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218 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART C: APPLICATIONS AND REVIEWS, VOL. 38, NO. 2, MARCH 2008

A Genetic-Algorithm-Based Optimization Modelfor Solving the Flexible Assembly LineBalancing Problem With Work Sharing

and Workstation RevisitingZ. X. Guo, W. K. Wong, S. Y. S. Leung, J. T. Fan, and S. F. Chan

AbstractThis paper investigates a flexible assembly line bal-ancing (FALB) problem with work sharing and workstation revis-iting. The mathematical model of the problem is presented, andits objective is to meet the desired cycle time of each order andminimize the total idle time of the assembly line. An optimizationmodel is developed to tackle the addressed problem, which involvestwo parts. A bilevel genetic algorithm with multiparent crossover

is proposed to determine the operation assignment to workstationsand the task proportion of each shared operation being processedon different workstations. A heuristicoperationrouting rule is thenpresented to route the shared operation of each product to an ap-propriate workstation when it should be processed. Experimentsbased on industrial data are conducted to validate the proposedoptimization model. The experimental results demonstrate the ef-fectiveness of the proposed model to solve the FALB problem.

Index TermsAssembly line balancing (ALB), genetic algo-rithms (GAs), optimization, work sharing, workstation revisiting.

I. INTRODUCTION

FACING ever-increasing global competition and unpre-

dictable demand fluctuations, more and more manufac-turing enterprises are seeking benefits from manufacturing flex-

ibility and effective assembly line management. This paper will

investigate the balancing problem of the assembly line with fea-

tures of flexible manufacturing so as to implement the effective

assembly line control.

A. Manufacturing Flexibility and Assembly Lines

Beach et al. [1] have provided a comprehensive review on

manufacturing flexibility. Manufacturing flexibility is of vari-

ous 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 fromone operation to another. If a workstation can process multiple

Manuscript received October 9, 2006. This work was supported by the In-novation and Technology Commission of the Government of the Hong KongSAR and Genexy Company, Ltd., under Project UIT/62. This paper was rec-ommended by Associate Editor R. Subbu.

Z. X. Guo is with the Institute of Textiles and Clothing, The Hong KongPolytechnic University, Kowloon, Hong Kong and also with the College ofInformation Science and Technology, Donghua University, 200051 Shanghai,China (e-mail: [email protected]).

W. K. Wong, S. Y. S. Leung, J. T. Fan, and S. F. Chan are with the Instituteof Textiles and Clothing, The Hong Kong Polytechnic University, Kowloon,Hong Kong (e-mail: [email protected]; [email protected]; [email protected]; [email protected]).

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 prod-ucts [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 semifin-

ished 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 worksta-

tion performs two or more operations that are not proximate in

the predetermined processing sequence.

Undoubtedly, allowing work sharing and workstation revis-

iting is helpful to improve both the machine and the routing

flexibility of the assembly line. Actually, the flexible assemblyline (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 bal-

ancing 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 as-

sumptions with respect to real-life assembly lines [5], [7]. In re-

cent years, a lot of research work has been done in order to solve

more realistic ALB problemsgeneralized 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 shar-

ing work in the assembly line have been presented, such as

bucket brigade [9], D-skill chaining [10], craft [11], etc. How-

ever, work sharing has received little attention in the existing

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GUO et al.: GENETIC-ALGORITHM-BASED OPTIMIZATION MODEL FOR SOLVING THE FALB PROBLEM 219

ALB literature. Furthermore, the ALB problem with worksta-

tion 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 nonop-

timal owing to the complexity of the ALB problem. Thus, amethodology to make the ALB decision better in the FAL is

needed.

In this paper, thebalancing problem of an FAL with work shar-

ing 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 al-gorithm [13], linear programming method [14], dynamic pro-

gramming 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 ap-

plied extensively, such as tabu search method [18], simulated

annealing method [19], immune algorithm [20], ant colony al-

gorithm [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 va-

riety of applications [26][28]. Furthermore, some researchers

have concluded that using multiparent crossover does increasethe performance of GA with binary or real-coded representa-

tion [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 multipar-

ent 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

several different machine types. Each workstation is a physi-

cal location that accommodates an operator, a machine, and a

buffer. Several production orders with given quantities repre-

senting 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. Inthe FAL, work sharing and workstation revisiting are allowed,

that is, one operation can be assigned to multiple worksta-

tions 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 typeof product will be produced. Oil denotes the lth operation oforderPi , Mkj denotes the jth machine (workstation) of the kthmachine type, and STil represents the standard time of operationOil , i.e., the time to complete operation Oil of one productwith 100% operative efficiency. We use the symbol ilkj (0 ilkj 1) to denote the task proportion (weight) of operation

Oil being performed on machine Mk j , i.e., the ilkj time of thetotal tasks of operation Oil is processed on machine Mkj . Onaverage, for operation Oil of each product, the task ofilkj STilshould be processed on machine Mkj . If operation Oil is onlyprocessed on machine Mkj , ilkj = 1; and if operation Oil isnot processed on machine Mkj , ilkj = 0. For each operationOil ,

k j ilkj = 1. The average assembly time MATk j of eachproduct on machine Mk j can be expressed as

MATkj =

il,O i l SO k j

ilkj STilEMilkj

(1)

where SOkj denotes the set of operations that can be processed

on machine Mk j and EMilkj denotes the operative efficiencyof operation Oil on machine Mk j .

A. Objective Function

The aim of ALB is to generate the optimal operation assign-

ment and routing Xilkj of each operation Oil . Xilkj indicatesthat if operation Oil is assigned to machine Mkj , Xilkj is equalto 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{Xi l k j }

Z(Xilkj )

with

Z(Xilkj ) =

p

i= 1

[ii (DCTi ACTi )

+ i (1 i )(ACTi DCTi )] (2)

where DCTi represents the desired cycle time of order Pi thatis the desired time interval of consecutive jobs entering the

assembly line, ACTi represents the actual cycle time of orderPi , i denotes the penalty weight for order Pi when its actual

cycle time is less than its desired cycle time, i denotes the

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penalty weight for orderPi when its actual cycle time is greaterthan its desired cycle time, and i indicates that if the actual

cycle time ACTi is less than the desired cycle time DCTi , iis equal to 1; otherwise it is equal to 0. Z is used to measurethe 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 willbe 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 asfollows:

min{Xi l k j }

IT(Xilkj )

with

IT(Xilkj ) =

p

i= 1

(ACTi Ni

kj,Mk j AMi

MATkj ) (3)

where AMi denotes the set of workstations processing orderPi ,and Ni denotes the number of workstations processing orderPi .

B. Assumptions and Constraints

In this research, the addressed problem satisfies the following

assumptions.

1) Each operators 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 fol-

lowing three basic types of constraints.

1) Allocation Constraint: Operation Oil can only be operatedon workstations that can handle it, i.e.,

kj,Mk j /SM i l

Xilkj = 0 (4)

where SMil denotes the set of machines that can handle opera-tion Oil .

Each workstation must process at least one operation, i.e.,

il

Xilkj 1. (5)

Each operation of a particular production order must be pro-

cessed, i.e.,

k j

Xilkj 1. (6)

2) Operation Precedence Constraint: For each product, an op-

eration cannot be started before the completion of its preceding

operation plus an elapsed time-out between the two operations,

i.e.,

Cil + ETil + 1 Sil , Oil PR(Oil ) (7)

where Cil is the completion time of operation Oil , ETil iselapsed time between operation Oil and its latter operation in-cluding the transportation time and the setup time, Sil is the

starting time of operation Oi

l

, and PR(Oi

l

) is the set of thepreceding operations of operation Oil .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

In order to solve the addressed FALB problem, a GA-based

optimization model is presentedin this section. In this model, the

BiMGA will be used firstlyto deal with theoperation assignment

of the FAL, i.e., assign operations to workstations and determinethe 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.

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 solutionfor 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 algo-

rithm 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 work-

stations 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 de-

termine 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 multi-

variate 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

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Fig. 1. BiMGA.

order-based representations tackling operation assignment have

been introduced, e.g., workstation-oriented representation [31],

operation-oriented representation [32], sequence-oriented rep-

resentation [33], etc. In the chromosome of these representa-

tions, 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 workstationrevisiting.

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 cor-

responding workstation processes. If the number of the machine

type is t(t 1), the genes in each chromosome will be dividedinto 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 prod-

uct, workstation 2 firstly processed operation 1, and then, the

semifinished product was transported to the other workstations

for further processing of operations 35. After operation 5 was

completed, the semifinished product revisited workstation 2 for

the processing of operation 6. Moreover, some shared operations

Fig. 2. Example of the chromosome representation.

existed in this solution. For example, the processing of opera-

tion 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 nQoperations, let nmil denote the number of machines that are al-located to process operation Oil and PSil denote the summationofnmil1 weights ofOil . The number of genes in each chro-mosome of GA-2 is the summation ofnmil minus nQ since thenmil th weight is equal to 1PSil .

2) Initialization: The GA operates on a population of chro-

mosomes. Either heuristic or random procedures can be used togenerate 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 assign-

ing each operation, from operations 1 to nOp, to the worksta-tions 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 sizePsize, population POP = {}, and a maximum quan-

tity mxQ of machines that an operation can be as-signed to.

Step 2. In light of procedure 2, randomly generate a string

chromosome CHRi , POP = POP CHRi .Step 3. Set i = i + 1. Stop ifi > Psize, 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 opera-

tion is selected to be processed.

Step 2. Generate randomly an integer k between 1 and thenumber of operations that can be processed on ma-

chine 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, theoperation cannot be selected.

Step 4. Assign the selected operation(s) to machine j. Foreach selected operation, let PRO = PRO 1/mxQ.

Step 5. Set j = j + 1. If j > nOp, go to Step 6, else go toStep 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 repre-

sents its probability to survive. The greater the fitness of a

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Fig. 3. 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 opti-

mized. 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(Xilkj

) = wZ

Z(Xilkj

) + wIT

IT(Xilkj

) (9)

where wZ and wI T are the relative weights placed upon theobjectives Z(Xilkj ) and IT(Xilkj ), respectively.

The less the weighted summation of the two objectives is, the

greater the fitness. Thus, the fitness function f t can be definedas

f t =100

OBJ(Xilkj ) + 1. (10)

4) Selection: The selection in GA, based on the natural law

of survival of the fittest, is the process to determine which chro-

mosomes are selected for the next generation in terms of their

fitness. Many selection schemes have been presented [34]. Thetournament 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 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 chromo-somes 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 . A larger value ofwill 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 initializedto 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 posi-

tions 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) ofchoosing the gene from parent spi can be

PR(i) =f t(i)

f t(i). (11)

Step 4. Put the chosen gene in the kth position of the childchromosome.

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 isgreater 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 mech-

anism works, in which the fitness of parents 13 are 0.90, 0.45,

and 0.45, respectively. The marked positions in parents are in-

dicated 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

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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 inver-

sion 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 pop-

ulation 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 thegenetic operations are performed, its nmil weights should bechanged to the corresponding values between 0 and 1 ifP Sil ofthe operation Oil is greater than 1. Firstly, randomly generatea real number between 0 and 1 as the nmil th weight. Then,normalize the nmil weights, and the normalized weights are thefinal 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 processof GA-1 or GA-2 is terminated.

B. Operation Routing

The proposed BiMGA can only obtain the optimized oper-

ation 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 sat-

isfy the optimized task proportion in each assigned workstation

during production.

Assume that operation Oil is assigned to n machines (Mk 1 ,

Mk 2 , . . ., Mkn ) according to the optimized operation assign-

TABLE IEXAMPLE OF OPERATION ROUTING TO PROCESS

OPERATION O11 OF TEN PRODUCTS

ment, ilkj denotes the optimized task proportion that operationOil should be processed on machine Mkj (

ilkj > 0),

ilkj de-

notes the task proportion that operation Oil has been processedon machine Mkj , and Qilkj denotes the number of operationOil that has been assigned to machine Mk j .

For shared operation Oil of a product, the heuristic operationrouting rule is described as the following procedure.

Procedure 6:

Step 1. Calculate ilkj = Qilkj /(n

l=1 Qilkj ) for each ma-chine Mkj (for the first product, set

ijk l = 0).

Step 2. Calculate ilkj/ilkj for each machine Mkj .

Step 3. Assign operation Oil of the current product to the ma-

chine Mkj with the minimum ilkj/ilkj . If multiplemachines 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 operationO11 is assigned to machines M11 , M12 , and M13 . The taskproportions of operation O11 to be processed on these threemachines are 0.4, 0.4, and 0.2, respectively, generated by the

proposed BiMGA. The row of ilkj/ilkj describes the current

value ilkj/ilkj of operation O11 of each product in the relevant

machine, and the shaded grid represents that the corresponding

machine is selected to process theoperation of thecorresponding

product. According to the result of operation routing shown inTable I, operation O11 of the first unit of product is assignedto 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; anal-

ysis 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

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 ex-

periments were conducted. This section highlights four out of

these experiments in detail. The FAL consists of 11 worksta-

tions 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 IIOPERATIVE EFFICIENCIES IN WORKSTATIONS OF EXPERIMENT 1

TABLE IIIOPERATIVE EFFICIENCIES IN WORKSTATIONS OF EXPERIMENT 2

known in advanced and included in the processing time. More-

over, 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 products 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 products 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 are70 and 225 s, respectively. The products 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 IIV. 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 IIV. The operative efficiency is set as 0 if

the operator cannot process the corresponding operation. The

processing time of operation Oil on workstation Mkj is equalto the standard time of this operation divided by its operative

efficiency on workstation Mkj

.

In experiments 24, 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 imple-

mented. In case 1, both work sharing and workstation revisiting

are allowed whereas both are not allowed in case 2 of experi-

ments 2 and 3. In case 2 of experiment 4, only work sharing is

implemented.

The optimized operation assignments and line balancing re-

sults of the four experiments generated by the proposed BiMGA

are shown in Tables VI and VII. In Table VI, the first col-

umn (Machine type) represents the machine type, the second

(Workstation No.) shows the workstation number, and other

columns show the optimized operation assignment of differ-

ent 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 ilkj of the operation be-ing 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

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TABLE IVOPERATIVE EFFICIENCIES IN WORKSTATIONS OF EXPERIMENT 3

TABLE VOPERATIVE EFFICIENCIES IN WORKSTATIONS OF EXPERIMENT 4

TABLE VIOPTIMIZED OPERATION ASSIGNMENT OF FOUR EXPERIMENTS

TABLE VIIOPTIMIZED 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 con-

sidering both work sharing and workstation revisiting. For ex-

ample, in case 1 of experiment 2, the processing of operation

9 was shared on workstations 2, 3, and 4 while workstation re-visiting 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 twoperformances were also inferior to the corresponding perfor-

mances in case 1. Obviously, the work sharing can improve the

performance of the assembly line.

In the optimization processes of these experiments, the evo-

lutionary trajectories of the maximum value of the fitness over

generations are shown in Fig. 5. The optimal results in this pa-

per 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 i and i of each order are10 and 100; and the relative weights wZ and wI T are both set

as 1. Moreover, in order to reduce the computation time of theoptimization process, we adjust probabilities of crossover and

mutation according to the fitness values of the population based

on the method developed by Syswerda [37].

B. Comparison Between GA-Based Optimization Model

and Industrial Practice

In industrial practice, the manager of shop floor usually bal-

ances the assembly line using precedence diagrams and trial-

and-error methods [38]. Considering case 1 of 4 experiments in

last section, their line balancing results based on industrial prac-

tice are shown in the rows of Industrial results of Table VIII.

The due dates of most orders could not be satisfied, and a largenumber 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 propor-

tions of the shared operation were the same on the worksta-

tions 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 propor-

tion of Table VIII. These results are also inferior to those of

Fig. 5. Trends of the chromosome fitness. (a) Experiment 1 and case 1 ofexperiments 24. (b) Case 2 of experiments 24.

TABLE VIIIRESULTS OF LINE BALANCING IN SECTIONS IV-BD

Section IV-A. That is because this assumption restricts the flex-

ibility 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, dif-

ferent operation routing rules can generate different balancing

performances. Here, we balance case 1 of the aforementioned

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four experiments based on the same operation assignment de-

scribed in Section IV-A and the following routing rule (ORR2).

ORR2: Let OSize denote the order size. Operation Oil ofilkj OSize products should be processed on machine Mk j . Weassign the first ilkj OSize operation Oil to machine Mk 1 , thenil k 2 OSize ones to machine Mk 2 , . . ., and the last

ilkn OSize

ones to machine Mk n .Assuming that OSize is equal to 3000, the final balancing re-sults 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 psy-chological 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 oper-ator absenteeism will increase the uncertainty of the ALB and

the processing time of the production order undoubtedly. In gen-

eral, 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 dif-

ferent 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 worksta-

tion by the proposed operation routing rule when it need be

processed.

The production data from the real-life FAL have been col-

lected to validate the proposed optimization model. The exper-

imental 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 multi-model 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 solv-

ing this problem. Further research will focus on it and the effects

of various uncertainties on the FALB, including machine break-

down, 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 incontrol theory andcontrol engineering fromDonghuaUniversity, Shanghai, China, in 2000 and 2003, re-spectively. He is currently working toward the Ph.D.degree at the Institute of Textiles and Clothing, TheHong Kong Polytechnic University, Hong Kong.

Since 2003, he has been an Assistant Lecturer inthe College of Information Science and Technology,Donghua University. His current research interestsinclude production planning and control and intelli-gent optimization techniques.

W. K. Wong received the Ph.D. degree from theInstitute of Textiles and Clothing, The Hong KongPolytechnic University, Kowloon, Hong Kong.

He has been with several southeast Asian coun-tries, specializing in production and quality manage-ment, industrial engineering, and productivity im-provement. In 1997, he joined The Hong KongPolytechnic University, where he is currently an As-sistant Professor. He is theauthor or coauthor of morethan 30 scientific articles published in refereed jour-nals and conference papers. His current research in-

terests include production planning and scheduling, modeling of manufacturingand management systems, and applications of artificial intelligence techniquesin the apparel manufacturing process.

S. Y. S. Leung received the M.Sc. (Clothing) degreein advancedmanufacture and the Ph.D.degree in sup-ply chain management from Manchester Metropoli-tan University, Manchester, U.K., 1992 and 1998,respectively.

He is currently an Assistant Professor at the Insti-tute of Textiles and Clothing, The Hong Kong Poly-technic University, Kowloon, HongKong,where he is

also the Deputy Chair of the Departmental Learningand Teaching Committee and the Deputy ProgrammeLeader 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 ar-tificial intelligent techniquesin fabric cutting, and utilization of radio-frequencyidentification in fashion cross-selling. He has authored or coauthored confer-ence and journal papers in these areas.

J. T. Fan received the B.Sc. degree in textile engi-neering fromChina Textile University (now DonghuaUniversity), Shanghai, in 1985, and the Ph.D. degreein clothing comfort from the University of Leeds,Leeds, U.K., in 1989.

He is currently a Professor in The Hong KongPolytechnic University (PolyU), Kowloon, HongKong, and is well known for his invention of theworlds first sweating fabric manikinWalter, thedevelopment of the worlds first and largest apparelknowledge portal (www.apparelkey.com), and his

contribution in clothing science and technology. He has authored or coauthoredextensively 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 Institutionof Textiles and Apparel. He is also the recipient of the 2001 PolyU Presi-dents Award 2001 for his outstanding performance/achievement in researchand scholarly activities, the 2003 Distinguished Achievement Award of the USFiber Society, and the Gold Medal Award from the International Invention Ex-hibition in Geneva in 2004.

S. F. Chan received the M.Sc. degree in fiber sci-ence and technology from Leeds University, Leeds,U.K., in 1978, and the D.B.A. degree in organiza-tional behavior from the Southern Cross University,Coffs Harbour, Australia, in 2002.

He hasworked in thetextile andapparel industriesfor17 years.In 1989,he joinedThe Hong KongPoly-technic University, Kowloon, Hong Kong, where heis currently an Assistant Professor. He is experiencedin production scheduling and quality system instal-lation. His current research interests include using

artificial intelligence in scheduling, total quality management, and curriculumissues.

Dr. Chan is an Associate Member of the Textile Institute, U.K., where he iscurrently the Deputy Programme Leader of the M.Sc. in Quality Management.

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