A data envelopment analysis method for optimizing multi response problem with censored data in the taguchi method

A data envelopment analysis method for optimizing multi-response
problem with censored data in the Taguchi method
Hung-Chang Liao
Department of Health Services Administration, Chung-Shan Medical University, No. 110, Sec.1, Jian-Koa N. Road,
Taichung 402, Taiwan, ROC
Available online 1 July 2004
Abstract
Taguchi method is an efﬁcient method used in off-line quality control in that the experimental design is combined
with the quality loss. This method including three stages of systems design, parameter design, and tolerance design
has been deeply discussed in Phadke [Quality engineering using robust design (1989)]. It is observable that most
industrial applications solved by Taguchi method belong to single-response problems. However, in the real world
more than one quality characteristic should be considered for most industrial products, i.e. most problems customers
concern about are multi-response problems. As a result, Taguchi method is not appropriate to optimize a multi-
response problem. At present, it is still necessary to rely on the engineering judgment to optimize the multi-response
problem; therefore uncertainty will be increased during the decision-making process. On the other hand, due to some
uncontrollable causes occurring, only a portion of experiment can be completed so that the censored data will be
produced. Traditional approaches for analysis of censored data are computationally complicated. In order to
overcome above two shortages, this article proposes an effective procedure on the basis of the neural network (NN)
and the data envelopment analysis (DEA) to optimize the multi-response problems. A case study of improving the
quality of hard disk driver in Su and Tong [ Total Quality Management 8 (1997) 409] is resolved by the proposed
procedure. The result indicates that it yields a satisfactory solution.
q 2004 Elsevier Ltd. All rights reserved.
Keywords: Taguchi method; Multi-response; Neural network; Data envelopment analysis; Censored data
1. Introduction
Taguchi method is a traditional approach for robust experimental design that seeks to obtain the best
combination offactors/levels with the lowest societal cost solution to achieve customers’ requirement. In
Taguchi’s design method the design parameters (factors can be controlled by designers) and noise
factors (factors cannot be controlled by designers, such as environmental factors) are considered
inﬂuential on product quality. Therefore, the Taguchi’s design method is to select the levels of design
0360-8352/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.cie.2004.05.012
Computers & Industrial Engineering 46 (2004) 817–835
www.elsevier.com/locate/dsw
E-mail address: huncliao@ms43.hinet.net

parameters and to reduce the effects of noise factors. That is, parameter setting should be determined
with the intention that the product response (quality characteristic) has minimum variation while its
mean is close to the desired target (Phadke, 1989). Nevertheless, so far the Taguchi method can only be
used for a single response problem; it cannot be used to optimize a multi-response problem.
Unfortunately, nowadays more than one quality response results from most industrial products and these
products’ quality characteristics are considered by customers. In the Taguchi method, to solve the multi-
response problem, engineering judgment is the primary method. But, without doubt an engineer’s
judgment will increase uncertainty during the decision-making process. In order to solve the multi-
response problem, an approach of assigning weight to each response is submitted (Antony (2000); Hung,
1990; Lin & Lin, (2002); Lin, Wang, Yan, & Tarng, 2000; Shiau, 1990; Tai, Chen, & Wu, 1992.)
However, it still remains difficult to determine and define a weight for each response in a real case.
Another method is the regression technique based approaches (Logothetis & Haigh, 1988; Pignatello,
1993). Whereas, such approaches increases the complexity of computational process, and the possible
correlations among the responses may still not be considered. Furthermore, the significant factor in a
single-response case may not be considered significant when considered in a multi-response case.
Therefore, in order to solve this multi-response problem, a principal component analysis (PCA) method
is considered (Su & Tong, 1997). The PCA method transforms a set of original responses into a set of
uncorrelated principal components in order to choose the principal component, whose eigen value is
greater than 1, as a multi-response performance index by which the optimal experimental conditions can
be determined. Nevertheless, there are still two obvious shortcomings in PCA method. First, when more
than one principal component is selected, that is, more than one eigen value is greater 1, how to trade-off
to select a feasible solution is unknown. Second, when the chosen principal component with only less
variation can be explained to total variation of multi-response, the chosen principal component is not
able enough to illustrate and solve the multi-response problem.
On the other hand, sometimes, due to some uncontrollable causes, for example, impaired facilities,
constrained time, and constrained cost, only a portion of experiment can be completed. Under these
circumstances, the experimental results will produce complete and censored data. Censored data only
contain less information and are harder to be analyzed. Traditional approaches for analysis of censored
data are computationally complicated. Several types of censored data have been studied (Nelson &
Hahn, 1972). Most researchers used statistical approaches to estimate censored data (Hamada & Wu,
1991; Hahn & Nelson, 1974; Schmee & Hahn, 1979; Taguchi, 1987; Tong & Su, 1997). But these
statistical approaches must be assumed that the population is in a normal distribution. Moreover, these
statistical approaches are complicated (Stern, 1996). Su and Miao (1998) proposed the neural network
(NN) procedures for experimental analysis with censored data. The main drawback about Su’s neural
network procedures is that they cannot distinguish significant and not significant experimental
controllable factors. Also, these censored data researches can only develop for single responses. If the
experiment belongs to multi-response problem with censored data, how to simultaneously overcome
censored data and to reach optimization in multi-response problem is an important issue.
In this article, the NN method and the data envelopment analysis (DEA) (Charnes, Cooper, & Rhodes,
1997) are proposed to effectively optimize the multi-response problem in Taguchi method. With the NN,
the signal-to-noise (SN) ratios of these multiple responses are estimated by the known experimental
complete data for each factors/levels combination, which is also named decision-making unit (DMU).
Then to use DEA to find each DMU’s relative efficiency, that is also to find each factors/levels
combination’s relative efficiency, so that the optimal factors/levels combination can be found by relative
H.-C. Liao / Computers & Industrial Engineering 46 (2004) 817–835818

efficiency value 100%. Using NN to estimate censored data, researchers do not need to require any
assumptions related to the relationship between inputs and outputs and can overcome the analysis
constraints of censored data. The proposed optimal procedure in solving the optimal multi-response
problem includes a series of steps capable of decreasing the uncertainty caused by engineering judgment
in Taguchi method, and it also becomes a universal approach to optimize the multi-response robust
design.
2. Neural network (NN)
The NN is composed of processing elements (nodes or neurons) and connections. The nodes are
interconnected layer-wise of interconnection among themselves. Each node in the successive layer
receives the inner product of synaptic weights, with the outputs of the nodes in the previous layer. The
operation of a neuron is shown in Fig. 1. Each node ðX1; X2; …; XnÞ has an output signal connected to
each of the other nodes. Each connection is assigned a relative weight. A node’s output depends on the
specified threshold and transfer function FðXÞ: The NN has been shown to be effective for addressing the
complex nonlinear problem. Two types of learning networks are, respectively, supervised and
unsupervised. For a supervised learning network, a set of training input vectors with a corresponding set
of target vectors is trained to adjust the weights in an NN. For an unsupervised learning network, a set of
input vectors is proposed; however, no target vectors are specified. In this study, a supervised learning
network is more suitable for the multi-response with censored data estimated problem. Among the
several well-known supervised learning NNs are back-propagation (BP), learning vector quantization,
and counter propagation network. The BP model is the most extensively used and can provide better
solutions for many applications (Dayhoff, 1990; Lippmann, 1987). Therefore, the BP model is selected
in this study.
A BP neural network consists of three or more layers, including an input layer, one or more hidden
layers, and an output layer. Fig. 2 illustrates a basic BP neural network with three layers. BP neural
network learning works on a gradient-descent algorithm (Funahashi, 1989). The BP neural network
initially receives the input vector and directly passes it into the hidden layer. Each element of the hidden
layer is used to calculate an activation value by summing up the weighted input, and the sum of the
weighted input will be transformed into an activity level by using a transfer function. Each element of
the output layer is then used to calculate an activation value by summing up the weighted inputs
Fig. 1. Operation of A neuron.
H.-C. Liao / Computers & Industrial Engineering 46 (2004) 817–835 819

attributed to the hidden layer. Next, a transfer function is used to calculate the network output. The actual
network output is then compared with the target value. BP neural network algorithm refers to the
propagation of errors of the nodes from the output to the nodes in the hidden layers. These errors are used
to update the weights of the network. The amount of weights to be added or subtracted to the previous
weight is governed by the delta rule. After the knowledge representation is determined, the BP neural
network will be trained to attempt the prediction behavior. The number of hidden layers and the number
of nodes in each hidden layer are determined at the training phase. In this study, a fully connected
feedforward neural network will be used, and also its network parameters and stopping criterion will
be set.
To be able to attempt the prediction behavior, a learning rule will be used in the BP neural network. In
the case of a multi-layer perception, this rule should also be able to adapt the weights of all connections
in order to model a nonlinear function. The learning rule used frequently most for this purpose is the BP
rule. It acts in two steps. First, the generalized difference Dp
i ðtÞ is calculated by
Dp
i ðtÞ ¼ ðAp
i ðtÞ 2 AiðtÞÞAiðtÞð1 2 AiðtÞÞ; ð1Þ
where Ap
i ðtÞ is the desired activation of output unit i; and AiðtÞ is the generated activation of this unit. In
order to obtain the generalized difference Dp
i ðtÞ; the calculated difference ðAp
i ðtÞ 2 AiðtÞÞ is multiplied by
the simpliﬁed derivative of the activation function AiðtÞp
ð1 2 AiðtÞÞ: Second, the generalized differences
of the units in the output layer are propagated back through the weighted connections to the units of the
hidden layer. The generalized difference collected from a hidden unit is multiplied by the simpliﬁed
derivative of the activation function of the unit to obtain the generalized difference of the hidden unit
Dp
j ðtÞ ¼
Xn
i¼1
ðWijðtÞDp
i ðtÞÞAjðtÞð1 2 AjðtÞÞ: ð2Þ
Using the generalized difference Dp
i ðtÞ; the weights are adjusted by
Wijðt þ 1Þ ¼ WijðtÞ þ CDp
i ðtÞAjðtÞ: ð3Þ
Fig. 2. A BP neural network.

The adaptation size of the weight WijðtÞ of the connection used to send information from unit j to unit i is
influenced by the existing weight WijðtÞ; the learning rate C; the generalized difference Dp
i ðtÞ; and the
actual activation AjðtÞ of unit j: To reduce the probability of weight change oscillation, a weight
momentum term is added to adjust the weight. The weight momentum term is constructed by the
previous adjustment of the weight DWijðtÞ and a constant value B; so
Wijðt þ 1Þ ¼ WijðtÞ þ CDp
i ðtÞAjðtÞ þ BDWijðtÞ: ð4Þ
If more hidden layers are implemented, the BP rule will use the generalized differences of the hidden
units for BP neural network to the hidden units of the layer closer to the input layer. To test the network,
the test set data are assigned to the networks, and the output will then be evaluated. The network should
be able to interpolate and, possibly, extrapolate.
3. Data envelopment analysis (DEA)
DEA is a linear programming based technique for measuring the relative efficiency of a set of
competing decision-making units (DMU) where the presence of multiple inputs and outputs makes the
comparisons difficult (Dyson, Thanassoulis, & Boussofiane, 1990). The relative efficiency of the
‘multiple inputs and outputs’ in DMU is typically defined as a ratio (weighted sum of the DMU’s outputs
divided by weighted sum of the DMU’s inputs). So, if the higher performance in the relative efficiency
can be obtained, the input data of ratio must have lower values and the output data of ratio must have
higher values. Or, when the input data are constrained to fixed values and the output data have higher
values, the relative efficiency also has a higher performance. In this article, we use DEA to solve the
multi-response problem in Taguchi method. A DMU is defined as a factors/levels combination, its input
data are set to value 1, and its output data are the value of responses’ SN ratios. When a higher output
value is got, the higher relative efficiency value of DMU will be obtained. Thus, it will be suitable for
solving the multi-response problem because the higher relative efficiency value implies that the
products’ quality characteristics are easy to stand out in relief. With the mathematical notation of Doyle
and Green (1994), the general efficiency measure used by DEA is summarized as the following Eq. (5)
Eks ¼
X
y
Osyvky
X
x
Isxukx
ð5Þ
where
Eks; the efficiency measure of DMU s, using the weights of the assessed DMU k;
Osy; the values of output y for DMU s;
Isx; the values of input x for DMU s;
vky; the weights assigned to trial DMU k for output y;
ukx; the weights assigned to trial DMU k for input x:
In the basic DEA ratio model (CCR model) developed by Charnes, Cooper, and Rhodes (1978), the
objective is to maximize the relative efficiency value of a trial DMU k from among a reference set of
DMU s; by selecting the optimal weights associated with the input and output measures. The maximum
H.-C. Liao / Computers & Industrial Engineering 46 (2004) 817–835 821

relative efficiencies are constrained to 1. The formulation is represented in expression (6)
max Ekk ¼
X
y
Okyvky
X
x
Ikxukx
s:t: Eks # 1 ; designs s
ukx; vky . 0
ð6Þ
This nonlinear programming formulation (6) is equivalent to the following linear programming (LP)
formulation (7) by setting its denominator equal to 1 and by maximizing its numerator.
max Ekk ¼
X
y
Okyvky;
s:t:
X
x
Ikxukx ¼ 1;
Eks # 1 ; designs s
ukx; vky . 0
ð7Þ
The result of formulation (7) is an optimal efficiency value ðEp
kkÞ that is at most 1. If Ep
kk ¼ 1; then no
other DMUs are more efficient than DMU k under its selected weights. That is, Ep
kk ¼ 1 has DMU k on
the optimal frontier and is not dominated by other DMU. If Ep
kk , 1; then DMU k does not lie on the
optimal frontier, and there is at least another DMU that is more efficient under the optimal set of weights
determined by (7). The formulation (7) is employed for each DMU to calculate DMU’s relative
efficiency with respect to its own optimal set of weights. For more details about DEA, please refer to
Charnes et al. (1997).
4. The proposed optimal procedure
In order to solve the information limitation problem of censored data and to obtain the optimal
factors/levels combination in a multi-response problem, an optimal procedure is proposed in this section.
The proposed optimal procedure including five steps is summarized as following:
Step 1. Use the BP model to estimate the SN ratios of all factors/levels combination: because the
experimental result includes censored data, the information is limited and is hard to be analyzed.
However, the BP model can estimate the incomplete data. In here, with the BP model, the knowledge
representation is to define the relationship between the control factors/ levels and the multi-response’s
SN ratios. The number of input nodes is equal to the number of control factors, and the input values are
the values of the control factor’s levels. In addition, the number of output nodes is equal to the number of
multiple responses, and the output values are the values of the multi-response’s SN ratios. The SN ratios
are obtained as following:
Let the SN ratio be Xij for the jth response at the ith trial, for i ¼ 1; …; m; j ¼ 1; …; n:
Xij ¼ 210 log10
1
l
Xl
k¼1
y2
ijk
" #
; 0 # yijk , 1; ð8Þ

(for the smaller-the-better response)
Xij ¼ 210 log10
1
l
Xl
k¼1
1
y2
ijk
" #
; 0 # yijk , 1; ð9Þ
(for the larger-the-better response), and
Xij ¼ 210 log10
y2
ij
S2
ij
#
; 0 # yijk , 1 ð10Þ
(for the nominal-the-best response)
where yijk ¼observed data in complete data for the jth response at the ith trial, the kth
repetition, yij ¼ 1
l
Pl
k¼1 yijk (the average observed data in complete data for the jth response at the ith
trial), S2
ij ¼ 1
l21
Pl
k¼1 ðyijk 2 yijÞ2
(the variation of observed data in complete data for the jth response at
the ith trial,) for i ¼ 1; …; m; j ¼ 1; …; n; and k ¼ 1; …; l:
The network parameter—learning rate and momentum—will be set to assist the trained network to
attempt the convergence and stabilization in prediction behavior. In order to stop the trained network, the
stopping criterion should be set to lower the root mean square error (RMSE) of training and lower the
RMSE of testing simultaneously. By selecting the RMSE of training and testing among the considered
estimation models, the best estimation model will be obtained.
Step 2: Normalize Xij: Xij is the SN ratio for the ith ði ¼ 1; 2; …; mÞ response in the jth ðj ¼ 1; 2; …; nÞ
experiment.
Now, Xij is normalized as Zijð0 # Zij # 1Þ by the following formula to avoid the effect of adopting
different units.
Zij ¼
Xij
max{Xij; j ¼ 1; 2; …; n}
; for i ¼ 1; 2; …; m; j ¼ 1; 2; …; n: ð11Þ
(To be used for responses with larger-the-better and nominal-the-best manners)
Zij ¼
max{Xij; j ¼ 1; 2; …; n}
Xij
; for i ¼ 1; 2; …; m; j ¼ 1; 2; …; n: ð12Þ
(To be used for response with smaller-the-better manners).
Step 3: Use formula (7) for each experiment to calculate its relative efficiency of each experiment.
(The calculation task can be fulfilled with the package software of Frontier Analyst or Warwick-
Windows-DEA.)
The input data is set to 1 and the output data is set to Zij for each experiment to calculate its relative
efficiency of each experiment (DMU).
Step 4: Rank the DMU according to its relative efficiency and obtain the optimal factors/levels
combination. The higher relative efficiency value implies the better product quality. The set of
factors/levels combination with relative efficiency 100% will be obtained. The quality engineer will select
the favorite and the best choice of factors/levels combination according to cost, tolerance, and so on.
Step 5. Obtain the significant and not significant controllable factors for these multiple responses.
Using ANOVA (analysis of variance) to analyze and test the estimated SN ratios, it can help quality
engineers to understand whether controllable factors are significant or not at the jth response.
H.-C. Liao / Computers Industrial Engineering 46 (2004) 817–835 823

5. Illustrations
In this section, one case study in Su and Tong (1997) is illustrated to confirm the feasibility of optimal
procedure in this article. This case study involves improving the quality of hard disk drive. It also
includes five controllable factors: (A) disk writability; (B) magnetization width; (C) gap length; (D)
coercivity of media; (E) rotational speed, and one uncontrollable factor: (N) flying height (noise). The
four desired responses are: (1) 50% pulse width (PW50, smaller-the-better); (2) peak shift (PS, smaller-
the-better); (3) high-frequency amplitude (HFA, larger-the-better), and (4) over write (OW, larger-the-
better).
In this case, controllable factor (A) has two levels and the others have three levels. In addition, the
standard array L18 is selected as an example. Under the circumstances of this experiment with all
complete data, the starting condition in this experiment is A1B1C2D2E3, the optimal combination of
factors/levels chosen in Taguchi method is A1B1C1D1E3, and that in PCA method is A1B1C1D3E3. Now,
if this experiment produces complete data and censored data, the data of PW50 and PS responses are
censored to right and HFA and OW responses are censored to left. PW50’s censored point is 70 and PS’s
censored point is 15. HFA’s censored point is 300 and OW’s censored point is 35.
The proposed optimal procedure is described as the following steps:
Step 1. Use the BP model to estimate the SN ratio of all factors/levels combination.
Table 1 shows the situation about the experimental complete data and censored data. Table 2 shows
the SN ratios obtained by using complete data for each response. Table 3 shows the more appropriate
results of BP trained by SN ratios for different responses. Appendix A shows the estimated SN ratios of
all factors/levels combinations. The total number of factors/levels combination is 162 (2 £ 34
).
Table 1
Data summary with complete data and censored data
Exp. no. L18 PW 50 HFA OW PS
A B C D E F G H N1 N2 N1 N2 N1 N2 N1 N2
1 1 1 1 1 1 1 1 1 63.5 66.0 * * 232.2 230.1 10.9 12.0
2 1 1 2 2 2 2 2 2 64.2 66.0 343.0 310.6 234.8 233.3 11.6 13.0
3 1 1 3 3 3 3 3 3 65.6 67.0 381.1 354.4 * * 13.6 14.7
4 1 2 1 1 2 2 3 3 54.5 56.6 328.1 * 233.5 231.5 9.2 10.8
5 1 2 2 2 3 3 1 1 56.2 57.8 368.3 333.0 236.2 234.9 10.2 11.2
6 1 2 3 3 1 1 2 2 * * * * * * * *
7 1 3 1 2 1 3 2 3 63.6 66.1 * * 231.7 229.5 10.1 11.8
8 1 3 2 3 2 1 3 1 64.3 66.1 335.8 304.9 * 233.9 10.7 12.1
9 1 3 3 1 3 2 1 2 65.6 66.9 312.7 * * * 14.4 *
10 2 1 1 3 3 2 2 1 47.7 49.5 451.0 393.8 215.6 222.3 11.0 11.8
11 2 1 2 1 1 3 3 2 * * * * 233.6 232.6 16.3 *
12 2 1 3 2 2 1 1 3 * * 346.8 312.4 * 233.8 * *
13 2 2 1 2 3 1 3 2 47.7 49.5 447.9 393.8 225.8 222.3 10.0 11.6
14 2 2 2 3 1 2 1 3 * * 312.8 * 229.7 228.9 14.6 *
15 2 2 3 1 2 3 2 1 * * * * * * * *
16 2 3 1 3 2 3 1 2 54.5 56.6 385.2 336.7 220.4 217.2 11.6 13.4
17 2 3 2 1 3 1 2 3 56.2 57.8 378.7 341.5 * 234.6 12.1 13.4
18 2 3 3 2 1 2 3 1 * * * * * * * *
H.-C. Liao / Computers Industrial Engineering 46 (2004) 817–835824

Step 2. Normalize Xij:
Appendix B lists the normalized ratios of SN ratios in Appendix A for further DEA analysis in order
to obtain the results of relative efficiency.
Step 3. Calculate relative efficiency for each factors/levels combination.
Table 4 shows the relative efficiency results of all factors/level combination.
Step 4. Rank the DMU according to its relative efficiency and obtain the optimal factors/levels
combination.
Table 5 lists the set of factors/levels combinations with their relative efficiency 100%. According to
the result of Table 5, the optimal condition can be set as any factors/levels combination shown in Table 5
by engineers’ consideration.
Step 5. Obtain the significant and not significant controllable factors for these four responses.
Table 6 performs the ANOVA based on estimated SN ratios. From Table 6, in response PW50, factors
C and E are significant when the significance level a is 0.05, and factors A, C, and E are significant when
Table 2
SN ratios for complete data
Exp no. L18 PW 50 SN ratio PS
A B C D E F G H HFA OW
1 1 1 1 1 1 1 1 1 236.23 * 29.85 221.19
2 1 1 2 2 2 2 2 2 236.27 50.25 30.64 221.81
3 1 1 3 3 3 3 3 3 236.43 51.29 * 223.02
4 1 2 1 1 2 2 3 3 234.90 * 30.23 220.03
5 1 2 2 2 3 3 1 1 235.12 50.86 31.01 220.60
6 1 2 3 3 1 1 2 2 * * * *
7 1 3 1 2 1 3 2 3 236.24 * 29.70 220.81
8 1 3 2 3 2 1 3 1 236.29 50.08 * 221.15
9 1 3 3 1 3 2 1 2 236.42 * * *
10 2 1 1 3 3 2 2 1 233.73 52.45 25.14 221.14
11 2 1 2 1 1 3 3 2 * * 30.39 *
12 2 1 3 2 2 1 1 3 * 50.32 * *
13 2 2 1 2 3 1 3 2 233.73 52.43 27.55 220.69
14 2 2 2 3 1 2 1 3 * * 29.33 *
15 2 2 3 1 2 3 2 1 * * * *
16 2 3 1 3 2 3 1 2 234.90 51.09 25.39 221.96
17 2 3 2 1 3 1 2 3 235.12 51.09 * 222.12
18 2 3 3 2 1 2 3 1 * * * *
Table 3
The selected BP model (learning rate 0.01–0.3, iteration 5000)
Response Architecture (nodes)
input-hidden-output
Moment RMSE (training) RMSE (testing)
PW50 5-4-1 0.75 0.004593 0.009895
HFA 5-4-1 0.8 0.01116 0.081980
OW 5-2-1 0.73 0.01395 0.054671
PS 5-3-1 0.75 0.014803 0.057959

Table 4
The relative efficiency for each DMU
DMU Efficiency DMU Efficiency DMU Efficiency DMU Efficiency
1 96.63 46 99.92 91 97.61 136 95.53
2 98.46 47 99.89 92 98.13 137 97.33
3 100.00 48 99.91 93 99.74 138 99.86
4 96.03 49 99.91 94 96.65 139 95.22
5 97.90 50 99.89 95 97.78 140 97.12
6 100.00 51 99.92 96 100.00 141 99.90
7 95.63 52 99.88 97 95.84 142 95.40
8 97.63 53 99.87 98 97.88 143 97.49
9 100.00 54 100.00 99 100.00 144 99.98
10 98.78 55 97.48 100 99.34 145 98.27
11 99.03 56 98.98 101 99.30 146 98.61
12 99.90 57 100.00 102 99.62 147 99.59
13 98.59 58 96.61 103 99.25 148 97.67
14 98.92 59 98.59 104 99.25 149 98.31
15 99.92 60 100.00 105 99.77 150 99.82
16 98.06 61 96.04 106 98.98 151 96.38
17 98.64 62 98.34 107 99.24 152 97.86
18 100.00 63 100.00 108 100.00 153 100.00
19 99.83 64 99.14 109 95.33 154 99.61
20 99.80 65 99.47 110 97.26 155 99.58
21 99.83 66 100.00 111 99.97 156 99.71
22 99.82 67 99.02 112 95.19 157 99.56
23 99.79 68 99.38 113 97.24 158 99.55
24 99.84 69 100.00 114 99.98 159 99.87
25 99.78 70 98.74 115 95.54 160 99.42
26 99.78 71 99.17 116 97.81 161 99.48
27 100.00 72 100.00 117 100.00 162 100.00
28 97.03 73 100.00 118 97.95
29 98.82 74 99.98 119 98.38
30 100.00 75 100.00 120 99.62
31 96.34 76 99.99 121 97.19
32 98.33 77 99.97 122 98.02
33 100.00 78 100.00 123 99.94
34 95.80 79 99.97 124 95.99
35 98.22 80 99.96 125 97.74
36 100.00 81 100.00 126 100.00
37 98.96 82 95.23 127 99.48
38 99.18 83 97.26 128 99.45
39 99.96 84 100.00 129 99.66
40 98.82 85 95.27 130 99.41
41 99.11 86 97.41 131 99.41
42 99.97 87 100.00 132 99.82
43 98.43 88 95.72 133 99.22
44 98.88 89 98.15 134 99.34
45 100.00 90 100.00 135 100.00

the significance level a is 0.1. In responses HFA and OW, all factors A, B, C, D, and E are significant
when the significance levels a are 0.05 and 0.1. In response PS, factors A, B, C, and E are significant
when the significance levels a are 0.05 and 0.1.
From Table 5, it is known that the obtained factors/levels combinations with all complete data from
Taguchi method and PCA method are included in the combination set with relative efficiency 100%. The
obtained factors/levels combination from Taguchi method is shown in DMU 3 of Table 5 and the
obtained factors/levels combination from PCA method is shown in DMU 9 of Table 5. So, in this case
study with complete data and censored data, the obtained factors/levels combination from Taguchi
method and the obtained factors/levels combination from PCA method are special cases in the proposed
optimal procedure in this paper. Additionally, the starting condition in Table 4 is in DMU 15 whose
relative efficiency is 99.92%. By using the proposed optimal procedure, the set of relative efficiency
100% of Table 5 will improve the quality of hard disk drive. In addition, from Fig. 3, the DEA also
suggests potentially enhanced total quality according to formulation (7). The first consideration is to
improve PS (the potential performance improvement is 65.35%) without considering improving OW
which is less worthy (the potential performance improvement is only 0.39%.)
6. Conclustion
In this article, the NN method and the DEA method are proposed to achieve the optimization of
multi-response problems with censored data. Five steps are included in the proposed optimal
procedure. (1) Use NN to estimate SN ratios of all factors/levels combination by complete data.
(2) Normalize SN ratios. (3) Calculate each DMU’s relative efficiency. (4) Rank DMU and obtain
Table 5
The DMU of relative efficiency 100%
DMU Combination DMU Combination
3 A1B1C1D1E3 73 A1B3C3D1E1
6 A1B1C1D2E3 75 A1B3C3D1E3
9 A1B1C1D3E3 78 A1B3C3D2E3
18 A1B1C2D3E3 81 A1B3C3D3E3
27 A1B1C3D3E3 84 A2B1C1D1E3
30 A1B2C1D1E3 87 A2B1C1D2E3
33 A1B2C1D2E3 90 A2B1C1D3E3
36 A1B2C1D3E3 96 A2B1C2D2E3
45 A1B2C2D3E3 99 A2B1C2D3E3
54 A1B2C3D3E3 108 A2B1C3D3E3
57 A1B3C1D1E3 117 A2B2C1D3E3
60 A1B3C1D2E3 126 A2B2C2D3E3
63 A1B3C1D3E3 135 A2B2C3D3E3
66 A1B3C2D1E3 153 A2B3C2D3E3
69 A1B3C2D2E3 162 A2B3C3D3E3
72 A1B3C2D3E3

the set of optimal factors/levels combinations. (5) Determine the significant controllable factors for
each response by ANOVA. One real case study from Su’s paper on hard disk drive’s quality process
has been performed to substantiate the proposed optimum procedure to indicate its feasibility and
effectiveness compared to PCA method and Taguchi method. Three significant contributions are
achieved in the proposed optimal procedure. First, in the traditional statistical method to estimate
censored data, the statistic assumption must be considered and these methods are complicated. The NN
can be regarded as a statistical method without requiring any assumptions concerning the relationship
between inputs and outputs. Secondly, the proposed optimal procedure reduces the uncertainty and
complexity of engineers’ judgment acquired in multi-response problem. Thirdly, this procedure is a
universal approach to optimize the multi-response robust design. Future research can be considered in
other complementary aspects and practical implications of NN theory and DEA theory. For instance,
with the extension of fuzziness, the proposed optimal procedure is made more powerful for practical
applications.
Table 6
NOVA of the estimated multiple SN ratios
Source Dependent variable Sum of squares Degrees of freedom Mean square F Sig.
A PW50 0.324 1 0.324 3.039 0.083
HFA 10.458 1 10.458 182.007 0.000
OW 56.478 1 56.478 85.379 0.000
PS 74.149 1 74.149 280.776 0.000
B PW50 0.03717 2 0.01858 0.174 0.840
HFA 1.030 2 0.515 8.963 0.000
OW 4.878 2 2.439 3.687 0.027
PS 5.075 2 2.538 9.609 0.000
C PW50 82.311 2 41.155 385.417 0.000
HFA 3.282 2 1.641 28.555 0.000
OW 319.197 2 159.599 241.269 0.000
PS 129.141 2 64.570 244.506 0.000
D PW50 0.02349 2 0.01175 0.110 0.896
HFA 9.427 2 4.714 82.032 0.000
OW 30.554 2 15.277 23.094 0.000
PS 0.445 2 0.222 0.842 0.433
E PW50 63.411 2 31.706 296.920 0.000
HFA 69.713 2 34.856 606.628 0.000
OW 4.690 2 2.345 3.545 0.031
PS 58.049 2 29.025 109.907 0.000
Error PW50 16.231 152 0.107
HFA 8.734 152 0.05746
OW 100.547 152 0.661
PS 40.141 152 0.264
Total PW50 208,759.368 162
HFA 412,241.962 162
OW 146,310.518 162
PS 80,861.436 162
a ¼ 0:05; 0.1 significance levels.

Appendix A
Fig. 3. The potential quality improvement.
Table A1
The estimated SN ratios of each factors/levels combination
No. A B C D E PW50 HFA OW PS
1 1 1 1 1 1 236.2168 49.6691 29.8496 221.1901
2 1 1 1 1 2 234.8881 49.8982 30.0386 220.0730
3 1 1 1 1 3 233.8281 50.8313 30.1368 219.6721
4 1 1 1 2 1 236.1990 49.7220 28.7785 221.3119
5 1 1 1 2 2 234.8624 50.1419 29.3413 220.0879
6 1 1 1 2 3 233.8140 51.3972 29.6946 219.6687
7 1 1 1 3 1 236.1810 49.8201 26.7678 221.4728
8 1 1 1 3 2 234.8378 50.5371 27.7412 220.1284
9 1 1 1 3 3 233.8011 51.8988 28.5687 219.6739
10 1 1 2 1 1 236.8017 49.6503 31.0159 223.2259
11 1 1 2 1 2 236.3293 49.8091 30.9945 221.8989
12 1 1 2 1 3 235.0729 50.5239 30.9651 220.3549
13 1 1 2 2 1 236.7961 49.6876 30.9378 223.3140
14 1 1 2 2 2 236.3134 49.9815 30.9461 222.0916
15 1 1 2 2 3 235.0469 51.0580 30.9354 220.4384
16 1 1 2 3 1 236.7902 49.7557 30.7174 223.3816
(continued on next page)

Table A1 (continued)
17 1 1 2 3 2 236.2971 50.2831 30.8094 222.2996
18 1 1 2 3 3 235.0217 51.6137 30.8503 220.5655
19 1 1 3 1 1 236.9343 49.6370 31.3903 223.6251
20 1 1 3 1 2 236.8275 49.7478 31.3715 223.4864
21 1 1 3 1 3 236.4268 50.2694 31.3508 222.7239
22 1 1 3 2 1 236.9324 49.6636 31.3856 223.6303
23 1 1 3 2 2 236.8226 49.8675 31.3690 223.5192
24 1 1 3 2 3 236.4127 50.7271 31.3498 222.8804
25 1 1 3 3 1 236.9304 49.7113 31.3710 223.6335
26 1 1 3 3 2 236.8174 50.0861 31.3603 223.5423
27 1 1 3 3 3 236.3981 51.2877 31.3448 223.0200
28 1 2 1 1 1 236.2361 49.6582 30.1044 220.9126
29 1 2 1 1 2 234.9622 49.8454 30.2308 220.0295
30 1 2 1 1 3 233.8471 50.6526 30.2895 219.6736
31 1 2 1 2 1 236.2181 49.7024 29.2852 220.9581
32 1 2 1 2 2 234.9365 50.0476 29.7096 220.0082
33 1 2 1 2 3 233.8332 51.2040 29.9621 219.6574
34 1 2 1 3 1 236.1999 49.7836 27.4929 221.0385
35 1 2 1 3 2 234.9119 50.3890 28.4060 220.0045
36 1 2 1 3 3 233.8204 51.7372 29.0923 219.6480
37 1 2 2 1 1 236.7856 49.6425 31.0839 222.8205
38 1 2 2 1 2 236.3395 49.7726 31.0606 221.3312
39 1 2 2 1 3 235.1463 50.3740 31.0309 220.1433
40 1 2 2 2 1 236.7794 49.6738 31.0272 222.9666
41 1 2 2 2 2 236.3233 49.9144 31.0256 221.4576
42 1 2 2 2 3 235.1203 50.8675 31.0096 220.1585
43 1 2 2 3 1 236.7729 49.7306 30.8673 223.0944
44 1 2 2 3 2 236.3067 50.16800 30.9263 221.6221
45 1 2 2 3 3 235.0952 51.4289 30.9480 220.2007
46 1 2 3 1 1 236.9234 49.6312 31.4196 223.5877
47 1 2 3 1 2 236.8114 49.7226 31.4022 223.2882
48 1 2 3 1 3 236.4295 50.1531 31.3833 222.0760
49 1 2 3 2 1 236.9212 49.6538 31.4163 223.6034
50 1 2 3 2 2 236.8059 49.8207 31.4006 223.3660
51 1 2 3 2 3 236.4149 50.5571 31.3828 222.2611
52 1 2 3 3 1 236.9189 49.6938 31.4057 223.6137
53 1 2 3 3 2 236.8001 50.0018 31.3944 223.4252
54 1 2 3 3 3 236.4000 51.0935 31.3793 222.4573
55 1 3 1 1 1 236.2446 49.6492 30.3129 220.8170
56 1 3 1 1 2 235.0564 49.8027 30.3930 220.0620
57 1 3 1 1 3 233.8952 50.4893 30.423 219.7032
58 1 3 1 2 1 236.2262 49.6864 29.6999 220.8102
59 1 3 1 2 2 235.0305 49.9699 30.0079 220.0157
60 1 3 1 2 3 233.8810 51.0101 30.1822 219.6757
61 1 3 1 3 1 236.2075 49.7541 28.2054 220.8289
62 1 3 1 3 2 235.0057 50.2604 28.9906 219.9837
63 1 3 1 3 3 233.868 51.5605 29.5262 219.6551
64 1 3 2 1 1 236.7617 49.6360 31.1446 222.3509
65 1 3 2 1 2 236.3381 49.7432 31.1205 221.0163

66 1 3 2 1 3 235.2336 50.2433 31.0913 220.0860
67 1 3 2 2 1 236.7547 49.6625 31.1034 222.5113
68 1 3 2 2 2 236.3213 49.8598 31.0953 221.0657
69 1 3 2 2 3 235.2074 50.6872 31.0762 220.0624
70 1 3 2 3 1 236.7472 49.7101 30.9873 222.6749
71 1 3 2 3 2 236.3042 50.0715 31.0232 221.1500
72 1 3 2 3 3 235.1822 51.2366 31.0315 220.0570
73 1 3 3 1 1 236.9084 49.6264 31.4460 223.4879
74 1 3 3 1 2 236.7878 49.7022 31.4301 222.9131
75 1 3 3 1 3 236.4203 50.0554 31.4128 221.4829
76 1 3 3 2 1 236.9058 49.6457 31.4437 223.5309
77 1 3 3 2 2 236.7815 49.7830 31.4291 223.0500
78 1 3 3 2 3 236.4050 50.4045 31.4126 221.6134
79 1 3 3 3 1 236.9030 49.6794 31.4360 223.5607
80 1 3 3 3 2 236.7749 49.9325 31.4246 223.1684
81 1 3 3 3 3 236.3894 50.9027 31.4103 221.7800
82 2 1 1 1 1 236.0942 49.7778 27.0965 223.3723
83 2 1 1 1 2 234.7474 50.4029 27.9642 222.4867
84 2 1 1 1 3 233.7523 51.7850 28.6329 220.8755
85 2 1 1 2 1 236.0742 49.9259 25.2590 223.4334
86 2 1 1 2 2 234.7229 50.9115 25.9945 222.6549
87 2 1 1 2 3 233.7402 52.1886 26.8953 220.9871
88 2 1 1 3 1 236.0540 50.1919 24.4326 223.4770
89 2 1 1 3 2 234.6996 51.4769 24.6908 222.8092
90 2 1 1 3 3 233.7292 52.4455 25.1401 221.1400
91 2 1 2 1 1 236.7504 49.7259 30.5086 223.6326
92 2 1 2 1 2 236.2156 50.1729 30.5634 223.5470
93 2 1 2 1 3 234.9278 51.4728 30.5784 223.0672
94 2 1 2 2 1 236.7432 49.8280 30.0216 223.6370
95 2 1 2 2 2 236.1972 50.5925 30.2595 223.5693
96 2 1 2 2 3 234.9024 51.9665 30.3888 223.1781
97 2 1 2 3 1 236.7356 50.0177 28.7583 223.6398
98 2 1 2 3 2 236.1787 51.1405 29.4316 223.5842
99 2 1 2 3 3 234.8781 52.3073 29.8646 223.2657
100 2 1 3 1 1 236.9143 49.6904 31.2229 223.6543
101 2 1 3 1 2 236.7814 50.002 31.2008 223.6468
102 2 1 3 1 3 236.3223 51.1292 31.1745 223.6135
103 2 1 3 2 1 236.9118 49.7607 31.1907 223.6546
104 2 1 3 2 2 236.7750 50.3225 31.1812 223.6480
105 2 1 3 2 3 236.3057 51.6877 31.1628 223.6199
106 2 1 3 3 1 236.9091 49.8926 31.0998 223.6548
107 2 1 3 3 2 236.7683 50.8035 31.1248 223.6486
108 2 1 3 3 3 236.2888 52.1216 31.1280 223.6239
109 2 2 1 1 1 236.1138 49.7474 27.7607 223.1088
110 2 2 1 1 2 234.8401 50.2680 28.5213 222.0253
111 2 2 1 1 3 233.7882 51.6133 29.0537 220.6529
112 2 2 1 2 1 236.0934 49.8693 25.7520 223.2201
113 2 2 1 2 2 234.8152 50.7282 26.6274 222.1841

114 2 2 1 2 3 233.7759 52.0663 27.5498 220.6902
115 2 2 1 3 1 236.0729 50.0917 24.6077 223.3071
116 2 2 1 3 2 234.7916 51.2861 24.9844 222.3530
117 2 2 1 3 3 233.7648 52.3673 25.5966 220.7616
118 2 2 2 1 1 236.7263 49.7050 30.6557 223.5988
119 2 2 2 1 2 236.2237 50.0717 30.6848 223.4072
120 2 2 2 1 3 235.0179 51.2817 30.6843 222.5967
121 2 2 2 2 1 236.7183 49.7889 30.2992 223.6120
122 2 2 2 2 2 236.2049 50.4358 30.4629 223.4636
123 2 2 2 2 3 234.9922 51.8135 30.5461 222.7595
124 2 2 2 3 1 236.7098 49.9458 29.3301 223.6208
125 2 2 2 3 2 236.1859 50.9485 29.847 223.5036
126 2 2 2 3 3 234.9678 52.2041 30.1608 222.9071
127 2 2 3 1 1 236.8983 49.6758 31.2698 223.6512
128 2 2 3 1 2 236.7573 49.9297 31.2482 223.6362
129 2 2 3 1 3 236.3210 50.9362 31.2231 223.5637
130 2 2 3 2 1 236.8954 49.7338 31.2465 223.6522
131 2 2 3 2 2 236.7501 50.1995 31.2341 223.6401
132 2 2 3 2 3 236.3037 51.5081 31.2149 223.5833
133 2 2 3 3 1 236.8922 49.8425 31.1802 223.6529
134 2 2 3 3 2 236.7425 50.6285 31.1931 223.6426
135 2 2 3 3 3 236.2863 51.9889 31.1897 223.5964
136 2 3 1 1 1 236.1229 49.7229 28.3787 222.7792
137 2 3 1 1 2 234.9503 50.1529 29.0007 221.7063
138 2 3 1 1 3 233.8522 51.4281 29.4048 220.5908
139 2 3 1 2 1 236.1019 49.8233 26.3605 222.9300
140 2 3 1 2 2 234.9248 50.5594 27.3086 221.8204
141 2 3 1 2 3 233.8394 51.9253 28.1702 220.5787
142 2 3 1 3 1 236.0809 50.0086 24.8597 223.0624
143 2 3 1 3 2 234.9007 51.0932 25.3900 221.9598
144 2 3 1 3 3 233.8278 52.2741 26.1699 220.5918
145 2 3 2 1 1 236.6939 49.688 30.7789 223.5183
146 2 3 2 1 2 236.2209 49.9879 30.7900 223.1518
147 2 3 2 1 3 235.1197 51.0866 30.7791 222.1200
148 2 3 2 2 1 236.6847 49.7574 30.5193 223.5525
149 2 3 2 2 2 236.2013 50.2985 30.6285 223.2596
150 2 3 2 2 3 235.0936 51.6436 30.6786 222.2801
151 2 3 2 3 1 236.6751 49.8870 29.7955 223.5759
152 2 3 2 3 2 236.1816 50.7651 30.1761 223.3434
153 2 3 2 3 3 235.0687 52.0832 30.3971 222.4490
154 2 3 3 1 1 236.8771 49.6638 31.3117 223.6428
155 2 3 3 1 2 236.7251 49.8709 31.2911 223.6068
156 2 3 3 1 3 236.3084 50.7503 31.2675 223.4389
157 2 3 3 2 1 236.8736 49.7120 31.2949 223.6458
158 2 3 3 2 2 236.7168 50.0957 31.2810 223.6187
159 2 3 3 2 3 236.2902 51.3179 31.2617 223.4904
160 2 3 3 3 1 236.8697 49.8019 31.2465 223.6479
161 2 3 3 3 2 236.708 50.4701 31.2512 223.6265
162 2 3 3 3 3 236.2719 51.8379 31.2435 223.5266

Appendix B
Table B1
Normalized data
No. PW50 HFA OW PS No. PW50 HFA OW PS
1 1.0738 0.9471 0.9492 1.0785 82 1.0701 0.9491 0.8617 1.1895
2 1.0344 0.9514 0.9552 1.0216 83 1.0302 0.9611 0.8893 1.1445
3 1.0029 0.9692 0.9584 1.0012 84 1.0007 0.9874 0.9105 1.0625
4 1.0732 0.9481 0.9152 1.0847 85 1.0695 0.9520 0.8033 1.1927
5 1.0336 0.9561 0.9331 1.0224 86 1.0295 0.9708 0.8266 1.1530
6 1.0025 0.9800 0.9443 1.0011 87 1.0003 0.9951 0.8553 1.0682
7 1.0727 0.9499 0.8512 1.0929 88 1.0689 0.9570 0.7770 1.1949
8 1.0329 0.9636 0.8822 1.0244 89 1.0288 0.9815 0.7852 1.1609
9 1.0021 0.9896 0.9085 1.0013 90 1.0000 1.0000 0.7995 1.0759
10 1.0911 0.9467 0.9863 1.1821 91 1.0896 0.9481 0.9702 1.2028
11 1.0771 0.9497 0.9856 1.1146 92 1.0737 0.9567 0.9719 1.1984
12 1.0398 0.9634 0.9847 1.0360 93 1.0355 0.9815 0.9724 1.1740
13 1.0909 0.9474 0.9838 1.1866 94 1.0894 0.9501 0.9547 1.2030
14 1.0766 0.9530 0.9841 1.1244 95 1.0732 0.9647 0.9623 1.1996
15 1.0391 0.9735 0.9838 1.0402 96 1.0348 0.9909 0.9664 1.1797
16 1.0908 0.9487 0.9768 1.1900 97 1.0891 0.9537 0.9145 1.2032
17 1.0761 0.9588 0.9798 1.1350 98 1.0726 0.9751 0.9359 1.2003
18 1.0383 0.9841 0.9811 1.0467 99 1.0341 0.9974 0.9497 1.1841
19 1.0950 0.9465 0.9982 1.2024 100 1.0944 0.9475 0.9929 1.2039
20 1.0919 0.9486 0.9976 1.1954 101 1.0905 0.9534 0.9922 1.2035
21 1.0800 0.9585 0.9970 1.1565 102 1.0769 0.9749 0.9914 1.2018
22 1.0950 0.9470 0.9981 1.2027 103 1.0944 0.9488 0.9919 1.2039
23 1.0917 0.9508 0.9976 1.1970 104 1.0903 0.9595 0.9916 1.2036
24 1.0796 0.9672 0.9969 1.1645 105 1.0764 0.9856 0.9910 1.2022
25 1.0949 0.9479 0.9976 1.2028 106 1.0943 0.9513 0.9890 1.2039
26 1.0916 0.9550 0.9973 1.1982 107 1.0901 0.9687 0.9898 1.2036
27 1.0791 0.9779 0.9968 1.1716 108 1.0759 0.9938 0.9899 1.2024
28 1.0743 0.9469 0.9573 1.0644 109 1.0707 0.9486 0.8828 1.1761
29 1.0366 0.9504 0.9614 1.0194 110 1.0329 0.9585 0.9070 1.1210
30 1.0035 0.9658 0.9632 1.0013 111 1.0017 0.9841 0.9239 1.0511
31 1.0738 0.9477 0.9313 1.0667 112 1.0701 0.9509 0.8189 1.1818
32 1.0358 0.9543 0.9448 1.0183 113 1.0322 0.9673 0.8468 1.1291
33 1.0031 0.9763 0.9528 1.0005 114 1.0014 0.9928 0.8761 1.0530
34 1.0733 0.9492 0.8743 1.0708 115 1.0695 0.9551 0.7825 1.1862
35 1.0351 0.9608 0.9033 1.0181 116 1.0315 0.9779 0.7945 1.1377
36 1.0027 0.9865 0.9252 1.0000 117 1.0011 0.9985 0.8140 1.0567
37 1.0906 0.9466 0.9885 1.1615 118 1.0889 0.9477 0.9749 1.2011
38 1.0774 0.9490 0.9877 1.0857 119 1.0740 0.9547 0.9758 1.1913
39 1.0420 0.9605 0.9868 1.0252 120 1.0382 0.9778 0.9758 1.1501
40 1.0904 0.9472 0.9867 1.1689 121 1.0886 0.9493 0.9635 1.2017
41 1.0769 0.9517 0.9866 1.0921 122 1.0734 0.9617 0.9687 1.1942
42 1.0412 0.9699 0.9861 1.0260 123 1.0374 0.9880 0.9714 1.1584
43 1.0902 0.9482 0.9816 1.1754 124 1.0884 0.9523 0.9327 1.2022
44 1.0764 0.9566 0.9835 1.1005 125 1.0728 0.9715 0.9492 1.1962
45 1.0405 0.9806 0.9842 1.0281 126 1.0367 0.9954 0.9591 1.1659

References
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ed). Dordrecht: Kluwer.
Dayhoff, J. E. (1990). Neural network architecture. New York: Van Nostrand Reinhold.
Table B1 (continued)
No. PW50 HFA OW PS No. PW50 HFA OW PS
46 1.0947 0.9463 0.9992 1.2005 127 1.0940 0.9472 0.9944 1.2037
47 1.0914 0.9481 0.9986 1.1853 128 1.0898 0.9520 0.9937 1.2030
48 1.0801 0.9563 0.9980 1.1236 129 1.0768 0.9712 0.9929 1.1993
49 1.0946 0.9468 0.9991 1.2013 130 1.0939 0.9483 0.9937 1.2038
50 1.0912 0.9500 0.9986 1.1892 131 1.0896 0.9572 0.9933 1.2032
51 1.0796 0.9640 0.9980 1.1330 132 1.0763 0.9821 0.9927 1.2003
52 1.0946 0.9475 0.9987 1.2018 133 1.0938 0.9504 0.9915 1.2038
53 1.0910 0.9534 0.9984 1.1922 134 1.0893 0.9654 0.9920 1.2033
54 1.0792 0.9742 0.9979 1.1430 135 1.0758 0.9913 0.9919 1.2010
55 1.0746 0.9467 0.9640 1.0595 136 1.0710 0.9481 0.9025 1.1594
56 1.0393 0.9496 0.9665 1.0211 137 1.0362 0.9563 0.9222 1.1048
57 1.0049 0.9627 0.9675 1.0028 138 1.0036 0.9806 0.9351 1.0480
58 1.0740 0.9474 0.9445 1.0592 139 1.0703 0.9500 0.8383 1.1670
59 1.0386 0.9528 0.9543 1.0187 140 1.0354 0.9640 0.8684 1.1106
60 1.0045 0.9726 0.9598 1.0014 141 1.0033 0.9901 0.8958 1.0474
61 1.0735 0.9487 0.8969 1.0601 142 1.0697 0.9535 0.7906 1.1738
62 1.0378 0.9583 0.9219 1.0171 143 1.0347 0.9742 0.8074 1.1177
63 1.0041 0.9831 0.9390 1.0004 144 1.0029 0.9967 0.8322 1.0480
64 1.0899 0.9464 0.9904 1.1376 145 1.0879 0.9474 0.9788 1.1970
65 1.0773 0.9485 0.9896 1.0696 146 1.0739 0.9531 0.9791 1.1783
66 1.0446 0.9580 0.9887 1.0223 147 1.0412 0.9741 0.9788 1.1258
67 1.0897 0.9469 0.9891 1.1457 148 1.0876 0.9487 0.9705 1.1987
68 1.0769 0.9507 0.9888 1.0722 149 1.0733 0.9591 0.9740 1.1838
69 1.0438 0.9665 0.9882 1.0211 150 1.0405 0.9847 0.9756 1.1340
70 1.0895 0.9478 0.9854 1.1541 151 1.0873 0.9512 0.9475 1.1999
71 1.0763 0.9547 0.9866 1.0764 152 1.0727 0.9680 0.9596 1.1881
72 1.0431 0.9770 0.9868 1.0208 153 1.0397 0.9931 0.9666 1.1426
73 1.0943 0.9462 1.0000 1.1954 154 1.0933 0.9470 0.9957 1.2033
74 1.0907 0.9477 0.9995 1.1662 155 1.0888 0.9509 0.9951 1.2015
75 1.0798 0.9544 0.9989 1.0934 156 1.0765 0.9677 0.9943 1.1929
76 1.0942 0.9466 0.9999 1.1976 157 1.0932 0.9479 0.9952 1.2035
77 1.0905 0.9492 0.9995 1.1731 158 1.0886 0.9552 0.9948 1.2021
78 1.0793 0.9611 0.9989 1.1000 159 1.0759 0.9785 0.9941 1.1956
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80 1.0903 0.9521 0.9993 1.1792 161 1.0883 0.9623 0.9938 1.2025
81 1.0789 0.9706 0.9989 1.1085 162 1.0754 0.9884 0.9936 1.1974

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