A two-stage heuristic method for balancing mixed ... - Semantic Scholar

int. j. prod. res., 2002, vol. 40, no. 6, 1405±1420

A two-stage heuristic method for balancing mixed-model assembly lines with parallel workstations PEDRO M. VILARINHOy* and ANA SOFIA SIMARIAy This work presents a new mathematical programming model for the mixed-model assembly line balancing problem with parallel workstations and zoning constraints. It allows the user to control the process to create parallel workstations. The model’s primary goal is to minimize the number of workstations along the line, for a given cycle time, and its secondary goal is to balance the workloads between and within workstations. A two-stage procedure, using a simulated annealing approach, was developed to tackle this complex problem. The ®rst stage of the procedure looks for a sub-optimal solution to the problem’s primary goal, whilst the second stage deals with the secondary goal. The procedure is illustrated with a numerical example and the results from computational experiments show that even for large-scale problems the proposed procedure performs very well.

1.

Introduction An assembly line is a set of sequential workstations linked by a material handling system. In each workstation a set of tasks is performed using a prede®ned assembly process, in which the following issues are de®ned: (i) the task time, i.e. the time required to perform each task; (ii) a set of precedence relationships, which determine the sequence in which the tasks can be performed; and (iii) a set of zoning constraints, which force or forbid the assignment of di erent tasks to the same workstation. In a paced assembly line (the type of line addressed in this work) each workstation has a prede®ned amount of time to complete all the tasks assigned to it: the cycle time. When this time is elapsed the sub-assembly must be moved to the next workstation along the line and the workstation receives a new sub-assembly from the previous workstation. Thus, the cycle time determines the production rate of the assembly line. In unpaced assembly lines, there is no ®xed time for a workstation to complete its workload, so bu ers between workstations are necessary in order to keep the in-process inventories. The assignment of tasks to workstations, in such a way that the assembly costs (labour costs, equipment costs, etc) are minimized, the demand is met and both the precedence and zoning constraints are satis®ed, is the main problem associated with the design of assembly lines and is termed the assembly line balancing problem (ALBP).

Revision received October 2001. { Departamento de Economia, GestaÄo e Engenharia Industrial, Universidade de Aveiro, Campo UniversitaÂrio de Santiago, 3810-193 Aveiro, Portugal. * To whom correspondence should be addressed. [email protected] International Journal of Production Research ISSN 0020±7543 print/ISSN 1366±588X online # 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00207540110116273

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P. M. Vilarinho and A. S. Simaria

An assembly line designed to assemble a single homogeneous product is called a simple assembly line and is used when the demand for that product is su ciently large to justify a dedicated assembly line. The mathematical formulation of the ALBP for simple assembly lines was ®rst stated by Salveson (1955) and, since then, extensive research has been done in the area. Comprehensive literature reviews on this subject are provided in Ghosh and Gagnon (1989) and Scholl (1999). The recent market trends show that there is a growing demand for customized products, increasing the pressure for manufacturing ¯exibility. Consequently, simple assembly lines are being replaced by mixed-model assembly lines, in which a set of similar models of a product can be assembled simultaneously. Several techniques have been proposed to tackle the mixed-model ALBP, namely by GoÈcken and Erel (1997, 1998), McMullen and Frazier (1997, 1998), Merengo et al. (1999) and Erel and GoÈcken (1999). Most of the techniques used to solve the ALBP require the assignment of each task to a single workstation and, consequently, the production rate is limited by the longest task time. This assumption can be relaxed by utilizing parallel workstations in such a way that two or more replicas of a workstation can perform the same set of tasks on di erent assemblies. The introduction of parallel workstations not only allows for cycle times shorter than the longest task time and thus an increase in the production rate, but also provides greater ¯exibility in designing the assembly line (Buxey 1974). However, when parallel workstations are introduced, the number of tasks performed by each worker increases (e.g. it is possible to design an assembly line with just one workstation and a su cient number of replicas to meet the demandÐin such a case, each worker has to perform all tasks), but this contradicts one of the main advantages of using an assembly line: the use of low-skilled labour that can be easily trained (due to the strict division of labour). Therefore, in order to maintain that advantage, it is necessary to control the process to create parallel workstations (i.e. the replication process) in such a way that workstations are replicated only when required, i.e. when a task’s processing time is larger than the cycle time and, as such, requires the replication of the workstation in which the task will be carried out. Most of the models for the ALBP with parallel workstations proposed in the literature base the decision to create parallel workstations on a trade-o between the incremental tooling/equipment cost of the duplicated workstation and the cost of hiring workers for the original line in order to satisfy the demand (e.g. Johnson 1983, Pinto et al. 1975, 1981, Bard 1989, Daganzo and Blumenfeld 1994, Askin and Zhou 1997). McMullen and Frazier (1997, 1998) allow the replication of a workstation as long as its utilization increases. Scho®eld (1979) and Sarker and Shantikumar (1983) de®ne a limit on the number of parallel workstations to control the replication process, while Buxey (1974) includes a limit on the number of tasks per workstation. In either of these approaches, tasks with processing times much shorter than the cycle time can trigger the replication of workstations, which can lead to an excessive number of parallel workstations. This work introduces a mathematical programming model for the mixed-model ALBP with parallel workstations and zoning constraints, which minimizes the number of workstations for a given cycle time. A secondary goal was introduced so that the model also balances the workload between workstations (i.e. for each model, the idle time is distributed across the workstations as equally as possible) and within workstations (i.e. the overall idle time for each workstation is distributed

Balancing mixed-model assembly lines

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across models as evenly as possible), thus ensuring that all the workers across the line perform approximately the same amount of work on each model being assembled and that approximately the same amount of work is carried out in each workstation regardless of the model being assembled. The model allows the decision-maker to de®ne a limit on the maximum number of replicas of a workstation and the conditions under which a workstation can be replicated. Owing to the combinatorial nature of the problem, a two-stage procedure, which uses the simulated annealing algorithm, was developed to solve it, thus providing near-optimal solutions with an acceptable computational e ort. 2.

The mathematical programming model The proposed model for the mixed-model assembly line balancing problem with parallel workstations has the following characteristics: (i) (ii) (iii) (iv) (v)

(vi) (vii)

(viii)

(ix)

(x)

(xi)

the planning horizon has a ®xed length P, a set of similar models (m ˆ 1; . . . ; M) can be simultaneously assembled, the forecast demand, over the planning horizon, for model P m is Dm , requiring the line to be operated with a cycle time C ˆ P= M mˆ1 Dm , the overall proportion of the number of units of model m being assembled P is then qm ˆ Dm = M D , pˆ1 p each model has its own set of precedence relationships, but there is a subset of tasks common to all models. Hence, the precedence diagrams for all the models can be combined and the resulting one has N tasks, the tasks (i ˆ 1; . . . ; N) are performed in a set of workstations (k ˆ 1; . . . ; S), the time required to perform task i on model m, tim , may vary among models (tim ˆ 0 means that model m does not require task i to be assembled), a task can be assigned to only one workstation and, consequently, the tasks that are common to several models need to be performed on the same workstation, the set of tasks that cannot be performed before task i is completed, Fi , (successors of task i) is given by the precedence constraints derived from the combined precedence diagram, the zoning constraints are de®ned in the assembly processÐZP is the set of task pairs that must be assigned to the same workstation (compatible tasks) and ZN is the set of task pairs that cannot be performed on the same workstation (incompatible tasks), a workstation can be duplicated up to a maximum of MAXP replicas, but only if the task time of one of the tasks assigned to it exceeds a prede®ned value (¬% of the cycle time) for at least one of the models.

The decision variables are de®ned as follows: ( 1; if task i is assigned to workstation k xik ˆ 0; otherwise ( 1; if workstation k can be replicated rk ˆ 0; otherwise


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skm ˆ idle time of workstation k due to model m: The mixed-model assembly line balancing problem with parallel workstations and zoning constraints that, for a given cycle time, minimizes the number of workstations and balances the workloads, can be modelled as the mathematical programming model shown in ®gure 1. The ®rst term in the objective function (1) minimizes the index of the workstation to which the last task is assigned, thus minimizing the number of workstations. The second term balances the workload between the workstations (in this term, S 0 is the actual number of workstations required to meet the demand in the assembly lineÐ S 0 ˆ k: xNk ˆ 1). P The third term balances the workload within each workstation (in this term Sk ˆ M mˆ1 qm ¢ skm ). As both the second and the third terms are within the value range [0,1], the ®rst term is dominant for S 0 > 2, which is always the case for real world applications. So, the model minimizes the number of workstations, before the secondary goal becomes active. The model constraints can be interpreted as follows: (2) constraints ensuring that each task is assigned to only one workstation of the station interval, (3) constraints ensuring that no successor of a task is assigned to an earlier station than that task, (4) compatibility zoning constraints, (5) incompatibility zoning constraints, 2

S

min Z = å k × x Nk k =1

subject to : S

åx

ik

=1

åx

ak

åx

ak

k =1 S k =1 S k =1

æ ö ç ÷ 2 S’ S’ M æ s q m s km S’ M 1 1 ö ç km - ÷ + M ç ÷ + q (1) å må åå S’- 1 m =1 k =1 ç S’ S’ ÷ S’ ( M - 1) k =1 m =1 çè S k M ÷ø ç å s lm ÷ è l =1 ø i = 1,..., N

( 2)

- å x bk £ 0

a Î N, b Î Fa

(3)

- å x bk = 0

(a , b) Î ZP

( 4)

(a , b) Î ZN , k = 1,..., S

(5)

k = 1,..., S, m = 1,..., M

(6 a )

k = 1,.., S, 0 £

£ 100%

(6 b )

k = 1,.., S, 0 £

£ 100%

(6 c )

S

k =1 S k =1

x ak + x bk £ 1 N

åt i =1

im

rk £

× xik + s km = C[1 + rk ( MAXP - 1)]

åx åx a

ik i:$t im >aC ; m =1,..., M

M × rk ³ s km ³ 0

ik i:$ t im > C ;m =1,..., M

k = 1,..., S, m = 1,..., M

(7 a )

x ik Î [ 0,1]

k = 1,..., S, i = 1,..., N

(7 b )

rk Î [ 0,1]

k = 1,..., S

(7 c )

Figure 1. Mathematical programming model for the mixed-model ALBP with parallel workstations.


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(6) the set of constraints ensuring that: each workstation time capacity is not exceeded, the maximum number of replicas of a workstation is not exceeded and only workstations where the processing time of the tasks assigned to it, for at least one model, exceeds a certain proportion (¬%) of the cycle time can be duplicated (M is a very large positive integer), (7) the set of constraints de®ning the decision variables domains. The complexity of the proposed model is high and it cannot be solved to optimality, at least for real world problems. Thus, a two-stage procedure that uses the simulated annealing technique was developed to tackle the problem. The following sections describe the general simulated algorithm and its application to the proposed twostage procedure. 3.

The general simulated annealing algorithm Simulated annealing was chosen, among other meta-heuristics, mainly because of its ¯exibility to respond to modi®cations in the objective functions or in the problem constraints. When these changes occur, the basic simulated annealing program remains unchanged. In addition, the ALBP solutions and neighbourhood structures can be easily de®ned using simulated annealing. The simulated annealing algorithm, which attempts to solve NP hard combinatorial optimization problems through controlled randomization, was introduced by Kirkpatrick et al. (1983). Since then, the algorithm has been applied to many optimization problems in a wide variety of areas, including the assembly line balancing problem (e.g. Suresh and Sahu 1994, Heinrici 1993, McMullen and Frazier 1998). The simulated annealing algorithm is a randomized search procedure that starts from an initial solution to the problem, S0 . A control parameter, T, is set to an initial `temperature’ value, T0 , and is systematically decreased according to an annealing schedule. In this schedule, the following issues are de®ned: (i) a temperature reducing function and (ii) the length of each temperature level, L, that determines the number of solutions generated at a certain temperature. At each temperature level, and as the temperature decreases, neighbouring solutions to the current solution are generated. A neighbouring solution, Sn , is accepted (i.e. replaces the current solution) if it is not worse than the current solution, S, ( f …Sn † µ f …S†, where f is the general objective function to minimize). If the neighbouring solution is worse than the current solution ( f …S n † > f …S†), it may still be accepted with a certain probability, p ˆ e¡¢=T where ¢ˆ

f …Sn† ¡ f …S† £ 100: f …Sn†

This probability of accepting inferior solutions allows the simulated annealing algorithm to escape from local minima. The performance of the algorithm depends on the de®nition of several of the annealing schedule parameters. (i) The initial temperature should be high enough so that, in the ®rst iteration of the algorithm, the probability of accepting worst solutions is, at least, 80% (Kirkpatrick et al. 1983). (ii) The most commonly used temperature reducing function is geometric: Ti ˆ ai Ti¡1 (ai < 1 and constant). Typically, 0:8 µ ai µ 0:99 (Eglese 1990). (iii) The length of each temperature level, L, determines the number of solutions generated at each temperature, T.

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(iv) The stopping criterion de®nes when the system has attained a desired energy level. Some of the most common criteria are based on: . the total number of solutions generated, . the temperature at which the desired energy level is attained (freezing temperature), . the acceptance ratio (ratio between the number of solutions accepted and the number of solutions generated). Naturally each of these control parameters must be re®ned according to the speci®c problem on hand. Two other important issues that need to be de®ned when adapting this general algorithm to a speci®c problem are the procedures to generate both the initial solution and the neighbouring solutions. The details of the proposed implementation of the simulated annealing algorithm to the problem on hand are presented in the next section. 4.

The two-stage procedure for the mixed-model ALBP with parallel workstations and zoning constraints The features of the proposed procedure are as follows: (i) the main goal is to minimize the number of workstations, for a given cycle time, (ii) the additional goals of balancing workloads between and within workstations are also envisaged, (iii) an upper bound on the maximum number of replicas of a workstation can be set by the decision-maker, (iv) zoning constraints are accounted for, (v) the decision-maker may de®ne the minimum task time (expressed as a percentage of the cycle time) that triggers the replication of the workstation performing that task (this time will be called the minimum replication timeÐ MRT). By default, this value is 100% of the cycle time, which means only workstations performing tasks whose duration is larger that the cycle time, for at least one of the models, can be duplicated.

The two stages of the proposed procedure are shown in ®gure 2. In the ®rst stage the procedure looks for a sub-optimal solution for the problem’s main goal. In the second stage, the additional goals are ful®lled. In both stages a simulated annealing approach is used. A common annealing schedule was used for both stages of the procedure, in which the following control parameters were de®ned. (i) Initial temperature (T0 ): the computational experience showed that the values of the objective functions never changed by more than 10% between two neighbouring solutions. So, for an initial temperature of 50 it is guaranteed that at least 80% of the inferior solutions are accepted. (ii) Temperature reduction function: the geometric function with a temperature reduction factor of 0.9 (Ti ˆ 0:9Ti¡1 ) was used on each stage. (iii) The length of each temperature level (L): a dominant factor on the computational e ort associated with the solution of the problem is the number of tasks (N). So, in order to restrict the computational e ort to the ®rst order of the dominant factor, the number of solutions searched at each tempera-

Balancing mixed-model assembly lines Initial solution Modified RPW heuristic

Obj: MIN Idle time STOP?

Y

1st stage Best solution 2 nd stage Initial solution

N Neighbouring solution

Obj: MIN Unbalance between and within stations STOP?

Current solution

Current solution

Neighbouring solution

Swap or Transfer

Swap or Transfer

Verify constraints? (precedence, zoning, capacity, first stage)

Y N

N

Solution in taboo list? FIRST STAGE

Figure 2.

2nd stage Best solution Heuristic Best solution

Y

N

Y Verify constraints? (precedence, zoning, capacity)

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Y

Y

Solution in taboo list?

N

N

SECOND STAGE

The two-stage procedure for the mixed-model ALBP.

ture level was set to KN, where K is a user-de®ned constant (K ˆ 1 is the value suggested by default). (iv) Stopping criteria: two alternative criteria were set. In the ®rst, a freezing temperature of 10 is set, which means that 16 temperature levels are used (T0 a15 i ˆ 50:…0:915† ˆ 10:29). In the second one, it is admitted that, if in ®ve consecutive temperature levels 85% of the generated solutions are rejected, then the probability of replacing the best solution found is very small and the procedure is then terminated. 4.1. The ®rst stage 4.1.1. Initial solution The initial solution is obtained using a version of the Rank Positional Weight (RPW) heuristic proposed by Helgeson and Birnie (1961). The original RPW version only addresses the simple assembly line balancing problem, where one single model is assembled and no parallel workstations are allowed. The positional weight of a task in a mixed-model assembly line is the cumulative average task time associated with itself and its successors. The average task time is the sum of the processing times of that task for each model weighted by the respective production share. Tasks are assigned to the lowest numbered feasible workstation by decreasing order of their positional weight and considering the individual task processing times for each model. In the original version of the RPW heuristic, the cumulative duration of the tasks in a workstation cannot exceed the cycle time (hence the concept of a feasible workstation) and thus does not account for parallel workstations. The version of the RPW heuristic used to obtain the initial solution in the ®rst stage of the procedure rede®nes the concept of a feasible workstation: if a workstation performs a task with processing time larger than MRT, for at least one model, its time capacity is C £ MAXP, otherwise it is C. The implemented version of the RPW heuristic also checks if the task to be assigned is not incompatible with any of the tasks already allocated to the workstation. The procedure also merges tasks that need to be processed in the same workstation previously, so that they are treated as only one task.

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4.1.2. Solution evaluation criterion In the ®rst stage, the procedure looks for the solution that minimizes the number of workstations in the assembly line. As the total idle time of the line decreases with the overall number of workstations in the line, one can attain the ®rst stage goal by minimizing the balance delay time (BDT), which is computed as follows: Á ! M X K X X BDT ˆ qm : SPk :C ¡ tim ; …8† mˆ1 kˆ1

i2Ok

where SPk is the number of replicas of station k and Ok is the set of tasks assigned to station k. This goal is equivalent to the one expressed by the ®rst term of the objective function of the mathematical programming model, and it was used instead to ease the programming e ort. 4.1.3. Neighbouring solutions A neighbouring solution can be generated by one of the following actions: (i) swapping two tasks in di erent workstations or (ii) transferring a task to another workstation. The tasks to be swapped, as well as the task and the workstation for the transfer are randomly chosen. For any of these actions to result in a new neighbouring solution the precedence, zoning and workstations time capacity constraints must be ful®lled. When this is not the case, a new swap or transfer must be attempted. Only transfer movements may contribute to reduce the number of workstations, thus minimizing the balance delay time. Nevertheless, swap procedures are also required to ease the generation of successful transfer movements. Consequently, the probability of performing a transfer procedure must be higher than for the swap procedure and, by default, probabilities of 75% and 25% were respectively set, although the user can set di erent values. In both stages of the proposed procedure, a taboo list is used to maintain information about the most recently generated neighbouring solutions, in order to avoid cycling.

4.2. The second stage The goal in the second stage is to balance simultaneously the between-station and within-station workloads, for the number of workstations obtained in the ®rst stage. The initial solution for the second stage is the ®nal solution found in the ®rst stage. The criterion used to evaluate the neighbouring solutions generated in this second stage derives directly from the second and third terms of the objective function of the mathematical programming model presented in section 2. 4.2.1. Neighbouring solutions The generation of neighbouring solutions in the second stage also employs swap and transfer movements, but the tasks and workstations involved in these movements are selected to foster improving solutions. As the goal in this second stage is to balance the workloads, swap movements are more likely to contribute towards this end. (Probabilities of 75% for swap and 25% for transfer moves are set as the default.)


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Swap movement Step 1. Let Z be a randomly selected workstation and X the model whose idle time for that station has the highest deviation from the workstation average idle time ³ » ¼´ SZ X : ¢sZX ˆ max s ¢ q ¡ : Zm m m M

Step 2. If sZX > SZ =M, then go to step 3, else go to step 5. Step 3. Select the task assigned to station Z with the lowest processing time for model X (T1 : tT1 X ˆ mini ftiX g ^ i 2 OZ ). Step 4. From the set of tasks performed on model X that are not assigned to station Z and whose task time is higher than the task time of T1 , randomly select one (T2 : tT2 X > tT1 X ^ T2 2 = OZ ). Go to step 7. Step 5. Select the task assigned to station Z with the highest processing time for model X (T1 : tT1 X ˆ maxi ftiX g ^ i 2 OZ ). Step 6. From the set of tasks performed on model X that are not assigned to station Z and whose task time is smaller than the task time of T1 , randomly select one (T2 : tT2 X < tT1 X ^ T2 2 = OZ ). Step 7. If precedence, zoning, capacity and number of workstations constraints are met, swap tasks T1 and T2 . Transfer movement Step 1. Let Z be a randomly selected workstation and X the model whose idle time for that station has the highest deviation from the workstation average idle time ³ » ¼´ SZ X : ¢sZX ˆ max s ¢ q ¡ : Zm m m M

Step 2. If sZX > SZ =M, then go to step 3 else go to step 5. Step 3. Select a task not assigned to station Z with processing time for model X higher than for the other models (T1 : tT1 X ˆ maxm ftT1 m g ^ i 2 = O Z ). Step 4. If precedence, zoning, capacity and number of workstations constraints are met, transfer task T1 to workstation Z. Step 5. Select the task assigned to station Z with the highest processing time for model X (T1 : tT1 X ˆ maxi ftiX g ^ i 2 OZ ). Step 6. Randomly select a workstation (W) where the workload for model X is lower than the station average idle time (W : sWX < SW =M). Step 7. If precedence, zoning, capacity and number of workstations constraints are met, transfer task T1 to workstation W. If, after a prede®ned number of attempts, neither swap nor transfer movements lead to a neighbouring solution, the tasks or workstations involved in these movements are selected randomly to force a new neighbouring solution.

4.3. Numerical illustration A numerical example, with the following characteristics, is used to illustrate the proposed procedure.


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(i) The models A and B are simultaneously assembled in a line and over a planning horizon of 480 t.u. (time units). The demand for each model is, respectively 20 and 28 units (then, the cycle time is C ˆ 10, qA ˆ 42% and qB ˆ 58%). (ii) the combined precedence diagram, with 25 tasks, is depicted in ®gure 3, where each node represents a task and each arc represents a precedence relation between a pair of tasks, (iii) the task times for the two models are shown in table 1, (iv) tasks 9 and 10 cannot be executed on the same workstation, (v) only workstations performing tasks with a processing time greater than the line cycle time can be replicated (MRT ˆ C), (vi) the maximum number of replicas of a workstation is two (MAXP ˆ 2). The initial solution is determined by the modi®ed version of the RPW heuristic described in section 4.1.1. The average processing times and average positional weights of each task are shown in table 1. Figure 4 illustrates some steps of the procedure applied to the numerical example. In each of the tables shown in the ®gure, a line balancing solution is shown. To simplify the schema, only the workstations where changes occurred are represented in the neighbouring solutions. The content of each column in these tables is the following: (1) workstation index, S, (2) set of tasks assigned to the workstation, and (3) the number of replicas of the workstation, R. The last line in each table

1

4

8

14

5

9

12

6

11

13

7

10

19 15

23

3 18 16

17

2

24 21

25

22

20

Figure 3.

Precedence diagram for the numerical example.

Task 1 2 3 tA 0 7.7 7.3 tB 2.0 7.7 7.3 Weighted average 1.2 7.7 7.3 task time Positional 115.4 54.4 114.2 weight Task 14 15 tA 1.3 5.5 tB 0 5.5 Weighted average 0.5 5.5 task time Positional weight 19.7 23.4

16 1.9 2.0 2.0

4 15.0 15.0 15.0

5 8.8 8.8 8.8

46.6 85.3 17 3.7 0 1.6

18 9.4 9.4 9.4

44.2 26.3 33.8

6 7 6.2 3.6 0 0 2.6 1.5

8 9 0 6.6 2.0 6.6 1.2 6.6

10 2.5 2.5 2.5

11 5.5 5.5 5.5

12 13 7.1 5.9 7.1 5.9 7.1 5.9

66.2 15.8 31.6 38.9 46.7 61.1 30.5 55.6 19 20 1.3 0 1.3 9.0 1.3 5.2

21 22 2.0 4.7 2.0 4.7 2.0 4.7

23 9.6 8.2 8.8

19.2 14.3 24.8 13.8 17.9

24 25 4.1 12.5 3.7 0 3.9 5.3 9.1

5.2

Table 1. Processing times and average positional weights for the numerical example.

Balancing mixed-model assembly lines Initial solution S Tasks R S Tasks 1 1,3 1 9 18 5 1 10 12,19 2 3 6,7 1 11 17,21 4 11 1 12 15 5 13 1 13 23 6 2 1 14 20 7 4,10,16 2 15 22,24 8 8,9,14 1 16 25 S’=18; Bb-s=0.08; Bw-s=0.17

R 1 1 1 1 1 1 1 2

...

Intermediate solution S Tasks R S Tasks 1 1,2 1 9 8,12 3 1 10 18 2 3 4 2 11 15 4 5 1 12 17 5 6,7 1 13 14,19,21 6 10,11 1 14 22 7 13 1 15 23 8 9,16 1 16 20,24,25 S’=18; Bb-s=0.04; Bw-s=0.15 HEURISTIC BEST SOLUTION S Tasks R S Task s R 1,3 1 8 4,8,7 2 1 6 1 9 18 1 2 3 5 1 10 12,21 1 4 9 1 11 7,15 1 5 2 1 12 14,19,22 1 6 10,11 1 13 23 1 7 13,16 1 14 20,24,25 2 S’=16; Bb-s=0.04; Bw-s=0.15

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Transfer task 15 to station 12 R 1 1 1 1 1 1 1 2

S 11

Tasks R S Tasks R 0 12 15,17 1 S’=17; Bb-s=0.04; Bw-s=0.19

Transfer 1st STAGE task 22 to station 13 BEST SOLUTION S Tasks R S Tasks R 14,19, 13 1 14 0 21,22 S’=16; Bb-s=0.05; Bw-s=0.22 Swap tasks 8 and 16 S 8

...

Tasks R S Tasks R 8,9 1 9 12,16 1 S’=16; Bb-s=0.05; Bw-s=0.21

Swap tasks 4 and 6 S 3

Tasks R S Tasks R 6 1 5 4,7 2 S’=16; Bb-s=0.05; Bw-s=0.20

Figure 4. Illustration of the proposed procedure applied to the numerical example.

shows the total number of workstations (including replicas) required by the solution, S 0 , and the between-station (Bb-s ) and within-station (Bw-s ) balancing values (computed from the expressions in the second and third term of the objective function of the mathematical programming model). The initial solution requires a total of 18 workstations (including all replicas). After a number of swap and transfer movements, starting from the initial solution, an intermediate solution is generated. From this solution, the heuristic is able to reduce the number of workstations performing the transfer procedures shown in the ®gure. The best solution found for the ®rst stage of the heuristic indicates that 16 workstations, including replicas, are required (for this solution Bb-s ˆ 0:05 and Bw-s ˆ 0:22). The best solution found on the ®rst stage of the procedure is used as the initial solution for the second stage. In this stage, the number of workstations remains constant, while the workload balancing value (Bb-s ‡ Bw-s ) is reduced. The ®nal solution shows an improvement of about 30%.

5.

Computational experience The procedure was coded in Visual C‡‡ on a 350 MHz Pentium II computer. It is not possible to compare directly the heuristic with others reported in the literature as none have the same underlying characteristics. Nevertheless, in order to test the performance of the proposed heuristic, a series of comparative tests were carried out by adapting the heuristic to the conditions under which the benchmark problems proposed by Scholl (1993) were originally set. Scholl’s test problems are for singlemodel balancing problems without parallel workstations, so the heuristic was run setting MAXP ˆ 1, also only the objective function of minimizing the number of workstations was tested. For the 168 instances analysed, the optimal solution is known for 161. The solutions obtained by the proposed heuristic for this data set are slightly inferior to the optimal solution in most cases (14% in the worst case) but the procedure manages to ®nd the optimal solution in 76 instances.


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In order to evaluate the real performance of the heuristic, a set of 20 assembly line balancing problems with parallel workstations and zoning constraints was tested, taking into account only the objective function (1). For a set of four problems, an optimal solution was obtained using the Cplex MIP solver, to solve the proposed mathematical programming model, which allowed verifying that the proposed procedure generates the optimal solution for these instances (see table 3 later). For larger sized problems it is not possible to ®nd an optimal solution within reasonable time limits. So, in order to evaluate the procedure performance, a lower bound for the number of workstations was derived.

5.1. A lower bound for the number of workstations A lower bound for the mixed-model assembly line balancing problem with parallel workstations, LBpmix , was derived using the following set of assumptions: (i) the maximum number of replicas per workstation is two (MAXP ˆ 2), (ii) a workstation can be duplicated only if the task time of one of the tasks assigned to it exceeds the cycle time (¬ ˆ 100%, MRT ˆ C), and (iii) the task time of the longest task does not exceed twice the cycle time (tmax µ 2C). The steps required to compute LBpmix are described as follows and illustrated for the numerical example introduced in section 4.3. Step 1. For each model, classify the tasks according to the corresponding task time, as shown in table 2. Step 2. For each model, compute LB 0 …m†. ¹³ 1 0 LB …m† ˆ 2…nA ‡ nB ‡ nC † ‡ y…nD ¡ nC † ‡ w…nE ¡ nB † 2 ´º 5 4 2 1 ‡ nF ‡ nG ‡ nH ‡ nI : …9† 3 3 3 3 Tasks Task type A B C D E F G H I J

Task time 5 3C 4 3C

< tA µ 2C < tB µ 53 C C < tC µ 43 C 2 3 C < tD µ C 1 2 3 C < tE µ 3 C 5 tF ˆ 3 C tG ˆ 43 C tH ˆ 23 C tI ˆ 13 C tJ < 13 C

Model A ± 4 25 2, 3, 5, 12, 18, 23 6, 7, 9, 11, 13, 15, 17, 22, 24 ± ± ± ± 1, 8, 10, 14, 16, 19, 20, 21

Model B

4 ± 2, 3, 5, 12, 18, 20, 23 9, 11, 13, 15, 22, 24 ± ± ± ± 1, 6, 7, 8, 10, 14, 16, 17, 19, 21, 25

Table 2. Classi®cation of the tasks to compute LBpmix .


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LB 0 …m† is the main term of LBpmix …m†. It is derived from one of the lower bounds, LB3 , de®ned by Scholl (1999) for the SALPB and adapted to the parallel workstations problem. LB 0 …m† is given by expression (9), where nX is the number of tasks of type X (X ˆ A; . . . ; J), y equals 1 if nD ¡ nC > 0 or zero otherwise and w equals 1 if nE ¡ nB > 0 or zero otherwise. The reasoning for this computation is as follows. The workstations performing tasks of types A, B or C (whose task time is longer than the cycle time) need to be duplicated. As two tasks of any of these types cannot share the same workstation, because the value of MAXP would be exceeded, a lower bound for the overall number of workstations (including replicas) is twice the number of tasks of types A, B and C. Each task of type D can be combined with a task of type C in a duplicated workstation; however, if there are not enough duplicated workstations of type C to accommodate the tasks of type D, each of these remaining tasks will require a workstation. The same reasoning applies to tasks of type E, that is, two tasks of type E require a single workstation, but can also be combined with a duplicated workstation performing tasks of type B. Finally, the tasks of types F, G, H and I have a ®xed task time and so they occupy a fraction of a workstation corresponding to the ratio between their task time and the cycle time. For the numerical example, the values of LB 0…A† and LB 0 …B† are computed as follows: ¹³ ´º 1 LB 0 …A† ˆ 2…0 ‡ 1 ‡ 1† ‡ …6 ¡ 1† ‡ …9 ¡ 1† ˆ 13; 2 ¹³ ´º 1 LB 0 …B† ˆ 2…0 ‡ 1 ‡ 0† ‡ …7 ¡ 0† ‡ …6 ¡ 1† ˆ 12: 2 Step 3. For each model, compute Z…m†. &" Á !#¿ ’ X X 0 Z…m† ˆ ti ¡ LB …m† ¢ C ¡ ti C : iˆJ

i6ˆJ

…10†

A second term is added to LB 0 …m† to compute the value of LBpmix . This term adds up the number of workstations needed to process tasks of type J, which in most realworld problems account for a large proportion of the workstations. Because these tasks can easily be included in workstations that perform tasks of the other types, it is necessary to verify if, after ®lling up these workstations, there are tasks of type J remaining to create new workstations. The minimum number of workstations (Z…m†) required to perform tasks of type J, after ®lling up the remaining capacity of the workstation assigned to other task types is then given by expression (10). For the numerical example, the values of Z…A† and Z…B† are computed as follows: Z…A† ˆ d‰9 ¡ …13 £ 10 ¡ 123:2†Š=10e ˆ 1; Z…B† ˆ d‰11:8 ¡ …12 £ 10 ¡ 104:4†Š=10e ˆ 0: Step 4. For each model, compute LBpmix …m† ˆ LB 0 ‡ Z…m†. For the numerical example, LBpmix …A† ˆ 14 and LBpmix …B† ˆ 12. Step 5. Select LBpmix for the problem. LBpmix ˆ maxm ‰LBpmix …m†Š. For the numerical example, LBpmix ˆ LBpmix …A† ˆ 14.


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5.2. Computational results The heuristic was tested on a set of 20 problems, whose main characteristics are shown in the ®rst columns of table 3, namely, the number of tasks of the combined precedence diagram (N), the number of models (M) and the assembly line cycle time (C). The precedence diagrams used for the test problems were taken from Scholl (1993), except for problems 7 and 8, where the one from the numerical example introduced in section 4.3 was used. The cycle time and the task processing times for each problem were randomly generated taking into account the di erent task types that might be present in a real world assembly process. The test problems were solved using the proposed procedure and the minimum, maximum and mean value of the solutions, for each of the test problems shown in table 3, results from ten runs of each instance of the problem. A comparison between these values and the ones obtained using the Cplex solver (Optimal Solution) and the lower bound (LBpmix ) is depicted in the tenth column (D(%)) where the di erence between the minimum value produced by the heuristic and either the LBpmix or the best solution is shown. For small-sized problems (for which an optimal solution was found) the proposed procedure gives the optimal solution. For large-sized problems, the worst performance is for problem 15, where the di erence between the solutions obtained and the lower bound is 20%. Nevertheless, as the calculation of the lower bound does not take into account the precedence and zoning constraints, one is led to consider that the results obtained from the heuristic are fairly good. This conclusion is reinforced by the values for the line e ciency shown in column 11 of table 3 (Weighted E (%)), where a high line usage rate can be perceived, particularly for the largest sized problems. The weighted line e ciency was computed using the following expression:

Problem

N

M

C

LBpmix

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

8 8 11 11 21 21 25 25 28 28 30 30 32 32 35 35 45 45 70 70

2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3

10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10

4 6 7 6 14 13 14 14 19 18 15 17 16 17 20 21 23 24 41 39

Optimal Heuristic solution Weighted CPU solution Mean Min Max D(%) E . (%) Time (s) 4 8 7 7 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

4 8 7 7 16 15 16 15 21 20 16.2 19.4 19 19 24 24 25.2 28.2 44 44.2

4 8 7 7 16 15 16 15 21 20 16 19 19 19 24 24 25 28 44 44

4 8 7 7 16 15 16 15 21 20 17 20 19 19 24 24 26 29 44 45

0 0 0 0 14.3 15.4 14.3 7.1 10.5 11.1 6.7 11.8 18.8 11.8 20.0 14.3 8.7 16.7 7.3 12.8

Table 3. Computational results for the test problems.

85.6 54.9 71.0 76.5 72.6 79.6 76.8 82.0 86.5 83.2 86.6 83.4 77.3 81.0 80.0 85.2 85.4 81.4 87.0 86.0

1 1 1.4 1 1 4.4 21.6 6.6 10.4 10 11.6 12.6 3.4 8.4 7.2 10 11.6 13.8 15 20

Balancing mixed-model assembly lines 2

N X

3

tim 7 M 6 X 6 iˆ1 7 6 7 WeightedEff ˆ 6qm S 0 C 7; 5 mˆ14

1419

…11†

where, for S 0 , the minimum value for the number of workstations given by the heuristic solutions was used. Finally, the average running times for each of the test problems are depicted in the last column of table 3 (CPU time(s)) and they show that solutions can be obtained in acceptable times, even for large problems. 6.

Conclusions and future research directions In this paper, a new mathematical programming model for the mixed-model assembly line balancing problem with parallel workstations and zoning constraints was presented. The model minimizes the number of workstations and allows the user to control the replication process. As a secondary goal, the model looks to obtain a good workload balance between and within the workstations. Due to the model complexity a two-stage heuristic procedure was developed to tackle the problem, which uses the simulated annealing algorithm. Computational experiments showed that the proposed heuristic performs very well, producing good quality solutions in reasonable running times. The major contribution of the reported research work derives from the use of a set of constraints to control the workstations’ replication process and from the use of a two-goal objective function. The issues are of crucial importance to solve realworld line balancing problems. In fact the proposed heuristic, combined with a simulation study, has been applied for balancing a mixed-model PC camera assembly line (Ramos et al. 2001) and very good results were obtained. The matching between theoretical procedures and practical applications is a very important area that needs further developments. In addition, more complex balancing problems, with di cult constraints present in real-world problems, must be addressed. Further research must be made on the mixed-model assembly line balancing problem, considering the actual market trends, where there is a growing demand for customized products, thus increasing the pressure for manufacturing ¯exibility. The areas of research to be pursued in the future, related to the present work are: (i) alternative procedures to control the workstations’ replication process, and (ii) the application of the same type of concepts reported in this work to solve the type-II MALBP (which attempts to minimize the cycle time, for a given number of workstations). Acknowledgements The authors are grateful for the constructive feedback they received from the anonymous referees. This feedback was very helpful in improving the paper. Reference Askin, R. G. and Zhou, M., 1997, A parallel station heuristic for the mixed-model production line balancing problem. International Journal of Production Research, 35, 3095±3106.

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