by the desirabil-ity function for the three objectives of interest. We use fivedifferent values for each scaling parameter and thus test 125different combinations. Parameter k1 is incremented from0.1 to 2.1 in steps of 0.4; k2 is incremented from 0.1 to 1.1in steps of 0.2; and k3 is incremented from 0.001 to 0.011in steps of 0.002. These values are chosen based on someempirical pilot runs; the values also draw upon research byChen et al. (2007a), in which parameterization of the com-posite dispatching rules is discussed at length.
Let l represent the total number of k1, k2, and k3 scal-ing parameter combinations evaluated over the grid spaceat the SSP level. Schedule i(i = 1 . . . l) is characterized byits corresponding objective function values for the three per-formance metrics of interest, Cmax(i),CT(i), and TWT(i).Further, let Cmax(min)(Cmax(max)),CT(min)(CT(max)), andTWT(min)(TWT(max)) denote the corresponding grid space’sminimum (maximum) objective function values over alll schedules evaluated at the SSP level. Let D(max) =maxi=1..lD
(i), where D(i) denotes the combined desirabil-ity of schedule i. We identify the schedule correspondingto D(max) as S∗. Finally, let rCmax, rct, and rtwt signify thedesirability weights for makespan, cycle time, and TWT,respectively.
Procedure SetGoals below determines the upper and goalvalues for each objective of interest by generating l sched-ules over the scaling parameter grid space for the toolgroupcurrently under study in the SSP. Then, Procedure Find-MostDesirable identifies the most desirable schedule withinthe set of l generated schedules.
Procedure SetGoalsFor i = 1 to l
Create schedule i using the i-th combination ofsmoothing parameters k1, k2, and k3 on thesubproblem.
Record Cmax(i),CT(i), and TWT(i).If α(i) < α(min) Then α(min) = α(i),
∀α ∈ {Cmax,CT,TWT}If α(i) > α(max) Then α(max) = α(i),
∀α ∈ {Cmax,CT,TWT}Next i
Goal value Gα = α(min), ∀α ∈ {Cmax,CT,TWT}Upper value Uα = α(max), ∀α ∈ {Cmax,CT,TWT}Procedure FindMostDesirableD(max) = 0
For i = 1 to l
Calculate D(i) using (1) and (2)If D(i) > D(max) Then
D(max) = D(i)
S∗ = i
End IfNext i
The upper (goal) limit is thus set to the worst (best) valueobserved for each objective over the l different schedulesconsidered at the SSP level. Clearly, the upper and goal lim-its could be fixed, pre-determined values, as knowledge ofa particular toolgroup and its performance may be known apriori in a real world setting, thereby making it easier to de-cide upon appropriate values for these variables. However,in the more general framework that is proposed in this re-search, it is intuitive to equate the worst result observed asthe upper limit for a given performance, and the best resultobserved as the goal value.
Considering the fact that combined desirability is thegeometric mean of all component desirabilities, we mandatethat the schedule with the worst (highest) objective functionvalue over all l schedules will not result in a component de-sirability of zero. Clearly, the schedule that performs worstfor one objective may not necessarily perform poorly for theother objectives of interest. Alternatively, we assign a desir-ability value of 0.0001. Letting any δi = 0 causes the com-bined desirability of schedule iD(i) to equal 0, which dis-qualifies schedule i for selection even when its performancefor other objectives may be quite good.
4.2 MCM level
At the MCM level, the critical toolgroup is identified andthen scheduled. One way of identifying the critical tool-group is to determine each toolgroup’s contribution to theoverall complex job shop’s TWT (“MCM-TWT”). Anothertechnique suggested by Pinedo and Chao (1999) considersthe deviation of job completion times if machine i werescheduled at the current iteration (“MCM-PC”):
n∑
k=1
wk(C′′k − C′
k) exp
(− (dk − C′′
k
)+
K
)
. (4)
In (4), C′k(C
′′k ) denotes the completion time of the job k
before (after) machine i is scheduled and K is a scaling pa-rameter. While the MCM-PC approach considers job com-pletion time deviations, it does not lend itself easily to theinclusion of multiple objectives using the desirability ap-proach. In addition, the K value used can dramatically af-fect which toolgroup is identified as critical at a given it-eration, depending on the scheduler’s sensitivity to job duedate. Therefore, we consider only MCM-TWT as our base
J Sched (2008) 11: 29–47 35
criticality measure and use the desirability approach to blendobjectives other than TWT into it.
Before the desirability function can be used at the MCMlevel, the upper limits and goal values for each of the threeobjectives of interest must be determined. We employ a pro-cedure similar to the one used at the SSP level. The only dif-ference is that at the MCM level, we are interested in iden-tifying and scheduling the toolgroup with the least (lowest)combined desirability value. Since the desirability value ag-gregates three different criteria, a low desirability value fora given toolgroup indicates that it is the most critical withrespect to all the criteria under consideration and thereforeshould be scheduled first.
We consider two different approaches for identifying thecritical toolgroup at the MCM level. First, each toolgroupschedule’s impact on the makespan, cycle time, and TWT ofthe entire complex job shop is assessed when identifying thecritical toolgroup (“Global MCM”) by inserting the sched-ule of the toolgroups into the disjunctive graph. Alterna-tively, the critical toolgroup can be identified using only SSPlevel performance metrics (“Local MCM”) (i.e., do not con-sider the toolgroup’s impact on the rest of the complex jobshop). Our goal in the proposition of these two approachesis to test whether a difference is noticeable in the global andlocal approaches. In a practical setting, if the MSBH proce-dure were to be used for scheduling, the computation timefor the Local MCM approach for large problem sizes wouldbe less that the time for the Global MCM approach. But intu-itively, it would seem that the Global MCM approach wouldreflect the critical machine more accurately, since it takesinto account conflicts between jobs in the entire wafer fab.Regardless of which MCM approach is used, the end resultof this step of the MSBH is the scheduling of the criticaltoolgroup via the insertion of arcs into the correspondingproblem’s disjunctive graph.
5 Testing and experimentation
Table 2 shows the different combination of approaches at theSSP and MCM levels that we investigate in this paper. Ap-proach 1 is the MSBH of Mason et al. (2002) that seeks tominimize TWT. The other five approaches use some combi-nation of desirability (“Des”) and TWT minimization. Weemploy the naming convention “SSP-γ _MCM-ω” to de-scribe an approach that uses γ ∈ {TWT, Des} at the SSPlevel and ω ∈ {TWT, Des(Local), Des(Global)} at the MCMlevel. For example, Approach 6 in Table 2 corresponds toSSP-Des_MCM-Des(Global). We also compare the six ap-proaches in Table 2 with the following pure dispatching-based approaches: Critical Ratio (CR), Earliest Due Date(EDD), and First In First Out (FIFO).
Table 2 Different approaches
No. SSP level MCM level
1 Only TWT Only TWT
2 Only TWT Desirability (Local MCM)
3 Only TWT Desirability (Global MCM)
4 Desirability Only TWT
5 Desirability Desirability (Simple MCM)
6 Desirability Desirability (Global MCM)
Fig. 2 The Minifab model (El Adl et al. 1996; Mason et al. 2002)
5.1 Experimental testbed
We examine two different complex job shop models in ourexperimental testbed. The first model, the “Minifab” modelof El Adl et al. (1996), is perhaps the most succinct repre-sentation of a wafer fab in the open literature. The Minifabconsists of three toolgroups and two job types that requirereentrant flow during their processing (Fig. 2). Toolgroup 1consists of two batch-processing machines with maximumbatch size of three jobs, operating in parallel (e.g., a dif-fusion oven). Toolgroup 2 consists of two identical serialprocessing machines operating in parallel (e.g., a photolitho-graphy stepper), while Toolgroup 3 consists of a single ma-chine characterized by sequence-dependent setups (e.g., anion implanter).
We first consider 20-job static instances of the Mini-fab wherein rj = 0 ∀j . There are two part types in thismodel: 10 jobs of Part Type A and 10 jobs of Part TypeB. The weights (priorities) of the jobs, regardless of parttype, are generated using a discrete uniform distributionwj ∼ DU[1,100]. The due date for job j is assigned us-ing the notion of a flow factor fj used to represent somemultiple of job j ’s theoretical (raw) processing time RPTj :dj = rj + fj RPTj , where rj = 0 in the Minifab instances.In order to generate reasonable values for fj , we exam-ine a single static instance using FIFO dispatching at alltoolgroups. From each job’s completion time Cj we esti-mate fj = (Cj −rj )/RPTj . Let fmax = maxj fj and fmin =minj fj . Since FIFO is independent of due-date and weightsettings, only one instance is necessary to obtain fmax andfmin values. For our experiments involving the MSBH,we generate each flow factor fj uniformly over the range[fmin, fmin +fmax/2]. We generate 20 different instances ofthe Minifab model, each with its own unique flow factor and
36 J Sched (2008) 11: 29–47
job weight values, using common random numbers (Lawand Kelton 2000) across the scheduling approaches to betterdiscriminate between each approach. In our initial Minifabexperiments, we focus on optimizing Cmax and TWT, disre-garding CT in an attempt to illustrate the difference betweeneach approach’s performance more clearly.
The second model we examine is based on TestbedDataset 1 of Fowler et al. (1995). The full dataset, whichcontains 83 toolgroups, is reduced by Mason et al. (2005)to an 11-toolgroup factory containing all bottleneck tool-groups of the original dataset with purchase prices in excessof $100,000 (“Modified Testbed Dataset 1”). Of the 11 tool-groups in Modified Testbed Dataset 1 (MTD1), three con-tain batch-processing machines, while two toolgroups arecharacterized by sequence-dependent setups. Two productstypes exist in MTD1: Product 1 and Product 2. Product 1requires 73 processing steps while the second product in-volves 97 steps. Twenty-five jobs of each product type areconsidered, but unlike the Minifab model, rj ≥ 0 ∀j . As wasthe case with the Minifab experiments, wj ∼ DU[1,100],while fj ∼ U [1,1.3] for the MTD1 instances under study.All three objectives of interest (Cmax, CT, and TWT) are in-vestigated for the MTD1 experimental instances.
5.2 Desirability function weights
Previously, we discussed the procedure for setting the upperand goal limits for the three different objectives at the MCMand SSP levels. However, we still must determine how de-sirability weights zα will be set for α ∈ (Cmax,CT,TWT).Since the three weights are blended together into a singleobjective function, their ratios relative to each other are im-portant. Experimentation with these weights involves testingthe weight values between zero and one, subject to the con-straint that
∑α zα = 1 (Myers and Montgomery 1995 and
Dabbas et al. 2003), i.e., this is a mixture experiment. Ta-ble 3 describes the desirability weight settings used in ourexperimentation. Note that the first combination of weightsin both the 2-criteria (part a) and 3-criteria (part b) optimiza-tion cases represents the typical TWT optimization associ-ated with the MSBH of Mason et al. (2002). These weightsetting combinations are obtained from an augmented sim-plex centroid design of either 2 or 3 variables in a mixtureexperiment (Myers and Montgomery 1995).
In the presentation of our results for the two models,we assume initially that the decision maker has placedequal emphasis on all the objectives being considered (i.e.,zCmax = zct = ztwt = 0.333). However, as the desirabilityfunction used to choose either the schedule at the SSP levelor the critical toolgroup at the MCM level is a just a heuristicprocedure, it is necessary to test a number of possible com-binations of weights to get a “good” final solution. If weperform an exhaustive search using all desirability weight
Table 3 Desirability weight settings
Cmax TWT
1 0
0 1
0.5 0.5
0.67 0.33
0.33 0.67
0.83 0.17
0.17 0.83
(a) 2-criteria weights
Cmax CT TWT
1 0 0
0 1 0
0 0 1
0.5 0.5 0
0.5 0 0.5
0 0.5 0.5
0.33 0.33 0.33
0.67 0.17 0.17
0.17 0.67 0.17
0.17 0.17 0.67
(b) 3-criteria weights
Fig. 3 Structure of the simulation and testing environment (Rose et al.2002)
combinations, the MSBH must be run each time a combina-tion of weights is tested at the SSP or MCM levels. Whilethe computational effort associated with running the MSBHfor the Minifab model is insignificant, it is extremely highfor the MTD1 model that contains 50 jobs and more than70 processing steps for each product type.
To counter this problem, we use the simulation environ-ment developed by Rose et al. (2002). The structure of theenvironment is shown in Fig. 3. The main purpose of thesimulation environment is to emulate the behavior of a realwafer fabrication facility under different schedules providedby the Shifting Bottleneck heuristic. The environment al-lows for jobs arriving continuously over time in a waferfab and the MSBH develops schedules for pre-defined timeintervals. In essence this method is similar to that used bySinger (2001), which decomposes large job shops instancesinto smaller instances that fit in time windows and appliesthe Shifting Bottleneck procedure independently each timewindow. In the MTD1 instances considered in this paper(where the number of jobs to be scheduled is fixed) weuse the Rose et al. (2002) environment as a temporal de-
J Sched (2008) 11: 29–47 37
composition to reduce computational effort. The MSBH isused to schedule the complex job shop every four hours un-til all jobs have completed their processing. Only the un-completed processing steps of both ready and in-processjobs are considered for scheduling during this time horizon.This rolling horizon-based decomposition procedure consid-erably reduces the computational effort involved with invok-ing the MSBH.
6 Experimental results
6.1 Bicriteria optimization for minifab model
For each of the 20 experimental instances, we use theseven different combinations of desirability weights in Ta-ble 3(a) at both the SSP and MCM levels. For the Mini-fab model, pilot runs suggested that SSP-TWT_MCM-Des (Global), SSP_TWT-MCM-Des (Simple), and SSP-TWT_MCM-TWT produced the same results over all 20instances (i.e., the sequence in which the toolgroups werescheduled in the MSBH was the same for all three ap-proaches). This is not surprising, as the Minifab containsonly three toolgroups. Toolgroup 3, which contains onlyone machine and is subject to sequence-dependent setups(Fig. 2) is always determined to be the most critical tool-group, followed by toolgroup 1 (the batch-processing tool-group), then toolgroup 2. Thus, the impact of using desir-ability at the MCM level is not significant for the Minifabmodel. The MTD1 model with its 11 toolgroups providesgreater potential for measuring the impact of desirability atthe MCM level.
However, using desirability at the SSP level producedsignificantly different results for the Minifab model in-stances. As the SSP-level results are independent of the ap-proach used at the MCM level, the approaches to be com-pared reduce to SSP-TWT and SSP-Des. The former rep-resents the first combination in Table 3(a), while the latterrepresents the “best” of the remaining six combinations ofdesirability weights. To determine the best combination, weuse the desirability function again, but now externally whenthe complex job shop has been fully scheduled and all ob-jective functions have been realized.
Let TWT(best) denote the best (i.e., lowest) value ofTWT obtained from using the following three dispatch-ing rules: CR, EDD, and FIFO. As these dispatchingrules are expected to perform poorly for TWT as theydo not explicitly consider job due dates and/or weights,we use TWT(best) as our upper bound on TWT. There-fore, let GCmax = Cmax(min),Gtwt = TWT(min),UCmax =Cmax(max), and Utwt = TWT(best). At the final output levelwe assume initially that the decision maker places equalemphasis on both objectives (zCmax = ztwt = 0.5). Table 4
Fig. 4 Solutions in the objective space (I) for instance 1 of the Minifabmodel. Numbers adjacent to each point indicate the weight of totalweighted tardiness criterion
shows the combined, Cmax, and TWT desirabilities foreach of the 20 Minifab model instances for three differ-ent scheduling approaches (SSP-TWT, SSP-Des (with thecombination of weights being chosen via pilot runs as de-scribed above), and SSP-Des* (the schedule generated usingzCmax = ztwt = 0.5)) and the three competing dispatchingrules (CR, EDD, and FIFO). The bold number in each rowis the best desirability value of the corresponding instance.
It is clear that while SSP-TWT is superior in terms ofTWT performance, its Cmax desirability is very poor whencompared to CR and SSP-Des. Therefore, its combined de-sirability is not good. CR and FIFO have high Cmax desir-ability, but their respective TWT desirabilities over all 20instances are poor. SSP-Des performs reasonably well forboth objectives, and therefore has the best combined de-sirability for the 20 Minifab problem instances. SSP-TWTshares the best desirability with SSP-Des in only one ofthe 20 instances, instance 17. A comparison between SSP-Des and SSP-Des* reveals that SSP-Des has a desirabilityroughly 9% better than SSP-Des*. Further, the former per-forms more consistently than the latter (see standard devi-ation results). However, SSP-Des is computationally moreexpensive due to the exhaustive desirability weight search.Therefore, if a quick solution that is “good” in both objec-tives is desired, SSP-Des* should be used for Minifab prob-lem instances.
Figures 4 and 5 show objective space plots of totalweighted tardiness and makespan for two experimentalMiniFab instances. All solution points that have been la-beled by a fraction between 0 and 1 are those obtained byusing the different weight combinations at the SSP levellisted in Table 4. The fraction indicates the weight of to-tal weighted tardiness used in the desirability function. Theweight of makespan, of course, is the difference between1 and the weight of total weighted tardiness. In Fig. 4 it ispossible to see the tradeoff between total weighted tardi-ness and makespan: while the SSP-TWT solution produces
38 J Sched (2008) 11: 29–47
Tabl
e4
Min
ifab
mod
el:d
esir
abili
tyva
lues
for
the
appr
oach
esw
ithw
eigh
tsr t
wt=
0.5
and
r cm
ax=
0.5
for
the
com
bine
dde
sira
bilit
y
Com
bine
dde
sira
bilit
yM
akes
pan
desi
rabi
lity
TW
Tde
sira
bilit
y
Inst
ance
SSP-
TW
TSS
P-D
esSS
P-D
es*
CR
ED
DFI
FOSS
P-T
WT
SSP-
Des
SSP-
Des
*C
RE
DD
FIFO
SSP-
TW
TSS
P-D
esSS
P-D
es*
CR
ED
DFI
FO
10.
5995
0.86
580.
8217
0.10
380.
0084
0.00
840.
3697
0.77
040.
7839
1.00
000.
7124
0.74
830.
9721
0.97
310.
8614
0.01
080.
0001
0.00
01
20.
8643
0.97
290.
9688
0.09
160.
0015
0.00
150.
7471
1.00
001.
0000
0.75
410.
0215
0.75
411.
0000
0.94
650.
9386
0.01
110.
0001
0.00
01
30.
8621
0.86
400.
8640
0.11
050.
0015
0.00
150.
7432
0.78
800.
7880
1.00
000.
0214
0.75
011.
0000
0.94
720.
9472
0.01
220.
0001
0.00
01
40.
9062
0.92
130.
9213
0.01
000.
0965
0.09
650.
8486
0.85
010.
8501
0.99
670.
8144
0.85
010.
9678
0.99
850.
9985
0.00
010.
0114
0.00
01
50.
1343
0.87
350.
7362
0.01
000.
0932
0.09
320.
0180
0.95
350.
6739
0.99
340.
8087
0.84
691.
0000
0.80
020.
8043
0.00
010.
0108
0.00
01
60.
7056
0.95
810.
9581
0.00
990.
0174
0.01
740.
4979
0.98
170.
9817
0.98
170.
0283
0.99
091.
0000
0.93
520.
9352
0.00
010.
0107
0.00
01
70.
6149
0.93
080.
7623
0.00
870.
0015
0.00
150.
3781
0.92
250.
7331
0.76
250.
0219
0.76
251.
0000
0.93
910.
7927
0.00
010.
0001
0.01
10
80.
1718
0.97
920.
9792
0.00
960.
0898
0.08
980.
0307
1.00
001.
0000
0.92
900.
7290
0.94
080.
9614
0.95
880.
9588
0.00
010.
0111
0.00
01
90.
5360
0.95
270.
9527
0.00
920.
0014
0.00
140.
2974
1.00
001.
0000
0.84
460.
0185
0.84
460.
9662
0.90
760.
9076
0.00
010.
0001
0.01
06
100.
5672
0.86
770.
8178
0.00
860.
0152
0.01
520.
3217
0.77
070.
7840
0.74
630.
0213
0.74
631.
0000
0.97
690.
8530
0.00
010.
0108
0.00
01
110.
8845
0.92
100.
8924
0.00
890.
0160
0.01
600.
8008
0.94
210.
7963
0.79
630.
0229
0.79
630.
9770
0.90
041.
0000
0.00
010.
0112
0.00
01
120.
1510
0.84
130.
1300
0.10
690.
0080
0.00
800.
0228
0.80
860.
0228
1.00
000.
6346
0.69
861.
0000
0.87
520.
7413
0.01
140.
0001
0.00
01
130.
1376
0.96
740.
8583
0.00
940.
0046
0.00
460.
0189
0.99
130.
8863
0.88
630.
2161
0.89
041.
0000
0.94
410.
8311
0.00
010.
0001
0.01
07
140.
1335
0.91
420.
6689
0.00
940.
0941
0.09
410.
0178
0.96
060.
8628
0.87
720.
8416
0.87
721.
0000
0.87
010.
5185
0.00
010.
0105
0.00
01
150.
7947
0.88
280.
8489
0.11
280.
0014
0.00
140.
6315
0.96
800.
8471
0.99
650.
0185
0.84
461.
0000
0.80
510.
8507
0.01
280.
0001
0.00
01
160.
7171
0.88
110.
8521
0.01
000.
0015
0.00
150.
7501
1.00
000.
7262
0.99
480.
0214
0.75
010.
6855
0.77
631.
0000
0.00
010.
0001
0.01
14
170.
8451
0.84
510.
8220
0.00
900.
0015
0.00
150.
7945
0.79
450.
6757
0.80
210.
0231
0.80
210.
8989
0.89
891.
0000
0.00
010.
0001
0.01
16
180.
1516
0.91
660.
8205
0.00
830.
0077
0.00
770.
0230
1.00
000.
6967
0.68
220.
5925
0.69
131.
0000
0.84
010.
9662
0.00
010.
0001
0.01
15
190.
5274
0.88
790.
8595
0.00
870.
0015
0.00
150.
2781
1.00
000.
7561
0.76
330.
0220
0.76
331.
0000
0.78
830.
9771
0.00
010.
0001
0.01
08
200.
6139
0.94
310.
8503
0.00
870.
0015
0.00
150.
3769
0.95
200.
7501
0.75
010.
0214
0.75
011.
0000
0.93
420.
9638
0.00
010.
0001
0.01
02
Ave
rage
0.54
590.
9093
0.81
920.
0332
0.02
320.
0232
0.39
830.
9227
0.78
070.
8778
0.28
060.
8049
0.97
150.
9008
0.89
230.
0030
0.00
390.
0045
Std.
Dev
0.29
230.
0436
0.18
020.
0428
0.03
640.
0364
0.31
060.
0885
0.20
810.
1125
0.34
770.
0790
0.07
170.
0675
0.11
780.
0052
0.00
530.
0055
J Sched (2008) 11: 29–47 39
Table 5 Minifab model:performance of the approachesfor a range of decision-maker’spriorities
(Cmax,TWT) SSP-Des SSP-TWT CR EDD FIFO SSP-DM*
(0,1) 7 13 0 0 0 13
(1,0) 17 0 3 0 0 15
(0.5,0.5) 19 1 0 0 0 18
(0.83,0.17) 18 2 0 0 0 14
(0.17,0.83) 20 0 0 0 0 20
(0.67,0.33) 20 0 0 0 0 16
(0.33,0.67) 18 2 0 0 0 14
avg. 17 2.57 0.43 0 0 15.71
Fig. 5 Solutions in the objective space (II) for instance 4 of the Mini-fab model. Numbers adjacent to each point indicate the weight of totalweighted tardiness criterion
the best total weighted tardiness, its makespan value is notvery good. The solutions generated by the different weightcombinations generate a variety of solutions, which produceslightly worse total weighted tardiness values (but still sig-nificantly better than CR or EDD solutions; FIFO resultsare not shown as they are really poor) but improve on themakespan. Figure 5 shows a graph where the using the de-sirability approach generates a solution better in both to-tal weighted tardiness and makespan than SSP-TWT. Thesetwo different instances shown in Figs. 4 and 5 roughly clas-sify the nature of solutions for all 20 of the instances: eithera trade-off exists between the two objective values or SSP-Des generates a solution better in both objective values.
The results for combined desirability in Table 4 as-sume that both performance measures are equally important.However, decision makers often have differing priorities foreach objective. To consider this reality, we record the num-ber of times the most desirable solution is generated by eachscheduling approach for a given a set of desirability weights(Table 5). In Table 5, (a, b) signifies the decision makerplaces a% priority on makespan and b = (1 − a)% priorityon TWT. It is clear that SSP-Des generates the most numberof desirable solutions for the 20 instances irrespective of thedecision-maker’s priorities except for the case of a = 0. In
this case, SSP-TWT produces the best solutions for 65% ofthe instances, while SSP-Des produces the best solutions forthe remaining seven instances. For all other cases in whicha > 0, SSP-Des produces the best solutions for a minimumof 85% of the Minifab problem instances. In addition weuse the notation SSP-DM* to represent the solution that usesthe same combination of desirability weights as that used bythe decision-maker, unlike the exhaustive search procedureof SSP-Des. For the each of the decision-maker’s 7 weightcombinations considered, we count the number of times thatSSP-DM* produces a better solution than SSP-TWT, CR,EDD, and FIFO. These results are shown in the last columnof Table 5. SSP-DM* produces a better solution than SSP-TWT, CR, EDD, and FIFO in more than 80% of the twentycases. It was also observed that the desirability of the so-lution generated by SSP-DM* (obtained from a single runof the MSBH) for each of the decision-maker’s priorities isvery close to the desirability of the solution generated fromSSP-Des (obtained by exhaustive search).
6.2 3-criteria optimization for MTD1 model
The MTD1 model with 11 toolgroups provides an oppor-tunity to test both the MCM approaches proposed in thispaper and the approaches where desirability optimization atthe MCM level is used in conjunction with desirability op-timization at the SSP level. For setting the upper and goallimits for the desirability calculations, we use the same tech-niques employed for the Minifab model described above.Initially, we again assume equal importance of all threeobjectives. For SSP-TWT_MCM-Des (Global) and SSP-TWT_MCM-Des (Local), we test the nine of the 10 differentcombinations of weights in Table 3(b), as the first combina-tion is again excluded as it represents the SSP-TWT_MCM-TWT case. Our goal in this testing is to identify the com-bination of γ (approach at the SSP level) and ω (approachat the MCM level) that produce the most desirable results.Similarly, nine different desirability weight combinationsare tested for SSP-Des_MCM-TWT to identify the most de-sirable schedules.
40 J Sched (2008) 11: 29–47
Clearly, the three approaches mentioned above are com-putationally intensive considering the exhaustive desirabil-ity weight search. Therefore, we also check the perfor-mance of these approaches when only one combination ofweights (i.e., equal importance to all objectives) is usedat the SSP and MCM levels. These cases are denotedby SSP-TWT_MCM-Des (Global)*, SSP-TWT_MCM-Des(Local)*, and SSP-Des_MCM-TWT*. For SSP-Des_MCM-Des (Global) and SSP-Des_MCM-Des (Local) desirabilityoptimization is used at both levels. Ideally, therefore, an ex-haustive search would mean 100 (10 × 10) runs would berequired before all weight combinations can be explored.Even a set of 5 desirability combinations at each level wouldrequire 25 runs for each instance before the best can be cho-sen. Since the number of runs is prohibitive, for these twoapproaches, at both SSP and MCM levels we use the com-bination (0.333,0.333,0.333). This combination of equalweight importances is also motivated by our initial pilot runcomparisons in which we assume equal emphasis is placedon all objectives.
Table 6 shows both the average and standard deviationof the combined desirability values of each approach whileTables 7, 8, and 9 show the individual desirability valuesof makespan, cycle time, and total weighted tardiness, re-spectively, for each of the 20 instances of the MTD1 model,assuming that all objectives have equal importance to thedecision maker (i.e., zCmax = zct = ztwt = 0.333). It is clearthat SSP-Des_MCM-TWT, SSP-Des_MCM-TWT*, SSP-Des_MCM-Des (Global), and SSP-Des_MCM-Des (Local)perform well for all three measures. In fact, the TWT-desirability of SSP-TWT_MCM-TWT, though much betterthan that of CR on average, is still significantly lower thanthe approaches that use desirability at the SSP level. The re-sults for combined desirability follow a similar pattern: SSP-Des_MCM-TWT has the highest mean combined desirabil-ity, closely followed by SSP-Des_MCM-Des (Global) andSSP-Des_MCM-Des (Local). Further, all three approacheshave low standard deviations, an indication of their consis-tency. The approaches that use desirability optimization onlyat the MCM level perform well in one or two instances, butfall short in their combined mean desirabilities. Therefore,using desirability merely to choose a critical machine doesnot appear to be effective consistently unless the scheduleimplemented at the SSP level is also chosen using a desir-ability approach.
It is also clear from Table 6 that SSP-TWT_MCM-Des (Global)*, SSP-TWT_MCM-Des (Local)*, and SSP-Des_MCM-TWT* have solution qualities close to theircomputationally intensive counterparts (i.e., SSP-TWT_MCM-Des (Global), SSP-TWT_MCM-Des (Local),and SSP-Des_MCM-TWT). It is particularly encouragingto notice the performance of SSP-Des_MCM-TWT*, SSP-Des_MCM-Des (Global), and SSP-Des_MCM-Des (Local),
as their solutions are generated by a single MSBH run re-quiring 25 minutes on a 2.0 GHz Pentium IV computer with1 GB of RAM. On average, these three “single pass” ap-proaches’ desirabilities are within 6% of the desirability ofSSP-Des_MCM-TWT, which requires 225 minutes to gen-erate the best solution for a single problem instance.
Table 10 compares the performance of SSP-TWT_MCM-TWT (i.e., the original MSBH of Mason et al. 2002) withthe approach that produced D(max) for each of the 20 MTD1problem instances via a performance ratio. For each probleminstance, objective α ∈ (Cmax,CT,TWT) resulting from theapproach that produced D(max) is divided by the correspond-ing objective of the SSP-TWT_MCM-TWT approach forthe same problem instance. This performance ratio allowsus to analyze the quality of the most desirable schedule interms of percentage gain/loss with respect to each objectiveα. Surprisingly, the average gain in TWT over the TWT-based MSBH is 28%. Further, the average gain in Cmax per-formance is 4–5%, while cycle time is decreased by 2%on average using a desirability-based approach. Therefore,in the case of the MTD1 model, the desirability-based ap-proaches generated schedules superior in all three objectivesas compared to the MSBH. Although the gains in Cmax andcycle time are relatively small, these small gains have con-tributed significantly to the large gains in TWT performancesince these measures are not independent of each other.
Table 11 displays the performance of the desirability ap-proaches for the MTD1 model when the decision maker mayor may not place equal importance on each objective. In or-der to evaluate the performance of each approach for dif-ferent priorities weightings, we count the number of timesthe best solution is produced by each approach for a givenset of priorities. Table 11 shows 10 possible combinations ofdecision maker priorities (a, b, c) and the number of timesthat each approach produces the best solution for a given(a, b, c) weighting scheme. Under this weighting scheme,the decision maker places a% weight on Cmax, b% weighton cycle time, and c = (1−a−b)% weight on TWT. Experi-mental results suggest that SSP-Des_MCM-TWT is the rulethat performs well most often for a wide range of decisionmaker priorities, followed by SSP-Des_MCM-Des (Local)and SSP-Des_MCM-Des (Global). Additionally, it was alsoobserved, just as in the Minifab cases, that for all approachesthat use desirability at the SSP level, if only the correspond-ing decision-maker’s weights were used (therefore requiringonly 1 run of the MSBH), the desirability of the solutionswere close in most cases to the desirability of the solutionsgenerated by SSP-Des (which requires exhaustive search).Therefore, the clear conclusion from these experimental re-sults is that it is important to implement schedules at thetoolgroup level that take into account multiple criteria.
J Sched (2008) 11: 29–47 41
Tabl
e6
Des
irab
ility
Res
ults
for
the
MT
D1
Mod
elA
cros
s20
Prob
lem
Inst
ance
sfo
rr C
max
=r c
t=
r tw
t=
0.33
3
Inst
ance
SSP-
TW
TSS
P-D
esSS
P-D
esSS
P-T
WT
SSP-
TW
TSS
P-T
WT
SSP-
TW
TSS
P-D
esSS
P-D
esC
RE
DD
FIFO
MC
M-T
WT
MC
M-T
WT
MC
M-T
WT
*M
CM
-Des
(G)
MC
M-D
es(G
)*M
CM
-Des
(L)
MC
M-D
es(L
)*M
CM
-Des
(G)
MC
M-D
es(L
)
10.
759
0.75
90.
044
0.04
10.
026
0.03
90.
039
0.88
30.
982
0.41
80.
041
0.01
8
20.
851
0.96
80.
968
0.94
40.
889
0.89
70.
897
0.88
70.
937
0.04
40.
431
0.01
1
30.
800
0.97
80.
978
0.83
40.
816
0.92
00.
774
0.99
50.
984
0.39
30.
043
0.01
1
40.
730
0.97
90.
970
0.85
90.
797
0.86
50.
781
0.95
30.
995
0.04
30.
403
0.01
1
50.
721
0.97
30.
973
0.87
40.
776
0.62
80.
469
0.99
90.
925
0.34
10.
038
0.01
0
60.
599
0.99
20.
929
0.74
50.
609
0.99
90.
938
0.97
30.
864
0.37
80.
044
0.01
1
70.
041
1.00
00.
989
0.04
10.
041
0.03
80.
021
0.95
80.
959
0.36
60.
040
0.01
8
80.
040
1.00
01.
000
0.65
80.
040
0.04
00.
040
0.89
90.
894
0.37
00.
041
0.01
0
90.
041
0.87
60.
876
0.89
60.
878
0.90
00.
900
1.00
00.
917
0.38
40.
039
0.01
0
100.
699
0.96
20.
937
0.76
30.
681
0.71
60.
645
0.98
30.
984
0.35
60.
039
0.01
1
110.
826
0.99
80.
990
0.79
30.
793
0.72
60.
671
0.90
80.
966
0.04
10.
337
0.01
8
120.
799
1.00
00.
970
0.90
10.
808
0.66
20.
662
0.86
30.
739
0.35
70.
039
0.01
7
130.
041
0.99
80.
997
0.04
10.
041
0.04
20.
042
0.97
10.
952
0.35
80.
041
0.01
1
140.
733
0.93
30.
933
0.87
80.
799
0.95
20.
712
0.89
30.
870
0.03
90.
349
0.01
1
150.
039
0.95
00.
946
0.04
30.
022
0.04
20.
042
0.95
20.
973
0.04
20.
396
0.01
5
160.
823
0.94
40.
926
0.83
60.
656
0.82
70.
827
0.97
80.
931
0.40
40.
041
0.01
1
170.
798
0.99
20.
987
0.85
20.
852
0.83
60.
825
0.99
80.
915
0.04
10.
351
0.01
0
180.
717
0.90
20.
902
0.67
40.
640
0.81
70.
817
0.95
50.
963
0.04
10.
356
0.01
8
190.
772
1.00
01.
000
0.94
40.
885
0.89
50.
832
0.97
40.
896
0.37
00.
043
0.01
1
200.
722
0.86
80.
857
0.77
70.
777
0.78
70.
787
0.90
81.
000
0.03
90.
304
0.01
0
Ave
0.57
80.
954
0.90
80.
670
0.59
10.
631
0.58
60.
947
0.93
20.
241
0.17
10.
013
Std.
Dev
0.32
30.
061
0.20
80.
331
0.33
90.
362
0.34
20.
045
0.06
10.
168
0.16
50.
003
42 J Sched (2008) 11: 29–47
Tabl
e7
MT
DM
odel
:Mak
espa
n-D
esir
abili
tyof
the
appr
oach
es
Inst
ance
SSP-
TW
TSS
P-D
esSS
P-D
esSS
P-T
WT
SSP-
TW
TSS
P-T
WT
SSP-
TW
TSS
P-D
esSS
P-D
esC
RE
DD
FIFO
MC
M-T
WT
MC
M-T
WT
MC
M-T
WT
*M
CM
-Des
(G)
MC
M-D
es(G
)*M
CM
-Des
(L)
MC
M-D
es(L
)*M
CM
-Des
(G)
MC
M-D
es(L
)
10.
791
0.79
10.
889
0.74
20.
198
0.64
40.
644
0.92
70.
948
1.00
00.
844
0.56
1
20.
773
0.99
80.
998
0.84
00.
824
0.81
00.
810
0.98
80.
926
0.95
20.
993
0.12
5
30.
619
0.98
60.
986
0.75
40.
714
0.91
10.
705
1.00
00.
978
0.93
60.
902
0.11
8
40.
692
0.98
90.
986
0.78
40.
771
0.79
60.
762
0.91
31.
000
0.85
70.
873
0.11
6
50.
731
0.93
00.
930
0.85
50.
741
0.50
80.
263
1.00
00.
949
0.81
50.
653
0.11
1
60.
558
1.00
00.
918
0.83
30.
674
0.99
70.
855
0.98
90.
887
0.81
50.
947
0.12
3
70.
775
1.00
00.
992
0.78
40.
781
0.63
20.
102
0.95
50.
940
0.85
40.
759
0.56
4
80.
718
1.00
01.
000
0.72
70.
712
0.74
40.
744
0.93
90.
909
0.84
80.
847
0.10
9
90.
754
0.78
90.
789
0.82
00.
827
0.92
50.
925
1.00
00.
956
0.80
40.
723
0.11
3
100.
630
0.94
00.
880
0.81
50.
662
0.68
80.
609
0.97
01.
000
0.74
60.
685
0.12
1
110.
844
0.99
50.
980
0.78
00.
780
0.71
40.
662
0.88
31.
000
0.85
30.
786
0.58
2
120.
865
1.00
00.
937
0.93
40.
795
0.83
80.
838
0.87
30.
844
0.83
80.
750
0.46
3
130.
837
1.00
00.
996
0.82
90.
822
0.81
20.
812
0.98
50.
953
0.76
90.
778
0.11
9
140.
782
0.94
40.
943
0.96
10.
927
0.86
30.
397
0.95
01.
000
0.72
30.
761
0.12
3
150.
661
0.91
30.
911
0.83
50.
119
0.79
70.
797
0.86
31.
000
0.78
90.
854
0.30
1
160.
711
0.85
50.
810
0.75
80.
679
0.76
50.
765
0.95
71.
000
0.91
10.
762
0.11
5
170.
752
0.97
70.
964
0.87
40.
874
0.84
90.
772
1.00
00.
985
0.80
00.
789
0.11
2
180.
567
0.93
60.
936
0.46
30.
379
0.73
90.
739
0.87
01.
000
0.82
20.
806
0.55
6
190.
673
1.00
01.
000
0.97
00.
809
0.87
80.
823
0.95
80.
887
0.86
90.
939
0.11
8
200.
661
0.84
70.
830
0.69
10.
691
0.68
90.
689
0.83
11.
000
0.72
20.
555
0.10
5
Ave
0.72
00.
944
0.93
40.
803
0.68
90.
763
0.66
80.
943
0.95
80.
836
0.80
00.
233
Std.
Dev
0.08
70.
071
0.06
50.
109
0.21
30.
131
0.20
80.
053
0.04
70.
073
0.10
50.
191
J Sched (2008) 11: 29–47 43
Tabl
e8
MT
Dm
odel
:CT-
Des
irab
ility
ofth
eap
proa
ches
Inst
ance
SSP-
TW
TSS
P-D
esSS
P-D
esSS
P-T
WT
SSP-
TW
TSS
P-T
WT
SSP-
TW
TSS
P-D
esSS
P-D
esC
RE
DD
FIFO
MC
M-T
WT
MC
M-T
WT
MC
M-T
WT
*M
CM
-Des
(G)
MC
M-D
es(G
)*M
CM
-Des
(L)
MC
M-D
es(L
)*M
CM
-Des
(G)
MC
M-D
es(L
)
10.
987
0.98
70.
940
0.94
90.
926
0.94
90.
949
0.97
61.
000
0.97
70.
820
0.10
6
20.
977
0.97
80.
978
1.00
00.
982
0.98
80.
988
0.93
70.
959
0.88
80.
938
0.10
6
30.
992
0.99
20.
992
0.98
20.
985
0.99
70.
953
0.99
60.
975
0.91
20.
856
0.10
4
40.
951
1.00
00.
997
0.98
80.
971
0.99
00.
967
0.97
50.
984
0.92
70.
919
0.10
9
50.
898
1.00
01.
000
0.94
30.
923
0.88
40.
867
0.99
80.
961
0.82
90.
828
0.10
1
60.
941
0.97
80.
961
0.94
50.
937
1.00
00.
995
0.96
00.
922
0.91
10.
873
0.10
5
70.
874
1.00
00.
996
0.90
00.
881
0.87
50.
884
0.98
30.
996
0.88
90.
821
0.10
2
80.
911
1.00
01.
000
0.92
10.
893
0.88
50.
885
0.94
40.
963
0.90
80.
838
0.10
1
90.
929
0.98
20.
982
0.98
20.
974
0.96
20.
962
1.00
00.
946
0.89
50.
838
0.10
1
100.
937
1.00
00.
994
0.93
10.
921
0.91
80.
909
0.97
80.
975
0.90
20.
884
0.10
9
110.
899
1.00
00.
998
0.90
10.
901
0.87
90.
880
0.93
90.
957
0.82
40.
826
0.09
9
120.
931
1.00
00.
995
0.97
00.
938
0.95
70.
957
0.92
60.
878
0.86
40.
816
0.10
1
130.
853
1.00
00.
997
0.82
70.
826
0.88
80.
888
0.97
40.
954
0.90
60.
909
0.10
6
140.
896
0.95
50.
955
0.93
40.
920
1.00
00.
987
0.94
10.
868
0.79
40.
855
0.10
5
150.
924
0.99
00.
983
0.95
20.
924
0.91
50.
915
1.00
00.
955
0.90
80.
921
0.10
8
160.
978
1.00
01.
000
0.97
10.
954
0.95
70.
957
0.97
90.
944
0.93
60.
925
0.10
5
170.
922
1.00
00.
998
0.93
10.
931
0.92
80.
940
0.99
70.
946
0.85
80.
841
0.10
2
180.
902
0.90
50.
905
0.91
20.
921
0.90
20.
902
1.00
00.
954
0.86
20.
847
0.10
3
190.
935
0.99
90.
999
0.96
90.
954
0.96
00.
946
0.98
20.
939
0.87
80.
865
0.10
2
200.
895
0.92
80.
925
0.92
80.
928
0.93
40.
934
0.96
91.
000
0.84
80.
842
0.10
0
Ave
0.92
60.
985
0.98
00.
942
0.92
90.
938
0.93
30.
973
0.95
40.
886
0.86
30.
104
Std.
Dev
0.03
80.
026
0.02
80.
040
0.03
70.
044
0.03
90.
024
0.03
50.
043
0.03
90.
003
44 J Sched (2008) 11: 29–47
Tabl
e9
MT
DM
odel
:TW
T-D
esir
abili
tyof
the
appr
oach
es
Inst
ance
SSP-
TW
TSS
P-D
esSS
P-D
esSS
P-T
WT
SSP-
TW
TSS
P-T
WT
SSP-
TW
TSS
P-D
esSS
P-D
esC
RE
DD
FIFO
MC
M-T
WT
MC
M-T
WT
MC
M-T
WT
*M
CM
-Des
(G)
MC
M-D
es(G
)*M
CM
-Des
(L)
MC
M-D
es(L
)*M
CM
-Des
(G)
MC
M-D
es(L
)
10.
5611
0.56
110.
0001
0.00
010.
0001
0.00
010.
0001
0.76
101.
0000
0.07
450.
0001
0.00
01
20.
8148
0.92
880.
9288
1.00
000.
8684
0.90
180.
9018
0.75
470.
9252
0.00
010.
0860
0.00
01
30.
8329
0.95
670.
9567
0.78
380.
7730
0.85
720.
6902
0.99
041.
0000
0.07
120.
0001
0.00
01
40.
5920
0.94
910.
9290
0.81
730.
6757
0.82
030.
6467
0.97
191.
0000
0.00
010.
0818
0.00
01
50.
5712
0.99
120.
9912
0.82
940.
6823
0.55
050.
4523
1.00
000.
8673
0.05
860.
0001
0.00
01
60.
4091
0.99
770.
9096
0.52
560.
3578
1.00
000.
9716
0.96
950.
7896
0.07
270.
0001
0.00
01
70.
0001
1.00
000.
9774
0.00
010.
0001
0.00
010.
0001
0.93
730.
9422
0.06
440.
0001
0.00
01
80.
0001
1.00
001.
0000
0.42
500.
0001
0.00
010.
0001
0.81
930.
8156
0.06
550.
0001
0.00
01
90.
0001
0.86
690.
8669
0.89
270.
8415
0.81
940.
8194
1.00
000.
8525
0.07
890.
0001
0.00
01
100.
5802
0.94
720.
9390
0.58
550.
5184
0.58
170.
4840
1.00
000.
9765
0.06
730.
0001
0.00
01
110.
7424
1.00
000.
9919
0.70
900.
7090
0.61
000.
5189
0.90
210.
9410
0.00
010.
0589
0.00
01
120.
6330
1.00
000.
9799
0.80
720.
7066
0.62
220.
6222
0.79
560.
5446
0.06
280.
0001
0.00
01
130.
0001
0.99
410.
9987
0.00
010.
0001
0.00
010.
0001
0.95
440.
9488
0.06
560.
0001
0.00
01
140.
5630
0.90
130.
9018
0.75
470.
5974
1.00
000.
9205
0.79
640.
7577
0.00
010.
0655
0.00
01
150.
0001
0.94
840.
9441
0.00
010.
0001
0.00
010.
0001
1.00
000.
9652
0.00
010.
0790
0.00
01
160.
8008
0.98
490.
9797
0.79
520.
4349
0.77
170.
7717
1.00
000.
8547
0.07
740.
0001
0.00
01
170.
7331
1.00
000.
9987
0.75
990.
7599
0.74
110.
7741
0.99
850.
8217
0.00
010.
0653
0.00
01
180.
7198
0.86
620.
8662
0.72
410.
7517
0.81
840.
8184
1.00
000.
9342
0.00
010.
0659
0.00
01
190.
7319
1.00
001.
0000
0.89
680.
8981
0.84
940.
7409
0.98
340.
8629
0.06
610.
0001
0.00
01
200.
6365
0.83
060.
8193
0.73
170.
7317
0.75
760.
7576
0.92
941.
0000
0.00
010.
0599
0.00
01
Ave
0.49
60.
936
0.89
90.
602
0.51
50.
585
0.54
50.
928
0.89
00.
041
0.02
80.
000
Std.
Dev
0.31
10.
103
0.21
80.
334
0.33
20.
367
0.35
00.
090
0.11
10.
035
0.03
60.
000
J Sched (2008) 11: 29–47 45
7 Model parameters
Our model and solution approach assume information on thedue-dates and weights of the jobs. It also requires specifyingthe scaling parameters for the ATCSR heuristic at the SSP
Table 10 MTD model: Comparison of SSP-TWT_MCM-TWT withthe best of the desirability approaches
Instance Ratios
MkSp CT TWT
1 0.9703 0.9964 0.7904
2 0.9930 0.9940 0.9206
3 0.9525 0.9988 0.8689
4 0.9584 0.9884 0.8761
5 0.9543 0.9714 0.5509
6 0.9556 0.9844 0.7510
7 0.9474 0.9665 0.5322
8 0.9500 0.9739 0.6144
9 0.9594 0.9792 0.7803
10 0.9664 0.9851 0.7687
11 0.9616 0.9696 0.6352
12 0.9673 0.9791 0.6614
13 0.9747 0.9664 0.6452
14 0.9908 0.9740 0.6845
15 0.9786 0.9820 0.7230
16 0.9820 0.9939 0.8918
17 0.9569 0.9785 0.7453
18 0.9627 0.9735 0.7450
19 0.9579 0.9811 0.7547
20 0.9368 0.9687 0.6040
Ave 0.9638 0.9802 0.7272
Std. Dev 0.0144 0.0101 0.1114
level and desirability weights that need to be used at bothSSP and MCM levels. In this section, we discuss the settingof these parameters.
7.1 Job due-dates and weights
The due-dates could be vendor’s promises or the customer’srequirements; in either case, these due-dates can be used inour model. Companies also have customers that are moreimportant than others, and this information can be used toset weights for the products such that they reflect the relativeimportance of a product to another.
7.2 Desirability weights
It is important to make a distinction between the weights thedecision maker chooses to assign to each criterion and theweights that are assigned at the SSP level or MCM level.We refer to Sect. 5.2 for a description of how these sets ofweights are identified. While it is intuitive to set the desir-ability weights at the SSP and MCM level to be equal tothe decision maker’s desirability weights, for better solu-tions a more exhaustive search for desirability weight com-binations may be required at the SSP and MCM levels. Pro-vided that there is a simulation model of the fab, it wouldbe reasonable to state that this process would take 1–3 daysof computing time and analysis. We expect that the resultswould have to be reassessed and the experimentation car-ried out again when there are changes in fab with respect totechnology, processes, and product mix.
7.3 Scaling parameters for ATCSR
At the SSP level, we use a grid approach to set the scal-ing parameters. The scaling parameters are dependent on
Table 11 MTD model: Performance of the approaches for a range of decision-maker’s priorities, indicated weights are (Cmax,CT,TWT)
(Cmax,CT,TWT) SSP-TWT SSP-Des SSP-TWT SSP-TWT SSP-Des SSP-Des CR EDD FIFO
MCM-TWT MCM-TWT MCM-Des(G) MCM-Des(L) MCM-Des(G) MCM-Des(L)
(0,0,1) 0 7 1 2 6 4 0 0 0
(0,1,0) 0 7 0 0 4 8 1 0 0
(1,0,0) 0 12 1 2 3 2 0 0 0
(0,0.5,0.5) 0 8 0 1 5 6 0 0 0
(0.5,0,0.5) 0 8 1 2 6 3 0 0 0
(0.5,0.5,0) 0 9 0 1 4 5 1 0 0
(0.33,0.33,0.33) 0 7 0 2 5 6 0 0 0
(0.17,0.17,0.66) 0 6 1 2 8 3 0 0 0
(0.17,0.66,0.17) 0 8 0 1 5 6 0 0 0
(0.66,0.17,0.17) 0 8 0 2 8 2 0 0 0
Average 0 8 0.4 1.5 5.4 4.5 0.2 0 0
46 J Sched (2008) 11: 29–47
the characteristics of the scheduling instance such as due-date tightness, weights, release date range, and setup sever-ity factor. Several different approaches can be used to setthese parameters. Lee and Pinedo (1997) use curve-fittingmethods to set the scaling parameters for a similar compos-ite dispatching rule; they present expressions for the scal-ing parameters in terms of the characteristics of the probleminstance. More recently, regression based approaches havebeen used to set the parameters. Gadkari et al. (2007) presenta regression model to determine the values of the scaling pa-rameters. Chen et al. (2007a) improve upon this approach toreduce significantly the number of experiments needed fordata collection. Finally, Chen et al. (2007b) present an effi-cient approach to determine the robust scaling parameters—i.e., the parameters that work for well for all scheduling in-stances. Such techniques can help find the ranges for thescaling parameters for our problem.
Formulas and values suggested in aforementioned paperscould be used directly, or the analyst could redo the exper-imentation proposed in them. As with desirability weights,this will be part of a pilot study using a fab simulation modeland historical data. The computational effort should be lessthan one day, given that the composite dispatching rule canbe easily implemented and computes schedules quickly. Asbefore, a periodic reassessment of these parameters is war-ranted with changes in the wafer fab with respect to technol-ogy, processes, and product mix.
8 Conclusions and future research
Semiconductor wafer fabrication is a complex process typi-cally requiring hundreds of steps with unique features suchas re-entrant flows, batch machines, and sequence depen-dent setups. These features can be modeled as a complex jobshop. The MSBH heuristic of Mason et al. (2002) considersthe minimization of TWT in a complex job shop. We buildon this and propose a methodology for multicriteria opti-mization using the desirability function. Given the stronglyNP-hard nature of this multicriteria problem, our approachprovides a computationally feasible way of accommodatingmultiple criteria.
We use the desirability approach at two different lev-els of the MSBH, the SSP level and the MCM level, andpropose five new approaches for scheduling complex jobshops. Using two representative complex job shop modelsfrom the literature, we compare our approaches to the origi-nal MSBH (approach “SSP-TWT_MCM-TWT”) as well asthree dispatching rules. Twenty problem instances were gen-erated and analyzed for each of the two representative com-plex job shop models (20-job instances of the Minifab and50-job instances of MTD1). External to the MSBH, we usethe desirability function to compare the schedules generated
by the different competing approaches. Experimental resultsshow that when equal emphasis is placed on all three objec-tives, the desirability approach performs significantly betterthan both the MSBH and dispatching rules. While a tradeoffbetween Cmax and TWT was observed in the Minifab ex-periments, the desirability approaches performed the best inall three objectives in the MTD1 experiments. An importantconclusion from our experimentation is that the use of a de-sirability approach at the SSP level consistently producessuperior results. We are available by email (see correspond-ing author’s contact information) to answer any questionsthat readers may have regarding the algorithms in this paperas well as the experimental results.
In the future, we plan to explore the use of the desirabil-ity approach to approximately generate the efficient frontierfor the complex job shop environment, as this informationcould prove quite useful to decision makers to help to under-stand the inherent trade-off between competing objectives.Further, we will extend the approaches proposed in this pa-per to the practical case wherein dynamic job arrivals arepresent.
Acknowledgements This research was partially supported by theSemiconductor Research Corporation and International Sematechthrough Factory Operations Research Center (FORCe) grant 2001-NJ-880.
References
Adams, J., Balas, E., & Zawack, D. (1988). The Shifting Bottleneckprocedure for job shop scheduling. Management Science, 34,391–401.
Balas, E., Lancia, G., Serafini, P., & Vazacopoulos, A. (1998). Job shopscheduling with deadlines. Journal of Combinatorial Optimiza-tion, 1, 329–353.
Balas, E., Lenstra, J. K., & Vazacopoulos, A. (1995). The one-machineproblem with delayed precedence constraints and its use in jobshop scheduling. Management Science, 41(1), 94–109.
Balas, E., & Vazacopoulos, A. (1998). Guided local search with shift-ing bottleneck for job shop scheduling. Management Science,44(2), 262–275.
Balas, E., Simonetti, N., & Vazacopoulos, A. (2005). Job shop schedul-ing with set-up times, deadlines, and precedence constraints. InProceedings of the 2nd multidisciplinary international conferenceon scheduling: theory and applications (pp. 520–532). New York:MISTA.
Balasubramanian, H., Fowler, J. W., & Pfund, M. E. (2006). Singlemachine bicriteria scheduling using the apparent tardiness costheuristic. In Industrial engineering research conference, Orlando,FL, USA.
Chen, Y., Fowler, J. W., Pfund, M. E., & Montgomery, D. C. (2007a).Methodologies for parameterization of composite dispatchingrules ASUIE-ORPS-2007-019. Industrial Engineering, ArizonaState University. http://ie.fulton.asu.edu/research/workingpaper/wps.php
Chen, Y., Pfund, M. E., Montgomery, D. C., Fowler, J. W., & Callar-man, T. E. (2007b). Robust scaling parameters for compositedispatching rules ASUIE-ORPS-2007-020. Industrial Engineer-ing, Arizona State University. http://ie.fulton.asu.edu/research/workingpaper/wps.php
J Sched (2008) 11: 29–47 47
Cochran, J. K., Horng, S. M., & Fowler, J. W. (2003). A multi-population genetic algorithm to solve multi-objective schedulingproblems for parallel machines. Computers and Operations Re-search, 30, 1087–1102.
Dabbas, R., Fowler, J., Rollier, D., & McCarville, D. (2003). Mul-tiple response optimization using mixture-designed experimentsand desirability functions in semiconductor scheduling. Interna-tional Journal of Production Research, 41, 939–961.
Dauzère-Pérès, S., & Lasserre, J. B. (1993). A modified Shifting Bot-tleneck procedure for the job shop scheduling. International Jour-nal of Production Research, 31, 923–932.
Derringer, G., & Suich, R. (1980). Simultaneous optimization of sev-eral response variables. Journal of Quality Technology, 12, 214–219.
El Adl, M. K., Rodriguez, A. A., & Tsakalis, K. S. (1996). Hierarchicalmodeling and control of re-entrant semiconductor manufacturingfacilities. In Proceedings of the 35th conference on decision andcontrol, Kobe, Japan.
Esquivel, S. C., Ferrero, S. W., & Gallard, R. H. (2002). Parametersettings and representations in Pareto-based optimization for jobshop scheduling. Cybernetics and Systems, 33, 559–578.
Foote, B. L., Ravindran, A., & Lashine, S. (1988). Computationalfeasibility of multi-criteria models of production, planning andscheduling. Computers and Industrial Engineering, 15, 129–138.
Fowler, J., & Pfund, M. (2001). State of the art scheduling survey re-sults. Semiconductor Research Corporation Publication NumberP003240
Fowler, J. W., Feigin, G., & Leachman, R. (1995). Semiconductor man-ufacturing Testbed: data sets. Arizona State University
Gadkari, A., Pfund, M. E., Fowler, J. W., & Chen, Y. Scheduling jobson parallel machines with setup times and ready times, Computersand Industrial Engineering, to appear (2007)
Graham, R. L., Lawler, E. L., Lenstra, J. K., & Rinnooy Kan, A. H.G. (1979). Optimization and approximation in deterministic se-quencing and scheduling: a survey. Annals of Discrete Mathemat-ics, 5, 287–326.
Hoogeveen, H. (2005). Multicriteria Scheduling, European Journal ofOperational Research, 167(3).
Holtsclaw, H. H., & Uzsoy, R. (1996). Machine criticality measuresand subproblem solution procedures in shifting bottleneck meth-ods: a computational study. Journal of the Operational ResearchSociety, 47(5), 666–677.
Iima, H., Hara, T., Ichimi, N., & Sannomiya, N. (1999). Autonomousdecentralized scheduling algorithm for a job-shop schedulingproblem with complicated constraints. In Proceedings of the 19994th international symposium on autonomous decentralized sys-tems (pp. 366-369), Tokyo, Japan.
Itoh, K., Huang, D., & Enkawa, T. (1993). Twofold look-ahead searchfor multi-criterion job shop scheduling. International Journal ofProduction Research, 31(9), 2215–2234.
Ivens, Ph., & Lambrecht, M. (1996). Extending the Shifting Bottleneckprocedure to real-life applications. European Journal of Opera-tional Research, 90, 252–268.
Kacem, I., Hammadi, S., & Borne, P. (2002). Pareto-optimality ap-proach for flexible job-shop scheduling problems: hybridizationof evolutionary algorithms and fuzzy logic. Mathematics andComputers in Simulation, 60, 245–276.
Law, A. M., & Kelton, W. D. (2000). Simulation modeling and analy-sis. Boston: Mcgraw-Hill.
Lee, Y. H., & Pinedo, M. Scheduling jobs in parallel machines with se-quence dependent setup times, European Journal of OperationalResearch, 100, 464–474
Mason, S. J., Fowler, J. W., & Carlyle, W. M. (2002). A modified Shift-ing Bottleneck heuristic for minimizing total weighted tardinessin complex job shops. Journal of Scheduling, 5(3), 247–262.
Mason, S. J., Fowler, J. W., Carlyle, W. M., & Montgomery, D. C.(2005). Heuristics for minimizing total weighted tardiness in com-plex job shops. International Journal of Production Research,43(10), 1943–1963.
Myers, R., & Montgomery, D. (1995). Response surface methodology.New York: Wiley.
Nagar, A., Haddock, J., & Heragu, S. (1995). Multiple and bicriteriascheduling: a literature survey. European Journal of OperationalResearch, 81, 88–104.
Neacy, E., Brown, S., & McKiddie, R. Measurement and improvementof manufacturing capacity (MIMAC) survey and interview results.SEMATECH Technology Transfer #94052374A-XFR (1994)
Ovacik, I. M., & Uzsoy, R. (1992). A Shifting Bottleneck algorithmfor scheduling semiconductor testing operations. Journal of Elec-tronic Manufacturing, 2, 119–134.
Pinedo, M. L., & Chao, X. (1999). Operations scheduling with applica-tions in manufacturing and services. New York: Irwin/McGraw-Hill.
Pinedo, M. L., & Singer, M. (1999). A Shifting Bottleneck heuristicfor minimizing the total weighted tardiness in a job shop. NavalResearch Logistics, 46, 1–17.
Rose, O., Mönch, L., & Sturm, R. (2002). Testing, comparison andimplementation issues. In Proceedings of the 12th internationalconference on flexible automation and intelligent manufacturing,Dresden, Germany.
Schutten, M. (1998). Practical job shop scheduling. Annals of Opera-tional Research, 83, 161–177.
Semiconductor Industry Association (SIA) (2005). The internationaltechnology roadmap for semiconductors. http://public.itrs.net/.
Singer, M. (2001). Decomposition methods for large job shops. Com-puters and Operation Research, 28, 193–207.
T’kindt, V. T., & Billaut, J.-C. (2006). Multicriteria scheduling: theory,models and algorithms (2nd ed.). Berlin: Springer.