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International Journal of Production
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Procedure for supplier selection
based on C
pm
applied to super
twisted nematic liquid crystal
display processes
W. L. Pearn a , C.-W. Wu b & H. C. Lin a
a
Department of Industrial Engineering & Management , National Chiao Tung University , Hsinchu, Taiwan
b
Department of Business Administration , Feng Chia University , Taichung, Taiwan
Published online: 21 Feb 2007.
To cite this article: W. L. Pearn , C.-W. Wu & H. C. Lin (2004) Procedure for supplier selection
based on Cpm applied to super twisted nematic liquid crystal display processes, International
Journal of Production Research, 42:13, 2719-2734, DOI: 10.1080/0020754042000203876
To link to this article: http://dx.doi.org/10.1080/0020754042000203876
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Procedure for supplier selection based on C
pmapplied to super twisted
nematic liquid crystal display processes
W. L. PEARNy*, C.-W. WUz and H. C. LINy
The loss-based process capability index Cpm, sometimes called the Taguchi index,
has been proposed to the manufacturing industry as a method to measure process performance. The index Cpmtakes into account the targeting degree of the
pro-cess, which essentially measures process performance based on average process loss. Based on the Cpm index, a mathematically complicated approximation
method was developed previously for selecting a subset of processes containing the best supplier from a given set of processes. The present paper implements this method and develops a practical step-by-step procedure to aid supplier selection decisions. The accuracy of the selection method is investigated using a simulation technique. The accuracy study provides useful information about the sample size required for a designated selection power. A two-phase selection procedure is developed to select a better supplier and to examine the magnitude of the differ-ence between the two suppliers. Also investigated is a real-world case on the super twisted nematic liquid crystal display manufacturing process, and it is applied to the selection procedure using actual data collected from the factories to reach a decision in supplier selection.
1. Introduction
The aim of process capability indices is to provide numerical measures of whether or not the reproduction ability of a manufacturing process meets a predetermined level of production tolerance. Process capability indices have received considerable research attention and increased use in process assessments and purchasing decisions in the automotive and other industries during the last decade. Examples include Pearn et al. (1992, 1998), Pearn and Chen (1997), Kotz and Lovelace (1998), Palmer and Tsui (1999), Kotz and Johnson (2002), Pearn and Lin (2003a, b), Pearn and Shu (2003a–c), Spiring et al. (2003) and many others. Those indices are effective tools for process capability analysis and quality assurance, and the formulae for those indices are easy to understand and straightforward to apply. The Cpindex
was developed by Kane (1986) and it considers the overall process variability relative to the manufacturing tolerance to measure process precision (product consistency). Due to simplicity of the design, Cpcannot reflect the tendency of process centring
(targeting):
Cp¼USL LSL 6 : Revision received November 2003.
yDepartment of Industrial Engineering & Management, National Chiao Tung University, Hsinchu, Taiwan.
zDepartment of Business Administration, Feng Chia University, Taichung, Taiwan. *To whom correspondence should be addressed. e-mail: [email protected]
International Journal of Production ResearchISSN 0020–7543 print/ISSN 1366–588X online # 2004 Taylor & Francis Ltd http://www.tandf.co.uk/journals
DOI: 10.1080/0020754042000203876
In order to reflect the deviations of process mean from the target value, several indices similar in nature to Cphave been proposed. Those indices attempt to take
the magnitude of process variance as well as process departures from the target value into consideration. One of those indices is Cpkdefined as:
Cpk¼min USL 3 , LSL 3 ,
where USL is the upper specification limit, LSL is the lower specification limit, is the process mean, is the process standard deviation and T is the target value. The index Cpk was developed because Cp does not adequately deal with cases
where process mean is not centred. However, Cpkalone still cannot provide
ade-quate measure of process centring. That is, a large Cpkdoes not really say anything
about the location of the mean in the tolerance interval. The Cpand Cpkindices are
appropriate measures of progress for quality improvement paradigms in which reduction of variability is the guiding principle and process yield is the primary measure of success. However, they are not related to the cost of failing to meet customers’ requirement. Taguchi, on the other hand, emphasizes the loss in a product’s worth when one of its characteristics departs from the customers’ ideal value T.
To help account for this, Hsiang and Taguchi (1985) introduced the index Cpm,
which was also proposed independently by Chan et al. (1988). The index is related to the idea of squared error loss, loss (X ) ¼ (X T )2, and this loss-based process capability index Cpm, sometimes called the Taguchi index. The index emphasizes on
measuring the ability of the process to cluster around the target, which therefore reflects the degrees of process targeting (centring). The index Cpm incorporates
with the variation of production items with respect to the target value and the specification limits preset in the factory. The index Cpmis defined as:
Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þ ð T Þ2 q ¼USL LSL 6 ,
where 2¼2þ( T )2¼E[(X T )]2is the major part of the denominator of Cpm,
which incorporates two variation components: (1) variation to the process mean and (2) deviation of the process mean from the target. For on target processes, Cpmindex
reaches its maximum, implying that the process capability runs under the desired condition. On the other hand, smaller values of Cpmmean higher expected loss and
poorer process capability. Therefore, the index Cpmis considered to be more sensitive
than Cpand Cpkin reflecting the process loss. Boyles (1991) has provided a definitive
analysis of Cpm and its usefulness in measuring process centring. He notes that
both Cpk and Cpm coincide with Cp when ¼ T and decrease as moves away
from T. However, Cpk<0 for >USL or <LSL, whereas Cpm of process
with | T | ¼ >0 is strictly bounded above by the Cp value of a process with
¼.
In the initial stage of production setting, the decision-maker usually faces the problem of selecting the best manufacturing supplier from several available manu-facturing suppliers. There are many factors, such as quality, cost, service and so on, that need to be considered in selecting the best suppliers. Process yield is currently defined as the percentage of the processed product units passing the inspections.
Traditionally, the fraction of non-conformities for manufacturing processes has been calculated by counting the number of non-conforming items in the sample. With the fraction non-conforming now commonly less than 0.01%, which is often expressed in parts per million (ppm), those traditional methods for estimating the fraction non-conforming no longer work since all reasonably sized samples would contain zero defective items. Several selection rules have been proposed for selecting the means or variances in analysis of variance (see Gibbons et al. 1977, Gupta and Panchapakesan 1979, Gupta and Huang 1981 for more details). Process capability indices are useful management tools, particularly in the manufacturing industry, which provide common quantitative measures on manufacturing capability and production quality. In the situation of the manufacturing process being control, it is assumed that the quality characteristic X is normally distributed, USL and LSL are usually fixed and determined in advance, the larger Cpis equivalent to looking for the smallest 2.
Tseng and Wu (1991) considered the problem of selecting the best manufacturing process from k available manufacturing processes based on the precision index Cp
and a modified likelihood ratio selection rule is proposed. Chou (1994) developed three one-sided tests (Cp, CPU, CPL) for comparing two process capability indices in
order to choose between competing processes when the sample sizes are equal. Based on the Cpmindex, a mathematically complicated approximation method is developed
by Huang and Lee (1995) for selecting a subset of processes containing the best supplier from a given set of processes.
Under the circumstance, to search the larger Cpm’s which are used to provide a
unitless measure of the process performance is equivalent to looking for a smaller 2. The present paper implements this method and develops a practical step-by-step procedure for practitioners to use in making supplier selection decisions. In practice, the process mean and process variance are unknown. To calculate the index, sample data must be collected, and a great degree of uncertainty may be introduced into capability assessments due to sampling errors. Thus, the distributional properties of the estimated index Cpm are then introduced and the unbiased estimator of loss
function, ^2, is considered. Accuracy of the selection method is investigated using a simulation technique. The accuracy study provides useful information about the sample size required for designated selection power. Subsequently, also investigated is a real-world case on the super twisted nematic liquid crystal display (STN-LCD) manufacturing process and the selection procedure is applied using actual data collected from the factories to reach a decision in supplier selections.
2. Distribution of the estimated Cpm
Since the process mean and the process variance 2must be estimated from the sample, the estimated index ^CCpm is obtained by replacing and 2 by their
estimators. Chan et al. (1988) and Boyles (1991) proposed two different estimators of Cpm, respectively, defined as the following:
^ C CpmðCCSÞ¼ d 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2þ ½n=ðn 1Þð xx T Þ2 q and ^CCpmðBÞ¼ d 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2 nþ ðxx T Þ 2 q :
where d ¼ (USL LSL)/2 is the half width of the specification interval, x x ¼Pni¼1xi=n, s2¼ Pn i¼1ðxixxÞ2=ðn 1Þ and s2n¼ Pn
i¼1ðxixxÞ 2=n: In fact, the
two estimators, ^CCpmðCCSÞ and ^CCpmðBÞ, are asymptotically equivalent. Assuming that
the process data are normally distributed and T ¼ M, Chan et al. (1988) derived the probability density function of ^CCpmðCCSÞ¼Y as:
fYðyÞ ¼ a 2n=21y3exp 1 2 a y2þl X1 j¼1 ljða=y2Þn=2þj1 j!ðn=2 þ jÞ22j, y > 0,
where a ¼ Cpm2 ð1 þ l=nÞðn 1Þ and l ¼ n( T)2/2. Note that ^CCpmðCCSÞ can be
shown to be functions of the inverse moments of a non-central chi-square distribution. An alternative but equivalent formula was provided by Pearn et al. (1992).
The distributional properties of ^CCpmðCCSÞare intractable for asymmetrical
speci-fications ((USL þ LSL)/2 6¼ T). When the case of (USL þ LSL)/2 ¼ T, ^CCpmðCCSÞ is
a biased estimator of Cpm, but is asymptotically unbiased. For detailed descriptions
and proofs of the properties of ^CCpmðCCSÞ, see Chan et al. (1988). On the other hand,
Boyles (1991) considered that it would be more appropriate to replace the factor n 1 by n in the denominator since the term ^ðBÞ¼s2nþ ðxx TÞ 2in the denominator
of ^CCpmðBÞ is the uniformly minimum variance unbiased estimator (UMVUE) of the term 2þ( T)2. Note that xx and s2n are the maximum likelihood estimators
(MLEs) of and 2, respectively. Hence, the estimated ^CCpmðBÞ is also the MLE
of Cpm.
The approach by simply looking at the calculated values of the estimated indices and then making a conclusion on whether the given process is capable is highly unreliable as the sampling errors have been ignored. As the use of the capability indices grows more widespread, users are becoming educated and sensitive to the impact of the estimators and their sampling distributions on constructing confidence intervals and performing hypothesis testing. Under the assumption of normality, Kotz and Johnson (1993) obtained the rth moment and calculated the first two moments, the mean and variance of ^CCpm. Cheng (1994) developed a
hypothesis-testing procedure where tables of the approximate p values were provided for some commonly used capability requirements, using the natural estimator of Cpm.
The practitioners can use the obtained results to determine if their process satisfies the targeted quality condition. However, Cheng’s approach requires further estima-tion of the distribuestima-tion characteristic ( T )/ when calculating the p values, which introduces additional sampling errors, thus making the decisions less reliable. Zimmer and Hubele (1997) provided tables of exact percentiles for the sampling distribution of the estimator ^CCpm. Zimmer et al. (2001) proposed a graphical
proce-dure to obtain exact confidence intervals for Cpm, where the parameter ( T)/ is
assumed to be a known constant. On the other hand, using the method similar to that presented in Va¨nnman (1997), Pearn and Shu (2003a) obtained an exact form of the cumulative distribution function of ^CCpm. Under the assumption of normality, the
cumulative distribution function of ^CCpm can be expressed in terms of a mixture of
the chi-square distribution and the normal distribution: FCC^pmðxÞ ¼1 Zbpffiffin=ð3xÞ 0 G b 2 n 9x2t 2 ! ðt þ pffiffiffinÞ þðt pffiffiffinÞ dt, ð1Þ for x>0, where b ¼ d/, ¼ ( T )/, G(() is the cumulative distribution function of the chi-square distribution with degree of freedom n 1, 2n1, and () is the
probability density function of the standard normal distribution N(0, 1). Note that
one would obtain an identical equation if was substituted by in equation (1) for fixed values of x and n.
3. Selecting a better supplier with a smaller c2
Huang and Lee (1995) considered the supplier selection problem based on the index Cpm and developed a rather complicated method for supplier selection
appli-cations. The method essentially compares the average loss of a group of candidate processes and selects a subset of these processes with small process loss 2, which with a certain level of confidence contains the best process. Because the specification limits are usually fixed and determined in advance, searching the largest Cpmis the
equivalent to looking for the smallest 2. The selection rule of Huang and Lee is retaining the population i in the selected subset if and only if ^2i c min1 jkj6¼i ^
2 j,
where c is determined by a function of parameters, which can be determined by calculating from collected samples. Note that the choice of c must be larger than 1 but as small as possible. The method, however, provides no indication on how one could proceed further with selecting the best population among those chosen from the subset of populations. This method is investigated for cases with two candidate processes. Let i be the population 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. When the
populations are ranked in terms of i2, one wants to select the better process with a smaller value of 2. A correct selection is denoted as CS, and the ordered 2as ½12 < ½22 is assumed.
Denote (i)as the population associated with ½i2, i ¼ 1, 2. The better population
is then (1). We wish to define a procedure with selection rule R such that the
probability of a correct selection is no less than a pre-assigned number p* and 0.5<p*<1, i.e. Pr(CS|R) p*. This requirement is referred to as the p* condition. The selection rule R based on the unbiased and consistent estimators ^i2 of ^i2,
i ¼1, 2, and ^i2 is defined as follows:
^ 2i ¼ Pni j¼1 ðxijTÞ2 ni ¼ðni1ÞS 2 i þniðxxiTÞ2 ni , S2i ¼ 1 ni1 Xni j¼1 ðxijxxiÞ2, xxi¼ 1 ni Xni j¼1 xij:
For cases with two candidate processes, comparing ^CCpm1 and ^CCpm2 is equivalent to
comparing ^12 and ^22. Hence, by the result of Pearn et al. (1992) are:
^ i2 i2 ni 2niðliÞ, li¼ni iT i 2 ,
where 2niðliÞis the non-central chi-squared distribution with degrees of freedom and
non-centrality parameter li.
3.1. Selection rule R
Consider the problem of selecting two populations with the smaller ^2. The selection rule R is as follows: consider i as the better supplier if and only if
^
i2c ^j2 and ^j2> c ^2i, i ¼ 1, 2 and i 6¼ j. To satisfy the p* condition, then:
c1¼exp 2A1 ffiffiffiffiffiffi 1 ^vv½1 s þ 1 ^vv½1 1 ^vv½2 ffiffiffiffiffiffi ^vv½2 ^vv½1 s ( ) , c2¼exp 2A2 ffiffiffiffiffiffi 1 ^vv½1 s þ 1 ^vv½1 1 ^vv½2 ffiffiffiffiffiffi ^vv½2 ^vv½1 s ( ) :
Choose the value of c that is larger than 1 and choose it as small as possible, so that: c ¼minfc1, c2g, if c1>1 and c2>1 c ¼ c1, if c1>1 and c21; c ¼ c2, if c2>1 and c11, where A1¼ d2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 24d1d3 q 2d1 , A2¼ d2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 2 4d1d3 q 2d1 d1¼a 1 þ a2 a1 þa2a 2 a2 a1þa2 a1 , d2¼b ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þa2 a1 r þab a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1þa2 p a1 a2 ffiffiffiffiffi a1 p d3¼ b2a2 4aa 1 ln 2p pffiffiffiffiffiffiffi2a , a ¼0:5 a a2 a1 , a1¼ 1 ^vv½1, a2¼ 1 ^vv½2 b ¼ 0:513277, a ¼ 0:085514 ^vv1¼ n1þ ^ll1 2 n1þ2^ll1 , ^vv2¼ n2þ ^ll2 2 n2þ2^ll2 , ^ll1 ¼n1 x x1T S1 2 , ^ll2¼n2 x x2T S2 2 , where ^vvi is used to estimate vi, i ¼ 1, 2, and ordered ^vvi are denoted by ^vv½1 ^vv½2.
3.2. Selection procedure
The selection procedure is based on a mathematically complicated approxima-tion method developed by Huang and Lee (1995) for selecting a problem. To make this method practical for in-plant applications, the selection procedure can be summarized and expand as follows:
Step 1. Input the original sample data of size ni, i ¼ 1, 2, set the specification
limits USL, LSL, target value T, the probability p*, and the constants a ¼ 0.085514, b ¼ 0.513277.
Step 2. Calculate the sample mean xxi, sample standard deviation Si, ^lli and ^i2,
i ¼1, 2: x xi¼ 1 ni Xni j¼1 xij, Si¼ 1 ni1 Xni j¼1 ðxijxxiÞ2 " #1=2 , ^lli¼ni x xiT Si 2 ^ i2¼ Pni j¼1 ðxijT Þ2 ni ¼ðni1ÞS 2 i þniðxxiT Þ2 ni :
Step3. Calculate ^vvi, ajðaj¼1= ^vv½j Þ, and a*: ^vv1¼ n1þ ^ll1 2 n1þ2^ll1 , ^vv2¼ n2þ ^ll2 2 n2þ2^ll2 , a1¼ 1 ^vv½1 , a2¼ 1 ^vv½2 , a¼0:5 a2 a1 : Step4. Calculate d1, d2, d3, and A1, A2:
d1¼a 1 þ a2 a1 þa2a 2 a2 a1þa2 a1 , d2¼b ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þa2 a1 r þab a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1þa2 p a1 a2 ffiffiffiffiffi a1 p d3¼ b2a2 4aa 1 ln 2p pffiffiffiffiffiffiffi2a , A 1¼ d2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 2 4d1d3 q 2d1 , A2¼ d2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 2 4d1d3 q 2d1 : Step5. Calculate c1and c2, then choose the value c from c1and c2:
c1¼exp 2A1 ffiffiffiffiffiffi 1 ^vv½1 s þ 1 ^vv½1 1 ^vv½2 ffiffiffiffiffiffi^vv ½2 ^vv½1 s ( ) c2¼exp 2A2 ffiffiffiffiffiffi 1 ^vv½1 s þ 1 ^vv½1 1 ^vv½2 ffiffiffiffiffiffi ^vv½2 ^vv½1 s ( ) :
Then choose the value c that is greater than 1 but as small as possible: c ¼minfc1, c2g, if c1>1 and c2 >1
c ¼ c1, if c1>1 and c21; c ¼ c2, if c2>1 and c1 1:
Step6. Conclude which supplier is better using the following rule R:
If ^12c ^22 and ^22 > c ^12 then we conclude that 1is the better supplier.
If ^22c ^12 and ^21 > c ^22 then we conclude that 2is the better supplier.
If ^12c ^22 and ^22 c ^12, then there is not enough information to make a
supplier selection.
4. Selection power analysis
Huang and Lee (1995) proposed a mathematically complicated approximation method for selecting a subset of processes containing the best supplier from a given set of processes based on the index Cpm. The method essentially compares the
aver-age loss of a group of candidate processes and selects a subset of these processes with small process loss 2, which with a certain level of confidence contains the best process. The accuracy of the selection method is investigated by using the simulation technique. The accuracy analysis provides useful information about the sample size required for the designated selection power.
4.1. Sample size required for designated selection power
In practice, if a new supplier II wants to compete for the orders by claiming that its capability is better than the existing supplier I, then the new supplier must furnish convincing information justifying the claim with a prescribed level of confidence. Thus, the sample size required for a designated selection power must be determined to collect actual data from the factories. The method, however, applies some approx-imating results and provides no indication on how one could further proceed to
select the best population among those chosen subsets of populations. This method was investigated for cases with two candidate processes. If the minimum requirement of Cpm values for two candidate processes, Cpm0, and the minimal difference
¼ Cpm2Cpm1 are determined, then the sample size required needs to sample
such that the suppliers must be differentiated with designated selection power. Thus, based on the proposed selection procedures, if ^22c ^12 and ^12> c ^22,
then it is concluded that the Cpmof 2is better than 1. Otherwise, one would believe
that the existing supplier I is better than the new supplier II since there is not sufficient information to reject the null hypothesis. The selection method and accu-racy analysis were investigated using a simulation technique with a simulated 10 000 numbers. For users’ convenience in applying the procedure in practice, the sample sizes required for various designated selection power were tabulated as 0.90, 0.95, 0.975, 0.99. The selection power calculates the probability of rejecting the null hypothesis H0: Cpm1Cpm2, while actually Cpm1< Cpm2is true, using the simulation
technique. Tables 1–4 summarize the sample size required for various capability requirements Cpm¼1.00, 1.33, 1.50, 1.67 and difference ¼ 0.05(0.05)1.00 under
the p* condition ¼ 0.95, respectively. For example, if the capability requirement of
Cpm1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Cpm2 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 0.90 3408 898 414 240 165 118 90 71 59 50 0.95 4351 1120 520 347 204 151 115 91 73 63 0.975 5130 1356 640 371 250 180 137 109 91 76 0.99 6131 1631 785 451 303 220 171 135 110 93 Cpm1 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Cpm2 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 0.90 43 37 32 29 26 24 22 22 19 18 0.95 53 48 41 37 33 31 29 28 27 26 0.975 65 57 50 45 40 37 34 30 28 27 0.99 80 70 61 56 49 45 40 38 35 33
Table 1. Sample size required for power ¼ 0.90, 0.95, 0.975, 0.99 under p* ¼ 0.95, with Cpm1¼1.00, Cpm2¼1.05(0.05)2.00. Cpm1 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 Cpm1 1.38 1.43 1.48 1.53 1.58 1.63 1.68 1.73 1.78 1.83 0.90 5900 1520 694 400 269 194 147 115 94 79 0.95 7493 1297 896 530 343 246 191 149 119 102 0.975 9014 2350 1060 622 401 301 231 178 147 120 0.99 10999 2859 1315 765 499 368 272 222 175 149 Cpm1 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 1.33 Cpm2 1.88 1.93 1.98 2.03 2.08 2.13 2.18 2.23 2.28 2.33 0.90 67 59 52 45 41 36 33 32 29 26 0.95 85 73 65 59 52 46 43 39 35 33 0.975 103 90 78 69 64 56 51 48 43 39 0.99 127 109 95 85 76 70 64 56 52 49
Table 2. Sample size required for power ¼ 0.90, 0.95, 0.975, 0.99 under p* ¼ 0.95, with Cpm1¼1.33, Cpm2¼1.38(0.05)2.33.
suppliers Cpmis set to 1.00 and ¼ 0.30, one would need to collect 151 samples to
satisfy the designated selection power ¼ 0.95.
Note that the sample size required is a function of Cpm, the difference between
two suppliers and the designated selection power. Tables 1–4 show that the larger the difference between two suppliers, the smaller the sample size required for a fixed selection power. For fixed and Cpm, the sample size required increases as the
desig-nated selection power increases. This phenomenon can be explained easily, since the smaller the difference and the larger the designated selection power, the more the collected sample is required to account for the smaller uncertainty in the estimation. 4.2. Phase I: Supplier selection
In most applications, the supplier selection decisions would be based solely on the hypothesis testing comparing the two Cpm values: H0: Cpm1Cpm2 versus
H1: Cpm1<Cpm2. If the test rejects the null hypothesis H0: Cpm1Cpm2, then there
is sufficient information to conclude that the new supplier II is superior to the original supplier I, and the decision of the replacement would be suggested.
For the Phase I of the supplier selection problem, the practitioner should input the preset minimum requirement of Cpmvalues, and the minimal difference that must
Cpm1 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 Cpm2 1.72 1.77 1.82 1.87 1.92 1.97 2.02 2.07 2.12 2.17 0.90 9291 2360 1091 630 408 292 223 173 141 115 0.95 12004 3034 1387 807 531 371 282 220 177 151 0.975 14297 3700 1650 970 629 448 338 260 218 180 0.99 17990 4400 2000 1163 765 544 400 325 255 220 Cpm1 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 1.67 Cpm2 2.22 2.27 2.32 2.37 2.42 2.47 2.52 2.57 2.62 2.67 0.90 100 85 75 66 60 52 49 43 39 38 0.95 125 108 95 85 75 67 63 55 51 48 0.975 154 130 115 102 91 82 75 66 63 56 0.99 185 159 140 120 112 99 91 83 74 69
Table 4. Sample size required for power ¼ 0.90, 0.95, 0.975, 0.99 under p* ¼ 0.95, with Cpm1¼1.67, Cpm2¼1.72(0.05)2.67. Cpm1 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 Cpm2 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 0.90 7394 1941 891 513 338 245 184 145 118 96 0.95 9506 2460 1120 657 430 308 232 180 151 125 0.975 11503 3001 1338 801 515 376 283 220 180 151 0.99 13502 3540 1634 974 627 457 340 268 221 177 Cpm1 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 Cpm2 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 0.90 83 71 62 55 49 45 39 38 35 32 0.95 106 91 79 71 63 56 51 48 44 40 0.975 125 109 95 85 75 69 63 57 53 50 0.99 155 134 115 103 92 83 75 71 65 60
Table 3. Sample size required for power ¼ 0.90, 0.95, 0.975, 0.99 under p* ¼ 0.95, with Cpm1¼1.50, Cpm2¼1.55(0.05)2.50.
be differentiated between suppliers with designated selection power. The practitioner alternatively might check tables 1–4 for the sample size required for p* con-dition ¼ 0.95, with designated selection powers ¼ 0.90, 0.95, 0.975, 0.99. In this case, one only needs to compare the test statistic ^i2, i ¼ 1, 2, with the selection
value c based on the selection procedure corresponding to the preset capability requirement and required sample sizes.
4.3. Phase II: Magnitude outperformed measurement
In Phase I of the supplier selection problem, the supplier selection decisions would be based solely on the hypothesis testing comparing the two Cpm values
without investigating further the magnitude of the difference between the two sup-pliers. In other applications, the supplier selection decisions would be based on the hypothesis test comparing the two Cpm values: H0: Cpm1þh Cpm2 versus
H1: Cpm1þh<Cpm2, where h>0 is a specified constant. If the test rejects the null
hypothesis H0: Cpm1þh Cpm2, then there is sufficient information to conclude that
supplier II is significantly better than supplier I by a magnitude of h, and the replace-ment would then be made due to the high cost of the supplier replacereplace-ment. In this case, one would have to compare the test statistic ^i2, i ¼ 1, 2, with the selection value
ccorresponding to the preset capability requirement for a given sample and desig-nated selection power to ensure that the magnitude of the difference between the two suppliers exceeds h. Note that Cpm1must be greater than the preset capability
requirement, and Cpm2¼Cpm1þh, where h ¼ max{h0| test rejects Cpm1þh0Cpm2}.
The basic problem of checking whether or not the two suppliers meeting the preset capability requirement could be solved by finding the lower confidence bounds on their process capabilities.
4.4. Comments on the classical approach
The classical approach for estimating the fraction of defectives is to take a sample of size n and calculate the proportion D/n, where D is the number of defective items in the sample. Note that for processes with a very low fraction of defectives, the classical approach requires a large sample size for the sample to contain at least one defective item. Table 5 shows the sample sizes required for the sample to include at least one defective item with a probability of 95% for various fractions of defectives in ppm. For capability requirement Cpm¼1.33 (33.04 ppm), the classical approach
requires the sample size>90 000. Therefore, the classical approach is not feasible for real applications with a lower fraction of defectives.
5. Application example STN-LCD
Liquid crystals have been used for display applications with various configura-tions. Most of the recently produced displays involve the use of either twisted nematic (TN) or super twisted nematic (STN) liquid crystals. The technology for the latter was introduced recently to improve the performance of LCD without using
p 100 000 10 000 1000 100 10 1 0.1
n 29 299 2996 29 957 299 572 2 995 731 29 957 322
Table 5. Sample sizes required for various p (ppm).
the TFT. A larger twist angle results in a significantly larger electro-optical distor-tion. This leads to substantial improvement in the contract and viewing angles over TN displays. The STN-LCD products are popularly used to make personal digital assistants (PDAs), notebook personal computers, word processors and other periph-erals. Due to the advancement of modern manufacturing technology in making STN-LCD and relatively low production costs, STN-LCD has remained at a com-petitive advantage in the marketplace. A typical assembly drawing for the STN-LCD product is shown in figure 1, and the custom glass and modules of the STN-LCD product are shown in figure 2.
An increasing number of personal computers are now network-ready and multi-media capable and are equipped with CD-ROM drives. Due to advances in tele-communications’ technology, simple monochromatic displays are no longer in popular demand. The next generation of telecommunication products will require displays with rich, graphic-quality images and personal interfaces. Therefore, future displays must be clearer and sharper to meet these demands. Until this point, STN-LCD have been used mainly to display still images, and because of the slow response time needed to process still images, STN-LCD have not been able to reproduce animated images at an adequate contrast level. Thus, with the growing popularity
Figure 1. Assembly drawing for the STN-LCD product.
Figure 2. Custom glass and modules of the STN-LCD product.
of multimedia applications, there is a need for PCs equipped with colour STN-LCD capable of processing animated pictures instead of still images. The space between the glass substrate is filled with liquid crystal material and the thickness of the liquid crystal is kept uniform with glass fibres or plastic balls as spacers. Therefore, the STN-LCD is sensitive to the thickness of the glass substrates.
To illustrate which of the two suppliers has a better process capability, a case study on STN-LCD manufacturing processes in a science-based industrial park manufacturer in Taiwan is presented. These factories manufacture various types of LCD. For a particular model of the STN-LCD investigated, the USL of a glass sub-strate’s thickness was set to 0.77 mm, the LSL of a glass subsub-strate’s thickness was set to 0.63 mm, and the target value was set to T ¼ 0.70 mm. If the characteristic data do not fall within the tolerance (LSL, USL), the lifetime or reliability of the STN-LCD will be discounted.
5.1. Data analysis and supplier selection
For Phase I of the Supplier Selection problem, the practitioner should input the preset minimum requirement of Cpmvalues, and the minimal difference that must be
differentiated between suppliers with a designated selection power. If the minimum requirement of an STN-LCD product is Cpm¼1.00, ¼ 0.25 with a selection power
of 0.95. By checking table 1, the sample size required for estimation is 204. Thus, the glass substrate’s thickness data taken from two LCD suppliers are shown in table 6. To confirm if the data of both suppliers are normally distributed, a Shapiro–Wilk test for normality is performed (figures 3 and 4). Because the p>0.05, the null hypothesis is not rejected because the data are normally distributed. Histograms of the data are shown in figures 5 and 6.
5.2. Phase I: Supplier selection
The aim is to determine if supplier II has a better process capability than supplier I, i.e. hypothesis testing must be performed to compare the two Cpm values,
H0: Cpm1Cpm2versus H1: Cpm1<Cpm2. First, calculate the sample means, sample
standard deviations, sample estimators of ^CCpm, ^2, and ^vv for suppliers I and II
(table 7). Based on the selection procedure, c1¼1.241426 and c2¼1.478218. Choose
the value of c that is larger than 1 and as small as possible, so c ¼ min{c1,
c2} ¼ 1.241426. In this case, one only needs to compare the test statistic ^i2,
i ¼1, 2, with the selection value c. Since ^22c ^12 and ^12> c ^22, it is concluded
that 2is a better supplier with a larger process capability Cpm.
5.3. Phase II: Magnitude outperformed measurement
To investigate further the magnitude of the capability difference between the two suppliers, the supplier selection decisions would find a magnitude of h such that Cpm2¼Cpm1þh, where h ¼ max{h0| test rejects Cpm1þh0Cpm2}. From the
estima-tion of Phase I, the obtained selecestima-tion values c and the decision based on the selec-tion procedure for h ¼ 0.01, 0.05, 0.10, 0.12(0.01)0.15 are shown in table 8. Therefore, from the analysis of magnitude outperformed detection based on sample statistics, the magnitude of the difference between the two suppliers is h ¼0.14. That is, it is concluded that Cpm2>Cpm1þ0.14.
Supplier I Supplier II 0.688 0.719 0.666 0.698 0.707 0.709 0.709 0.698 0.695 0.693 0.692 0.683 0.725 0.706 0.679 0.731 0.697 0.715 0.697 0.716 0.690 0.697 0.708 0.695 0.711 0.701 0.706 0.696 0.699 0.685 0.690 0.698 0.719 0.750 0.693 0.695 0.712 0.702 0.697 0.679 0.698 0.700 0.708 0.717 0.729 0.693 0.749 0.729 0.698 0.717 0.683 0.688 0.691 0.706 0.730 0.718 0.706 0.717 0.712 0.741 0.687 0.699 0.730 0.709 0.708 0.710 0.741 0.727 0.713 0.698 0.724 0.698 0.712 0.702 0.695 0.716 0.679 0.677 0.715 0.717 0.699 0.713 0.710 0.718 0.679 0.694 0.700 0.695 0.700 0.708 0.730 0.697 0.678 0.719 0.733 0.710 0.707 0.723 0.711 0.693 0.670 0.723 0.694 0.728 0.709 0.708 0.705 0.721 0.691 0.713 0.680 0.719 0.691 0.680 0.696 0.747 0.707 0.739 0.721 0.688 0.686 0.684 0.727 0.705 0.685 0.670 0.711 0.730 0.715 0.696 0.715 0.709 0.714 0.695 0.685 0.696 0.733 0.710 0.702 0.735 0.728 0.728 0.735 0.688 0.679 0.673 0.715 0.680 0.691 0.706 0.726 0.709 0.727 0.678 0.737 0.707 0.684 0.691 0.708 0.716 0.679 0.718 0.723 0.690 0.705 0.710 0.710 0.721 0.705 0.704 0.729 0.698 0.716 0.689 0.726 0.711 0.729 0.722 0.704 0.730 0.709 0.711 0.719 0.678 0.669 0.711 0.729 0.727 0.685 0.684 0.692 0.704 0.684 0.713 0.691 0.731 0.691 0.710 0.713 0.710 0.710 0.734 0.691 0.723 0.688 0.708 0.670 0.693 0.696 0.703 0.715 0.711 0.713 0.726 0.704 0.714 0.676 0.685 0.728 0.713 0.685 0.697 0.709 0.690 0.694 0.694 0.698 0.718 0.693 0.699 0.710 0.699 0.711 0.681 0.715 0.682 0.703 0.713 0.701 0.748 0.696 0.698 0.691 0.693 0.700 0.720 0.742 0.697 0.702 0.735 0.662 0.711 0.677 0.669 0.690 0.724 0.690 0.685 0.639 0.698 0.712 0.705 0.691 0.764 0.704 0.712 0.690 0.716 0.693 0.714 0.717 0.721 0.706 0.700 0.723 0.725 0.736 0.721 0.679 0.713 0.728 0.730 0.720 0.736 0.699 0.722 0.686 0.698 0.707 0.683 0.700 0.683 0.715 0.723 0.722 0.705 0.740 0.691 0.709 0.716 0.676 0.711 0.702 0.714 0.701 0.702 0.693 0.720 0.704 0.716 0.696 0.704 0.675 0.697 0.685 0.695 0.740 0.697 0.712 0.715 0.684 0.714 0.692 0.733 0.705 0.691 0.699 0.716 0.701 0.681 0.691 0.705 0.724 0.704 0.744 0.716 0.687 0.714 0.688 0.706 0.702 0.695 0.695 0.717 0.711 0.680 0.696 0.685 0.682 0.685 0.727 0.686 0.712 0.717 0.702 0.680 0.680 0.711 0.725 0.734 0.688 0.728 0.694 0.701 0.715 0.687 0.712 0.712 0.741 0.696 0.687 0.742 0.702 0.713 0.677 0.731 0.708 0.677 0.723 0.724 0.714 0.703 0.708 0.718 0.692 0.669 0.710 0.708 0.704 0.686 0.702 0.681 0.713 0.720 0.713 0.672 0.688 0.713 0.687 0.715 0.670 0.697 0.715 0.710 0.699 0.706 0.716 0.715
Table 6. Sample data collected from the two suppliers.
-3 -2 -1 0 1 2 3 0. 66 0. 68 0. 70 0. 72 0. 74 0. 76
Quantiles of Standard Normal
data2
Figure 3. Normal probability plot for thickness data of supplier I.
0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0 10 20 30 40
Figure 6. Histogram for supplier II.
0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0 10 20 30 40 50
Figure 5. Histogram for supplier I.
-3 -2 -1 0 1 2 3 0. 68 0. 70 0. 72 0. 74
Quantiles of Standard Normal
data1
Figure 4. Normal probability plot for thickness data of supplier II.
6. Conclusions
In the initial stage of production setting, the decision-maker usually faces the problem of selecting the best manufacturing supplier from several available manu-facturing suppliers. According to today’s modern quality improvement theory, reduction of the process loss is as important as increasing the process yield. The use of loss functions in quality assurance settings has grown with the introduction of Taguchi’s philosophy. The index Cpmincorporates with the variation of production
items with respect to the target value and the specification limits preset in the factory. Huang and Lee (1995) proposed a mathematically complicated approximation method for selecting a subset of processes containing the best supplier from a given set of processes based on the index Cpm. The present paper implements this
method and develops a practical step-by-step procedure for practitioners to use in making supplier selection decisions. Accuracy of the selection method is investigated by using a simulation technique. The accuracy analysis provides useful information about the sample size required for designated selection power. A two-phase selection procedure is developed to select a better supplier and further to examine the mag-nitude of the difference between the two suppliers. To make this method practical for in-plant applications, an application example of STN-LCD manufacturing processes, under a specific power, was also presented illustrating the sample size information to distinguish which supplier has better process capability.
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