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. LIUx yDepartment 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
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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