1 s2.0-s122631921400091 x-main

Journal of the Korean Statistical Society 44 (2015) 366–375
Contents lists available at ScienceDirect
Journal of the Korean Statistical Society
journal homepage: www.elsevier.com/locate/jkss
Allocation of the equipment path in a multi-stage
manufacturing process
YongBin Lima
, Jonghee Chunga
, Changsoon Parkb,∗
a
Department of Statistics, Ewha Womans University, Seoul, 120-750, Republic of Korea
b
Department of Statistics, Chung-Ang University, Seoul, 156-756, Republic of Korea
a r t i c l e i n f o
Article history:
Received 11 May 2014
Accepted 21 October 2014
Available online 11 November 2014
AMS 2000 subject classifications:
primary 62P30
secondary 62K05
Keywords:
Multi-stage manufacturing process
Equipment path
Fractional factorial design
Orthogonal array
Product design
a b s t r a c t
The allocation of equipment in a multi-stage process is discussed in this article. In most
of the multi-stage manufacturing processes, multiple equipment are operated to minimize
the waiting times between stages. Thus, the allocation of the equipment path becomes an
issue in choosing the equipment for the next stage. In solving the allocation problem of
the multi-stage process, it is assumed that main effects and two-way interaction effects
for the two adjacent stages are significant. The efficient allocation problem for the multi-
stage process for a given historical data is solved by the general linear model approach, and
then the predicted responses are ordered to choose the subsequently optimal equipment
paths. The effectiveness of the proposed allocation strategy is evaluated in terms of the
probabilities for detecting all true effects and detecting optimal equipment path for three
cases of precisions: baseline, precise errors and noisy errors. It turns out that the noisy
error case is less efficient than the others. When it is possible to use pilot experiments, the
efficiency of the product design of orthogonal arrays for two-level and three-level fractional
factorial designs is compared to that of the random selection of factorial design points. It is
shown that the former is more efficient than the latter in a case study.
© 2014 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.
1. Introduction
There have been numerous statistical process control (SPC) methods for quality and productivity improvement. Examples
of such activities are the control charting methods and gauge R & R, etc. Most of such methods are designed for single-stage
processes. In modern manufacturing processes, such as semiconductor manufacturing, electric device assembly, and so on,
it is quite common that multi-stage processes are used for the production of the item. A multi-stage process is referred as
a manufacturing process where multiples of unit processes are serially connected and final products can be manufactured
when materials pass all the unit processes. A famous example of the multi-stage process is the manufacturing process of
IC chips that involves hundreds of operations being executed layer by layer onto a silicon wafer. The whole IC chip making
process includes several group processes such as insulating, placing, patterning, and so on. The SPC methods for multi-stage
processes have also been studied recently, but not as much as the single-stage processes. The SPC of the multi-stage process
is complicated and difficult since it requires understanding of the effects of the current process to adjacent processes in
consideration of the serially connected processes.
∗ Corresponding author.
E-mail address: cspark@cau.ac.kr (C. Park).
http://dx.doi.org/10.1016/j.jkss.2014.10.003
1226-3192/© 2014 The Korean Statistical Society. Published by Elsevier B.V. All rights reserved.

Y. Lim et al. / Journal of the Korean Statistical Society 44 (2015) 366–375 367
Fig. 1. The flow of three batches for case 1.
Fig. 2. The flow of three batches for case 2.
Besides the SPC, another problem occurring in the multi-stage process is the allocation of equipment at each stage of the
whole process. Equipment in a stage is defined here as a tangible property that is used to accomplish the same operations
of the manufacturing. Examples of equipment in a stage of a multi-stage process include devices, machines, and tools used
for manufacturing in each stage.
In most of the multi-stage processes, there are multiple of equipment at most of the stages, and the raw materials are
fed to each stage by the dispatching rule. The dispatching rule of the manufacturing process is to keep operating the process
by minimizing the dead time, the waiting time between unit processes. Such dispatching rule makes the raw materials flow
smoothly through the whole process, and thus maximize the productivity. Consider a multi-stage process with three batch
processes (say, A, B, and C) whose lead times, times required to pass through the process, are 20, 60, and 40 min, respectively.
Consider case 1 where there is only a single equipment at each process. The flow of three batches for case 1 is described
in Fig. 1. In case 1, batch 1 passed through process A can be put into process B at 20 min, and put into process C at 1 h and
20 min without any waiting time. On the other hand, batch 2 will be out from process A at 40 min, but should wait 40 min to
be put into process B. Moreover, batch 3 should wait 80 min to be put into process B. The total lead time of case 1 is 240 min.
Consider case 2 where there are 1, 3, and 2 equipment for process A, B, and C, respectively. Then the flow of three batches
for case 2 is described in Fig. 2. In case 2, it can be easily seen that there is no waiting time between processes and the total
lead time is only 160 min.
As an example of the multi-stage process with multiple equipment, the layer process of the PCB manufacturing process
can be considered. In the layer process, four processing units, such as PCB preprocessing process, D/F lamination process,
D/F exposure, and DES process (circuit development, PCB etching, D/F stripping), are connected in serial. Equipment for the
layer process are typically very expensive, for example, D/F exposure machine costs around couple of million dollars or more.
Despite the high price, several equipment are provided to satisfy the dispatching rule. Fig. 3 shows the operation of multi-
equipment in the layer process. In Fig. 3, there is a single equipment for preprocessing and lamination process, while there
are three and two equipment for exposure process and DES process, respectively. All the final products, the manufactured
PCBs, are inspected and accessed as pass (conforming) and fail (nonconforming) through several inspection procedures such
as AOI (automatic optical inspection) and E-check, and then the yield of the manufacturing process is calculated.
If some equipment is occupied at a certain stage during the material flow, then one of the remaining equipment can
be used for the operation of the stage. An unexpected problem occurred in such multiple equipment situations is that
equipment between stages may have significantly related effects to the quality of the final product. That is, some specific
allocations of equipment produce products with better quality than the other allocations of equipment. It has been reported
by engineers that certain equipment combinations produce high yield, whereas some others low yield. Such harmonious
and disharmonious equipment combinations can be revealed through the study of the equipment allocation. Although
the optimal equipment combination is most preferable, the equipment corresponding to the current stage of the optimal
equipment combination may not be available because it is still being operated for previously allocated materials. In such
cases, the subsequent best equipment combinations will be considered. For such allocation of subsequent equipment
combinations, all the possible equipment combination will be ordered according to the estimated yield. Then the materials

368 Y. Lim et al. / Journal of the Korean Statistical Society 44 (2015) 366–375
Fig. 3. Flow of equipment path for layer process in the PCB manufacturing process.
at the current stage will be fed to the next best available equipment considering the equipment passed through up to the
previous stages. An allocation of equipment for the whole multi-stage process is called an equipment path. Hence, the task
for allocation of equipment is trying to use some harmonious sets of equipment for production and not to use disharmonious
sets of equipment. In fact, such problem has been a long and annoying problem to engineers, but they just relied on their
own personal experience or only the main effect of equipment without considering the related effects by adjacent processes.
In Section 3, a case study is developed. In Section 4, approaches for the missing value problem are explained. In Section 5,
a case study is followed by the simulation results for three possible cases of design.
2. Design and analysis for equipment path
Suppose that there is a k-stage process. Then the final product quality can be expressed as a linear function of stage effects
and inter-stage effects. The stage effect implies the main effect and the inter-stage effect implies the interaction effect of
two or more than two stages. It is often true that higher order interaction effects tend to become negligible and can properly
be disregarded. For the parsimony of the response model, only the main effects and the two-way interaction effect for the
two adjacent stages are considered to be possible significant effects. That is, interaction effects for more than two adjacent
stages are not significant to the final product quality. Let the ordered set of equipment be (i1, i2, . . . , ik), that is, equipment
il is used at stage l for l = 1, 2, . . . , k. Then the final product quality can be expressed as the response of a linear model with
the main effects and the two-way interaction effects, that is,
yi1i2···ikj = µ +
k
l=1
βil
+
k−1
l=1
βil,il+1
+ εi1i2···ikj (1)
where µ denotes the overall mean, βil
the main effect of equipment il at stage l, βil,il+1
the interaction effect of equipment il
and il+1 used in stages l and l + 1, respectively, and j the replication for the specific equipment path. It is assumed that the
error term ε follows a normal distribution with mean zero and variance σ2
ε .
In a k-stage process, each stage works as a factor and each equipment in a stage as a level of the factor. Thus the model
in Eq. (1) can be treated as a linear model with k factors with mixed levels. Since the diversity of number of levels is not
preferred in the analysis of linear models, it is assumed that there are only two or three levels at each stage. Actually there
can be more than three equipment, but they can be classified as two or three groups of equipment according to similarity
of their mechanical properties. Then equipment in a group are treated as the same level. The number of runs required by a
full three-level factorial design 3k
increases geometrically as k is increased. Running a 3−f
fraction of a 3k
design is called a
three-level fractional factorial design 3k−f
. In general, a three-level fractional factorial design 3k−f
with estimable effects of
interest could be used. To assign a two-level factor A to a 3k−f
design, the dummy level technique is used. A two-level factor
A is formally turned into a three-level factor by repeating one of the levels. The third level of A is a duplicate of one of the
actual two levels, which could be chosen as the more frequently used equipment.
Now, a special case is considered. Suppose that the number of equipment at two adjacent stages are different and let the
numbers of two-level factors and three-level factors be k∗
. Since there are many factors with two or three levels, fractional
factorial design should be used for the analysis of the linear model. We are interested in how to get a fractional factorial
design with those mixed-level factors where all the main effects and two-way interaction effects between the two adjacent
stages are estimable.
A design of the resolution III does not confound main effects with another but does confound main effects with two-
factor interactions. Since main effects are interested, we choose the resolution III design with smallest fraction for the sake
of small number of experiments. Also let 2
k∗−f
III and 3
k∗−f
III denote the resolution III two-level fractional factorial design with
fraction 1/2f
and the resolution III three-level fractional factorial deign with fraction1/3f
, respectively. Then the product

Table 1
Number of equipment in a six-stage manufacturing process.
Stage (factor) A B C D E F
Equipment (level) 2 3 2 3 2 3
Table 2
The 23−1
III × 33−1
III product design.
23−1
III × 33−1
III = 23−1
III × 33−1
III
A C E B D F A B C D E F
1 1 1 1 1 1 1 1 1 1 1 1
1 2 2 1 2 2 1 1 1 2 1 2
2 1 2 1 3 3 ...
2 2 1 2 1 2 1 3 1 3 1 2
2 2 3 1 1 2 1 2 1
2 3 1 1 1 2 2 2 2
3 1 3 ...
3 2 1 1 3 2 3 2 2
3 3 2 ...
2 3 2 3 1 2
design Dk = 2
k∗−f
III ×3
k∗−f
III where all runs of a 3
k∗−f
III are conducted at each run of a 2
k∗−f
III is used for the analysis of the model
in order to make two-way interaction effects for the two adjacent stages estimable.
Since the data are obtained from the manufacturing history of the multi-stage process, the number of replication varies
drastically depending on the equipment path. Certain equipment paths will have a large number of replication, while some
others will have a small number. Moreover, some equipment path will have no observation, which corresponds to the
missing case. The average response can be expressed from Eq. (1) as
¯yi1i2···ik· = µ +
k
l=1
βil
+
k−1
l=1
βil,il+1
+ ¯εi1i2···ik·. (2)
The variance of the average error ¯ε is σ2
ε /r for the number of replications, r.
3. A case study
Suppose that a multi-stage process has three two-level factors and three three-level factors given in Table 1. Here, the
number of stage corresponds to the number of factors and the number of equipment in each stage corresponds to the number
of levels in each factor.
Equipment path implies the allocation of specific equipment combination that the product materials pass through during
the whole manufacturing process. The data used for the analysis are the equipment paths and the corresponding responses.
It is of interest to find the equipment path for the optimum response as well as the subsequently sorted equipment path in
the order of responses.
The data to be analyzed are mostly historical, not experimental. The historical data contains the quality characteristic y
and all the information about the equipment paths through which materials have passed. It is assumed that the response
is affected by all the main effects and two-way interaction effects between the two adjacent stages. The historical data is
sorted according to the equipment paths. It suffices to get the average response ¯y and the number of replications for each
equipment path. Since the number of replications in each path are different in general, the heteroscedasticity of ¯y should be
considered and thus, the weighted least squares method is used in estimating the effects. Since the data are obtained from
the manufacturing history of the multi-stage process, some equipment paths corresponding to design points in 23
× 33
factorial design have no observation, which corresponds to the missing cases.
In order to detect the significant effects in model (2), we may use all historical data or part of them corresponding to a
fractional factorial design to maintain the balance of the design points.
A practical method is discussed on how to get a fractional factorial design with those mixed-level factors where all the
main effects and two-way interaction effects between the two adjacent stages are estimable.
First, find the product design between two-level fractional factorial design 23−1
III and three-level fractional factorial design
33−1
III . Then all the two way interaction effects between two level factors and three level factors are estimable. Thus all the
two-way interaction effects between the two adjacent stages are estimable. Table 2 shows the product design between 23−1
III
with defining contrast ACE and 33−1
III with defining contrast BDF2
. It can be easily checked that at each run of 23−1
III all nine
runs of 33−1
III are conducted.
This idea is implemented in the computation algorithm in R program developed by Isaac Newton Institute in order to
construct fractional or block factorial designs in mixed levels (Monod, Bouvier, & Kobilinsky, 2013). In R-program, inputs are

Table 3
Size of effects (multiple of σε) in the simulation model.
Level/factor ai bj dk el a1bj a2bj
1 −1 −2 −1.5 −1 −3 3
2 1 0 0 1 0 0
3 – 2 1.5 – 3 −3
Overall mean effect µ = 65.
the factors with number of levels, the model with interested effects, and the unit size (see Appendix A.1). Then the output
from the R-program is a relevant experimental design.
By analyzing the historical data or part of them, it is expected to detect the vital effects for the response. The popular
graphical method to identify vital effects is to use the half normal probability plot by decomposing the effects with more than
two degrees of freedom into the effects with single degree of freedom. In order to eliminate the subjectivity of the graphical
procedures for detecting significant effects, many quantitative procedures have been proposed in the recent literature. In
this article, the vital effects are selected based on the LGB method by Lawson, Grimshaw, and Burt (1998). They suggested
a hybrid method based on the half normal probability plot, which is a blend of Lenth’s (1989) and Loh’s (1992) methods. It
is known that LGB method is uniformly more powerful than Lenth’s. The method consists of fitting a simple least squares
line and calculating prediction limits to the half normal probability plot. The effects fall outside the prediction limits are
significant.
The proper model for the quality characteristic y being found, it is straightforward to order all the possible equipment
paths according to the predicted responses. In order to shorten the cycle time and increase the productivity of the
manufacturing process, it is not a good way to insist the optimal equipment path since the equipment designated by the
optimal equipment path is often occupied for processing previous materials. Thus it is essential to find the subsequently
optimal equipment paths. Such ordering of equipment paths can be found by evaluating the predicted responses.
4. Missing value treatment
If all the design points in the factorial design or fractional factorial design have at least one observation, then the design
has no missing value. Otherwise, the design points with no observation correspond to missing value cases. Missing value
cases occur often in historical data, and thus the remedy for such cases should be prepared. There are two general approaches
to the missing value problem. The first is the exact analysis where the model is estimated based on the unbalanced data.
The second is the approximate analysis where the missing observations are estimated by the predicted value of the quality
characteristic y and then, their predicted values are used as imputed values. Then we proceed with the analysis as usual just
as if the imputed observations are real data. The first approach reduces the number of design points used in the analysis and
the second approach maintains the balance of the design points.
5. A simulation study
Consider a six-stage manufacturing process whose number of equipment is given by Table 1. A set of historical data is
generated by simulation for the six-stage process. Out of 23
× 33
factorial design points we randomly select one third of
them to construct a historical data.
Without loss of generality, a large value of the response is preferred. It is assumed that four main effects A, B, D, E and
one two-way interaction effect A ×B are significant to the response. Since the sample average of the quality characteristic y
is calculated for given equipment paths, the error terms for that sample average are assumed to follow N(0, 1/nijkl) for the
number of replications nijkl. The number of replication in each equipment path is generated by the discrete uniform distri-
bution between 1 and 10. Thus the simulation model for the generation of the sample average of the quality characteristic
y is given as the following;
¯yijkl = µ + ai + bj + (ab)ij + dk + el + ¯εijkl, i, l = 1, 2 j, k = 1, 2, 3 (3)
where ¯εijkl ∼ N(0, σ2
ε /nijkl), and nijkl ∼ discrete Uniform {1, 2, . . . , 10}. The size of effects in Eq. (3) is given in Table 3. It
can be easily checked that the optimal equipment path is A1B3D3E2 at which the mean response is 71.5.
In order to screen the vital few effects from the assumed model (2) and then find the proper model, we consider three
cases: using all the historical data, using only the part of the historical data corresponding to the fractional factorial design
and using the pilot experimental data.
5.1. Using all the historical data
For the historical data, a data set Historical_72 is generated by the model in Eq. (3) and the vital effects are detected using
half normal probability plot based on the LGB method. Then, all the possible equipment paths are sorted in the order of the
predicted responses.

Fig. 4. Half normal probability plot based on the LGB method.
Table 4
The estimates of the screened effects, A, B, D, E and AxB.
Level/factor ˆai
ˆbj
ˆdk ˆel
a1bj
a2bj
1 −0.9811 −1.9886 −1.4597 −1.0831 −3.0545 3.0545
2 0.9811 −0.0264 −0.0119 1.0831 0.0980 −0.0980
3 – 2.0150 1.4716 – 2.9565 −2.9565
Overall mean effect ˆµ = 64.9238.
To draw the half normal probability plot, treatment effects in the model need to be estimated and then, those effects with
more than one degrees of freedom to be decomposed into the effects with single degree of freedom. The GLM procedure in R
performs this process by projecting the data onto the successive orthogonal subspaces generated by the QR decomposition.
To screen the significant effects, those effects are inserted into the LGB function, which is implemented in package ‘daewr’
developed by John Lawson (2012) in order to obtain the LGB test statistic. The R code is presented in Appendix A.2.
One case of the half normal plots is shown in Fig. 4. In the figure, it is shown by the LGB method that effects A1B1, B1,
D1, E1, A1, D2 and B2 are significant. Thus, all true effects, A, B, D, E and AxB are detected in this case.
It is proposed that the hierarchy of effects are corrected automatically in screening the significant effects, which means
that relevant main effects are included in the model if interaction effects are screened (see Appendix A.3 for the R code.).
Table 4 shows the estimates of the screened effects, A, B, D, E and AxB. It can be easily checked that the optimal equipment
path is A1B3D3E2 with the predicted response, 71.47. Also, the second best equipment path is A1B3D2E2, with the predicted
response, 69.99. The equipment paths are sorted in decreasing order of the predicted response. The sorted results are
given in Table 5. This information can be used in deciding which equipment at the next stage should be assigned to the
materials.
From Table 5, it is seen that the optimal equipment path is chosen as A1B3D3E2. If equipment D3 is occupied at stage 4,
then the next optimal path is A1B3D2E2 in Table 5. This simulation process is repeated 10,000 times. The case of variance of
errors being 1/nijkl is called the baseline case. Also two other cases of different size of the variance of error are considered,
which are 0.25/nijkl for the precise error and 9/nijkl for the noisy error.
In assessing the performance of the proposed method, let PT denote the proportion of detecting all true effects, A, B, D, E
and AxB and let PO denote the proportion of detecting optimal path A1B3D3E2. The simulation results for the three cases are
summarized in Table 6(a).
PT for the baseline case is 0.9980. Even though all the true effects are screened, the proportion of detecting one more false
effect and two more false effects in addition to the true effects are 0.4124 and 0.5012, respectively. A half normal probability
plot for the former case is given in Fig. 5. Main effect F is the smallest significant effect detected by LGB method, which may
not be screened by eyeballs checking of a half normal probability plot. So there is a good chance of reducing the proportion of
detecting one false effect further. Also PO for the baseline case is 0.9980. The proportion of the designed optimal path being
the true optimal path A1B3D3E2 is 0.0708 and the proportion that the designed optimal path identify two indifferent stages
in addition to the true path A1B3D3E2 is 0.5106. Also PO for the precise error case is 0.9996, but the proportion of identifying

Table 5
The equipment paths in decreasing order of the predicted response.
A B D E ˆy
1 3 3 2 71.47
1 3 2 2 69.99
2 1 3 2 69.53
1 3 3 1 69.30
1 3 1 2 68.54
2 2 3 2 68.34
2 1 2 2 68.04
1 3 2 1 67.82
2 3 3 2 67.52
2 1 3 1 67.36
2 2 2 2 66.85
2 1 1 2 66.59
1 2 3 2 66.57
1 3 1 1 66.37
2 2 3 1 66.17
2 3 2 2 66.03
2 1 2 1 65.88
2 2 1 2 65.40
2 3 3 1 65.35
1 2 2 2 65.09
2 2 2 1 64.69
2 3 1 2 64.59
2 1 1 1 64.43
1 2 3 1 64.40
2 3 2 1 63.87
1 2 1 2 63.64
2 2 1 1 63.24
1 2 2 1 62.92
2 3 1 1 62.42
1 2 1 1 61.47
1 1 3 2 61.45
1 1 2 2 59.97
1 1 3 1 59.29
1 1 1 2 58.52
1 1 2 1 57.80
1 1 1 1 56.36
Fig. 5. A half normal probability plot in the case of detecting one more false effect.
false effects are even higher, for example the proportion that the designed optimal path identify two indifferent stages in
addition to the true path A1B3D3E2 is 0.7682. The efficiency of identifying true effects and optimal path gets a little worse

Table 6
Simulation results on the efficiency of the proposed strategy.
Precise error Baseline Noisy error
(a) Simulation results using all the historical data
Historical_72
PT 0.9996 0.9980 0.9220
No false 0.0092 0.0707 0.5428
One false effect 0.2203 0.4124 0.3130
Two false effects 0.7544 0.5012 0.0605
PO 0.9996 0.9980 0.9211
True optimal path 0.0092 0.0708 0.5443
One indifferent stage 0.2222 0.4166 0.3252
Two indifferent stages 0.7682 0.5106 0.0516
(b) Simulation results using only the historical data corresponding to the fractional factorial design
Historical_FFD (imputed)
PT 0.9884 0.9696 0.7027
No false 0.0094 0.0677 0.2966
One false effect 0.2056 0.3597 0.2863
PO 0.9884 0.9696 0.6826
(c) Simulation results using the pilot experimental data
Random_36
PT 0.9631 0.9049 0.4564
No false 0.0174 0.1036 0.2966
One false effect 0.2638 0.4127 0.1376
PO 0.9631 0.9048 0.4464
FFD_36
PT 1.0000 0.9990 0.7117
No false 0.1137 0.4637 0.6576
One false effect 0.4841 0.4275 0.0428
PO 1.0000 0.9990 0.7068
in the noisy error case. PT and PO in the noisy error case are 0.9220 and 0.9211, respectively. Surprisingly, the proportion of
the designed optimal path being the true optimal path is 0.5443 and the proportion that the designed optimal path identify
two indifferent stages in addition to the true path is 0.0516. Even though the efficiency of screening all true effects and
identifying the true optimal path for the noisy error case is not as good as the baseline and precise error cases, the efficiency
in identifying few false effects is best in the case of noisy errors.
Traditionally, engineers have used to sort the equipments path by the decreasing order of the mean responses and the
equipments path corresponding to the maximum response is the candidate for the optimal equipments path. PO for this
method in the precise error, the baseline, and the noisy error cases are 0.9142, 0.8929, and 0.5318, respectively. PO for
the proposed method in those three cases are 0.9996, 0.9980, and 0.9211, respectively from Table 6(a). Thus the proposed
method is doing well for identifying true optimal path especially in the noisy error case.
5.2. Using only the historical data corresponding to the fractional factorial design
As discussed in Section 3, the product design between 23−1
III with defining contrast being ACE and 33−1
III with defining
contrast being BDF2
is generated. Only historical data whose equipment paths corresponding to the product design points in
Table 2 are used and the rest of them are disregarded. Since one third of 23
×33
factorial design points are selected randomly
to construct a historical data, there exist about twenty-four missing observations in the product design. The approximate
analysis is adopted where the missing observations are estimated by the predicted value of the quality characteristic y
based on all the historical data and then, their predicted values are used as imputed values. The same analysis has been
done as in Section 5.1 to find PO and PT in the precise error, the baseline, and noisy error cases. The simulation results for
the three cases are summarized in Table 6(b). PO in those three cases are 0.9696, 0.9884 and 0.6826. Thus, the efficiency of
screening all true effects and identifying the true optimal path with part of the historical data is not as good as that with all the
historical data. Especially that is worse in the nosy error case, where it is important to have more data to predict the response
well.

5.3. Using the pilot experimental data
Suppose it is possible to use pilot experiments. A set of data generated by the model in Eq. (3) at the product design
points is called FFD_36 and that at randomly selected thirty six factorial design points is called Random_36. The simulation
results in the precise error, the baseline, and the noisy error cases are summarized in Table 6(c). It is interesting to check
the efficiency of screening all true effects and identifying the true optimal path with FFD_36, Random_36 and Historical_72.
Even though the size of FFD_36 is the half of Historical_72, PO with FFD_36 is a little better than PO with Historical_72 in the
precise error and the baseline cases. It is interesting to note that the proportion of the designed optimal path being the true
optimal path with FFD_36 in the baseline case is 0.4639. On the other hand that with Historical_72 is 0.0708. In the noisy
error case, PO with FFD_36 is 0.7068 and PO with Historical_72 is 0.9211, which implies that taking more data is critical in
getting higher efficiency. Note that PO with FFD_36 is much better than PO with Random_36 regardless of size of errors.
6. Conclusions and further studies
The allocation of equipment in a multi-stage process is discussed in this article. Usually, there are multiple equipment in
each stage of the multi-stage process to provide the smooth flow of the materials through the process. When multiple
equipment are available at a certain stage, a specific equipment should be chosen among them. Also when a specific
equipment is occupied at a certain stage, the next best available equipment should be chosen. Such preparations for the
smooth flow of the product material make the production process efficient, and subsequently make the production cost low.
In the multi-stage process, it is assumed that main effects and two-way interaction effects for the two adjacent stages are
significant. The efficient allocation problem for the multi-stage process for a given historical data is solved by the general
linear model approach, and then the predicted responses are ordered to choose the subsequently optimal equipment paths.
The effectiveness of the proposed allocation strategy is evaluated in terms of the probabilities for detecting all true effects
and detecting optimal equipment path for three cases of precisions. It turns out that noisy error case is less efficient than
the others. When it is possible to use pilot experiments, the efficiency of the product design of orthogonal arrays especially
for 2
k∗−f
III and 3
k∗−f
III fractional factorial designs is compared to that of the random selection of factorial design points. It is
shown that the former is more efficient than the latter in a case study.
The number of equipment available at each stage is constrained up to three to make the analysis simple in this article.
Designs for stages with more than three equipment can also be developed by taking similar steps proposed in this article. It is
assumed by the dispatching rule that the next available equipment is searched immediately when the designated equipment
by the selected equipment path is occupied, regardless of expected waiting time for the specific equipment. It would be
more appealing to process engineers if the equipment allocation is determined by the cost-based approach than the risk-
based approach proposed in this article. Such considerations are left as further studies for improvement of the equipment
allocation strategy for multi-stage processes.
Acknowledgments
The authors would like to thank the associate editor and the referees for their valuable comments and suggestions.
Works of YongBin Lim and JongHee Chung were supported in part by Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2014R1A1A2002032)
and BK21 Plus Project through the National Research Foundation of Korea (NRF) funded by the Ministry of Education
(22A20130011003), respectively. Changsoon Park’s work was supported by the Chung-Ang University research grant in
2014.
Appendix. R codes
A.1. R code for the generation of a product design given in Table 2
A.2. R code for drawing the half normal probability plot based on the LGB method

A.3. R code for respecting the hierarchy of effects and getting weighted LSE of effects
References
Lawson, J. (2012). daewr: Design and Analysis of Experiments with R, R package version 1.0–10.
Lawson, J., Grimshaw, S., & Burt, J. (1998). A quantitative method for identifying active contrasts in unreplicated factorial designs based on the half-normal
plot. Computational Statistics and Data Analysis, 26, 425–436.
Lenth, R. V. (1989). Quick and easy analysis of unreplicated factorials. Technometrics, 31, 469–473.
Loh, W. L. (1992). Identification of active contrasts in unreplicated factorial experiments. Computational Statistics and Data Analysis, 14, 135–148.
Monod, H., Bouvier, A., & Kobilinsky, A. (2013). A quick guide to PLANOR, an R package for the automatic generation of regular factorial designs. Technical report.
MIA Unit, INRA Jouy en Josas.

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