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Simultaneous balancing and buffer allocation decisions for the design of mixed-model assembly lines with parallel workstations and stochastic task times

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The buffer allocation problem (BAP) and the assembly line balancing problem (ALBP) are amongst the most studied problems in the literature on production systems. However they have been so far approached separately, although they are closely interrelated. This paper for the first time considers these two problems simultaneously. An innovative approach, consisting in coupling the most recent advances of simulation techniques with a genetic algorithm approach, is presented to solve a very complex problem: the Mixed Model Assembly Line Balancing Problem (MALBP) with stochastic task times, parallel workstations, and buffers between workstations. An opportune chromosomal representation allows the solutions space to be explored very efficiently, varying simultaneously task assignments and buffer capacities among workstations. A parametric simulator has been used to calculate the objective function of each individual, evaluating at the same time the effect of task assignment and buffer allocation decisions on the line throughput. The results of extensive experimentation demonstrate that using buffers can improve line efficiency. Even when considering a cost per unit buffer space, it is often possible to find solutions that provide higher throughput than for the case without buffers, and at the same time have a lower design cost.
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Simultaneous balancing and buffer allocation decisions for the design of
mixed-model assembly lines with parallel workstations and stochastic
task times
Lorenzo Tiacci*
Università degli Studi di Perugia - Dipartimento di Ingegneria, Via Duranti, 93 06125 Perugia, Italy
Abstract
The buffer allocation problem (BAP) and the assembly line balancing problem (ALBP) are amongst the most studied
problems in the literature on production systems. However they have been so far approached separately, although they
are closely interrelated. This paper for the first time considers these two problems simultaneously. An innovative
approach, consisting in coupling the most recent advances of simulation techniques with a genetic algorithm approach,
is presented to solve a very complex problem: the Mixed Model Assembly Line Balancing Problem (MALBP) with
stochastic task times, parallel workstations, and buffers between workstations. An opportune chromosomal
representation allows the solutions space to be explored very efficiently, varying simultaneously task assignments and
buffer capacities among workstations. A parametric simulator has been used to calculate the objective function of each
individual, evaluating at the same time the effect of task assignment and buffer allocation decisions on the line
throughput. The results of extensive experimentation demonstrate that using buffers can improve line efficiency. Even
when considering a cost per unit buffer space, it is often possible to find solutions that provide higher throughput than
for the case without buffers, and at the same time have a lower design cost.
Keywords: mixed-model assembly line; unpaced lines; asynchronous lines; buffer allocation; balancing; stochastic task times;
paralleling; genetic algorithm; discrete event simulation.
1 Introduction
Mixed model assembly lines have received a growing attention in the last years, due to their capability of
producing a variety of different product models simultaneously and continuously. This feature has become
fundamental with the increasing of customized products demand, and the design of such manufacturing
systems has nowadays a considerable industrial importance.
When different models are assembled in a line, the realized task times assigned to each station can vary, not
only for the variations in the speed of manual labour (stochastic task times), but also for the different
completion times required by different models in each station. In such conditions the adoption of
paced/synchronous lines gives rise to some difficulties. In fact, it is difficult to impose a production rate
given by a fixed cycle time for each station, because realized task times, and so the realized cycle time in
each station, are considerably dependent on the model types that are being assembled. For this reason the
adoption of un-paced asynchronous lines is usually preferred when different models have to be assembled.
In un-paced assembly lines, the stations can be decoupled by buffer stocks. If a station is generally faster
than another one, the buffer storage will soon be filled to capacity and lose its function. However, temporary
* Corresponding Author: Tel.: +39-075-5853743
E-mail address: lorenzo.tiacci@unipg.it
Web-site: www.impianti.dii.unipg.it/tiacci
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deviations in realized task times, due to the model mix and/or to non-deterministic completion times, can be
compensated by allocating buffers among stations.
From these considerations it is clear that two fundamental aspects have to be taken into consideration in the
design of mixed model assembly lines: line balancing, to achieve a similar production rate in each station,
and buffers allocation, to compensate task times variations due to model mix and stochastic completion
times.
So far, the assembly line balancing problem (ALBP) and the buffer allocation problem (BAP) have been
treated in literature in a separate way, although they are closely interrelated. The majority of studies on the
BAP refer to flow-lines production systems, consisting in a linear sequence of unreliable machines, where
production rates in each stage are assumed to be known. The same problem can be transposed to assembly
lines, with the difference that production rates in each station depend on which tasks are assigned to it, that
is, on the result of the balancing procedure. Because even slight changes in the work content at a station
might lead to a more efficient buffer allocation and improve the system’s overall performance, the successive
planning of line balance and buffer allocation most likely will not lead to a global optimum of the whole
system (Boysen et al., 2008). As far as studies related to ALBP are concerned, they usually never consider
buffers between stations.
In this paper for the first time the line balancing problem and the buffer allocation problem are
simultaneously tackled. The problem considered is one of the most complexes in assembly line balancing
literature: mixed model lines, stochastic task times of completion, and parallel workstations.
While searching for a solution for this kind of problem one of the main issues is the performances evaluation
of such a complex line typology, considering also the extra-complexity due to the presence of buffers
between stations. To estimate the realized throughput is difficult, but it is needed in order to compare
different solutions.
We propose a genetic algorithm approach in which a parametric object oriented simulator is embedded in the
genetic algorithm structure to evaluate the fitness function (which include the realized throughput) of the
individuals at each iteration. This approach allows overcoming the limit of using indirect measures of
throughput, that can be calculated in an analytical way, but that are often not correlated to the realized
throughput.
The paper is organized as follows. In the next section a literary review on assembly line balancing problems,
buffers allocation problems and performances evaluation of assembly systems is carried out. In Section 3 the
particular problem taken into consideration is described in details. The genetic algorithm approach, coupled
to the parametric simulator for the objective function evaluation, is described in Section 4. Section 5 is
dedicated to the design of experiment for the evaluation of the proposed approach. In Section 6 results are
reported and discussed.
2 Literary Review
The literary review is organized in three sections. Section 2.1 is related to assembly line balancing problems,
with a particular focus on works related to the line features considered in this work: mixed model lines,
parallel workstations, and stochastic task times. Section 2.2 is focused on buffer allocation problems,
considering assembly systems in which the production rate in each station is not given a-priori, but is the
result of the balancing procedure. Section 2.3 is focused on the performances evaluation of assembly lines,
that is a critical and fundamental issue when different design alternatives have to be compared.
2.1 Line balancing
The Assembly Line Balancing Problem (ALBP) consists in assigning tasks to workstations, while optimizing
one or more objectives without violating tasks precedence constraints and any other restriction imposed on
the line. Scholl and Becker (2006) gave an overview of exact methods and heuristics developed to solve the
basic version of the problem, the so called Simple Assembly Line Balancing Problem (SALBP), which
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provides single-model, paced line with fixed cycle time and deterministic task times. The SALBP can be
classified with respect to the objective type: in ‘type-1 problem one tries to minimize the number of stations
for a given cycle time, while in type-2 problem one tries to minimize the cycle time for a given number of
workstations. So usually the two basic objectives (performances and costs) of the problem are treated in a
separate way, that is fixing one of the two as a constraint, and trying to optimize the other one. If both,
number of stations and the cycle time, can be altered, the problem is of ‘type E’, i.e. the line efficiency E can
be used to determine the quality of a balance. It is possible to maximize the line efficiency by simultaneously
minimizing the cycle time and the number of workstations, for example through a unique objective function
by applying objective-specific weighting factors.
Real problems are much more complicated than the SALBP, and research has recently evolved towards
formulating and solving generalized problems (namely Generalized Assembly Line Balancing Problem,
GALBP) with different additional characteristics, such as cost functions, equipment selection, paralleling,
stochastic task times and others. For a comprehensive classification of the possible features of the GALBP
see Becker and Scholl (2006) and Boysen et al. (2007).
The Mixed-model Assembly Line Balancing Problem (MALBP) can be seen as a particular case of the
GALBP. Here a set of similar models, that are variations of the same base product and only differ in specific
customizable product attributes, can be assembled simultaneously. Set-up times between models can be
reduced sufficiently enough to be ignored (Boysen et al., 2008). Studies published in the last years utilize
different approaches to solve it, such as: Simulated Annealing (McMullen and Frazier, 1998; Simaria and
Vilarinho, 2001; Vilarinho and Simaria, 2002), Ant techniques (McMullen and Tarasewich, 2003; McMullen
and Tarasewich, 2006; Yagmahan, 2011), Genetic Algorithms (Akpinar and Bayhan, 2011; Haq et al., 2006;
Simaria and Vilarinho, 2004; Tiacci et al., 2006; Zhang and Gen, 2011), and other heuristics (Askin and
Zhou, 1997; Bukchin et al., 2002; Jin and Wu, 2003; Karabati and Sayin, 2003; McMullen and Frazier, 1997;
Merengo et al., 1999).
Paralleling is another feature that characterizes many real assembly line configurations. There are different
forms of parallelism such as: parallel lines (Gokcen et al., 2006; Scholl and Boysen, 2009); parallel
workstations performing the same task set (Akpinar and Bayhan, 2011; Cakir et al., 2011); parallel two sided
lines (Ozcan et al., 2010; Roshani et al., 2012). When parallel workstations perform the same task set, it is
possible to perform tasks with processing time larger than the desired cycle time. In general, the possibility
to implement paralleling enlarges the solution space of the problem, so that feasible and potentially better
balanced configurations can be found (Tiacci et al., 2006; Vilarinho and Simaria, 2002).
The great majority of the studies on assembly line balancing problem consider deterministic task times.
Stochastic task times complicate very much the problem, because the blocking and starvation phenomena are
accentuated with respect to the deterministic case. For this reason, it becomes harder to predict line
performances , especially if different models are assembled. In fact, the variability of task times of different
models assigned to a workstation contributes to amplify blocking and starvation phenomena. The only papers
in literature that take into consideration stochastic task times, multiple products produced in a mixed-model
way, parallel workstations (without limits regarding the maximum number of replicas per work centre) are
the works from McMullen and Frazier (1998), McMullen and Tarasewich (2003), McMullen and Tarasewich
(2006) and Tiacci et al. (2006).
2.2 Buffer allocation
The buffer allocation problems have been extensively studied in the last decades, but they mostly refer to
transfer lines in which unreliable machines are assigned, and processing times, time between failures and
repair times are stochastic. In this context the inclusion of buffers increases the average production rate by
limiting the propagation of distributions, but at the cost of additional capital investments, floor space of the
line and inventory (Amiri and Mohtashami, 2012). In this type of problem, the average production rate in
each workstation is usually assigned, and it is not the result of a not trivial balancing procedure, which is on
the contrary requested in case of assembled products, where precedence constraints play a decisive role in
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assigning tasks to workstations. Boysen et al. (2008) outlined that as even slight changes in the work content
at a station might lead to a more efficient buffer allocation and improve the system’s overall performance,
the successive planning of line balance and buffer allocation will most likely not lead to a global optimum of
the whole system. For example, an interesting attribute of asynchronous lines with no buffers is the so called
“bowl phenomenon” (Hillier and Boiling, 1966; Hillier and So, 1993), according to which the throughput of
a line can be improved by assigning smaller station loads to central stations of the line. A similar concept,
known as the “storage bowl phenomenon”, applies to buffer allocation, if buffer storages in the center (or at
bottleneck stations) are increased in size (Conway et al., 1988; Harris and Powell, 1999). It is difficult to
evaluate the interactions among these two effects, but it is clear that a sequential approach regarding
balancing and buffer allocation decisions may not lead to an optimal solution. Two studies by Hillier and
Hillier (2006) and Hillier (2013) approached the problem of simultaneously assigning the production rates
and the buffers size in unpaced reliable production lines, with no parallel stations. In these studies optimal
solutions are found for small instances (up to 4 stations) and for particular distributions (exponential or
Erlang) of processing times. In both these studies it is assumed that there is a fixed total amount of work to
be done, but that this work can be divided among the stations arbitrarily. That is, the complexity of the
Assembly Line Balancing Problem, where stations workload is determined by tasks assignment, that is in
turn constrained by precedence diagrams, is not considered.
On the other hand, there are no works in the ALBP literature that consider the extra complication of defining
the capacity of buffers between workstations while assigning tasks to workstations to obtain a balanced
solution. Battini et al. (2009) proposed an ‘indirect’ approach, that provides a step by step balancing-
sequencing procedure for mixed model assembly systems that aims to optimize the assembly line
performance and tries to contain the buffer dimensions. They divided the problem into two sequential sub-
problems: a long-term term balancing problem, and a short term sequencing problem. The tasks are first
assigned to each workstation (long term problem); then, to limit the use of buffers, the smoothness of the line
is optimized by choosing the most opportune sequence of models entering the line (short term problem). The
buffers capacity needed between each workstation is calculated ex-post through a simulation study. The
approach is indirect because the assignment of buffers capacity is not explicitly considered during the
balancing procedure of the line.
2.3 Performances evaluation
The design of an assembly line is a complex problem that, as many other industrial problems, has to take into
account two fundamental aspects: performances and costs. Performances of an assembly line are mainly
related to its throughput, i.e. to the number of products that can be completed in the unit time. One of the
main issues while solving ALBPs is that it is difficult to evaluate the throughput of a complex assembly line,
but it is necessary to compare different design alternatives. While costs of a determined line configuration
can be easily calculated from the amount of resources employed, it is often not easy to calculate its
throughput with an acceptable degree of precision.
Throughput analysis is important for the design, operation and management of production systems. A
substantial amount of research has been devoted to developing analytical methods to estimate the throughput
of production systems with unreliable machines and finite buffers. These models however are often meant to
be approximate models, and they often cannot mimic the manufacturing system dynamics very well and pose
restrictive assumptions (such as exponential distributions of processing times). Furthermore, they are often
complex in nature, and hard to develop (Li et al., 2009). Currently, analytical approaches for throughput
analysis are not able to take into consideration the same realistic line features (such as mixed models, parallel
workstations, non-exponential distribution etc.) that, on the contrary, algorithms developed for line balancing
provide.
For this reason most of the algorithms developed to solve mixed model assembly lines problems use some
‘indirect’ measure of throughput to compare the performances of different solutions during the searching
procedure. For example, methods that are developed to seek station assignments that lead to more balanced
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workloads across stations and across products, like the above mentioned study by Battini et al. (2009), are
motivated to limit the effect, on the realized cycle time, of the sequencing of different models on the
assembly line. For an overview and a comparison among objectives to smoothen workload see Emde et al.
(2010). However one should remind that workload smoothing it is not an objective in itself, but a (supposed)
mean to achieve a high and stable throughput (Tiacci, 2012). Workload smoothing objectives are
substantially utilized, instead of throughput, because the throughput of a mixed model line is difficult to
estimate, while measures related to workload smoothing, for given tasks assignments, can be calculated
straightforwardly. But using workload smoothing as objective, as outlined by Karabati and Sayin (2003),
remains an approximate approach, for the very reason that the effects of the sequencing decision on the line
throughput are not incorporated explicitly.
Bukchin (1998) and Venkatesh and Dabade (2008) studied the correlation among different kinds of indirect
measures of throughput and the simulated throughput. From their works it is possible to conclude that is
difficult to find a measure of general validity for all the possible typologies of the problem and that a loss of
correlation of the same measures with the simulated throughput is expected as the number of features trying
to capture the complexity of real cases increases.
The only way to accurately evaluate the line throughput of a complex assembly line would be to build a
simulation model. However, as Bukchin (1998) and Altiparmak et al. (2007) outlined, the use of simulation
is not always practical, due to the need of evaluating a very large number of alternative system
configurations.
In this paper we adopt an approach that, thanks to the latest advances of the research on simulation
architectures and techniques, allows overcoming the limit to use, for the objective function evaluation,
indirect measures of throughput instead of the simulated one. This is performed by coupling a genetic
algorithm and an event/object oriented parametric simulator. The simulator, named Assembly Line Simulator
(Tiacci, 2012) and synthetically described in section 4.6, has some characteristics, like modularity and
fastness of execution, that allow it to be effectively coupled to those algorithms and procedures where
numerous variants of line configurations have to be simulated, and the evaluation of a fitness function (which
includes a line performances indicator, such as the throughput) has to be carried out several times.
3 The problem
The problem is to assign a certain number of tasks to a number of work centers, to define how many workers
there are in each work centre, to define the buffers capacity among WCs, while respecting precedence
constraints and trying to maximize the objective function.
3.1 Notation
i the task index (i = 0, …, N-1)
j the model index (j = 0, …, M-1)
k the work centre index (k = 0, …, P-1)
αj the demand proportion for model j
ct the imposed cycle time;
cteff the realized average cycle time;
N the number of tasks of the problem;
Nk the number of tasks assigned to work centre k;
M the number of models;
P the total number of work centres in the line;
tij the average time required to execute the task i on model type j;
σij the standard deviation of the time required to execute the task i on model type j;
cv the coefficient of variation of tij;
Wk the number of workers (or workstations) in work centre k;
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Ek the number of pieces of equipment in work centre k (Ek = Wk . Nk);
Bk the buffer size between work centre k-1 and k;
CW annual cost of a worker;
CE annual cost of a piece of equipment;
CB annual cost per unit buffer space.
3.2 Tasks
A set of N tasks (numbered with i = 0,…, N-1) has to be performed in the line in order to complete each
product. Because we are dealing with mixed model lines, the number of models (types of product) to be
assembled can be higher than one, and it is indicated by M (numbered with j = 0, …, M-1). Input data are
thus represented by an N x M matrix tij whose elements represent the average completion time of task i on
model type j. If the completion of a model does not require the execution of a certain task, this would result
in a 0 in the corresponding matrix element. Task times are considered to be independent normal variates,
which is considered to be realistic in most cases of human work. The standard deviation
ij of the completion
time of task i for model j is taken equal to its mean value (tij) multiplied by the coefficient of variation cv (
ij
= cv . tij).
3.3 Line features
In the line, each operator has a WorkStation (WS) where he performs one or more tasks. An annual cost CW
is associated to each operator. Each Work Centre (WC) consists of either one WS, for the case of non-
paralleling, or multiple parallel WSs (see Fig. 1).
Figure 1. An assembly line with parallel workstations.
‘Paralleling’ means that when a WC consists of two or more workstations, all the tasks assigned to the WC
are not shared among the WSs, but each WS performs all of them. Thus an increment of production capacity
of the WC is obtained through the addition of one (or more) WS and the related operator who performs the
same set of tasks. Each task requires a different piece of equipment to be performed, to which a cost CE is
associated . So if one station is parallelized there is an extra cost due to equipment duplication.
The line is asynchronous, that is blockage as well as starvation are possible. Pieces are retrieved from buffers
following a First In First Out rule. If a buffer is placed before a WC with parallel WSs, pieces are retrieved
from the same buffer by all the WSs.
One WC with multiple WSs is considered busy if every WS inside is busy. If a WS finishes its work on a
workpiece while the subsequent WC is still busy, the workpiece cannot move on, and remains in the current
WS keeping it busy (blocking after processing policy); the WS will be released (i.e. it will be able to
process another workpiece) only when the workpiece leaves the WS. The first WC is never starved (there is
buffers
work centre
work station
operator
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always raw material for the first WC) and the last station is never blocked (there is always storage space for
the finished product).
Models enter the line with a fixed sequence, able to comply with the demand proportion for each model. Set-
up times between models are considered negligible.
3.4 Constraints
The way tasks are assigned to WC is constrained by the precedence relations among tasks (which impose a
partial ordering, reflecting which task has to be completed before the others).
3.5 Objective (Fitness function)
The objective to minimize is the Normalized Design Cost (NDC), introduced by Tiacci et al. (2006). The
NDC takes into consideration the two fundamentals aspects between the trade-off exists: the economic aspect
and the performance aspect.
The economic aspect is evaluated through the Design Cost, that is in our case the total annual cost for labour,
equipment and buffers costs of the line configuration:
 
11
01
PP
k k k
kk
DC CE E CW W CB B


 

(1)
As shown in eq. (1), a cost per unit buffer space is considered, as in (Hillier and Hillier, 2006).
The performance aspect is taken into consideration introducing the realized cycle time (cteff), i.e. the actual
average cycle time of the line, that also corresponds to the throughput inverse. The realized cycle time can be
different from the imposed one (ct), but if cteff is higher than the ct a penalty in the evaluation of the line
configuration is considered. This is attained by increasing the Design Cost value if
eff
ct ct
:
2
1
eff
eff
eff
DC if ct ct
NDC ct ct
DC z if ct ct
ct



 




(2)
where z is a penalty factor that can be assigned according to the required observance of ct. The NDC reflects
the fact that in real cases a small deviation on the average cycle time with respect to the theoretical one is
admitted, especially if this implies consistent savings in terms of line configuration. Note that for z tending to
infinity the constraint related to the imposed cycle time becomes strict (type-I problem), while for z tending
to 0 the problem becomes equivalent to design costs minimization, with little regard for the imposed cycle
time. So in the latter case the algorithm could return solutions with effective cycle time quite distant from the
imposed ones.
4 The genetic algorithm approach
In this section a genetic algorithm approach to solve the defined problem is described. The genetic algorithm
is coupled with an object event/object oriented simulator that is used to calculate the individuals fitness
(Section 4.6).
4.1 Representation
Each individual is represented through a triple of chromosomes (see Fig. 2). Each chromosome is formed by
a number of genes, and can be seen as a vector formed by a number of elements. The first chromosome
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contains an ordered sequence of all tasks, i.e. a number of elements equal to N. Their order corresponds to
the same order they are assigned to WCs.
The second chromosome is also a vector with N elements, and establishes the WC the task is given to. In this
chromosome the WCs are numbered in ascending order starting from the first one in the line, to the last one.
So when a task is assigned to a different WS with respect to the preceding task, the corresponding value in
the second chromosome will be incremented by one. Taking for example the individual depicted in Figure 2,
tasks 0,3 and 1 will be assigned to WC#0, tasks 6 and 4 to WC#1, tasks 2 and 5 to WC#2 and tasks 7,8,9 to
WC#3. In this way we chose to separate the sequence of tasks and the assignments to the stations into two
chromosomes. Alternatively, one could use separators in a unique variable length chromosome to split up a
sequence of tasks in different stations, as in Chica et al. (2011). However, we have preferred a separate
chromosome to represent task assignments because the use of separators implies the introduction of repair
mechanisms to be applied after the crossover operator, in order to redistribute spare tasks among available
stations and removing empty stations.
Figure 2. Chromosomal representation.
The third chromosome has a variable length (i.e. a variable number of genes), equal to the number of WCs in
the line decreased by 1. The ordered sequence represents the capacity of the buffers between each WC of the
line. In the example of Figure 2, the capacity of the buffer between WC#0 and WC#1 is 2, there is no buffer
between WC#1 and WC#2, and a buffer of 4 units between WC#2 and WC#3.
4.2 Decoding
The decoding phase is performed as follows. Tasks assignment to each WC is performed, as above
mentioned, by comparing the first two chromosomes, and assigning the task in the n-th position of CH1 to
the corresponding WC indicated in the n-th position of CH2. Once all tasks have been assigned to WCs, it
remains to calculate the number of workers (i.e. WSs) in each WC. This is performed by imposing a certain
probability of completion, pc, of all tasks assigned to each WC. To do this the so called average model is
considered. First of all the weighted average
, and the weighted standard deviation
ˆi
of the time needed
to execute each task i are calculated:
ˆ
i j ij
j
tt
,
2
ˆi j ij
j
 
. (3)
Then, the total time needed in a WC to perform all the tasks assigned to it will follow a normal distribution
N
2
ˆˆ
,
ii
ii
t




with mean value equal to the sum of average task times assigned to it,
ˆ
i
it
, and
standard deviation equal to
2
ˆi
i
. Using the inverse normal distribution function, it is possible to
calculate the time needed to complete all tasks assigned to a WC with a probability pc. Indicating with Tlimit
0 3 1 6 4 2 5 7 8 9
0 0 0 1 1 2 2 3 3 3
CH1
CH2
sequence with which tasks will be given
N° of work centre the task is given to
2 0 4
CH3
Buffer size between WCs
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this time, the number of workers Wk assigned to the WC will be equal to the minimum integer value so that
ct .Wk Tlimit.
Table 1 shows how the number of workers Wk is calculated for the individual represented in Figure 2,
considering an imposed cycle time ct = 12, and pc = 80%.
Finally, the buffer capacity between each WC is indicated, as described in the preceding section, in the third
chromosome. Figure 3 shows the final line configuration related to the individual depicted in Figure 2. Note
that a buffer capacity equal to 0 means that there is no buffer between two WCs.
WCk
task
Tlimit
Wk
0
0
7
1.4
22
2.16
23.82
2
3
6
1.2
1
9
1.8
1
6
7
1.4
17
2.44
19.05
2
4
10
2
2
2
6
1.2
10
1.44
11.21
1
5
4
0.8
3
7
10
2
21
2.57
23.16
2
8
4
0.8
9
7
1.4
Table 1. Calculating the number of workers Wk in each WC (ct=12; pc=80%)
Figure 3. The line configuration corresponding to the individual depicted in fig. 2.
4.3 Initial population
The initial population is created randomly, respecting the precedence constraints. The first chromosome is
generated following the procedure adopted by Kim et al. (1996):
Step 1. Form an initial available set of tasks having no predecessors, and create an empty string (i.e. an
empty chromosome).
Step 2. Terminate, if the available set is empty. Otherwise go to Step 3.
Step 3. Select a task from the available set at random, and append it to the string.
Step 4. Update the available set by removing the selected task and by adding every immediate successor of
the task if all the immediate predecessors of the successor are already in the string. Go to Step 2.
2
4
tasks assigned
WC#0
0,3,1
WC#1
6,4
WC#2
2,5
WC#3
7,8,9
buffer size
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Note that in Step 4 the available set is updated with tasks satisfying precedence constraints so that it always
ensures the generation of a feasible sequence.
To generate the second chromosome, a value equal to 1 is initially assigned to the first position of the vector.
Then the subsequent position can assume the same value as the previous one, or a value increased by 1. This
option is made at random, with a probability of 50%.
The generation of the third chromosome is done together with the second one: each time the value of the
gene in the second chromosome is increased by 1 (i.e. a new WC is created), a new gene is added to the third
chromosome, whose value is an integer uniformly distributed between 0 and b1 and represents the buffer
capacity before the corresponding WC.
4.4 Crossover
The Crossover operator is applied during the reproduction phase. Two individuals, namely p1 and p2
(parent1 and parent2), generate two children s1 and s2. The crossover operator involves all the three
chromosomes.
A single point crossover operator is applied for the first chromosome CH1 (Fig. 4a). A cut point c is chosen
randomly in one parent (p1), and the elements after the cut point are copied into the same position of the
offspring s1. These elements are removed from the other parent p2, and the remaining elements of p2 are
copied into the initial positions of s1 in the same order as they appear in p2. In this way a feasible sequence
that satisfies precedence restrictions is generated, avoiding also duplication or omissions of tasks.
For the second chromosome (CH2) the following method is adopted (Fig. 4b). The chosen cut point c is the
same as the one of the first chromosome. The values in the preceding positions are copied in the same order
from p1 to s1. Let d represent the difference between the last copied value of p1 and the corresponding value
of p2. The values of the positions to the right of the cut point of p2 are copied in the same positions in s1
after being increased by d. Note that in this way the first value after the cut-point in s1 can assume or a value
equal to the preceding gene (position), or at most increased by 1.
To generate the third chromosome (CH3) of the offspring it is necessary to gather information from both the
second and the third chromosomes of the parents (see Fig. 4c).The procedure starts by taking into
consideration the second chromosome (CH2) of each parent, where the same chosen cut point c of the
preceding steps is considered. Then the value in the preceding positions with respect to c is taken into
consideration to generate the cut points c’ and c’’ in the third chromosome (CH3) of the respective parents.
For example in the case of the individual depicted in Fig. 4c, suppose that the random cut point for CH2 of
p1 is c=5, that is the cut point between the 5th and the 6th element of the chromosome. So the value of the 5th
element has to be taken into consideration to generate the cut point in the CH3 of p1. In this case the value of
the 5th element is 2, so a cut point c’=2 (between the 2nd and the 3rd element) is generated in CH3 of p1. The
same procedure stands for p2: in this case the value of the 5th element of CH2 is 1, so a cut point c’’ = 1
(between the 1st and the 2nd element) is generated in CH3 of p1.
When the cut points are generated in the third chromosomes of both parents, the elements of p1 before the
cut point c’ are copied in the initial position of the third chromosome of s1. Then the elements after the cut
point c’’ in p2 are appended to the third chromosome of s1. In this way the correct number of genes in the
third chromosome of s1 is always guaranteed.
The second offspring s2 is generated inverting p1 and p2 roles.
4.5 Mutation
When a mutation occurs, the operator is applied to all the three chromosomes of the individual. For the first
chromosome the feasible insertion method is adopted (Kim et al., 1996). A task is randomly selected (eg.
task 5 in Fig. 5a) and its position is swapped with another feasible position in the chromosome. As in a
feasible sequence the selected task must follow all of its immediate predecessors and precede all of its
- 11 -
immediate successors, the potential positions are consecutive, forming a substring in the parent. Among the
potential insertions positions, one is randomly chosen.
Regarding the second chromosome (Fig. 5b), the mutation takes place in the following way: a position in the
chromosome is randomly chosen. Let p represent the value contained in this position, and l the value of the
preceding position. Naturally will be l p. If l = p, then all the values from p (included) onwards are
increased by one. This means that one WC has been divided into two. If l < p then all the values from p
onwards are decreased by one. This means that two WCs have been united.
The mutation of the third chromosome (Fig. 5c) is done by randomly selecting one of the element, and then
randomly increasing or decreasing by one unit the buffer capacity.
a. b.
c.
Figure 4. Crossover operator.
1 2 5 3 6
142 7 5 3 6 8 9 10
2 3 6 1 5 4 8 7 9 10
random cut point
p2
p1
s1
4 8 7 9 10
CH1
CH1
CH1
c
6
4 1 0 2 3
CH3
1 2 3 3 3
0 0 1 1 2 2 2 2 3 3 0 0 0 1 1
p2
p1
CH2 CH2
CH3
random cut point
4 1 63
CH3
2 3 4 4 40 0 1 1 2
s1
CH2
c’ = 2 c’’ = 1
c = 5 c = 5
1 2 3 3 3
2 3 4 4 40 0 1 1 2
0 0 1 1 2 2 2 2 3 3
0 0 0 1 1
d
= 2 1 = +1
random cut point
p2
p1
s1
CH2
CH2
CH2
+1 +1 +1 +1 +1
c
- 12 -
a. b.
c.
Figure 5. Mutation operator.
4.6 Fitness evaluation: the Assembly Line Simulator (ALS)
The fitness of an individual is the Normalize Design Cost (eq. (2)). When an individual is decoded, all the
info related to the number of WCs in the line, the number of workers in each WC, the tasks (and thus the
pieces of equipment) assigned to each WC, the buffers capacities between WCs are known. We will use the
term ‘line configuration’ to refer to this set of information. So the DC of the line configuration associated to
the individual can be calculated straightforwardly through eq. (1).
The evaluation of the effective cycle time cteff of the line configuration (needed to calculate the NDC) is
performed by exploiting the integration capabilities of an Assembly Line Simulator (ALS) (Tiacci, 2012).
ALS is a parametric event-oriented simulator, developed in Java, capable to immediately calculate the
average cycle time of a complex line by simply receiving as inputs the task times durations, the line
configuration, and the sequence of models entering the line. Thanks to its object-oriented architecture,
different lines configurations can be created by the mean of very simple array representations, so that the
time required for building the model is zeroed. ALS has been developed following an event oriented
paradigm: the simulation run is performed only by events scheduling, rescheduling, cancelling etc.; by
avoiding the use of ‘processes’, execution times are kept very low with respect to the analogous process-
oriented version (Tiacci and Saetta, 2007), arriving to outperform Arena. These characteristics allow ALS to
be effectively coupled to those algorithms and procedures where numerous variants of line configurations
have to be simulated, and the evaluation of a fitness function (which includes a line performances indicator,
such as the cycle time in our case) has to be performed several times. ALS allows to overcome the limit of
1 4 2 7 5 3 6 8 9 10
CH1
possible insertion points
1 4 7
10
2 5
8 9
3 6
-1
before
mutation
after
mutation
4 1 0
CH3
4 0 0
CH3
2 3 4 4 4
1 1 1 2 2
CH2
3 4 5 5 51 1 2 2 3
+1 +1 +1 +1 +1+1
CH2
before
mutation
after
mutation
- 13 -
using traditional measures (not simulation-based) of the performance parameter (the line throughput or,
equivalently the effective cycle time) that are poorly correlated to its real value. To embed ALS, the genetic
algorithm procedure has been developed using the same programming environment of the simulator (Java).
ALS has then been imported as a library (for details refer to ALS documentation). ALS is contained in a
package named lineSimulator, which provides the public class Simulation, whose constructor’s
arguments represent the line configuration that has to be simulated and the other inputs needed to perform
the simulation (task times, models sequence etc,). When a Simulation object is created, the simulation model
of the assembly line is created, the simulation is performed and outputs (f.e. average cycle time, average flow
time, average WIP) are stored in the object attributes. To evaluate the average cycle time cteff of each line
configuration related to an individual, we set the simulation run performed by ALS to end when 3000 loads
have been completed.
It is noteworthy that annual inventory costs (equal to WIP multiplied by annual unit holding cost) could have
been considered, in the objective function, instead of buffer space cost. In fact, as above mentioned, ALS is
able to provide the average WIP as one of the outputs of the simulation. However, the amount of WIPs and
buffer spaces are expected to be highly correlated.
The methodology to generate different sequences of models entering the line has been chosen to reflect the
different nature of the balancing and the sequencing problems. The balancing problem has a long-term
nature, concerning the design phase of the assembly line. The sequencing problem has a short-term nature,
concerning the daily production. In other words, the exact production sequence is not usually already known
during the design stages of the assembly line (balancing problem), but only a few days before production
starts. For this reason the different design alternatives during the searching procedure are evaluated just
assuming a determined demand proportion among models, without specifying any exact sequence. This is
carried out by exploiting the possibility provided by ALS to generate random sequences of models while
respecting a determined demand proportion among models.
Once obtained cteff it is possible calculate the individual fitness function through eq. (2).
4.7 Genetic algorithm iterations and resulting general structure
The genetic algorithm structure is depicted in Fig. 6. After creating the initial population of NP individuals,
during the reproduction all the individuals are randomly coupled. Each couple of parents generates a couple
of children (as described in Section 4.4) that are added to the initial population. Each individual has now a
certain probability mr (mutation rate) of undergoing a mutation (Section 4.5). Each mutated individual is
added to the population, but the original one is also maintained in the population.
After terminating the mutation procedure, the fitness function of each individual of the resulting population
is calculated (fitness evaluation), and only the best NP individuals survive and pass the next generation. After
a number G of iterations the algorithm stops and the individual with the best fitness is considered the final
solution.
Figure 6. Genetic algorithm iterations and general structure.
initial population
selection
reproduction
(crossover)
mutation
new generation
iterations
- 14 -
5 Design of experiment
5.1 Experiment 1
The proposed approach has been tested on 5 problems (Bartholdi, 1993; Gunther et al., 1983; Rosenberg and
Ziegler, 1992; Scholl, 1993; Wee and Magazine, 1981) of different size, varying the number of tasks
between a minimum of 25 and a maximum of 297. With the aim of exploring a larger number of cases, each
of the 5 problems has been solved for 4 different cycle times (see Table 4). Furthermore, each of the 20
problems obtained has been solved for two different mixes (M = 2 products and M = 4 products), obtaining a
total number of 40 different explored cases.
The problems precedence diagrams can be downloaded from the dataset of Scholl (2012), in which the
number of tasks and precedence constraints among tasks are defined for a single model. Figure 7 shows for
example the precedence diagram of the 35 tasks problem from Gunther.
Figure 7. Precedence diagram of Gunther 35.
The task times tij for both the 2 and the 4 model instances that have been created for our study are available
at (Tiacci, 2015b). A coefficient of variation cv = 0.3 has been utilized to obtain task times standard
deviations. As it was mentioned in Section 4.6, during the searching procedure only the demand proportion
among models has been assumed to be known. In 2 models problems, the demand has been equally
distributed with 50% for m#1 and 50% for m#2. In 4 models problems, the following mix has been assumed:
50% m#1, 20% m#2, 20% m#3, 10% m#4. Sequences of models have been randomly generated while
respecting these demand proportions. The annual costs of a worker (CW) of a piece of equipment (CE) and
for unit buffer space (CB) have been considered respectively equal to 30000 and 3000 and 300 €/y. The
values of the genetic algorithm parameters utilized in the study are: NP = 40; mr = 0.55; G = 100, pc = 0.5,
b1 = 0.
The final solution obtained by a single run of the genetic algorithm is evaluated on the basis of the
Normalized Design Cost, considering a penalty coefficient z = 20 (see eq. (2)). To obtain the NDC value the
line configuration corresponding to the final solution is simulated again 10 times using ALS, for a length
equivalent to the completion of 5000 loads, and the average value of the cycle time is then used, together
with the number of workers, pieces of equipment, and buffer sizes, to calculate the NDC trough eq.(2). The
simulations carried out through ALS, for the final evaluation of solutions, consider deterministic sequences
of models entering the line: the sequence 1212121212 for 2 models problems; 1213141213 for 4 models
problems (McMullen and Frazier, 1998).
There is no other algorithm in literature which can be compared to ours. However it is interesting to evaluate
the potential advantage of using buffers (we will refer to the approach presented here in as GA+BUF) with
respect to an analogous genetic algorithm approach (indicated later on simply as GA) in which it is
0
1
4
6
9
11
2 3 10
57
13
17
812
13
27
32
14 15 20
16 19
18
21
24
29
22 23 26
25
33
28
30 31
34
- 15 -
assumed that there are no buffers between WCs. This is obtained, with respect to the approach proposed here
in, considering just the first two chromosomes for the individual representation, and applying the crossover
and mutation operators just to these two chromosomes. GA has already demonstrated its superiority with
respect to the other algorithms, in particular Simulated Annealing and Genetic algorithms approaches,
presented in literature, able to solve the same problem type without buffers (mixed-model, parallel stations,
stochastic task times of completion) (Tiacci, 2015a).
All the 40 instances above described have been solved following both the approaches. Because of the stochastic nature
of the genetic algorithm and the simulator, the solution found for an instance changes if the procedure is repeated using
a different random seed. For this reason, each of the 40 instances has been solved 20 times, and the mean and standard
deviation values of the NDC for each instance have been calculated (results are shown in Table 4). A comparison of the
best solutions found by the two approaches (GA+BUF and GA) has been made (Tables 4-7).
Table 2 summarizes the values of the parameters utilized in the experiment.
Parameter
Value
Basic GA parameters
Population size (NP)
40
Number of iterations (G)
100
Mutation rate (mr)
55%
Crossover rate (cr)
100%
Representation & other data
Penalty coefficient (z)
20
Probability of completion (pc)
50%
Worker cost (WC)
30000 €/year
Piece of equipment cost (EC)
3000 €/year
Buffer space cost (BC)
300 €/year
Objective function evaluation during each iteration
Simulation length (to calculate cteff)
Sequence of models
3000 loads
Random
Demand proportion of models:
2 models problems:
4 models problems:
m#1, m#2, m#3, m#4
50%, 50% - -
50%, 20%, 20%, 10%
Objective function evaluation of the final solution
Simulation length (to calculate cteff)
5000 loads
Number of replication
10
Deterministic sequence of models:
2 models problems:
4 models problems:
…1,2,1,2…
…1,2,1,3,1,4,1,2,1,3…
Design of experiment
Number of instances
40
Number of replications for each instance
20
Table 2. Experiment 1 settings.
- 16 -
5.2 Experiment 2
The structure of experiment 2 is the same of experiment 1, but refers to a more realistic scenario, in which
many of the simplifying assumptions of the first experiment have been relaxed, such as a limited number of
models, the same pieces of equipment costs and the same coefficient of variation for all tasks. To do this a
new instance has been generated, taking the same precedence diagram of the most complex instance in
literature, the one from Scholl with 297 tasks.
Firstly a coefficient of variation cvi for each task i has been randomly generated from a uniform distribution
with range [0.1, 0.5]. Analogously, different pieces of equipment costs CEi have been randomly generated
from a uniform distribution with range [100, 10.000].
Then, average completion times for 200 different models have been randomly generated, taking the task time
ti of the Scholl’s instance as reference. To do this, different degrees of customization and optional equipment
have been considered. Basically, tasks fall into three different categories (Golz et al., 2012):
common tasks have to be performed into every model. In this case the average task time is the same
for all models, and equal to ti;
OR-tasks are optionally performed depending on the customer specifications. This kind of task are
modelled assuming that they are performed only for one model (randomly chosen), with an average
completion time ti. For all the other models the completion time is set equal to 0;
XOR-tasks represent variants that are performed in every model. These tasks have been generated by
randomly assigning to each model an average completion time uniformly distributed in the range
[0.7ti, 1.3ti].
In this experiment, the degree of variability is defined by the fraction of the three different task categories as
low, medium and high. In the low task variability instances the distribution of tasks is defined as 70% for
common and 15% each for OR and XOR-tasks, while these fractions are 50 and 25% in the medium and 30
and 35% in the high part variability instances.
Low
Medium
High
70% common tasks
50% common tasks
30% common tasks
15% OR tasks
25% OR tasks
35% OR tasks
15% XOR tasks
25% XOR tasks
35% XOR tasks
Table 3. Degree of tasks’ variability
The demand proportion of each model and the exact model sequences for the final evaluation has been
generated for each instance in the following way. An exact sequence is firstly generated by appending each
model to the sequence a number of times drawn from a uniform distribution with range [1, 4]. Then, the
obtained sequence is randomly shuffled. The demand proportion for each model is calculated considering the
same proportion observed in the deterministic sequence. All the other settings are the same of experiment 1.
The instances data can be downloaded from (Tiacci, 2015b).
6 Results
Table 4 shows that GA+BUF obtains always mean values of NDC lower than GA. Considering a null
hypothesis H0: NDC(GA+BUF) NDC(GA), the P-value of the statistical test (difference in means of two
normal distributions, variances unknown and not necessarily equal) resulted lower than 0.001 in 34 out of the
40 instances, indicating that in these cases the null hypothesis has to be rejected with a probability of 99.9%,
and that the alternative hypothesis H1: NDC(GA+BUF) < NDC(GA) has to be accepted.
Decrease of NDC (indicated as %diff) reaches the 8% in some instances.
- 17 -
It can be observed that using buffers improves NDC mostly for small and medium sized instances, while for
problems from Barthold and Scholl (respectively 148 and 297 tasks), improvements are less important and
the significance levels of the statistic test worsen, especially in the Barthold’s problem.
Table 5 shows in details the results related to the best solutions found by GA and GA+BUF for each
instance.
In almost all the instances GA+BUF obtains a lower NDC. The reduction of NDC is obtained both through a
reduction of the effective cycle time cteff, which is achieved in almost all the instances (35 out of 40), and
through a reduction of the design cost DC (in 31 cases out of 40). It is noteworthy that in 26 cases out of 40
the use of buffers allows to achieve both these objectives at the same time, providing simultaneously more
efficient and economic solutions than in the case without buffer. Table 6 and Table 7 show the best solution
found respectively through GA and GA+BUF for the problem Gunther 35, 4 models, ct=30 (the precedence
diagram for this problem is depicted in Figure 7). In this example it is clear how the use of buffers allows
enlarging the solution space. In fact, although the number of WCs increase from 12 to 14, the use of buffers
allows decreasing paralleling (WSs decreases from 20 to 19), improving at the same time the performance
related to the effective cycle time of the line.
The use of buffers in production lines is usually linked, in the literature, to two concepts that are often
considered necessarily connected: performances improvement and increasing design costs. From this point of
view, there are 4 interesting cases in which a counter intuitive behaviour can be observed: the use of buffers
brings to a solution with a higher cycle time and a lower design cost than without buffers. It is noteworthy
that, however, even in these cases consistent reductions of NDC are obtained. This means that the possibility
to use buffers allows enlarging the solution space of the problem, and that the proposed algorithm is able to
explore this space to find solutions that would be hardly considered by standard constructive heuristics.
Table 8 shows in summary other characteristics of the best solutions found by GA and GA+BUF related to
all the 40 instances: the total number of WCs in the line, P; the total number of workers (or equivalently
WSs) in the line, WTOT=Wk
P-1
k=0 ; the total number of pieces of equipment in the line, ETOT=k
P-1
k=0 ; the
paralleling degree,
= WTOT/P; the total capacity of buffers in the line, BTOT=k
P-1
k=1 ; the greatest capacity of
a buffer in the line, BMAX = maxk (Bk); the percentage of WCs in the line that are preceded by a buffer,
B%=( βk
P-1
k=1 /(P-1), where βk = 1 if Bk > 0, βk = 0 otherwise. Off course the latter three characteristics are only
given for GA+BUF. The columns under the label (GA+BUF)vs(GA)” indicate if the parameters P, WTOT,
ETOT,
are lower (“<”) or higher (“>”) in the solution provided by GA+BUF than in the corresponding one
provided by GA.
It is possible to outline how the solutions provided by GA+BUF are characterized by the use of small
buffers, that are rather distributed along the line. In fact, most of the solutions show a consistent percentage
(often higher than 50%) of WCs preceded by a buffer, while the maximum capacity of each buffer is lower
than or equal to 2 units in more than a half of the instances, and never exceeds 4 units.
In most of the cases (29 out of 40 instances) a reduction of the number of workers is obtained, while in 18
cases a reduction of the total number of pieces of equipment is achieved. There are 13 cases in which both
workers and pieces of equipment are reduced.
The GA+BUF algorithm seems to exploit the possibility to utilize buffers to find solutions with different
paralleling degree with respect to GA. However, it does not seem that the use of buffers is correlated to
solutions with higher or lower paralleling degree: in 17 cases out of 40 the paralleling degree decreases, in 13
cases increases, while in the remaining cases remains the same. The best solutions found by both methods for
all the 40 problems are published at (Tiacci, 2015c).
- 18 -
Table 4. Results of experiment 1
NDC (GA)
NDC (GA+BUF)
Problem
size
M
ct
Mean
St. Dev.
Mean
St. Dev.
%diff
P-Value
Roszieg
25
2
18
349,340
6,755
337,737
11,588
-3.32%
0.000
14
456,719
6,741
432,344
11,170
-5.34%
0.000
11
556,614
8,868
519,825
6,109
-6.61%
0.000
9
668,380
5,922
629,705
10,129
-5.79%
0.000
Gunther
35
2
54
461,962
6,398
444,195
13,317
-3.85%
0.000
41
617,048
7,011
590,459
16,492
-4.31%
0.000
36
678,793
6,684
650,360
12,946
-4.19%
0.000
30
793,423
8,628
739,983
12,788
-6.74%
0.000
Wee-mag
75
2
34
2,131,852
17,412
2,061,980
17,777
-3.28%
0.000
28
2,551,227
19,350
2,360,792
19,581
-7.46%
0.000
24
3,061,146
29,515
2,805,611
42,700
-8.35%
0.000
21
3,366,757
13,510
3,334,198
14,762
-0.97%
0.000
Barthold
148
2
470
949,227
20,431
924,543
23,876
-2.60%
0.001
403
1,039,616
22,178
1,017,833
25,158
-2.10%
0.003
378
1,070,935
19,380
1,065,328
24,568
-0.52%
0.214
351
1,125,704
28,487
1,112,551
18,864
-1.17%
0.047
Scholl
297
2
1515
2,906,985
30,937
2,842,294
51,014
-2.23%
0.000
1394
3,146,223
38,956
3,041,026
54,694
-3.34%
0.000
1301
3,276,713
61,707
3,207,868
35,550
-2.10%
0.000
1216
3,489,616
36,080
3,378,069
42,471
-3.20%
0.000
Roszieg
25
4
18
355,171
11,722
340,177
9,252
-4.22%
0.000
14
459,014
6,309
430,499
10,992
-6.21%
0.000
11
557,234
6,416
526,835
10,584
-5.46%
0.000
9
672,520
4,113
630,576
15,199
-6.24%
0.000
Gunther
35
4
54
462,558
7,573
440,614
6,588
-4.74%
0.000
41
622,918
10,751
583,174
17,631
-6.38%
0.000
36
679,911
8,942
648,196
18,679
-4.66%
0.000
30
792,238
7,152
741,642
11,321
-6.39%
0.000
Wee-mag
75
4
34
2,137,600
16,907
2,061,492
25,831
-3.56%
0.000
28
2,558,510
12,271
2,355,863
21,930
-7.92%
0.000
24
3,052,327
18,366
2,806,067
28,742
-8.07%
0.000
21
3,365,087
10,589
3,329,051
11,883
-1.07%
0.000
Barthold
148
4
470
955,869
22,222
930,774
20,408
-2.63%
0.000
403
1,045,238
19,436
1,016,547
19,921
-2.74%
0.000
378
1,081,947
23,898
1,062,538
20,798
-1.79%
0.005
351
1,130,987
31,944
1,117,308
21,365
-1.21%
0.060
Scholl
297
4
1515
2,920,733
49,685
2,864,793
41,930
-1.92%
0.000
1394
3,130,775
51,055
3,036,704
55,506
-3.00%
0.000
1301
3,285,752
29,528
3,223,923
57,774
-1.88%
0.000
1216
349,340
6,755
337,737
11,588
-2.58%
0.000
- 19 -
Table 5. Comparison of the best solutions found by GA and GA+BUF in each instance of experiment 1.
GA
GA+BUF
improvements
Problem
size
M
ct
cteff
DC
NDC
cteff
DC
NDC
%diff
cteff
DC
cteff &
DC
Roszieg
25
2
18
17.98
345,000
345,000
17.94
316,200
316,200
-8.35%



14
13.99
441,000
441,000
14.00
418,500
418,500
-5.10%



11
11.18
540,000
542,744
10.99
505,500
505,500
-6.86%



9
9.15
654,000
657,699
8.97
605,400
605,400
-7.95%



Gunther
35
2
54
56.09
435,000
448,064
53.74
424,500
424,500
-5.26%



41
41.91
597,000
602,822
40.33
560,700
560,700
-6.99%



36
37.09
654,000
665,928
36.13
623,100
623,263
-6.41%



30
31.14
759,000
780,768
30.20
714,000
714,662
-8.47%



Wee-mag
75
2
34
34.51
2,100,000
2,109,629
33.92
2,033,700
2,033,700
-3.60%



28
28.78
2,478,000
2,516,692
28.01
2,329,800
2,329,812
-7.43%



24
24.67
2,973,000
3,019,682
24.18
2,740,200
2,743,263
-9.15%



21
20.99
3,345,000
3,345,000
21.04
3,296,100
3,296,305
-1.46%



Barthold
148
2
470
482.81
894,000
907,274
470.51
894,300
894,321
-1.43%



403
406.32
996,000
997,353
399.72
967,200
967,200
-3.02%



378
385.91
1,020,000
1,028,934
377.90
1,023,600
1,023,600
-0.52%



351
364.41
1,053,000
1,083,743
352.09
1,072,800
1,073,007
-0.99%



Scholl
297
2
1515
1546.63
2,811,000
2,835,501
1517.99
2,754,000
2,754,215
-2.87%



1394
1431.39
3,042,000
3,085,764
1418.20
2,939,100
2,956,811
-4.18%



1301
1339.82
3,033,000
3,087,020
1301.85
3,159,900
3,159,927
2.36%



1216
1241.51
3,390,000
3,419,831
1243.32
3,239,400
3,272,097
-4.32%



Roszieg
25
4
18
17.98
345,000
345,000
18.05
316,200
316,251
-8.33%



14
14.15
441,000
441,991
14.06
416,400
416,540
-5.76%



11
11.29
537,000
544,404
11.10
511,800
512,724
-5.82%



9
9.08
663,000
664,103
9.11
599,400
601,162
-9.48%



Gunther
35
4
54
55.91
435,000
445,860
53.98
435,900
435,900
-2.23%



41
42.12
597,000
605,841
41.19
545,700
545,941
-9.89%



36
36.42
660,000
661,839
35.84
617,100
617,100
-6.76%



30
30.74
771,000
780,491
30.30
714,300
715,716
-8.30%



Wee-mag
75
4
34
34.49
2,100,000
2,108,627
34.13
2,008,200
2,008,793
-4.73%



28
28.64
2,508,000
2,533,907
27.99
2,313,000
2,313,000
-8.72%



24
24.66
2,964,000
3,008,564
24.03
2,756,100
2,756,181
-8.39%



21
21.23
3,339,000
3,346,990
21.16
3,302,400
3,306,435
-1.21%



Barthold
148
4
470
484.22
894,000
910,357
471.07
894,300
894,392
-1.75%



403
416.26
984,000
1,005,301
403.62
984,900
984,946
-2.02%



378
382.65
1,041,000
1,044,150
378.08
1,032,900
1,032,901
-1.08%



351
357.37
1,077,000
1,084,102
348.91
1,084,500
1,084,500
0.04%



Scholl
297
4
1515
1549.48
2,802,000
2,831,021
1500.76
2,814,600
2,814,600
-0.58%



1394
1422.23
3,003,000
3,027,623
1409.21
2,896,500
2,903,395
-4.10%



1301
1320.20
3,225,000
3,239,047
1304.55
3,126,600
3,127,067
-3.46%



1216
1247.12
3,372,000
3,416,162
1225.03
316,200
316,200
-3.70%



tot
35
31
26
- 20 -
WC#
WS
assigned
(Wk)
tasks assigned
0
1
2
3
4
5
6
7
8
9
10
11
2
2
1
2
1
2
1
2
1
2
2
2
0, 11
9, 1, 2
16, 4, 5, 6
3, 10
17, 7, 18
8, 12
13, 19
14, 15
20, 24, 25, 29
21, 22, 23
26, 27, 33
30, 28, 31, 32, 34
cteff = 30.74; DC = 771000; NDC = 780491
Table 6. The best GA solution for Gunther 35, 4 models, ct = 30.
WC#
WSs
assigned
(Wk)
Buffer size
(Bk)
tasks assigned
0
1
2
3
4
5
6
7
8
9
10
11
12
13
2
1
1
2
1
1
1
1
2
1
1
1
2
2
0
3
2
1
1
0
1
0
0
0
1
1
1
0
0, 11
4, 1, 16, 5, 7
2, 8
9, 12
3, 6
17, 18, 13
19
14
10, 15
20, 29, 24, 30
21, 31, 22
25, 23
26, 33, 27
28, 32, 34
cteff = 30.29; DC = 714300; NDC = 715716
Table 7. The best GA+BUF solution for Gunther 35, 4 models, ct = 30.
- 21 -
GA
GA+BUF
(GA+BUF)vs(GA)
Problem
size
M
ct
P
WTOT
ETOT
P
WTOT
ETOT
BTOT
BMAX
B%
P
WTOT
ETOT
Roszieg
25
2
18
9
9
25
1.000
8
8
25
4
1
57%
1.000
<
<
14
11
12
27
1.091
10
11
29
5
1
56%
1.100
<
<
>
>
11
9
14
40
1.556
9
13
38
5
1
63%
1.444
<
<
<
9
12
18
38
1.500
10
16
41
8
1
89%
1.600
<
<
>
>
Gunther
35
2
54
11
11
35
1.000
9
10
41
5
1
63%
1.111
<
<
>
>
41
11
15
49
1.364
9
13
56
9
3
75%
1.444
<
<
>
>
36
13
17
48
1.308
10
15
57
7
1
78%
1.500
<
<
>
>
30
14
20
53
1.429
15
19
47
10
2
57%
1.267
>
<
<
<
Wee-mag
75
2
34
49
59
110
1.204
48
57
106
19
2
36%
1.188
<
<
<
<
28
32
66
166
2.063
43
64
133
36
2
71%
1.488
>
<
<
<
24
31
78
211
2.516
30
71
199
44
3
86%
2.367
<
<
<
<
21
26
86
255
3.308
24
83
267
17
3
52%
3.458
<
<
>
>
Barthold
148
2
470
15
15
148
1.000
15
15
148
1
1
7%
1.000
403
17
18
152
1.059
16
17
152
4
1
27%
1.063
<
<
>
378
18
19
150
1.056
18
19
151
2
1
12%
1.056
>
351
19
20
151
1.053
19
20
157
6
2
28%
1.053
>
Scholl
297
2
1515
64
64
297
1.000
62
62
297
10
2
13%
1.000
<
<
1394
68
70
314
1.029
68
68
297
27
2
34%
1.000
<
<
<
1301
65
69
321
1.062
71
74
311
23
2
27%
1.042
>
>
<
<
1216
79
82
310
1.038
72
76
318
18
2
21%
1.056
<
<
>
>
Roszieg
25
4
18
9
9
25
1.000
8
8
25
4
1
57%
1.000
<
<
14
11
12
27
1.091
10
11
28
8
2
78%
1.100
<
<
>
>
11
9
14
39
1.556
9
13
40
6
1
75%
1.444
<
>
<
9
11
18
41
1.636
11
16
39
8
1
80%
1.455
<
<
<
Gunther
35
4
54
11
11
35
1.000
11
11
35
3
1
30%
1.000
41
13
15
49
1.154
12
14
41
9
2
73%
1.167
<
<
<
>
36
12
17
50
1.417
13
16
45
7
1
58%
1.231
>
<
<
<
30
12
20
57
1.667
14
19
47
11
3
62%
1.357
>
<
<
<
Wee-mag
75
4
34
52
60
100
1.154
43
55
117
24
2
50%
1.279
<
<
>
>
28
29
65
186
2.241
49
65
117
40
3
67%
1.327
>
<
<
24
27
76
228
2.815
27
70
214
47
4
92%
2.593
<
<
<
21
26
86
253
3.308
23
82
279
18
2
64%
3.565
<
<
>
>
Barthold
148
4
470
15
15
148
1.000
15
15
148
1
1
7%
1.000
403
18
18
148
1.000
18
18
148
3
1
18%
1.000
378
18
19
157
1.056
18
19
154
3
1
18%
1.056
<
351
19
20
159
1.053
20
21
151
5
1
26%
1.050
>
>
<
<
Scholl
297
4
1515
60
62
314
1.033
64
64
297
12
2
16%
1.000
>
>
<
<
1394
66
69
311
1.045
65
66
302
35
3
41%
1.015
<
<
<
<
1301
74
77
305
1.041
68
72
320
22
2
31%
1.059
<
<
>
>
1216
74
80
324
1.081
74
78
313
24
2
30%
1.054
<
<
<
Tot <
19
29
18
17
Tot >
8
3
14
13
Table 8. Experiment 1, details of the best solutions found by GA and GA+BUF in each instance.
- 22 -
Table 9 summarizes the results related to experiment 2 in terms of NDC, showing both mean values with
standard deviations on 20 replications and the best solutions found by GA and GA+BUF. The improvements
achievable through GA+BUF are always significant, even if in one instance (Tasks variability = High, ct =
1216) the best solution obtained by GA is slightly better than the one obtained by GA+BUF. The full
factorial design of experiment 2 has been analysed, considering all the 20 replications made for each
combination of factors. The factors considered are: the algorithm utilized (“alg”), the task times variability
(“variability”) and the imposed cycle time (“ct”). From the Analysis of Variance (Table 10) it is possible to
see that all the three factors have a significant impact on NDC. The nature of this impact can be deduced
from Figure 8, which shows that NDC increases when:
lower values of the imposed cycle time ct are adopted
the tasks variability is high
buffers are not allowed
The ANOVA also indicates that the two-way interaction between variability and cycle time is significant at a
level of 0.002. In particular, from the interaction plot reported in Figure 9, it is possible to observe that high
tasks variability has a negative impact on NDC especially when a lower ct is imposed.
20 replications
Best solutions
Tasks
variability
NDC (GA)
NDC (GA+BUF)
ct
Mean
St. Dev.
Mean
St. Dev.
%diff
P-Value
GA
GA+BUF
%diff
Low
1515
3,252,055
54,168
3,168,811
47,278
-2.56%
0.000
3,172,208
3,066,152
-3.34%
1394
3,412,357
41,870
3,331,053
39,212
-2.38%
0.000
3,313,580
3,261,335
-1.58%
1301
3,572,955
44,494
3,480,201
38,251
-2.60%
0.000
3,483,070
3,419,274
-1.83%
1216
3,747,311
47,716
3,619,870
29,880
-3.40%
0.000
3,652,285
3,569,738
-2.26%
Medium
1515
3,231,903
48,768
3,146,019
51,669
-2.66%
0.000
3,166,244
3,058,143
-3.41%
1394
3,401,777
38,622
3,313,782
35,476
-2.59%
0.000
3,323,044
3,244,134
-2.37%
1301
3,548,929
46,674
3,440,471
34,601
-3.06%
0.000
3,478,291
3,375,989
-2.94%
1216
3,718,324
45,393
3,581,791
41,511
-3.67%
0.000
3,576,205
3,501,839
-2.08%
High
1515
3,251,814
34,893
3,183,692
38,482
-2.09%
0.000
3,187,740
3,128,529
-1.86%
1394
3,473,200
67,565
3,366,519
28,414
-3.07%
0.000
3,365,786
3,324,772
-1.22%
1301
3,621,428
71,115
3,536,865
53,318
-2.34%
0.000
3,497,546
3,458,438
-1.12%
1216
3,770,309
53,361
3,685,377
38,035
-2.25%
0.000
3,608,296
3,614,561
0.17%
Table 9. Results of experiment 2.
Source DF Seq SS Adj SS Adj MS F p
variability 2 3.24729E+11 3.24729E+11 1.62365E+11 77.48 0.000
ct 3 1.52809E+13 1.52809E+13 5.09363E+12 2430.60 0.000
alg 1 1.09808E+12 1.09808E+12 1.09808E+12 523.99 0.000
variability*ct 6 45334086505 45334086505 7555681084 3.61 0.002
variability*alg 2 6967859364 6967859364 3483929682 1.66 0.191
ct*alg 3 21434613066 21434613066 7144871022 3.41 0.018
variability*ct*alg 6 16452201916 16452201916 2742033653 1.31 0.252
Error 456 9.55604E+11 9.55604E+11 2095622183
Total 479 1.77495E+13
Table 10. Experiment 2, Analysis of Variance on NDC.
- 23 -
Figure 8. Experiment 2, main Effects for NDC.
Figure 9. Experiment 2, interaction plot for NDC
7 Summary
With this work a decisive step forward in the literature is done by tackling for the first time simultaneously
the assembly line balancing problem and the buffer allocation problem. Due to the amplitude of the solution
space, it is very hard to approach the problem as a whole. However, to solve sequentially the balancing and
the buffer allocation problems (or vice versa) means not to take into consideration the interrelations between
the two problems. In fact, the structure of precedence constraints has a direct influence on the order in which
tasks can be assigned to each station, i.e. on tasks assignment to workstations. This in turn determines
stations workloads and their smoothness. Stations workloads and their smoothness have an influence on
starving and blocking phenomena, that are in turn influenced by buffer spaces among workstations.
An innovative approach, consisting in coupling the most recent advances of simulation techniques with a
genetic algorithm approach, has been presented to solve a very complex problem: the Mixed Model
Assembly Line Balancing Problem (MALBP) with stochastic task times, parallel workstations, and buffers
between workstations. An opportune chromosomal representation allows to explore the solutions space very
efficiently, varying simultaneously tasks assignments and buffers capacity among workstations. Our
va ria bilit y
cycl eT
alg
1515139413011216 G A +B UFGA
3600000
3400000
3200000
3600000
3400000
3200000
L
M
H
variability
1216
1301
1394
1515
cy cleT
Interaction Plot (data means) for NDC
Mean of NDC
HML
3700000
3600000
3500000
3400000
3300000
3200000
1515139413011216 GA +BUFGA
variability
cycleT
alg
Main Effects Plot (data means) for NDC
- 24 -
approach allows not only to simultaneously consider line balancing and buffer allocation beyond simple
work division in a line, but also to take into consideration complicated line structures.
A parametric simulator, named ALS, has been used to calculate the objective function of each individual,
evaluating at the same time the effect of tasks assignment and buffer allocation decisions on the line
throughput. ALS has been built following the event/object oriented paradigm. Its fastness and flexibility
allow its utilization in those algorithms and procedures where the evaluation of a fitness function (which
includes some performance parameters, like the throughput or the cycle time) has to be carried out a large
number of times. It allows overcoming the limit of using others measures of throughput (or cycle time),
presented in literature, that are poorly correlated to its real value when the complexity of the line increases.
The approach has been compared to an analogous approach in which buffers between workstations are not
allowed. The results of an ample experiment on different issues of the problem allow to draw some
conclusions: even considering a cost per unit buffer space, it is often possible to find solutions that provide
higher throughput than in the case without buffers, and at the same time have a lower design cost; using
buffers gives major benefits for small and medium sized instances; the solutions are generally characterized
by the use of small buffers, that are rather distributed along the line.
The promising results achieved in this work suggest continuing to exploit the latest advances in discrete
event simulation techniques to solve more complex problems or take into consideration different objectives.
For example, due to the computational complexities involved, the assembly line balancing problem and the
sequencing problems are usually addressed in literature independently from each other, although they are
closely interrelated. A possible improvement in the genetic algorithm approach presented here-in could be to
find a representation that allows considering the sequence of models entering the line as a variable itself, and
not as an input of the problem. By using ALS for the evaluation of the fitness function one could explicitly
consider the effect of different sequencing policies on the line throughput (and thus on the objective
function), taking at the same time the best sequencing, balancing and buffers allocation decisions.
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... [BUKC98, p. 2669] Resulting drawbacks concerning the system performance comprise increasing process time spreads, increasing set-up and reconfiguration times. [TIAC15,p. 202] Considering these performance issues that arise from today's market conditions, several new organizational approaches are developed to improve the overall efficiency of assembly systems. Conservative approaches are dedicated or mixed-model lines with a focus on intermediate buffers. ...
... [WILS14;BOYS07] The corresponding problems are frequent line balancing procedures and the complex allocation of buffers. [TIAC15,p. 201] The adaptability of such assembly lines is limited, due to the linear transfer of mixed-model assembly lines. Due to the implementation of buffers, certain deviations from planned process times can be absorbed, but critical deviations such as breakdowns will yet affect the entire assembly line. ...
Thesis
Full-text available
Increasing product individuality, globalization of markets, shorter product life cycles, and disrupted supply chains are global trends that demand adaptable industrial assembly systems. Dynamically interconnected assembly systems (DIAS) fulfill this demand by en- abling individual job routes and by detaching from cycle time and linear transfer. A central component of DIAS is the control system that requires new online scheduling algorithms for efficient operation. In the computer science domain, the artificial intelligence (AI) al- gorithm AlphaZero showed groundbreaking results in playing strategy board games. Since online scheduling in DIAS is comparable to decision-making in board games, AlphaZero is a transferable AI solution that could significantly increase scheduling performance. This dissertation presents an AlphaZero online scheduling agent to investigate the perfor- mance potential. The agent uses AlphaZero’s Monte-Carlo tree search and deep artificial neural networks trained by reinforcement learning. Various auxiliary software components and models were created for enabling AlphaZero online scheduling. An automated scenario analysis workflow, that incorporates a simulation environment for modeling DIAS, was created for training the agent. For evaluating DIAS states during training, an additional simulation software module was specifically developed. The training included hyperparameter optimization and was conducted in multiple cycles with large data sets based on two industrial use cases. It resulted in significant improvements of the Monte-Carlo tree search and the neural network. A modular online scheduling architecture was elaborated for the adaptive communication of the agent with simulation models and control systems via standardized interfaces and data models. This architecture facilitates the seamless deployment and updating of the AlphaZero agent into an operating production system. These newly created auxiliary components and models were successfully verified and validated with established techniques. The validation of the AlphaZero agent was performed with sensitivity analyses showing plausible algorithm behavior. Comparing the AlphaZero agent with heuristic rule-based reference scheduling agents and a mathematical mixed- integer linear programming model revealed heterogeneous performance improvements, depending on the DIAS scenario characteristics. On average, the AlphaZero agent could improve the scheduling performance with AI methods in DIAS.
... Another take on this problem is parallel assembly lines [14,15], which increase the reliability and flexibility of the lines, allow better balancing due to superior cycle times and lower number of operators and, therefore, increased productivity at the expense of larger equipment investments and space required. Combining both approaches-WWALs and parallel assembly lines-can provide important benefits in contexts of high-mix lowvolume demand. ...
Article
Full-text available
Demand trends towards mass customization drive the need for increasingly productive and flexible assembly operations. Walking-worker assembly lines can present advantages over fixed-worker systems. This article presents a multiproduct parallel walking-worker assembly line with shared automated stations, and evaluates its operational performance compared to semiautomated and manual fixed-worker lines. Simulation models were used to set up increasingly challenging scenarios based on an industrial case study. The results revealed that semiautomated parallel walking-worker lines could achieve greater productivity (+30%) than fixed-worker lines under high-mix low-volume demand conditions.
... Widely used analytical methods in solving the BAP are decomposition method (Gershwin and Schor, 2000, Helber, 2001, Shi and Men, 2003, Nahas et al., 2006, Shi and Gershwin, 2009, Massim et al., 2010, Demir et al., 2011, Papadopoulos et al., 2013, Nahas et al., 2014, Su et al., 2017, Liberopoulos, 2020, Lopes et al., 2020, aggregation method (Dolgui et al., 2002, Han and Park, 2002, Diamantidis and Papadopoulos, 2004, Enginarlar et al., 2005, Dolgui et al., 2007, Wang et al. 2016, and generalized expansion method (Cruz et al. 2008, Smith and Cruz 2005, Narasimhamu et al., 2015, Xi et al., 2020. In the case of the production system cannot be modelled by an analytical model, simulation (Sabuncuoglu et al. 2006, Costa et al. 2015, Kose and Kilincci, 2015, Tiacci 2015, Zandieh et al. 2017, Motlagh et al. 2019, Renna 2019, Kose and Kilincci 2020, Lin et al. 2021, Shaaban and Romero-Silva 2021, Zhang et al. 2022) is very useful tool to model the complex and real-life problems. For more detailed explanation on evaluation algorithms and other related references, the interested reader can refer to a recent survey study of Weiss et al. (2019). ...
Article
Designing production lines is an important research issue for both academy as well as industry since it should consider both system efficiency and production costs. In this study, the buffer allocation problem (BAP) is solved to maximize the profit of unreliable lines. The profit of a production line is a function of the throughput and various costs, such as WIP, holding and buffer space costs. In this respect, a crucial decision-making problem, i.e., BAP, is how to allocate the finite buffers to deal with the trade-off between maximizing the throughput of the line and production costs. In this study, profit maximization problem is formulated considering two different objective functions and solved under different constraint sets. An adaptive hybrid variable neighborhood search algorithm that incorporates large neighborhood search as a part of the diversification strategy is proposed to solve the problem. Moreover, a new initialization procedure based on the buffer location providing more profit is proposed to reduce the search effort. The efficiency of the proposed algorithm is tested on existing benchmark problems as well as the newly introduced large-sized data sets. In addition to these experiments, a comprehensive experimental design is conducted to determine the influencing factors on the problem at hand. The experimental study reveals that the proposed solution algorithm is capable to solve the profit maximization problem for large production lines, and the number of machines and the reliability parameters are the most influential factors in solving the BAP for profit maximization. Moreover, it has been observed that the proposed initialization procedure significantly reduces the search effort.
... As for solution methods applied to MMAL balancing and design problems, researchers mostly rely on mathematical programming models (Lopes et al., 2018;Battaïa et al., 2015;Biele and Mönch, 2018;Kucukkoc and Zhang, 2014a;Giard and Jeunet, 2010), various exact methods (Bukchin and Rabinowitch, 2006;Li and Gao, 2014;Alghazi and Kurz, 2018;Delorme et al., 2019;Choi, 2009), and (meta-)heuristics (Tiacci, 2015;Dolgui et al., 2018;Özcan et al., 2010;Samouei and Fattahi, 2018;AkpıNar et al., 2013;Kucukkoc et al., 2018;Saif et al., 2019;Samouei and Fattahi, 2018). The Markov Decision Process (MDP) is a mod-6 elling framework commonly used in reinforcement learning which is one of the main machine learning paradigms (Bengio et al., 2020). ...