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International Journal of Production
Research
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Bootstrap approach for supplier
selection based on production yield
C.-W. Wu a , M. H. Shu b , W. L. Pearn c & K. H. Liu c aDepartment of Industrial Engineering and Systems Management , Feng Chia University , Taiwan
b
Department of Industrial Engineering and Management , National Kaohsiung University of Applied Sciences , Taiwan c
Department of Industrial Engineering and Management , National Chiao Tung University , Taiwan
Published online: 02 Sep 2008.
To cite this article: C.-W. Wu , M. H. Shu , W. L. Pearn & K. H. Liu (2008) Bootstrap approach for supplier selection based on production yield, International Journal of Production Research, 46:18, 5211-5230, DOI: 10.1080/00207540701278414
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International Journal of Production Research, Vol. 46, No. 18, 15 September 2008, 5211–5230
Bootstrap approach for supplier selection based on production yield
C.-W. WUy, M. H. SHU*z, W. L. PEARNx and K. H. LIUxyDepartment of Industrial Engineering and Systems Management, Feng Chia University, Taiwan
zDepartment of Industrial Engineering and Management, National Kaohsiung University of Applied Sciences, Taiwan
xDepartment of Industrial Engineering and Management, National Chiao Tung University, Taiwan
(Revision received February 2007)
Current manufacturing industries have increased their level of out-sourcing and relied more heavily on their supply chain as a source of competitive advantage. Supplier selection decisions have become an important component of production management. Those decisions have a significant impact on a firm’s marketing competition, and suppliers may account for a large portion of the production cost. Production quality is one of the key factors in supplier evaluation. The manual of supplier certification includes a discussion of process capability analysis, which recommends a procedure for evaluating the most prevalent process capability index Cpk. However, the recommended procedure is applicable only when evaluating an individual supplier’s performance. In this paper, we apply the bootstrap method to the supplier selection problem. We construct lower confidence intervals for the capability difference and ratio between two given suppliers. Performance comparisons are made among various bootstrap methods in terms of error probability and selection power. For convenience of applications, the sample sizes required for various designated selection power are also tabulated.
Keywords: Bootstrap resampling; Error probability; Lower confidence bound; Production yield; Supplier selection
1. Introduction
Manufacturers purchase components from suppliers or hire contract manufacturers to produce necessary parts, and they assemble these parts to deliver the finished products to customers. The major considerations when choosing a supplier or a contract manufacturer include quality, cost, goodwill, service, delivery, and so on. According to research conducted by Dickson (1966), quality and delivery are two of the most demanded items by component suppliers. Twenty five years after Dickson’s research, Weber et al. (1991) still considered quality to be of ‘extreme importance’ and delivery to be of ‘considerable importance’. According to Weber’s research on the just-in-time (JIT) model, the importance of quality and delivery remains the same. Pearson and Ellram (1995) surveyed 210 members of the National Association
*Corresponding author. Email: [email protected]
International Journal of Production Research
ISSN 0020–7543 print/ISSN 1366–588X onlineß 2008 Taylor & Francis
http://www.tandf.co.uk/journals DOI: 10.1080/00207540701278414
of Purchasing Management (NAPM), who were randomly selected from the listings of electronic firms in the two-digit SIC code 38, and they indicated that quality is the most important criterion in the selection and evaluation of suppliers for both the small and large electronic firms that were surveyed. Moreover, according to the survey of current and potential outsourcing end-users by the Outsourcing Institute (2003), the top 10 factors in vendor selection are commitment to quality, price, reference/reputation, flexible contract terms, scope of resources, additional value-added capability, cultural match, existing relationship, location, and others. Quality is still the most important factor of all. Furthermore, Olhager and Selldin (2004) investigated supply chain management strategies and practices in a sample of 128 Swedish manufacturing firms and concluded that many aspects are important when companies choose supply chain partners, but quality is the single most important criterion. Kane (1986) stated that the quantification of the process mean and variation is central to understanding the quality of the units produced from a manufacturing process. Process capability indices (PCIs) can also be used to measure process potential at the initial stage of the production setting. These facts bring the issue of supplier selection based on PCIs into the main focus.
The first PCI appearing in the literature was the precision index Cp and it is
defined as (see Juran 1974 and Kane 1986): Cp ¼
USL LSL
6 , ð1Þ
where USL is the upper specification limit, LSL is the lower specification limit, and is the process standard deviation. The index Cpmeasures process precision (product
quality consistency), and does not consider whether the process is centred. To measure the degree of process centring, Pearn et al. (1998) introduced the following accuracy index Ca:
Ca¼1
j mj
d , ð2Þ
where is the process mean, d ¼ USL LSLð Þ=2, and m ¼ USL þ LSLð Þ=2. The index Cameasures the centring tendency, which alerts the user if the process mean
deviates from its midpoint. The Cpk index considers process variation and the
location of process mean, Cpk¼min Cpu, Cpl ¼min USL 3 , LSL 3 ¼d j mj 3 : ð3Þ
Obviously, we have Cpk¼CpCa. Taguchi, on the other hand, emphasizes the
product loss when one of its characteristics departs from the target value T. Hsiang and Taguchi (1985) introduced the index Cpm, which was also proposed
independently by Chan et al. (1988). The index Cpm incorporates the variation of
production items with the target value and the specification limits preset in the factory. It is defined as:
Cpm ¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ ð TÞ2 q : ð4Þ
In practice, process mean and process variance 2are usually unknown. Since sample data must be collected to calculate the index value, sampling errors are introduced into the capability assessments. Consequently, lower confidence bounds (LCBs) or capability testing must be performed using their sampling distributions. Many authors have promoted the use of various PCIs for evaluating a supplier’s process capability. Examples include Boyles (1991), Pearn et al. (1992), Kushler and Hurley (1992), Kotz and Johnson (1993), Va¨nnman and Kotz (1995), Va¨nnman (1997), Kotz and Lovelace (1998), Pearn et al. (1998), Kotz and Johnson (2002), Spiring et al. (2003), Pearn and Shu (2003), Pearn et al. (2005), and references therein. However, one area that has received little attention in the literature is the comparison between two suppliers’ PCIs. In a review of the problem of selecting the best manufacturing process based on PCIs, Tseng and Wu (1991) considered the problem for multiple available manufacturing processes based on the precision index Cpunder a modified likelihood ratio (MLR) selection rule. Chou (1994) developed a
test for comparing two one-sided processes and choosing a better supplier when the sample sizes are equal. Hubele et al. (2005) applied a Wald statistic for testing the equality of multiple Cpuor Cplindices. Huang and Lee (1995) considered the supplier
selection problem based on the index Cpm, and developed a mathematically
complicated approximation method for selecting a subset of processes containing the best supplier from a given set of processes. The method essentially compares the average loss of a group of candidate processes and selects a subset of these processes with smaller process loss 2þ( T)2, which, with certain level of confidence, contains the best process. Pearn et al. (2004) provided additional useful information regarding the sample size required for various designated selection power. A two-phase selection procedure was developed to select a better supplier. Chen and Chen (2004) offered four approximate confidence interval methods, one based on the statistical theory given in Boyles (1991) and three based on the bootstrap method, for selecting the better one of two suppliers. A comparison of the coverage percentage of the four methods was investigated by simulation. Although statistical tests have been developed to compare two Cp, Cpm, Cpu, and Cpl capability indices of normal
processes, a statistical test for comparing two Cpkvalues has not been developed due
to the complexity of the sampling distribution of ^Cpk2 ^Cpk1 or ^Cpk2= ^Cpk1. In this
paper, we apply the bootstrap method to compare two processes based on Cpk in
terms of error of probability and selecting power. The obtained confidence intervals provide information regarding actual process performance, which is useful in making reliable decisions for capability testing (H0: Cpk1Cpk2versus H1: Cpk15Cpk2).
2. Process yield measure based on Cpk
2.1 Fraction of nonconformities (NC)
Process yield is traditionally defined as the percentage of the product units that pass the inspections. Units are inspected according to specification limits placed on various key product characteristics and sorted into two categories: passed (conforming) or rejected (non-conforming). Process yield has long been the most common and standard criteria used in the manufacturing industries for judging process performance. In the past, fraction nonconforming were calculated by
counting the number of nonconforming items in a sample, then extrapolating the results. With the fraction nonconforming now commonly less than 0.01%, often expressed in parts per million (ppm), traditional methods for calculating the fraction nonconforming no longer work since all reasonably sized samples will probably have no defective items. Capability indices are alternatives for measuring fraction nonconforming.
Suppose that the proportion of conforming items is the primary concern then the most natural measure is the proportion itself called the yield, which we define as:
Yield ¼ ZUSL
LSL
dFðxÞ ¼ FðUSLÞ FðLSLÞ ð5Þ
where F(x) is the cumulative distribution function (CDF) of the measured characteristic X. If the process characteristic X follows N(, 2), then the fraction of nonconformities NC is: NC ¼ 1 USL þ LSL , ð6Þ
where () is the CDF of the standard normal distribution N(0, 1).
2.2 Yield assurance based on Cpk
The index Cpkcan be used to fill such a purpose for normally distributed processes.
Given a fixed value of Cpk, we have 2(3Cpk) 1 yield (3Cpk). For Cpk¼1.00,
one would expect that the fraction of defectives, is no more than 2700 ppm. The exact number of non-conformities can be expressed as a function of Cpkand Caor Cpkand
Cptogether as follows: NC ¼ 3Cpk þ 3Cpkð2 CaÞ=Ca , NC ¼ 3Cpk þ 3ð2CpCpkÞ : For most manufacturing factories, reducing the fraction of non-conformities is the primary concern and the guiding principle for quality improvement. Montgomery (2001) recommended some minimum capability requirements for processes running under certain designated quality conditions. In particular, Cpk1.33 is for existing processes, and Cpk1.50 is for new processes; Cpk1.50
is also for existing processes on safety, strength, or critical parameter, and Cpk1.67
is for new processes on safety, strength, or critical parameter. Finley (1992) also found that required Cpk values on all critical supplier processes are 1.33 or higher
and Cpk values of 1.67 or higher are preferred. Many companies have recently
adopted criteria for evaluating their processes that include more stringent process capability objectives. Motorola’s Six Sigma program essentially requires the process capability to be at least 2.0 to accommodate the possible 1.5 process shift (see Harry 1988), and no more than 3.4 ppm are defectives.
3. Selecting a better supplier by comparing two Cpk
We investigate the selection problem for cases with two candidate processes based on the Cpk index. Let i be the population assumed to be normally distributed with
mean i and variance i2, i ¼ 1, 2, and xi1, xi2, . . . , xini are the independent random
samples from i, i ¼ 1, 2. In most applications, if a new supplier no. 2 (S2) wants to
compete for the orders by claiming that its capability is better than the existing supplier no. 1 (S1), then the new S2 must furnish convincing information justifying the claim with a prescribed level of confidence. Thus, the decision of supplier selection would be based on the hypothesis testing comparing the two Cpk values,
H0: Cpk1Cpk2 versus H1: Cpk15Cpk2. If the test rejects the null hypothesis
H0: Cpk1Cpk2, then one has sufficient information to conclude that the new S2 is
superior to the original S1, and the decision of the replacement would be suggested. Equivalently, this test hypothesis problem can be rewritten as H0: Cpk2Cpk10
versus H1: Cpk2Cpk140 (difference testing), or H0: Cpk1/Cpk21 versus H1:
Cpk2/Cpk141 (ratio testing). Thus, if the LCB for the difference between two PCIs
Cpk2Cpk1is positive, then S2 has a better process capability than S1. Otherwise, we
do not have sufficient information to conclude that the S2 has a better process capability than S1. In this case, we would believe that Cpk1Cpk20 is true, i.e.
Cpk1Cpk2. Similarly, if the LCB for the ratio between two PCIs Cpk1/Cpk2is greater
than 1, then S2 has a better process capability than S1. Otherwise, if the LCB of the ratio statistic is less than 1, then we conclude that S1 has a better process capability than S2.
The assessment of values requires knowledge of i, and i. From the definition
of Cpkexpressed in equation (3), the natural estimator ^Cpkiis obtained by replacing
the process mean i and the process standard deviation i by their conventional
estimators xi and si, which may be obtained from a process that is demonstrably
stable (under statistical control). ^ Cpki ¼min USL xi 3si ,xiLSL 3si ¼d j ximj 3si ¼ 1 jximj d ^ Cpi, ð7Þ where xi¼ Pni j¼1xij=ni, si¼ Pni j¼1ðxijxiÞ2=ðni1Þ h i1=2 and ^Cpi¼d=3si.
Numerous methods for constructing approximate confidence intervals of Cpk
have been proposed. Examples include Chou et al. (1990), Zhang et al. (1990), Franklin and Wasserman (1992a, b), Kushler and Hurley (1992), Nagata and Nagahata (1994), Tang et al. (1997), Hoffman (2001), and many others. Under the assumption of normality of the estimated particular ^Cpki defined in equation (7), ^Cpi
is distributed as ðni1Þ1=2Cpið1ni1Þ, and n
1=2
i jximj=i is distributed as the folded
normal distribution with parameter n1=2i jimj=i(see Leone et al. 1961 for details
about this distribution). Thus, single ^Cpkiis a mixture of 1ni1and the folded normal
distribution (Pearn et al. 1992). Furthermore, using the integration technique similar to that presented in Va¨nnman (1997), an exact and explicit form of the CDF of the individual natural estimator ^Cpki can be expressed as (see Pearn and Lin 2003):
FC^pkiðyÞ ¼1 Z bi ffiffiffini p 0 G ðni1Þðbi ffiffiffiffi ni p tÞ2 9niy2 ! ðt þ i ffiffiffiffini p Þ þðt i ffiffiffiffini p Þ dt, ð8Þ for y40, where bi¼d/i, i¼(im)/, G() is the CDF of the chi-square
distribution with degree of freedom ni1, 2ni1, and ð Þ is the probability density
function (PDF) of the standard normal distribution N(0, 1). Based on the CDF of ^
Cpki, Pearn and Lin (2003) implemented the statistical theory of the hypotheses
testing. Pearn and Shu (2003) further developed an efficient algorithm with the
Matlab computer program to find the reliable LCBs conveying critical information regarding the true process capability. However, their investigations are all developed for evaluating whether a single supplier’s process conforms to a customer’s requirements. Due to the complexities of the sampling distributions of ^Cpk2 ^Cpk1
or ^Cpk2= ^Cpk1, constructions of exact confidence intervals for Cpk2Cpk1 or Cpk2/
Cpk1are difficult.
3.1 Bootstrap methodology
The bootstrap, a data-based simulation technique for statistical inference introduced by Efron (1979, 1982), is a non-parametric, computationally intensive, but also effective, estimation method. It can be applied whenever the construction of confidence intervals for parameters using the standard statistical techniques becomes intractable. An overview of this topic in bootstrap confidence intervals can be found in Hall (1988), Efron and Tibshirani (1993). Moreover, 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 the statistic, is asymptotically normally distributed. Unfortunately, many real world processes are not normally distributed and this departure from normality could potentially affect these estimates. The bootstrap approach is far more general. It does not rely on any distributional assumptions about the underlying population. The more ambiguous the information is to the researcher regarding the underlying population distribution, the more likely it is that the bootstrap may prove useful. Rather than using distribution frequency tables to compute approximate probability values, the nonparametric bootstrap method generates a unique sampling distribution based on the actual sample rather than the analytic methods. Due to the advantage of the bootstrap simulation technique, many studies of process capability analyses used the bootstrap approach to calculate confidence intervals for process capability indices, dating back at least to Franklin and Wasserman (1992). Also see Choi et al. (1996), Chen and Chen (2004), and the references therein. Most of them concluded that the performance of such bootstrap confidence limits is quite satisfactory in the majority of the cases. Therefore, we apply bootstrap re-sampling method to construct confidence intervals on Cpk2Cpk1 and Cpk2/Cpk1 for selecting a better supplier,
which has never been done in the literature.
In the following, four bootstrap confidence limits are employed to determine the LCBs of difference and ratio statistics and the results are used to select the better supplier of the two candidates. For n1¼n2¼n, let two bootstrap samples of
size n drawn with replacement from the two original samples be denoted by fx
11, x21, . . . , x1ng fx21, x22, . . . , x2n g. The bootstrap sample statistics x1, s1, x2, and s2
are computed, as well as ^C
pk1, and ^Cpk2. A random sample of n
n
possible re-samples is drawn, the statistic is calculated for each of these, and the resulting empirical distribution is referred to as the bootstrap distribution of the statistic. Due to the overwhelming computation time, it is not of practical interest to choose nn such samples. Empirical work (Eforn and Tibshirani 1986) indicated that a minimum of roughly 1000 bootstrap re-samples is usually sufficient to compute reasonably accurate confidence interval estimates for population parameters. For the purpose of accuracy, we consider B ¼ 5000 bootstrap re-samples (rather than 1000). Thus, we
take B ¼5000 bootstrap estimates ^¼ ð ^C
pk2 ^Cpk1Þ or ð ^Cpk2= ^Cpk1Þ of
¼ Cpk2Cpk1 or Cpk2/Cpk1, respectively, then order them from the smallest to
the largest ^
ðlÞ¼ð ^Cpk2 ^Cpk1ÞðlÞ or ð ^Cpk2= ^Cpk1Þðl Þ where l ¼ 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) method introduced by Efron (1981) and Efron and Tibshiraniwill (1986) are conducted in this paper. The generic notations ^and ^ will be used to denote the
estimator of and the associated ordered bootstrap estimate. Construction of a two-sided 100(1 2)% confidence limit will be described. We note that a lower 100(1 )% confidence limit can be obtained by using only the lower limit. The formulation details for the four types of confidence intervals are displayed as follows. A. Standard bootstrap (SB) method. From the B bootstrap estimates ^
ðlÞ,
l ¼1, 2, . . . , B, the sample average and the sample standard deviation can be obtained as ^ ¼1 B XB l¼1 ^ ðlÞ, S¼ 1 B 1 XB l¼1 ½ ^ðlÞ^2 !1=2 : The quantity S
is an estimator of the standard deviation of ^if the distribution
of ^is approximately normal. Thus, the 100(1 2)% SB confidence interval for can be constructed as ½^
zS, ^
þzS, where ^is the estimated 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 ^ ðlÞ,
l ¼1, 2, . . . , B, the percentage and 1 percentage points are used to obtain the 100(1 2)% PB confidence interval for , ½ ^ðBÞ, ^ðð1 ÞBÞ.
C. Biased-corrected percentile bootstrap (BCPB) method. While the percentile confidence interval is intuitively appealing, it is possible that due to sampling errors, the bootstrap distribution may be biased. In other words, it is possible that the bootstrap distributions obtained using only a sample of the complete bootstrap distribution may be shifted higher or lower than would be expected. A three-step procedure is suggested to correct for the possible bias (Efron 1982). First, using the ordered distribution of ^, we calculate the probability
p0¼P½ ^ ^0. Second, we compute the inverse of the CDF of a standard
normal based upon p0 as z0¼1(p0), pL¼(2z0z), and pU¼(2z0z).
Finally, we execute these steps to obtain the 100(1 2)% BCPB confidence interval, ½ ^ðp
LBÞ, ^ðpUBÞ.
D. Percentile-t bootstrap (PT ) method. By using bootstrapping to approximate the distribution of a statistic of the form ð ^ Þ=S^, 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,
and hence finding the bootstrapped T-values T ¼ ð ^ ^Þ=S
. The hope is
then that the generated distribution will mimic the distribution of T. The 100(1 2)% PT confidence interval for may constitute as ½ ^t
S^, ^ t
1S^, where t
and t1 are the upper and 1 quantiles
of the bootstrap t-distribution respectively, i.e. by finding the values that satisfy
the two equations P½ð ^ ^Þ=S
> t ¼ and P½ð ^ ^Þ=S > t1 ¼1 , for
the generated bootstrap estimates.
4. Performance comparisons of four bootstrap methods 4.1 Simulation layout setting
When focusing on the capability of a process, there are two important characteristics, the process location relative to its specification limits and the process spread. The closer the process output is to the mid-point of the specification limits and the smaller the process spread, the more capable the process. Based on the relationship Cpk¼CpCa, it is worth noting that there are several combinations of
Cpand Cafor an equivalent Cpkvalue by trading-off between the degree of process
centring and the magnitude of process variation. Table 1 displays various Cavalues
and the corresponding ranges of the departure magnitude of .
Figure 1 plots four processes with different combinations of (Ca, Cp) with
Cpk¼1.00, i.e. (Ca, Cp) ¼ (0.25, 4) for process A, (Ca, Cp) ¼ (0.50, 2.00) for process B,
(Ca, Cp) ¼ (0.75, 4/3) for process C, and (Ca, Cp) ¼ (1.00, 1.00) for process D (from
left to right in plot). These four processes are equivalent according to Cpk
Figure 1. Four processes with Cpk¼1.00. Table 1. Cavalues and ranges of .
Cavalue Range of Ca¼1.00 ¼ m 0.755Ca51.00 05j mj5d/4 0.505Ca50.75 d/45j mj5d/2 0.255Ca50.50 d/25j mj53d/4 0.005Ca50.25 3d/45j mj5d Ca¼0.00 ¼LSL or ¼ USL Ca50.00 5LSL or 4USL
(i.e. Cpk¼1.00 for all four processes) and all have yields exceeding 99.73%, but they
differ substantially with respect to centring. Hence, in order to make a comparative study among four bootstrap confidence limits, a series of simulations are undertaken to investigate the error probability and selection power of difference and ratio testing statistics for the performance comparisons of the four bootstrap methods. The sets of parameter values for two manufacturing suppliers used in the simulation study are given in table 2. The selected parameters are chosen so as to investigate the performance of the methods for a wide range of index values and for on-target or off-target processes. For each combination, 5000 random samples are generated and, for each of these samples, the corresponding bootstrap confidence intervals are assessed in section 4.
4.2 Error probability analysis
The error probability is the proportion of times that the null hypothesis H0:
Cpk1Cpk2 is rejected, when actually H0: Cpk1Cpk2 is true. That is, we will
calculate the proportion of times that the LCB of Cpk2Cpk1 is positive and the
LCB of Cpk1/Cpk2is larger than 1. For each case given in table 2, a sample of size
n ¼100 was drawn with B ¼ 5000 bootstrap re-samples, and the single simulation was then replicated N ¼ 3000 times. Figures 2 and 3 show the error probability of those four bootstrap methods for the difference and the ratio statistics, respectively, with 16 combinations tabulated in table 2. Usually, it is required that the probability of the error selection be less than a maximum value
, generally referred to as the
-condition. The frequency of error selection is a binomial random variable with N ¼3000 and
¼0.05. Thus, a 99% confidence interval for the error probability is Z 0:005 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 Þ N r ¼0:05 2:576 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð0:05 0:95Þ 3000 r ¼0:05 0:0103:
Table 2. Parameter values for two manufacturing suppliers used in the simulation study under Cpk1¼Cpk2¼1.00.
Cases Cpk1 Cp1 Ca1 Cpk2 Cp2 Ca2 1 1 4 0.25 1 4 0.25 2 1 4 0.25 1 2 0.50 3 1 4 0.25 1 4/3 0.75 4 1 4 0.25 1 1 1.00 5 1 2 0.50 1 4 0.25 6 1 2 0.50 1 2 0.50 7 1 2 0.50 1 4/3 0.75 8 1 2 0.50 1 1 1.00 9 1 4/3 0.75 1 4 0.25 10 1 4/3 0.75 1 2 0.50 11 1 4/3 0.75 1 4/3 0.75 12 1 4/3 0.75 1 1 1.00 13 1 1 1.00 1 4 0.25 14 1 1 1.00 1 2 0.50 15 1 1 1.00 1 4/3 0.75 16 1 1 1.00 1 1 1.00
That is, one could be 99% confident that a ‘true 0.05% error probability’ would have a proportion of range from 0.0397 to 0.0610.
In fact, for the difference statistic, there are six occurrences out of the 16 cases that are outside the interval (0.0397, 0.0610) for the SB, PB, and PT methods. In contrast with the BCPB method, three out of the 16 cases are beyond these limits. As for the ratio statistic, there are six occurrences out of the 16 cases that are outside
0 2 4 6 8 10 12 14 16 0 0.02 0.04 0.06 0.08 0.1 0.12 n = 100 ratio P UCL CL LCL Error probability Cases SB PB BCPB BT
Figure 3. The error probability of four bootstrap methods for ratio statistic under Cpk1¼Cpk2¼1.00. 0 2 4 6 8 10 12 14 16 0 0.02 0.04 0.06 0.08 0.1 0.12 n = 100 diff P UCL CL LCL Error probability Cases SB PB BCPB BT
Figure 2. The error probability of four bootstrap methods for difference statistic under Cpk1¼Cpk2¼1.00.
the interval (0.0397, 0.0610) for the SB and PB methods. For the BT method, there are 13 occurrences out of the 16 cases outside the interval (0.0397, 0.0610). However, the BCPB method has only three out of the 16 cases beyond these limits. In addition, an average LCB and the standard deviation of the LCB are calculated based on the N ¼3000 different trials. Table 3 also displays the average LCB and the standard deviation of the LCB for each of the four bootstrap confidence intervals.
4.3 Selection power analysis
To compare the performance of those four bootstrap methods, further simulations of selection power analysis are conducted with sample sizes n ¼ 10(10)200 for Cpk1¼1.00 and Cpk1¼1.05(0.05)1.50. The selection power computes the probability
of rejecting the null hypothesis H0: Cpk1Cpk2while actually H1: Cpk15Cpk2is true.
For the difference statistic, the selection power computes the proportion of times that the LCB of Cpk2Cpk1 is positive in the simulation. Similarly, for the ratio
statistic, the selection power computes the proportion of times that the LCB of Cpk2/Cpk1is larger than 1. Figures 4 and 5 display the power of the four bootstrap
methods for the difference and ratio statistic with sample size n ¼ 10(10)200, Cpk1¼1.00, Cpk1¼1.50, respectively.
According to figures 4 and 5, we find that the PB and BCPB methods have smaller required sample size with fixed selection power. By contrast, the SB and BT methods have larger required sample size with fixed selection power. In terms of error probability analysis described above and selection power analysis, the BCPB method has more correct error probability and better selection power with fixed sample size. Therefore, we recommend that the best of those four bootstrap methods in our approach is the BCPB method.
5. Supplier selection based on BCPB method
5.1 Sample size determination with designated selection power
In practice, if a new S2 wants to compete for the orders by claiming that its capability is better than the existing S1, the new S2 must furnish convincing information justifying the claim with a prescribed level of confidence. Thus, 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 ¼ 5000 bootstrap re-samples, and the single simulation was then replicated N ¼ 3000 times. For users’ convenience in applying our procedure in practice, we tabulate the sample size required for various designated selection power ¼ 0.90, 0.95, 0.975, 0.99. The selection power computes the probability of rejecting the null hypothesis H0:
Cpk1Cpk2while actually H1: Cpk15Cpk2is true. Tables 4 and 5 display the sample
size required of the BCPB method for the difference with Cpk1¼1.00 and
Cpk2¼1.10(0.05)1.50 and ratio statistic with Cpk2/Cpk1¼1.10(0.05)1.50. From
tables 4 and 5, it can be seen that the larger the value of the difference ¼ Cpk2Cpk1 between two suppliers, the smaller the sample size required for
fixed selection power. For fixed and Cpk1, the sample size required increases as
designated selection power increases. This phenomenon can be explained easily, since
Table 3. The simulation results of the error probability of four bootstrap methods for the difference statistic and ratio statistic with 16 combinati ons of (C a 1 , Cp 1 ) and (C a 2 , Cp 2 ) under Cpk1 ¼ Cpk2 ¼ 1.00. USL ¼ 3, LSL ¼ 3, d ¼ 3, m ¼ 0 n ¼ 100 (Difference statistic) n ¼ 100 (Ratio statistic) Cpk1 Cp 1 Ca 1 1 1 Cpk2 Cp 2 Ca 2 2 2 Bootstrap method Error probability Average LCB Standard deviation of LCB Error probability Average LCB Standard deviation of LCB 1 4 0.25 2.25 0.25 1 4 0.25 2.25 0.25 SB 0.0580 0.1871 0.1186 0.0493 0.8280 0.0959 PB 0.0593 0.1868 0.1199 0.0593 0.8392 0.0970 BCPB 0.0567 0.1867 0.1184 0.0580 0.8393 0.0969 BT 0.0533 0.1867 0.1161 0.0340 0.8121 0.0946 1 4 0.25 2.25 0.25 1 2 0.5 1.5 0.5 SB 0.0570 0.1871 0.1187 0.0503 0.8280 0.0960 PB 0.0590 0.1869 0.1199 0.0590 0.8392 0.0970 BCPB 0.0573 0.1868 0.1183 0.0597 0.8392 0.0968 BT 0.0530 0.1867 0.1161 0.0343 0.8121 0.0945 1 4 0.25 2.25 0.25 1 1.33 0.75 0.75 0.75 SB 0.0573 0.1871 0.1187 0.0500 0.8280 0.0959 PB 0.0593 0.1869 0.1199 0.0593 0.8392 0.0970 BCPB 0.0577 0.1868 0.1184 0.0587 0.8392 0.0969 BT 0.0533 0.1867 0.1161 0.0350 0.8120 0.0945 1 4 0.25 2.25 0.25 1 1 1 0 1 S B 0.0270 0.2190 0.1139 0.0240 0.7999 0.0892 PB 0.0277 0.2200 0.1151 0.0277 0.8098 0.0901 BCPB 0.0410 0.1973 0.1156 0.0417 0.8278 0.0932 BT 0.0327 0.2059 0.1123 0.0190 0.7963 0.0896 1 2 0.5 1.5 0.5 1 4 0.25 2.25 0.25 SB 0.0557 0.1871 0.1186 0.0490 0.8280 0.0959 PB 0.0580 0.1869 0.1199 0.0580 0.8391 0.0970 BCPB 0.0573 0.1870 0.1185 0.0583 0.8390 0.0969 BT 0.0543 0.1866 0.1161 0.0353 0.8121 0.0946 1 2 0.5 1.5 0.5 1 2 0.5 1.5 0.5 SB 0.0573 0.1871 0.1187 0.0497 0.8280 0.0960 PB 0.0597 0.1868 0.1199 0.0597 0.8392 0.0970 BCPB 0.0577 0.1869 0.1183 0.0593 0.8391 0.0969 BT 0.0540 0.1866 0.1161 0.0343 0.8121 0.0945
1 2 0.5 1.5 0.5 1 1.33 0.75 0.75 0.75 SB 0.0563 0.1871 0.1187 0.0490 0.8280 0.0960 PB 0.0593 0.1869 0.1200 0.0593 0.8392 0.0970 BCPB 0.0570 0.1867 0.1185 0.0583 0.8392 0.0970 BT 0.0527 0.1866 0.1160 0.0353 0.8122 0.0945 1 2 0.5 1.5 0.5 1 1 1 0 1 S B 0.0277 0.2190 0.1140 0.0237 0.7999 0.0892 PB 0.0273 0.2200 0.1153 0.0273 0.8099 0.0903 BCPB 0.0413 0.1973 0.1156 0.0410 0.8278 0.0932 BT 0.0320 0.2059 0.1123 0.0190 0.7962 0.0895 1 1.33 0.75 0.75 0.75 1 4 0.25 2.25 0.25 SB 0.0563 0.1870 0.1187 0.0510 0.8281 0.0960 PB 0.0587 0.1868 0.1199 0.0587 0.8392 0.0970 BCPB 0.0550 0.1868 0.1182 0.0570 0.8391 0.0967 BT 0.0543 0.1866 0.1162 0.0353 0.8122 0.0945 1 1.33 0.75 0.75 0.75 1 2 0.5 1.5 0.5 SB 0.0570 0.1871 0.1187 0.0477 0.8280 0.0960 PB 0.0597 0.1869 0.1200 0.0597 0.8392 0.0971 BCPB 0.0573 0.1867 0.1185 0.0587 0.8392 0.0969 BT 0.0540 0.1867 0.1162 0.0350 0.8121 0.0945 1 1.33 0.75 0.75 0.75 1 1.33 0.75 0.75 0.75 SB 0.0573 0.1871 0.1186 0.0483 0.8280 0.0959 PB 0.0590 0.1868 0.1199 0.0590 0.8392 0.0970 BCPB 0.0580 0.1867 0.1184 0.0603 0.8392 0.0969 BT 0.0537 0.1866 0.1160 0.0340 0.8121 0.0944 1 1.33 0.75 0.75 0.75 1 1 1 0 1 S B 0.0273 0.2191 0.1139 0.0240 0.7999 0.0892 PB 0.0283 0.2201 0.1152 0.0283 0.8098 0.0902 BCPB 0.0407 0.1974 0.1156 0.0403 0.8276 0.0932 BT 0.0320 0.2059 0.1122 0.0190 0.7963 0.0894 1 1 1 0 1 1 4 0.25 2.25 0.25 SB 0.0987 0.1427 0.1111 0.0843 0.8620 0.0966 PB 0.1047 0.1412 0.1123 0.1047 0.8737 0.0976 BCPB 0.0747 0.1635 0.1129 0.0743 0.8549 0.0959 BT 0.0753 0.1551 0.1092 0.0507 0.8341 0.0939 (continued )
Table 3. Continued. USL ¼ 3, LSL ¼ 3, d ¼ 3, m ¼ 0 n ¼ 100 (Difference statistic) n ¼ 100 (Ratio statistic) Cpk1 Cp 1 Ca 1 1 1 Cpk2 Cp 2 Ca 2 2 2 Bootstrap method Error probability Average LCB Standard deviation of LCB Error probability Average LCB Standard deviation of LCB 1 1 1 0 1 1 2 0.5 1.5 0.5 SB 0.0980 0.1427 0.1112 0.0833 0.8620 0.0967 PB 0.1030 0.1412 0.1124 0.1030 0.8737 0.0977 BCPB 0.0737 0.1635 0.1128 0.0733 0.8548 0.0958 BT 0.0760 0.1551 0.1094 0.0523 0.8342 0.0942 1 1 1 0 1 1 1.33 0.75 0.75 0.75 SB 0.0973 0.1426 0.1111 0.0843 0.8620 0.0965 PB 0.1027 0.1412 0.1122 0.1027 0.8737 0.0975 BCPB 0.0727 0.1635 0.1128 0.0717 0.8548 0.0958 BT 0.0770 0.1551 0.1092 0.0540 0.8342 0.0939 1 1 1 0 1 1 1101 S B 0.0517 0.1743 0.1063 0.0437 0.8328 0.0897 PB 0.0517 0.1741 0.1074 0.0517 0.8431 0.0907 BCPB 0.0583 0.1745 0.1104 0.0577 0.8430 0.0924 BT 0.0497 0.1742 0.1055 0.0310 0.8179 0.0891
0 20 40 60 80 100 120 140 160 180 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Selection power Sample size SB PB BCPB BT
Figure 5. The selection power of the four bootstrap methods for the ratio statistic with sample size n ¼ 10(10)200. 0 20 40 60 80 100 120 140 160 180 200 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Selection power Sample size SB PB BCPB BT
Figure 4. The selection power of the four bootstrap methods for the difference statistic with sample size n ¼ 10(10)200.
the smaller the difference and the larger the 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 order to satisfy the user’s need and distinguish which supplier has better process capability, the minimum required Cpkvalues for the two candidate processes and the
minimal difference are determined, then the sample size required with designated selection power need to be sampled. Thus, based on the BCPB method if the LCB of
^
Cpk2 ^Cpk1 is positive or the LCB of ^Cpk2= ^Cpk1 is larger than 1, then we conclude
that the S2 is better than the S1. Otherwise, we would believe that the existing S1 is better than the new S2 since we don’t have sufficient information to reject the null hypothesis H0: Cpk1Cpk2.
6. Application example: PCB supplier selection
Printed circuit boards (PCBs) are widely used in the microelectronic manufacturing industry, making computers and peripherals, digital phones, fax machines, channel switch devices, remote controls, and many others. Factories producing various PCBs and related products generally are classified as ‘the PCB industry’ because the core components inside those products are the PCBs. The PCB manufacturing process mainly consists of a series of chemical related operations, and the chemical operations determine the functions of a PCB. PCBs are laminates. This means that they are made from two or more sheets of material stuck together, often copper and fibreglass. Unwanted areas of the copper are etched away to form conductive lands
Table 5. Sample size required of BCPB method for the ratio statistics under ¼ 0.05, with power ¼ 0.90, 0.95, 0.975, 0.99, Cpk1¼1.00, Cpk2¼1.10(0.05)1.50. Cpk1 1 1 1 1 1 1 1 1 1 Cpk2 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 90.0% 1045 475 289 191 139 101 84 63 55 95.0% 1340 625 358 239 170 126 107 84 73 97.5% 1600 738 424 286 203 161 122 94 84 99.0% 1975 895 549 391 268 211 162 124 105
Table 4. Sample size required of BCPB method for the difference statistics under ¼ 0.05, with power ¼ 0.90, 0.95, 0.975, 0.99, Cpk1¼1.00, Cpk2¼1.10(0.05)1.50. Cpk1 1 1 1 1 1 1 1 1 1 Cpk2 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 90.0% 1045 478 285 185 133 103 79 65 55 95.0% 1328 601 357 233 168 130 103 81 72 97.5% 1567 757 432 283 204 156 127 98 84 99.0% 1972 875 497 356 233 191 156 124 104
or tracks, which replace the wires carrying the electric currents in other forms of construction.
Some parts of the side with copper tracks are coated with solder resist (usually green in colour) to prevent solder sticking to those areas where it is not required. This avoids unwanted solder bridges between tracks. The solder resist is an important operation in the post-process for PCB manufacturing, which is chemically unrelated. The effects of the solder resist are to protect the metal-ingredients inside the circuits from oxidizing, and also to protect the board itself from exterior damaging when embedding specific electronic components for various applications. The uniformity smooth surface of the PCB is an essential quality characteristic considered in all PCB quality control schemes. The operation of the solder-resist is the key to surface coating in the PCB manufacturing industry. The simple method to judge whether the PCBs satisfy the uniformity flat requirement after the solder resist, is to measure its thickness. It particularly checks the uneven parts including the caves and towers of a PCB. By measuring the thickness, one can obtain the degrees of the uniformity for a PCB’s surface, which is used for PCBs capability measures on thickness.
The example investigated is taken from a company, located in Tao-Yuan Industrial Park in Taiwan. The company has two competing suppliers manufactur-ing multi-layer PCBs for the company’s orders. The company would like to determine which supplier provides better PCBs. The nominal-the-better character-istic thickness is the key measurement for the comparison. For a particular model of PCBs, the USL, LSL, and the target value of a PCB’s thickness are set to 28.5 mm, 13.5 mm, and 21.0 mm, respectively.
6.1 Data analysis and supplier selection
For the supplier selection problem, we begin by setting two factors, (1) the minimum requirement of the Cpkvalue, and (2) , the minimal difference of Cpkbetween these
two suppliers, and then we can decide the required sample size for preset selection power. In this example, the upper specification limit is 28.5, the lower specification limit is 13.5, and the target value is 21.0. The minimum requirement for the PCB product is 1.00 and ¼ 0.30, with selection power 0.95. Then, we have to take 168 samples for the difference statistics and 170 samples for the ratio statistics (by checking tables 4 and 5). In this study, we take 170 samples for S1 and S2 respectively.
The histogram and the normal probability plot of the 170 samples for S1 and S2 are used to check whether the sample data is normal. The statistic W of Shapiro–Wilk test is found to be 0.9967 with p-value40.1 for S1, and 0.9965 with p-value40.1 for S2. Thus, we conclude that the sample data for the two suppliers can be regarded as taken from near normal processes. We calculate the sample means, sample standard deviations, and the sample estimators ^Cpk for S1 and S2, as
summarized in table 6. Based on the selection procedure, we execute the Matlab program and determine the LCB for the difference between the two processes
^
Cpk2 ^Cpk1 to be 0.1821 and the LCB for the ratio ^Cpk2= ^Cpk1 to be 1.1479.
Therefore, we conclude that S2 is a better supplier than S1.
7. Conclusions
Supplier’s performance variability is a key issue that needs to be considered in the evaluation process. It provides the buyer with effective alternative choices within suppliers. Process capability indices are useful management tools that provide common quantitative measures on manufacturing capability and production quality. The manual of supplier certification includes a discussion of process capability analysis, which recommends a procedure for evaluating the most prevalent process capability index Cpk. In this paper, we implemented the bootstrap re-sampling
approach and developed a practical procedure for practitioners to use in making supplier selection decisions between two given suppliers. Performance of the various selection methods is investigated in terms of the error probability and the selecting power by using a simulation technique. For user’s convenience in applying our procedure, we provide the sample size required with designated selection power. To make the proposed method practical for in-plant applications, a real example of PCB manufacturing processes is presented to demonstrate the applicability of the proposed method.
The study of making reliable supplier decisions in comparing i 2 available production yields of manufacturing processes, the performance of the bootstrap approach methods, and the sample size determination for various designed selection power under different distributional assumptions that usually arise in applications, would be an interesting issue for further research.
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