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IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
A Genetic-Algorithm-Based Optimization Model
for Solving the Flexible Assembly Line
Balancing Problem With Work Sharing
and Workstation Revisiting
Z. X. Guo, W. K. Wong, S. Y. S. Leung, J. T. Fan, and S. F. Chan
Abstract—This paper investigates a flexible assembly line balancing (FALB) problem with work sharing and workstation revisiting. The mathematical model of the problem is presented, and
its objective is to meet the desired cycle time of each order and
minimize the total idle time of the assembly line. An optimization
model is developed to tackle the addressed problem, which involves
two parts. A bilevel genetic algorithm with multiparent crossover
is proposed to determine the operation assignment to workstations
and the task proportion of each shared operation being processed
on different workstations. A heuristic operation routing rule is then
presented to route the shared operation of each product to an appropriate workstation when it should be processed. Experiments
based on industrial data are conducted to validate the proposed
optimization model. The experimental results demonstrate the effectiveness of the proposed model to solve the FALB problem.
Index Terms—Assembly line balancing (ALB), genetic algorithms (GAs), optimization, work sharing, workstation revisiting.
I. INTRODUCTION
ACING ever-increasing global competition and unpredictable demand fluctuations, more and more manufacturing enterprises are seeking benefits from manufacturing flexibility and effective assembly line management. This paper will
investigate the balancing problem of the assembly line with features of flexible manufacturing so as to implement the effective
assembly line control.
F
A. Manufacturing Flexibility and Assembly Lines
Beach et al. [1] have provided a comprehensive review on
manufacturing flexibility. Manufacturing flexibility is of various types, such as machine flexibility, routing flexibility, etc.
Machine flexibility is measured by the number of operations
that a workstation processes and the time needed to switch from
one operation to another. If a workstation can process multiple
Manuscript received October 9, 2006. This work was supported by the Innovation and Technology Commission of the Government of the Hong Kong
SAR and Genexy Company, Ltd., under Project UIT/62. This paper was recommended by Associate Editor R. Subbu.
Z. X. Guo is with the Institute of Textiles and Clothing, The Hong Kong
Polytechnic University, Kowloon, Hong Kong and also with the College of
Information Science and Technology, Donghua University, 200051 Shanghai,
China (e-mail: zx.guo@polyu.edu.hk).
W. K. Wong, S. Y. S. Leung, J. T. Fan, and S. F. Chan are with the Institute
of Textiles and Clothing, The Hong Kong Polytechnic University, Kowloon,
Hong Kong (e-mail: tcwongca@inet.polyu.edu.hk; tcleungs@inet.polyu.edu.
hk; tcfanjt@inet.polyu.edu.hk; tcchansf@inet.polyu.edu.hk).
Digital Object Identifier 10.1109/TSMCC.2007.913912
operations, the machine flexibility is high. Routing flexibility
is the ability of a production system to manufacture a product
using several alternative routes in the system, and it is usually
determined by the number of such potential routes.
Assembly lines are flow-oriented production systems that are
still attractive means of large-scale series production, and even
gain importance in low-volume production of customized products [2]. In the traditional assembly line, work sharing and
workstation revisiting are not permitted. Work sharing means
that one operation (task) is assigned to multiple workstations
for processing. Workstation revisiting occurs when the semifinished product (uncompleted product) revisits the workstation
for another operation to be processed after the product has been
processed by other workstations. In other words, the workstation performs two or more operations that are not proximate in
the predetermined processing sequence.
Undoubtedly, allowing work sharing and workstation revisiting is helpful to improve both the machine and the routing
flexibility of the assembly line. Actually, the flexible assembly
line (FAL) with these two features is widely adopted in some
manufacturing industries, and a typical example is the apparel
assembly line in the apparel industry.
B. Assembly Line Balancing Problem
The first published analytical statement of the assembly line
balancing (ALB) problem can be traced back to the middle of
the twentieth century [3], [4]. Since then, the topic of line balancing has been of great interest to researchers and practitioners,
and their research has been expanded greatly. With the growth of
knowledge on this subject, many studies have also been reported
to review the published literature comprehensively [2], [5]–[7].
Most of the existing ALB literature focuses on modeling and
solving the simple ALB problem that has some restricting assumptions with respect to real-life assembly lines [5], [7]. In recent years, a lot of research work has been done in order to solve
more realistic ALB problems—generalized ALB problem [2],
which considers some realistic features of assembly lines, such
as parallel workstations, U-shaped line layout, mixed-model or
multimodel assembly environment, etc.
Mcclain et al. [8] have pointed out that work sharing can
improve the efficiency of the assembly line. Some ways of sharing work in the assembly line have been presented, such as
bucket brigade [9], D-skill chaining [10], craft [11], etc. However, work sharing has received little attention in the existing
1094-6977/$25.00 © 2008 IEEE
GUO et al.: GENETIC-ALGORITHM-BASED OPTIMIZATION MODEL FOR SOLVING THE FALB PROBLEM
ALB literature. Furthermore, the ALB problem with workstation revisiting has also not been reported so far.
In the FAL with work sharing and workstation revisiting, the
line balancing activity mainly relies on managers/supervisors’
experience, and subjective and ad hoc assessment. However,
the human decision tends to be late, inconsistent, and nonoptimal owing to the complexity of the ALB problem. Thus, a
methodology to make the ALB decision better in the FAL is
needed.
In this paper, the balancing problem of an FAL with work sharing and workstation revisiting, i.e., the FAL balancing (FALB)
problem, will be investigated, which considers two objectives
including meeting the desired cycle time of each production
order and minimizing the total idle time of the assembly line.
C. Techniques for Assembly Line Balancing
A large variety of techniques have been developed to solve
the ALB problem [2], [5], [7]. Some classical optimization
techniques can provide optimal or near-optimal solutions, for
example, shortest-path technique [12], branch and bound algorithm [13], linear programming method [14], dynamic programming method [15], and integer programming method [16].
It is well known that the ALB problem belongs to NP-hard
class of combinatorial optimization problems [17]. In recent
years, various intelligent algorithms have been studied and applied extensively, such as tabu search method [18], simulated
annealing method [19], immune algorithm [20], ant colony algorithm [21], [22], and genetic algorithm (GA) [23]–[25] in
which GA is the most commonly used, and has been proven to
be very powerful in finding heuristic solutions from a wide variety of applications [26]–[28]. Furthermore, some researchers
have concluded that using multiparent crossover does increase
the performance of GA with binary or real-coded representation [29], [30]. However, the GA with multiparent crossover has
not been developed to solve the ALB problem.
In the FALB problem with work sharing and workstation
revisiting, it is significant to determine the flexible operation
assignment, find the shared work (operation), and assign the
shared operation to different workstations. In this paper, a
GA-based optimization model will be presented to solve them.
First, a bilevel GA with multiparent crossover [bilevel multiparent GA (BiMGA)] will be proposed to determine the operation
assignment to workstations and the task proportion of the shared
operation to be processed at different workstations. Second, an
operation routing rule will be presented to route each shared
operation of each product to an appropriate workstation.
The rest of this paper is organized as follows. In Section II,
the FALB problem is formulated. The GA-based optimization
model is described in detail to solve the addressed problem in
Section III. Experiments and detailed discussions are presented
to validate the effectiveness of the proposed optimization model
in Section IV. Finally, the paper is summarized and further
research is suggested in Section V.
II. PROBLEM FORMULATION
In this section, the ALB problem in an FAL is formulated.
The FAL is composed of a number of workstations including
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several different machine types. Each workstation is a physical location that accommodates an operator, a machine, and a
buffer. Several production orders with given quantities representing different product types will be produced in the FAL.
Each order comprises a series of manual operations. According
to a predetermined processing sequence, operations involved in
each order must be processed on corresponding workstations. In
the FAL, work sharing and workstation revisiting are allowed,
that is, one operation can be assigned to multiple workstations and one workstation can also process multiple operations
simultaneously.
In this paper, we let Pi represent the ith prodution order
(1 ≤ i ≤ p). In each order, a certain quantity of identical type
of product will be produced. Oil denotes the lth operation of
order Pi , Mk j denotes the jth machine (workstation) of the kth
machine type, and STil represents the standard time of operation
Oil , i.e., the time to complete operation Oil of one product
with 100% operative efficiency. We use the symbol ηilk j (0 ≤
ηilk j ≤ 1) to denote the task proportion (weight) of operation
Oil being performed on machine Mk j , i.e., the ηilk j time of the
total tasks of operation Oil is processed on machine Mk j . On
average, for operation Oil of each product, the task of ηilk j STil
should be processed on machine Mk j . If operation Oil is only
processed on machine Mk j , ηilk j = 1; and if operation Oil is
not processed
on machine Mk j , ηilk j = 0. For each operation
Oil , k j ηilk j = 1. The average assembly time MATk j of each
product on machine Mk j can be expressed as
ηilk j STil
MATk j =
(1)
EMilk j
il,O i l ∈SO k j
where SOk j denotes the set of operations that can be processed
on machine Mk j and EMilk j denotes the operative efficiency
of operation Oil on machine Mk j .
A. Objective Function
The aim of ALB is to generate the optimal operation assignment and routing Xilk j of each operation Oil . Xilk j indicates
that if operation Oil is assigned to machine Mk j , Xilk j is equal
to 1, otherwise it is equal to 0. In this paper, the objective of the
FALB problem includes two folds. The first one is to satisfy the
desired cycle time of each order, whereas the second one aims
at minimizing the total idle time in each cycle. The objective of
satisfying the desired cycle time can be described as
min Z(Xilk j )
{X i l k j }
with
Z(Xilk j ) =
p
[αi λi (DCTi − ACTi )
i=1
+ βi (1 − λi )(ACTi − DCTi )]
(2)
where DCTi represents the desired cycle time of order Pi that
is the desired time interval of consecutive jobs entering the
assembly line, ACTi represents the actual cycle time of order
Pi , αi denotes the penalty weight for order Pi when its actual
cycle time is less than its desired cycle time, βi denotes the
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penalty weight for order Pi when its actual cycle time is greater
than its desired cycle time, and λi indicates that if the actual
cycle time ACTi is less than the desired cycle time DCTi , λi
is equal to 1; otherwise it is equal to 0. Z is used to measure
the degree how the actual cycle time is close to desired cycle
time. The smaller the value of Z, the better the actual cycle
time satisfies the desired cycle time. The delivery dates will
be delayed and tardiness penalties generate if the actual cycle
time is greater than the desired cycle time, whereas the storage
cost arise and earliness penalties generate if the actual cycle
time is less than the desired cycle time.
The second objective of the FALB problem is to minimize
the total idle time IT in each cycle, which can be expressed as
follows:
min IT(Xilk j )
i.e.,
with
In order to solve the addressed FALB problem, a GA-based
optimization model is presented in this section. In this model, the
BiMGA will be used firstly to deal with the operation assignment
of the FAL, i.e., assign operations to workstations and determine
the task proportions of the shared operation to be processed
on different workstations. Then, a heuristic operation routing
process (operation routing rule) will be used to route the shared
operation of each product to an appropriate workstation. The
two processes are described in detail as follows.
{X i l k j }
IT(Xilk j ) =
p
(ACTi Ni −
MATk j )
(3)
k j,M k j ∈A M i
i=1
where AMi denotes the set of workstations processing order Pi ,
and Ni denotes the number of workstations processing order Pi .
B. Assumptions and Constraints
In this research, the addressed problem satisfies the following
assumptions.
1) Each operator’s efficiency is constant during production.
2) Once an operation of the product is started, it cannot be
interrupted.
3) There is no shortage of materials, workstation breakdown,
and operator absenteeism in the FAL.
4) The FAL discussed is empty initially, in other words, there
is no work-in-progress (WIP) in each workstation.
Furthermore, the real-life manufacturing environment has
many peculiar characteristics and is subject to some constraints.
A feasible solution of the FALB problem must satisfy the following three basic types of constraints.
1) Allocation Constraint: Operation Oil can only be operated
on workstations that can handle it, i.e.,
Xilk j = 0
(4)
k j,M k j ∈
/ SM i l
where SMil denotes the set of machines that can handle operation Oil .
Each workstation must process at least one operation, i.e.,
Xilk j ≥ 1.
(5)
il
Each operation of a particular production order must be processed, i.e.,
Xilk j ≥ 1.
(6)
kj
2) Operation Precedence Constraint: For each product, an operation cannot be started before the completion of its preceding
operation plus an elapsed time-out between the two operations,
Cil + ETil + 1 ≤ Si l , Oil ∈ PR(Oi l )
(7)
where Cil is the completion time of operation Oil , ETil is
elapsed time between operation Oil and its latter operation including the transportation time and the setup time, Si l is the
starting time of operation Oi l , and PR(Oi l ) is the set of the
preceding operations of operation Oi l .
3) Processing Time Requirements: Operation Oil must be
assigned with processing time, i.e.,
Cil = Sil + Til − 1
(8)
where Til is the time for processing operation Oil .
III. GA-BASED OPTIMIZATION MODEL FOR FALB
A. Bilevel Multiparent Genetic Algorithm
The operation assignment of the addressed FALB problem
can be considered as a two-stage optimization problem where
the first stage is to assign operations to workstations, while
the second one is to determine the task proportions of each
operation assigned to different workstations. Since the solution
for the second-stage subproblem has to depend on the solution
for the first-stage subproblem, the complexity of the addressed
problem is increased greatly. The BiMGA is proposed to solve
the two-stage FAL optimization problem.
Fig. 1 illustrates the steps involved in the BiMGA. The algorithm comprises two genetic optimization processes where the
second-level GA (GA-2) is nested in the first-level GA (GA-1).
The GA-1 generates the optimal operation assignment to workstations using the order-based representation. The chromosome
in the GA-1 represents the operation assignment of the FALB
problem. Based on each chromosome of GA-1, GA-2 will determine the task proportion (weight) of the operation that is
assigned to different workstations. If an operation is assigned
to multiple workstations, the weights on these workstations will
be optimized. Seeking the optimal weights is a first-order multivariate function optimization problem, which can be optimized
by a real-coded GA.
The following sections describe the detailed mechanism of
GA-1 and GA-2 of the BiMGA.
1) Representation: The first step of the GA is to define an
appropriate genetic representation. A representation that can
well describe problem-specific characteristics is crucial since it
significantly affects all the subsequent steps of the GA.
In GA-1, each chromosome represents a feasible solution
of assigning each operation to different workstations. Various
GUO et al.: GENETIC-ALGORITHM-BASED OPTIMIZATION MODEL FOR SOLVING THE FALB PROBLEM
Fig. 2.
Fig. 1.
BiMGA.
order-based representations tackling operation assignment have
been introduced, e.g., workstation-oriented representation [31],
operation-oriented representation [32], sequence-oriented representation [33], etc. In the chromosome of these representations, the gene represents one operation or one workstation.
Work sharing implies that one operation will be assigned to
multiple workstations, and workstation revisiting implies that
one workstation will process multiple operations. Obviously,
these existing representations cannot deal with the operation
assignment considering both work sharing and workstation
revisiting.
In GA-1, each chromosome is composed of a sequence of
genes whose length is equal to the number of workstations to
which operations can be assigned. In a chromosome, each gene
represents a workstation, and the value of each gene represents
the operation number(s) of one or more operations that the corresponding workstation processes. If the number of the machine
type is t(t ≥ 1), the genes in each chromosome will be divided
into t parts in turn. Each part represents one type of machines.
Each operation can only be assigned to the workstations that
can handle it. Fig. 2 shows an example of this representation
that considers a problem with 11 operations to be assigned to
11 workstations. These workstations are divided into two types,
type 1 including machines 1 through 8, and type 2 including
machines 9 through 11. Operations 1, 2, 3, 5, 6, 7, 9, and 11
must be processed on the machines of type 1, while operations
4, 8, and 10 must be operated on the machines of type 2. A
feasible solution, represented as an array of length 11, could be
[5 (1,6) 9 3 11 7 6 2 (4,10) 4 (8,10)]. In this solution, workstation
revisiting occurred in workstations 2, 9, and 11. For each product, workstation 2 firstly processed operation 1, and then, the
semifinished product was transported to the other workstations
for further processing of operations 3–5. After operation 5 was
completed, the semifinished product revisited workstation 2 for
the processing of operation 6. Moreover, some shared operations
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Example of the chromosome representation.
existed in this solution. For example, the processing of operation 6 was shared on workstations 2 and 7, and the processing
of operation 4 was shared on workstations 9 and 10.
In GA-2, the real-coded representation is adopted. Each gene
represents the task proportion of an operation assigned to the
corresponding workstation. Considering the assignment of nQ
operations, let nmil denote the number of machines that are allocated to process operation Oil and PSil denote the summation
of nmil −1 weights of Oil . The number of genes in each chromosome of GA-2 is the summation of nmil minus nQ since the
nmil th weight is equal to 1−PSil .
2) Initialization: The GA operates on a population of chromosomes. Either heuristic or random procedures can be used to
generate the initial population comprising a specified number
of chromosomes. Anderson and Ferris [32] have mentioned that
the performance of the GA is not so good from the preselected
initial population as it is from a random start.
In GA-1, each chromosome is randomly initialized by assigning each operation, from operations 1 to nOp, to the workstations that can handle it. The initialization process can, thus, be
described as procedure 1.
Procedure 1:
Step 1. Initialize parameters: index i = 1, a population size
P size, population POP = {φ}, and a maximum quantity mxQ of machines that an operation can be assigned to.
Step 2. In light of procedure 2, randomly generate a string
chromosome CHRi , POP = POP ∪ CHRi .
Step 3. Set i = i + 1. Stop if i > P size, else go to Step 2.
Procedure 2:
Step 1. Set index j = 1. For each operation, let PRO = 1,
where PRO represents the probability that an operation is selected to be processed.
Step 2. Generate randomly an integer k between 1 and the
number of operations that can be processed on machine j.
Step 3. Randomly select k operation(s) that can be processed
on this machine. The operation with greater PRO will
be selected with a greater probability. If PRO = 0, the
operation cannot be selected.
Step 4. Assign the selected operation(s) to machine j. For
each selected operation, let PRO = PRO − 1/mxQ.
Step 5. Set j = j + 1. If j > nOp, go to Step 6, else go to
Step 2.
Step 6. Stop if all operations are assigned, else go to Step 1.
In GA-2, the initial population is generated by initializing
randomly each task proportion (weight) in the chromosome
between 0 and 1 based on the condition of PSil ≤ 1.
3) Fitness: The fitness of a particular chromosome represents its probability to survive. The greater the fitness of a
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Fig. 3.
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008
Example of the modified-fitness-based scanning crossover operator.
chromosome is, the greater the probability to survive. The value
of the fitness is relevant to the objective function to be optimized. The fitness of GA-1 is the same with that of GA-2,
which is described as follows.
In this research, two objective functions described in
Section II are optimized, which can be combined as the equation
OBJ(Xilk j ) = wZ Z(Xilk j ) + wIT IT (Xilk j )
(9)
where wZ and wI T are the relative weights placed upon the
objectives Z(Xilk j ) and IT(Xilk j ), respectively.
The less the weighted summation of the two objectives is, the
greater the fitness. Thus, the fitness function f t can be defined
as
ft =
100
.
OBJ(Xilk j ) + 1
(10)
4) Selection: The selection in GA, based on the natural law
of survival of the fittest, is the process to determine which chromosomes are selected for the next generation in terms of their
fitness. Many selection schemes have been presented [34]. The
tournament selection [35] is commonly utilized because it is
simple to implement and provides good solutions. In this study,
this scheme is applied in GA-1 and GA-2, and its procedure can
be described as follows.
Procedure 3:
Step 1. Set a tournament size δ ≥ 2.
Step 2. Generate a random permutation of the chromosomes
in the current population, which is a feasible solution
of operation assignment or task proportions.
Step 3. Compare the fitness value of the first δ chromosomes listed in the permutation, and copy the best one
into the next generation. Discard the chromosomes
compared.
Step 4. If the permutation is exhausted, generate another
permutation.
Step 5. Repeat steps 3 and 4 until no more selections are
required for the next generation.
The scheme can control the population diversity and selective
pressure by adjusting the tournament size δ. A larger value of δ
will increase the selective pressure but decrease the population
diversity.
5) Genetic Operators: To improve the adaptability of the
population, two basic operators, crossover and mutation, are
used to modify the chromosome. The detailed descriptions of
the two genetic operators are as follows.
a) Crossover: Crossovers are deterministic operators that
capture the features of the parents and pass it to a new offspring.
The population is recombined according to a probability of
crossover that ranges typically between 0.6 and 1.0.
In GA-2, the center of mass crossover operator [29] is used.
In GA-1, the fitness-based scanning crossover [30] is modified
to suit the proposed representation, which is described as next.
Procedure 4:
Step 1. Let sp1 , sp2 , . . .,spr be the selected parents with L
genes.
Step 2. Initialize parameters: position markers i1 = · · · =
ir = 1, i.e., the position markers are all initialized
to the first position in each of the parents; the gene
position in the child chromosome k = 1.
Step 3. Choose a gene from the r genes in the marked positions of the parents, which is based on the rule that
the probability of the gene of the parent being chosen
is proportional to the fitness values of the parent. For
example, for a maximization problem where parent
spi has a fitness of f t(i), the probability PR(i) of
choosing the gene from parent spi can be
f t(i)
.
PR(i) = f t(i)
(11)
Step 4. Put the chosen gene in the kth position of the child
chromosome.
Step 5. Update position markers i1 , . . . , ir . For each parent,
if the gene in the current position is the same with
the chosen gene, increase its marker until it denotes a
value that has not already been added to the child or
equals L.
Update k = k + 1.
Step 6. Repeat steps 3, 4, and 5 until the gene position k is
greater than L.
Step 7. Stop if each operation in the parent is assigned to
machines, else go to Step 2.
Fig. 3 shows an example of how the proposed crossover mechanism works, in which the fitness of parents 1–3 are 0.90, 0.45,
and 0.45, respectively. The marked positions in parents are indicated by shaded grids.
b) Mutation: After crossover, the offspring undergoes
mutation according to the probability of mutation (the typical
value is between 0.0015 and 0.03). The mutation operation is
important to the success of the GA since it diversifies the search
GUO et al.: GENETIC-ALGORITHM-BASED OPTIMIZATION MODEL FOR SOLVING THE FALB PROBLEM
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TABLE I
EXAMPLE OF OPERATION ROUTING TO PROCESS
OPERATION O 1 1 OF TEN PRODUCTS
Fig. 4.
Example of the modified inversion mutation operator.
direction and prevents a population prematurely converging at
local minima.
In GA-2, the nonuniform mutation operator [36] is adopted.
In GA-1, a modified mutation operation being similar to inversion mutation operator [35] is developed, which is described
detailedly as procedure 5.
Procedure 5:
Step 1. Inverse the genes between two randomly selected
genes of a chromosome.
Step 2. The gene with two or more operations is separated
according to a suitable probability (between 0.6 and
1).
Step 3. The separated operation is recombined randomly with
its proximate genes.
In this procedure, steps 2 and 3 are helpful to increase the population diversity of GA and avoid the premature convergence.
Fig. 4 shows an example of this mutation operator.
In GA-1, the crossover and mutation operations can only
be performed among genes with the same machine type since
each operation must be processed on machines of a certain
type. Therefore, for the genes of each machine type, the genetic
operations should be performed separately. In GA-2, after the
genetic operations are performed, its nmil weights should be
changed to the corresponding values between 0 and 1 if P Sil of
the operation Oil is greater than 1. Firstly, randomly generate
a real number between 0 and 1 as the nmil th weight. Then,
normalize the nmil weights, and the normalized weights are the
final weights.
6) Termination Criterion: The GA in this study is controlled
by a specified number of generations and by using a diversity
measure to stop the algorithm. The diversity of the algorithm
is defined by the standard deviation of the fitness values of all
chromosomes of a population in a certain generation. If either
of these two termination criteria is satisfied, the cycled process
of GA-1 or GA-2 is terminated.
ment, ηilk
j denotes the optimized task proportion that operation
Oil should be processed on machine Mk j (ηilk
j > 0), ηilk j denotes the task proportion that operation Oil has been processed
on machine Mk j , and Qilk j denotes the number of operation
Oil that has been assigned to machine Mk j .
For shared operation Oil of a product, the heuristic operation
routing rule is described as the following procedure.
Procedure 6:
n
Step 1. Calculate ηilk
j = Qilk j /(
l=1 Qilk j ) for each ma
chine Mk j (for the first product, set ηij
k l = 0).
Step 2. Calculate ηilk j /ηilk j for each machine Mk j .
Step 3. Assign operation Oil of the current product to the ma
chine Mk j with the minimum ηilk
j /ηilk j . If multiple
machines have the same minimum value, one of these
machines will be chosen randomly.
Table I shows an example of the operation routing to process
operation O11 of 10 units of identical product. The operation
O11 is assigned to machines M11 , M12 , and M13 . The task
proportions of operation O11 to be processed on these three
machines are 0.4, 0.4, and 0.2, respectively, generated by the
proposed BiMGA. The row of ηilk
j /ηilk j describes the current
value ηilk j /ηilk j of operation O11 of each product in the relevant
machine, and the shaded grid represents that the corresponding
machine is selected to process the operation of the corresponding
product. According to the result of operation routing shown in
Table I, operation O11 of the first unit of product is assigned
to M11 , that of the second unit of product is assigned to M13 ,
etc. After the 10 units of product are completed, the actual task
proportion processed on each machine is equal to the optimized
task proportion.
IV. EXPERIMENTAL RESULTS AND DISCUSSIONS
This section will present the validation of the effectiveness
of the proposed optimization model; performance comparison
between the proposed model and the industrial practice; analysis of the effects of task proportion, operation routing, and
violation of assumption (presented in Section II-B) on the FALB
performance.
A. Validation of GA-Based Optimization Model
B. Operation Routing
The proposed BiMGA can only obtain the optimized operation assignment and task proportion of the shared operation
on different workstations. After the previous operations of the
shared operation of each product are completed, the operation
should be then routed to an appropriate workstation so as to satisfy the optimized task proportion in each assigned workstation
during production.
Assume that operation Oil is assigned to n machines (Mk 1 ,
Mk 2 , . . ., Mk n ) according to the optimized operation assign-
To evaluate the performance of the GA-based optimization
model, real industrial data were collected from an FAL of a
Hong Kong-owned manufacturing company, and a series of experiments were conducted. This section highlights four out of
these experiments in detail. The FAL consists of 11 workstations with two types of machines. The workstations of type 1
machines include eight workstations numbered as 1 to 8 and
those of type 2 machines include three workstations numbered
as 9 to 11. In these experiments, the transportation time of
semifinished products and the setup time of each operation are
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TABLE II
OPERATIVE EFFICIENCIES IN WORKSTATIONS OF EXPERIMENT 1
TABLE III
OPERATIVE EFFICIENCIES IN WORKSTATIONS OF EXPERIMENT 2
known in advanced and included in the processing time. Moreover, each production order is available for processing starting
from time zero.
In each experiment, two different production orders are
scheduled. Some basic data of these experiments are as
follows.
Experiment 1: The desired cycle times of orders 1 and 2 are
both 400 s. The product’s assembly process of order 1 is from
operations 1 to 7, and order 2 is from operations 8 to 12.
Experiment 2: The desired cycle times of orders 1 and 2 are
55 and 130 s, respectively. The product’s assembly process of
order 1 is from operations 1 to 6, and order 2 is from operations
7 to 11.
Experiment 3: The desired cycle times of two orders are both
50 s. The assembly processes of two orders are the same with
those in experiment 2.
Experiment 4: The desired cycle times of orders 1 and 2 are
70 and 225 s, respectively. The product’s assembly process of
order 1 is from operations 1 to 5, and order 2 is from operations
6 to 10.
The standard time of each operation in these experiments is
shown in the last row of Tables II–V. The operative efficiency
of each workstation depends on the type of the machine and
the skill level as well as recent performance of the operator,
as shown in Tables II–V. The operative efficiency is set as 0 if
the operator cannot process the corresponding operation. The
processing time of operation Oil on workstation Mk j is equal
to the standard time of this operation divided by its operative
efficiency on workstation Mk j .
In experiments 2–4, the number of workstations is equal to
or greater than the number of operations. In order to evaluate
the effect of work sharing and workstation revisiting on the
FALB performance, different assignment strategies are implemented. In case 1, both work sharing and workstation revisiting
are allowed whereas both are not allowed in case 2 of experiments 2 and 3. In case 2 of experiment 4, only work sharing is
implemented.
The optimized operation assignments and line balancing results of the four experiments generated by the proposed BiMGA
are shown in Tables VI and VII. In Table VI, the first column (Machine type) represents the machine type, the second
(Workstation No.) shows the workstation number, and other
columns show the optimized operation assignment of different experiments to the workstation, in which the first value of
each cell represents the operation number and the value in the
bracket represents the task proportion ηilk j of the operation being processed in the corresponding workstation. For example,
the value 12(1) in the column of “Experiment 1” describes that
workstation 1 processes all (100%) operation 12, and the value
[7(0.67), 9(0.15)] in the column of “Experiment 2” shows that
workstation 2 processes 67% tasks of operation 2 and 15% tasks
of operation 9. In Table VII, the rows of “Actual cycle time”
show the optimized actual cycle time (seconds) of orders 1 and
2 in four experiments whereas the rows of “Idle time” and “Line
efficiency” show the optimized average idle time (seconds) in
each cycle and the optimized line efficiencies of orders 1 and
2 in four experiments, respectively. The line efficiency of order
Pi is defined as the average processing time of workstations
GUO et al.: GENETIC-ALGORITHM-BASED OPTIMIZATION MODEL FOR SOLVING THE FALB PROBLEM
TABLE IV
OPERATIVE EFFICIENCIES IN WORKSTATIONS OF EXPERIMENT 3
TABLE V
OPERATIVE EFFICIENCIES IN WORKSTATIONS OF EXPERIMENT 4
TABLE VI
OPTIMIZED OPERATION ASSIGNMENT OF FOUR EXPERIMENTS
TABLE VII
OPTIMIZED RESULTS OF LINE BALANCING OF FOUR EXPERIMENTS
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processing this order in each cycle divided by the actual cycle
time of this order.
As shown in Table VI, the proposed genetic optimization
algorithm can implement flexible operation assignments considering both work sharing and workstation revisiting. For example, in case 1 of experiment 2, the processing of operation
9 was shared on workstations 2, 3, and 4 while workstation revisiting occurred on workstation 2. Moreover, in the optimized
operation assignment of case 1 of experiment 4, some parallel
workstations existed, which processed the same operation set,
such as workstations 1 and 3, workstations 2 and 6, workstations
4 and 8, etc. It indicates that the proposed algorithm can also
handle the ALB problem with parallel workstations.
As shown in Table VII, since the desired cycle times of orders
1 and 2 were achieved in experiment 1 and case 1 of experiment
3, and the actual cycle times of two orders were very close to
the desired cycle time in case 1 of experiments 2 and 4, the
proposed BiMGA can solve the FALB problem effectively.
Moreover, in case 2 of experiments 2, 3, and 4, the actual
cycle times went beyond the desired cycle times; the other two
performances were also inferior to the corresponding performances in case 1. Obviously, the work sharing can improve the
performance of the assembly line.
In the optimization processes of these experiments, the evolutionary trajectories of the maximum value of the fitness over
generations are shown in Fig. 5. The optimal results in this paper are obtained based on the settings: the population sizes of
GA-1 and GA-2 are 200 and 100, respectively; the maximum
numbers of generations of GA-1 and GA-2 are 100 and 50,
respectively; the penalty weights αi and βi of each order are
10 and 100; and the relative weights wZ and wI T are both set
as 1. Moreover, in order to reduce the computation time of the
optimization process, we adjust probabilities of crossover and
mutation according to the fitness values of the population based
on the method developed by Syswerda [37].
Fig. 5. Trends of the chromosome fitness. (a) Experiment 1 and case 1 of
experiments 2–4. (b) Case 2 of experiments 2–4.
TABLE VIII
RESULTS OF LINE BALANCING IN SECTIONS IV-B–D
B. Comparison Between GA-Based Optimization Model
and Industrial Practice
In industrial practice, the manager of shop floor usually balances the assembly line using precedence diagrams and trialand-error methods [38]. Considering case 1 of 4 experiments in
last section, their line balancing results based on industrial practice are shown in the rows of “Industrial results” of Table VIII.
The due dates of most orders could not be satisfied, and a large
number of earliness and tardiness penalties occurred that are
inferior to the optimized results shown in Section IV-A.
C. Effect of Task Proportion on FALB Performance
In the previous studies, it was assumed that the task proportions of the shared operation were the same on the workstations processing the operation. For example, if one operation
is assigned to four workstations, the task proportion on each
workstation should be 0.25. In light of this assumption, the
optimized balancing results of case 1 of the aforementioned
four experiments are shown in the rows of “Same task proportion” of Table VIII. These results are also inferior to those of
Section IV-A. That is because this assumption restricts the flexibility of the operation assignment and shrinks the search space
of the possible ALB solutions.
D. Effect of Operation Routing on FALB Performance
The previous studies on ALB only focused on the operation
assignment and did not pay attention on the operation routing
based on the optimized operation assignment. However, different operation routing rules can generate different balancing
performances. Here, we balance case 1 of the aforementioned
GUO et al.: GENETIC-ALGORITHM-BASED OPTIMIZATION MODEL FOR SOLVING THE FALB PROBLEM
four experiments based on the same operation assignment described in Section IV-A and the following routing rule (ORR2).
ORR2: Let OSize denote the order size. Operation Oil of
ηilk
j OSize products should be processed on machine Mk j . We
assign the first ηilk
j OSize operation Oil to machine Mk 1 , then
ηilk 2 OSize ones to machine Mk 2 , . . ., and the last ηilk
n OSize
ones to machine Mk n .
Assuming that OSize is equal to 3000, the final balancing results are shown in the rows of “ORR2” of Table VIII. The actual
cycle times are much greater than the desired cycle times and
the line efficiencies are comparatively low. The results indicate
that the effectiveness of an operation routing rule is important
for the performance of the FALB.
E. Discussion on Assumption Relaxation
In the real-life manufacturing environment, the assumptions
described in Section II-B are often violated.
The operative efficiency is often variable. Thus, the factors
affecting operative efficiencies should be considered, such as
effects of learning and forgetting, and physiological and psychological effects. The variable operative efficiency will lead
to the fluctuation of the actual cycle time and increases the
complexity of the FALB.
Once one operation is preempted and another operation is
processed, more additional time should be spent on adjusting
the machine setup. If the processing time of an operation is not
very long and the time precision of the ALB is not too high, the
operation preemption has little influence on the performance of
the ALB. Contrariwise, operation preemption can lead to the
decrease of the ALB performance owing to the additional setup
time of machine.
The shortage of materials, workstation breakdown, and operator absenteeism will increase the uncertainty of the ALB and
the processing time of the production order undoubtedly. In general, the ALB solution can be obtained if the occurrence of these
uncertain factors is assumed with certain probabilities.
V. CONCLUSION
In this paper, we investigated an FALB problem with work
sharing and workstation revisiting. The mathematical model
for the problem has been proposed. Besides the objective of
meeting the desired cycle time of each order, the model also
minimizes the total idle time of the FAL. These objectives are
particularly useful to help manufacturing enterprises to meet the
due dates, and also to improve the efficiency of the assembly
line by optimizing the use of limited resources.
A GA-based optimization model was developed to deal with
the proposed FALB problem, in which a BiMGA and a heuristic
operation routing rule were presented. The BiMGA generates
the optimal operation assignment to workstations and the task
proportion of each shared operation being processed on different workstations. In the BiMGA, the fitness-based scanning
crossover and the inversion mutation are modified to suit the
representation of the flexible operation assignment. The shared
operation of each product is routed to an appropriate workstation by the proposed operation routing rule when it need be
processed.
227
The production data from the real-life FAL have been collected to validate the proposed optimization model. The experimental results have demonstrated that the optimization model
can solve the FALB problem effectively. Moreover, since the
FAL investigated contains the features of multimodel and the
mixed-model assembly line, the proposed optimization model
can be extended to solve the balancing problem of the multimodel assembly line or the mixed-model assembly line.
This paper also showed that the GA with multiparent
crossover can be used in tackling the operation assignment of
the ALB problem. However, the performance of the multiparent
GA has not been compared with that of two-parent GA on solving this problem. Further research will focus on it and the effects
of various uncertainties on the FALB, including machine breakdown, operator absenteeism, shortage of materials, and learning
effects on operative efficiency, etc.
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Z. X. Guo received the B.Sc. and M.Sc. degrees in
control theory and control engineering from Donghua
University, Shanghai, China, in 2000 and 2003, respectively. He is currently working toward the Ph.D.
degree at the Institute of Textiles and Clothing, The
Hong Kong Polytechnic University, Hong Kong.
Since 2003, he has been an Assistant Lecturer in
the College of Information Science and Technology,
Donghua University. His current research interests
include production planning and control and intelligent optimization techniques.
W. K. Wong received the Ph.D. degree from the
Institute of Textiles and Clothing, The Hong Kong
Polytechnic University, Kowloon, Hong Kong.
He has been with several southeast Asian countries, specializing in production and quality management, industrial engineering, and productivity improvement. In 1997, he joined The Hong Kong
Polytechnic University, where he is currently an Assistant Professor. He is the author or coauthor of more
than 30 scientific articles published in refereed journals and conference papers. His current research interests include production planning and scheduling, modeling of manufacturing
and management systems, and applications of artificial intelligence techniques
in the apparel manufacturing process.
S. Y. S. Leung received the M.Sc. (Clothing) degree
in advanced manufacture and the Ph.D. degree in supply chain management from Manchester Metropolitan University, Manchester, U.K., 1992 and 1998,
respectively.
He is currently an Assistant Professor at the Institute of Textiles and Clothing, The Hong Kong Polytechnic University, Kowloon, Hong Kong, where he is
also the Deputy Chair of the Departmental Learning
and Teaching Committee and the Deputy Programme
Leader of Fashion and Textile Studies. His current
research interests include discrete event simulation for clothing manufacture,
apparel supply chain management, lean and agile production, application of artificial intelligent techniques in fabric cutting, and utilization of radio-frequency
identification in fashion cross-selling. He has authored or coauthored conference and journal papers in these areas.
J. T. Fan received the B.Sc. degree in textile engineering from China Textile University (now Donghua
University), Shanghai, in 1985, and the Ph.D. degree
in clothing comfort from the University of Leeds,
Leeds, U.K., in 1989.
He is currently a Professor in The Hong Kong
Polytechnic University (PolyU), Kowloon, Hong
Kong, and is well known for his invention of the
world’s first sweating fabric manikin—“Walter,” the
development of the world’s first and largest apparel
knowledge portal (www.apparelkey.com), and his
contribution in clothing science and technology. He has authored or coauthored
extensively with more than 180 academic papers or patents.
Prof. Fan is a Fellow of the Royal Society for the Encouragement of Arts,
Manufacture & Commerce, the Textile Institute, and the Hong Kong Institution
of Textiles and Apparel. He is also the recipient of the 2001 PolyU President’s Award 2001 for his outstanding performance/achievement in research
and scholarly activities, the 2003 Distinguished Achievement Award of the US
Fiber Society, and the Gold Medal Award from the International Invention Exhibition in Geneva in 2004.
S. F. Chan received the M.Sc. degree in fiber science and technology from Leeds University, Leeds,
U.K., in 1978, and the D.B.A. degree in organizational behavior from the Southern Cross University,
Coffs Harbour, Australia, in 2002.
He has worked in the textile and apparel industries
for 17 years. In 1989, he joined The Hong Kong Polytechnic University, Kowloon, Hong Kong, where he
is currently an Assistant Professor. He is experienced
in production scheduling and quality system installation. His current research interests include using
artificial intelligence in scheduling, total quality management, and curriculum
issues.
Dr. Chan is an Associate Member of the Textile Institute, U.K., where he is
currently the Deputy Programme Leader of the M.Sc. in Quality Management.