A genetic algorithm for scheduling dual flow shops

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A genetic algorithm for scheduling dual ﬂow shops

Chie-Wun Chiou

⇑

, Wen-Min Chen, Chin-Min Liu, Muh-Cherng Wu

Department of Industrial Engineering and Management, National Chiao Tung University, Hsin-Chu 300, Taiwan

a r t i c l e

i n f o

Keywords: Scheduling Dual ﬂow shop Setup time Due date

Genetic algorithm (GA)

a b s t r a c t

This study examines a dual-flow shop-scheduling problem that allows cross-shop processing. The sched-uling objective is to minimize the coefficient of variation of slack time (lateness), where the slack time (ST) of a job denotes the difference between its due date and total completion time. This scheduling prob-lem involves two decisions: job route assignment (assigning jobs to shops) and job sequencing. This study develops a genetic algorithm (GA) embedded with the earliest due date (EDD) dispatching rule for making these decisions. Numerical experiments with the GA algorithm indicate that the performance of adopting a cross-shop production policy may significantly outperform that of adopting a single-shop production policy. This is particularly true when the two flow shops are asymmetrically designed.

This study develops a grouping heuristic algorithm to reduce setup time and due-date-based demand simultaneously. This study uses the proposed genetic algorithm (GA) to prove that the grouping heuristic algorithm performs well. Obtaining an approximate optimal solution makes it possible to decide the route assignment of jobs and the job sequencing of machines.

1. Introduction

Some manufacturing companies must build two plants to fulﬁll customer demand. This dual-plants strategy arises from two rea-sons: rapid capacity expansion and capacity sharing mechanisms. In the case of rapid capacity expansion, it is often more difﬁcult to acquire land than equipment. Therefore, many companies will initially build a space large enough for two plants, and then grad-ually purchase equipment based on market demand.

For capacity sharing reasons, the dual-plants strategy can adopt a cross-plant production policy in which two plants mutually sup-port each other in capacity. This policy increases single plant capacity because it allows one plant to utilize its equipment capac-ity fully by ﬁlling orders from the other plant.

The cross-plant production policy involves two major sequenc-ing decisions: (1) route assignment—how to allocate jobs to each plant, and (2) job sequencing within a plant, as well as how to se-quence allocated jobs for each plant. Most studies on dual-plant scheduling are developed under a route assignment assumption (Toba, Izumi, Hatada, & Chikushima, 2005; Wu & Chang, 2007; Wu, Chen, & Shih, 2009; Wu, Shih, & Chen, 2009).

This paper considers the problem of scheduling a dual-flow shop that allows cross-flow shop processing. Each flow shop has three stages that can process any jobs from the previous stage of a dual flow shop. Each stage has two machines capable of processing

one job at a time. Each stage in the two ﬂow shops is functionally comprehensive. That is, a job can be completely processed in one

stage in either ﬂow shop.Fig. 1shows that all jobs follow the same

sequential processing route: Stage 1, Stage 2, and Stage 3. The pur-pose of cross-shop production is to increase the total throughput of both ﬂow shops and reduce the average job cycle time.

This study presents a genetic algorithm (GA) embedded with the earliest due date (EDD) dispatching rule (a GA-EDD algorithm) for scheduling dual flow shops. All jobs have two sequencing decisions to be made: (1) route assignment (assigning jobs to stages), and (2) job sequencing within each stage. This study uses the single-shop production policy as a benchmark to determine the effectiveness of the proposed dual flow shop algorithm. Single-shop production means that each job can be processed in only one plant (a flow shop). Numerical experiments show that GA-EDD for a dual-flow shop is much better than the production of two single shops.

The remainder of this paper is organized as follows. Section2

reviews the relevant literature. Section 3 explains the research

problem in detail. Section4describes how to compute the

coefﬁ-cient of slack time variation for a job sequence. Section5presents

the solution architecture of the proposed algorithms. Section6

re-ports numerical experiments, while Section7provides concluding

remarks.

2. Relevant literature

Previous studies on the dual-ﬂow shop-scheduling problem gen-erally fall into two groups: product-level and operational-level. In

⇑Corresponding author.

E-mail address:cw_chiou@yahoo.com.tw(C.-W. Chiou).

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the product-level group, a product can be processed only within sin-gle plant. In the operational-level group, each plant can perform a group of operations, distributing its operations among different plants.

Most studies in the product level group-prohibiting any cross-plant routes category assume that each cross-plant manufactures

prod-ucts separately, which prohibits cross-plant production.Wu, Erkoc,

and Karabuk (2005)provided a comprehensive survey of this topic,

and recent studies have extended their ﬁndings (Chiang, Guo,

Chen, Cheng, & Chen, 2007; Lee, Chung, Lee, & Kang, 2006). Most of these studies use the linear programming (LP) technique to solve the dual ﬂow shop scheduling problem. These production-level studies consider one factor: job sequencing within a plant.

Most studies in the operational-level group, which allow some cross-plant routes, assume that each plant can obtain mutual

capac-ity support.Toba et al. (2005)studied the route-planning problem

in a real-time environment.Wu and Chang (2007)examined the

route-planning problem on a weekly schedule. Wu, Shih, et al.

(2009) and Wu, Chen, et al. (2009) studied route planning in dual-plant scenarios with different limited transportation

capaci-ties and varying transportation times.Paolo et al. (2010)proposed

a game theoretic protocol of cooperation and multiagent architec-ture to share capacity among different plants. These operational-level studies consider one factor between two plants—route assign-ment—.

Most studies consider either job sequencing within a plant or route assignment decisions. However, this paper considers both of these decisions simultaneously.

3. Problem statement

This section explains the dual-plant problem in detail, where the two plants called Plant_A and Plant_B. The following discussion ﬁrst explains the assumptions in this study and then describes the problem.

Assumption 1. The machine in each dual-plants stage is functionally comprehensive. Each plant is equipped with the same machine functionality in each stage of job production.

Assumption 2. Each job has eight routes. The processing routes of each job fall into three segments. The break point between two stages of the processing route is called a cut-off point. A job has eight possible routes: 1 ? 1 ? 1, 1 ? 1 ? 2, 1 ? 2 ? 1, 1 ? 2 ?

2, 2 ? 2 ? 2, 2 ? 1 ? 1, 2 ? 1 ? 2, 2 ? 2 ? 1. The route

i ? j ? k indicates that the ﬁrst segment i is processed in Stage 1, the second one is processed in Stage 2, and the third one is pro-cessed in Stage 3. The number 1 indicates that a job is manufac-tured in Plant_A, while 2 denotes Plant_B.

The problem addressed in this study consists of a set of n jobs (j 2 J) to be processed in a dual-ﬂow shop, and each plant has three machines. Each machine m can process one job at a time. The route of each job requires three sequential stages (three machines) to be

completed. Each job j has a processing time pjmon machine m. The

scheduling objective is to minimize the coefﬁcient of variation of slack time, in which the slack time of a job denotes the difference between its due date and total completion time.

4. Slack time evaluation for job sequences

Given a job sequence, this study uses a procedure to evaluate the slack time for completing all the jobs. This procedure emulates the processing ﬂow of each job, and makes it possible to obtain the slack time (i.e., the difference between its due date and total com-pletion time). The following section presents the notation and de-tails of the procedure in this study.

Notation

j index of job, j = 1, 2, . . . , n

i index of stage, i = 1, 2, . . . , m

f index of ﬂow shop, f = 1, 2

M total number of stages

pi,f,j processing time required for stage i in ﬂow shop f to pro-cess job j

ti1,i transportation time for shipping a job from the i 1 stage to the next i stage

p

i,f a job sequence for n jobs at the stage i of ﬂow shop f,

p

i,x= [

p

i,x(1), . . . ,

p

i,x(n)]

p

i,f(j) the job in the jth position of a sequence at the stage i of

ﬂow shop f

dj the due date for job j

Ci;pi;f ðjÞ the completion time of job

p

i,f(j) at stage i of the ﬂow shop f

Ai,f,t the time epoch in which machine in stage i of ﬂow shop f becomes available; that is, when the machine (i, f) is free at t, Ai,l,tis the last job-completion-epoch before t; while the

machine (i, f) is in operation at t, Ai,f,t is the ﬁrst job

-completion-time after t

Sj slack time of job j

r

s the standard deviation of slack time

Xs the average slack time

CVs the coefﬁcient of variation of slack time, CVs¼r_Xs

s

4.1. Evaluation procedure

The following ﬁve equations govern the procedure of evaluating the coefﬁcient of variation of slack time:

Ci;pðjÞ¼ max

16f 62fmaxfAi;f ;t;Ci1;f ;pi1ðjÞþ ti1;ig þ pi;f ;piðjÞg ð1Þ

where t = Ci,p(j1)for 1 6 i 6 M Sj¼ dj CM;pðjÞ ð2Þ Xs¼ PJ j¼1Sj J ð3Þ

r

s¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PJ j¼1ðSj XsÞ2 J s ð4Þ CVs¼

r

s Xs ð5Þ

Eq.(1)indicates the completion time of job j in job

p

(j) at stage

i. The term presents the time epoch when machine at stage i is

ready for processing job j; and the term Ai,f,tdenotes the time epoch

when job j is ready to be processed at stage i. Eq.(2)indicates the

slack time of job

p

(j) at stage M. Eq.(3)indicates the average slack

time of a job sequence. Eq.(4)indicates the standard deviation of

slack time of a job sequence. Eq.(5)indicates the coefﬁcient of

var-iation of slack time of a job sequence.

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5. Algorithms

This study proposes four algorithms (EDD-C, EDD-S, GA-FIFO-C, and GA-FIFO-S) to solve the dual-flow shop-scheduling problem. Two proposed cross-flow algorithms are respectively called GA-EDD-C and GA-FIFO-C, while two single flow shop sched-uling algorithms are called GA-EDD-S and GA-FIFO-S—without any cross-shop routes. The single flow shop algorithm (GA-FIFO-S) serves as a benchmark to determine the effectiveness of the other algorithms. While all four algorithms follow the solution architec-ture of a typical GA and chromosome representation, they vary in their solution encoding schemes. This section first describes the solution architecture of a typical GA, and then describes the details of GA-EDD-C, GA-EDD-S, GA-FIFO-C, and GA-FIFO-S.

5.1. Solution architecture of a typical GA

A possible solution to a genetic algorithm (GA) is called a chro-mosome. A chromosome typically represents a string of integers, each of which is called a gene. The solution quality of a chromo-some is called its ﬁtness, which represents the coefﬁcient of

varia-tion of slack time in this study (Holland, 1975). A GA aims to ﬁnd a

near-optimum chromosome using the evolutionary mechanism described below.

Procedure Typical_GA_Evolution Step 1: Initialization

Set t = 0

Randomly create a set of N chromosomes to form initial

population P0

Step 2: Generate new chromosomes

Use genetic operators to create Ncnew chromosomes

randomly

Create a set S = Nc[ Pt Step 3: Update the population

Set t = t + 1

Adopt a strategy to select N chromosomes out of S to form Pt

Step 4: Termination check If (Termination = ‘‘yes’’)

then Output the best chromosome in Ptand STOP.

Else go to Step 2.

Two common terminating conditions in Step 4 of the procedure above are (1) the best solution in P(t) remains the same for more

than Tb iterations, or (2) more than Tfiterations have been

per-formed (i.e., t = Tf).

In Step 3, after completing genetic operations, the GA produces

Q = Nc+ N chromosomes in P(t) and newly generated ones, but only

N of them are selected to form P(t + 1). To make this selection, ﬁrst sort the Q chromosomes in descending order based on their ﬁtness

values. Denote the sorted chromosomes as

p

1, . . . ,

p

Q. In forming

P(t + 1), the algorithm always selects

p

1and selects the other

p

i

based on probability by applying the roulette wheel selection meth-od (Goldberg, 1989; Michalewicz, 1996).

5.2. GA-EDD-C algorithm

The GA-EDD-C algorithm represents the job sequence sorted by due date, where manufacturing operations adopt a cross-plant mechanism. This GA algorithm can be described as follows. 5.2.1. Chromosome representation and decoding

In the GA algorithm, each chromosome represents a job se-quence that must be decoded to model a dual-ﬂow shop

scheduling solution. A chromosome is denoted by

p

= [

p

1, . . . ,

p

i],

i 6 3, where

p

irepresents the job sequence at stage i. Each job

se-quence

p

iis denoted by

p

i= [

p

i(1), . . . ,

p

i(n)], where

p

irepresents the job in the jth job sequence at stage i. The length of the chromo-some equals the product of the total number of jobs and the total number of stages. For the chromosome inFig. 2,

p

= [

p

1,

p

2,

p

3],

p

2= [J5, J3, J7, . . . , J2], and

p

2(3) = 7 indicate that job J7is in the third job sequence at stage 2. The chromosome is 24 integers (3 8) long.

The chromosome

p

represents two set of sequencing decisions:

(1) sequencing among stages (route assignment, assigning jobs to stages of dual-flow shop), and (2) job sequencing within each stage of a dual-flow shop. The decoding procedure consists of the two phases described below. In the first phase, the job_allocation pro-cedure decodes the sequence of the stages of dual-flow shop. In the second phase, after the initial decoding, the due date-based decoding scheme decodes job sequencing within each stage i.

For job

p

i(j), the term pi;f ;piðjÞrepresents the processing time for a

job j at stage i in ﬂow shop f. Let Ti,fdenote the total processing

time at the stage i in ﬂow shop f for job

p

i(j). The procedure for

interpreting the job allocation decision from a given chromosome is described below.

Procedure Job_Allocation

Step 1: Form the job groups in stage n

l = M, /M is a large number⁄/

k = 1, /⁄index of job group⁄/

For i = 1 to N

Tn;1¼Pi_j¼1pi;1;pðjÞ; /⁄total processing time of the N jobs in

stage n of Plant_A⁄/

Tn;2¼PNj¼iþ1pi;2;pðjÞ; /⁄total processing time of the N jobs in

stage n of Plant_B⁄/

e

¼ Tn;1 Tn;2

; /⁄the variance between total processing

time in two plants⁄/ If (

e

< l) then;

k = i; /⁄a cut-point of job group⁄/

l =

e

; /⁄update the least variance⁄/

End if If (i = N) then

go to Step 2 /⁄check if job group formation

ﬁnished⁄/ Endif Endfor

Step 2: Output job allocation results Output C(k), 1 6 k 6 f 1

Given the job allocation decision C(k), the procedure for deter-mining the job sequence decision for each stage is relatively easy. The job sequence for stage k (1 6 k 6 f 1) is

p

ik= [

p

i(s), . . . ,

p

i(e)],

where s = C(k 1) + 1 and e = C(k), whereC(0) = 0 and C(f) =

p

(N).

Fig. 3(a) illustrates the chromosome-decoding scheme using a two-phase example. In the ﬁrst phase, the job allocation of chromosome-decoding scheme, there are eight jobs to be scheduled

5 3 7 1 6 4 8 2 Stage 1 5 3 7 1 6 4 8 2 Stage 2 5 3 7 1 6 4 8 2 Stage 3 5 3 7 1 6 4 8 2 5 3 7 1 6 4 8 2 Stage 1 5 3 7 1 6 4 8 2 5 3 7 1 6 4 8 2 Stage 2 5 3 7 1 6 4 8 2 5 3 7 1 6 4 8 2 Stage 3 Sequence of Jobs

at stage 1 Sequence of Jobs at stage 2 Sequence of Jobs at stage 3

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on two machines in Stage 2. Support that the processing times for these jobs in Plant 1 are p1,i= {1, 2, 3, 4, 5, 6, 7, 8}, and the processing time in Plant 2 are p2,i= {2, 3, 1, 5, 8, 6, 7, 4}. The accumulated

pro-cessing time in Plant 1 and 2 are Ti,1= {1, 3, 6, 10, 15, 21, 28, 36},

and Ti,2= {36, 34, 31, 30, 25, 17, 11, 4}, respectively. The variance

processing time between two plants is

e

= {35, 31, 25, 20,

10, 4, 17, 32}. The job cutoff point is located between

p

(J6) and

p

(J4). The set of jobs allocated to machine 1 and the job sequences

are

p

21= [J5, J3, J7, J1, J6], while those for the other machine are

p

22= [J4, J8, J2].

In the second phase, the due-date decode scheme, the earlier a due date appears in

p

ik, the higher is its sequencing priority. For in-stance, the job sequence at Stage 2 is J5 J3 J7 J1 J6 J4 J8 J2 and the due-dates are di= {3, 2, 4, 9, 5, 6, 7, 1}. Applying the due date-sequence decoded scheme sorts the job sequence at Stage 2 in Fig. 3(b) in due-date order, i.e.,

p

21= [J3, J5, J7, J6, J1],

p

22= [J2, J8, J4]. As a result, the post-decoding result of chromosome

p

is a due date-based job sequence. This job sequence ﬁrst imposes capacity constraints on the sequence among stages and then deter-mines a job sequence within each stage.

5.2.2. Genetic operators

We used two types of genetic operators to generate the new chromosomes. The ﬁrst genetic operator is a crossover operator, and the second is a mutation operator. A crossover operator gener-ates a new pair of chromosomes from an existing pair of chromo-somes, while a mutation operator generates a new chromosome from an existing one.

The one-crossover operators involve C1 (Reeves, 1995). The

one-mutation operators are Swap (Wang & Zheng, 2003). As stated

in Procedure Typical_GA_Evolution, each iteration generates Ncnew

chromosomes. These new chromosomes are generated as follows:

N Pc1ones through C1, N Pm1ones through Swap, where Nc= N

(Pc1+ Pm1).

The following subsection explains the mechanism of the one-crossover operators, where parent-1 and parent-2 represent the parent chromosomes and child-1 and child-2 represent the created chromosomes.

C1 operator: Fig. 4 shows that one randomly selected point

splits each parent into two sections (head and tail). To generate

an offspring (say, child-2), the head is copied from the head of parent-2—a string (3, 5, 6) in this case. The tail is determined by sequentially referring to the genes of parent-1; only the gene val-ues not in the head of child-2 appear in the tail. This yields a string (1, 4, 9, 8, 7, 2) as the tail of child-2.

The following subsection describes the one-mutation operator,

where

p

adenotes the parent chromosomes and

p

bdenotes the

child chromosomes.

Fig. 3. GA-EDD-C chromosome and decoding schemes.

Parent 1 1 6 4 9 3 5 8 7 2 Parent 2 3 5 6 1 8 2 7 9 4 x-child-1 x-child-2 Parent 1 Child-2 Child 1 Parent 2 1 6 4 3 5 6 1 6 4 9 3 5 8 7 2 3 5 6 1 4 9 8 7 2 1 6 4 3 5 8 2 7 9 3 5 6 1 8 2 7 9 4

Fig. 4. Crossover operators: C1.

3

SWAP 1 6 4 5 8 2 7 9

3

1 6 8 5 4 2 7 9

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Swap operator:Fig. 5 shows that we randomly selected two

distinct genes in

p

a, and then swapped their gene values to

gener-ate

p

b.

5.3. GA-EDD-S algorithm

The GA-EDD-S algorithm sorts the job sequence by due-date, but only a single plant can perform manufacturing operations. The GA algorithm can be described as follows. The GA-EDD-S algo-rithm adopts the same chromosome as GA-EDD-C, but adopts a dif-ferent decoding scheme. Refer to the GA-EDD-S chromosome in Fig. 6(a). In the ﬁrst phase, job allocation of chromosome-decoding scheme, the ﬁrst segment denotes that the job sequence in Stage 1 is J5?J3?J7?J1?J6?J4?J8?J2. Applying the Job_Alloca-tion procedure to split the jobs into two plants produces the decoding results of

p

11= [J5, J3, J7, J1, J6] and

p

12= [J4, J8, J2] for this chromosome. In other words, all the jobs in Stage 1 should be com-pleted before proceeding to jobs in Stage 2 or Stage 3.

In the second phase, the job sequence in three segments of the chromosome is the same because the operation of a job in each plant cannot occur in the other plant using the cross-plant mechanism.

The GA-EDD-S algorithm creates a new chromosome by apply-ing a genetic operator to each chromosome segment indepen-dently, and then joining the newly generated segments to form a new chromosome. The two genetic operators (C1 and Swap) are

similar to those in the GA-EDD-C in creating new GA-EDD-S chromosomes.

5.4. GA-FIFO-C algorithm

The GA-FIFO-C algorithm sorts the job sequence by in ﬁrst-out, and allows the cross-plant manufacturing mechanism. The chromosomes in the GA-FIFO-C algorithm are similar to those in GA-EDD-C, but vary in the second phase of the decoding procedure. Fig. 7shows that we applied the job_allocation procedure to split the job sequence into two plants at three stages. Thus, a GA-FIFO-C chromosome represents a job sequence and imposes no due-date based on the sequence among machines.

Fig. 7shows a GA-FIFO-C chromosome, which is much like the pre-coding chromosome of GA-EDD-C. This chromosome produces a scheduling solution at three stages by applying the Job_Allocation procedure. To generate new chromosomes in The GA-FIFO-C gener-ates new chromosomes by applying a genetic operator to a whole chromosome, like GA-EDD-C. The GA-FIFO-C algorithm uses the two genetic operators mentioned above.

5.5. GA-FIFO-S algorithm

The chromosomes in the GA-FIFO-S algorithm are similar to those in the GA-EDD-S algorithm, but vary in the second phase of

the decoding procedure. Fig. 8 shows that we applied the

Fig. 6. GA-EDD-S chromosome and decoding schemes.

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job_allocation procedure to split the job sequence into two plants at Stage 1. In other words, a GA-FIFO-S chromosome directly repre-sents a job sequence and imposes no due-date based on the se-quence among machines.

Fig. 8(a) shows a chromosome in GA-FIFO-S algorithm, which is

similar to the pre-coding chromosome of GA-EDD-S.Fig. 8shows

that the chromosome at Stage 1 represents a scheduling solution by applying the Job_Allocation procedure. The job sequences in Stage 2 and Stage 3 then follow the job sequence in Stage 1. The GA-FIFO-S algorithm generates new chromosomes by applying a genetic operator to the whole chromosome, much like the GA-EDD-S algorithm, and uses the two genetic operators men-tioned above.

6. Numerical experiments

Numerical experiments were conducted to compare the four algorithms (GA-EDD-C, GA-EDD-S, GA-FIFO-C, GA-FIFO-S). Personal computers with PENTIUM Dual-Core 2.8 GHz CPU and 1 Gb mem-ory were used to run the programs, which were coded in Visual C++. The parameters of the four genetic algorithms were set as fol-lows: N = 100, Pc= 0.8, Pm= 0.2, Tb= 1000, Tf= 100,000.

6.1. Experiment design

Table 1shows that the dual-ﬂow shop considered in this study had two plants, six stages, and six machines. The operation time of

each machine (stage) i is a uniform distribution [ai, bi]. The due date of each job j is a uniform distribution [ai, bi] shown asTable 2.

We used (P, N, T) to represent a test case, where P represents the proportion of processing time at the same stage of plant A to that of plant B, N represents the number of jobs, T represents the ratio of

transportation time to processing time.Table 2shows that P has

four options (1:1, 1:1.5, 1:2, 1:3), N has ﬁve options (20, 40, 60, 80, 100 jobs), and T has three options (0.1:1, 0.5:1, 2:1). Thus, each algorithm had 4 5 3 = 60 test cases and each test case was ob-tained by averaging the experimental results of 15 replicates.

The average coefﬁcients of variation of slack time for the four

algorithms were deﬁned as CVGAEDDC, CVGAEDDS, CVGAFIFOC,

and CVGAFIFOS. Experimental results indicate that the GA-EDD-C

algorithm outperformed the other algorithms in most cases. A per-formance metric illustrates the degree of variation in solution

qual-ity,

c

x= (CVx CVGAEDDC)/CVx, to compare the solution quality

between the GA-EDD-C and a benchmark algorithm (say, x). A

po-sitive

c

xindicates that GA-EDD-C is better, while a negative value

indicates that GA-EDD-C is worse.

6.2. Experimental results

Table 3compares the solution quality of the four algorithms by averaging the experimental results obtained under the four pro-cessing time ratios and three transportation time ratio options mentioned above. This table indicates that the proposed dual-flow shop algorithm (GA-EDD-C) outperforms the three other algo-rithms (GA-EDD-S, GA-FIFO-C and GA-FIFO-S) in most cases. This is because GA-EDD-C adopts the cross-flow shop and due-date based scheduling approach to reduce the total setup number. This in turn reduces the total setup time and alleviates the effects of machine capacity loss. This finding advocates the use of the cross flow shop-based approach to solve the dual-flow shop-scheduling problem.

Table 3 shows the experimental results of

c

x. The GA-EDD-C

algorithm outperforms the other three algorithms in terms of

c

x

in most cases. Of the 60 test cases,

c

x ranged from 0% to 65%.

Fig. 9shows the average of

c

xfor each algorithm, indicating that the GA-EDD-C algorithm outperforms the other algorithms.

Fig. 10shows that a lower P (process time ratio between two

plants) increases the average of

c

x. When P reached 1:3, the

aver-age of

c

xreached 55%. This is due to the large gap in process time

Fig. 8. GA-FIFO-S chromosome and decoding schemes.

Table 1

Process times of three stages in a dual-ﬂow shop.

Plant Stage 1 Stage 2 Stage 3

A U[1, 5] U[1, 5] U[1, 5]

B U[1, 5] U[1, 5] U[1, 5]

Table 2

Due date of each job under the different number of jobs.

20 jobs 40 jobs 60 jobs 80 jobs 100 jobs Due

date

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between two plants, which beneﬁts the cross-ﬂow shop mechanism.

Fig. 11shows the average of

c

xfor different transportation time ratios for the 60 test cases, indicating that the ratio of

transporta-tion time to processing time is not obvious to

c

x. In terms of

com-putation time, Fig. 12 shows that all four algorithms are quite

efﬁcient, requiring less than eight minutes for each test case.

7. Concluding remarks

This study examines a dual-ﬂow shop-scheduling problem by comprehensively considering the processing time and transporta-tion time features. We propose four algorithms ( EDD-C,

GA-EDD-S, GA-FIFO-C, and GA-FIFO-S) to solve the dual-ﬂow

shop-scheduling problem. The GA-EDD-C algorithm serves as a

Table 3

Solution qualities of the four algorithms under different J, P, and T.

Jobs P T (%) CVGAEDDC CVGAEDDS rGAEDDS(%) CVGAFIFOC rGAFIFOC(%) CVGAFIFOS rGAFIFOS(%)

20 1:1 1 0.112 0.135 17 0.517 78 0.111 1 50 0.108 0.144 25 0.503 79 0.111 3 200 0.134 0.124 8 0.679 80 0.123 9 1:1.5 1 0.112 0.144 23 0.395 72 0.125 11 50 0.123 0.142 13 0.523 76 0.128 4 200 0.156 0.126 24 0.726 79 0.132 19 1:2 1 0.140 0.170 18 0.389 64 0.156 10 50 0.143 0.172 17 0.562 75 0.149 4 200 0.144 0.165 13 0.612 77 0.150 4 1:3 1 0.177 0.348 49 0.653 73 0.218 19 50 0.182 0.353 49 0.627 71 0.221 18 200 0.201 0.467 57 0.709 72 0.222 9 40 1:1 1 0.098 0.107 9 0.249 61 0.097 1 50 0.097 0.109 10 0.234 58 0.097 1 200 0.097 0.114 15 0.272 64 0.097 1 1:1.5 1 0.117 0.144 18 0.132 11 0.121 3 50 0.116 0.138 16 0.129 10 0.121 4 200 0.113 0.143 21 0.159 29 0.121 7 1:2 1 0.137 0.215 36 0.194 29 0.147 7 50 0.140 0.228 39 0.195 28 0.147 5 200 0.139 0.243 43 0.173 20 0.147 5 1:3 1 0.157 0.516 69 0.346 54 0.695 77 50 0.160 0.546 71 0.417 62 0.547 71 200 0.179 0.575 69 0.568 68 0.308 42 60 1:1 1 0.089 0.101 11 0.157 43 0.090 1 50 0.089 0.100 11 0.168 47 0.090 2/0 200 0.087 0.101 13 0.164 47 0.090 3 1:1.5 1 0.112 0.135 17 0.135 17 0.117 4 50 0.112 0.134 16 0.133 15 0.117 4 200 0.109 0.150 27 0.136 19 0.117 6 1:2 1 0.128 0.213 40 0.176 27 0.137 6 50 0.132 0.190 31 0.176 25 0.137 3 200 0.130 0.206 37 0.176 26 0.137 5 1:3 1 0.149 0.432 66 0.205 27 0.640 77 50 0.150 0.450 67 0.207 28 0.659 77 200 0.157 0.458 66 0.212 26 0.664 76 80 1:1 1 0.094 0.103 9 0.163 42 0.089 5 50 0.094 0.104 10 0.165 43 0.089 5 200 0.093 0.103 10 0.196 53 0.089 4 1:1.5 1 0.111 0.129 14 0.117 6 0.108 2 50 0.111 0.133 17 0.130 15 0.108 2 200 0.108 0.148 27 0.118 9 0.108 1 1:2 1 0.126 0.196 36 0.147 14 0.125 1 50 0.129 0.190 32 0.163 21 0.125 3 200 0.128 0.183 30 0.147 13 0.125 2 1:3 1 0.136 0.413 67 0.191 29 0.581 77 50 0.137 0.390 65 0.202 32 0.569 76 200 0.140 0.366 62 0.195 28 0.605 77 100 1:1 1 0.094 0.103 9 0.159 41 0.090 4 50 0.094 0.103 9 0.206 54 0.090 4 200 0.093 0.103 10 0.227 59 0.090 3 1:1.5 1 0.114 0.142 19 0.126 9 0.113 1 50 0.114 0.148 23 0.126 9 0.113 1 200 0.112 0.143 22 0.126 11 0.113 1 1:2 1 0.134 0.214 37 0.231 42 0.132 2 50 0.134 0.212 36 0.200 33 0.131 2 200 0.131 0.206 36 0.191 32 0.132 0 1:3 1 0.143 0.370 61 0.196 27 0.555 74 50 0.146 0.380 62 0.197 26 0.571 74 200 0.158 0.377 58 0.200 21 0.570 72

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Fig. 9. Average ofcxfor various algorithms.

Fig. 10. Average ofcxat various processing time ratios between two plants.

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benchmark. Numerical experiments indicate that the proposed GA-EDD-C algorithm outperforms the other algorithms. These results imply that a dual-ﬂow shop approach should be used to solve the scheduling problem.

The experiments in this study indicate that the GA-EDD-C algo-rithm outperforms other algoalgo-rithms when there is a large gap in processing times between two plants. This performance difference may be due to by the advantages of the cross-ﬂow shop.

An extension to this study is the scheduling of three or more ﬂow shops. Such an extension would require another decision-making criterion: how to allocate jobs to each stage among different ﬂow shops.

Acknowledgements

This research was sponsored by National Science Council, Tai-wan under a research contract NSC99-2221-E-009-110-MY3. References

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