Content uploaded by Lorenzo Tiacci
Author content
All content in this area was uploaded by Lorenzo Tiacci on Oct 25, 2017
Content may be subject to copyright.
Coupling a genetic algorithm approach and a discrete event simulator to
design mixed-model un-paced 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
In the paper, an innovative approach to deal with the Mixed Model Assembly Line Balancing Problem (MALBP)
with stochastic task times and parallel workstations is presented. At the current stage of research, advances in solving
realistic and complex assembly line balancing problem, as the one analyzed, are often limited by the poor capability to
effectively evaluate the line throughput. Although algorithms are potentially able to consider many features of realistic
problems and to effectively explore the solution space, a lack of precision in their objective function evaluation (which
usually includes a performance parameter, as the throughput) limits in fact their capability to find good solutions.
Traditionally, algorithms use indirect measures of throughput (such as workload smoothness), that are easy to calculate,
but whose correlation with the throughput is often poor, especially when the complexity of the problem increases.
Algorithms are thus substantially driven towards wrong objectives. The aim of this paper is to show how a decisive step
forward can be done in this filed by coupling the most recent advances of simulation techniques with a genetic
algorithm approach. A parametric simulator, developed under the event/object oriented paradigm, has been embedded
in a genetic algorithm for the evaluation of the objective function, which contains the simulated throughput. The results
of an ample simulation study, in which the proposed approach has been compared with other two traditional approaches
from the literature, demonstrate that significant improvements are obtainable.
Keywords: mixed-model assembly line; balancing; un-paced lines; asynchronous lines; stochastic task times; paralleling; discrete
event simulation.
1 Introduction
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. Costs are
related to the amount of resources (labor and equipment) needed to complete all the tasks. The assembly line
balancing problem is fundamentally a trade-off problem between these two factors. There are many ways to
formulate Assembly Line Balancing Problems (ALBPs), as it will be described in the next paragraph.
However, both these aspects have to be taken into account in some way, or in the objective function or in the
problem constraints.
One of the main issues while solving ALBPs is that is difficult to evaluate the throughput of a complex
assembly line. While costs of a determined line configuration can be easily calculated from the amount of
* Corresponding Author: Tel.: +39-075-5853741; fax +39-075-5853736.
E-mail address: lorenzo.tiacci@unipg.it (L.Tiacci).
resources employed, it is often not easy to calculate its throughput with an acceptable degree of precision.
This difficulty is correlated to the features considered in the ALBP. It is low for simple lines, with a limited
number of tasks, deterministic completion times, and where only one product can be produced. However,
when more realistic features are considered it becomes much higher. So for example if a large number of
tasks have to be considered, with stochastic times of completion, and/or when multiple products have to be
assembled in a mixed-model way, with a specified sequence, or with the possibility to utilize parallel
workstations in each workcentre, then it becomes very challenging to predict the throughput of a determined
line configuration. In this context, it becomes consequently very difficult to compare different design
alternatives on the basis of their performances.
As Bukchin (1998) outlined in his study, the only way to accurately evaluate the throughput of complex
assembly lines would be to perform a simulation study. However, this is unfortunately very time consuming,
because it requires to build a simulation model each time a different design alternative has to be evaluated.
This does not fits with the most part of solving methods presented in literature, which provide the evaluation
of a large number of possible solutions to find the final one.
For this reason until now, researchers have utilized indirect ‘measures’ of the throughput, which can be
easily calculated from a determined line configuration, without the need to perform a simulation run. These
measures are based upon the assumption that the way workloads are allocated to workstations (in terms of
variability inside each workstation and among different workstations) can have a direct influence on the line
throughput. Unfortunately, the correlation between the effective throughput and these measures are often
poor. Furthermore, the more the complex is the line, the more is expected this correlation to be low.
So at this stage of research, advances in solving more realistic and complex assembly line balancing problem
are limited by the poor capability to effective evaluate the line throughput. Although algorithms are
potentially able to consider many features of realistic problems and to effectively explore the solution space,
a lack of precision (or a sort of bias) in their objective function evaluation limits in fact their capability to
find good solutions. In summary, algorithms are driven towards wrong objectives.
The aim of this paper is to show how a decisive step forward can be done in this filed by coupling the most
recent advances of simulation techniques with a genetic algorithm approach. In particular, the adoption of
event and object oriented simulation approaches has recently allowed to build parametric simulation models
that can be embedded in genetic algorithms procedures for the effective evaluation of their objective function
(Tiacci, 2012).
In the paper one of the most complex problem in assembly lines is considered: mixed-model lines, with
stochastic task times of completion and parallel workstations allowed. There are few algorithms presented in
literature capable to deal with all these features at the same time. In this work we introduce a new genetic
algorithm approach in which a parametric simulator is embedded, and compare it with other two methods,
namely a Simulation Annealing (SA) approach and a Genetic Algorithm (GA) approach, that on the contrary
utilize indirect measures of throughput in their objective functions. The results of an ample simulation study,
reported in Section 6, demonstrate the radical improvements obtainable by the proposed approach.
The paper is organized as follows. In the next section, a literary review on assembly line balancing problems
and the related operational objectives is carried out. In Section 3 the particular problem taken into
consideration is described in detail. The GA 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, which is also compared with other two approaches from the
literature. In Section 6 results are reported and discussed.
2 Literary Review
In this paper we focus our attention to a very complex assembly line balancing problem (described in detail
in Section 3), which provides the following features: mixed-model un-paced line, with parallel workstations,
stochastic task times of completion; the objective takes into consideration equipment and labor cost. The
literary review is organized in two sections. Section 2.1 is related to assembly line balancing problems that
fit with our case. Section 2.2 is related to how the objectives of this type of problem have been approached in
literature.
2.1 Assembly line balancing problems
The Assembly Line Balancing Problem (ALBP) consists in assigning tasks to workstations, while optimizing
one or more objectives without violating any restriction imposed on the line (e.g. precedence constraints
among tasks). The basic version is the so-called Simple Assembly Line Balancing Problem (SALBP) and
provides a single-model, paced line with fixed cycle time and deterministic task times. For an overview of
exact methods and heuristics developed to solve the SALBP see Scholl and Becker (2006). One of the most
restricting assumptions of SALBP is related to the production of a single model. In fact, pressure for
manufacturing flexibility, due to the growing demand for customized products, has led to a gradual
replacement of simple assembly lines with mixed-model assembly lines, in which a set of similar models of a
product can be assembled simultaneously.
The related problem that arises is the so called MALBP (Mixed-model Assembly Line Balancing Problem),
that is much more complicated because of the variability of assembly times of different models assigned to a
specific workstation. In mixed-model production, set-up times between models could be reduced sufficiently
enough to be ignored and usually all models are variations of the same base product and only differ in
specific customizable product attributes (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).
The MALBP can be seen as a particular case of the more general GALBP (Generalized Assembly Line
Balancing Problem), which embrace many features related to more realistic problems, such as cost functions,
equipment selection, paralleling, stochastic task times. For a comprehensive classification of the possible
features of the GALBP see Becker and Scholl (2006) and Boysen et al. (2007).
Stochastic task times in particular are a difficult issue to deal with. Tasan and Tunali (2008), in their survey
on genetic algorithms applied to ALBP, outlined that only one article (Suresh and Sahu, 1994), out of the 29
analysed, dealt with stochastic processing times. Non deterministic task times in fact complicate the
prediction of line performances, because blocking and starvation phenomena are accentuated. Furthermore,
blocking and starvation are also induced by the variability of task times of different models assigned to a
specific workstation. Thus, the combination among mixed-model lines and stochastic task times is
particularly challenging.
Another feature characterizing many real assembly lines configuration is the possibility to implement some
type of parallelism, for example utilizing parallel workstations performing the same task set. The aim of
using parallel stations is often to perform tasks with processing time larger than the desired cycle time.
However, also if any given task time does not exceed cycle time, the possibility to replicate workstations
may be desirable, because it enlarges the space of feasible solutions of the balancing problem, including
many feasible and potentially better balanced configurations (Tiacci et al., 2006; Vilarinho and Simaria,
2002).
There are not many algorithm in literature that takes into consideration the same amount of modeling options
as our case at the same time, such as, in particular, stochastic task times, multiple products produced in a
mixed-model way, parallel workstations (without limits regarding the maximum number of replicas per work
centre), and a fitness function that considers cycle time performance as well as total labour and equipment
costs. In fact, the one considered is a realistic industrial setting, which differs significantly from the
simplified theoretical problem. Such particularities are very difficult to include in mathematical models and
generally increase significantly the complexity of the problem to be solved. This can explain why they are
generally not considered together in the literature (Lapierre et al., 2006). Of the above mentioned studies, the
only ones that address all these issues at the same time are the works from McMullen and Frazier (1998),
McMullen and Tarasewich (2003), McMullen and Tarasewich (2006) and Tiacci et al. (2006).
2.2 Operational objectives and their measures
As already mentioned in the introduction, the two basic objectives that have to be taken into consideration in
the assembly line balancing problems are performances and costs. In the existing assembly line literature, the
performance and the cost objective are usually treated by fixing one of the two as a constraint, and trying to
optimize the other one. Thus in type-1 problem one tries to minimize the number of stations for a given cycle
time, while in the so called type-2 problem one tries to minimize the cycle time for a given number of
workstations. The most general problem version is addressed as the type-E problem, which tries to maximize
the line efficiency by simultaneously minimizing the cycle time and the number of workstations. This can be
achieved for example through a unique objective function by applying objective-specific weighting factors.
These types of problem versions, which have been originally addressed to the SALBP, can be extended to
the MALBP, where mixed models production is considered. However, in this case, the sequence of models
that enter the line, together with the variability of task times of different models assigned to a specific
workstation, can cause work overload in many stations. To avoid these difficulties, different objectives for
assembly line balancing for workload smoothing have been introduced in the literature. For an overview and
a comparison among objectives to smoothen workload see Emde et al. (2010).
Methods that are developed to seek station assignments that lead to more balanced workloads across stations
and across products are motivated to limit the effect, on the realized cycle time, of the sequencing of
different models on the assembly line. 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. However using workload smoothing as objective, as
outlined by Karabati and Sayin (2003), remain to be an approximate approach, for the very reason that the
effects of the sequencing decision on the line throughput are not incorporated explicitly.
Most of the studies in literature that deal with mixed model assembly lines use some of these ‘indirect
measures’ of throughput. However, their correlation with the throughput should be verified. The only two
studies that deal with this issue in literature are those by Bukchin (1998) and by Venkatesh and Dabade
(2008). From their works, it is possible to conclude that is difficult to find a measure of general validity for
all the possible issues of the problem. Furthermore, a loss of correlation of the same measures with the
simulated throughput is expected as the number of features that try to capture the complexity of real cases
increases (e.g. mixed models, sequencing policies, parallel workstations, stochastic task times, etc.) (Tiacci,
2012).
How anticipated in the introduction, in this paper we present an innovative approach that, thanks to the last
advances of the research on simulation architectures and techniques, allow to overcome 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.
3 The problem
The problem is to assign a certain number of tasks to a number of WC, 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)
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;
Ek the number of pieces of equipment in work centre k (Ek = Wk . Nk);
CW annual cost of a worker;
CE annual cost of a piece of equipment;
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.
In order to take into account another important feature of real assembly lines, stochastic task times have to be
considered. As many authors in literature, task times are considered normally distributed, 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. A 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 may consist of two or more workstations. In this case, the tasks assigned
to the WC are not shared among the WSs, but each WS performs all of them. Thus, an increment of
work centre
work station
operator
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 is associated a cost CE. So if one station is parallelized there is an extra cost due to equipment
duplication.
The line is asynchronous, that is as well as blockage and starvation are possible. There are no buffers
between WCs. 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. will be
able to process another workpiece) only when the workpiece leaves the WS. The first WC is never starved
(there is 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 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 the total annual cost for labour and
equipment costs of the line configuration:
1
0
P
kk
k
DC CE E CW W
(1)
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 correspond 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 it the evaluation of the line
configuration has to be 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. Note that for z
equal to infinity the constraint related to the imposed cycle time becomes strict.
4 Coupling GA and event-oriented simulation
In this section, a genetic algorithm approach to solve the defined problem is described. The GA is coupled
with an object event/object oriented simulator that is used to calculate the individuals’ fitness (Section 4.5).
4.1 Representation and decoding
Each individual is represented through a couple of chromosomes. They both are vectors having a number of
elements equal to N. The first chromosome contains an ordered sequence of all tasks. Its order corresponds to
the same order they are assigned to WCs. The second chromosome 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 (see Fig. 2).
Figure 2. Chromosomal representation.
Once all tasks have been assigned to WCs, it remains to calculate the number of workers (i.e. WSs) in each
WC. This is obtained by imposing a certain probability of completion, pc, of all tasks assigned to each WC.
By summing the average task times and variances of tasks assigned to a WC, and using the inverse normal
distribution function, it is possible to calculate the time needed to complete all tasks with a probability pc.
Indicating with Tlimit this time, the number of workers Wi assigned to the WC will be equal to the minimum
integer value so that ct .Wi ≥ Tlimit .
4.2 Initial population
The initial population is created randomly, respecting the precedence constraints. The first chromosome is
generated following the 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.
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%.
4.3 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 both chromosomes.
1 4 2 7 5 3 6 8 9 10
1 1 1 2 2 3 3 4 4 4
CH1
CH2
sequence with which tasks will be given
N° of work centre the task is given to
tasks:
1,4,2
WC1
Tasks
positions
within WCs
tasks:
7,5
WC2
tasks:
3,6
WC3
tasks:
8,9,10
WC4
A single point crossover operator is applied for the first chromosome (Fig. 3a). A cut point 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 its remaining elements 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. 3b). The chosen cut point 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 the first chromosome 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 preceding gene (position), or a value increased by 1.
The second offspring s2 is generated inverting p1 and p2 roles.
a. b.
Figure 3. Crossover operator.
4.4 Mutation
The mutation operator is also applied to both chromosomes. For the first chromosome the feasible insertion
method is adopted (Kim et al., 1996). A task is randomly selected (eg. task 5 in Fig. 4a) and its position is
swapped with another feasible position in the chromosome. Because in a feasible sequence the selected task
must follow all of its immediate predecessors and precede all of its 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, 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 (Fig. 4b). 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.
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
2 3 4 4 4
3 4 5 5 51 1 2 2 2
1 1 2 2 3 3 3 3 4 4
1 1 1 2 2
1 1 2 2 3
d
= 3 –2 = +1
random cut point
p2
p1
s1
CH2
CH2
CH2
+1 +1 +1 +1 +1
a. b.
Figure 4. Mutation operator.
4.5 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, and the tasks (and thus the
pieces of equipment) assigned to each WC 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 (number of WCs, of parallel workstations in each WC, of tasks assigned to each WC), 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 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
some line performances indicator, such as the cycle time in our case) has to be performed several times. ALS
allows to overcome the limit of 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 (as f.e. the approaches that will be described in Sections 5.1.1 and 5.1.2).
To embed ALS in the GA, 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. 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.
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
1 4 2 7 5 3 6 8 9 10
CH1
possible insertion points
1 4 7
10
2 5
8 9
3 6
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
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).
4.6 GA iterations and resulting general structure
The GA structure is depicted in Fig. 5. After creating the initial population of N 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.
In order to maintain high population diversity, an elimination strategy has been implemented during the
selection phase. Let define the ‘distance’ between two individuals as the absolute difference between their
fitness values (Section 4.3). This phase provides a spacing procedure among individuals so that the distance
between any two individuals in the resulting population is always higher than a threshold value ∆f. At this
purpose, the population is sorted. Firstly, the individual with the best fitness is considered as reference, and
all the individuals whose distance from that one is lower than ∆f are eliminated from the population. The
closer individual that satisfies the threshold is then taken as reference, and the procedure continue iteratively
until all the population have been spaced.
After terminating the spacing procedure, only the first N individuals of the resulting population 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 5. GA iterations and general structure.
In our implementation we adopted a strategy to dynamically set the threshold value ∆f at each iteration. For
the first iteration, ∆f is set equal to a percentage d1 of the fitness of the best individual. Than it decreases
linearly in such a way that equals zero after a certain percentage d2 of the total number of iterations G, and
remains zero for the residual iterations:
∆f (iter)=[bestfit .d1. (1 - iter/(d2.G))]+ (3)
where bestfit is the fitness value of the best individual in the current population, and iter is the current
iteration number. In this way, high population diversity is initially guaranteed, in order not to be too early
trapped in local optima. On the contrary, a local search behavior is induced in the last iterations by not
discarding clones and very similar individuals, and refining the best solution by searching among the closest
neighbors.
Initial
population
Selection
(fitness evaluation and
clones elimination)
Reproduction
(crossover) mutation new generation
iterations
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 3). 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 between tasks are defined for a single model. The task times tij
for both the 2 and the 4 model instances that have been created for our study are available at (Tiacci, 2014).
A coefficient of variation cv = 0.3 has been utilized to obtain task times standard deviations. As it was
mentioned in Section 4.5, 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 cost of a worker (CW) and of a piece of equipment (CE) have been considered
respectively equal to 30000 and 3000 €/y. The values of the genetic algorithm parameters utilized in the
study are N = 40; mr = 0.55; G = 100, pc = 0.5, d1 = 0.03%, d2 = 70%.
The algorithm has been compared, on the same problems instances, with other two approaches proposed in
literature, namely a Simulated Annealing approach (McMullen and Frazier, 1998) and a Genetic Algorithm
approach (Tiacci et al., 2006).
The final solutions of each approach have been evaluated on the basis of their Normalized Design Cost,
considering a penalty coefficient z = 20 (see eq. (2)). To obtain the NDC value the final solutions has been
simulated 10 times using ALS, and the average value of the cycle time has been used, together with the
number of workers and pieces of equipment, 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). This is to reflect the fact that it is practically impossible to have the exact information about
the models sequence at the time of the assembly line design decision, but that an exact sequence is
determined when production starts. These sequences have been chosen so that the products are not being
introduced into the production system at a steady rate, as would be desirable in many JIT systems. Using
these specific sequences for the evaluation of the final solutions, instead of the random sequences used
during the searching procedure, allows drawing more meaningful conclusions about the robustness of the
solutions found by the algorithm.
Because of the stochastic nature of the algorithms, 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 10 times, and results here reported are related to the best performance chosen among these five
solutions.
The difference that we want to outline between the approach presented here in and the two studies from the
literature is related to the objective function evaluation. In these studies, indirect measures of the cycle time
have been utilized to compare different solutions during the search procedures. This forced to create
objective functions that do not contain explicitly the cycle time value, and thus cannot be equal to the real
objective of the problem as defined by eq. (2). It is noteworthy that, on the contrary, by embedding ALS in
the genetic algorithm proposed here in, it is possible to calculate during the procedure any desired objective
function, as NDC, that explicitly contains the effective cycle time value.
In the next two subsections the objective function utilized in the two studies are reported.
5.1.1 Simulated annealing approach
We have implemented all the six different versions of the Simulated Annealing algorithm proposed by
McMullen and Frazier (1998), which differ from one another in the objective function that are minimized
(see Table 1).
type
name
description
single objective
minimize
DC
design cost
S
smoothness index
L
probability of lateness across all work centers
multi objective
minimize
DC&L
= DC + 1.67 * L
DC&3L
= DC + 3 * 1.67 . L
3D&L
= 3 * DC + 1.67 * L
Table 1. Objective functions of the Simulated Annealing algorithm
The first three are single-objective functions, whereas the second three are multi-objective and weigh
differently (1:1, 1:3, 3:1) the cost parameter (DC) and the performance parameters (L and S). For more
details about S (which is an index intended to distribute work into the workstations as evenly as possible), L
(representing the probability of lateness across all WCs) and the normalizing weight equal to 1.67, the latter
which appears in the three multi-objective functions, refer to the McMullen's publication. The number of
iterations for each temperature has been set equal to the number of tasks of the problem. All the other values
of the setting parameters of Simulated Annealing that are not quoted by us (such as for instance annealing
temperature, cooling rate, etc.) are the same assumed by McMullen and Frazier (1998).
5.1.2 Genetic Algorithm with traditional fitness function
In the study presented by Tiacci et al. (2006) the declared objective function of the problem was the NDC.
However, due to the impossibility to correctly estimate the cycle time of a solution when calculating its
fitness, a different function was used to evaluate each individual during the procedure:
fit = k . LEp + MVp (4)
This fitness function, that has to be maximized, takes into account two indexes, which are concerned with the
two different aspects of the solution: the line efficiency (LE) and the model variability (MV). LE, introduced
by Driscoll and Thilakawardana (2001), represents, with a range from 0 to 100%, the attainment of a good
utilization of the line, being a direct measure of the economic convenience of a solution. MV has been
introduced by Bukchin (1998), and it is a measure of the variability of assembly times of a certain model
assigned to different WCs. Bukchin (1998) showed that the model variability is one of the indexes with the
highest (inverse) correlation to simulated throughput in a mixed model assembly line. In their original
formulation, both LE and MV did not take into account paralleling. LEp and MVp in eq. (4) are new
formulations, proposed by Tiacci et al. (2006), that take into account paralleling .
Because MVp is an a-dimensional parameter that may not necessarily have a range between 0 and 100%, as
LEp does have, they also introduced the coefficient k, which allows to weigh the two indexes in the most
coherent way. After some tuning operations on the algorithm, they assigned to k the value of 4.
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 2. 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, 2014)
6 Results
Table 3 shows the results obtained with the GA coupled with ALS for the 40 instances of the experiment 1.
The CPU time needed to solve each instance, with respect to algorithm that use non-simulation-based
measures of throughput, is higher. This is due substantially to the extra-time required to simulate the line
configuration each time that the fitness function of an individual is evaluated. However, the total CPU times
required to solve an instance ranges from about 100 to 1600 sec., depending on the instance complexity. This
amount of time is acceptable considering the long-term nature of the problem and the expensiveness of the
decisions that have to be taken (design of an assembly line).
It is possible to note as the effective cycle time obtained is quite near to the imposed one, and always slightly
higher (3.89% on the average). This means that the algorithm takes advantage of the low penalty applied to
the DC for very little ct overshooting, to find very efficient solutions.
Table 4 shows the solution found by the algorithm for the 35 tasks, 2 models problem by Gunther with an
imposed cycle time equal to 30. The line configuration provides 12 WCs, in three of which paralleling is
utilized. All the solutions related to each of the 40 problems are available at (Tiacci, 2014).
Tables 5, 6 and 7 show the comparison between the approach presented here-in (that is indicated as
GA+ALS), the Simulated Annealing (SA) approach in all its 6 variants related to the objective functions
described in section 5.1.1, and the Genetic Algorithm with traditional fitness function described in section
5.1.2 (indicated as GA+Trad.).
The superiority of the GA+ALS approach is evident from Table 5, in which the NDC is reported. The last
column of the table (%diff) shows the percentage difference between the NDC obtained by the GA+ALS
approach and the best result (highlighted in bold) obtained by the other two approaches (SA and GA+Trad.)
for the corresponding problem. The GA+ALS approach obtains always the lower values, with improvements
that reach almost the 20% in some instances.
Table 6 is related to the effective cycle times (cteff) of the solutions. Defining the Absolute Percentage
Deviation (APD) between the effective cycle time and the imposed cycle time, APD =|(cteff – ct)/ct|, results
show that the GA+ALS approach obtains, on the average, the lower values than all the other approaches.
Furthermore, the cteff values obtained by GA+ALS are close to ct for almost all the 40 considered issues, as
demonstrated by the maximum value of the APD (3.44%). On the contrary, the other approaches provide in
at least some cases higher deviations values, as the higher maximum value of their APD demonstrates. It is
noteworthy that in some cases SA provides effective cycle times that are far below the planned ct, but the
corresponding solutions are very costly (Table 7).
Regarding the cost parameter DC (Table 7), the better results are almost always obtained from GA+trad and
SA+3DC&L. However, the effective cycle time achieved in the corresponding instances by these two
algorithms largely exceeds the planned one, and this penalizes the evaluation of the solution in terms of
NDC.
The superiority of the GA+ALS approach with respect to the GA+Trad is remarkable. As described in
section 5.1.2, the GA+Trad has been developed declaring the NDC as objective to minimize. Thus, the
fitness function tuning (i.e. the determination of the weighting factor k in eq. (4) through a trial and error
procedure) has been performed with this scope. Despite this, GA+ALS outperforms the traditional approach
in all the instances. This fact outlines one of the main drawbacks of the traditional approaches: using indirect
measures of cycle time forces the adoption of fitness function that does not coincide with the real objective;
and this in turns requires preliminary trial and error tuning procedures to ‘align’ the fitness function as much
as possible to the real objective. However, of course, this alignment will never bring to better result than
using a fitness function that corresponds to the real objective.
Tables 8-10 show the results related to experiment 2. The values here refer, as in the preceding experiment,
to the best solutions obtained by each algorithm. Figures 6-7 shows the results related to the full factorial
design of experiment 2, considering all the repetitions 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”). We reported the main effect plots of each factors on NDC, APD and DC (Figure 6), and the
interaction effect plot between the alg and ct (Figure 7). All the other interaction effects among factors
resulted non-significant.
Analyzing the results some considerations can be drawn. Even in this case, the lower values of the NDC are
obtained in all the instances by GA+ALS, which also outperforms all the other algorithms in terms of
average APD. Only SA+L obtains similar performances in terms of APD, but through much more costly line
configurations.
The tasks variability does not seem to have an evident influence on the quality of the solution. This is valid
both in terms of the main performance indicator, the NDC, and in terms of its components, i.e. the DC and
the respect of the imposed cycle time (measured through the APD).
On the contrary, the factor that seems to have the higher impact on performances is the imposed cycle time.
Lower values of the imposed ct lead to worst performances in terms of NDC, APD and DC. A possible
explanation for this behavior is that, when cycle time is lower, the combinations of tasks that can be
allocated into a WorkCentre in a feasible way decrease. The consequent reduction of the solution space may
be overcome using parallel workstations. Paralleling on the other hand introduces extra-cost due to
equipment duplications, and increases the difficulty to evaluate the effective cycle time of the line through
non-simulation based metrics.
If this is true, due to the simulation-based nature of the objective function evaluation, GA+ALS is expected
to perform better than the other algorithms in terms performance decrement. This is exactly what can be
observed from the interactions plot between ct and the alg for NDC (Figure 7). It is clear from the graphic
that the performance decrement, observed when ct decreases, is much less evident when GA+ALS is
utilized, with respect to the all the other algorithms.
Problem
size
M
ct
cteff
DC
(x1000)
NDC
(x1000)
CPU time*
(sec)
Roszieg
25
2
18
18.0
345
345
102
14
14.0
441
441
115
11
11.2
540
545
113
9
9.2
654
658
126
Gunther
35
2
54
55.6
447
455
140
41
41.8
609
614
141
36
37.1
654
666
159
30
31.1
759
781
166
Wee-mag
75
2
34
34.8
2109
2134
456
28
28.8
2478
2517
398
24
24.6
3015
3053
381
21
21.2
3360
3366
395
Barthold
148
2
470
482.8
894
907
537
403
406.3
996
997
544
378
376.0
1050
1050
558
351
358.8
1092
1103
568
Scholl
297
2
1515
1549.3
2871
2900
1513
1394
1423.6
3084
3112
1542
1301
1323.2
3198
3217
1539
1216
1236.2
3429
3448
1605
Roszieg
25
4
18
18.1
345
345
104
14
14.1
441
442
116
11
11.3
543
551
112
9
9.2
657
667
126
Gunther
35
4
54
55.6
453
461
140
41
41.4
609
610
160
36
36.8
669
675
160
30
30.9
771
784
159
Wee-mag
75
4
34
34.5
2118
2128
492
28
28.6
2532
2554
400
24
24.9
2973
3048
409
21
21.2
3345
3353
385
Barthold
148
4
470
469.5
924
924
542
403
416.3
984
1005
541
378
383.1
1068
1072
530
351
358.9
1092
1103
540
Scholl
297
4
1515
1560.0
2901
2952
1501
1394
1429.6
3066
3106
1520
1301
1328.2
3231
3259
1536
1216
1254.5
3423
3492
1567
*The experiment has been performed on a computer with an Intel Core Processor i5-2400 (3.1 GHz), with Java 7 Update 25,
running under Windows 7 Professional Edition.
Table 3. Experiment 1: results for the 40 instances of the GA approach coupled with ALS (GA+ALS)
Table 4. GA+ALS solution for Gunther 35, 2 models, ct = 36.
WC#
WS assigned (Wi)
tasks assigned
0
1
16, 0
1
2
9, 11
2
1
1, 4, 5, 6
3
1
7, 13, 8
4
1
17, 14, 2
5
1
12
6
1
3
7
2
18, 19, 10
8
1
15
9
1
20, 21, 24, 25
10
1
22, 29, 30
11
1
31, 23, 26, 33
12
3
32, 27, 34, 28
cteff = 37.09; DC = 654000; NDC = 665928
Problem
size
M
ct
GA+
ALS
Simulated Annealing
GA+
Trad.
%diff
DC
S
L
DC&L
DC&3L
3DC&L
Roszieg
25
2
18
345
397
486
404
379
377
347
364
-0.7%
14
441
478
558
512
516
518
536
583
-8.3%
11
545
571
717
665
617
615
581
667
-4.7%
9
658
696
834
757
734
731
813
857
-5.8%
Gunther
35
2
54
455
499
648
541
492
491
487
519
-7.0%
41
614
675
756
740
750
706
690
842
-10.0%
36
666
848
849
771
775
774
768
979
-15.3%
30
781
886
993
906
1058
1054
1056
1086
-13.4%
Wee-mag
75
2
34
2134
2686
2460
2403
2791
2783
2918
2863
-12.6%
28
2517
3768
2793
2896
3395
3386
3268
4088
-11.0%
24
3053
4126
3375
3379
4068
3976
4611
5362
-10.6%
21
3366
3711
3756
3646
3549
3548
4005
3970
-5.4%
Barthold
148
2
470
907
1194
1497
1305
1134
1194
1224
1030
-13.5%
403
997
1284
1809
1404
1284
1240
1344
1168
-17.1%
378
1050
1252
1539
1458
1320
1146
1233
1302
-9.1%
351
1103
1203
1854
1522
1271
1394
1335
1196
-8.5%
Scholl
297
2
1515
2900
3321
4437
5291
3268
3329
3214
3367
-10.8%
1394
3112
3461
5010
5498
3480
3442
3681
3609
-10.6%
1301
3217
3911
5088
6007
3754
3990
3909
3552
-10.4%
1216
3448
4068
5766
6831
4152
5075
4545
4137
-18.0%
Roszieg
25
4
18
345
388
486
405
383
384
381
370
-7.2%
14
442
488
612
513
559
560
543
586
-10.5%
11
551
572
714
666
624
628
590
686
-3.8%
9
667
698
834
758
755
755
774
924
-4.7%
Gunther
35
4
54
461
512
657
541
522
520
537
565
-11.1%
41
610
763
798
741
762
760
657
899
-7.7%
36
675
776
849
771
812
809
919
974
-14.2%
30
784
890
993
907
1110
1117
1075
1143
-13.5%
Wee-mag
75
4
34
2128
2707
2418
2400
2537
2535
2822
3155
-12.8%
28
2554
3806
2830
2901
3424
3425
3363
4752
-10.8%
24
3048
4191
3385
3389
4023
4017
4181
5440
-11.1%
21
3353
3718
3756
3643
3463
3459
4056
4045
-3.1%
Barthold
148
4
470
924
1068
1359
1173
965
985
970
973
-4.4%
403
1005
1064
1386
1382
1040
1087
1208
1107
-3.5%
378
1072
1194
1584
1434
1137
1260
1187
1124
-4.9%
351
1103
1103
1683
1509
1155
1199
1195
1194
0.0%
Scholl
297
4
1515
2952
2959
4245
5291
3117
3055
3272
3224
-0.2%
1394
3106
3447
4860
5547
3513
3614
3456
3697
-11.0%
1301
3259
3819
5250
6005
4410
4473
4013
3870
-17.2%
1216
3492
4127
5772
6836
3868
4161
4455
4333
-10.8%
Table 5. Experiment 1: Normalized Design Cost, NDC.
Problem
size
M
ct
GA+
ALS
Simulated Annealing
GA+
Trad.
DC
S
L
DC&L
DC&3L
3DC&L
Roszieg
25
2
18
18.0
20.0
14.0
18.3
19.5
19.5
18.3
19.6
14
14.0
14.9
12.9
14.2
15.4
15.5
15.6
16.1
11
11.2
11.4
9.7
11.2
12.1
12.1
11.7
12.4
9
9.2
9.5
7.9
9.5
9.7
9.7
10.1
10.3
Gunther
35
2
54
55.6
58.4
47.5
56.8
57.8
57.8
58.2
59.9
41
41.8
44.8
35.9
43.8
45.7
44.5
44.6
47.7
36
37.1
40.7
8.0
35.4
39.1
39.0
39.6
42.4
30
31.1
32.1
26.6
30.7
34.5
34.4
34.6
35.0
Wee-mag
75
2
34
34.8
38.4
30.6
35.3
38.8
38.8
39.3
39.4
28
28.8
32.8
27.4
29.2
31.9
31.9
32.0
33.9
24
24.6
27.9
25.3
24.7
27.9
27.5
28.6
29.8
21
21.2
22.6
20.9
21.7
22.2
22.2
23.1
23.2
Barthold
148
2
470
482.8
471.2
310.9
452.2
442.0
467.8
455.9
516.0
403
406.3
402.0
320.2
391.9
402.0
413.4
396.6
449.3
378
376.0
386.6
284.5
376.7
364.2
388.6
351.1
424.5
351
358.8
354.8
274.4
353.1
361.6
369.5
369.4
379.6
Scholl
297
2
1515
1549.3
1690.9
1427.4
1563.6
1691.9
1683.8
1704.1
1695.6
1394
1423.6
1535.3
1328.0
1449.1
1559.5
1535.2
1600.4
1560.1
1301
1323.2
1474.3
1224.3
1314.7
1440.6
1471.1
1491.1
1429.4
1216
1236.2
1362.5
1141.7
1311.3
1368.2
1436.9
1426.6
1375.8
Roszieg
25
4
18
18.1
19.7
14.0
18.3
19.6
19.6
19.6
19.7
14
14.1
15.1
11.9
14.2
15.7
15.8
15.6
16.1
11
11.3
11.4
9.3
11.2
12.1
12.1
11.8
12.4
9
9.2
9.5
8.0
9.5
9.8
9.8
9.9
10.5
Gunther
35
4
54
55.6
58.9
47.4
56.8
59.4
59.4
60.1
60.6
41
41.4
45.8
34.7
43.8
45.9
45.9
44.1
48.5
36
36.8
39.2
32.9
35.4
40.2
40.2
41.6
42.0
30
30.9
32.2
26.6
30.7
34.8
34.8
34.7
35.4
Wee-mag
75
4
34
34.5
38.4
31.6
35.3
37.9
37.9
38.8
40.1
28
28.6
32.9
28.2
29.3
32.0
32.0
32.1
35.0
24
24.9
28.0
25.3
24.7
27.7
27.6
28.1
29.9
21
21.2
22.6
20.9
21.7
21.9
21.9
23.3
23.3
Barthold
148
4
470
469.5
521.1
363.4
471.9
505.9
509.4
512.4
507.4
403
416.3
424.9
311.0
419.4
414.7
430.8
453.0
443.1
378
383.1
413.0
288.9
374.4
410.2
423.4
415.3
408.0
351
358.9
361.6
325.6
327.3
371.3
375.6
370.5
378.7
Scholl
297
4
1515
1560.0
1615.2
1405.0
1563.6
1647.5
1620.6
1717.3
1670.8
1394
1429.6
1531.9
1329.9
1456.8
1546.8
1566.2
1567.3
1572.9
1301
1328.2
1456.5
1228.0
1313.7
1500.8
1518.3
1501.1
1466.3
1216
1254.5
1365.0
1174.9
1311.6
1348.5
1367.0
1424.2
1393.2
Avg. APD from ct
2.02%
8.92%
13.28%
3.37%
9.84%
10.12%
11.20%
13.87%
Max. APD from ct
3.79%
17.38%
77.81%
7.86%
16.05%
18.17%
19.14%
24.91%
Table 6. Experiment 1: effective cycle time, cteff
Problem
size
M
ct
GA+
ALS
Simulated Annealing
GA+
Trad.
DC
S
L
DC&L
DC&3L
3DC&L
Roszieg
25
2
18
345
315
486
402
330
330
345
315
14
441
438
558
510
426
426
426
405
11
540
558
717
660
519
519
534
510
9
654
654
834
717
654
654
630
600
Gunther
35
2
54
447
441
648
513
447
447
435
420
41
609
576
756
678
594
615
597
546
36
654
630
849
771
678
678
639
603
30
759
804
993
897
732
732
720
693
Wee-mag
75
2
34
2109
2019
2460
2334
1986
1986
1968
1905
28
2478
2376
2793
2787
2442
2442
2331
2166
24
3015
2682
3192
3327
2685
2772
2661
2469
21
3360
3339
3756
3567
3330
3330
3324
3231
Barthold
148
2
470
894
1194
1497
1305
1134
1194
1224
864
403
996
1284
1809
1404
1284
1224
1344
924
378
1050
1239
1539
1458
1320
1128
1233
999
351
1092
1200
1854
1521
1248
1320
1266
1056
Scholl
297
2
1515
2871
2616
4437
5184
2568
2667
2451
2622
1394
3084
2871
5010
5331
2715
2856
2559
2811
1301
3198
2886
5088
5994
3051
2973
2739
2973
1216
3429
3153
5766
6084
3162
3057
2841
3075
Roszieg
25
4
18
345
330
486
402
330
330
330
315
14
441
438
612
510
426
426
435
405
11
543
558
714
660
522
522
534
510
9
657
654
834
717
657
657
651
603
Gunther
35
4
54
453
441
657
513
435
435
429
435
41
609
597
798
678
594
594
588
537
36
669
672
849
771
636
636
618
624
30
771
804
993
897
735
735
720
690
Wee-mag
75
4
34
2118
2019
2418
2334
2010
2010
2010
1917
28
2532
2373
2826
2787
2442
2442
2349
2121
24
2973
2682
3192
3327
2748
2748
2634
2481
21
3345
3339
3756
3567
3330
3330
3294
3240
Barthold
148
4
470
924
864
1359
1173
864
864
834
864
403
984
1005
1386
1338
1023
993
924
924
378
1068
1020
1584
1434
993
978
993
999
351
1092
1083
1683
1509
1083
1092
1125
1062
Scholl
297
4
1515
2901
2721
4245
5184
2703
2784
2412
2661
1394
3066
2883
4860
5331
2832
2769
2640
2781
1301
3231
2970
5250
5994
2997
2871
2724
2925
1216
3423
3174
5772
6084
3126
3180
2808
3042
Table 7. Experiment 1: Design Cost, DC.
Tasks
Variability
size
M
ct
GA+
ALS
Simulated Annealing
GA+
Trad.
%diff
DC
S
L
DC&L
DC&3L
3DC&L
low
297
200
1515
3227
3911
5570
6452
3839
3489
3684
3336
-3.28%
1394
3380
3962
6243
6851
3849
4157
3957
4008
-12.17%
1301
3559
4504
7375
7348
4405
3898
4607
4072
-8.72%
1216
3721
4993
6345
7582
4849
4508
5039
4378
-15.01%
medium
297
200
1515
3195
3584
6193
6399
3674
3588
3693
3609
-10.85%
1394
3351
4020
5902
7271
4122
4111
4188
4066
-16.64%
1301
3514
4350
6648
7636
4331
4040
4657
3993
-11.98%
1216
3727
4299
8050
7998
4382
4432
4301
4198
-11.23%
high
297
200
1515
3235
3660
5479
6606
3787
3687
3829
3411
-5.15%
1394
3429
4255
6079
7186
4071
4204
4146
4075
-15.78%
1301
3567
4432
6868
7416
4530
4226
4856
3858
-7.54%
1216
3787
4666
6816
7940
4477
4379
5133
4364
-13.22%
Table 8. Experiment 2: Normalized Design Cost, NDC.
Tasks
Variability
size
M
ct
GA+
ALS
Simulated Annealing
GA+
Trad.
DC
S
L
DC&L
DC&3L
3DC&L
low
297
200
1515
1556
1713
1401
1537
1701
1671
1702
1634
1394
1429
1564
1283
1410
1551
1582
1579
1567
1301
1335
1493
1180
1336
1474
1436
1508
1439
1216
1226
1400
1126
1250
1397
1381
1424
1369
medium
297
200
1515
1537
1678
1447
1577
1697
1683
1708
1681
1394
1426
1559
1287
1435
1569
1583
1596
1565
1301
1337
1480
1240
1325
1474
1454
1517
1443
1216
1246
1358
1170
1266
1364
1364
1380
1352
high
297
200
1515
1544
1684
1399
1550
1695
1678
1712
1635
1394
1420
1582
1276
1380
1569
1579
1588
1563
1301
1319
1467
1248
1282
1493
1464
1521
1413
1216
1276
1384
1167
1262
1360
1363
1427
1354
Avg. APD from ct
2.30%
12.87%
6.42%
2.46%
12.73%
12.08%
14.76%
10.72%
Max. APD from ct
4.94%
15.12%
9.28%
4.08%
14.86%
13.61%
17.37%
12.61%
Table 9. Experiment 2: effective cycle time, cteff
Tasks
Variability
size
M
ct
GA+
ALS
Simulated Annealing
GA+
Trad.
DC
S
L
DC&L
DC&3L
3DC&L
low
297
200
1515
3181
2911
5570
6424
2948
2881
2821
2971
1394
3338
3054
6243
6833
3066
3047
2926
3061
1301
3512
3136
7375
7245
3257
3209
3063
3326
1216
3716
3426
6345
7466
3363
3289
3177
3321
medium
297
200
1515
3181
2913
6193
6195
2851
2881
2791
2911
1394
3316
3136
5902
7147
3137
3004
2949
3121
1301
3462
3152
6648
7586
3200
3166
2997
3229
1216
3683
3379
8050
7741
3377
3420
3148
3362
high
297
200
1515
3211
2929
5479
6538
2951
2994
2863
3031
1394
3406
3121
6079
7186
3099
3107
2990
3151
1301
3554
3340
6868
7416
3154
3213
3089
3361
1216
3611
3378
6816
7723
3500
3388
3201
3473
Table 10. Experiment 2: Design Cost, DC.
Figure 6. Main effects plots for the factorial experiment 2.
Mean of NDC
M
L
H
8000000
7000000
6000000
5000000
4000000
3000000
1515
1394
1301
1216
GA+
trad
SA+3DC&L
SA+DC&3L
SA+DC&L
SA+S
SA+L
SA+DC
GA+ALS
Variability
ct
alg
Main Effects Plot (data means) for NDC
Mean of APD
M
L
H
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
1515
1394
1301
1216
GA+trad
SA+3DC&L
SA+DC&3L
SA+
DC&L
SA+S
SA+L
SA+
DC
GA+ALS
Variability
ct
alg
Main Effects Plot (data means) for APD
Mean of DC
M
L
H
8000000
7000000
6000000
5000000
4000000
3000000
1515
1394
1301
1216
GA+
trad
SA+3DC&L
SA+DC&3L
SA+DC&L
SA+S
SA+L
SA+DC
GA+ALS
Variability
ct
alg
Main Effects Plot (data means) for DC
Figure 7. Interaction plot (alg*ct) for NDC for the factorial experiment 2.
7 Conclusions
In the paper, an innovative approach to solve the Mixed Model Assembly Line Balancing Problem
(MALBP) with stochastic task times and parallel workstations is presented.
A genetic algorithm that considers all the features of this complex problem has been presented. To evaluate
the fitness function of each individual in each generation, a parametric simulator, named ALS, has been
embedded in the genetic algorithm procedure. ALS has been built undo 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, as the throughput or the cycle time) has to be
performed 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 proposed approach has been compared with other two approaches presented in literature (a Simulated
Annealing and a Genetic Algorithm approaches), that use indirect measures of the cycle time, not simulation
based, to compare different solutions during the search procedures. This forced to create objective functions
that do not contain explicitly the cycle time as variable, and thus cannot be identical to the real objective of
the problem as defined by eq. (2).
The benefits of the proposed approach, in terms of achievement of the final objective, are relevant, as
demonstrated by the results of a comparative study performed on an ample set of problems. Another
important advantage is that there is no more need to perform trial and error procedures to tune the weighting
coefficients of traditional objective functions that contains indirect measures of the cycle time, and that for
this reason have to be ‘aligned’ with the real objective.
The promising results achieved in this work suggest continuing to exploit the latest advances in discrete
event simulation techniques to solve more complex problems.
For example, due to the computational complexities involved, the assembly line balancing problem and the
sequencing problems are usually addressed in literature independently of each other, although they are
closely interrelated. A possible improvement in the GA approach presented here in could be to find a
representation that allows considering the sequence of models entering the line as a variable itself. 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 and balancing decisions.
alg
Mean
GA+
trad
SA+3DC&L
SA+DC&3L
SA+
DC&L
SA+S
SA+L
SA+
DC
GA+
ALS
9000000
8000000
7000000
6000000
5000000
4000000
3000000
1216
1301
1394
1515
ct
Interaction Plot (data means) for NDC
Similarly, exploiting the capabilities of ALS to simulate line configurations with buffers among WCs, it
could be possible to approach simultaneously the balancing and buffer allocation problems, that are usually
treated separately in literature but are in fact closely interrelated.
8 References
Akpinar, S., Bayhan, G.M., 2011. A hybrid genetic algorithm for mixed model assembly line balancing
problem with parallel workstations and zoning constraints. Eng Appl Artif Intel 24, 449-457.
Askin, R.G., Zhou, M., 1997. A parallel station heuristic for the mixed-model production line balancing
problem. International Journal of Production Research 35, 3095-3105.
Bartholdi, J.J., 1993. Balancing 2-Sided Assembly Lines - a Case-Study. International Journal of Production
Research 31, 2447-2461.
Becker, C., Scholl, A., 2006. A survey on problems and methods in generalized assembly line balancing.
European Journal of Operational Research 168, 694-715.
Boysen, N., Fliedner, M., Scholl, A., 2007. A classification of assembly line balancing problems. European
Journal of Operational Research 183, 674-693.
Boysen, N., Fliedner, M., Scholl, A., 2008. Assembly line balancing: Which model to use when?
International Journal of Production Economics 111, 509-528.
Bukchin, J., 1998. A comparative study of performance measures for throughput of a mixed model assembly
line in a JIT environment. International Journal of Production Research 36, 2669-2685.
Bukchin, J., Dar-El, E., Rubinovitz, J., 2002. Mixed model assembly line design in a make-to-order
environment. Computers & Industrial Engineering 41, 405-421.
Driscoll, J., Thilakawardana, D., 2001. The definition of assembly line balancing difficulty and evaluation of
balance solution quality. Robotics and Computer-Integrated Manufacturing 17, 81-86.
Emde, S., Boysen, N., Scholl, A., 2010. Balancing mixed-model assembly lines: a computational evaluation
of objectives to smoothen workload. International Journal of Production Research 48, 3173-3191.
Golz, J., Gujjula, R., Gunther, H.O., Rinderer, S., Ziegler, M., 2012. Part feeding at high-variant mixed-
model assembly lines. Flex Serv Manuf J 24, 119-141.
Gunther, R.E., Johnson, G.D., Peterson, R.S., 1983. Currently practiced formulations for the assembly line
balance problem. Journal of Operations Management 3, 209-221.
Haq, A.N., Rengarajan, K., Jayaprakash, J., 2006. A hybrid genetic algorithm approach to mixed-model
assembly line balancing. Int J Adv Manuf Tech 28, 337-341.
Jin, M.Z., Wu, S.D., 2003. A new heuristic method for mixed model assembly line balancing problem.
Computers & Industrial Engineering 44, 159-169.
Karabati, S., Sayin, S., 2003. Assembly line balancing in a mixed-model sequencing environment with
synchronous transfers. European Journal of Operational Research 149, 417-429.
Kim, Y., Kim, J., Kim, Y., 1996. Genetic algorithms for assembly line balancing with various objectives.
Computers & Industrial Engineering 30, 397-409.
Lapierre, S., Ruiz, A., Soriano, P., 2006. Balancing assembly lines with tabu search. European Journal of
Operational Research 168, 826-837.
McMullen, P., Frazier, G., 1998. Using simulated annealing to solve a multiobjective assembly line
balancing problem with parallel workstations. International Journal of Production Research 36, 2717-2741.
McMullen, P., Tarasewich, P., 2003. Using ant techniques to solve the assembly line balancing problem. Iie
Transactions 35, 605-617.
McMullen, P.R., Frazier, G.V., 1997. A heuristic for solving mixed-model line balancing problems with
stochastic task durations and parallel stations. International Journal of Production Economics 51, 177-190.
McMullen, P.R., Tarasewich, P., 2006. Multi-objective assembly line balancing via a modified ant colony
optimization technique. International Journal of Production Research 44, 27-42.
Merengo, C., Nava, F., Pozzetti, A., 1999. Balancing and sequencing manual mixed-model assembly lines.
International Journal of Production Research 37, 2835-2860.
Rosenberg, O., Ziegler, H., 1992. A comparison of heuristic algorithms for cost-oriented assembly line
balancing. Zeitschrift für Operations Research 36, 477-495.
Scholl, A., 1993. Data of assembly line balancing problems. Schriften zur Quantitativen
Betriebswirtschaftslehre 16, 1-28.
Scholl, A., 2012. Data sets for SALBP. http://www.assembly-line-balancing.de.
Scholl, A., Becker, C., 2006. State-of-the-art exact and heuristic solution procedures for simple assembly line
balancing. European Journal of Operational Research 168, 666-693.
Simaria, A.S., Vilarinho, P.M., 2001. The simple assembly line balancing problem with parallel workstations
- A simulated annealing approach. International Journal of Industrial Engineering-Theory Applications and
Practice 8, 230-240.
Simaria, A.S., Vilarinho, P.M., 2004. A genetic algorithm based approach to the mixed-model assembly line
balancing problem of type II. Computers & Industrial Engineering 47, 391-407.
Suresh, G., Sahu, S., 1994. Stochastic Assembly-Line Balancing Using Simulated Annealing. International
Journal of Production Research 32, 1801-1810.
Tasan, S.O., Tunali, S., 2008. A review of the current applications of genetic algorithms in assembly line
balancing. J Intell Manuf 19, 49-69.
Tiacci, L., 2012. Event and object oriented simulation to fast evaluate operational objectives of mixed model
assembly lines problems. Simulation Modelling Practice and Theory 24, 35-48.
Tiacci, L., 2014. Coupling a genetic algorithm approach and a discrete event simulator to design mixed-
model un-paced assembly lines with parallel workstations and stochastic task times - SUPPLEMENTARY
RESOURCES. www.researchgate.net/profile/Lorenzo_Tiacci.
Tiacci, L., Saetta, S., 2007. Process-oriented simulation for mixed-model assembly lines, Proceedings of the
2007 Summer Computer Simulation Conference. Society for Computer Simulation International, San Diego,
USA, pp. 1250-1257.
Tiacci, L., Saetta, S., Martini, A., 2006. Balancing mixed-model assembly lines with parallel workstations
through a genetic algorithm approach. International Journal of Industrial Engineering-Theory Applications
and Practice 13, 402-411.
Venkatesh, J.V.L., Dabade, B.M., 2008. Evaluation of performance measures for representing operational
objectives of a mixed model assembly line balancing problem. International Journal of Production Research
46, 6367-6388.
Vilarinho, P., Simaria, A., 2002. A two-stage heuristic method for balancing mixed-model assembly lines
with parallel workstations. International Journal of Production Research 40, 1405-1420.
Wee, T.S., Magazine, A., 1981. An efficient branch and bound algorithm for an assembly line balancing
problem - part II: maximize the production rate. Working Paper No. 151, University of Waterloo, Waterloo.
Yagmahan, B., 2011. Mixed-model assembly line balancing using a multi-objective ant colony optimization
approach. Expert Syst Appl 38, 12453-12461.
Zhang, W.Q., Gen, M., 2011. An efficient multiobjective genetic algorithm for mixed-model assembly line
balancing problem considering demand ratio-based cycle time. J Intell Manuf 22, 367-378.
