sequencing problem. Former research has focused mainly on leveling heuristic for the case of color{batch{sequencing.

Transcription

1 Objectives for Order{Sequencing in Automobie Production Christoph Enge and Jurgen Zimmermann University of Karsruhe Afons Steinho IBM Informationssysteme GmbH Abstract: The probem of sequencing units on a mixed{mode assemby ine can be viewed with severa objectives in mind. This paper presents dierent optimization criteria and objectives for the order{ sequencing probem. Former research has focused mainy on eveing procedures for mode{sequencing and has emphasized materia suppy. In contrast, we provide a poynomia heuristic for order{sequencing by eveing the workoad. In the context of automobie production we investigate dierent sequencing poicies and introduce an extended heuristic for the case of coor{batch{sequencing. For dierent types of objectives the performance of the heuristics presented is anayzed, taking known heuristics into consideration. 1 Introduction Goba competition forces enterprises, particuary in automobie industry, to increasing customer orientation and product diversication. With respect to the production system, this resuts in buid{to{order production, out{sourcing of capacities, and the integration of pre{manufactured sub{systems. Due to that deveopment, there arise new requirements to materia suppy systems and sequencing procedures. In a buid{to{ order production the conguration of each product is determined by an individua seection of options corresponding to a customer order. In a buid{to{pan production ony a few mode types are produced repeatedy. The variation in workoad per order increases, if we consider buid{to{order production of individua products instead of a buid{to{pan production of a few mode types.

2 In order to achieve a smooth workoad distribution at the assemby stations, the extension of procedures for mode{sequencing to the case of order{sequencing is necessary. A mode{sequence is a production sequence where each unit represents a mode type. In an order{sequence each unit of the sequence corresponds to a customer order and, therefore, is individua in its conguration. With respect to automobie production a batch{sequence may denote, e.g., a sequence where orders of a uniform coor are combined to severa coor batches. For the basic concepts of assemby ine sequencing we refer to [1, 12, 13, 23]. Increasing integration of sub{systems and buid{to{order production resut in an order{based component fabrication. In connection with Just{in{Time (JIT) production systems, this requires a sucient ook ahead of the order{sequence. In the case of order{sequencing, the underying mode demands, in genera, are equa to one. However, many agorithms (approximatey) soving the sequencing probem (cf. [10, 17, 24]) consider production rates of the underying modes by determining a mode{ sequence and, therefore, cannot be appied to the case of order{ sequencing. Agorithms that consider production rates, which are not based on mode types, can easiy be adapted to the case of order{sequencing. In genera, we can distinguish between agorithms for mode{ and order{sequencing or, with respect to the objective, between workoad{ and component{based approaches. One of the rst approaches considering the workoad of modes was presented by Thomopouos [28], who treated the sequencing of assemby ines in combination with the baancing probem. Open and cosed stations are described and four kinds of ineciencies termed ideness, deciency, congestion, and utiity work are introduced. The unit to be schedued next is determined by comparing penaty cost that are incurred by these ineciencies. A drawback of this proceeding is the accumuation of modes with high workoad at the end of the sequence. [28] motivates further research onworkoad based sequencing by Gorke & Lentes [9] and Macaski [16]. Macaski aternatey schedues a mode with owest penaty cost and a mode with highest workoad, which does not incur any utiity work.

3 Dar{E [4] and Dar{E & Cother [5] studied the minimization of the ine ength by buiding mode{sequences with minima operator dispacements from the eft station border. These approaches are ony usefu in combination with the design and the baancing of assemby ines, since a change in the mode{mix impies a change in the ine ength. Okamura&Yamashina [21] proposed an improvement method for the minimization of the maxima operator dispacement from the eft station border, which is considered to be equivaent to the risk of stopping the conveyor. Tsai [29] introduced an optima agorithm for minimizing the maxima operator dispacement and the tota utiity work for the singe station case. Finay, Sumichrast et a. [26, 27] transformed the agorithm of Monden [20] to an agorithm for workoad eveing instead of eveing the usage of components. Leveing the variation in component usage is the objective of a second category of agorithms. This idea has been introduced by Monden [20], who describes two scheduing agorithms used by Toyota. The rst aternative, known as Goa Chasing I, consecutivey schedues the mode that incurs the minima mean squared deviation between the expected accumuated component usage and the actua accumuated component usage. A simpied approach, termed Goa Chasing II, ony takes the few critica parts into consideration and schedues the mode that woud, if not schedued, incur the maxima deviation from the expected component usage. Mitenburg [17] adopted the idea of eveing the usage of components and suggested three improved scheduing heuristics. Mitenburg & Sinnamon [18, 19] generaized the previous approach to the case of muti{eve production systems by eveing the production rates of the corresponding sub{assembies. Apart from the aforementioned priority{rue{based heuristics, various soution techniques for the sequencing probem on mixed{ mode assemby ines are discussed in the open iterature. Bard et a. [2] suggest a tabu search agorithm which seeks to minimize the tota ine ength and to eve the component usage by a muti{criteria objective. Branch and bound techniques have been used by Scho [23] to minimize work overoad, as we as by

4 Boat [3], who additionay considered setup costs. Kim et a. [11] present a genetic agorithm for the minimization of the tota ine ength and Rachamadugu & Yano [22] propose a Markov process approach to minimize work overoad. Moreover, Steiner & Yeomans [24, 25] investigated a graph{ theoretic procedure to minimize the deviation between actua and expected production rate of modes. McCormick et a. [15] devise a transformation to a network ow formuation, whereas Kubiak & Sethi [14] introduce a transformation to the we{known assignment probem. Decker [6] considered a transformation to the traveing saesman probem. As mentioned in Decker [6] and Domschke et a. [7], the sequencing probem of mixed{mode assemby ines shows a cose reationship to the permutation ow shop probem incuding eariest and atest start times. Nevertheess, this probem cass cannot easiy be adapted to the probem of sequencing mixed{mode assemby ines because minimizing the makespan, as the most common objective in permutation ow shop, is not of crucia interest in the context of mixed{mode assemby ine sequencing. In the foowing section, we describe the sequencing probem on mixed{mode assemby ines. We discuss possibe objectives, before we present a probem formuation. In the third section, we devise an agorithm for the assemby ine order{sequencing (AOS) based on the eveing of workoad. We iustrate the sequencing procedure by an exampe and present an extended assemby ine batch sequencing agorithm (ABS) with respect to the requirements of automobie production. In the fourth section, we briey describe experimenta resuts concerning the performance of the AOS{agorithm and evauate dierent poicies of buiding sequences in automobie production. Finay, we give concusions of this study, evauate the operative usefuness of our agorithms, and give an outook to further deveopments of the suggested approach.

5 2 Probem formuation In this section we consider the eect of order{sequencing on the performance of mixed{mode assemby ines. We discuss dierent criteria and objectives to evauate the performance of an assemby ine. Then, we briey cassify procedures for the assemby ine sequencing probem described in the iterature and give a motivation for the concept of workoad eveing. Finay, we provide an integer programming formuation for the sequencing probem in question. In the context of panning and running assemby ines we can distinguish between two main probem types [28]: Assemby ine baancing Determination of an order{sequence In what foows, we dea with the atter probem, the determination of a \good" order{sequence. In the open iterature, the sequencing probem is either considered in a rather short{term context [23] or as part of the process of ine baancing [16, 28]. We consider the sequencing probem in reation to the baancing probem for the foowing reason: the optimization of the ine baance as we as the optimization of the order{sequence shoud improve the eciency of an assemby ine. In order to evauate the performance of an assemby ine in process, we suggest the foowing criteria: Utiity work Labor utiization Component usage Utiity work denotes the work overoad which cannot be performed by the reguar operators of each station. We distinguish between the distribution and the maximum of utiity work per station. The distribution of utiity work to the stations and over the time determines the number and aocation of necessary \utiity workers". Reducing the maximum utiity work is considered to be equivaent to reducing the risk of stopping the conveyor [21]. A high abor utiization obviousy corresponds to a high productivity of the assemby ine. The abor utiization UT at station

6 ( = 1 ;:::;s) can be dened as i o2oi Pn Tn 0 U k=1 UT := nw s where Tn denotes the accumuated workoad at station for a sequence of ength n. Uk denotes the utiity work incurred at station by the unit in sequence position (stage) k. The abor capacity of station can be computed by nws, where s denotes the number of units in station. We assume, that station has a xed rate aunch interva and that each unit in the station is processed by w operators. The abor utiization UT can be increased by reducing the abor capacity nws or by decreasing the accumuated utiity work Pn k=1 Uk of the sequence. Determining the abor capacity of each station is part of the ine Pn baancing. The accumuated utiitywork k=1 Uk depends on the sequence and the abor capacity. Thus, the utiization UT of each station can ony be determined, if the ine baance is known and a sequencing agorithm is avaiabe for a given order set. Therefore, sequencing does not ony represent a short{term probem, but has to be considered in the context of ine baancing, too. Obviousy, the same sequencing procedure shoud be used to evauate the ine baance as to determine the order{sequence. This fact is not appropriatey covered in recent research. Finay, with respect to materia suppy, a constant rate of component usage reduces the eort of materia suppy and eads to a smooth production in the underying sub{assembies. We nowintroduce three objectives for the order{sequencing probem reated to the evauation criteria mentioned above. Leveing the deviation of the actua from the expected accumuated workoad unti stage k eads to uniform workoad over the sequence. Due to that, the dispacement of operators in their stations is reduced and utiitywork becomes ess probabe. Thus, we rst consider the objective of eveing the workoad in the dierent stations over the sequence. Let order i possess options o 2 Oi. Then, the workoad X t := p o k

7 is required to assembe order i at station, where po denotes the workoad caused by option o at station. The order i assigned to sequence position k is denoted by unit ik. The accumuated workoad k n s XX k X iq q=1 Pn 2 Min. ( kt 0 T ) (2: 1) k=1 =1 i i=1 represents the tota squared deviation of the accumuated workoad Tk from the expected accumuated workoad kt of each station and of each stage k. The second objective toevauate the quaity of a sequence considers the minimization of tota utiity work. In order to determine the tota utiity work some detais of the underying assemby ine are required. According to [16, 28] we assume a paced assemby ine with a xed rate aunchinterva and open stations u with upstream aowance time t and downstream aowance time d t. Concurrent work is assumed to be not aowed, which means that two options assigned to dierent stations cannot be assembed simutaneousy. The ength s of station is indicated by the number of units assigned to the station at the same time. Furthermore, the number of operators w assigned to each unit at station may dier from station to station. The arriva time ak and the departure time dk of the unit in position k at station ( = 1 ;:::;s) are given by X0 1 a k 0 s k T := performed at station unti position (stage) k depends on the currenty schedued units iq( q = 1 ;:::;k). The average workoad t performed at station per unit is cacuated by such that the objective t := t t n ; := ( 1) + k q=1 q

8 and d := a + s : k k The eariest possibe start time to process the unit of stage k at u station is equa to ak 0 t, i.e., the point in time when the unit of stage k reaches the upstream imit. The areas between station boundary and upstream imit aswe as downstream imit are caed overap areas of the station. The station ength enarged by these overap areas is termed working area. With respect to utiity work, we assume that the atest possibe nish time of the d unit in position k at station is equa to the time dk + t when the unit in question reaches the downstream imit. The conguration of a station incuding the working area as we as the corresponding overap areas is depicted in Fig. 1. Figure 1: Conguration of station The part of the workoad that cannot be performed within the working area is assumed to be utiity work. Therefore, utiity work occurs, i the expected nish time sk + t i =w k exceeds the d atest possibe nish time d + t and is dened by k t U := max 0 ;s + ( d + t ) : k ik d f k 0 k g w Here, the start time sk of unit ik at station depends on the eariest possibe start time ak 0 t when unit i u k enters the working area, the nish time fk;01 of unit ik at station 01 and the avaiabiity of operators at station. In order to determine whether operators are avaiabe to process the unit in question, we assign

9 the operators to the units at station foowing this poicy: the s sw operators avaiabe at station are divided into s teams of w operators, which are assigned to one unit each. Within the station preemption of workoad is assumed to be aowed. Let ik be the unit entering station next. If the w operators of team s assigned to unit ik0shave nished their workoad, they continue with unit ik0s Simutaneousy, team s 1 shifts to unit ik0s+2, and so on. Team 1 of station is now without any unit and ready to process the unit ikentering the station. In consequence, we consider the nish time fk0s ; of the unit ik0s in order to determine the avaiabiity of operators to process the unit ik that enters station. Start time sk and nish time fk of unit ik at station s fa 0t;f ;f g n s XX u k0s ; k;01 d f fs t =w ; d t g k k ik k k=1 =1 k +2; k+2 k +2;01 k+2 can be cacuated by k := max k := min + + where we set a11 := 0, f0 := 0 ( = 1 ;:::;s), and fk0 := 0 ( k= 0s ;:::;n). The minimization of tota utiity work can be formuated as Min. U : (2: 2) The start and nish times at station are iustrated by Fig. 2, which depicts station with a station ength of s = 2 units. The horizonta bars represent the unit ik of that stage k, given on the ordinate. The ength of station is represented by the gray area; dotted ines parae to the station boundary mark the aowance imits of the stations as mentioned in the egend. The intersections of the boundary ines of station and the bottom ine of unit{bar ik represent arriva time ak as we as departure time dk of unit ik at station ; the start time of unit ik at station is determined by the nish time fk 02; of unit ik02 at station. The handing of utiity work is shown by unit i. That part of the workoad k01 d exceeding the downstream imit dk 01; + t is eiminated and is assumed to be performed by utiity workers. The start time s of unit i at station is determined by the nish time f of unit i at the previous station 01. k

10 Figure 2: Start times of orders at station The third criteria to evauate a sequence is a smooth component usage. In the case of automobie production a product option consists of severa components, whereas each component beongs to exacty one product option. A uniform distribution of options over the sequence is considered to be equivaent to a uniform usage of components. Dierent options show dierent frequencies in the order set. Thus, a measure to make dierent options comparabe with respect to their distribution over the sequence has to be dened. Therefore, we cacuate the variation coecient s/ o xo of the distances k2 0 k1 between two consecutive orders ik and i 1 k2 with option o. Mean distance x o and standard deviation s o can be cacuated by x o := 1 n n Xo o jo=2 k ojo 0 k o;jo01 and s := o vu u t n o n Xo jo=2 ( k 0 k 0 x) 2 ojo o;jo01 o

11 respectivey, where no denotes the number of orders with option o in the order set and koj the position of the j o o{th order with option o. Given x o and s o, we can formuate the third objective, i.e. the minimization of the mean variation coecient of a N options as N Xo 1 s Min. N o o=1 o x : : o o (2 3) In this paper the main emphasis is on the eveing of workoad as done by Sumichrast [26], whereas most sequencing agorithms known from iterature consider one of the foowing objectives to determine a \good" order{sequence: Minimization of work overoad [23, 30] Leveing occurrence of modes types [17] Leveing component usage [20] As mentioned above, the minimization of tota work overoad is equivaent to the minimization of tota utiitywork. However, the minimization of tota work overoad does not necessariy ead to an even distribution of workoad, which is desired for reasons of ergonomics and continuity of work, as we as for quaity reasons. In order to avoid this drawback, the eveing of workoad has to be considered for each station. The rea objective of eveing the occurrence of modes over the sequence or eveing the usage of components is to achieve an even materia suppy. Impicity, these approaches intend to eve the workoad and to minimize the variation in dispacement of the operators. In the case of buid{to{order production, eveing of modes is impossibe, since the mode demand is equa to one. In practica appications the determination of an order{sequence is often done by eveing the usage of components. Hereby, components are cassied with respect to their inuence on materia suppy and the variation in workoad on the ine. We distinguish between: basic components optiona components order{dependent components

12 Basic components are required for each unit in an identica manner and ead to a constant workoad and a constant eort of materia suppy at the stations. Optiona components are used ony in some units, depending on the order conguration, and cause additiona workoad and suppementary eort of materia handing. Components required for each unit in an order{dependent conguration dier in workoad, but do not cause an additiona eort of materia handing. Thus, ony optiona components need to be considered, if the focus is on the additiona eort of materia handing. Dierent components ead to dierent workoads at the stations, such that the eveing of components does not necessariy ead to a uniform workoad. Furthermore, dierent components, that incur workoads at the same station, may resut in a high utiity work at that station. Thus, high utiity work is possibe, even if each component shows a uniform distribution over the sequence. Due to the reationship between the usage of a component and the workoad p at a station, we expect i o 1. an even distribution of components causing intensive workoad 2. an even distribution of a modest tota utiity work over the time horizon in eveing the workoad. The above considerations indicate that eveing the workoad is a reasonabe objective to order{sequencing on mixed{mode assemby ines. Using (2.1) we present the workoad eveing probem (WLP) for order{sequencing on a mixed mode assemby ine. The objective of WLP is to minimize the mean squared deviation of the actua accumuated workoad from the expected accumuated workoad of each stage at each station. The decision variabe xik indicates whether an order i has aready been schedued at position k k of the sequence and is given by x ik = ( 1; if order i is schedued in position ki k ; 0 otherwise

13 Thus the WLP can be formuated as PP n s Min. ( kt 0x t ) k=1 =1 Pn i=1 ik ik k ik 2 s.t. x = k ( k = 1 ;:::;n) (2: 4) 0 x x 1 ( i = 1 ;:::;n; k = 2 ;:::;n) (2: 5) i;k01 ik x 2f ; g i ;:::;n k ;:::;n : ik 01 ( = 1 ; =1 ) (2 6) Restriction (2: 4) and (2: 6) ensure that there are exacty k units schedued unti stage k and restriction (2: 5) guarantees that each order i is ony schedued once. 3 Sequencing agorithm In this section weintroduce an agorithm for the order{sequencing probem WLP on mixed{mode assemby ines. We describe the basic ideas of the approach and give a forma representation of the basic workoad eveing agorithm. Then, we iustrate the appication of the agorithm by an exampe. Next, we discuss probems that arise in the practica appication of sequencing agorithms in automobie production. With regard to that case, we suggest an extended workoad eveing agorithm computing a batch{sequence. The agorithm to be proposed is an iterative greedy heuristic. The two characteristic features are the eveing of workoad and the determination of an order{sequence. Recent research in the ed of sequencing on mixed{mode assemby ines emphasizes the eveing of the production rate of outputs and the determination of mode{sequences [13]. A workoad based sequencing agorithm was proposed by Sumichrast [26], who used, as the rate to eve, the accumuated workoad Tn of the order set at station divided Ps by the tota workoad =1 Tn of the order set at a stations. In contrast to Sumichrast, we consider the average workoad t at

14 station over the sequence as the expected workoad of station at each stage. Thus, the accumuated workoad expected to be performed unti stage k at station is given by kt. The agorithm consecutivey schedues order i ;:::;i where at each stage k the 1 n 3 eigibe order i with minima priority vaue vi3k is schedued. The priority vaue vik of each eigibe order i 2 E at stage k is given by the minima squared deviation of the expected accumuated workoad kt from the actua accumuated workoad Tk 01; + ti. 3 If there is more than one eigibe order i with minima priority vaue vi3k, the order with smaest index is chosen. A representation in pseudo code of the agorithm, approximatey soving the 2 WLP with time compexity Ons ( ), is given in the foowing: Agorithm [AOS] Step 1: Initiaization k:= 1 E := f1; :::; ng For a 2f1 ;:::;sg: T := 0 and t := Step 2: Sequencing the orders Whie E 6 = ; Do Ps For a i 2E : v := ( kt 0T 0t ) =1 i fi 2 Ev j v g k ik ik k := min = min j2e P k k 01; ik n t i i =1 n 2 k 01; i E:= Enfikg For a 2f1 ;:::;sg: T := T + t k := k+1 End (Whie) jk In order to iustrate the proceeding of the AOS{agorithm, we consider an exampe with ve stations and six orders. Again, each order consists of severa options and each option may incur workoad at severa stations. Therefore, the set of options Oide- termines the workoad t of order i at station. Tabe 1 shows i

15 the workoads ti of orders i =1;:::; 6 at stations 1;:::;5. These quantities ead to a constant average workoad of t = 2: 7 time units at each station, depicted in the bottom ine of Tabe 1. Assuming a xed{rate aunch interva of three minutes, we obtain an expected abor utiization of 90% at each station. order i n station : 42 : 13 : 43 : 17 : 2 14 : 18 : 13 : 31 : 17 : 3 52 : 24 : 49 : 31 : 45 : 4 34 : 18 : 37 : 31 : 33 : 5 14 : 42 : 13 : 19 : 17 : 6 34 : 18 : 37 : 07 : 33 : t 27 : 27 : 27 : 27 : 27 : Tabe 1: Aocation of workoad t i Appying the AOS{agorithm, we compute the priority vaues v of each order i. The priority vaues vi1 ( i = 1;:::; 6) are shown in the second row oftabe 2. i1 stage k norder i : 562 : 14: : 754 : 666 : : 456 : 28: : 13: : : 890 : 626 : : 816 : 22: : : 6 0 Tabe 2: Priority vaue v ik of orders For exampe, with T0 := 0 for each station, the priority vaue v of order i= 1 at the stage k = 1 can be cacuated by (2: 7 1: 4) + (2: 7 4: 2) + (2: 7 1: 3) + (2: 7 4: 3) + (2: 7 17) : =946 :

16 Since order i = 4 has the minima priority vaue vi 3 1,we set i1 = 4. Atevery further stage k, the priority vaues vik of units, that are not yet schedued, are cacuated. For exampe, the priority vaue v of order i = 1 at stage k = 2 can be computed by (5: 4 34 : 1: 4) + (5: 4 18 : 4: 2) + (5: 4 37 : 13) : (5: 4 31 : 4: 3) + (5: 4 33 : 17) : =504 : Continuing with the AOS{agorithm, we obtain the order{sequence(4; 5; 6; 1; 3; 2)with an objective function vaue equa to 20.7, whereas an optima sequence is (4; 1; 6; 5; 3; 2)with an objective function vaue equa to With regard to automobie production, an agorithm for the determination of an order{sequence has to consider, additionay, the structure of the production system and the constraints of materia suppy. Before we investigate sequencing poicies in context of automobie production, we give a short overview of the organization of the production system considered. In automobie production the three sub{systems body shop, paint shop and assemby shop are distinguished. Each sub{system has dierent production and scheduing restrictions. Body shop as we as assemby shop are organized as a mixed{mode{system, whereas the paint shop is typicay a muti{mode ine. With respect to sequencing, the \modes" are dened by specic shop{ reated options, which dier from shop to shop. For instance, in the body shop the number of doors and the sunroof may determine the mode type of an order, whereas in the paint shop modes are dened by the coor, in genera. The basic idea of the new approach is to dene modes in the assemby shop with respect to the options of an order. Since it is unikey that two cars possess the same set of options, a mode demand equa to one has to be assumed in the assemby shop. Since in dierent shops dierent options are considered to dene mode types, the mode types dier from shop to shop. Therefore, an optima sequence for the paint shop in genera does not correspond to an optima sequence for the other shops.

17 In what foows, we investigate the probem \how to provide the assemby shop with a good order{sequence". Here, the procedure of order{sequencing for the assemby shop depends on: the sequencing poicy the quaity of the painting process with respect to sequencing the performance of the sorting buer providing the assemby shop Sequencing poicies dier in the point in time at which the sequence is determined, the ocation in the production system where the sequence is buit physicay, the orders eigibe for each position of the sequence, and other technoogica constraints. The quaity of the paint process with respect to the sequencing probem depends on the probabiity of rework and the ength of rework cyces. The order{sequence can be changed in sorting buers between the shops. The performance of a sorting buer depends on its size, the type of accessing stored units and the veocity of providing an expected unit. We nowinvestigate three sequencing poicies with respect to the resuting order{sequences in the assemby shop: 1. resorting a batch{sequence disturbed in the painting process 2. scheduing a buer{sequence on the basis of the units avaiabe in the sorting buer 3. scheduing an \optima" sequence in the assemby shop, assuming that each unit can be provided by the sorting buer in time Appying the rst poicy, we suppose that the sequences in the paint shop and in the assemby shop are identica. In determining the batch{sequence, we have to consider the size of coor batches in the paint shop as we as the workoad of orders at the assemby stations. The advantage of this poicy is a ong{term ook ahead of the order{sequence to assembe. In the paint shop the sequence of

18 units is disturbed by rework cyces. The abiity to resort a units disturbed in the paint process depends on the performance of the sorting buer. The consideration of the coor batch restriction, generay, resuts in a reduced quaity of the sequence with respect to the eveing of workoad. Providing a buer{sequence according to the second poicy resuts in a short ook ahead of the orders entering the assemby shop next. That is unfavorabe with regard to materia suppy. The third poicy entais a ong ook ahead of the sequence in the assemby shop. However, the sequence of units eaving the paint shop is not deterministic due to the possibiity of rework in the paint shop. Therefore, this poicy is of more theoretica signicance but it can be used as a reference for the quaity of the sequences according to the poicies 1 and 2, respectivey. In order to determine the batch{sequence of poicy 1, we propose an extended workoad eveing agorithm termed as assemby ine batch sequencing agorithm (ABS). For the determination of order{sequences according to poicy 2 and 3 we use the AOS{ agorithm. Here, we can appy the AOS{agorithm for poicy 2, if we consider the orders avaiabe in the sorting buer to be the set of eigibe orders at each stage. The basic idea of the ABS{agorithm is to determine a sequence of batches and then to schedue orders within each batch with respect to the eveing of workoad. In doing so, we rst 3 3 choose the coor c of the orders of the next batch. Coor c is the coor with the maximum positive deviation of the actua from the expected amountofschedued orders with coor c. If there is more 3 than one coor c with that property, the coor with smaest index is chosen. The actua size of the coor batch AB is initiay set to the minimum of batch size B and the amount of eigibe orders 3 with coor c. Then, we schedue AB orders of the current coor 3 c according to the AOS{agorithm where ci denotes the coor of oder i. This procedure is repeated unti a orders are schedued. A forma representation of the ABS{agorithm with time com- 2 pexity Ons ( )isgiven as foows:

19 Agorithm [ABS] Step 1: Initiaization k := 1; AB := 0 E := f1; :::; ng For a c 2f1;:::;N gdo E fj ic cg c ACc := S := 0 c := = jecj n 0 2f1;:::;Ncg c3 Ps k f 2 c3j ik jkg j2ec3 Pn 2f ;:::;sg T t t =n E 6 ; i For a 1 : := 0 and := Step 2: Sequencing the orders Whie = AB := min ( B; je j) c 2 2 c3 ik 0 k 01; 0 i =1 i i=1 If AB = 0 Then Do 3 c := min fce j c 6 = ; ^( kacc 0Sc) = max ( kac 0S ) g For a i E : v := ( kt T t ) i := min i E v = min v E := E nfikg Ec3 := Ec3 nfikg Sc3 := Sc3 +1 For a 2f1 ;:::;sg: T := T + t k := k+1; AB := AB 01 k k 01; ik End (Whie) In genera, the ABS{agorithm can be used if the workoad ti of each order i at each station is given and if a mode type can be assigned to each order.

20 4 Experimenta performance anaysis We briey report on an experimenta anaysis of the agorithms introduced in Section 3. First, detais of the underying assemby ine are presented. Then, the AOS{agorithm is compared with the two workoad eveing heuristics of [13] and [26]. Finay, an evauation of the sequencing poicies proposed in the previous section is given and further experiments are briey discussed. For a detaied view, we refer to Enge [8]. The experimenta anaysis was part of a recent research, initiated by IBM Informationssysteme GmbH, Germany, concerning production panning of mixed{mode assemby ines in automobie production. The described sequencing poicies have been used for the evauation of mixed{mode assemby ines in automobie production. Sequences are evauated by the eveing of workoad WL, the tota amount of utiity work U and the eveing of options OL according to the objectives (2.1), (2.2) and (2.3), respectivey. With respect to the characteristics of automobie production, we consider a paced assemby ine with s = 30 stations, where the station engths s of three specic stations are equa to 6, 3, and 4 units. The remaining stations obtain a station ength of s = 2 units. w = 2 operators are aocated to each unit at each station ( = 1 ;:::;s). The xed aunch rate is given by three minutes. u d Upstream aowance time t and downstream aowance time t of each station are uniformy set to 50% of the xed aunch rate. Frequencies of No = 20 options and workoad po incurred by option o in station are given simiar to those used in practica appications. The AOS{agorithm can be appied to the case of order{sequencing as we as to the case of mode{sequencing. For the case of order{sequencing, we generated 100 sets containing 100 orders where each order possesses an individua conguration of options. The set of orders was generated by a random procedure, so that the set of orders contains the xed frequencies No for each option o. The order{sequence determined by the AOS{agorithm was compared with a random sequence and with the sequence determined by the Time Spread heuristic devised in [26].

21 For the case of mode{sequencing, we generated 10 sets of 100 orders, where 5 or 10 mode types are distinguished and a orders of a mode type possess the same conguration of options. The mode{sequence determined by AOS was compared with a random sequence and, additionay, with an agorithm described by Kubiak [13]. Tabe 3 and 4 show the mean objectives WL, U, and OL over a sequences of the test set. Considering Tabe 3, we see that the AOS{agorithm is markedy superior to the Time Spread heuristic with respect to a objectives. The poor performance of the Time Spread heuristic can be expained by its scheduing criteria. The Time Spread heuristic prefers orders that have a sma tota workoad and show a very even distribution of workoad over the stations. With regard to objective (2.1) of workoad eveing WL this eads to a high deviation of the accumuated workoad from the expected workoad for stages in the midde of the sequence. Since the Time Spread{ heuristic, obviousy, does not ead to a eveing of workoad, it seems to be inadequate for the case of order{sequencing. sequencenobjective WL U OL Random : : 19 0: 74 Time Spread : : 09 0: 80 AOS 15846: : 31 0: 54 Tabe 3: Comparison between Random,Time Spread, and AOS Considering the case of mode{sequencing the AOS{agorithm outperforms the agorithm of Kubiak with respect to the objectives WL and U, whereas the atter one shows a better performance in OL (see Tabe 4). Kubiak determines the average workoad for each station and for each mode. At each stage the mode is schedued, which obtains the maximum deviation between expected and accumuated workoad at the previous stage. In order to evauate the three sequencing poicies for the assemby shop, mentioned in Section 3, we generated 10 sets containing 300 dierent orders. A specic coor is assigned to each order. The number of coors avaiabe was given by Nc = 10. With respect to sequencing poicy 1, we generate a batch{sequence with

22 sequencenobjective WL U OL Random : : 41 0: 71 Kubiak 42836: : 07 0: 30 AOS 32766: : 77 0: 33 Tabe 4: Comparison between Random, Kubiak, and AOS coor batch size B = 5by appying the ABS{agorithm. The determined batch{sequence is assumed to be the input sequence of the paint shop. With no rework cyces in the paint shop, this sequence theoreticay passes to the assemby shop. In genera, the input sequence of the paint shop is disturbed by rework cyces. Therefore, the batch{sequence was randomy disturbed with a disturb factor of 10%, which means that on the average 90% of the units eave the paint shop without running through a rework cyce. The ength of the rework cyce was uniformy chosen between 10 and 60 units per deay. The disturbed batch{ sequence is caed disturbed sequence. Since a ong ook ahead of the assemby sequence is desired, we seek to resort the disturbed sequence to the origina batch{sequence. The abiity to resort the batch{sequence depends on size and accessibiity of the sorting buer. In the considered probems, a buer size of 40 units and random buer access are assumed. The resut of resorting the disturbed sequence is denoted by resorted sequence. Sequencing poicy 2 provides a sequence on the set of units actuay eaving the paint shop. Considering the units of the disturbed sequence that are in the sorting buer, a buer{sequence can be computed with the AOS{agorithm. Finay, the AOS{ sequence is generated on basis of the tota order set. The AOS{ sequence is ony of theoretica signicance because, in genera, this sequence can ony be provided to the assemby shop, if the buer is of size n.tabe 5 shows the objectives of the dierent sequences. Loosey speaking, the batch{sequence is not as good as the buer{sequence or the AOS{sequence, because the consideration of batches reduces the set of eigibe jobs at each stage. Due to a reative sma buer size the resorted sequence is generay not as good as the batch{sequence. Obviousy, the AOS{sequence

23 shows the best performance, but with respect to practica appications the buer sequence outperforms each avaiabe sequence. poicynobjective WL U OL Batch{Sequence : : 19 0: 69 Disturbed Sequence : : 87 0: 72 Resorted Sequence : : 59 0: 70 Buer{Sequence 66324: : 38 0: 60 AOS{Sequence 56247: : 56 0: 58 Tabe 5: Sequencing poicies Further tests to determine the performance of AOS with regard to variations in the frequency of options, ength of the sequence, size of the coor batch or ength of the aowance imits have been done. For further detais we refer to Enge [8]. 5 Concusions In this paper we discussed the performance anaysis of mixed{ mode assemby ines with given ine baance. We presented three objectives for the sequencing probem. We motivated a workoad eveing approach and introduced an integer programming formuation for the workoad eveing probem WLP. We devised two 2 poynomia heuristics AOS and ABS with time compexity Ons ( ) for the WLP. Thereby, AOS provides an order{sequence and ABS computes batch{sequences of orders. We proposed three sequencing poicies for practica appications in automobie production. In an experimenta performance anaysis we compared the AOS{ agorithm with two workoad eveing agorithms for order{ and mode{sequencing, respectivey. The AOS{agorithm outperforms the heuristic of [26] for the probem of order{sequencing as we as the heuristic of [13] for the probem of mode{sequencing. Finay, we evauated the proposed sequencing{poicies of automobie production. Important areas of further research are eveing the variation in workoad at the stations over the sequence as we as resource constraints of options over time, which is important for practica appications.

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