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Performance analysis of re-entrant flow shop with single-job and batch machines using mean value analysis

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Keywords re-entrantow shop, single-job machine, batch machine, mean value analysis. Abstract. We propose an approximate method based on the mean value analysis for estimating the average performance of re-entrantow shop with single-job machines and batch machines. The main focus is on the steady-state averages of the cycle time and the throughput of the system. Characteristics of the re-entrantow and inclusion of the batch machines com- plicate the exact analysis of the system. Thus, we propose an approximate analytic method for obtaining the mean waiting time at each buÄ er of the workstation and a heuristic method to improve the result of the analytic method. We compare the
PRODUCTION PLANNING & CONTROL, 2000, VOL. 11, NO. 6, 537±546
Performance analysis of re-entrant ¯ ow shop with
single-job and batch machines using mean value
analysis
YOUNGSHIN PARK, SOOYOUNG KIM and CHI-HYUCK JUN
Keyw ords re-entrant ¯ ow shop, single-job machine, batch
machine, mean value analysis.
Abstract. We propose an approximate method based on the
mean value analysis for estimating the average performance of
re-entrant ¯ ow shop with single-job machines and batch
machines. The main focus is on the steady-state averages of
the cycle time and the throughput of the system. Characteristics
of the re-entrant ¯ ow and inclusion of the batch machines com-
plicate the exact analysis of the system. Thus, we propose an
approximate analytic method for obtaining the mean waiting
time at each bu er of the workstation and a heuristic method to
improve the result of the analytic method. We compare the
Authors: Y. Park, S. Kim and C. -H. Jun, Departm ent of I ndustrial Engineering, Pohang University
of Science and Technology, San 31 Hyoja-dong, Pohang, 790-784, South Korea, e-mail: sookim@
postech.ac.kr
YOUNGSHIN PAR K is a PhD candidate in Industrial Engineering at Pohang University of Science
and Technology in Korea. She has received Bachelor and Master of Science in Industrial
Engineering from Pohang University of Science and Technology. Her current research activities
focus on performance analysis of production systems.
SOOYOUNG KIM is Associate Professor of the Department of Industrial Engineering at Pohang
University of Science and Technology (`POSTECH’ ) in Korea. He received his BS in Mechanical
Engineering from Seoul National University, MS in Manufacturing Engineering from Korea
Advanced Institute of Science and Technology, and his PhD in IE from University of California
at Berkeley, USA in 1988. From 1989 to 1993, he was Assistant Professor of Industrial Engineering
at Rutgers University in New Jersey, USA. His research interests are in production planning,
scheduling and control, especially in semiconductor manufacturing processes. He has published
in II E T rans acti ons, I nter nation al J our nal o f P rod uction R ese arch ,Euro pean J o urnal of OR ,Interna tional
J our nal of C IM , etc.
CHI-HY UCK JUN is Associate Professor in Industrial Enginering at Pohang University of Science
and Technology. He received a BS in Mineral and Petroleum Engineering from Seoul National
University in 1977, MS in Industrial Engineering from Korea Advanced Institute of Science and
Technology in 1979, and PhD in Operations R esearch from University of California, Berkeley, in
1986. He is interested in performance analysis of communication systems and production systems.
He has published in P rob abili ty in the Eng ineering and Informational Sciences,M icro elect ron ics &
Reliability,I EICE T ra nsacti ons on C ommu nicat ions, etc.
Pr oduction P lann ing & Control ISSN 0953±7287 print/ISSN 1366±5871 online #2000 T aylor & Francis Ltd
http://www.tandf.co.uk/journals
results of the proposed approach with a simulation study using
some numerical examples.
1. Introduction
In this paper, we propose an approach to analyse the
steady-state performance of the re-entrant ¯ ow shop with
batch machines using the mean value analysis ( MVA).
The re-entrant line here is a ¯ ow shop in which parts visit
some machines more than once at di erent stages of pro-
cessing ( Kumar 1993) . A w ell-known example of the re-
entrant manufacturing systems is the semiconductor
manufacturing line, especially the wafer fabrication pro-
cess ( `fab’ ) . The fab process is also characterized by the
usage of the batch machines. For example, some
machines at the deposition process and the furnaces accu-
mulate jobs in the queue and then process jobs as a batch
of predetermined size.
Chaudhry and Templeton (1983) summarized the
general theory concerning batch arrival and batch ser-
vice queues, and Bailey (1954) , Neuts ( 1967) , and Powell
(1986) studied several rules and control policies of batch
processing. Connors et al. ( 1996) developed an open
queueing network model for a rapid performance analy-
sis of semiconductor manufacturing systems considering
the variability of lot sizes and compared the results to the
simulation study. It is di cult to analyse exactly the
system that consists of both single-job and batch
machines under re-entrant ¯ ow of jobs. Such systems
cannot be modelled by a product-form queueing net-
work, and therefore they are mostly analysed using
simulation. However, simulation of a large-scale semicon-
ductor fab often requires too much computing time. In
our experience during simulation of the fabs, it could take
a 1-year simulation run just to reach the steady-state. In
lieu of the simulation, we present the MVA approach to
analyse the system based on the work of Narahari and
Khan ( 1996) where they only consider single-job
machines.
This paper is organized as follows. In section 2, the
system under consideration is described. Our proposed
approximate method is explained in section 3. In section
3.1, we propose the method of obtaining the average
waiting time of jobs at each bu er of the workstation
having batch machines, which is the major contribution
of this paper. Section 3.2 presents the method of comput-
ing the mean waiting time in the system. T o solve a
problem raised in the analytic method at the single-job
machine, we present the `cuto point’ and smoothing
technique in section 3.3. For further compensation of
the underestimated cycle time from the MVA, we pro-
pose some adjustment heuristics in section 3.4. In section
4, we present some numerical tests obtained by compar-
ing the analytical results and the simulation experiments
for the sample systems.
2. Re-entrant  ow shops
The system that we consider has several workstations,
where each workstation consists of a single-job machine
or batch machine. Single-job machines process jobs one
at a time, and batch machines process jobs in a batch of
speci® ed size. We do not consider a workstation that has
both a single-job machine and batch machine. A single
type of jobs is assumed to be ¯ owing through a ® xed
routing. Jobs may visit the same workstation more than
once at di erent stages of processing. Each workstation
has several queues, each representing a di erent stage of
the process, for incoming job ¯ ow a separate bu er is
assumed for each queue. Figures 1 and 2 are examples
of our system, which will be used later for numerical
experiments in section 4. Figure 1 is a system that has
10 workstations and 30 bu ers. A job enters workstation
1, ¯ ows in a predetermined route as depicted by the arcs,
and ® nally exits from workstation 10. Figure 2 depicts
the system considered by Lu et al . (1994) as a model of a
full-scale semiconductor fab, which consists of 12 work-
stations and a total of 60 bu ers.
3. Mean value analysis
Dispatching or scheduling policies to decide which job
to process next when a machine becomes available are
important factors that have a signi® cant e ect on the
performance of re-entrant lines. Lu and Kumar ( 1991)
and Kumar ( 1993) investigated a few scheduling policies
based on bu er priorities and due dates. In our
approach, we assume the last bu er ® rst serve ( LBFS)
policy, and propose an approximate method based on
MVA to ® nd the mean cycle time and the throughput
of the system described above.
The original MVA is an iterative method based on the
following tw o princ iples ( Reiser and Lavenberg 1980) .
(1) Arrival theorem. The queue length distribution
observed by a job upon its arrival to a given
station is the same as the overall distribution
seen by an outside observer when one less job is
in the system.
(2) Little’s law can be used to relate the mean queue
length with the arrival rate and the mean waiting
time.
These principles hold when the system can be described
as a closed and product-form queueing network. The
538 Y. Park et al.
closed queueing network is a system in which the jobs
neither enter nor depart from the network and the num-
ber of jobs circulating among the workstations is ® xed.
This is not exactly the case in our system, but the system
can be assumed to be a closed network when it is in
steady-state and so the MVA is adopted as an approx-
imation methodology.
Using these principles, the performance measures of
the system when there are k1 jobs can be evaluated
by the value of those when there are kjobs. The perform-
ance measures to be considered are the mean cycle time
and the mean steady-state throughput rate for a given
® xed WIP ( work in process) in the system. T o clarify the
terms, we use the cycle time for representing the total
time spent for a job from the beginning of the process
to the end. For the time taken by a job from entering a
bu er at a workstation until it leaves the station, we use
the term `waiting time’. First, we compute the mean
waiting time at each bu er and then compute the
mean cycle time in the system. Using the mean cycle
time in the system, the throughput rate of the system
and the queue length at each bu er are obtained.
3.1. M e an wa iti ng ti me at ea ch b uåer
Narahari and Khan ( 1996) studied re-entrant lines
with workstations consisting of single-job machines
only. They considered the system to be closed and
applied the arrival theorem to obtain the mean waiting
times at bu ers. Further, they classi® ed the bu ers into
`internal’ bu ers ( ones connected from station ito itself)
and `external’ bu ers (ones connected from another
station to station i). We extend their methodology to
the case including the workstations of batch machines.
Here we consider external bu ers only and a workstation
that has only one batch machine. It is assumed that the
batch size for workstation iis xi(1). A batch is to be
composed of jobs from a single bu er, i.e. no mix allowed
M ea n va lue anal ysis 539
or s a ons an
Work Statio n 1
Work Statio n 2
Work Sta tion 3
Work Sta tion 4
Work S tation 5
Work S tation 6
Work Statio n 7
Work Statio n 8
Work Statio n 9
Work Statio n 10
Figure 1. Example system with 10 workstations and 30 bu ers.
ple system with 12 work stations and 60 buffer
Work Station 2 3.5
Work Station 6 0.5
Work S tation 1
Work S tation 2
Work Statio n 3 Work
Station 4
Work S tation 8
Work S tation 9
Work S tation 10
Work Statio n 6
Work S tation 11
Work Station 5
Work S tation 7
Work S tation 12
Figure 2. Example system with 12 workstations and 60 bu ers
(Lu et al. 1994) .
from several bu ers. Note that the case of xiˆ1 reduces
to the single-job machine. T he notation used in this
paper is as follows.
mnumber of workstations
iindex for workstation, i2f
1;2;. . . ;mg
ninumber of bu ers in workstation i
xibatch size of workstation i
bij jth bu er of workstation i;j2f1;2;. . . ;nig
pij processing time of one batch/unit in bu er bij
Nmaximum number of jobs in the system to be
considered
knumber of jobs in the system, kˆ1;... ;N
Lij kmean steady-state number of jobs in bij when
the system has kjobs
Wij kmean waiting time of a job in bij including the
processing time when the system has kjobs
Wkmean cycle time in the entire system when the
system has kjobs
kmean steady-state throughput ( or arrival) rate
of jobs when the system has kjobs
rspan of moving average
When a job (referred to as a distinguished job) arrives
at bu er bij , it sees a certain number of jobs ahead in
several bu ers. T he ordered set of the integers represent-
ing these jobs, S, denotes the state of the system at the
arrival instant of the job. Sincludes all the jobs that are
ahead of the distinguished job in bij and jobs in the other
bu ers having higher priority than bij . The distinguished
job must ® rst wait until all jobs in Sare serviced and
leave the station i. Also, it must wait for the service com-
pletion of those jobs that arrive in higher-priority bu ers
during its waiting time in bu er bij . Finally, it gets pro-
cessed and enters the next bu er. T he total waiting time
observed by the distinguished job is considered as a mean
value under the assumption of steady-state of the system.
Hence, the mean total waiting time of a job at bu er bij is
obtained by the sum of three components, W1
ij ,W2
ij and
W3
ij , de® ned as follows.
W1
ij mean total time elapsed until all jobs in Sare
serviced and leave station i
W2
ij mean total time required to process all the jobs
which arrive at the higher-priority bu er dur-
ing the stay of the distinguished job in the queue
at bij
W3
ij mean processing time of the distinguished job or
the batch including it
In the case of batch machines, the distinguished job
can be processed ahead of some higher-priority jobs if the
latter cannot be formed in a batch as the number of the
current jobs is less than xiin the bu er.
In bu er bij , an arriving job, according to the arrival
theorem, would see Lit k¡1jobs in the bu ers bit ,
tˆ1;2;. . . ;ni. Because the LBFS scheduling policy is
being used, the arriving job needs only to wait for the
processing of jobs ahead of it in bu ers bit,tˆj,
j1;. . . ;ni. Note that the number of batches to be pro-
cessed among the jobs in bit is expressed by Li tk¡1
xi
,
where bxcis the largest integer smaller than x. Therefore
W1
ij ˆX
ni
tˆj
Lit k¡1
xi
¢pit 1
W2
ij is the mean time required to process all the batches
in the higher-priority bu ers made by the remaining jobs
plus jobs to arrive while the distinguished job waits. The
number of remaining jobs that are not formed in bit at the
arrival instance of the distinguished job is
Rit ²Lit k¡1¡Litk¡1
xi
¢xi
To compute the expected number of jobs to arrive
during the wait of the distinguished job, ® rst consider a
job staying in bu er bi j . The mean delay of a job in bu er
bij , excluding processing time, is Wij k¡pij . According
to the arrival theorem, k¡1can be taken as the rate
at which jobs are ¯ owing in the system and therefore the
expected number of jobs to arrive into bit during the stay
in the bu er is given as follows.
Mit ²Wij k¡pijk¡1
Then, W2
ij is
W2
ij ˆX
ni
tˆj1
Mit Rit
xi
¢pit 2
For the distinguished job to be processed, it should be
included in a batch. It may be included in a batch being
formed with jobs in front of it, or wait until the next
batch to be formed. Hence,
W3
ij ˆmax 0;xi¡Mij Rij 1††Š
k¡1pij 3
It can be noticed that the arrival rate is involved in
calculating W3
ij . Because the initial values (when the
system is empty) for the arrival rate and the queue length
are all zeros, we cannot calculate W3
ij by an iterative
method. Further, W3
ij becomes unreasonably large
when the arrival rate and kare small. This is due to
the fact that the waiting time for forming a batch can
even be in® nite if kis less than the minimum batch size.
Of course, such a case cannot actually occur, but to apply
the iterative calculations it is necessary. Thus, we propose
a heuristic method of using BIG M instead of using equa-
tion (3) for calculating W3
ij in this case. We suggest using
the following value of BIG M for W3
ij if the arrival rate of
system is less than 1/BIG M:
540 Y. Park et al.
BIG M ˆ10 £X
m
iˆ1X
ni
jˆ1
pij 4
The total mean waiting time in bu er bij for batch
machines is given by the sum of the three terms above.
Wij kˆW1
ij W2
ij W3
ij 5
where W3
ij should be calculated by either equation (3) or
BIG M in equation (4) .
For the single-job machine case, i.e. for xiˆ1, equa-
tions (1) ±( 3) are reduced to the following, as appeared in
Narahari and Khan ( 1996) .
W1
ij ˆX
ni
tˆj
Litk¡1¢pit
W2
ij ˆWij k¡pijk¡1X
ni
tˆj1
pit
W3
ij ˆpij
3.2. M ean cycle time in system and throug hput
The mean cycle time in the entire system is the sum of
the waiting times in all the bu ers. Therefore,
WkˆX
m
iˆ1X
ni
jˆ1
Wij k† …6
If we compute this mean cycle time in the system, we
obtain the arrival rate or equivalently the throughput
rate of the system and also the mean queue length of
each bu er by applying L ittle’s law ( Little 1961 ) , e. g.
kˆk
Wk7
Lij kˆkWijk† …8
The initial conditions are as follows:
Lij 0ˆ0;jˆ1;...;ni;iˆ1;. . . ;m
0ˆ0
Using the initial conditions above and the recurrent
relationships of Wijk,k,Lijk, we can compute the
mean cycle time in the system, the average queue length
of the bu er and the throughput rate for kˆ1;2;...;N,
where Ndenotes the speci® ed total WIP of the system.
3.3. M odi cation of mean cycle time by moving-average
smoothing
As mentioned earlier, the MVA is employed as an
approximation technique to compute a non-product-
form network in this study. T herefore, some errors are
inevitable in computing the mean waiting times, and the
use of recursions for the evaluation of the performance
measures may increase the error. Especially, one can
expect some major errors in computing the mean waiting
times in the bu ers when the number of bu ers is large or
the processing times of the jobs are long, and the popula-
tion of the system is large in the case of the single-job
machine. For example, equation ( 2) can generate nega-
tive values when xiis 1. In order to solve such problems,
we propose an adjustment of the MVA calculations using
a moving-average smoothing technique.
With the moving span size of r, we suggest revising
Wkwith the following:
new W kˆ1
r¢X
k
lˆk¡r1
Wl† …9
If the system has batch machines and the BIG M is
employed, i.e. for small k, the total mean waiting time
by equation ( 9) might be unreasonably overestimated.
Therefore it is more reasonable to ignore these values
for relatively small kuntil some `cuto point’ for kis
reached, and to take the values after this cuto point,
K. Because it is in some sense similar to the `warm-up’
period in simulation, it may be di cult to determine the
generic form of the `good’ formula for obtaining this cut-
o value. Thus, we later present one example of it in our
computational tests.
3.4. Further correction of mean cycle time
The mean cycle time is obtained by adding up the
mean waiting time in each bu er. However, we need to
consider a sort of correction to account for the `batching’
e ect for the waiting times not explicitly accounted for in
the above calculations. If a batch of jobs arrive at a
single-job machine just after the completion of service
at a batch machine, the waiting time of the individual
job from the batch must be calculated di erently from
that of a job arriving from a single-job machine. T his is
because a distinguished job out of a batch sees not only
the jobs already in the bu er but also jobs brought
together with it in the same batch. The above calcula-
tions do not count such jobs ahead of it. Also, if a batch of
jobs arrive at a batch machine after being processed at
another batch machine that may have a di erent batch
size, there is some additional waiting time that is not
M ea n va lue anal ysis 541
represented by the three terms de® ned above. We thus
propose a correction term to re¯ ect this missing waiting
time by adding it to Wde® ned as:
CO ˆ0 if P<P0
WK¢Potherwise
(10
where Pis the proportion of batch workstation and P0is
the minimum value of Pthat is required to add CO.
Thus, CO is added to the cycle time smoothed only if
the portion of the batch machines exceeds the certain
amount. From our computational experience, it was
noted that adding CO was not necessary when the
batch stations are not signi® cant in the system. Our
experience with some numerical experiments suggests
that P0ˆ0:2 would be a suitable choice.
The mean cycle time and the ® nal throughput rate of
the system with the WIP of Nare as follows:
Mean cycle time ˆne w W NCO 11
Throughput rate ˆN
mean cycle time 12
We propose the following steps to calculate the mean
cycle time and the throughput rate.
Given m;N;r;ni;pij ;xi;jˆ1;. . . ;ni;iˆ1;... ;m
Step 1. Initialize variables.
Lij 0ˆ0;Wij 0ˆ0;jˆ1;...;ni;iˆ1;. . . ;m
0ˆ0
Determine the cuto point Kand BIG M.
Step 2. Increase kby 1.
Step 3. Compute Wij kusing equation (5) and compute
Wkusing equation (6).
Step 4. If kis larger than K, modify total waiting time
using equation (9).
Step 5. Compute kand Lijkusing equations (7) and
(8) .
Step 6. If kis N, go to step 7, else go to step 2.
Step 7. Find the ® nal mean cycle time using equation
(11) .
Step 8. Find the ® nal throughput rate using equation
(12) .
4. Numerical experiments
To see how close the proposed approach can predict
the performance as compared to the actual performance
of the re-entrant lines, two example systems in ® gures 1
and 2 were analysed. Example system 1 has 10 work-
stations, each workstation’s processing time per job is
shown in table 1, while the example system 2 has 12
workstations with processing times given in table 3.
Transfer times between workstations are not considered.
For the example system 1 in ® gure 1, ® ve cases (from case
1-1 to case 1-5) are considered. Table 2 shows the work-
station numbers with batch machine and their batch sizes
in parentheses by cases. We consider two cases (case 2-1,
case 2-2) for the example system 2 in ® gure 2. Table 4
indicates the workstation numbers with batch machine
542 Y. Park et al.
Table 1. Processing time of example system 1.
Processing time
Workstation 1 0.5
Workstation 2 3.5
Workstation 3 1.2
Workstation 4 3
Workstation 5 0.8
Workstation 6 0.5
Workstation 7 1
Workstation 8 1.7
Workstation 9 0.3
Workstation 10 0.9
Table 2. Workstation having batch machine in
example system 1.
Workstation # having
Case batch machine (batch size)
Case 1-1 2(5)
Case 1-2 2(5) , 6(3)
Case 1-3 2(5) , 6(3) , 7( 5)
Case 1-4 2(5) , 3(4) , 6( 3) , 7( 5)
Case 1-5 2(5) , 3( 4), 6(3) , 7( 5), 9( 7)
Table 3. Processing time of example system 2.
Processing time
Workstation 1 0.125
Workstation 2 0.125
Workstation 3 0.25
Workstation 4 1.8
Workstation 5 0.9
Workstation 6 0.6
Workstation 7 1.8
Workstation 8 0.2
Workstation 9 0.6
Workstation 10 1.665
Workstation 11 0.6
Workstation 12 1.25
Table 4. Workstation having batch machine in
example system 2.
Workstation # having
Case batch machine (batch size)
Case 2-1 2( 3) , 10( 5)
Case 2-2 2(3) , 10( 5), 11( 4)
and their batch sizes for these cases. Batch processing
times are obtained by multiplication of batch size and
the processing time per job. For example, the batch pro-
cessing time of the workstation 2 in case 1-1 is
5£3:5ˆ17:5. For the moving span size, rˆ5 is used
in both examples.
From our experimental results, we used the following
cuto point:
Kˆtotal number of buffers in batch
machine stations r£2
Our computational tests indicated that the number of
BIG Ms applied ( i.e. the number of batch station bu ers)
and the moving-average span a ect the selection of K.
The tests also suggest that any su ciently large Kwould
do unless it is too close to N.
We used a commercial software package named `Simul
8’ for simulation, and obtained results after 10 000 warm-
up periods and 30 000 periods for collecting results. We
compared the results when the system has various WIP
levels, Nˆ30;35 ;40 ;... ;200. When the proposed ap-
proach is used to obtain performance measures, we only
need to consider the case of Nˆ200 as the results for the
other cases (for N<200) would be obtained from the
case of Nˆ200 during the recursion. However, simula-
tion must be performed separately whenever the target
WIP is changed.
As an overall performance index for the comparison
between the MVA and simulation, we use the following
absolute relative error ( ARE ) in percentage :
ARE ˆ
jCycle time from MVA)
¡Cycle time from Simulationj
Cycle time from Simulation £100%
We ® rst compute the ARE for every case of N considered
and then take the average of them.
4.1. Results f or example system 1
For each case, we present the results of the cycle times
and the throughput in pairs. Figures 3 and 4 show the
cycle time and the throughput for case 1-1, respectively,
® gures 5 and 6 for case 1-3, and ® gures 7 and 8 for case
1-5.
Figure 3 shows that the cycle time obtained from the
proposed approach is almost identical to that from simu-
lation. Our simulation starts when the number of jobs is
30 as Kof case 1-1 is 55£2ˆ20. The throughput
rate is shown to be in almost steady-state from the start as
shown in ® gure 4.
Figure 5 shows the cycle time of case 1-3. When the
number of jobs is larger than 50, the results of the pro-
posed approach are congruous to those of simulation.
When the number of jobs is small, the performance
results of simulation are irregular, which is partly because
a batch has a di erent batch size in each workstation.
But, the cycle time of the proposed approach is constantly
increasing after cuto point K,5335£2ˆ
32. As shown in ® gure 6, the throughput rate of simula-
tion reaches steady-state slower than those of the pro-
posed approach because of the irregularity.
Comparison in case 1-5 starts when the number of jobs
is 45. In this case, the shape of the graph in ® gure 7 is
M ea n va lue anal ysis 543
Figure 3. Cycle time of case 1-1
0
200
400
600
800
1000
1200
1400
30 45 60 75 90 105 120 135 150 165 180 195
No. of Jobs
Cycle time
Simul.
MV A
Figure 3. Cycle time of case 1-1.
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
30 45 60 75 90 105 120 135 150 165 180 195
No. of jobs
Throughput
Simul.
MVA
Figure 4. Throughput rate of case 1-1.
Figure 5. Cycle ti me of case 1-3
0
200
400
600
800
1000
1200
1400
35 50 65 80 95 110 125 140 155 170 185 200
No. of Jobs
Cycle time
Simul.
MV A
Figure 5. Cycle time of case 1-3.
similar to the case of 1-1. The number of jobs being 45
may be large enough to make batch in each batch pro-
cessing workstation in case 1-5. The throughput rate of
case 1-5 does not seem to reach steady-state as shown in
® gure 8. T able 5 shows the comparison results of cycle
time as a per cent. The throughput results are identical to
the cycle time case.
The error in case 1-3 is largest due to irregularity of
simulation in which the number of jobs is small. But all
the errors are ¹1% showing that our proposed approach
is very e cient in the sense of time and e ort.
4.2. Results f or e xample system 2
For the system in ® gure 2, we tested the cases for two
batch machines (case 2-1) and three batch machines
( case 2-2) . T his system was considered by Lu et al.
(1994) as a model of a semiconductor fab and consists
of 12 workstations and a total of 60 bu ers. Figures 9 and
10 show the results of cycle time and throughput rate of
case 2-1, respectively.
From ® gure 9 it can be seen that there are some di er-
ences between the MVA and simulation when the popu-
lation of the system is small. Due to these di erences, the
average of ARE from Nˆ45 is 2.5% which is larger
than the results of the ® rst system. Considering the per-
formances only over the populations of 80, the average of
ARE is 0.6%. It is not so surprising to observe that for
the system with many machines and bu ers, the results of
the analytic technique show some big di erence from the
actual results. Especially for the small WIP case, as the
system involves more machines and bu ers, the error
from the MVA approximation becomes bigger. Figures
11 and 12 show the results of case 2-2 and the average of
ARE is 1.5% if it is considered only over the population
544 Y. Park et al.
Figure 8. Thr oughput rate of case 1-5
le 5. Overall performance of each case for the example system
Case Error of cycle ti me
in percentage
Case 1-1 0.76%
0
0.02
0.04
0.0
6
0.08
0.1
0.12
0.14
0.16
0.18
0.2
35 50 65 80 95 110 125 140 155 170 185 200
No. of Jobs
Throughput
Simul.
MVA
Figure 6. Throughput rate of case 1-3.
Figure 7. Cycle time of case 1-5
0
200
400
600
800
1000
1200
1400
1600
45 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195
No. of Jobs
Cycle time
Simul.
MV A
Figure 7. Cycle time of case 1-5.
Figure 8. Throughput rate of case 1-5
Table 5. Overall performance of each case for the example syst 1
Case Error of cycle time
in percentage
Case 1-1 0.76%
Case 1-2 0.94%
Case 1-3 1.47%
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
45 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195
No. of Jobs
Throughput
Simul.
MV A
Figure 8. Throughput rate of case 1-5.
Table 5. Overall performance of each case for the
example system 1.
Error of cycle time
Case as a per cent
Case 1-1 0.76
Case 1-2 0.94
Case 1-3 1.47
Case 1-4 0.79
Case 1-5 0.86
Figure 9. Cycle time of case 2-1
0
50
100
150
200
250
300
350
400
45 55 65 75 85 95 105 115 125 135 145 155 165 175 185 195
No. of Jobs
Cycle time
Simul.
MV A
Figure 9. Cycle time of case 2-1.
of 80. T able 6 shows the comparison results of cycle time
as a per cent.
As a result, we can conclude that for a large system the
proposed approach must be applied with some care for
the relatively small WI P situation. However, such a case
may rarely be encountered, as for a real-world memory
fab the WIP population often goes up to thousands to
tens of thousands. For such a case, our approach may
take some considerable calculation time due to its itera-
tive nature, but still it is a much faster and convenient
alternative to the simulation.
5. Conclusions
We proposed a method based on the MVA approxima-
tion and some heuristic adjustments for estimating the
performance of re-entrant lines with batch machines and
single-job machines. This method has been found to be
e cient and quite accurate through experiments in section
4, even though it shows some errors in the case of small
population of WIP. Overall results represented by the
average of error for the system with 10 workstations are
¹1% or less. These results are thought to be very close to
the exact values and this technique is little a ected by the
proportion of the batch stations. For the more complex
model of a semiconductor manufacturing system, the
averages of di erences are 0.6%, 1.5% when the propor-
tion of the batch stations are 17% and 25%, respectively.
From these results, we conclude that the proposed
technique may be applied to the real system to ® nd a
reasonably accurate estimation of the cycle time and
the throughput quickly in lieu of the time-consuming
and high-cost simulation. In this paper, we limited our-
selves to the ideal shop conditions, e.g. no breakdowns of
the machines, no random reworks, etc. More realistic
and/or extended conditions can be included in modelling
as a further study. It is also required to extend the
approach when there are multiple job types, which is
part of our on-going research issues. The proposed tech-
nique may be applied to some other types of system
including a job shop, for example.
Acknowledgement
We would like to express our appreciation for valuable
comments from the anonymous referees that helped us to
enhance the presentation of the paper.
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