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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.

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