1s 6 balanceamento - artigo 5

8/11/2019 1S 6 Balanceamento - Artigo 5

1/22

Invited review

A survey on problems and methods in generalized assemblyline balancing

Christian Becker, Armin Scholl *

Friedrich-Schiller-Universitat Jena, Fakultat fur Wirtschaftswissenschaften, Carl-Zei-Strae 3, D-07743 Jena, Germany

Available online 11 September 2004

Abstract

Assembly lines are traditional and still attractive means of mass and large-scale series production. Since the earlytimes of Henry Ford several developments took place which changed assembly lines from strictly paced and straightsingle-model lines to more flexible systems including, among others, lines with parallel work stations or tasks, cus-tomer-oriented mixed-model and multi-model lines, U-shaped lines as well as unpaced lines with intermediate buffers.

In any case, an important decision problem, called assembly line balancing problem, arises and has to be solvedwhen (re-) configuring an assembly line. It consists of distributing the total workload for manufacturing any unit ofthe product to be assembled among the work stations along the line.

Assembly line balancing research has traditionally focused on the simple assembly line balancing problem (SALBP)which has some restricting assumptions. Recently, a lot of research work has been done in order to describe and solvemore realistic generalized problems (GALBP). In this paper, we survey the developments in GALBP research. 2004 Elsevier B.V. All rights reserved.

Keywords:Assembly line balancing; Mass-production; Literature survey; Production process

1. Introduction

Assembly lines are flow oriented production

systems which are still typical in the industrial pro-duction of high quantity standardized commodi-ties and even gain importance in low volume

production of customized products. Among thedecision problems which arise in managing suchsystems, assembly line balancing problems are

important tasks in medium-term productionplanning.

An assembly line consists of (work) stationsk= 1, . . . ,m arranged along a conveyor belt or asimilar mechanical material handling equipment.The workpieces (jobs) are consecutively launcheddown the line and are moved from station tostation. At each station, certain operations are

0377-2217/$ - see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.ejor.2004.07.023

* Corresponding author. Tel.: +49 3641 943171.E-mail addresses: [email protected] (C. Becker),

[email protected](A. Scholl).

European Journal of Operational Research 168 (2006) 694715

www.elsevier.com/locate/ejor
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


2/22

repeatedly performed regarding the cycle time(maximum or average time available for eachworkcycle). The decision problem of optimally

partitioning (balancing) the assembly work amongthe stations with respect to some objective isknown as the assembly line balancing problem(ALBP).

Manufacturing a product on an assembly linerequires partitioning the total amount of work intoa set of elementary operations named tasksV= {1, . . . , n}. Performing a task j takes a tasktime tjand requires certain equipment of machinesand/or skills of workers. Due to technological andorganizational conditions precedence constraintsbetween the tasks have to be observed.

These elements can be summarized and visual-ized by a precedence graph. It contains a nodefor each task, node weights for the task timesand arcs for the precedence constraints. Fig. 1shows a precedence graph withn= 10 tasks havingtask times between 1 and 10 (time units). The prec-edence constraints for, e.g., task 5 express that itsprocessing requires the tasks 1 and 4 (direct prede-cessors) and 3 (indirect predecessor) be completed.The other way round, task 5 must be completedbefore its (direct and indirect) successors 6, 8, 9,

and 10 can be started.Any type of ALBP consists in finding a feasible

line balance, i.e., an assignment of each task to astation such that the precedence constraints andfurther restrictions are fulfilled (see Section 2).The set Sk of tasks assigned to a station k(=1, . . . , m) constitutes its station load, the cumu-lated task time tSk

Pj2Sk

tj is called stationtime. When a fixed common cycle time c is given(paced line; cf. Section 2), a line balance is feasibleonly if the station time of neither station exceeds c.

In case of t (Sk) < c, the station khas an idle timeofct (Sk) time units in each cycle.

For the example ofFig. 1, a feasible line bal-ance with cycle time c= 11 and m= 5 stations isgiven by the station loads S1= {1,3}, S2= {2,4},

S3= {5,6}, S4= {7,8}, S5= {9,10}. While no idletime occurs in stations 2 and 5, stations 1, 3, and 4show idle times of 1, 2, and 5, respectively.

The installation of an assembly line is a long-term decision and usually requires large capitalinvestments. Therefore, it is important that sucha system is designed and balanced so that it worksas efficiently as possible. Besides balancing a newsystem, a running one has to be re-balanced peri-odically or after changes in the production processor the production program have taken place. Be-cause of the long-term effect of balancing deci-sions, the used objectives have to be carefullychosen considering the strategic goals of the enter-prise. From an economic point of view cost andprofit related objectives should be considered (cf.Section 4). However, measuring and predictingthe cost of running a line over months or yearsand the profits achieved by selling the productsassembled is rather complicated and error-prone.A usual surrogate objective consists in maximizingthe line utilization which is measured by the lineefficiency as the productive fraction of the lines

total operating time and directly depends on thecycle time c and the number of stations m (cf.Section 3).

2. Characteristics of assembly line systems

Because of very different conditions in indus-trial manufacturing, assembly line systems andcorresponding ALBPs are multifaceted. In the fol-lowing, we shortly characterize the most relevant

properties for classifying assembly lines. For moredetailed classifications and overviews on balancingissues we refer to, e.g.,Buxey et al. (1973),Baybars(1986),Shtub and Dar-El (1989),Ghosh and Gag-non (1989), Erel and Sarin (1998), Scholl (1999,Chapter 1) as well asRekiek et al. (2002b). Fur-thermore, seeRekiek and Delchambre (2001).

In case of apaced assembly line, the station timeof every station is limited to the cycle time c as amaximum value for each workpiece. Since tasksare indivisible work elements, ccan be no smaller

16

26

82

74

65

54

45

34

101

910

Fig. 1. Precedence graph.

C. Becker, A. Scholl / European Journal of Operational Research 168 (2006) 694715 695


3/22

than the largest task time tmax= max{tjjj= 1,. . .,n}. Due to the cycle time restriction, pacedassembly lines have a fixed production rate (recip-

rocal of the cycle time).In the absence of a common cycle time, i.e., allstations operate at an individual speed, workpiecesmay have to wait before they can enter the nextstation and/or stations may get idle when theyhave to wait for the next workpiece. These difficul-ties are partially overcome by buffers between thestations. In this case of an buffered (unpaced)assembly line, the ALBP is accompanied by theadditional decision problem of positioning anddimensioning buffers (cf., e.g.,Buzacott, 1968;Su-hail, 1983;Baker et al., 1990;Hillier and So, 1991;Hillier et al., 1993;Malakooti, 1994;Powell, 1994;Dolgui et al., 2002a).

If only one product is assembled, all workpiecesare identical and a single-model line is present. Ifseveral products (models) are manufactured onthe same line, the ALBP is connected to a sequenc-ing problemwhich has to decide on the sequence ofassembling the model units (cf. Yano and Bolat,1989; Sumichrast and Russell, 1990; Sumichrastet al., 1992; Bard et al., 1992; Merengo et al.,1999). The sequence is important with respect to

the efficiency of a line, because the task timesmay differ considerably between the products.

Depending on the type of intermixing the unitstwo variants arise: amixed-model lineproduces theunits of different models in an arbitrarily inter-mixed sequence (cf.Bukchin et al., 2002), whereasa multi-model line produces a sequence of batches(each containing units of only one model or a

group of similar models) with intermediate setupoperations. Therefore, balancing and sequencingare connected to a lot sizing problem in the latter

case (cf., e.g.,Burns and Daganzo, 1987;Dobsonand Yano, 1994). The different line types are char-acterized inFig. 2, where different models are sym-bolized by different geometric shapes. Dependingon these line types, single-model, mixed-modeland multi-model versions of ALBP have to be con-sidered and solved (cf. Section 10).

A further important characteristic defining dif-ferent versions of ALBP is the variability of tasktimes. Whenever the expected variance of tasktimes is sufficiently small, as in case of, e.g., simpletasks or highly reliable automated stations, thetask times are considered to be deterministic (cf.,e.g., Johnson, 1983). Considerable variations,which are mainly due to the instability of humanswith respect to work rate, skill and motivation aswell as the failure sensitivity of complex processes,require considering stochastic task times (cf. Sec-tion 9 andBuzacott, 1990;Robinson et al., 1990;Hillier and So, 1991, 1993; Pike and Martin,1994). Besides stochastic time variations, system-atic reductions are possible due to learning effectsor successive improvements of the production

process (cf., e.g., Boucher, 1987; Chakravarty,1988).

Because of the rigid process orientation, thelay-outof flow-line production systems is partially pre-determined by the flow of materials. Nevertheless,some layout possibilities exist. Traditionally, anassembly line is organized as a serial line, wheresingle stations are arranged along a (straight) con-

a. single-model line

b. mixed-model line

c. multi-model line

setupsetup

Fig. 2. Assembly lines for single and multiple products.

696 C. Becker, A. Scholl / European Journal of Operational Research 168 (2006) 694715


4/22

veyor belt. Such serial lines are rather inflexibleand have other disadvantages which might beovercome by a U-shaped assembly line(cf. Section

7). Both ends of the line are closely together form-ing a rather narrow U. Stations may work attwo segments of the line facing each other simulta-neously (crossover stations). Besides improvementswith respect to job enrichment and enlargementstrategies, a U-shaped line design might result ina better balance of station loads due to the largernumber of task-station combinations (cf. Milten-burg and Wijngaard, 1994;Monden, 1998;Scholland Klein, 1999).

Further improvements in flexibility and failuresensitivity of an assembly line system may beachieved by introducing some type of parallelism(cf. Section 6): In a multi-model context installingcomplete parallel lineseach designed for one prod-uct or family of related products often allows bet-ter balances and increased productivity. Then theALBP is accompanied by the additional decisionproblems concerning the number of lines to be in-stalled and assigning products and work forces tolines (cf. Lehman, 1969; Geoffrion and Graves,1976;Globerson and Tamir, 1980;Ahmadi et al.,1992). Even with a single line the advantages of

parallelization can be utilized by installing parallelstations, i.e., the workpieces are distributed amongseveral operators who perform the same tasks. Asis the case with parallel lines, the equipment has tobe installed several times. Parallel stations allowthe reduction of the (global) cycle time of the sys-tem if certain tasks have task times longer than thedesired cycle time (cf. Freeman and Jucker, 1967;Buxey, 1974;Pinto et al., 1981;Sarker and Shant-hikumar, 1983;Bard, 1989;Daganzo and Blumen-feld, 1994). Another possibility of reducing the

global cycle time below the largest task time isthe concept ofparallel tasks(cf.Arcus, 1966;Pintoet al., 1975; Inman and Leon, 1994). Respectivetasks are assigned to several stations of a serial linewhich cyclically perform them completely on dif-ferent workpieces.

Whenever there are process alternatives, i.e.,tasks may be performed by different equipmentand/or by using different technologies, the balanc-ing problem is connected to an equipmentor proc-ess selection problem (cf. Section 5).

For the assembly of heavy workpieces it may benecessary to operate atwo-sided linewhich consistsof two connected serial lines in parallel. Instead of

single stations, pairs of opposite stations on eitherside of the line (left-hand side and right-hand sidestations) work in parallel, i.e., they work simulta-neously at opposite sides of the same workpieces(cf. Section 8 andHautsch et al., 1972;Bartholdi,1993).

In order to perform a task assigned, the stationmust be equipped by operators and machineswhich have the skills and technological capabilitiesrequired. Especially in case of complex products itis usually not possible to have all stations equippedequally resulting in station related assignmentrestrictions (cf. Section 8 and Kilbridge and Wester,1961;Bukchin and Tzur, 2000). Additionally, theassignment of tasks may be restricted by task re-latedconstraints such as incompatibilities betweentasks, minimum or maximum distances (in terms oftime or space) between stations performing a pair(or subset) of tasks (cf. Ignall, 1965; Deckro,1989; Agnetis et al., 1995). Furthermore, positionrelated constraints are relevant for workpieceswhich are heavy, large or fixed at the conveyor beltsuch that they cannot be turned in any position

which is required for performing a task in a certainstation (cf. Buxey and Sadjadi, 1976; Wang andWilson, 1986). Another type of assignment restric-tions is operator related, because operators havedifferent levels of skill such that only certain taskcombinations are possible when an operator is as-signed to a particular station. Furthermore, aspectsof job satisfaction have to be observed (cf. Agra-wal, 1985;Iskander and Chou, 1990).

3. Simple versus generalized assembly line balancing

Most of the research in assembly line balancinghas been devoted to modelling and solving the sim-ple assembly line balancing problem (SALBP)which has the following main characteristics (cf.Baybars, 1986; Scholl, 1999, Chapter 2.2; Scholland Becker, this issue):

mass-production of one homogeneous product;given production process;



5/22

paced line with fixed cycle timec; deterministic (and integral) operation timestj; no assignment restrictions besides the prece-

dence constraints; serial line layout with m one-sided stations; all stations are equally equipped with respect to

machines and workers; maximize theline efficiency E=tsum/(m c) with

total task time tsumPn

j1tj.

Several problem versions arise from varying theobjective as shown inTable 1.

The versions of SALBP may be complementedby a secondary objective which consists ofsmooth-ing station loads (vertical balancing; cf. Merengoet al., 1999). For example, one may minimize the

smoothness index SX ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiPm

k1ctSk2

q pro-

vided that the combination (m, c) is optimal withrespect to line efficiency (see, e.g., Moodie andYoung, 1965;Rachamadugu and Talbot, 1991).

Since SALBP-F is an NP-complete feasibilityproblem, the optimization versions of SALBP areNP-hard (cf. Wee and Magazine, 1982; Scholl,1999, Chapter 2.2.1.5). All the more, the same istrue for any relevant generalized problem.

The assumptions of SALBP are very restrictingwith respect to real-world assembly line systems.Therefore, researchers have recently intensifiedtheir efforts to identify, formulate and solve morerealistic problems which consider the characteris-tics described in Section 2. Following a classifica-tion scheme ofBaybars (1986)such problems areembraced by the term generalized assembly linebalancing problem (GALBP). Because most of themodels and procedures are directly based on theirSALBP counterparts, we refer toScholl and Bec-ker (this issue)for state-of-the-art solution proce-

dures for SALBP and concentrate here on

surveying the way of generalizing the problemrather than describing all methodical aspects insolving these problems.

4. Cost- and profit-oriented objectives

As stated in Section 1, the installation of anassembly line requires large (long-term) capitalinvestments. Furthermore, operating the linecauses short-term operating costs such as wages,material, set-up, inventory and incompletioncosts (cf. Scholl, 1999, p. 20). In case of a non-fixed production rate and different levels of pro-duction quality, these costs have to be contrastedwith the profit attained by the line (cf. Zapfel,1975).

The installation and operating costs as well asthe profits mainly depend on the cycle time andthe number of stations (cf. Deckro, 1989), suchthat cost-oriented models are strongly related toSALBP-E. The latter problem is usually solvedby iterating on SALBP-1 or SALBP-2 instances,respectively (seeScholl and Becker, this issue, Sec-tion 4.3). Thus, the same procedures can likewisebe used for cost or profit oriented objectives on

principle. However, in some situations it is neces-sary to consider models which incorporate costsand/or profits explicitly. This is especially truewhen the balancing problem is connected withthe decision problem of selecting processing orequipment alternatives (cf. Section 5).

4.1. Cost-oriented models

Rosenberg and Ziegler (1992) assume that theoperation of a station k causes a wage rate wk

per time unit that is equal to the maximum wagerate of all tasks that are assigned to that station.The background of this assumption is that themost demanding task assigned to a station definesthe level of qualification the operator(s) must have(for other definitions of station wage rates seeStef-fen, 1977). The objective is to minimize the aggre-gate wage rate over all stations, while the numberof stations is a variable. Production costs per prod-uct unit are obtained by multiplying that rate withthe given cycle time. The considered objective is

Table 1Versions of SALBP

Cycle time c

Given Minimize

No.m of stationsGiven SALBP-F SALBP-2Minimize SALBP-1 SALBP-E



6/22

equivalent to minimizing the number of stations, ifall tasks have the same wage rate. Hence, the prob-lem is a direct generalization of SALBP-1. Rosen-

berg and Ziegler describe and evaluate priorityrule based heuristics, where some of the rulesavailable for SALBP-1 (Scholl and Becker, thisissue, Section 5) are extended to allow for smooth-ing the wage rates within each station.

Amen (2000a)extends the problem by addition-ally considering station related costs of capital, i.e.,each station is assumed to require a constant pre-specified investment.Amen (1997, 2000a)presentsan exact branch-and-bound procedure which ex-tends respective procedures for SALBP-1 (cf.Scholl and Becker, this issue, Section 3) for thisproblem which uses a station-oriented construc-tion scheme and a laser search strategy based ona topological task labeling. The enumeration is re-stricted by means of (global and local) lowerbounds extending such for SALBP- 1 and domi-nance rules, where the maximal load rule whichis essential for solving SALBP-1 is shown to beinappropriate for the cost-oriented problem.Therefore, only weaker versions of this rule andsome other SALBP-1 based rules are applied (cf.Scholl and Becker, 2003).

For the same problem, Amen (2000b, 2001)develops station-oriented priority rule based pro-cedures with cost-oriented dynamic priority rulesand compares them to existing ones using a largeset of randomly generated problem instances.The new rule which controls the idle time andthe difference of wage rates in a station (bestchange of idle cost) performs best. Furtherimprovements are obtained by approaches whichuse several priority rules. The best results are re-ported for a restricted version of the branch-and-

bound procedure outlined above which is basedon successively solving small problems each repre-senting a feasible subset of remaining tasks. How-ever, the latter procedure takes much longercomputation times.

Malakooti (1991, 1994) and Malakooti andKumar (1996) consider a multi-objective ALBPwith capacity- and cost-oriented objectives andpropose different solution approaches includinggeneration of efficient alternatives, interactive ap-proaches and goal programming.

4.2. Profit-oriented models

The cost-oriented models may be extended by

additionally considering profits. Zapfel (1975, pp.31)proposes a model with the objective of maxi-mizing the total contribution margin per shift. Itconsiders operating expenses, idle time costs,material costs, and advertising expenses as wellas constant selling prices. The model of Klenke(1977, pp. 30)includes fixed selling prices, materialcosts as well as wages and equipment costs. A sim-ilar model which considers individual fixed costsfor the different stations is proposed byRosenblattand Carlson (1985). This model is extended byMartin (1994) for the case of unpaced lines withbuffers, where inventory related cost componentsare relevant.

5. Equipment selection and process alternatives

The cost-oriented models described in Section 4assume that the equipment of the stations is givenand that the production process is fixed. However,selecting the equipment should be related to therequirements induced by the tasks assigned to a

station. Furthermore, the way of manufacturinga product depends on the equipment (machines,manpower). When these decisions are connectedto the balancing problem, the term assembly linedesign problem (ALDP) is frequently used in theliterature (cf.Baybars, 1986).

Pinto et al. (1983)consider a model which com-bines the balancing problem with the decision onprocess alternatives. The model considers a basicprocess which may be complemented by one ormore optional process alternatives each of which

reduces some task times or even removes certaintasks completely such that each combination ofalternatives defines a precedence graph. Thesealternatives cause fixed costs per time unit. As fur-ther cost category, wage rates per time unit areconsidered. The total wage depends on the numberof stations, each of which is manned by one oper-ator, and the cycle time, which may vary in a givenrange. Due to a desired production rate, a lowerbound on the cycle time is given. Taking a largercycle time causes a certain percentage of overtime



7/22

which has to be paid by higher wage rates. Pintoet al. propose a branch-and-bound procedurewhich branches by selecting/discarding single

alternatives and computes lower and even upperbounds by solving respective SALBP instances(cf.Domschke et al., 1997, Chapter 4.3.6.3).

Graves and Lamar (1983)as well asGraves andHolmes Redfield (1988)consider models for one orseveral products, where the stations to be installedat an assembly line are chosen from a set of non-identical station types with different equipments.However, the balancing problem is simplified byassuming a fixed task sequence (serial precedencegraph).

Bukchin and Tzur (2000) consider equipmentalternatives and minimize the total equipmentcosts for a given cycle time. Every station is pro-vided with one equipment chosen from a set ofequipment types. Each type has individual costsand an individual influence on the task times. Sotwo problems arise: (1) A variable number of sta-tions need to be installed and provided with equip-ment. (2) The tasks have to be assigned to thestations considering station related assignmentrestrictions because some tasks can only be per-formed with a subset of the equipment types (cf.

Section 8). Bukchin and Tzur present an exactand a heuristic algorithm for solving the problem.The first one is a branch-and-bound procedurewhich is based on the task-oriented constructionscheme and uses the MLB strategy (cf. Scholland Becker, this issue, Sections 3.2 and 3.6).Accompanying the assignment of the first task toa new station the equipment for this station is se-lected. Lower bounds are computed by discardingthe precedence and integrality constraints and con-sidering aggregate cycle time restrictions. The

branch-and-bound procedure is capable for solv-ing problems of moderate size with up to 30 tasksand at most 10 equipment types. Therefore, a heu-ristic version of the procedure is developed whichskips nodes of the branch-and-bound tree control-led by a user-specified parameter.

The same problem is examined byNicosia et al.(2002) who propose a DP procedure (based onSchrage and Baker, 1978; Kao and Queyranne,1982; cf. Scholl and Becker, this issue, Section3.5) and a branch-and-bound procedure (based

on FABLE and OptPack; cf. Scholl and Becker,this issue, Section 3.6).Falkenauer (1997)presentsa grouping genetic algorithm (cf. Scholl and Bec-

ker, this issue, Section 5.2) for a quite similar prob-lem with resource dependent task times.Rubinovitz and Bukchin (1993) consider a re-stricted version of the problem, where all equip-ment types have identical installation costs, i.e.,the number of stations is minimized. Pinnoi andWilhelm (1997, 1998)propose branch and cut pro-cedures for basic and generalized ALDP.

An ALDP where an operating mode definingthe task times and equipment costs has to be cho-sen for each task is considered by Rekiek et al.(2002a).

Bukchin and Rubinovitz (2002) show that theparallel station problem (cf. Section 6) is a specialcase of the above mentioned equipment selectionproblem (p stations in parallel correspond to anequipment which is p times as fast as the basicequipment in a single station). Therefore, the par-allel station problem can be solved with the meth-ods outlined above and can be combined with theequipment selection problem without changing themodel.

An extension of the balancing problem to trans-

fer lines is described inDolgui et al. (1999)and re-fined in Dolgui et al. (2001a). A (synchronized)transfer line consists of a sequence of automatedstations which perform blocks of tasks sequentiallyand the tasks within each block in parallel usingspecialized equipment (spindle heads). Combiningtasks to blocks is subject to assignment restrictions(cf. Section 8). The block time is determined by themaximal task time in the block and an additionaltime for transfer operations. The objective func-tion is to minimize the life cycle line cost per part,

which is composed of fixed costs per station andadditional costs per block, given the cycle time.The problem is solved via shortest constrainedpath problems (Dolgui et al., 1999, 2001c, 2003),by a decomposition method (Dolgui et al.,2001a), based on a mixed integer program (Dolguiet al., 2001b,c) and by stochastic heuristic proce-dures (Dolgui et al., 2002b). Dolgui et al. (thisissue) consider the problem variation with theblocks of a station being performed simultaneouslyand adapt the shortest constrained path approach.



8/22

The equipment selection problem is equivalentto aworker selection problem, where workers withdifferent qualifications in terms of production

speed or quality are available and are paid accord-ing to their qualifications (cf. Akagi et al., 1983;Wilson, 1986;Lutz et al., 1994).

6. Parallel stations and tasks

Due to the indivisibility of tasks, the maximaltask time tmax is a lower bound on the cycle timec. If there are one or more tasks with task timesgreater than the desired cycle time, paralleling ofstations can resolve this conflict (cf. Buxey,1974). In the simplest form of paralleling, theduplication of stations, there are two identical sta-tions, that execute the same tasks and are providedwith the same equipment. Duplicated stations havea local cycle time of twice the regular cycle timeand are fed with workpieces and release themalternately. Due to the increased local cycle time,the number of feasible loads is greatly enlargedfor parallel stations.

Considering the example inFig. 1and the feasi-ble solution in Section 1 withc= 11 andm= 5, for

SALBP-2 there is no way for decreasing c. Whenparallel stations are available, a feasible solutionwith m= 5 and c= 10 can be found (Fig. 3),decreasing the total idle time from 8 to 3 time unitsand increasing the production rate. Stations 4aand 4b show local cycle times of 20. Their stationtime is 18 and they are fed with workpieces by sta-tion 3 alternately.

Of course, the installation of parallel stationscauses additional fixed costs such that cost-ori-ented objectives are necessary to evaluate a certain

line design. Pinto et al. (1981) present an ap-proach, where duplicating of stations is allowed.The objective is to minimize labour costs, which

consist of fixed costs for duplicating a station, reg-ular wage costs and overtime costs, which arise ifthe realized cycle time exceeds a desired cycle time.

A branch-and-bound procedure for assigningtasks to stations and deciding whether a stationis to be duplicated or not is presented (for detailsseeDomschke et al., 1997, Chapter 4.3.6.1).

Pinto et al. (1975)consider the concept ofparal-lel tasks. Long-lasting tasks are assumed to bedecomposable into shorter tasks having the sameprecedence relations as the original one. Now theseparallel tasks are assigned to different stations inorder to get a feasible balance for the desired cycletime at all and/or to improve the line efficiency.Because of the indivisibility of tasks, however,the original task can be executed only by one sta-tion per cycle. This is dealt with by alternately per-forming this task in each of the respective stationsthereby accepting the local cycle times to varyfrom cycle to cycle. The objective is to minimizetotal costs, which consist of facility costs and la-bour costs defined as in the parallel station case.Facility costs arise though in the parallel task caseno additional stations are needed, because a trans-port system has to be installed, that supports tem-porary violations of the cycle time. Pinto et al. give

a mathematical model of the problem and solve itby a branch-and-bound procedure. In their algo-rithm, parallel tasks must not be assigned to morethan two stations and the task time is subdividedequally among the two parts of the task.

Bard (1989) considers parallel tasks and sta-tions as well as dead time, which is the time thatis needed for transporting workpieces from onestation to the next, meanwhile no tasks can be exe-cuted. For a serial line, dead time decreases theemployable cycle time. If, e.g., c is 10 and the dead

time is 1, workers have to be paid for 10 time units,whilet (Sk) must not exceed 9 for every serial sta-tion k. With parallel stations the unproductiveportion of the cycle time can be decreased. If weconsider the above example, duplicated stationseach have a local cycle time of 20 and the produc-tive time is 19. That is, in the example, the parallelstations use 95% and the serial stations only 90%of the cycle time in a productive manner. Bardproposes a DP procedure, which is based on DPapproaches to SALBP-1 (cf. Scholl and Becker,

{3,4} {1,5} {2,7}

Station 1 Station 2 Station 3

Station 4a

{6,8,9,10}

Station 4b

{6,8,9,10}

Fig. 3. Parallel stations.



9/22

this issue, Section 3.5), for solving the outlinedproblem.

Buxey (1974) considers a model with parallel

stations which passes on fixed costs for additionalequipment and aims at minimizing the total idletime. He proposes a generalized priority rule basedprocedure to solve the problem. A further heuristicprocedure is developed bySarker and Shanthiku-mar (1983).

Another approach of duplicating stations is toarrange them side by side in a serial line. This canbe useful due to space limitations or when a lesscomplex transport system is required. In this case,buffers are needed in front of and behind dupli-cated stations. Consider an example with fourserial stations 1, 2a, 2b, and 3 with 2a and 2b beingduplicates of each other. When station 1 finishesits work on a workpiece while the stations 2aand 2b are still busy, the workpiece enters the in-buffer. In the next cycle, 2b is fed with the work-piece from the in-buffer, while 2a is directlydelivered by station 1. The output of station 2bis carried to the following station 3 and that of2a moves through 2b to the out-buffer. Inmanand Leon (1994) present an stochastic approachto this problem, where random failures, repair

times and processing times are variates. In thestochastic case, duplicated stations often do notstart and end their processes at the same time.As long as the first of the stations is busy, thesecond station cannot be loaded with the nextworkpiece. Four policies of feeding incomingworkpieces to duplicated stations are comparedby a simulation.

Another type of paralleling is to assign morethan one operator to a station (multiple manning).Shtub (1984) considers an objective which mini-

mizes the number of operators given cycle timeand number of stations. He describes a heuristicsimilar to that of Buxey (1974). Wilson (1986)reformulates and reduces the problem and solvesit by a standard MIP solver. Chakravarty andShtub (1986b)consider the case of dynamic changein manning due to learning effects.

Finally, an effect similar to that of parallelingstations consists of combining stations to largerunits (aggregate stations) which are operated byteams of operators. The aggregate stations have a

multiple of the original cycle time available andoperators may rotate increasing the job satisfac-tion. Respective problems are considered byJohn-

son (1991),Bukchin et al. (1997)andBukchin andMasin (2004).Sometimes, the teams are responsi-ble for the complete product such that the linehas only one aggregate station left. That is,team-oriented assembly systems may remove maincharacteristics of traditional assembly lines suchas strict division of labour and pacing (cf.Bukchinet al., 1997).

7. U-shaped line layout

The U-line assembly line balancing problem(UALBP) is introduced and modelled byMilten-burg and Wijngaard (1994). In that problem theassembly line is arranged in an U-shape (cf.Fig. 4). Stations can be arranged so that duringthe same cycle two workpieces at different posi-tions on the line can be handled. In the exampleinFig. 4, station 1 will execute the first tasks onone workpiece, which is starting its productionprocess in that cycle, and the last tasks of anotherworkpiece, that is to be finished at the same cycle.

So the difference to SALBP is that a station kcancontain not only tasks whose predecessors are as-signed to one of the stations 1,. . . ,k, but alsotasks whose predecessors will be finished untilthe product returns to station k for the secondtime (cf.Monden, 1998). A station which handlesthe same workpiece in two different cycles iscalled a crossover station (stations 1 and 5 inFig. 4).

We can define three problem versions ofUALBP regarding to SALBP (cf. Section 3):

S1

S2 S4

S5

S7

S6

S3

Fig. 4. U-shaped assembly line.



10/22

UALBP-1. Given the cycle time c, minimize thenumber of stations m.

UALBP-2. Given the number of stations m, min-

imize the cycle time c.UALBP-E. Maximize the line efficiency E for candm being variable.

Of course UALBP can undergo the same gener-alizations as SALBP with respect to cost-orientedobjectives, paralleling, equipment selection etc.(see below and the respective sections).

Every solution feasible for SALBP is feasiblefor UALBP as well, because an U-line does notneed to include crossover stations. However, theoptimal UALBP solution may have an improvedline efficiency compared to the optimal SALBPsolution due to the increased possibilities of com-bining tasks to station loads.

Applying the example of Fig. 1 to UALBP-2with given m= 5, an optimal solution withc= 10 can be found, which is shown in Fig. 5.The first station starts executing tasks 3 and 4,but gets back every workpiece at the end of theproduction process to perform task 10. Hence,the optimal UALBP-2 solution has a better lineefficiency than the optimal SALBP-2 solution

which requires a cycle time of 11.Miltenburg and Wijngaard (1994) modify the

DO approach ofHeld et al. (1963)and the rankedpositional weight technique ofHelgeson and Bir-nie (1961)to UALBP-1.Ajenblit and Wainwright(1998)use a genetic algorithm to solve UALBP-1.Urban (1998) gives an integer programming for-mulation for UALBP-1 and solves problem in-stances with CPLEX. Scholl and Klein (1999)develop the procedure ULINO (U-Line Optim-izer) which is an extension of SALOME-1 for

SALBP-1 (cf. Scholl and Klein, 1997) and applyit to all versions of UALBP distinguished above.Erel et al. (2001) present a simulated annealingalgorithm.

Nakade et al. (1997) develop bounds andapproximations for the cycle time when task timesare random variables and the number of stations is

given (stochastic UALBP-2).Urban and Chiang (this issue)present a chance-constrained, piecewise linear, integer program forUALBP-1 with stochastic task times which canbe solved with CPLEX for small problems.Chiangand Urban (2002)propose a hybrid heuristic com-posed of priority rule based procedures and animprovement step.

Miltenburg (2000)investigates the effect of theshape of the production line on its effectivenesswhen breakdowns occur. He points out that anU-line is better or equal to a straight line whenbuffer inventories are arranged at all contactpoints between stations. The reason is that cross-over stations cause more contact points than regu-lar stations and so more buffer inventories can beinstalled. Straight lines are to be preferred whenthere are no buffer inventories. Admittedly, thecosts of installing and operating buffer inventoriesare not considered.

Miltenburg (1998) presents the N-UALBPwhere there exist a number N of U-lines with thegiven identical cycle time c. The objective is to as-

sign tasks to stations, while each station can in-clude tasks of at most two adjacent productionlines. With a DP algorithm optimal solutions forsmall problem instances are obtained. Sparling(1998)develops an algorithm to find multiline sta-tions which include tasks of up to three productionlines. Furthermore, he considers the general case,where production line locations are not fixedalready.

8. Assignment restrictions

As discussed in Section 2, several types ofrestrictions may curtail the assignment of tasksto stations. Position related restrictions induce theneed for left hand side and right hand side stations.Especially large workpieces, such as buses ortrucks, acquire stations that only execute tasks atone of their sides, because moving around theworkpiece would cost to much time. This leadsto task related restrictions because left hand side

{1,5}

{9} {6,8}

{2,7}{3,4,

10}

Fig. 5. U-shaped assembly line.



11/22

tasks must not be combined with right hand sideones. In general, tasks that cannot be assigned tothe same station are called incompatible.

Bartholdi (1993) presents two-sided assemblylines, where pairs of workstations are locatedopposite to each other on the left and on the righthand side of the line. Each pair of stations is work-ing on one item at the same time. Tasks aregrouped by the side of the vehicle they can be exe-cuted at. So there are right hand tasks, e.g. mount-ing the right wheel, left hand tasks, tasks whichcan be assigned to either side of the line, e.g.,mounting a radio, as well as tasks that have tobe executed by both paired stations simultane-ously, e.g., installing the rear seat.

Of course, a two-sided line may have unmannedstations. In that case some stations do not have anopposite companion. Bartholdi shows, that for agivencin some cases, depending on the precedenceconstraints, a two-sided line requires less stationsthan a traditional one-sided line, but never requiresmore stations. An important remark is that for apair of opposite stations the precedence constraintsof assigned tasks need to be respected. If we con-sider a task 3 with its predecessors 1 and 2, tasktimes t1


12/22

range constraints which allow for flexibility inprecedence relations, and develop a local searchheuristic.

In a robotic assembly line, certain tools andparts are required for performing tasks by therobot cells (stations). Due to limited space to storethe parts and tools, restrictions for the joint assign-ment of tasks to stations are imposed. Kim andPark (1995) propose a mathematical formulationand a cutting plane procedure for this extensionof SALBP-1.

Operator related restrictions concerning the lev-els of qualification are considered by Johnson(1983)who tries to combine tasks with similar lev-els of complexity in a station. Carnahan et al.(2001) incorporate physical demand into ALBP.Tasks differ in their grip strength demands, whileworkers differ in their grip strength capacity. Inaddition, performing a task fatigues a worker,causing a decrease of his capacity. Workers mustnot be physically overloaded by the tasks thatare assigned to their stations. The objective is tominimize a composite score, which consists ofthe weighted cycle time and a weighted fatiguemeasure for a given number of stations. The prob-lem is solved by a multi-pass heuristic and two dif-

ferent genetic algorithms.

9. Stochastic task times

In all previous sections, we assume the tasktimes to be deterministic. Though this is regularlythe case in highly automated lines (like transferlines, see Section 5), task times usually vary fromcycle to cycle, especially when human operatorsare engaged (cf. Buzacott, 1990; Scholl, 1999,

Chapter 1.3.3). In general, the variance of a task

stime increases with its complexity.Because the literature on stochastic ALBP is

wide, we give only an outline of the differentlines of research for the case of paced assemblysystems (for details, see Scholl, 1999, Chapter2.3.1).

Moodie and Young (1965) assume the tasktimes to be independent normal variates which isconsidered to be realistic in most cases of humanwork. Other distributions are, e.g., examined by

Kao (1976, 1979), Sniedovich (1981) and Nkasuand Leung (1995).

In some cases it is sufficient to use a task time t0j

which contains a safety factor such that the real-ized (stochastic) task time of taskjdoes not exceedthis value at a certain probability. Then, the stoch-astic problem can be (heuristically) transformedinto a deterministic one and solved by modifiedSALBP procedures (for different respective ap-proaches see, e.g., Kottas and Lau, 1976, 1981;Sphicas and Silverman, 1976;Henig, 1986;Carra-way, 1989).

Specific time-oriented objectives of stochasticmodels are the minimization of the probability ofexceeding the cycle time in any station (cf. Reeveand Thomas, 1973) or the stochastic variationsof station times (cf. Raouf and Tsui, 1982). Usu-ally, time-oriented objectives do not consider theconsequences of exceeding the cycle time. Since itis generally not possible to use a cycle time largeenough to obtain sufficiently small excess probabil-ities, one has to consider effects of incompletetasks explicitly:

The conveyor isstoppeduntil the incomplete ormissed operations have been performed online.

So one station can block the production processof all other stations and the production rate isdecreased.

The process continues, while the incompletetasks and all their successors in the precedencegraph are left out. These tasks are performedat special off-line stations after the incompleteunit has left the line or by mobile workers whilethe process continues with a workpiece that hasbeen repaired earlier and waits in a buffer.

Stations which tend to exceed the cycle time get

an additional capacity by employing additionaloperators which increase the work pace butincrease the costs per workpiece.

Regardless of how incompletion is dealt with,they cause additional costs. These incompletioncostsare reduced by decreasing the station utiliza-tions. This can be done by increasing the numberof stations or the cycle time. However, this raisesthe labour and equipment costs (cf. Kottasand Lau, 1973, as well as Sections 4 and 5).



13/22

Furthermore, the ordering of tasks inside the sta-tions influences the incompletion costs with respectto their value and probability (cf. Kottas and Lau,

1976). The higher the potential incompletion costsof a task are, the more idle time has to be intro-duced into a station in order to avoid actualincompletion. Therefore, the idle time should beconcentrated in early stations of the line, becausethe incompletion costs depend on the number ofaffected successor tasks.

Problems with off-line error handling are de-scribed by Kottas and Lau (1973, 1976, 1981),Carter and Silverman (1984). Shtub (1984) addi-tionally considers the possibility of dynamicallyassigning additional operators. Problems with linestoppages in case of task incompletion are exam-ined by Silverman and Carter (1986), Lau andShtub (1987). In either case, it is proposed to solvethe problem in a two-stage manner: (1) Generate anumber of feasible line balances. (2) Select the onewith minimal cost.

More recently,Sarin and Erel (1990) also con-sider a problem with off-line error handling andsolve it heuristically by a restricted DP approach.Sarin et al. (1999) present an improved heuristic.First, the problem is decomposed into subprob-

lems which are solved by the above mentionedDP procedure. After improving these initial solu-tions by applying a branch-and-bound procedurethey are appended forming the final solution.

Gokcen and Baykoc (1999)compare the off-linerepair with the following alternative (see above) bya simulation approach: Every time, a workpieceremains unfinished at a station, the missing tasksare performed by a mobile worker off the lineand then the item enters the buffer. On the produc-tion line the workpiece is replaced by an item from

the buffer which has been repaired earlier. So theobjective has to be expanded in order to includebuffer costs.

Sotskov et al. (this issue) perform a stabilityanalysis of SALBP-1 solutions, i.e., the effect of(ceteris paribus) variations of task times on the sta-bility of the optimal solution is examined. Consid-ering the sensitivity of such a solution one maydecide whether it is necessary to model the prob-lem in a stochastic manner or sufficient to solvesome deterministic problem.

A simulated annealing (SA) approach for astochastic variant of SALBP-1 is proposed bySur-esh and Sahu (1994),a genetic algorithm bySuresh

et al. (1996). McMullen and Frazier (1998) andMcMullen and Tarasewich (2003)develop an SAprocedure and an ant algorithm for a GALBPwith respect to parallel stations, stochastic tasktimes, mixed-model production and alternativeobjectives. SALBP-2 with fuzzy task times is con-sidered byTsujimura et al. (1995).

10. Mixed-model lines

Mixed-model assembly lines manufacture sev-eral models (versions) of a standardized commod-ity in an intermixed sequence (cf. Section 2). Themodels may differ from each other with respectto size, colour, used material, or equipment suchthat their production requires different tasks, tasktimes and/or precedence relations. As a conse-quence, finding a line balance whose station loadshave the same station time and equipment require-ments whatever model is produced is almostimpossible. Therefore, the line must be flexible en-ough with respect to the equipment and the quali-

fication of operators as well as local cycle timeviolations. In opposite to the (deterministic) sin-gle-model case, the cycle time is no longer the max-imum time available in each station to perform thetasks on a workpiece but the average time (definedon the basis of a desired production rate).

By analogy with SALBP, the (medium-term)mixed-model assembly line balancing problem

(MALBP) consists of finding a number of stationsand a cycle time as well as a line balance such thata capacity- or even cost-oriented objective is

optimized. Corresponding to the problem versionsSALBP-1 and SALBP-2, the cycle time or thenumber of stations may be given, respectively,such that different versions MALBP-1, MALBP-2, and MALBP-E arise (cf. Scholl, 1999, Chapter3.2.2). However, the problem is more difficult thanin the single-model case, because the station timesof the different models have to be smoothed foreach station (horizontal balancing; cf. Merengo etal., 1999) in order to avoid operating inefficiencieslike work overload or idle time. Hence, the goals



14/22

of single-model problems should be extended toregard this aspect adequately (see below).

The better this horizontal balancing works, the

better solutions are possible in the connectedshort-term mixed-model sequencing problem(MSP) which is to find a sequence of model unitswhich meets the demands of all models given bythe production program of a short-term planningperiod and optimizes some objective. Such objec-tives mostly correspond to inefficiencies arisingfrom variations of the station times of the models(which are set by the line balance). Usually, the de-mands result from individual orders with certaindelivery dates. Therefore, the sequencing problemarises per shift, day or week with particular de-mands for all models. Surveys of mathematicalmodels and solution procedures for MSP can,e.g., be found in Yano and Bolat (1989), Bardet al. (1992), Scholl et al. (1998), Scholl (1999,Chapters 3.3 and 6.2). Domschke et al. (1996)andScholl (1999, Chapter 3.4) define a hierarchi-cal planning approach connecting MALBP andMSP at several hierarchical levels.

10.1. Characterization of MALBP

MALBP relies on the same basic assumptionsas SALBP (deterministic task times, no assignmentrestrictions, serial line-layout, fixed rate launching;cf. Section 3). Additionally given are task times tjpof the tasks j= 1, . . . ,n for the models p= 1, . . . ,P(with tjp= 0 if j is not required for assembling p)and individual precedence relations for each modelwhich can be combined to a single joint precedencegraph. Furthermore, the expected model mix withdemand portions dp of the different models p=1, . . . , P with

PPp1dp 1 is given (cf. Scholl,

1999, Chapter 3.2). The model mix allows for (par-tially) anticipating the data of the short-term MSP.

For modelling and solving MALBP two basicapproaches can be distinguished: Reduction to sin-gle-model problems and horizontal balancing inthe multi-model context.

10.2. Reduction to single-model problems

A straightforward transformation of the mixed-model data to some version of SALBP consists in

computingaverage task times tjPP

p1dp tjp forthe tasks j= 1, . . . ,n (cf. Thomopoulos, 1970;Macaskill, 1972; McMullen and Frazier, 1997;

Scholl, 1999, Chapter 6.1.2). Solving the resulting(average) SALBP instance guarantees that thecycle time is sufficient to perform all tasks onaverage. However, even in case of an optimal solu-tion for the average model, considerable inefficien-cies may occur when operating the line.

By relaxing the assumption that identical taskshave to be assigned to the same station for allmodels, the mixed-model problem is decomposedinto P independent SALBP instances. A respectivegeneralization of the shortest-path formulation ofSALBP-1 (cf.Scholl and Becker, this issue, Section3.5) is presented by Roberts and Villa (1970) aswell as Rao (1971). However, the assignment ofidentical tasks to different stations is usually notdesired with respect to additional facility require-ments, loss of specialization effects, complicatedproduction control, and setup inefficiencies. Onlyin the case of multi-model production, wherebatches of models are processed, this relaxationof MALBP may be useful (cf. Section 2).

A rather restrictive problem is obtained byimposing thecycle time restrictions for every model

(cf.Deutsch, 1971). That is, the station times of allmodels in all stations must not exceed the (com-mon) cycle time c. This, however, may lead to apoor efficiency of the line, because compensationeffects between the models cannot be utilized.Due to these restrictive capacity constraints, nosequencing problem occurs. A MALBP-1 withmodel-dependent cycle times cp is modelled byGokcen and Erel (1998) and solved through ashortest-path based procedure byErel and Gokcen(1999).

10.3. Horizontal balancing

In any case where the cycle time restriction isformulated on an aggregate/average basis, consid-erable inefficiencies may occur when operating theline. This is due to the variations in the stationtimes spk

Pj2Sk

tjp over the models p= 1, . . . ,Pconsidering a certain station k. This imbalancehas the following impacts on the performance ofa mixed-model line:



15/22

Work overload. Whenever the operator of a sta-tion is not able to complete the assigned tasksbefore the workpiece leaves the station (due to

a restricted station length or due to the trans-port system),work overloadoccurs. Work over-load may be compensated by the temporaryemployment of utility workers, stopping the lineor another sanction (cf. Section 9). Whatever isselected, work overload is inefficient and expen-sive and should be minimized. Unfortunately,the amount of work overload which reallyoccurs cannot be computed for a solution ofthe balancing problem directly, because itdepends on the unknown short-term productionprograms and corresponding productionsequences. Therefore, it is essential to obtainbalances in which potentialwork overload situ-ations are minimized.

Idle time. As in the single-model case, idle timeoccurs when a station has completed its workon a workpiece and has to wait for the nextone arriving at the station. The idle times percycle are constant if only one model is pro-duced. If several models are assembled, the idletimes differ and depend on the sequence. Similarto work overload, not the realized idle times can

be known in advance, but the potential idletimes should be minimized.

In order to reduce horizontal imbalances, addi-tional secondary objectives given the number m ofstations and the cycle time c are proposed in theliterature.

Thomopoulos (1970) proposes the followingobjective function which minimizes the sum ofabsolute differences between the average stationtime sp

Pnj1tjp=m of model p= 1, . . . , P and

its realized station times s

pk in all stationsk= 1, . . . , m:

Minimize W1 Xmk1

XPp1

jspk spj: 1

Thomopoulos proposes a heuristic procedurefor solving some version of MALBP combinedwith the secondary objective(1)that is similar tothe heuristic of Hoffmann (1963) for SALBP- 1.Different heuristics for MALBP-1 including the

one of Thomopoulos are compared by VanZante-de Fokert and de Kok (1997).

The following objective function minimizes the

maximal deviation of a station time of any modelfrom the average station time per unit (cf.Decker,1993):

Minimize W2 maxfjspk spj with

k 1;. . .;m and p 1;. . .;Pg:

2

A third objective function minimizes the sum ofcycle time violations of all models in all stations,i.e., it considers only potential work overload (cf.Domschke et al., 1996):

Minimize W3 Xmk1

XPp1

maxf0; spk cg: 3

Each of the above objectives may also be usedon an aggregate basis by multiplying the single dif-ferences with the demand proportions dp of themodels p.

How to solve some version of MALBP with anarbitrary secondary objective by an exact solutionprocedure for SALBP-1 is explained by Scholl

(1999, Chapter 6.1.2).Theoretical and experimental comparisons of

the objectives by Domschke et al. (1996) andScholl (1999, Chapter 3.2.2.3)shows that the thirdobjective has the best potential in anticipatingand avoiding inefficiencies in operating the line.Another comparison of the first objective func-tion with further ones is performed by Bukchin(1998).

Bukchin et al. (2002)propose a three-stage solu-tion approach for MALBP-1 using different sec-

ondary objective functions as discussed byBukchin (1998). The first stage consists of solvingSALBP-1 for the average (combined) model deter-mining the number of stations and fixing the tasks,which are common to all models, to stations. Thesecond stage reassigns the tasks of each modelwhich are specific for this model preserving thefixed assignments made before and optimizingthe horizontal balancing objective used. The thirdstage is a local search procedure changing theassignment of the common tasks and applying



16/22

stage 2 for completing the solution by assigningspecific tasks as described above.

Merengo et al. (1999)define two horizontal bal-

ancing functions, which are based on weighted dif-ferences between the maximal station time (causedby any model) and the station times of all othermodels. In their station-oriented heuristic (similartoHoffmann, 1963), Merengo et al. use the hori-zontal balancing as primary goal while the minimi-zation of the number of stations (MALBP-1) istaken as secondary goal unlike most other ap-proaches mentioned above.

10.4. Generalized problems

Pastor et al. (2002)consider MALBP-2 with anadditional objective that tries to increase the uni-formity of tasks at the stations. The problem issolved using priority rule based and tabu searchheuristics.

Vilarinho and Simaria (2002) consider andmodel MALBP-1 with additional assignmentrestrictions and parallel stations. As secondaryobjective, terms for measuring vertical and hori-zontal imbalances are minimized. As solution pro-cedure a two-stage simulated annealing approach

is proposed.Karabati and Sayin (2003) consider an

MALBP-2 with a given production sequence con-taining several copies of each model. In everycycle, the maximum station time (considering themodels being currently performed in all stations)defines the realized cycle time. The objective is tominimize the sum of realized cycle times for theproduction sequence. The problem is modelled asa binary linear program and solved by a modifiedpriority rule based procedure. The special case of a

dominating main model is solved as a SALBP-2.Kim et al. (2000b)consider a combined mixed-model balancing and sequencing problem which isbased on avoiding utility work and solved by dif-ferent genetic algorithms. Bock et al. (this issue)also consider a combined balancing and sequenc-ing problem incorporating different problemextensions which is solved by a parallel computingapproach.

Askin and Zhou (1997)present a nonlinear inte-ger program for a cost-oriented extension of

MALBP with parallel stations and equipmentselection. For solving the problem a priority rulebased procedure using the average model is pro-

posed.Chakravarty and Shtub (1985, 1986a) de-scribe priority based and shortest path heuristicsfor another cost-oriented version of MALBP.

Sparling and Miltenburg (1998) describe andanalyse a mixed-model U-line balancing problemand provide a branch-and-bound based solutionprocedure. A combined balancing and sequencingproblem is formulated and solved with a geneticalgorithm byMiltenburg (2002). A further geneticalgorithm for a similar combined problem is pro-posed byKim et al. (this issue).

11. Conclusions and further research

The survey shows that assembly line balancingresearch which traditionally was focused uponsimple problems (SALBP) has recently evolved to-wards formulating and solving generalizedproblems (GALBP) with different additional char-acteristics such as cost functions, equipmentselection, paralleling, U-shaped line layout andmixed-model production. While a lot of relevant

problems have been identified and modelled, how-ever, the development of sophisticated solutionprocedures has just begun. Thus, additional re-search is necessary to adopt state-of-the-art solu-tion concepts like metaheuristics and highlydeveloped enumeration and bounding schemes toGALBP. Furthermore, standardized and realistictest beds are required for testing and comparingmethodical enhancements. Because research hasproduced a large variety of problem definitionswithout a clear direction (one might say by arbi-

trarily combining problem characteristics), itseems to be necessary to provide a classificationwhich facilitates distinguishing and referencingthose problem types.

Going a step further, it is required to developuser-friendly computer software that is flexible en-ough to be applied to real-world problems andcontains state-of-the-art solution procedures. Afirst attempt which tries to combine these needshas been made by the softwarelab Optimal De-sign (www.optimaldesign.com ) by developing

http://www.optimaldesign.com/http://www.optimaldesign.com/


17/22

the MS Windows software Optiline. This softwarecombines user-friendly input of data, interactiveand automatic optimization tools based on genetic

algorithms and tools for analysis of results andreporting. Another system which is based onstate-of-the-art SALBP procedures has been devel-oped byFidan et al. (2003). Most other softwareproducts available for managing assembly systemsdo not contain powerful procedures for assemblyline balancing but concentrate on data manage-ment. To conclude, there is a considerable demandfor further developments.

References

Agnetis, A., Ciancimino, A., Lucertini, M., Pizzichella, M.,1995. Balancing flexible lines for car components assembly.International Journal of Production Research 33, 333350.

Agrawal, P.K., 1985. The related activity concept in assemblyline balancing. International Journal of ProductionResearch 23, 403421.

Ahmadi, R.H., Dasu, S., Tang, C.S., 1992. The dynamic lineallocation problem. Management Science 38, 13411353.

Ajenblit, D.A., Wainwright, R.L., 1998. Applying geneticalgorithms to the U-shaped assembly line balancing prob-lem. In: Proceedings of the 1998 IEEE InternationalConference on Evolutionary Computation, Anchorage,

AK, pp. 96101.Akagi, F., Osaki, H., Kikuchi, S., 1983. A method for

assembly line balancing with more than one worker in eachstation. International Journal of Production Research 21,755770.

Amen, M., 1997. Ein exaktes Verfahren zur kostenorientiertenFliebandabstimmung. In: Zimmermann, U., et al. (Eds.),Operations Research Proceedings 1996. Springer, Berlin, pp.224229.

Amen, M., 2000a. An exact method for cost-oriented assemblyline balancing. International Journal of Production Eco-nomics 64, 187195.

Amen, M., 2000b. Heuristic methods for cost-oriented assemblyline balancing: A survey. International Journal of Produc-

tion Economics 68, 114.Amen, M., 2001. Heuristic methods for cost-oriented assembly

line balancing: A comparison on solution quality andcomputing time. International Journal of Production Eco-nomics 69, 255264.

Arcus, A.L., 1966. COMSOAL: A computer method ofsequencing operations for assembly lines. InternationalJournal of Production Research 4, 259277.

Askin, R.G., Zhou, M., 1997. A parallel station heuristic forthe mixed-model production line balancing problem.International Journal of Production Research 35, 30953105.

Baker, K.R., Powell, S.G., Pyke, D.F., 1990. Buffered andunbuffered assembly systems with variable processing times.Journal of Manufacturing and Operations Management 3,200223.

Bard, J.F., 1989. Assembly line balancing with parallel work-stations and dead time. International Journal of ProductionResearch 27, 10051018.

Bard, J.F., Dar-El, E., Shtub, A., 1992. An analytic frameworkfor sequencing mixed model assembly lines. InternationalJournal of Production Research 30, 3548.

Bartholdi, J.J., 1993. Balancing two-sided assembly lines: Acase study. International Journal of Production Research31, 24472461.

Bautista, J., Suarez, R., Mateo, M., Companys, R., 2000. Localsearch heuristics for the assembly line balancing problemwith incompatibilities between tasks. In: Proceedings of the2000 IEEE International Conference on Robotics andAutomation, San Francisco, CA, 24042409.

Baybars, I., 1986. A survey of exact algorithms for the simpleassembly line balancing problem. Management Science 32,909932.

Bock, S., Rosenberg, O., van Brackel, T., this issue. Controllingmixed-model assembly lines in real-time by using distributedsystems, European Journal of Operational Research.doi:10.1016/j.ejor.2004.07.035.

Boucher, T.O., 1987. Choice of assembly line design under tasklearning. International Journal of Production Research 25,513524.

Bukchin, J., 1998. A comparative study of performancemeasures for throughput of a mixed model assembly linein a JIT environment. International Journal of Production

Research 36, 26692685.Bukchin, J., Masin, M., 2004. Multi-objective design of teamoriented assembly systems. European Journal of Opera-tional Research 156, 326352.

Bukchin, J., Rubinovitz, J., 2002. A weighted approach forassembly line design with station paralleling and equipmentselection. IIE Transactions 35, 573585.

Bukchin, J., Tzur, M., 2000. Design of flexible assembly lineto minimize equipment cost. IIE Transactions 32, 585598.

Bukchin, J., Dar-El, E.M., Rubinovitz, J., 1997. Team orientedassembly system design: A new approach. InternationalJournal of Production Economics 51, 4757.

Bukchin, J., Dar-El, E.M., Rubinovitz, J., 2002. Mixed-model

assembly line design in a make-to-order environment.Computers and Industrial Engineering 41, 405421.

Burns, L.D., Daganzo, C.F., 1987. Assembly line job sequenc-ing principles. International Journal of Production Research25, 7199.

Buxey, G.M., 1974. Assembly line balancing with multipleStations. Management Science 20, 10101021.

Buxey, G.M., Sadjadi, D., 1976. Simulation studies of con-veyor-paced assembly lines with buffer capacity. Interna-tional Journal of Production Research 14, 607624.

Buxey, G.M., Slack, N.D., Wild, R., 1973. Production flow linesystem design - A review. AIIE Transactions 5, 3748.

http://dx.doi.org/10.1016/j.ejor.2004.07.035http://dx.doi.org/10.1016/j.ejor.2004.07.035


18/22

Buzacott, J.A., 1968. Prediction of the efficiency of productionsystems without internal storage. International Journal ofProduction Research 6, 173188.

Buzacott, J.A., 1990. Abandoning the moving assembly

line: Models of human operators and job sequencing.International Journal of Production Research 28, 821839.

Carnahan, B.J., Norman, B.A., Redfern, M.S., 2001. Incorpo-rating physical demand criteria into assembly line balanc-ing. IIE Transactions 33, 875887.

Carraway, R.L., 1989. A dynamic programming approach tostochastic assembly line balancing. Management Science 35,459471.

Carter, J.C., Silverman, F.N., 1984. A cost-effective approachto stochastic line balancing with off-line repairs. Journal ofOperations Management 4, 145157.

Chakravarty, A.K., 1988. Line balancing with task learningeffects. IIE Transactions 20, 186193.

Chakravarty, A.K., Shtub, A., 1985. Balancing mixed modellines with in-process inventories. Management Science 31,11611174.

Chakravarty, A.K., Shtub, A., 1986a. A cost minimizationprocedure for mixed model production lines with normallydistributed task times. European Journal of OperationalResearch 23, 2536.

Chakravarty, A.K., Shtub, A., 1986b. Dynamic manning oflong cycle assembly lines with learning effect. IIE Transac-tions 18, 392397.

Chiang, W.-C., Urban, T.L., 2002. A hybrid heuristic for thestochastic U-line balancing problem, Working Paper, Uni-versity of Tulsa, Oklahoma, USA.

Daganzo, C.F., Blumenfeld, D.E., 1994. Assembly systemdesign principles and tradeoffs. International Journal ofProduction Research 32, 669681.

Decker, M., 1993. Capacity smoothing and sequencing formixed-model lines. International Journal of ProductionEconomics 3031, 3142.

Deckro, R.F, 1989. Balancing cycle time and workstations. IIETransactions 21, 106111.

Deutsch, D.F., 1971. A branch and bound technique for mixed-product assembly line balancing. Ph.D. Dissertation, Ari-zona State University.

Dobson, G., Yano, C.A., 1994. Cyclic scheduling to minimizeinventory in a batch flow line. European Journal ofOperational Research 75, 441461.

Dolgui, A., Ereemev, A., Kolokolov, A., Sigaev, V., 2002a. Agenetic algorithm for allocation of buffer storage capacitiesin production line with unreliable machines. Journal ofMathematical Modelling and Algorithms 1, 89104.

Dolgui, A., Finel, B., Guschinsky, N., Levin, G., Vernadat, F.,2002b. A heuristic approach for transfer lines balancing.Working Paper, University of Technology of Troyes,France.

Dolgui, A., Guschinski, N., Levin, G., 1999. On problem ofoptimal design of transfer lines with parallel and sequentialoperations. In: Fuertes, J.M. (Ed.), Proceedings of the 7thIEEE International Conference on Emerging Technologies

and Factory Automation, Barcelona, Spain, vol. 1, 1999,pp. 329334.

Dolgui, A., Guschinsky, N., Levin, G., 2001a. Decompositionmethods to optimize transfer line with parallel and sequen-

tial machining. In: Binder, Z. (Ed.), Management andcontrol of production and logistics, Proceedings of the 2ndIFAC Conference, Grenoble, France, 2000, vol. 3. Elsevier,Amsterdam, pp. 983988.

Dolgui, A., Guschinsky, N., Levin, G., 2001b. A mixed integerprogram for balancing of transfer line with groupedoperations. In: Proceedings of the 28th InternationalConference on Computer and Industrial Engineering, Flor-ida, USA, 2001, pp. 541547.

Dolgui, A., Guschinsky, N., Levin, G., Harrath, Y., 2001c.Optimal design of a class of transfer lines with paralleloperations. In: Groumpos, P.P., Tzes, A.A. (Eds.), Manu-facturing, modeling, management and control, A Proceed-ings volume from the IFAC Symposium, Patras, Greece,2000, Elsevier, Amsterdam, pp. 3641.

Dolgui, A., Guschinsky, N., Levin, G., this issue. A special caseof transfer lines balancing by graph approach. EuropeanJournal of Operational Research. doi:10.1016/j.eor.2004.07.025.

Dolgui, A., Guschinsky, N., Levin, G., 2003. Optimal design ofautomated transfer lines with blocks of parallel operations.In: Camacho, E.F., Basanez, L., De la Puente, J.A. (Eds.),Proceedings of the 15th IFAC World Congress, Barcelona,Spain, 2002, Elsevier, Amsterdam (on cd-rom).

Domschke, W., Klein, R., Scholl, A., 1996. AntizipativeLeistungsabstimmung bei moderner Variantenflieferti-gung. Zeitschrift fur Betriebswirtschaft 66, 14651490.

Domschke, W., Scholl, A., Vo, S., 1997. Produktionsplanung,2nd ed. Springer, Berlin.Erel, E., Gokcen, H., 1999. Shortest route formulation of

mixed-model assembly line balancing problem. EuropeanJournal of Operational Research 116, 194204.

Erel, E., Sabuncuoglu, I., Aksu, B.A., 2001. Balancing ofU-type assembly systems using simulated annealing.International Journal of Production Research 39, 30033015.

Erel, E., Sarin, S.C, 1998. A survey of the assembly linebalancing procedures. Production Planning and Control 9,414434.

Falkenauer, E., 1997. A grouping genetic algorithm for linebalancing with resource dependent task times. In: Proceed-

ings of the Fourth International Conference on NeuralInformation Processing 1997, University of Otago, Dun-edin, New Zealand, pp. 464468.

Fidan, G., Kandiller, L., Oguztuzun, H., 2003. A decisionsupport system for assembly line balancing, Working paper,Middle East Technical University, Ankara, Turkey.

Freeman, D.R., Jucker, J.V., 1967. The line balancing problem.Journal of Industrial Engineering 18, 361364.

Geoffrion, A.M., Graves, G.W., 1976. Scheduling parallelproduction lines with changeover costs: Practical applica-tion of a quadratic assignment/LP approach. OperationsResearch 24, 595610.

http://dx.doi.org/10.1016/j.eor.2004.07.025http://dx.doi.org/10.1016/j.eor.2004.07.025http://dx.doi.org/10.1016/j.eor.2004.07.025http://dx.doi.org/10.1016/j.eor.2004.07.025http://dx.doi.org/10.1016/j.eor.2004.07.025


19/22

Ghosh, S., Gagnon, R.J., 1989. A comprehensive literaturereview and analysis of the design, balancing and schedulingof assembly systems. International Journal of ProductionResearch 27, 637670.

Globerson, S., Tamir, A., 1980. The relationship between jobdesign, human behavior and system response. InternationalJournal of Production Research 18, 391400.

Gokcen, H., Baykoc, O.F., 1999. A new line remedial policy forthe paced lines with stochastic task times. InternationalJournal of Production Economics 58, 191197.

Gokcen, H., Erel, E., 1998. Binary integer formulation formixed-model assembly line balancing problem. Computersand Industrial Engineering 34, 451461.

Graves, S.C., Holmes Redfield, C., 1988. Equipment selectionand task assignment for multiproduct assembly systemdesign. International Journal of Flexible ManufacturingSystems 1, 3150.

Graves, S.C., Lamar, B.W., 1983. An integer programmingprocedure for assembly system design problems. OperationsResearch 31, 522545.

Hautsch, K., John, H., Schurgers, H., 1972. Taktabstimmungbei Fliearbeit mit dem Positionswert-Verfahren. REFA-Nachrichten 25, 451464.

Held, M., Karp, R.M., Shareshian, R., 1963. Assembly linebalancingDynamic programming with precedence con-straints. Operations Research 11, 442459.

Helgeson, W.B., Birnie, D.P., 1961. Assembly line balancingusing the ranked positional weight technique. Journal ofIndustrial Engineering 12, 394398.

Henig, M.I., 1986. Extensions of the dynamic programmingmethod in the deterministic and stochastic assemblyline

balancing problems. Computers and Operations Research13, 443449.Hillier, F.S., So, K.C., 1991. The effect of machine breakdowns

and interstage storage on the performance of productionline systems. International Journal of Production Research29, 20432055.

Hillier, F.S., So, K.C., 1993. Some data for applying the bowlphenomenon to large production line systems. InternationalJournal of Production Research 31, 811822.

Hillier, F.S., So, K.C., Boling, R.W., 1993. Toward character-izing the optimal allocation of storage space in productionline systems with variable processing times. ManagementScience 39, 126133.

Hoffmann, T.R., 1963. Assembly line balancing with a prece-

dence matrix. Management Science 9, 551562.Ignall, E.J., 1965. A review of assembly line balancing. Journal

of Industrial Engineering 16, 244254.Inman, R.R., Leon, M., 1994. Scheduling duplicate serial

stations in transfer lines. International Journal of Produc-tion Research 32, 26312644.

Iskander, W.H., Chou, J., 1990. Unbalanced production linescheduling with partial job specialization. Naval ResearchLogistics 37, 789805.

Johnson, R.V., 1983. A branch and bound algorithm forassembly line balancing problems with formulation irregu-larities. Management Science 29, 13091324.

Johnson, R.V., 1991. Balancing assembly lines for teams andwork groups. International Journal of Production Research29, 12051214.

Kao, E.P.C., 1976. A preference order dynamic program for

stochastic assembly line balancing. Management Science 22,10971104.

Kao, E.P.C., 1979. Computational experience with a stochasticassembly line balancing algorithm. Computers and Opera-tions Research 6, 7986.

Kao, E.P.C., Queyranne, M., 1982. On dynamic programmingmethods for assembly line balancing. Operations Research30, 375390.

Karabati, S., Sayin, S., 2003. Assembly line balancing in amixed-model sequencing environment with synchronoustransfers. European Journal of Operational Research 149,417429.

Kilbridge, M.D., Wester, L., 1961. The balance delay problem.Management Science 8, 6984.

Kim, H., Park, S., 1995. A strong cutting plane algorithmfor the robotic assembly line balancing problem. Inter-national Journal of Production Research 33, 23112323.

Kim, Y.K., Kim, Y., Kim, Y.J., 2000a. Two-sided assembly linebalancing: a genetic algorithm approach. Production Plan-ning and Control 11, 4453.

Kim, Y.K., Kim, J.Y., Kim, Y., 2000b. A coevolutionaryalgorithm for balancing and sequencing in mixed modelassembly lines. Applied Intelligence 13, 247258.

Kim, Y.K., Kim, J.Y., Kim, Y., this issue. An endosymbi-otic evolutionary algorithm for the integration of balanc-ing and sequencing in mixed-model U-lines. European

Journal of Operational Research. doi:10.1016/j.eor.2004.07.032.Klenke, H., 1977. Ablaufplanung bei Fliefertigung. Gabler,

Wiesbaden.Kottas, J.F., Lau, H.-S., 1973. A cost-oriented approach to

stochastic line balancing. AIIE Transactions 5, 164171.Kottas, J.F., Lau, H.-S., 1976. A total operating cost model for

paced lines with stochastic task times. AIIE Transactions 8,234240.

Kottas, J.F., Lau, H.-S., 1981. A stochastic line balancingprocedure. International Journal of Production Research19, 177193.

Lau, H.-S., Shtub, A., 1987. An exploratory study on stoppinga paced line when incompletions occur. IIE Transactions 19,

463467.Lee, T.O., Kim, Y., Kim, Y.K., 2001. Two-sided assembly line

balancing to maximize work relatedness and slackness.Computers and Industrial Engineering 40, 273292.

Lehman, M., 1969. On criteria for assigning models to assemblylines. International Journal of Production Research 7, 269285.

Lutz, C.M., Davis, K.R., Turner, C.F., 1994. Development ofoperator assignment schedules: A DSS approach. OMEGA22, 5767.

Macaskill, J.L.C., 1972. Production-line balances for mixed-model lines. Management Science 19, 423434.

http://dx.doi.org/10.1016/j.eor.2004.07.032http://dx.doi.org/10.1016/j.eor.2004.07.032http://dx.doi.org/10.1016/j.eor.2004.07.032http://dx.doi.org/10.1016/j.eor.2004.07.032http://dx.doi.org/10.1016/j.eor.2004.07.032


20/22

Malakooti, B., 1991. A multiple criteria decision makingapproach for the assembly line balancing problem.International Journal of Production Research 29, 19792001.

Malakooti, B., 1994. Assembly line balancing with buffers bymultiple criteria optimization. International Journal ofProduction Research 32, 21592178.

Malakooti, B., Kumar, A., 1996. An expert system forsolving multi-objective assembly line balancing problems.International Journal of Production Research 34, 25332552.

Martin, G.E., 1994. Optimal design of production lines.International Journal of Production Research 32, 9891000.

McMullen, P.R., Frazier, G.V., 1997. A heuristic for solvingmixed-model line balancing problems with stochastic taskdurations and parallel stations. International Journal ofProduction Economics 51, 177190.

McMullen, P.R., Frazier, G.V., 1998. Using simulated anneal-ing to solve a multiobjective assembly line balancingproblem with parallel workstations. International Journalof Production Research 36, 27172741.

McMullen, P.R., Tarasewich, P., 2003. Using ant techniques tosolve the assembly line balancing problem. IIE Transactions35, 605617.

Merengo, C., Nava, F., Pozetti, A., 1999. Balancing andsequencing manual mixed-model assembly lines. Inter-national Journal of Production Research 37, 28352860.

Miltenburg, J., 1998. Balancing U-lines in a multiple U-linefacility. European Journal of Operational Research 109, 1

23.Miltenburg, J., 2000. The effect of breakdowns on U-shapedproduction lines. International Journal of ProductionResearch 38, 353364.

Miltenburg, J., 2002. Balancing and scheduling mixed-model U-shaped production lines. International Journal of FlexibleManufacturing Systems 14, 119151.

Miltenburg, J., Wijngaard, J., 1994. The U-line line balancingproblem. Management Science 40, 13781388.

Monden, Y., 1998. Toyota production systemAn integratedapproach to just-in-time, 3rd ed. Kluwer, Dordrecht.

Moodie, C.L., Young, H.H., 1965. A heuristic method ofassembly line balancing for assumptions of constant orvariable work element times. Journal of Industrial Engi-

neering 16, 2329.Nakade, K., Ohno, K., Shanthikumar, J.G., 1997. Bounds and

approximations for cycle times of a U-shaped productionline. Operations Research Letters 21, 191200.

Nicosia, G., Pacciarelli, D., Pacifici, A., 2002. Optimallybalancing assembly lines with different workstations. Dis-crete Applied Mathematics 118, 99113.

Nkasu, M.M., Leung, K.H., 1995. A stochastic approach toassembly line balancing. International Journal of produc-tion research 33, 975991.

Park, K., Park, S., Kim, W., 1997. A heuristic for an assemblyline balancing problem with incompatibility, range, and

partial precedence constraints. Computers and IndustrialEngineering 32, 321332.

Pastor, R., Corominas, A., 2000. Assembly line balancing withincompatibilities and bounded workstation loads. Ricerca

Operativa 30, 2345.Pastor, R., Andres, C., Duran, A., Perez, M., 2002. Tabu search

algorithms for an industrial multi-product and multi-objec-tive assembly line balancing problem, with reduction of thetask dispersion. Journal of the Operational ResearchSociety 53, 13171323.

Pike, R., Martin, G.E., 1994. The bowl phenomenon inunpaced lines. International Journal of ProductionResearch 32, 483499.

Pinnoi, A., Wilhelm, W.E., 1997. A family of hierarchicalmodels for assembly system design. International Journal ofProduction Research 35, 253280.

Pinnoi, A., Wilhelm, W.E., 1998. Assembly system design: Abranch and cut approach. Management Science 44, 103118.

Pinto, P.A., Dannenbring, D.G., Khumawala, B.M., 1975. Abranch and bound algorithm for assembly line balancingwith paralleling. International Journal of ProductionResearch 13, 183196.

Pinto, P.A., Dannenbring, D.G., Khumawala, B.M., 1981.Branch and bound and heuristic procedures for assemblyline balancing with paralleling of stations. InternationalJournal of Production Research 19, 565576.

Pinto, P.A., Dannenbring, D.G., Khumawala, B.M., 1983.Assembly line balancing with processing alternatives: anapplication. Management Science 29, 817830.

Powell, S.G., 1994. Buffer allocation in unbalanced three-

station serial lines. International Journal of ProductionResearch 32, 22012217.Rachamadugu, R., Talbot, B., 1991. Improving the equality of

workload assignments in assembly lines. InternationalJournal of Production Research 29, 619633.

Rao, D.M., 1971. Single and mixed-model assembly linebalancing methods for both deterministic and normallydistributed work element times. M.S. Thesis, IndustrialEngineering Department, Oregon State University.

Raouf, A., Tsui, C., 1982. A new method for assembly linebalancing having stochastic work elements. Computers andIndustrial Engineering 6, 131148.

Reeve, N.R., Thomas, W.H., 1973. Balancing stochasticassembly lines. AIIE Transactions 5, 223229.

Rekiek, B., Delchambre, A., 2001. Assembly line balancing andresource planning: What is done and what is still missing.In: Proceedings of CARS and FOF, Durban, South Africa,2001, pp. 8693.

Rekiek, B., de Lit, P., Delchambre, A., 2002a. Hybrid assemblyline design and users preferences. International Journal ofProduction Research 40, 10951111.

Rekiek, B., Dolgui, A., Delchambre, A., Bratcu, A., 2002b.State of art of optimization methods for assembly linedesign. Annual Reviews in Control 26, 163174.

Roberts, S.D., Villa, C.D., 1970. On a multiproduct assemblyline-balancing problem. AIIE Transactions 2, 361364.



21/22


22/22

Wilson, J.M., 1986. Formulation of a problem involvingassembly lines with multiple manning of work stations.International Journal of Production Research 24, 5963.

Yano, C.A., Bolat, A., 1989. Survey, development, and

application of algorithms for sequencing paced assembly

lines. Journal of Manufacturing and Operations Manage-ment 2, 172198.

Zapfel, G., 1975. Ausgewahlte fertigungswirtschaftliche Opti-mierungsprobleme von Fliefertigungssystemen. Beuth,

Berlin.


1s 6 balanceamento - artigo 5

Documents