Computers & Industrial Engineering 41 (2002) 405±421
www.elsevier.com/locate/dsw
Mixed model assembly line design in a make-to-order environment
Joseph Bukchin a,*, Ezey M. Dar-El b, Jacob Rubinovitz b
a
Department of Industrial Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel
Faculty of Industrial Engineering and Management, Technion Ð Institute of Technology, Haifa 32000, Israel
b
Accepted 31 August 2001
Abstract
Mixed model assembly lines can be found today in many industrial environments. With the growing trend for
greater product variability and shorter life cycles, they are replacing the traditional mass production assembly
lines. In many cases, these lines follow a `make-to-order' production policy, which reduces the customer leadtime, and is expressed in a random arrival sequence of different model types to the line. Additional common
characteristics of such mixed model lines in a make-to-order environment are: small numbers of work stations, a
lack of mechanical conveyance, and highly skilled workers. The design problem of mixed model assembly lines in
a make-to-order environment is addressed in this paper. A mathematical formulation is presented which considers
the differences between our model and traditional models. A heuristic that minimizes the number of stations for a
predetermined cycle time is developed consisting of three stages: the balancing of a combined precedence
diagram, balancing each model type separately subject to the constraints resulting from the ®rst stage, and a
neighborhood search based improvement procedure. q 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Mixed model assembly lines; Make-to-order; Design problem
1. Introduction
In the early days of automation, high productivity was achieved in mass production assembly systems,
by designing and balancing a dedicated assembly line for a single product manufactured in very large
quantities. The balancing problem was studied extensively for single models, and is generally known as
the `single model assembly line balancing problem' (SALB-P). Comprehensive related surveys are
introduced by Baybars (1986) and Ghosh and Gagnon (1989).
But long gone are the days when one could purchase any low priced car so long as it was a black
* Corresponding author. Tel.: 1972-3-640-7941; fax: 1972-3-640-7669.
E-mail address: [email protected] (J. Bukchin).
0360-8352/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved.
PII: S0 3 6 0 - 8 3 5 2 ( 0 1 ) 0 0 06 5 - 1
406
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
colored model T Ford. In a market environment where product life cycles are short, and product variety
demands are high, many different models of a product must be produced in relatively small lot sizes and
reach the customer in a short lead-time. Nevertheless, assembly systems must still achieve high productivity, uniform quality and low assembly cost. Flexibility is also an essential requirement in order to
respond to shorter product life cycles, low to medium production volumes, changing demand patterns
and a higher variety of product models and options.
Mass production assembly systems achieved high productivity by using the principles of specialization and work division between assembly stations. As a result, rapid learning took place, even with
unskilled workers. The downside of this approach was high employee turnover due to low work
satisfaction.
Mixed model line balancing and sequencing was the early approach used for handling increasing
product variety. The balancing problem was ®rst solved, the objective being to evenly distribute the total
daily or shift workload of a given model-mix between the stations. The balancing procedure is similar to
SALB-P solution procedures, but it assumes a stable and de®ned model-mix for which the combined
workload is balanced for the duration of the entire shift, and not on a basis of station cycle times (Dar-El
& Nadivi, 1981; Macaskill, 1972; Thomopoulos, 1967). The problem was optimally solved by Erel and
Gokcen (1999) and a binary formulation was developed by Gokcen and Erel (1998). Solving this model
usually resulted in uneven work¯ow along the line and unbalanced station work-content for individual
models launched. Major efforts were then needed at a second stage to sequence the model release into the
line, so as to minimize the damage resulting from blockage and starvation of individual stations due to
work content variations between the different models. Mixed model sequencing studies attempted to
resolve the problem by suggesting sequencing procedures that optimize various system measures, such
as: throughput, cycle time, number of stations, idle time, ¯ow time, line length, work-in-process and raw
material demand deviation (Bard, Dar-El & Shtub, 1992; Dar-El, 1978; Dar-El & Cother, 1975; Dar-El
& Cucuy, 1977). Merengo, Nava and Pozzetti (1999) developed heuristics for the balancing±sequencing
problem, while solving sequentially the two problems.
However, these approaches are inadequate in today's environment, which tries to achieve high
productivity while being ¯exible and very responsive to changes in the demand for different models.
While the model-mix for production may be relatively stable, and is determined ahead of time based on
long-range (say, annual) forecast, the sequence of launching of products to the line must be determined
by actual short range demand patterns and customer orders (make-to-order policy). One approach to
maintain ¯exibility while improving line balance and productivity is the `bucket brigade' manufacturing, ®rst commercialized by Toyota in the `Toyota Sewn Products Management System' Ð TSS
(Bartholdi & Eisenstein, 1996). In such system, workers are allowed to move along the line, and perform
tasks allocated initially to consecutive stations. The result is a self-balancing system, in which the initial
partition of work on the line is improved by a self-adjusting partition of work among the workers.
In this work we propose a new and different approach to the mixed model line-balancing problem, by
exploiting the characteristics of modern assembly lines found in many assembly operations today. In this
assembly environment workers are expected to be more versatile and have better skills than those
working in traditional systems. Due to the fact that workers are cross-trained, it becomes admissible
to assign the same work element to different workers for different models. As a result, a much better
work balance between the assembly stations can be achieved, and work is balanced for the individual
station. All stations have a workload close to the line cycle time, which is relatively insensitive to the
particular demand-driven model sequence. Hence we divide the assembly tasks into two sets: the ®rst
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407
contains tasks that should be assigned to a single station for all model types requiring this task, and the
second contains tasks that can be assigned to different stations for different model types. Consequently,
the solution procedure consists of three main modules: in the ®rst, an initial solution is obtained to set the
locations of the tasks of the ®rst set. Next, the balancing of each model type is performed separately, by
assigning the tasks of the second set to stations, subject to already given locations of the other tasks. In
the last module, the solution is improved by a neighborhood search, which performs incremental
changes of the locations of the tasks of the ®rst set. Due to the stochastic nature of the objective function,
the solutions generated are evaluated by using the `Bottleneck' measure, developed by Bukchin (1998).
When a local optimum is reached, an estimate of the real average cycle time is obtained from a
simulation run, and a decision is made whether to accept this solution or to increment the number of
stations and perform another iteration of the algorithm (in case that the resultant cycle time is larger than
the required cycle time). A description of the problem environment is given in Section 2, and a short
discussion of performance measures follows in Section 3. A balancing procedure is developed in Section
4, which includes a fully solved example. Conclusions of the paper are summarized in Section 5.
2. Model description
2.1. Problem domain and assumptions
In new design approaches of modern assembly systems, the long traditional assembly line is replaced
by a modular, semi-autonomous assembly system based on shorter lines (Bukchin, Dar-El & Rubinovitz,
1997; Burbidge, 1989). These lines consist of about ®ve to ten workstations (workers), which make the
system simpler from all aspects; fewer operations, lower WIP, and non-mechanical conveyance. In
modern assembly lines, workers are expected to be more versatile and have better skills than those
working in traditional systems. One can even assume that each worker is able to perform any operation
in the line. This assumption is admissible in view of the small size of the assembly line due to the
relatively small amount of work allocated to it, together with a policy of intensive worker training.
Since the arrival sequence of models to the line is determined by customer order, the proposed design
approach focuses on the balancing procedure, and uses similar principles as in the traditional mixed
model assembly line. The assembly line balancing (ALB) problem is to assign tasks of several models to
stations, while the objective is to reduce the number of stations for a given cycle time. The task
allocation must not violate precedence relations, represented by the precedence diagram (Prenting &
Battaglin, 1964). The differences between the characteristics of our model and traditional models are
expressed in the assumptions outlined below:
1.
2.
3.
4.
5.
6.
7.
Precedence diagrams of all model types can be accumulated into a single combined precedence diagram.
Each task of the combined precedence diagram is performed for at least one model.
Task duration is known and depends on the model type.
Asynchronous line pace, as well as blockage and starvation are possible.
The ®rst station is never starved and the last station is never blocked.
The line production policy is `make-to-order'.
A task that is common to several models can be restricted in certain cases to the same station for all
models.
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J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
Fig. 1. Precedence diagrams of two models.
The ®rst assumption states that a family of similar models is assembled on the line. Hence it is
reasonable to assume that there are no contradictions between task precedence in different models,
and that all precedence relations can be described in a single precedence diagram. The second assumption de®nes the combined precedence diagram as the uni®cation of all tasks required by all models. The
third assumption relates to the deterministic character of the model, where the task duration is known
and can be different for different models. Assumption 4 relates to the line topology. Products move along
the line in an asynchronous pace, meaning that station times are not bounded by any external cycle time,
with blockage and starvation likely to occur. Assumption 5 enables the line to operate without external
disturbances; there is always raw material for the ®rst station and there is always storage space for the
®nished product. According to assumption 6 the arrival sequence of models to the line is determined by
customer orders. This assumption is widely discussed later on. The main difference between the
proposed and traditional models appears in assumption 7. Traditional mixed model assembly linebalancing assumes that a task, common to several models is always performed at the same station
(Scholl, 1995). In the proposed model, this constraint is not valid for all tasks, but restricted to certain
tasks, according to some factors discussed later. The reasons for applying this constraint in traditional
systems are:
1. Specialization: the division of work among workers, where each performs a small part of the work.
The implication of this constraint is that the work content of each station be kept constant (as far as
possible).
2. Learning: the number of assembly tasks for each worker is kept small so that improved performances
can be achieved in a relatively short time since learning is more rapid.
3. Equipment cost: assignment of common assembly tasks to a single station eliminates the duplication
of assembly equipment.
Assignment of common assembly tasks to the same station causes a strong interdependency between
models, and may actually result in low quality solutions. For example, consider the two precedence
diagrams shown in Fig. 1. The circled numbers are task identi®ers. The common assembly tasks are: 3, 4,
6, 7, 8, and 10. If each common task is to be assigned to the same workstation for all models, then
assignment of task 6 may pose a problem: the work content preceding task 6 is larger in the second
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
409
model, while work content of succeeding tasks is larger in the ®rst model. Assuming that task 6 is
assigned to the same station for both models, the balanced solution would result in high idle times. In the
®rst model, idle time is introduced in stations preceding the station to which task 6 is allocated, and in the
second model, in stations succeeding this station. However, if the two models were balanced separately,
task 6 would probably be allocated to a station closer to the start of the line in the ®rst model, and to a
later station in the second model. Still, one should note that the assignment of task 6 (or any other task) to
two different stations depends on other factors as in Fig. 1.
The environment of modern lines also differs from the traditional assembly line in other respects.
Since modern lines are usually shorter than traditional lines, they are manned by small number of highly
skilled workers, and the assignment of common tasks to the same stations is not always necessary. The
assignment of common assembly tasks to different stations requires different workers to perform the
same tasks, and as a consequence, each worker will have wider range of tasks to perform. Workers on
modern assembly lines, as opposed to traditional assembly line workers, have higher skills and are
capable of performing a wider range of assembly tasks. In a traditional assembly line learning is mostly
mechanical while in the environment of modern lines, learning is likely to be more cognitive and based
on understanding of the assembly process by each team member.
The proposed methodology is based on the classi®cation of tasks into two sets, set E and set E c.
Task from set E have to be assigned to a single station for all model types requiring this task. Tasks
of set E are mainly associated with the high cost equipment required for doing these tasks. Assigning
such tasks to different stations for different models would result in high additional costs due to
duplicated machines for the different stations. Tasks from set E c, on the other hand, can be assigned
to different stations for the relevant different model types. This set will mainly include manual tasks
or tasks that require low cost equipment and tooling, which can be duplicated to the different stations
without increasing the total cost signi®cantly. This relaxation allows for assigning tasks of set E c to
stations separately for each model type. As a result, the design process is much more ¯exible and
yields much more balanced solutions.
2.2. Problem formulation
Notation
duration of task k …k ˆ 1¼K† of model j …j ˆ 1¼J†
djk
duration of model j in station i
pij
P
set of model duration times at the different stations P ˆ {pij }
set of immediate predecessors of task k
IPk
E
set of tasks in which each task has to be assigned to a single station for all model types
M
high value parameter
a zero±one variable which equals to 1 if task k of model j is assigned to station i …i ˆ 1¼I†; and
xijk
zero otherwise
TP
assembly line throughput
The problem is formulated as follows:
Max{TP ˆ f …P†}
…1†
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subject to:
I
X
iˆ1
xijk ˆ 1
xijk #
pij ˆ
I
X
iˆ1
K
X
kˆ1
j ˆ 1; ¼; J; k ˆ 1; ¼; K
xijh
j ˆ 1; ¼; j; k ˆ 1; ¼; K; l ˆ 1; ¼; I; ;h [ IPk
xijk djk
M…1 2 xijg † 2
xijk [ …0; 1†
i ˆ 1; ¼; I; j ˆ 1; ¼; J
J X
X
f ˆ1 l±i
xifg $ 0
;g [ E; i ˆ 1; ¼; I; j ˆ 1; ¼J;
…2†
…3†
…4†
…5†
…6†
The objective function (1) maximizes line throughput, which is an implicit function of the balancing
solution (subject to given demand proportions). We can easily see that under the assumption of random
arrival sequence (subject to the demand proportions) the model is a stochastic programming type.
Constraint set (2) assures the assignment of each task to a single station. Constraint set (3) is a precedence constraint, which guarantees that if a certain task, k, is assigned to station l, each of its predecessors has already been assigned to station l or to previous stations. In constraint set (4), each station
time of each model type is calculated. Constraint set (5) is the assignment constraint, which deals with
the assignment of tasks of set E. Each of these tasks should be assigned to the same single station for all
model types that require this task. According to this constraint, if task g of set E is assigned to station k
for model type j, it would not be assigned to any other station for other model types.
The objective function maximizes throughput, which is a function of two elements: the balance
procedure outcome, namely, the assignment of the work elements to stations, and the model arrival
sequence to the line. Since we assume make-to-order production policy in our model, the sequence is
randomly distributed according to given demand proportions. In other words, the probability of each
model to be the next model to enter the line equals the average periodic demand for this model type
divided by the total demand for all models. Our purpose is to evaluate the objective function as a
function of the balance solution, for constant demand proportions. The objective function evaluation
is needed by the balancing procedure for evaluation of line performance, and for comparison of feasible
solutions. The lack of an exact method for evaluating line throughput in such environment suggests the
need for a simulation study in order to explicitly evaluate the throughput. On the other hand, when
evaluating different solutions, a good performance measure would be much easier and faster to use than
simulation. In this case, the quality of the performance measure would have great importance. Section 3
introduces the performance measure used in this study.
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
411
3. Performance measures of mixed model assembly line balancing problem
Bukchin (1998) examines the quality of ®ve performance measures used with mixed model assembly
lines. One of the ®ve was found to be signi®cantly better than the others, and showed superior performance for relatively short assembly lines. The following performance measures were considered in the
study:
1.
2.
3.
4.
5.
`Smoothed Station' measure (Thomopoulos, 1970).
`Minimum Idle Time' measure (Macaskill, 1972).
`Station Coef®cient of Variation' measure (Fremerey, 1991).
`Model Variability' measure (Bukchin, 1998).
`Bottleneck' measure (Bukchin, 1998).
Extensive simulation experiments indicated that the `Bottleneck' measure outperformed the other
measures in showing a high signi®cant correlation with the operational objective (the throughput).
Consequently, only the `Bottleneck' measure is described in detail.
3.1. The `Bottleneck' measure
The `Bottleneck' measure (Bukchin, 1998) is a performance measure of the line cycle time, which is
the inverse of the line throughput. It is obtained by estimating the expected value of the assembly time at
the bottleneck station. The expected value is attained by summing the products of the probability of each
model assembled at each station to be the largest station time with the model assembly time.
The stationary probability of model l assembled in station k,pkl, to be the largest station time is:
Pr…model l is served by station k† Pr…pkl $ max …ti ††
i;i±k
…7†
where ti is the current assembly time of the product served in station i. Under the assumption of
independence between station times:
Y
Pr…pkl $ ti †
…8†
Pr…pkl $ max …ti †† ˆ
i;i±k
i;i±k
and, if a j is the probability of model j to be the next model to enter the line, we assume a j also to be the
stationary probability of model j to be found in station i. Then:
Pr…pkl $ ti † ˆ
J
X
jˆ1
aj bij
…9†
where,
(
bij ˆ
1
if pij , pkl
0
otherwise
…10†
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J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
Table 1
Example data of the `bottleneck measure'
Model
St1
St2
St3
Sum
St1
St2
St3
M1
M2
M3
Weighted average
31
15
16
23.2
27
14
18
21.3
17
31
11
20
75
60
45
0.500
0.018
0.012
0.175
0.000
0.070
0.075
0.150
0.000
The expected value of the system cycle time, or the system TBD (Time Between Departures) is then:
E…TBD† ˆ
I X
J
X
iˆ1 jˆ1
Pr…pij is the largest time in system†pij
…11†
A small example of how to calculate the `Bottleneck' measure for a three-model, three-station
problem is presented as follows. The assembly times of model types at different stations and the total
assembly time of each model type are presented in the left ®ve columns of Table 1. The stationary
probability of each of the models 1, 2, and 3 to be the next model that enters the line is equal to 0.5, 0.3
and 0.2, respectively. In each cell in the last three columns of Table 1 we can see the probability of each
station time (nine combinations of models to stations) to be the largest station time in the line. Then, the
expected value of the TBD (e.g. the average cycle time) is calculated using Eq. (11), with a resultant
value of 0.175.
In Section 4, the `Bottleneck' measure is incorporated into the balancing algorithm in order to
compare the quality of solutions obtained by the proposed algorithm.
4. Heuristic mixed model line balancing algorithm
4.1. The heuristic procedure
The proposed algorithm aims to minimize the number of stations for a required production rate, or for
a given cycle time. Fig. 2 shows the ¯owchart of the algorithm. The balancing procedure attempts
initially to balance the line for the lowest required number of stations, while trying to minimize the line
cycle time. This procedure involves a modi®ed RALB algorithm (Rubinovitz & Bukchin, 1993), which
is discussed later, and a neighborhood search based improvement procedure. At the end of the balancing
procedure, a simulation is performed to evaluate the average cycle time, and the resultant cycle time, Cs,
is compared with the required cycle time, C. If the resultant cycle time is smaller or equal to the required
one, the procedure is completed with the obtained solution. Otherwise, the number of stations is
increased by one, and another iteration is performed, until a feasible solution is reached.
One should note that the predetermined cycle time, C, is the inverse of the required throughput rate of
the line. Since an asynchronous line is considered, the station time may in some cases exceed the given
cycle time, but the average value of the simulated cycle time, Cs should be smaller or equal to C.
4.1.1. The balancing procedure
The balancing procedure consists of three main modules. In the ®rst module, a single model line
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
413
Fig. 2. General ¯ow chart of the algorithm.
balancing procedure is applied on the combined precedence diagram to determine the locations of the
tasks in set E. In the second module, the balance solution, for each model separately, without contradicting the allocation constraints is obtained. The third module involves a neighborhood search, which
aims to improve the solution, obtained by the ®rst two modules.
The balancing procedure steps are presented in Fig. 3. In step 1, the tasks of the combined precedence
Fig. 3. The balancing procedure.
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J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
Fig. 4. The neighborhood search.
diagram are balanced for a ®xed number of stations, in order to minimize the cycle time. The combined
diagram task duration is calculated as follows:
dk ˆ
J
X
jˆ1
aj djk
…12†
where, dk is the average task duration for all models, weighted by the demand proportions. The
purpose of the assignment procedure in step 1 is to determine the set of locations (stations) of tasks
that belong to set E…l1 ; l2 ; ¼ln [ L†; prior to the subsequent step. In step 3, each model is balanced
separately, subject to the above constraints, and the objective is to minimize the cycle time for a given
number of stations. The complexity of the single model assembly line balancing problem is NP-Hard
(Gutjahr & Nemhauser, 1964), and many heuristic procedures are introduced in the literature, along
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
415
with few limited (by size) optimal algorithms. The algorithm used in this stage is also the RALB
algorithm (Rubinovitz & Bukchin, 1993), which was originally developed for the design of robotic
assembly lines. The single model assembly line balancing problem is a special case for this algorithm,
when the number of robot types allocated to the line are equal to one. The algorithm is based on a
frontier search Branch and Bound (B and B) approach. It is an optimal algorithm, while some ¯exible
heuristic rules are utilized for large size problems. The original algorithm was modi®ed in order to ®t
the current problem, by inclusion of the allocation constraints (set L). A general description of the
algorithm is presented in Appendix 1.
In step 4, the value of the appropriate performance measure is evaluated in order to initiate the
improvement procedure using a neighborhood search algorithm (step 5). This is discussed in Section
4.1.2.
4.1.2. Solution improvement
The quality of the balancing procedure is highly dependent on the location constraint obtained in step
1. The neighborhood search algorithm is used to examine alternative locations for the tasks of set E,
namely, alternatives for set L. For each alternative assignment, steps 3 and 4 of the balancing procedure
are performed, and a solution is obtained along with its performance measure value. A comparison
between alternatives is done using the `Bottleneck' measure (Section 3) and the process continues until
no further improvement is possible.
Consider the neighborhood search algorithm. Let e1 ¼en denote the tasks of set E. Let Ls denote
the seed, which holds the locations of the tasks of set E…ls1 ; ls2 ; ¼; lsn [ Ls †; and Lj denote solution j,
received during the algorithm execution. The neighborhood procedure operates as follows (see the
neighborhood search ¯ow chart in Fig. 4):
Step 5.1: De®ne the initial seed: Ls ˆ L:
Step 5.2: Set initial values to variable: i ˆ 1; j ˆ 0:
Step 5.3: If task ei is assigned to the last station, or, there is k, in which task ek succeeds task ei,
and, lsk ˆ ls1 ; then go to Step 5.8.
Step 5.4: Create a new feasible solution: j ˆ j 1 1; Lj ˆ Ls :
Step 5.5: Shift task ei in solution j forward by one station: lji ˆ lji 1 1:
Step 5.6: Balance each model separately, subject to Lj constraints.
Step 5.7: Calculate the value of the `Bottleneck' measure associated with the obtained solution
(Section 3): Oj ˆ f …Lj †:
Step 5.8: If task ei is in the ®rst station, or, there is k, in which task ek precedes task ei, and, lsk ˆ
ls1 ; then go to Step 5.13.
Step 5.9: Create new feasible solution: j ˆ j 1 1; Lj ˆ Ls :
Step 5.10: Shift task ei in solution j backward one station: lji ˆ lji 2 1:
Step 5.11: Balance each model separately, subject to Lj constraints.
Step 5.12: Calculate the `Bottleneck' measure associated with the obtained solution: Oj ˆ f …Lj †:
Step 5.13: If i # n : i ˆ i 1 1; go to step 5.3.
Step 5.14: If minj {Oj } $ Os : End procedure with Ls and Os.
Step 5.15: De®ne the best neighbor as the new seed: Ls ˆ {Lk uOk ˆ minj {Oj }}
Step 5.16: De®ne a new value for the performance measure: Os ˆ Ok ; go to step 5.2.
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J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
Fig. 5. Combined precedence diagram of the three models.
4.2. Numerical example
For demonstrating the algorithm, a ®ve-station line for three model types is de®ned. The demand
proportions are 50% for the ®rst model, 30% for the second model, and 20% for the third model.
The following incorporates the balancing procedure and neighborhood search improvement process.
After completing this step, a simulation should be executed in order to compare the obtained cycle time
with the given cycle time. If the current cycle time is too high, a worker should be added and the whole
procedure repeated. The combined precedence diagram for the three models is presented in Fig. 5. The
diagram consists of two types of activities, which are denoted by squares and circles. The squares
represent tasks that should be assigned to a single station for different models, namely, set E, while
the circles represent tasks that may be assigned to different stations for different models. The input data
Table 2
Task duration of the various models
Task no.
Model 1
Model 2
Model 3
Weighted average
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
6
4
2
7
5
0
6
8
0
0
6
0
4
0
0
5
0
8
2
5
7
4
5
0
4
3
0
5
0
0
2
4
5
3
6
7
0
2
3
4
4
1
2
3
2
4.9
2.8
4.4
4.7
5.2
3.5
4.2
5.9
0.6
2
4.7
0.2
3.9
0.6
0.4
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
417
Table 3
Balancing solution of the combined precedence diagram
Acc. diagram
E
ST 1
C1
ST 2
C27
ST 3
C3
ST 4
C4
ST 5
C5
1,4
9.6
2,5,9,12,14
2
9.4
3,6,10
6
9.9
7,8
8
10.1
11,13,15
13
9
for the algorithm is introduced in Table 2. It includes task duration for all models, where zero duration
means that this task is not required by the speci®c model. Table 2 also includes the average duration of
each task weighted by the demand proportions, which are used for step 1 of the balancing algorithm. The
data in Table 2 enables the precedence diagram for each model to be derived from the combined diagram
simply by eliminating the zero time tasks from the combined precedence diagram.
The ®rst stage of the algorithm includes the balancing of the combined precedence diagram, in order
to determine the location of tasks belonging to E, namely, tasks: 2, 6, 8 and 13. The result of this stage is
presented in Table 3, which contains the activity assignment and the accumulated time of each station.
The relevant data for the consequent stage is the locations of the elements of set E, namely, task 2 is
assigned to station 2, task 6 to station 3, task 8 to station 4 and task 13 to station 5. After creating set L,
each model is separately balanced subject to these constraints using the modi®ed RALB algorithm
(Rubinovitz & Bukchin, 1993). The solution is shown in Table 4, and includes task allocation and
time assignment to stations. This solution is the initial seed of the neighborhood search procedure, with
the `Bottleneck' measure value equal to 10.9.
The next stage is the improvement procedure based on neighborhood search. Neighbors of the initial
seed are created by shifting tasks 2, 6, 8, and 13 one station downstream and upstream, subject to
precedence constraints. Each shift provides a new neighbor. Table 5 ®xes a group of seven neighbors
for the ®rst seed. Tasks: 2, 6 and 8 are shifted on either side, and task 13 is shifted only downstream,
because it was initially located at the last station. For each neighbor, the balancing procedure for each
model is performed and the value of the performance measure is evaluated. As can be seen in Table 5,
the performance measure of the initial seed is better than six of the seven neighbors, with relatively high
differences. The ®rst neighbor has a performance measure value of 10.8, slightly better than the seed
value (10.9) and therefore becomes the new seed. For the new seed, task 2 is assigned to station 1, ®ve
neighbors are developed and the results shown in Table 6. This group of solution is smaller than the
previous group since task 2 cannot be shifted backward, and one of the neighbors is identical to the
previous seed, and therefore eliminated. None of the ®ve performance measures is better than the seed,
and accordingly, the procedure is ended with the last seed selected as the best solution.
Table 4
First feasible solution
Model 1
Model 2
Model 3
E
Os
ST 1
C1
ST 2
C2
ST 3
C3
ST 4
C4
ST 5
C5
1,3
1,5
1,3,4
8
10
10
2,4
3,4
2,5
2
11
10
10
5,7
6,7
6,9
6
11
11
10
8
8,10
8,10,11
8
8
9
10
11,13
11,13
12,13,14,15
13
10
8
8
10.9
418
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
Table 5
Series of neighbors to the ®rst seed
Model
Model
Model
E
O1
Model
Model
Model
E
O2
Model
Model
Model
E
O3
Model
Model
Model
E
O4
Model
Model
Model
E
O5
Model
Model
Model
E
O6
Model
Model
Model
E
O7
ST 1
C1
ST 2
C2
ST 3
C3
ST 4
C4
ST 5
C5
10
10
10
3,4
3,4
1,3,4
9
10
10
5,7
6,7
6,9
6
11
11
10
8
8,10
8,10,11
8
8
9
10
11,13
11,13
12,13,14,15
13
10
8
8
1
2
3
1,2
1,5
2,5
2
10.8
1,4
1,5
1,4
13
10
5
3,7
3,4
3
8
10
5
9
11
11
8
8,10
5,8,10,12
8
8
9
13
11,13
11,13
9,11,13,14,15
13
10
8
14
1
2
3
12.38
1,3
1,3
1,3,4
2,5
6,7
2,6
2,6
8
13
10
11
13
11
5,7
5,10
5,10,12
11
9
11
8
8,11
8,9,11
8
8
8
9
11,13
13
13,14,15
13
10
5
7
1
2
3
12.02
1,3
1,5
1,4
2,4
4,6,7
2,6
2,6
8
10
5
11
12
13
5,7
4
3
11
2
5
8
6,8
6,8,10
6,8
8
12
13
11,13
10,11,13
11,12,13,14,15
13
10
12
12
1
2
3
12.24
1,4
1,4,5
1,3,4
2,4
3,7
2,5,9
2
13
12
10
12
12
10
5,8
6,8
6,8
6,8
13
12
9
11
10,11
10,11,12
6
7
9
13
13
9,13,14,15
13
4
5
10
1
2
3
12.73
1,3,4
1,3
1,3,4
2,3,7
3,7
2,5
2
15
13
10
10
11
10
5
6,10
6,9
6
5
11
10
10,12,14
0
0
8
8,11,13
8,11,13
8,11,13,15
8,13
18
13
10
1
2
3
15.9
1,3,4
1,3
1,3,4
2,7
4,5,7
2,5
2
15
13
10
2,7
4,5,7
2,5
2
10
11
10
5
6,10
6,10
6
5
11
11
1
2
3
15.92
8,11,13
8,11,13
8,9,11,13
8,13
18
13
11
12,14,15
0
0
6
5. Summary and conclusions
The design of mixed model assembly lines is discussed in this paper. The model assumed a `make-toorder' production policy, and hence the arrival sequence is randomly distributed according to demand
proportions. Therefore the balancing procedure becomes the most important aspect of the design
process. Modern assembly systems are based in many cases on short lines, which are less complex
than traditional assembly line. These lines are manned by highly skilled workers, capable of performing
a wide range of activities. Consequently, one of the main constraints of traditional assembly line
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
419
Table 6
Series of neighbors to the second seed
Model
Model
Model
E
O1
Model
Model
Model
E
O2
Model
Model
Model
E
O3
Model
Model
Model
E
O4
Model
Model
Model
E
O5
1
2
3
1
2
3
1
2
3
1
2
3
1
2
3
ST 1
C1
ST 2
C2
ST 3
C3
ST 4
C4
ST 5
C5
1,2
1,3
1,2,3
2
11.85
1,2
1,4,5
1,2,5
2
11.87
1,2,3
1,4,5
1,2,3
2
11.99
1,2,3,5
1,3
2,5
2
16.45
1,2,3,5
1,3
1,2,3
2
16.46
10
13
11
3,4
4,6,7
4,6
6
9
13
10
5,7
5,10
5,9,12
11
9
10
8
8,11
8,10,11
8
8
8
10
11,13
13
13,14,15
13
10
5
7
10
12
12
3,4
3,7
3,4,9,12
9
12
12
5,7
11
0
3
8
6,8
6,8
6,8
8
12
9
11,13
10,11,13
10,11,13,15
13
10
12
12
12
12
11
4,5
3,7
4,5
12
12
9
8
12
9
7,11
10,11
9,10,11
12
7
11
13
13
12,13,14,15
13
4
5
8
17
13
10
4,7
4,5,7
1,3,4
13
11
10
10,12,14
0
0
8
8,11,13
8,11,13
8,11,13,15
8,13
18
13
1
17
13
11
4,7
4,5,7
4,5
13
11
9
14
8
6,8
6,8
6,8
6,10
6,9
6
6,10
6,10
6
0
11
10
0
10
11
8,11,13
8,11,13
8,9,11,13
8,13
18
13
11
12,14,15
0
0
6
balancing, the assignment constraint, is partially relaxed in this model, in that some speci®c tasks can be
performed at different stations for different models.
In this paper, a mathematical formulation of the problem is presented and a heuristic, which
takes into consideration the above relaxation, is developed. The heuristic minimizes the number
of stations for a predetermined cycle time. It consists of three stages: the balancing of the
combined precedence diagram, balancing each model separately subject to the constrained
tasks (resulting from the preceding stage), and an improvement procedure based on neighborhood
search which uses an appropriate performance measure in order to compare solutions. The cycle
time of each ®nal solution is then received from simulation, and compared to the required cycle
time.
Appendix A. The Line balancing modi®ed heuristics
The line balancing heuristics is based on a Branch-and-Bound algorithm using the Frontier
Search method. The algorithm is used for the assembly line balancing of the combined precedence
diagram, as a single model assembly line balancing problem, as well as for the balancing of each
model separately, while pre-assigning the work elements belong to set E. The algorithm creates a
420
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
search tree, by assigning tasks to stations, until a ®nal solution is reached. The main stages in this
search process are:
1. Creation of the ®rst (top) level of the search tree. At this level, each node contains a task
without preceding elements (based on the precedence matrix).
2. Selection of a node to be extended. This selection is based on a calculation of a lower bound on
the number of assembly stations for each node, as follows:
(
if Fin ˆ 1
N 1 …T p =C†
FLB ˆ
N 1 ……T p 2 Sc †=C† if Fin ˆ 0
P
where: T p ˆ i[s ti ; ti is the assembly time of task i, N number of stations already assigned, C
cycle time for which the assembly line is being balanced, s set of tasks which have not yet been
assigned to stations at the current stage, Sc the slack time at the current station and Fin is an
index with the value of 1 if the current station is closed (fully assigned) and a new station will
have to be open at the next stage, or 0 otherwise.
In the case of a station which has not been completely assigned …F ˆ 0†; T p is adjusted by the
slack value at the station Sc, assuming that tasks which will be assigned to this open station will
fully utilize the slack time. Since the value of FLB is a real number, and the lower bound on the
number of stations has to be integer, the actual lower bound is the smallest integer value, which
is larger than FLB: LB ˆ [FLB] 1.
The node with the smallest LB value is extended. However, in a case of a tie, additional rules
are applied for selection of the node to be extended, as follows:
(a) select the node which is at the lowest level in the search tree,
(b) if there are several nodes at the same lowest level, select the one with the lowest FLB
value.
3. Node extension. The selected node is expanded to new nodes, by adding one different feasible
task to each new node. A feasible task is a work element with all predecessors already assigned
(including tasks of set E), and ti # Sc :
4. If the node, which has been extended, contains all the assembly tasks, the optimal solution has
been found. Otherwise, return to stage 2.
In order to solve large and complex problems, a heuristic rule, which limits the search space,
was introduced. This rule eliminates nodes, which have relatively low probability to lead to an
optimal solution. The rule is activated by limiting the number of levels of the search tree for which
open nodes are kept. This heuristic rule can be tuned dynamically, as to the number of levels kept,
taking into account the available storage space, and problem complexity.
Hence, the heuristic rule used is a limit on the number of levels for which open nodes are kept. Large
values of this limit can provide better and closer to optimum solutions, but at the expense of longer
search time, and larger memory space for storage of the open nodes. Using a heuristic parameter of l
results in a fast depth search, in which the node with the lowest bound is extended at each level.
Application of this rule removed the storage space constraints, and allowed the solution of dif®cult
problems, which could not be solved by the optimal algorithm. Further details can be found in Rubinovitz and Bukchin (1993).
J. Bukchin et al. / Computers & Industrial Engineering 41 (2002) 405±421
421
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