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Balancing mixed-model assembly lines to reduce workoverloadSITTICHAI MATANACHAI a & CANDACE ARAI YANO ba The Boston Consulting Group , Bangkok, Thailandb Department of Industrial Engineering and Operations Research , University of California ,Berkeley, CA, 94720-1777, USA E-mail:Published online: 27 Apr 2007.

To cite this article: SITTICHAI MATANACHAI & CANDACE ARAI YANO (2001) Balancing mixed-model assembly lines to reducework overload, IIE Transactions, 33:1, 29-42, DOI: 10.1080/07408170108936804

To link to this article: http://dx.doi.org/10.1080/07408170108936804

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lIE Transactions (2001) 33, 29-42

Balancing mixed-model assembly lines to reduce workoverload

SITTICHAI MATANACHAI 1 and CANDACE ARAI YAN02

I The Boston Consulting Group, Bangkok, Thailand2 Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720-1777. USAE-mail: yano@ieor.berkeley.edu

Received October 1997 and accepted December 1999

We propose a new line balancing approach for mixed-model assembly lines with an emphasis on how the assignment of tasks tostations affects the ability to construct daily sequences of jobs (customer orders) that provide stable workloads (in a minute-to­minute sense) on the assembly line, while also achieving reasonable workload balance among the stations. The issueof short-termworkload stability has received little attention in the assembly line balancing literature. Such stability allows assembly workers tocomplete their tasks without being rushed and thereby contributes to product quality. We propose a new objective for assemblyline balancing that helps to achieve better short-term workload stability and develop a heuristic solution procedure based onfiltered beam search for this new objective. Computational results show that for small problems (which can be solved optimally),this approach provides near optimal solutions, and for larger problems, it provides significantly better results than traditionalassembly line balancing methods.

1. Introduction

An important part of automobile production is the finalassembly process, which historically has been labor­intensive. Labor productivity on the assembly line isdriven largely by how tasks are assigned to stations,commonly referred to as Assembly Line Balancing (ALB).When only a single product is produced, as was the casefor the Model T Ford, the objectives are relatively clear,and they all relate, in some way, to minimizing the totallabor and/or capital cost.

Conventional assembly lines were designed to producea single high-volume model, but since the time of theModel T, product proliferation has been increasing due todiverse customer tastes. For a typical mid-size automobilemodel, for example, a customer may choose among mil­lions of different combinations of features (Weiner, 1985).In many instances, this means that each customer order(or job) is essentially unique. Most automobile manufac­turers currently produce many variations of one or morebasic models on a "mixed-model" assembly line. In thevast majority of cases, the assembly line is paced (i.e., aconstant-speed conveyor) or semi-paced (e.g., sections ofthe line are paced). Such configurations were designed for,and operate well in, single-model environments. Whenthey are used for a variety of products, assembly linebalancing objectives need to go beyond the traditionalgoal of workload balancing.

0740-817X © 2001 "liE"

In this paper, we propose a new line balancing approachfor mixed-model assembly lines. Our focus is on assigningtasks to stations so that: (i) workloads are reasonably wellbalanced; and (ii) it is relatively easy to construct dailysequences of jobs that provide stable workloads(in a minute-to-minute sense) on the assembly line. Theissue of short-term workload stability has received littleattention in the assembly line balancing literature. Suchstability allows assembly workers to complete their taskswithout being rushed and thereby contributes to productquality. Several plant managers in the automobile indus­try have suggested that quality-related implicationsof assembly line balancing and sequencing are as signifi­cant as the labor cost effects, and the latter are, in them­selves, substantial, as we explain below,

To compensate for the usually unstable workloadpatterns that occur in high-variety assembly plants, as­sembly lines are balanced using lower labor utilizationtargets. For example, in plants with extremely highproduct variety, we have observed that some assemblystations facing particularly high variety have 50 to 60%utilization targets, with average utilization targets acrossall stations in the neighborhood of 70 to 80% (versus 85to 90% utilization in a low- to moderate-variety plant).The lower utilization levels provide slack to partiallycompensate for instability of the workloads. By consid­ering how workload instability affects the assembly linewhen deciding task assignments, it is possible to simul-

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30

taneously improve the target utilization levels and thestability of the workloads.

The mixed-model problem is significantly more difficultthan the single-model ALB problem, and has a number offeatures that are qualitatively significantly different. Inparticular, the line can be re-balanced only periodically(usually only once or twice a year), and thus, must berobust enough to handle daily and seasonal changes inthe mix of models (or job types). In single-product set­tings, the model mix does not change so robustness is nota concern.

In this paper, we focus on closed-station, paced lineswith Fixed-Rate Launching (FRL). This is the mostcommon type of assembly line used in the US automobileindustry for the so-called "trim" operations, where partsarc assembled onto the vehicle after the body has beenwelded and painted. The jobs move on a constant speedconveyor and are equally spaced on the line. Because eachstation is closed, the operator at the station cannot moveoutside some specified limits along the assembly line. Theamount of work that remains incomplete when he/shereaches the downstream boundary is referred to as workoverload. When work overload occurs, either: (i) the op­erator must rush to finish his work; (ii) the remainingwork is completed at intermediate repair stations or at theend of the line; (iii) the line is stopped for the operator tofinish the work; or (iv) utility workers are assigned tofinish the work. In all four cases, work overload has ad­verse effects on costs, quality, or both. A reduction inwork overload not only improves efficiency, but alsoimproves the quality of the product, which has a long-runimpact on market share and profitability.

We assume that the number of stations and the cycletime have been specified in advance. The cycle time typ­ically is chosen to provide the desired annual output rate.The specified number of stations is often dictated orconstrained by the existing physical infrastructure (con­vcyors, etc.), but our method can be applied for anynumber of stations that is realistic for the situation.

The objective is to develop a line balancing method­ology that not only performs well on the average, but isalso more robust to daily model mix changes. We ac­complish this by considering the effects of line balancingdecisions on our ability to construct high-quality (lowwork overload) sequences of jobs on a day-to-day basis.

Decisions regarding the physical design of the line alsoaffect one's ability to sequence well. For instance, if thestation lengths are sufficiently long, it is easy to constructa sequence with low work overload. Similarly, if the taskassignments are such that all jobs require similar pro­cessing times, almost any sequence will result in goodperformance. On the other hand, if the station lengths aretoo short or if tasks are poorly assigned, even the bestsequence may cause difficulties. In this paper, we con­centrate on the line balancing aspect of the problem as­suming that the station lengths are given, but our

Afatanachai and Yano

approach is designed to improve performance for anystation length configuration.

In the next section, we present a survey of the literatureon mixed-model assembly line balancing. In Section 3, wepropose a line balancing metric for mixed-model assem­bly lines and explain why it facilitates the construction ofjob sequences with improved workload stability. The newobjective consists of the traditional balancing objective(minimizing the deviation of respective station workloadsfrom the average workload per station) and terms thataim at minimizing two different measures of processingtime diversity. In Section 4, we present a heuristic to solvethe problem with the new objective. Computational re­sults are presented in Section 5 and conclusions appear inSection 6.

2. Literature review

There is a substantial literature on the single-productALB problem, which involves assigning tasks to an or­dered sequence of stations subject to precedence relationsand other constraints. One popular objective is to mini­mize the number of stations for a fixed cycle time (theType I problem), where the emphasis is on reducing laborcosts and space requirements. Another common objectiveis to minimize the cycle time for a fixed number of sta­tions (Type 2 problem) where the emphasis is on in­creasing throughput.

Traditional mixed-integer formulations for the single­product problem are based on the assumptions that eachtask must be assigned to exactly one station, and that taskprocessing times are additive and independent of thestations to which they are assigned. No layout or zoningrestrictions exist for the assignments. Mixed-model ALBis more complicated because the performance of the linecannot be measured by workload balance alone. Someimportant performance metrics depend on the quality ofjob sequences (an issue that does not arise in the single­model context), and the sequences are influenced andconstrained by the line balancing decisions.

There is a large and growing literature on mixed-modelassembly line sequencing. Indeed, a search of the litera­ture led us to over 60 papers on mixed-model sequencing,with the vast majority having been published within thepast few years. (A list is available from the authors uponrequest). Survey articles include Ghosh and Gagnon(1989), Yano and Bolat (1989), Gagnon and Ghosh(1991), Kubiak (1993) and Lima-Fernandes and Groover(1995). Most articles published within the past few yearsare concerned with determining job sequences to supportjust-in-time delivery of component parts.

Although several authors (Bhattacharjee and Sahu,1987; Ghosh and Gagnon, 1989) have pointed out theneed to consider aspects of real-world assembly lines,such as multiple models, much less work has been done

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Balancing mixed-model assembly lines

on the mixed-model ALB problem. There are very fewpapers on this subject, and most of them are based on atransformation of the mixed-model problem into a vari­ant of the single-model problem. These approaches usetask workloads (i.e., average workload per job imposedby a task) in place of task processing times, and the line isbalanced accordingly. One important feature that dis­tinguishes mixed-model from single-model line balancingis the presence of tasks that are common to multiplemodels. In most cases, researchers have assumed thatsuch tasks must be performed on all models at the samestation. There are a number of practical reasons for this,including efficiencies in worker training, parts storagenear the line, and parts delivery. The mixed-model ALBproblem with this constraint and with the objective ofminimizing the idle time for a fixed number of stationswas first formulated as an integer program by Robertsand Villa (1970). Berger et al. (1992) develop an efficientmethod for solving the problem of minimizing the num­ber of stations for a fixed cycle time. van Zante-deFokkert and de Kok (1997) compare several heuristics forminimizing the number of stations. Many procedures forsingle-model ALB can be adapted to solve both versionsof the problem, if one is concerned only with workloadbalance. Macaskill (1972) assumed the work is dividedinto "task groups", where each task groups is essentiallya (task, job type) pair. These task group are assigned tostations. Raouf et al. (1983) consider a slight variant inwhich similar tasks for different models can be groupedand assigned to the same station. Research has also beendone to consider multiple objectives (see Gokcen andErel, 1997).

Thomopoulos (1967, 1970) was among the first to il­lustrate how line balancing decisions affect the quality ofthe achievable job sequences and later he demonstrates,by way of a case example, how to achieve more balancedworkloads across stations for each job type. If it is pos­sible to accomplish this, then this effectively reduces thesequencing problem to a single-station problem. Thereappears to have been a dearth of research on mixed­model ALB that considers the impact on sequencing untilabout 20 years later. The recent work does not considerthe sequencing aspects directly, but instead uses proba­bilistic representations of the processing times to capturethe effects of the model mix.

Erlebacher et al. (1991) develop a model to evaluatetask assignments by treating each job type as a combi­nation of tasks. If there are k tasks assigned to a station,there are 2k possible task combinations at that station,and the distribution of the job processing time at eachstation is determined accordingly. Using this representa­tion of job processing times, they show that, for mixed­model lines, balancing a line based on the expected tasktimes may not necessarily lead to good performance.Rather, the probability that the job processing time willexceed the cycle time must be taken into account.

31

Hsu (1992) develops a balancing methodology formixed-model lines with deterministic task times with theobjective of minimizing the total cost of stations (essen­tially the regular time labor cost) and work overload.First, assuming that the processing times of the tasks arenormally distributed, she finds the limiting probability ofhaving an unfinished job at a single station for a givenstation length. She then extends this single-sta tion modelto the multiple-station case, with the goal of finding taskassignments that minimize the same objective, given a setof station lengths. For the mixed-model case with deter­ministic and stochastic (possibly correlated) processingtimes, she concludes that it is better to group stochastictasks together with as few deterministic tasks as possible,or to group deterministic tasks together with as few sto­chastic tasks as possible.

Erlebacher and Singh (1999) study how to allocate afixed total processing time variance among multiple sta­tions to minimize the total expected work overload. Intheir model, it is assumed that each station faces a ran­dom sequence of cars. They assume that the stationlength is set so that the sojourn time of each job within astation is equal to the cycle time. The main result is that ifthe total variance is less than a critical level, the optimalsolution is to allocate variance equally among the sta­tions. On the other hand, if the total variance is greaterthan the critical level, the optimal solution is to allocateall of the variance to one station (i.e., a spike-shapedallocation). These conclusions result from the convex­concave form of the objective function.

As we can see, to date, researchers have captured theeffects of line balancing decisions on work overload onlyin an approximate way, and have effectively assumed thatjobs are sequenced randomly. In reality, more systematicsequencing decisions are made, so the successive pro­cessing times that operators encounter are not completelyuncorrelated, and hence, cannot be described completelyby the representations used in the literature.

3. New objective function

As mentioned above, the research literature has not yetincorporated the full effect of task assignments on se­quence-related performance into mixed-model line bal­ancing methodologies. Thomopoulos (1970) pointed outthe benefit of balancing the workload across stations foreach model, and we include this effect as one component ofour objective function, which we call between-station di­versity. If the workload for each model is (nearly) balancedacross the stations, the sequencing problem is much easierto solve because a sequence which is good for one station isalso good for the other stations. However, when manytasks are common to multiple models and we require thateach task be assigned to a single station, it may be difficultto balance the workload for each model. We introduce the

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32 Matanachai and Yano

idea of assigning tasks to reduce the diversity of processingtimes among different job types within each station. If thisdiversity is small, any sequence will have low work over­load and the assembly line will be robust to model mixchanges. We include this within-station diversity as anothercomponent of our objective function. The third compo­nent of the objective captures traditionalll'orkioad balanceconsiderations. It is important to point out that the goal ofbalancing the workload across stations for each model(between-station diversity) is not inconsistent with, andindeed, contributes to, overall workload balancing. On theother hand, the goal of reducing the diversity of processingtimes among job types within each station may contributeto a less balanced overall workload, but with the benefit ofreducing work overload.

In addition to the assumptions presented earlier, weassume the following: (i) a single precedence diagram canbe used for all models, although not all tasks are requiredon all job types; (ii) each task is performed (on all jobsthat require it) at a single station, which is commonpractice; and (iii) the number of stations is specified ex­ogenously. Our procedure can be applied to differentnumbers of stations to determine the best alternative.

Before introducing the proposed objective function, weformally define some notation and terminology. A task isthe smallest indivisible processing element. Each productoption (e.g., power seats versus manually-adjusted seats)may consist of several tasks and a task may be common toseveral options. Eachjob type is defined by its combinationof options and an option may be common to several jobtypes. A task's workload is defined as the product of thetask time (expressed as a fraction of the cycle time) and thefraction of jobs that require this task. In our formulationand solution approach, we relate each job type directly toits constituent tasks, but in our computational studies,each job type is defined by its combination of options.

Notation:

nN=r=

WiT={IIy =

number of stations (specified in advance);number of tasks;number of job types;fraction of cars that are of job type y;

{I if job type y requires task j;o otherwise

processing time of task j;workload of task j, expressed as a fraction of thecycle time

tj 2:;=1 Illy lyj;

workload of station i;

total workload = 2:J=1 Wj;average utilization of the line = T In;average processing time per station for a job oftype y

(2:J=1 lyjtj) / n;

S, set of immediate successors of task j;P.i set of immediate predecessors of task j;L, lowest station index to which task j can be as­

signed;U, highest station index to which task j can be as­

signed.

The traditional balancing objective, i.e., minimizing thedeviation of workloads from the average workload, doesnot guarantee low work overload. We can illustrate thisby the following example:

N = 6 tasks;n = 3 stations;{I = 0.7;F = 4 job types;tl = 1,12 = 0.67,13 = 0.75,14 = 0.5,15 = 0.7,t6 = 0.7.Job type I consists of tasks I, 3, and 5.Job type 2 consists of tasks I, 3, and 6.Job type 3 consists of tasks 2, 4, and 5.Job type 4 consists of tasks 2, 4, and 6.IIII = 0.2, mi = 0.2, m3 = 0.3, m4 = 0.3;WI = 0.4,11'2 = 0.4, 11'3 = 0.3, 1~'4 = 0.3,11'5 = 0.35,11'6 = 0.35.

In this example, there are no precedence constraints.Figure I depicts a situation where tasks I and 3 are as­signed to station I, tasks 2 and 4 are assigned to station 2,and tasks 5 and 6 are assigned to station 3. The line isperfectly balanced. However, at station 2, over 50% of thejobs have a processing time greater than the cycle time,which will cause difficulty in sequencing. We cannot avoidsequencing two or more long jobs in a row, and this maycause work overload if the station length is insufficient.

Figure 2 illustrates an alternate assignment with tasks Iand 2 assigned to station I; tasks 3 and 4 assigned tostation 2; and tasks 5 and 6 assigned to station 3. In thiscase, the utilization levels at stations 1,2 and 3 are 0.8, 0.6and 0.7, respectively. Although the workloads are notwell balanced either in the aggregate or by job type, withthis assignment, any sequence of jobs will result in zerowork overload. Thus, an unbalanced line can still provideexcellent results.

Based on the observations of Thomopoulos (1970) andour observations discussed above, we include severalterms in our objective function. The first term is the tra­ditional balancing objective, minimizing the sum of theabsolute deviation of the actual utilization at each stationfrom the average utilization. The second term is designedto address within-station processing time diversity; theobjective is to minimize the sum of the absolute deviationof the actual utilization generated by each job type at eachstation from the overall average utilization level. Finally,the third term is concerned with the between-stationprocessing time diversity for each job type. The objectiveis to minimize the sum, across all job types, of the absolutedeviation of the processing time of a job type at each

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Balancing mixed-model assembly lines 33

Number of jobs Number of jobs Number of jobs

o P 1.7

Station 1

Processingtime

o P 1.17

Station 2

Processingtime

p

Station 3

Processingtime

D job type t mjob type 2 !LI job type 3 • jobtype4

Fig. 1. Assignment with balanced workloads.

station from the average processing time per station ofthat job type, fy• The formulation can be written as:

subject to

n

LXi) = 1 vi.i=l

N

LWjXi) < 1 Vi,j=l

(5)

(4)

Xi) E {a, I} Vi and j.

i

Xi):S; LXvr Vi,j and Vr E~,v=1

Constraints (2) ensure that the total workload assigned toeach station is less than the capacity. Constraints (3) statethat each task must be performed at a single station.Constraints (4) are the precedence constraints, andconstraints (5) ensure that the assignments are binary.Observe that our proposed objective reduces to thetraditional balancing objective when there is one type ofjob to produce.

We weight the first term of the objective function by D,which allows us to adjust the influence of the first termrelative to the entire objective. Observe that the first sumcontains n terms, while the other sums contain n r terms,

(2)

(I)

(3)

n r N

+ LL LlyjtjXi) - P;=1 y=1 j=l

n r N

+ L L L IyjtjXi) - t, ,;=1 y~1 j=1

n N

min DL LWjXi) - P;=1 j=l

Number of jobs Number of jobs Number of jobs

Processingtime

Station t

o 0.5 P 0.75 Processingtime

Station 2

p

Station 3

Processingtime

D job type 1 FiiiI job type 2 f2I job type 3 • job type 4

Fig. 2. Assignment with unbalanced workloads.

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34

and each absolute value is a small fraction. Thus, if r islarge, the second and third terms would dominate theobjective. This is the primary reason for weighting thefirst term by Q. 1n instances where the overall utilizationof the line is high, it may be sensible to use Q > r so as tokeep the maximum station utilization close to the aver­age, which facilitates job sequencing. On the other hand,workload balance is less critical in a lightly loaded sys­tem, so smaller values of Q may be used.

We note that for the within-station diversity (i.e., thesecond) term, we use p rather than I:N=I wjXij = Wi, theworkload for station i, for two reasons.JFirst, the value ofWi cannot be computed until the task assignments forstation i have been fully decided, so one needs to con­struct the complete assignment for station i before stationi's contribution to the objective can be computed. Thisincreases the computational burden. Second, and moreimportant, is that preliminary tests showed that using prather than Wi led to better performance in over 95% ofour preliminary test problems. The reason for this resultappears to be that p is a static workload target, making iteasier to match, whereas Wi changes as tasks or combi­nations of tasks are assigned, or as tasks are interchangedbetween stations (as is done in the improvement stage ofour heuristic).

It is important to point out, however, that our pro­posed objective function does not capture all that wewould like. For example, neither a reduction of between­station diversity nor of within-station diversity is guar­anteed to improve the performance of the line. Followingis a simple example to illustrate this point.

N = 12 tasks;11 = 3 stations;fI = 0.897;r = 4 job types;tl = 0.1, t: = 0.2, t3 = 0.2, 14 = 1.5, ts = 004, 16 = 1.0,t7 = 1.0, Is = 0.2, t9 = 0.7,110 = 0.1, III = 0.6, 112 = 0.6;Job type I consists of tasks I, 2, 3, 5, 7, 8, 9, and 10.Job type 2 consists of tasks I, 2, 3, 5, 9, 10, and 12.Job type 3 consists of tasks I, 2, 3, 4, 5,9, 10, and 11.Job type 4 consists of tasks 1,2,3, 5, and 6./Ill = 0.25,/Il2 = 0.4,m3 = 0.2,m4 = 0.15;WI = 0.1,11'2 = 0.2,11'3 = 0.2,11'4 = 0.3, Ws = 0.4,11'6 = 0.15,11'7 = 0.25, Ws = 0.05,11'9 = 0.595, WIO = 0.085,wu = 0.12, WI2 = 0.24.

In this example, there are no precedence relationsamong tasks. Two different assignments are shown inFig. 3. In assignment I, tasks 1,2, 3, 7, and 12 are as­signed to station I; tasks 4, 5, and 8 are assigned to sta­tion 2; and tasks 6, 9, 10, and II are assigned to station 3.In assignment 2, tasks 2, 4, 7, and 12 are assigned tostation I; tasks I, 3, 5, 6, and 8 are assigned to station 2;and tasks 9, 10, and II are assigned to station 3.

The two assignments are such that the sum of thelongest processing times across all stations is the same for

Matanachai and Yano

both assignments. That is, the minimum line length is thesame for both assignments. The fraction of jobs with longprocessing times (greater than the cycle time) is 0045for assignment 1 and 0.35 for assignment 2. The work­load deviation and the within- and between-station di­versity of processing times of each assignment is shown inTable I.

The workload deviation of assignment I is higher thanthat of assignment 2. Both assignments have the samewithin-station diversity of processing times, but assign­ment I has higher between-station diversity than that ofassignment 2. By examining the various performancemeasures, assignment 2 seems to be superior. We per­formed further analysis by constructing for each assign­ment, optimal sequences (to minimize work overload) for200 jobs, setting the length of each station to the longestprocessing time at that station. (The total line length isthe same in both cases). The optimal sequence was foundfor assignment 1, but due to the complexity of theproblem, we were able to obtain only a lower bound onthe work overload due to assignment 2. This was ob­tained by solving for the optimal sequence at each stationseparately using a generalization of the dynamic pro­gramming approach presented in Yano and Rachama­dugu (1991), then summing the work overload valuesacross all stations. We discovered that the optimal se­quence for assignment I gives zero work overload, whilethe (loose) lower bound on the work overload for as­signment 2 is 0.6, which seems counter-intuitive. It isuseful to point out that the spans of the processing timesat stations I and 3 are greater in assignment 2 than theyare in assignment 1; this may be a reason for thesecounter-intuitive results.

The example above demonstrates that an assignmentthat looks intuitively superior is not necessarily betterbecause it is difficult to fully consider all of the implica­tions on the final job sequence. Real-world problems havelarger numbers of stations, tasks and job types, making iteven more difficult to understand and quantify howmeasures of balance and diversity affect the performanceof the line. Our hope is that the second and third terms inthe new objective capture enough of the important effectsthat the solution provides a substantial improvement overa good or optimal solution that ignores the workloadstability effects.

Table 1. Workload deviation and diversity of processing timesof the alternative task assignments in Fig. 3

Assignment Deviation of Within-station Between-stationworkload diversity diversity(Q = J)

1 0.293 4.693 4.0672 0.193 4.693 3.867

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Balancing mixed-model assembly lines

Number of jobs Number of jobs Number of jobs

35

IIIIIII,

0.5 P 1.1 1.5 Processingtime

workload =0.99

Station 1

1.9 Processingtime

workload = 0.75

Station 2

Assignment 1

o 0.8 P 1 1.4 Processingtime

workload = 0.95

Station 3

Number of jobs Number of jobs Number of jobs

0.2 0.8 P 1.2 1.7 Processingtime

workload =0.99

0.7 P 0.9 J.7 ProcessingtIlDe

workload =0.9 workload =0.8

Processingtime

Station 2

Assignment 2Station 1

o job type 1

Fig. 3. Comparison of alternative assignments.

iJ job type 2 ~ job type 3

Station 3

• jobtype4

4. Solution approach

The single-model ALB problem with deterministic tasktimes is an NP-hard problem (Garey and Johnson, 1981).With the new objective and multiple models, the problemis also NP-hard (proof omitted). The typical problem inpractice has several hundred stations, several thousandtasks and potentially millions of option combinations.Hence, optimal solutions cannot be obtained within rea­sonable amounts of CPU time. Furthermore, the form ofthe objective function makes it hard to obtain goodbounds, thus also making it difficult to eliminate domi­nated solutions. It is also difficult to derive properties ofoptimal solutions because good solutions may have verydifferent characteristics, and because job sequences must

be constructed in order to evaluate the line balancingdecisions.

Due to the difficulty of obtaining optimal solutions forthe proposed model, we develop a heuristic to solve theproblem. To reduce the number of decision variables, weemploy a method similar to that of Patterson and Albracht(1975). The processing time ofeach taskj, lj, is replaced byits workload Wj' Hence, the range of stations to which thetask can possibly be assigned is [Lj , Uj ] where

L j = rW j + L wr] and u, = n + I - rWj + L ". ,rEA) rEB}

where Aj is the set of all predecessors of j and B, is the setof all successors of j.

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36 Afatanachai and Yano

4.1. Ol'en'iew of the heuristic

The proposed heuristic procedure is similar to a filteredbeam search (Ow and Morton, 1989). The solution isconstructed one station at a time, starting from the firststation. For each station, we attempt to construct severalpotential task assignments ("feasible subsets"). For thepurposes of the heuristic, we consider a subset of tasks tobe feasible for a station if they are precedence-feasible,and they permit all tasks that must be assigned to thesubsequent station to be assigned to that station. Weinitially branch from the feasible subset with the bestobjective value, but retain the others for later usc ifbacktracking is required. After a complete solution (forall stations) is constructed, we implement an improve­ment procedure which involves transferring tasks fromstation to station. A flow chart of the heuristic appears inFig. 4. Detailed descriptions of the steps appear below.Pseudocode is available from the authors upon request.

4.2. Generating feasible subsets

We define a trial as an attempt to construct a feasiblesubset. In constructing a potential subset of tasks for astation, tasks that must be assigned to station i (tasks jsuch that U, = i) are assigned first. Then further assign­ments are made by randomly selecting tasks for consid­eration one at a time from the pool of eligible tasks, S,using a biased weighting scheme. Task j is eligible if: (i)station i is in its feasible range (Lj ::; i ::; Uj); (ii) it is stillunassigned; (iii) all of its immediate predecessors havebeen assigned (i.e., it is precedence-feasible); and (iv) as­signing tasks j does not cause the workload of station i toexceed one. Task j is selected for consideration withprobability OJj, where

Be. +- CumulativeI

workload upto station i

No

Backtracking procedure

Assign theremaining tasks

to station n

No

Generatefeasiblesubsets

Pick the feasiblesubsetat stationi with

best Objective value

;+-;+1

Starti+-I

Be..... o It;I

Fig. 4. Flow chart of the heuristic.

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Balancing mixed-model assembly lines

and

BP~ = Wj + L W r ·

rEB}

(Recall that B, is the set of all successors of j). The weightWj is a function of the Back Positional Weight (BPW). (Indefining the Back Positional Weight in the mixed-modelcontext, we use the workloads instead of the processingtimes). This weighting scheme causes a high probabilityto be associated with tasks that have many successorsand/or successors with large workloads. By selecting suchtasks early, the number of tasks eligible for assignment atany point in time tends to increase, leading to more al­ternative feasible subsets and ultimately, to a greaterchance of constructing a feasible completion of the partialsolution.

Provided the resulting cumulative workload is less thanone, tasks are randomly selected from S and assigned tothe station until the cumulative workload is greater thanmax (ip, BC;), where ip is the average cumulative work­load for i stations and BC; is the cumulative workloadthrough station i in the partial solution obtained justprior to backtracking (if applicable). These constraintslimit the number of alternative subsets, but more im­portantly, they help to balance the workloads acrossstations and to eliminate partial solutions that are likelyto lead to infeasibility later (because too much work re­mains to permit a feasible completion). Once a task isconsidered, it is eliminated from consideration and theWjS are updated accordingly.

Once a subset that satisfies the cumulative workloadconstraint is generated, this subset is stored if: (i) it isdistinct and (ii) it is possible to assign all unassigned tasksj for which U, = i + I to station i + I without violatingthat station's capacity constraint. This entire subsetconstruction process is repeated for y trials at the stationunder consideration.

Beyond these y trials, we also consider subsets forwhich the cumulative capacity does not satisfy the cu­mulative workload constraints mentioned above. If, us­ing the procedure above, we have not constructed acapacity-feasible subset with a cumulative workload ex­ceeding the threshold, we also consider the subset thatexisted immediately before the unsuccessful attempt toassign a task so as to bring the cumulative workloadabove the threshold (but below capacity). Moreover,because of the discreteness of the tasks and the resulting"lumpiness" of the workloads, for each of the feasiblesubsets satisfying the cumulative workload threshold, wealso consider a related solution that does not strictlysatisfy the constraints on cumulative workload, but has adiscrepancy of less than the workload of a single task. Inparticular, we also consider the potential subset that ex­isted just before the addition of the task that caused thecumulative workload to exceed ip, A subset not satisfyingthe standard cumulative workload threshold is stored if

37

all of the following conditions hold: (i) the remainingworkload is smaller than n - i; (ii) the cumulativeworkload is greater than BC;; (iii) the subset is distinct;and (iv) it is possible to assign all unassigned tasks j forwhich U, = i + I to station i + I without violating thatstation's capacity constraints. Because these subsets aregenerated along the way with no additional computa­tional effort and generally have cumulative workloadsclose to ip, they provide a means of diversifying the set ofpotential solutions while still adhering to the basic prin­ciples on which our heuristic is based. They also providealternatives in situations where construction of solutionssatisfying the cumulative workload thresholds is difficultat some stations because of the magnitudes of the taskworkloads.

Finally, the feasible subset (if any) with the best ob­jective value is selected for branching and we proceed tothe next station. If no feasible subset has been constructedby the foregoing procedure, the backtracking procedure(below) is executed.

4.3. Backtracking

Whenever it is not possible to continue, we backtrack tothe previous station, take the next best feasible subsetsatisfying certain conditions, and branch from there. Inparticular, when backtracking, we consider only solutionswith a higher cumulative workload than that of the pre­vious partial solution. This tends to increase the chancesof constructing a feasible solution in the next iteration. Ifno feasible subset satisfies this condition, we backtrack tothe next previous station, and so forth. Finite terminationof the algorithm is guaranteed because after backtrack­ing, the algorithm considers only feasible subsets withhigher cumulative workload than that before backtrack­ing. However, because we do not consider all feasiblesubsets at each station, the heuristic is not guaranteed tofind a feasible solution if one exists. This property iscommon to all heuristic ALB procedures, but we have notfound this to be a practical problem.

4.4. Improvement procedure

We execute an improvement procedure after constructinga complete solution. The improvement procedure startsfrom the last station. If the workload of the station is lessthan the average utilization level, we consider the possi­bility of transferring a task from another station with anabove-average workload to this station while maintainingfeasibility. Among all such potential transfers (if any), weselect the one that leads to the greatest improvement inthe objective value. Then, we move to the subsequentstation and repeat the process, and continue this processuntil we reach the first station. This process is repeateduntil there is no transfer that leads to an improvement inthe objective value.

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The performance of the heuristic depends on thenumber of feasible subsets generated and their quality.Wc considered including other subsets with smallerworkloads. However, if Q is small, the heuristic tends toselect subsets with small workloads at early stations,thereby Icading to solutions with highly uneven work­loads across stations. Furthermore, when this happens,backtracking is more likely, which increases the compu­tation time unnecessarily.

5. Computational results

In this section, we present a summary of the results of ourcomputational study. The study is divided into two mainparts. In the first part, we solve small problems for whichoptimal solutions can be obtained (for both objectives)and compare the results with solutions from our heuristicusing the new objective. Our goal is to quantify the ben­cfits of using the new objective function and the penaltyfrom using the heuristic. Because of the computationalburden of finding optimal solutions, the first part of ourstudy is limited to small problems. In the second part,solutions from our heuristic are compared with solutionsobtained from an adaptation of the heuristic of Racha­madugu and Talbot (1991) for large problems. (This is theonly procedure of which we are aware that could beadapted to our objective of balancing the workload for afixed number of stations. Others, e.g., Johnson (1988),Hoffman (1992) and van Zante-de Fokkert and de Kok(1997), minimize the number of stations for a fixed cycletime. Klein and Scholl (1996) minimize the cycle time for afixed number of stations, but minimizing the cycle timedoes not necessarily optimize workload balance. Thisportion of the study is designed to provide an indicationof the benefits of using our new approach versus an ex­isting approach (as a benchmark) in practical settings.

The performance of the line depends not only on the linebalancing decisions, but also on the sequencing and stationIcngth decisions, Consequently, in order to compare theperformance of different line balancing approaches, theinfluence of the other two decisions must be either con­trolled for or factored out. We discuss each in turn.

5.1. Sequencing procedure

To evaluate the performance of the line balancing pro­cedure, we must "simulate" the resultant effect on theday-to-day performance of the line, measured in terms ofwork overload. (This need has been observed by otherresearchers, including Driscoll and Abdel-Shafi (1985)and Carlson and Yao (1992). To do so, we need to em­ploy a sequencing procedure that can quickly provide areasonable estimate of the work overload that wouldactually be observed. In practice, the sequencing prob­lems are much too large to solve optimally, so heuristic

Matanachai and Yano

procedures must be used. Thus, although we could solvesmall problems optimally, this would not accurately re­flect what would happen in practice. There are a fewheuristic sequencing procedures in the literature, but veryfew work efficiently for a large number ofjobs and a largenumber of job types.

For this reason, we develop a simple sequencing rulebased on fundamental concepts from the literature onmixed-model assembly line sequencing. It is a greedyheuristic which considers two major types of inefficien­cies: work overload and idle time. In the heuristic, jobsare appended to the end of the partial sequence, one byone, based on a ranking scheme. The job type with theminimum value of ex (work overload) + idle time, sum­med over all stations, is selected for assignment. Consid­eration of idle time prevents us from being too myopic,i.e., picking jobs with short processing times at the be­ginning which will lead to a concentration of jobs withlong processing times at the end. From preliminary ex­periments, we found that weighting the work overloadhigher than the idle time, with ex = 3.0, provided verygood performance (relative to lower bounds on workoverload) across a range of problem parameters.

5.2. Station length allocation

Increasing the length of the stations reduces work over­load for any given sequence ofjobs and task assignments.Station lengths are constrained due to physical spacelimitations, and thus are changed infrequently. Hence,even if we could identify an optimal line length for aparticular model mix, we would not be able to vary thelength when the model mix changes from day to day. Ourintent is not to find an optimal way to allocate stationlengths, but to develop a reasonable method for makingthis decision so that it does not obscure the impact of theline balancing decisions, which are our primary concerns.

In practice, the station length must be at least as long asthe longest processing time of any job type at that station,multiplied by the conveyor speed. Therefore in our ex­periments, we initially allocate this length to each station.For the given model mix, we generate a sequence usingour simple sequencing heuristic, then compute the corre­sponding work overload at each station. We increase thelength of the station with the highest work overload by asmall increment. d, and construct a sequence consistentwith the new length. This process is repeated until the linegives zero work overload. In all of our experiments, we setd to equal to 0.3 (i.e., 30% of the spacing between con­secutive jobs). For any given total line length, we use thestation lengths generated via this procedure.

5.3. Performance measure

We needed to devise a standard measure to compare theperformance of different procedures. This was not

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straightforward, because the performance of each pro­cedure depends not only on the sequence but also on thelength of the line. If we plot work overload versus linelength (using the station length allocation procedure de­scribed earlier), typical results from our experiments areroughly as illustrated in Fig. 5. In some unusual cases, thetwo curves cross once or more, generally in the region oflonger line lengths. We decided to use the area under therespective curves, denoted by A, as our performancemeasure. It is an aggregate measure of performance overa range of line lengths but still captures the relative per­formance of the two procedures. Because we solve theproblem only for a finite number of line lengths, A isactually the sum of work overload values for all linelengths considered, but conceptually, it is useful to view itas an integral or area. Our measure of the relative per­formance of different line designs is based on these ag­gregate statistics.

5.4. Problem generation

Line balancing problems in real life vary widely fromfactory to factory. For this reason, we designed our ex­periments to explore the performance of the line balanc­ing methods under a variety of circumstances. PracticalALB problems often involve products for which there arecorrelations among the options selected by an individualcustomer. Some correlations are due to "options pack­ages" offered by the manufacturers, but other correlationpatterns also arise.

The procedures available in the literature for generat­ing problem parameters apply only to the single-modelcase. In the mixed-model case, problem generation ismore complicated because the detailed relationships be­tween job types, options, and tasks must be specified. Wedeveloped a method for generating problem data usingcontrol parameters that allow us to represent the range ofvalues observed in practice. The primary control para­meters relate to the workloads of the tasks, correlationsamong the options, density of the precedence constraints,and the relations among the job types, options, and tasks.

Line Length

Fig. 5. Typical results.

39

Typical "small" problems in our experimental set haveeight stations, 10 job types, 10 product options, and 30 to40 tasks, while the "large" problems have 30 stations, 20job types, 20 product options, and 300 tasks. Processingtimes of the tasks are generated so that the average uti­lization is roughly 70, 80, or 90%, which are consistentwith the range of values observed in practice. For each setof control parameters, we randomly generate three mixesof job types that differ slightly from the mix assumed forthe line balancing; each of these consists of 200 specificcustomer orders. Because of space limitations, we cannotdescribe the problem generation process fully; details canbe found in Matanachai (1996). We should note, how­ever, that whenever the objective in (I) is employed tofind a (heuristic or optimal) solution, we consider differ­ent values of Q (from one up to 30r) and choose the valuethat provides the best overall performance for the threedifferent job mixes (for a given set of other problem pa­rameters). In practice, it would be relatively easy to fine­tune the choice of Q, or to invest the time to solve the taskassignment problem for different values of Q in order toobtain an array of solutions. Below, we present anoverview of the results.

5.5. Results for small problems using the traditional andnew objectives

Optimal solutions (for either objective) were obtainedusing CPLEX, a commercial mixed-integer optimizer. Tounderstand the benefit of using the new objective, wecompare the average work overload for three slightlydifferent mixes of 200 jobs based on optimal task as­signments for the new objective with the average workoverload (for the same mixes of jobs) obtained based onoptimal solutions for the traditional objective. The resultsshow average improvements (due to the new objective) inthe range of 22 to 41% depending on the average utili­zation of the line and the variability of the task processingtimes, with greater improvements for cases of higherutilization and/or greater processing time variability.These results suggest that the new objective has the po­tential for providing a substantial reduction in workoverload.

We are also interested in understanding the magnitudeof the loss from using heuristic rather than optimal so­lutions for the new objective. To do so, we compare re­sults obtained from our heuristic procedure againstoptimal solutions, both based on the new objective. Re­call that our new objective function cannot capture all ofthe effects of task assignments on work overload, soheuristic task assignments may perform as well or betterthan so-called "optimal" task assignments. Indeed, theheuristic solutions lead to work overload values that are,on the average, only 4.2% worse than the correspondingresults based on the optimum task assignment for the newobjective and perform as well or better than the optimum

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40

tasks assignments in over 80% of the problems. Thedifferences vary widely from problem to problem, and wedid not discern any patterns of the effects of problemparameters on the results. This not surprising because ofthe "lumpiness" that arises in small assembly line bal­ancing problems with only a few tasks (in this case, lessthan four tasks) per station. Such effects tend not to occurin larger problems where there is more latitude in as­signing tasks. (This effect is analogous to what happens inbin-packing problems with small versus large numbers ofbins).

We also compare solutions from our heuristic for thenew objective to optimal solutions for the traditionalobjective, again using the same method as describedabove. The average improvement from using the heuristicwith the new objective is over 21%, with strict improve­ments occurring in 74% of the problems and no changeoccurring in the remaining 26%. Thus, on small prob­lems, even if solutions are obtained heuristically, usingthe new objective leads to substantial improvements overthe traditional objective, in terms of the resultant workoverload incurred on the assembly line.

5.6. Results for large problems

For our set of large problems, it is not possible to obtainoptimal solutions for either objective within a reasonableamount of CPU time. Therefore, we compare heuristicsolutions for the new objective obtained using ourproposed procedure with heuristic solutions for thetraditional objective obtained using a straightforwardadaptation of the Rachamadugu and Talbot (1991)algorithm, (We usc task workloads rather than processingtimes to account for the mixed-model nature of ourproblem). Solutions based on the new objective led toimprovements in work overload averaging 29%, withsubstantial improvements occurring in the vast majority(» 90%) of the problems. These improvements are siz­able enough to have a measurable impact on daily as­sembly line operations. Not only will fewer jobs incurwork overload, but those that do will incur smalleramounts, thereby contributing to improved productquality and worker morale.

We observed several patterns regarding the effect of theparameters on the solutions. First, the improvement inwork overload is strongly influenced by the density (D) ofthe precedence diagram and the average utilization of thesystem. The greatest improvements, averaging over 45%,were found with D = 0.3 and p = 0.9, which are close tothe parameters observed in automobile assembly lines(when one is lucky enough to operate at p = 0.9). At theother end of the spectrum, improvements averaging 12%were observed for D = 0.7 and p = 0.7, where precedencerequirements greatly constrain the task assignments, andsequencing is relatively easy provided that the line ismoderately well balanced. The average percentage

Aiatanachaiand Yano

improvement also increased with the variability of thetask workloads. With high variability of task workloads,balancing is more difficult because of the lumpiness of thetasks, and careful choices of the assignments help to re­duce the impact of this lumpiness on the within- andbetween-station diversity. We also found that usinghigher values of Q for heavily loaded systems tended toimprove performance slightly, but caution should betaken when choosing Q so as not to over-shadow thebenefits inherent from considering the within- and be­tween-station diversity effects. Other problem parametersdid not cause wide or consistent differences in the per­formance of the heuristic.

6. Conclusions

Balancing mixed-model assembly lines is complicated bydifferences of processing times among job types. Even theproblem of balancing workloads is difficult, but if linebalancing is not performed also to facilitate the con­struction of good sequences, the short-term performanceof the line will be poor. Balancing the line by minimizingthe deviation of workloads across stations (the traditionalbalancing objective) does not guarantee good sequencingperformance. In order to improve the performance of theline, other factors related to processing times of job typesat the various stations must be considered while balanc­ing the line. For this reason, we propose a line balancingobjective which consists of the traditional objective andterms that represent the within- and between-stationdiversity of processing times.

We propose a heuristic filtered beam search algorithmin which feasible subsets are constructed at each station.Several feasible subsets with the best objective values areretained for each station, and we initially branch off theone with the best objective value to construct potentialsubsets at the subsequent station. Backtracking is em­ployed when the process cannot continue due to infeasi­bilities. After a feasible solution has been found, animprovement procedure is implemented to transfer tasksfrom station to station in order to improve the objectivevalue. This heuristic is shown to perform well, both in anabsolute sense (on small problems), and relative to heu­ristic solutions for the traditional objective (on largerproblems). Because the performance of the heuristic de­pends on the number of different feasible subsets con­sidered, it can be improved, if desired, by increasing thenumber of subsets retained for each station.

Our proposed heuristic was designed to solve a fairlygeneral version of the mixed-model assembly line bal­ancing problems. In practice, there may be additionalconstraints, due to restrictions on the equipment, layoutetc., which vary from plant to plant. It is easy to extendour heuristic to handle these extra constraints simplyby considering them in the subset generation process.

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Balancing mixed-model assembly lines

Examples include restrictions on the locations of tasks(zoning restrictions), station length constraints (con­straints on the maximum processing time for a job at astation), and constraints related to combinations of tasksthat should be or cannot be assigned to the same station.

More research is needed to fully understand the broad­ranging consequences of assembly line balancing deci­sions in mixed-model settings. There is also opportunityto refine our objective function and to generalize ourapproach to consider other factors such as task times thatdepend upon the assembly sequence, and other types ofassembly lines (see Ben-Arieh, 1995). We hope that thisresearch encourages further work in these directions.

Acknowledgement

This research has been supported in part by NationalScience Foundation Grant HRD 93-96288 to the Uni­versity of California at Berkeley.

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Matanachai, S. (1996) Balancing objectives for mixed-model, pacedassembly lines. Ph.D. Dissertation, Department of IndustrialEngineering and Operations Research, University of California,Berkeley, CA.

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Biographies

Sittichai Matanachai is a Consultant with The Boston ConsultingGroup (BCG) in Bangkok, Thailand. He earned a B.S. in ElectricalEngineering at Chulalongkorn University in Bangkok (1991), anM.Eng. in Operations Research and Industrial Engineering at CornellUniversity (1992) and an M.S. and a Ph.D. in Industrial Engineeringand Operations Research at the University of California, Berkeley(1993 and 1996). His recent work with BCG has been on restructuringof non-performing loans for Asian banks and deregulation strategiesfor utility companies in Southeast Asia and Europe.

Candace Arai Yano is Professor and Department Chair of IndustrialEngineering and Operations Research at the University of California,Berkeley, California, where she currently holds a Chancellor's Profes­sorship. She holds an A.B. in Economics (1977), a M.S. in OperationsResearch (1979), and a M.S. (1980) and Ph.D. (1981) in Industrial

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Engineering from Stanford University. Prior to joining the Universityor California, she held positions as a Member of the Technical Staff atBell Telephone Laboratories and as a faculty member in the Depart­mcnt or Industrial and Operations Engineering at the University orMichigan. Her primary research interests are productionand inventorycontrol, and interdisciplinary problems related to manufacturing anddistribution. She has authored or co-authored over 50 articles on these

Afatanachai and Yano

subjects and is the recipient or several National Science Foundationgrants. She serves on the Editorial Boards or /IE Transactions, Inter­faces. International Journal of Production Economics, Journal of Man­ufacturing and Service Operations Management, Naval ResearchLogistics, Operations Research. and Technology and Operations Review.

Contributed by the Manufacturing Systems Modeling Department

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