3. Selection Method
3.2 Bootstrap Methodology
Traditionally, statistical research work has relied on the central limit theorem and normal approximations to obtain standard errors and confidence intervals.
These techniques are valid only when the statistic, or some known transformation of statistic, is asymptotically normal distribution. Unfortunately, most process data in real world are not normal distributed. More than that, the distribution of data is usually unknown. A major motivation for the traditional reliance on normal-theory methods has been computational tractability. Access to powerful computation enables the use of statistics in new and varied way. Idealized models and assumptions can now be replaced with more realistic modeling or by virtually model-free analyses. Efron (1979, 1982) introduced a nonparametric, computational intensive but effective estimation method, called the “Bootstrap”, which is a data based simulation technique for statistical inference. One can use the nonparametric bootstrap method to estimate the sampling distribution of a statistic, while assuming only that the observations are independent and identically distributed. The merit of the nonparametric bootstrap approach is that it does not rely on any assumptions regarding the underlying distribution. Rather than using distribution frequency tables to compute approximate p probability vales, the bootstrap method generates a unique sampling distribution based on the actual sample rather than the analytic method.
Most of PCIs literature concluded that the performance of bootstrap limits for PCIs are quite satisfactory in the majority of the cases. After Efron (1979, 1982) introduced the bootstrap, Efron and Tibshirani (1986) further developed three bootstrap confidence intervals: the standard bootstrap (SB) confidence interval, the percentile bootstrap (PB) confidence interval, and the biased-corrected percentile bootstrap (BCPB) confidence interval. Franklin and Wasserman (1991) proposed an initial study of these three methods for obtaining confidence intervals for Cpk when the process was normal distributed. Franklin and Wasserman (1992) also offered three bootstrap lower confidence limits for index Cp, Cpk, and Cpm. They compared the confidence interval from bootstrap
and from parametric estimates. The simulation results show that, the bootstrap confidence limits perform as good as the lower confidence limits derived by the parametric method in the normal process environment. (see Chou et al (1990) for Cp, Bissell (1990) for Cpk, and Boyles (1991) for Cpm). These studies indicate that the bootstrap limits for PCIs are satisfactory in the cases.
In this thesis, the following four bootstrap confidence limits are employed to determine the lower confidence bounds of difference and ratio statistics and the results are used to select the better supplier of the two candidates. For
n
1=n
2=n, let two bootstrap samples of size n drawn with replacement from the two original sample be denoted by
x x11 21*, ,...,* x1*n
and
x x21 22*, ,...,* x2*n
. The bootstrap sample statisticsx1*, s1*, x2*, and s2* are computed, as well as ˆ* 1S
pk , and ˆ* 2S
pk . A random sample of nn possible resamples are drawn, the statistic is calculated by each of these, and the resulting empirical distribution is referred to as the bootstrap distribution of statistic. Due to the overwhelming computation time, it is not of practical interest to chose nn such samples. Empirical work (Eforn and Tibshirani (1986)) indicated that a roughly minimum of 1,000 bootstrap resamples is usually sufficient to compute reasonable accurate confidence interval estimates for population parameters. For accuracy purpose, we consider B=3,000 bootstrap resamples (rather than 1,000). Thus, we take B=3,000 bootstrap estimates qˆ
* = (S
ˆ*pk2S
ˆ*pk1) or (S
ˆ*pk2/S
ˆ*pk1) of θ = Spk2
Spk1 or Spk2/
Spk1 , respectively, then order them from the smallest to the largest
ˆ( )*l (S
ˆ*pk2S
ˆ*pk1 ( ))l or
ˆ( )*l (S
ˆ*pk2/S
ˆ*pk1 ( ))l where l 1,2, ,
B.Four types of bootstrap confidence intervals, including the standard bootstrap confidence interval (SB), the percentile bootstrap confidence interval (PB), the biased corrected percentile bootstrap confidence interval (BCPB), and the bootstrap-t (BT) methods introduced by Efron (1981), and Efron and Tibshirani (1986) are conducted in this paper. The generic notation ˆ
and are
ˆ* the estimator of θ and the associated ordered bootstrap estimate. Construction of a two-sided100(1 2 )%
confidence limit will be described. We note that a lower100(1 )%
confidence limit can be obtained by using only a lower limit.The formulation details for the four types of confidence intervals are displayed as follows.
[A] Standard Bootstrap (SB) Method
Form the B bootstrap estimates qˆ( )*l , l
1,2, ,
B, the sample average and the sample standard deviation can be obtained asˆ*
distribution of qˆ is approximately normal. Thus, the 100(1 2 )% a SB confidence interval for θ can be constructed as
q ˆ* a q*
[ z S , q ˆ* z S ,a q*]
where qˆis the estimator of θ for the original sample, and z is the upper quantile of the standard normal distribution.
[B] Percentile Bootstrap (PB) Method From the ordered collection of ˆ( )*
,
l l 1,2, ,
B, the percentage and 1- percentage points are used to obtain the100(1 2 )%
PB confidence interval forθ,* ( )
[
ˆB ,
ˆ*((1) )B ].[C] Biased-Corrected Percentile Bootstrap (BCPB) Method
While the percentile confidence interval is intuitively appealing, it is possible that cause sampling errors, the bootstrap distribution may be biased. In other words, it is possible that bootstrap distributions using only a sample of the complete bootstrap distribution may be shifted higher or lower than would be expected. A three steps procedure is suggested to correct for the possible bias (Efron, 1982). First, using the ordered distribution of
, calculate the
ˆ* probability p0 P q[ˆ*qˆ]0 . Second, we compute the inverse of the cumulative distribution function of a standard normal based upon p0 as z0 1( )p0 ,(2 0 )
pL z za pU (2z0 za). Finally, executing these steps to obtain the
100(1 2 )%
BCPB confidence interval* ( )
[
ˆp BL ,
ˆ ]*(p BU ) .[D] Bootstrap-t (BT) method
By using bootstrap method to approximate the distribution of a statistic of the form (q qˆ )/Sqˆ, the bootstrap approximation in this case is obtained by taking bootstrap samples from the original data values, calculating the corresponding estimates
and their estimated standard error, hence finding the
ˆ* bootstrapped T -values T (qˆ* qˆ)/Sq* . The hope is that the generated distribution will mimic the distribution of T. The100(1 2 )%
BT confidence interval for may constitute as* * *
ˆ ˆ
[
t S
,
ˆ*t S
1* *ˆ],where ta* and t1*a are the upper and
1
quantiles of the bootstrapt
-distribution respectively, i.e. by finding the values that satisfy the two equationsˆ*
[(
P q qˆ)/Sq* ta*]a and P q[(ˆ* qˆ)/Sq* t1*a] 1 a , for the generated bootstrap estimates.
4. Performance Comparisons of Four Bootstrap Methods
4.1 Simulation Layout Setting
There are mainly two important characteristics, the process location relative to its specification limits, and the process spread in process capability. The closer the process location is to the mid-point of the specification limits and the smaller the process spread, the more capable the process is. A mathematical relationship among the indices Cp,
C
a, and Spk can be established as:Based on this relationship, it is note that we can combine several different combinations of Cp and
C
a for the same Spk value by setting between the process centering and the magnitude of process variation. Table 2 displays many differentC
a values and the corresponding process spread of the magnitude of.
Figure 1. Four processes with Spk=1.00.
Figure 1 plots four process with difference combination of
(
C Cp,
a)
withSpk 1.00
, LSL=10, USL=20, and m=15. i.e.(
C Cp,
a) (1.00,1.00)
for process A,(
C Cp,
a) (1.23661662, 0.75)
for process B,(
C Cp,
a) (1.85478349, 0.5)
for process C,(
C Cp,
a) (3.70956682, 0.25)
for process D. These four processes have equivalent Spk 1.00
, and all have yields equal to 99.73%, but constructed with different and . Hence, in order to make a comparative study among four bootstrap confidence limits, we take series of simulations to investigate the error probability and the selection power of difference and ratio testing statistics for the performance comparisons of four bootstrap methods. The setting values of parameters for two manufacturing suppliers used in the simulation study are given in Table 3. We investigate the performance of the methods with selected parameters for a wide range of index values and in on-target and off-target processes. For each combination, we generate 3,000 random samples, and the corresponding bootstrap confidence intervals for each of these samples are assessed in section 4.2.Table 3. The parameter setting values for two manufacturing suppliers used in the
6 1 1.23661662 3/4 1 1.23661662 3/4
7 1 1.23661662 3/4 1 1.85478349 1/2
8 1 1.23661662 3/4 1 3.70956682 1/4
9 1 1.85478349 1/2 1 1 1
10 1 1.85478349 1/2 1 1.23661662 3/4
11 1 1.85478349 1/2 1 1.85478349 1/2
12 1 1.85478349 1/2 1 3.70956682 1/4
13 1 3.70956682 1/4 1 1 1
14 1 3.70956682 1/4 1 1.23661662 3/4
15 1 3.70956682 1/4 1 1.85478349 1/2
16 1 3.70956682 1/4 1 3.70956682 1/4
4.2 Error Probability Analysis
The error probability is the first step which we want to investigate. It is the proportion of times that reject the null hypothesis H0
:
Spk1
Spk2 , while0
:
pk1 pk2H S
S is true. Thus, we will calculate the proportion of times that the lower confidence bound of Spk2
Spk1 is positive and the lower confidence bound of Spk2/
Spk1 is larger than 1 for each case given in Table 2. We set sample sizen=100 drawn with replacement , the bootstrap resamples B=3,000, and the single
simulation is replicated N=3,000 times. We usually set that the probability of the error selection less than a maximum value , referred to the condition. The frequency of the error is a binomial random variable with N=3,000 and =0.05.Thus, the 99% confidence interval for the error probability is
* * *
0.005 (1 )/
Z N
a a a 0.05 2.576 (0.05 0.95)/ 3000 0.05 0.0103 . That is, one can have a 99% confidence that a “true 0.05 error probability”would have a range from 0.0397 to 0.061. Figure 2 and 3 show that the error probability of the four bootstrap methods for the difference and the ratio statistics with 16 different combination cases tabulated in Table 3.
Figure 2. Error probability of four bootstraps under Spk1
Spk2 1.00
.Figure 3. Error probability of four bootstraps under Spk2
/
Spk1 1.00
.We have some results from Figure 2 and 3. In Figure 2, it shows that there
difference statistic. There is only one occurrence out of the interval for SB method.
We can note that with BCPB and BT methods, there is no case out of the control limit. That is, for different combinations of Cp and
C
a for equal Spk value have no significant effect in error probability with BCPB and BT methods.As for the ratio statistic in Figure 3, there are three cases out of the control limit (0.0397, 0.061) for the PB method. For the BT method, all of these cases are behind the lower control limit. That is, BT is a conservative bootstrap method for ratio statistic. With the SB and BCPB methods, there is no occurrence out of the interval. Table 4 and 5 show the results of error probability analysis for difference and ratio test. It means that, for different combinations of Cp and
C
a for equal Spk value have no significant effect in error probability with SB and BCPB methods.Table 4. The results of error probability analysis for difference test.
Bootstrap
SB 0.053895 0.003684 1 3
PB 0.056896 0.004436 3 2,3,4
BCPB 0.054166 0.001966 0 None
BT 0.049917 0.001934 0 None
Table 5. The results of error probability analysis for ratio test.
Bootstrap
SB 0.045708 0.003297 0 None
PB 0.056896 0.004436 3 2,3,4
BCPB 0.055314 0.001595 0 None
BT 0.034917 0.001189 16 All
Besides that, an average lower confidence bound and the standard deviation of the lower confidence bound were calculated based on the N=3,000 different trials. Table 6 takes four of sixteen cases to show that the average lower confidence bound and the standard deviation of the lower confidence bound for each of the four different bootstrap methods. The results of all cases are summarized in Table 12.
Table 6. Simulation results of the four bootstrap methods for the
1 1 1 1 1 1 SB 0.05333 -0.16505 0.10305 0.04900 0.84596 0.08617
PB 0.05667 -0.16490 0.10415 0.05667 0.85486 0.08679 BCPB 0.05467 -0.16495 0.10370 0.05567 0.85475 0.08658 BT 0.05067 -0.16466 0.10118 0.03867 0.83317 0.08529
1 1.236617 3/4 1 1 1 SB 0.05033 -0.17027 0.10184 0.04233 0.84187 0.08462
PB 0.05333 -0.17004 0.10302 0.05333 0.85058 0.08527 BCPB 0.05633 -0.16281 0.10183 0.05700 0.85639 0.08556 BT 0.04967 -0.16656 0.09955 0.03367 0.83273 0.08385
1 1.854783 1/2 1 3.709567 1/4 SB 0.05400 -0.16182 0.10107 0.04500 0.84966 0.08423
PB 0.05500 -0.16168 0.10243 0.05500 0.85829 0.08501 BCPB 0.05400 -0.16179 0.10104 0.05467 0.85812 0.08467 BT 0.04967 -0.16148 0.09846 0.03467 0.83720 0.08297
1 3.709567 1/4 1 3.709567 1/4 SB 0.05233 -0.16188 0.10116 0.04467 0.84960 0.08431
PB 0.05467 -0.16167 0.10244 0.05467 0.85823 0.08507 BCPB 0.05167 -0.16167 0.10100 0.05467 0.85817 0.08466 BT 0.04833 -0.16150 0.09863 0.03467 0.83721 0.08316
4.3 Selection Power Analysis
After the error probability analysis, we can roughly ensure that, there is less effect for different combinations of Cp and
C
a for equal Spk value with difference and ratio statistic. In order to compare the performance of these four bootstrap methods, we conduct further simulations of selection power with different sample sizes n=30(10)200 for Spk1 1.00
, and Spk2 1.05(0.05)1.50
. The selection power calculates the probability of rejecting the null hypothesis0: pk1 pk2
H S S while actually H1:Spk1Spk2 is true. For the difference statistic, the selection power computes the proportion of times that the lower confidence bound of Spk2
Spk1 is positive in the simulation. Similarly, for the ratio statistic, the selection power computes the proportion of times that the lower confidence bound of Spk2/
Spk1 is larger than 1. Figures 4-5 show the power of the four bootstrap methods for the difference and ratio statistic with sample sizen=30(10)200 ,
Spk1 1.00
, Spk2 1.50
, respectively. The power curves for1
1.00
Spk
, Spk2 1.05(0.05)1.50
, and n=30(10)200 are showed in Figures 13-32.Figure 4. The selection power of the four bootstrap methods for the difference statistic with sample size n=30(10)200.
Figure 5. The selection power of the four bootstrap methods for the ratio statistic with sample size n= 30(10)200.
In Figure 4 and Figure 5, we find that PB and BCPB methods are much powerful under the same sample size. On the contrary, SB and BT methods have larger required sample size with fixed selection power. Under the two considerations of error probability above and selection power analysis, the BCPB method has more correct error probability and better selection power with fixed sample size. Consequently, we recommend the best of these four bootstrap methods is the BCPB method.
5. Supplier Selection Based on BCPB Method
5.1 Sample Size Determination with Designated Selection Power
In general, if a new supplier #2 (S2) wants to compete for the orders by claiming that its capability is better than the existing supplier #1 (S1), the new S2 has to convince purchaser with a prescribed confidence level information to justify the claim. Therefore, the sample size required for designated selection power must be determined to collect actual data from the factories. We investigate the BCPB method with B=3,000 bootstrap resamples, and the each simulation was then replicated with N=3,000 times. For convenience of applications, we tabulate the sample sizes required for various designated selection power = 0.90, 0.95, 0.975, and 0.99 under error probability
0.05
. The selection power calculates the probability of rejecting the null hypothesis H0:Spk1Spk2 while actually1: pk1 pk2
H S S is true. Tables 7-8 show the sample size required of the BCPB method for the difference with Spk1
1.00
and Spk2 1.10(0.05)1.50
and ratio statistics with Spk2 1.10(0.05)1.50
. We also calculate the sample size required for Spk1 1.30
and Spk2 1.40(0.05)1.80
for difference and ratio statistics in Table 9 and 10.Table 7. Sample size required of BCPB method for the difference statistics under
0.05
, with power = 0.90, 0.95, 0.975, 0.99, Spk1 1.00
, Spk2 1.10(0.05)1.50
.97.5%
1400 666 397 265 184 143 113 100 8199%
1777 807 463 333 228 184 133 115 93Table 8. Sample size required of BCPB method for the ratio statistics under
Table 9. Sample size required of BCPB method for the difference statistics under
0.05
, with power = 0.90, 0.95, 0.975, 0.99, Spk1 1.30
, Spk2 1.40(0.05)1.80
.95%
1916 891 521 354 251 190 155 125 10297.5%
2350 1088 643 413 338 230 176 147 11999%
2925 1350 763 525 382 279 227 178 150Table 10. Sample size required of BCPB method for the ratio statistics under
0.05
, with power = 0.90, 0.95, 0.975, 0.99, Spk1 1.30
,Spk2 1.40(0.05)1.80
.95%
1965 917 514 350 250 189 152 126 10297.5%
2397 1124 652 424 313 239 185 144 12499%
2829 1297 749 512 376 278 226 189 152For the convenience of observation, Figures 6-9 depict sample size curves based on the four sample size tables, respectively.
Difference Sample Size for S
pk1=1.00
0 1000 2000 3000 4000 5000 6000
1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 S
pk2SampleSize
0.99
0.975 0.95 0.90
Figure 6. The sample size curve for the difference statistic under
0.05
, with power = 0.90, 0.95, 0.975, 0.99, Spk1 1.00
, Spk2 1.10(0.05)1.50
.Ratio Sample size for S
pk1=1.00
0 1000 2000 3000 4000 5000 6000
1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 S
pk2SampleSize
0.99
0.975 0.95 0.9
Figure 7. The sample size curve for the ratio statistic under
0.05
, with power = 0.90, 0.95, 0.975, 0.99, Spk1 1.00
, Spk2 1.10(0.05)1.50
.Difference Sample Size for S
pk1=1.3
Figure 8. The sample size curve for the difference statistic under
0.05
, with power = 0.90, 0.95, 0.975, 0.99, Spk1 1.3
, Spk2 1.40(0.05)1.80
.Ratio Sample Size for S
pk1=1.3
0
Figure 9. The sample size curve for the ratio statistic under
0.05
, with power = 0.90, 0.95, 0.975, 0.99, Spk1 1.3
, Spk2 1.40(0.05)1.80
.From these figures, we can note that the larger the value of the difference
2 1
pk pk
S S
or the ratio
Spk2/
Spk1 between two suppliers, the smaller the sample size required for fixed selection power. For fixed or and Spk1, the sample size required increases as designated selection power increases. Besides,the sample size required is very similar either for the difference or the ratio statistics. This phenomenon can be explained easily, since the smaller of the difference and the larger designated selection power, the more collected sample is required to account for the smaller uncertainty in the estimation.
5.2 Selecting the Better Supplier
In this supplier selection problem, the practitioner should set the present minimum requirement of Spk values, and the minimal difference or the minimal ratio must be differentiated between suppliers with designated selection power. The practitioner alternatively might check Tables 7-10 for the sample size required under error probability
0.05
, with designated selection power = 0.90, 0.95, 0.975, 0.99. After that, based on the BCPB method, if the LCB of ˆ2 ˆ1pk pk
S
S
is positive or the LCB ofS
ˆpk2/S
ˆpk1 is greater than 1, then we can conclude that the supplier #2 is better than the supplier #1. Otherwise, we do not have sufficient information to reject the null hypothesis H :S0 pk1
Spk2. That is, we would believe that the existing supplier #1 is better than the new supplier #2.6. Application Example：Color Filter Supplier Selection
Thin-film transistor liquid-crystal display (TFT-LCD) is one of the potential module of the high-tech products in the communication, information and consumer electronics industries. The TFT-LCD consumes less energy and weighs less compared to a cathode-ray tube (CRT). Besides that, it has emerged as the most widely used display solution, due to its high reliability, viewing quality and performance, compact size and environment-friendly features.
The basic structure of a TFT-LCD panel may be thought of as two glass substrates sandwiching a layer of liquid crystal. The front glass substrate is fitted with a color filter, while the back glass substrate has transistors fabricated on it.
When voltage is applied to a transistor, the liquid crystal is bent, allowing light to pass through to form a pixel. A light source is located at the back of the panel and is called a backlight unit. The front glass substrate is fitted with a color filter, which gives each pixel its own color. Figure 10 shows the combination of the structure.
Figure 10. The combination of TFT-LCD structure.
The color filter is the most key component for a TFT-LCD. Many companies invest in producing a larger color filter to reduce the production cost. Competition in this market is very fierce. The thickness of the color filter is one of the most important quality characteristics. If the thickness of color filter is not in control, the TFT-LCD product may result in a certain degree of aberration.
The example is taken from a TFT-LCD manufacturing company, located in a science-based industrial park in Taiwan. The company would like to determine which of the two color filter suppliers has better process capability. For a particular model of the color filter investigated, the USL of a color filter thickness is set to 0.7mm, the LSL of a color filter thickness is set to 0.56mm,
and the target value of a color filter thickness is set to 0.63mm.
6.1 Data Analysis and Supplier Selection
For the supplier selection problem, the practitioner should input the minimal requirement of Spk value first. Second, the minimal difference of Spk between these two suppliers with a designated selection power has to be set. Then we could decide the sample size based on Tables 7-10. In this case, the upper specification limit is 0.7mm, the lower specification limit is 0.56mm, and the target value is 0.63mm. The minimal requirement for the color filter product is 1.00, and the minimal difference between these two suppliers is 0.3, with selection power 0.95.
By checking Tables 7-10, the sample size required for the difference statistics is 155, and for the ratio statistics is 151. We take 155 samples for S1 and S2, respectively. All sample data for two suppliers are showed in tables 15-16.
Figure 11. Histogram of data S1. Figure 12. Histogram of data S2.
Figures 11-12 show the histogram of the 155 samples for S1 and S2. We use Kolmogorov–Smirnov test to check if these two suppliers’data are normal distributed. The statistic d for S1 is 0.038, and the statistic d for S2 is 0.065.
Because both of these two p-values are greater than 0.05, we can not reject the null hypothesis. Thus, we conclude that the sample data for the two suppliers can be regarded as normal processes. We calculate the sample means, sample standard deviations, and the sample estimators ˆ
S
pk for S1 and S2, summarized in Table 11.Table 11. The calculated sample statistics for two suppliers.
x s ˆ
S
pkS1 0.630129 0.022558 1.0344
S2 0.633369 0.017689 1.2973
We execute the Matlab program to obtain the LCB for the difference between these two processes ˆ2 ˆ1
pk pk
S
S
is 0.09357, and the LCB for the ratioS
ˆpk2/S
ˆpk1 is 1.0865. Therefore, we can conclude that S2 is better than the present supplier S1.7. Conclusions
Supplier selection problem is an important issue in the manufacturing industry. The decision maker usually faces the problem of selecting the better supplier between two candidates. For most manufacturing factories, process yield is the fundamental criterion for supplier selection. The index Spk provides an exact measure on the process yield. However, the supplier selection problem based on index Spk has not been done.
In this thesis, we compared the performance of Spk2
-
Spk1 and Spk2/
Spk1 with four different bootstrap methods including the standard bootstrap (SB), theIn this thesis, we compared the performance of Spk2