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Supplier selection for one-sided processes with unequal sample sizes
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European Journal of Operational Research 195 (2009) 381–393 www.elsevier.com/locate/ejor
Stochastics and Statistics
Supplier selection for one-sided processes with unequal sample sizes W.L. Pearn a, H.N. Hung b, Ya Ching Cheng a,* a
Department of Industrial Engineering & Management, National Chiao Tung University, Taiwan b Institute of Statistics, National Chiao Tung University, Taiwan Received 13 September 2007; accepted 31 January 2008 Available online 9 February 2008
Abstract In this paper, we consider the supplier selection problem which deals with comparing two one-sided processes and selecting the one that has a significantly higher capability value. We present an exact approach to tackle the supplier selection problem. Testing hypotheses with two phases for comparing two processes are considered. Critical values of the tests are calculated to determine the selection decisions. Sample size required for a designated selection power and confidence level is also investigated. The results provide useful information to practitioners. The proposed approach generalizes the existing method by not requiring equal sample sizes for the two processes. An application example on comparing two wavelength division multiplexer (WDM) manufacturing processes is presented to illustrate the practicality of our approach to actual data collected from the factories. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Supplier selection; Process capability index; Nonconformities
1. Introduction Process capability indices have been widely used in the manufacturing industry, to measure the process capability of meeting the product quality requirement preset by the product designers or the process engineers. Several capability indices, including Cp, CPU, CPL, Cpk, Cpm, and Cpmk, are developed for this purpose (Kane, 1986; Pearn et al., 1992; Pearn et al., 1998). Those indices essentially compare the predefined manufacturing specifications with the actual production performance of the investigated quality characteristics and are defined as follows: USL LSL USL l l LSL USL l l LSL ; C PU ¼ ; C PL ¼ ; C pk ¼ min ; ; Cp ¼ 6r 3r 3r 3r 3r 9 8 > > = < USL LSL USL l l LSL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; C pm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; C pmk ¼ min > ; :3 r2 þ ðl T Þ2 3 r2 þ ðl T Þ2 > 6 r2 þ ðl T Þ2 where USL is the upper specification limit, LSL is the lower specification limit, l is the process mean, r is the process standard deviation (overall process variation), and T is the target value. While the indices Cp, Cpk, Cpm, and Cpmk are appropriate for normal processes with two-sided specification limits, the indices CPU and CPL are designed for processes with one-sided specification limit, and only one of USL and LSL is required in this case.
*
Corresponding author. Tel.: +886 3 5712121x57357. E-mail address: [email protected] (Y.C. Cheng).
0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.01.050
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The index Cpk measures actual process performance based on the process yield (fraction of product items conforming to the manufacturing specifications). With a fixed value of Cpk, the bounds of process yield are given by 2U(3Cpk) 1 6 yield 6 U(3Cpk), where U() is the cumulative distribution function of the standard normal distribution N(0, 1) (Boyles, 1991). For instance, if Cpk = 1.00, then it guarantees that the yield will be no less than 99.73%, or equivalently, the fraction of nonconformities is no greater than 2700 parts per million. Those indices have been used to establish the MPPAC (multi-process performance analysis chart) for factory defective control applications (Pearn and Chen, 1997; Pearn and Shu, 2003a,b), process performance analysis for multiple quality characteristics (Bothe, 1997; Wu and Pearn, 2005), better supplier selection (Chou, 1994; Huang and Lee, 1995), capability measures for multiple manufacturing streams (Bothe, 1999), variables acceptance sampling plans for lot sentencing (Pearn and Wu, 2006), and many others. Kotz and Johnson (2002) presented a thorough review on the development of process capability indices in the past 10 years. 2. Indices CPU, CPL and fraction of nonconformities For normally distributed processes with one-sided specification limit USL or LSL, formula of the process yield, Pr(X < USL), or Pr(X > LSL) in terms of CPU or CPL can be established as the following: X l USL l < ¼ PrðZ < 3C PU Þ ¼ Uð3C PU Þ; PrðX < USLÞ ¼ Pr r r l X l LSL < ¼ PrðZ > 3C PL Þ ¼ Uð3C PL Þ; PrðX > LSLÞ ¼ Pr r r where Z is the variable of standard normal N(0, 1). For convenience of presentation, we define CI = CPU or CPL. Therefore, the corresponding nonconformities in parts per million (NCPPM) for a well controlled normally distributed process can be calculated exactly, as NCPPM = 106 [1 U(3CI)]. Table 1 displays the corresponding NCPPM for CPU, CPL = 1.00, 1.25, 1.33, 1.45, 1.50, 1.60, 1.67, and 2.00. In current practice, a process is called inadequate if CI < 1.00; it indicates that the process is not adequate with respect to the production tolerances, either the process variation r2 needs to be reduced or the process mean l needs to be shifted closer to the target value. A process is called capable if 1.00 6 CI < 1.33; it indicates that caution needs to be taken regarding the process distribution and some process control is required. A process is called satisfactory if 1.33 6 CI < 1.50; it indicates that process quality is satisfactory, material substitution may be allowed, and no stringent quality control is required. A process is called good if 1.50 6 CI < 1.67. A process is called excellent if 1.67 6 CI < 2.00; it indicates that process quality exceeds satisfactory. Finally, a process is called super if CI P 2.00 (Pearn and Chen, 2002). Table 2 summarizes the above five conditions and the corresponding CI values.
Table 1 Corresponding NCPPM for some specific values of CI CI
NCPPM
1.00 1.25 1.33 1.45 1.50 1.60 1.67 2.00
1349.898 88.417 33.037 6.807 3.398 0.793 0.272 0.001
Table 2 Some commonly used capability requirement quality conditions Quality condition
CI values
Inadequate Capable Satisfactory Good Excellent Super
CI < 1.00 1.00 6 CI < 1.33 1.33 6 CI < 1.50 1.50 6 CI < 1.67 1.67 6 CI < 2.00 2.00 6 CI
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Table 3 Some minimal capability requirements of CPU and CPL for new and special processes Index value
Production process types
1.25 1.45 1.60
Existing processes New processes, or existing processes on safety, strength, or critical parameters New processes on safety, strength, or critical parameters
Montgomery (2001) recommended some minimum quality requirements on CPU and CPL, as presented in Table 3, for specific process types which must run under some designated capability conditions. It would be desirable to determine a bound so that practitioners could expect to find the true value of the process capability no less than the bound with certain level of confidence. Noting that for newly setup processes on safety, strength, or with critical parameters, the fraction of nonconformities should target no more than 0.8 PPM, a more stringent requirement set for possible 1.5r mean shift (Bothe (2002)). 3. WDM supplier selection Consider a company importing the couplers, splitters and wavelength division multiplexers (WDM). These devices are important passive components to the optical fiber communications, which divide route or combine multiple optical signals. Fiber-optic couplers either split optical signals into multiple paths or combine multiple signals on one path. Like electrical currents, a flow of signal carriers, in this case photons, comprise the optical signal. However, an optical signal does not flow through the receiver to the ground. Rather, at the receiver a detector absorbs the signal flow. Multiple receivers, connected in a series, would receive no signal passing the first receiver, which would absorb the entire signal. Multiple parallel optical output ports must be connected to divide the signal between the ports, which reduces the signal magnitude. The number of input and output ports characterizes a coupler. The most interesting type of fiber coupler is the wavelength division multiplexer (WDM). 3.1. Optical characteristic Polarization dependent loss (PDL) is one of the most important characteristics of those optical passive components. Polarization dependent loss (PDL) is a measure of the peak-to-peak difference during the signal transmission in an optical component or system with respect to all possible states of polarization. Components in fiber-optic telecommunications networks must be insensitive to changes in the polarization state of the signal, a requirement that is particularly challenging for surface-relief diffraction gratings since those with metallic coatings tend to polarize the light incident on them. The polarization of the constant, and fully polarized, input signal is varied. As the polarization of the incident light varies, the output signal shows a corresponding change in power. The measure of polarization dependent loss (PDL) is expressed in units of decibels (dB), with upper specification limit, USL, set to 0.08 dB. The company importing the couplers, splitters and WDM has two suppliers manufacturing the WDM and competing for the company’s orders. The company would like to determine which supplier provides better WDM products. The characteristic PDL is the one of the key measurements for the comparison. Table 4 displays the PDL data collected from Table 4 PDL data of suppliers I and II (in dB) Supplier I PDL data 0.058 0.051 0.065 0.058 0.064 0.060 0.059 0.059 0.058 0.068 0.061 0.056 0.060 0.061 0.069
0.064 0.059 0.058 0.054 0.064 0.061 0.056 0.051 0.059 0.059 0.063 0.054 0.072 0.068 0.062
Supplier II PDL data 0.066 0.060 0.055 0.061 0.060 0.064 0.067 0.061 0.063 0.067 0.064 0.064 0.061 0.055 0.060
0.062 0.051 0.067 0.070 0.058 0.061 0.066 0.058 0.054 0.064 0.067 0.062 0.058 0.058 0.070
0.057 0.060 0.058 0.056 0.057 0.068 0.064 0.060 0.065 0.061 0.067 0.070 0.056 0.049 0.062
0.057 0.064 0.054 0.068 0.051 0.054 0.064 0.054 0.064 0.061 0.064 0.052 0.062 0.055 0.068
0.062 0.057 0.064 0.058 0.062 0.061 0.058 0.061 0.055 0.062 0.062 0.057 0.062 0.071 0.064
0.052 0.049 0.041 0.052 0.052 0.057 0.056 0.049 0.051 0.045 0.051 0.042 0.048 0.052 0.053
0.045 0.060 0.055 0.041 0.045 0.049 0.048 0.048 0.045 0.048 0.055 0.055 0.049 0.055 0.048
0.058 0.042 0.054 0.042 0.053 0.052 0.048 0.054 0.050 0.048 0.038 0.047 0.046 0.055
0.049 0.041 0.053 0.062 0.048 0.059 0.050 0.052 0.049 0.061 0.046 0.052 0.045 0.056
0.054 0.047 0.057 0.052 0.049 0.054 0.057 0.040 0.046 0.060 0.049 0.045 0.047 0.047
0.054 0.046 0.048 0.050 0.057 0.052 0.047 0.051 0.046 0.053 0.044 0.050 0.045 0.054
0.051 0.053 0.055 0.048 0.055 0.052 0.044 0.053 0.050 0.053 0.049 0.050 0.051 0.048
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Supplier I with sample size n1 = 105, and Supplier II with n2 = 100. In current factory applications, most practitioners would take the intuitive and simple approach to just compare the sample estimates of CPU1 and CPU2 for the two suppliers, then make conclusion. Such approach ignores the sampling errors, which is considered highly unreliable. 3.2. Existing solution method Chou (1994) developed an approximation method for selecting better suppliers based on one-sided capability index CPU and CPL. A rule for testing H0: CPU1 P CPU2 versus H1: CPU1 < CPU2 is proposed. The rule, requiring equal sample sizes b PU 1 < C b PU 2 and satisfaction of the condition A < expfv2 ð1 2aÞ=2g, where n1 = n2 from the two suppliers, rejects H0 if C 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n b 2 þ 2 aC b PU 1 C b PU 2 b 2 þ 2 aC ; A ¼ 2= aC PU 1 PU 1 a = 9n/(n 1) and v21 ð1 2aÞ is the (1 2a)th quantile of a v2 distribution with one degree of freedom. The test statistic A is the log-likelihood ratio statistic approximating the exponential of the v2 distribution with one degree of freedom under the large-sample theory. The approximation method requires equal sample sizes, n1 = n2 = n, in the applications. Therefore, it cannot be applied to the above WDM example. In the following, we develop an exact method for selecting better suppliers, which is also applicable for cases with n1 – n2. 4. Development of the exact method To compare the two suppliers, we consider the hypothesis testing for comparing the two CPU values, H0: CPU1 P CPU2 versus H1: CPU1 < CPU2 (or equally, H0: CPU2/CPU1 6 1 versus H1: CPU2/CPU1 > 1). Under the assumption that the quality characteristic of the two processes, X1, and X2 are independent normally distributed, X 1 N ðl1 ; r21 Þ and X 2 N ðl2 ; r22 Þ, we have X 1 N ðl1 ; r21 =n1 Þ, X 2 N ðl2 ; r22 =n2 Þ. Therefore, b PU 1 C
USL X 1 USL X 1 ¼ ¼ = 3S 1 3r1
sffiffiffiffiffi S 21 ; r21
b PU 2 C
USL X 2 USL X 2 ¼ ¼ = 3S 2 3r2
sffiffiffiffiffi S 22 ; r22
and the testing statistic R, ratio of the two estimators, can be expressed as follows:
1 2 N USLl ; 9n12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N C PU 2 ; 9n2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3r2 b PU 2 = C b PU 1
F n1 1;n2 1 ¼
F n1 1;n2 1 : R¼C 1 ; 9n11 N USLl N C PU 1 ; 9n11 3r1 Thus, the distribution of the testing statistic R, is the convolution of the ratio of two normal distributions, and the square root of the F distribution, where the probability density
functions of the two distributions can be obtained below. 1 1 Probability density function of U N ðC PU 2 ; 9n2 Þ=N C PU 1 ; 9n1
Let X 1 N C PU 1 ; 9n11 , X 2 N C PU 2 ; 9n12 and U = X2/X1. Applying the well-known convolution formula fU ðuÞ ¼
Z
1
fX 1 ðx1 ÞfX 2 ðux1 Þjx1 j dx1 1
we can obtain the probability density function (PDF) of U as follows: " # pffiffiffiffiffiffiffiffiffi 2 9 n1 n2 9 9 ðn1 C PU 1 þ n2 C PU 2 uÞ 2 2 exp ðn1 C PU 1 þ n2 C PU 2 Þ exp fU ðuÞ ¼ 2p n1 þ n2 u2 2 2 ( " # 2 2 9 ðn1 C PU 1 þ n2 C PU 2 uÞ n1 C PU 1 þ n2 C PU 2 u : exp þ n1 þ n2 u2 9ðn1 þ n2 u2 Þ 2 n1 þ n2 u2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 2p 3ðn1 C PU 1 þ n2 C PU 2 uÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 2U 9ðn1 þ n2 u2 Þ n1 þ n2 u2 where 1 < u < 1.
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Probability density function of V F n1 1;n2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Assume that V F n1 1;n2 1 . By using the technique of change-of-variable, the probability density function (PDF) of V can be derived as follows:
n1 1 fV ðvÞ ¼ 2 n2 1
n121
2 C n1 þn 1 2 C n121 C n221
ðvÞn1 2 ;
n1 þn 2 2 1 n1 1 2 1 þ n2 1 v
where 0 < v < 1. b PU 1 b PU 2 = C Probability density function of C b PU 2 = C b PU 1 then can be expressed as R = U V, and by applying again the convolution The testing statistic R ¼ C formula Z 1
1
r r fU ðuÞfV fR ðrÞ ¼ I > 0
du; u u u 1
Fig. 1. PDF plots of R for sample sizes n1 = n2 = 30, 50, 100, 150, 200 (from bottom to top in plots).
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the probability density function of R then is as follows: " # r n1 2 Z 1 2
1 r 9 ðn C þ n C uÞ 1 PU 1 2 PU 2 u
I > 0 exp fR ðrÞ ¼ A
u 2 n1 þn n1 þ n2 u2 2 2 1 1 u n1 1 r 2 1 þ n2 1 u " # ( 2 2 9 ðn1 C PU 1 þ n2 C PU 2 uÞ n1 C PU 1 þ n2 C PU 2 u : exp þ n1 þ n2 u2 9ðn1 þ n2 u2 Þ 2 n1 þ n2 u2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 2p 3ðn1 C PU 1 þ n2 C PU 2 uÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du; 1 2U 9ðn1 þ n2 u2 Þ n1 þ n2 u2 where
n1 1 A¼2 n2 1
n121
pffiffiffiffiffiffiffiffiffi 2 C n1 þn 1 9 n1 n2 9 2 2 2 exp ðn1 C PU 1 þ n2 C PU 2 Þ ; 2p 2 C n121 C n221
U() is the cumulative density function of the standard normal distribution, I is the indicator function, and 1 < r < 1. Fig. 1 plots the probability density function of R for CPU1 = 1.0, 1.5, 2.0, CPU2 = 1.0, 1.5, 2.0, and n1 = n2 = 30, 50, 100, 150, 200 (from bottom to top in plots). The PDF plots of R reveal the following features: b PU 2 = C b PU 1 . 1. The larger the value of CPU2/CPU1, the larger the variance of R ¼ C 2. Distribution of R is single peaked and is almost symmetric to CPU2/CPU1. 3. The larger the sample sizes n1 and n2, the smaller the variance of R, which is certain for all sample estimators. 5. Supplier selection procedure 5.1. Phase I: Selecting a better process capability Assume that the minimum requirement of CPU values for all candidate processes is C, and the existing supplier, Supplier I, has achieved the requirement, i.e. CPU1 P C. If a new supplier, 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. To test whether the new supplier, Supplier II, has a significantly better capability than the existing Supplier I, we consider the hypothesis testing: H0 : C PU 2 6 C PU 1 ; H1 : C PU 2 > C PU 1 ; which is equivalent to H0 : C PU 2 =C PU 1 6 1; H1 : C PU 2 =C PU 1 > 1: b PU 2 = C b PU 1 and a given significance level a, the chance of incorrectly judging Based on the sampling distribution of R ¼ C b PU 2 = C b PU 1 P c0 , where c0 is CPU2 6 CPU1 as CPU2 > CPU1, the decision rule is to reject H0 if the testing statistic R ¼ C the critical value that satisfies PrfR P c0 jH0 : C PU 2 6 C PU 1 ; n1 ; n2 ; and C PU 1 P Cg 6 a: Since the larger the CPU2/CPU1 value, the larger the critical value, we calculate the critical value c0 with the conditions CPU1 = CPU2 = C, i.e. PrfR P c0 jC PU 1 ¼ C PU 2 ¼ C; n1 ; n2 g ¼ a: Table 5 shows the critical values for CPU1 = CPU2 = 1.0(0.1)2.0, n1 = n2 = n = 30(10)200 and a = 0.05. Note that our supplier selection procedure can be applied for cases with unequal sample sizes, n1 – n2. It is also noted that replacing CPU by CPL, the above procedure can also be used to test the hypothesis: H0 : C PL2 6 C PL1 ; H1 : C PL2 > C PL1 :
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Table 5 Critical values for rejecting CPU2 6 CPU1 with n1 = n2 = 30(10)200 and a = 0.05 n
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
CPU1 = CPU2 = C 1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
1.407 1.341 1.299 1.269 1.246 1.228 1.213 1.201 1.191 1.182 1.174 1.167 1.161 1.155 1.150 1.146 1.142 1.138
1.400 1.335 1.294 1.264 1.242 1.224 1.210 1.198 1.188 1.179 1.171 1.164 1.158 1.153 1.148 1.143 1.139 1.136
1.394 1.331 1.290 1.260 1.238 1.221 1.207 1.195 1.185 1.176 1.169 1.162 1.156 1.151 1.146 1.142 1.137 1.134
1.390 1.327 1.286 1.258 1.236 1.219 1.205 1.193 1.183 1.175 1.167 1.160 1.155 1.149 1.144 1.140 1.136 1.132
1.386 1.324 1.284 1.255 1.234 1.217 1.203 1.191 1.182 1.173 1.166 1.159 1.153 1.148 1.143 1.139 1.135 1.131
1.383 1.322 1.282 1.254 1.232 1.215 1.202 1.190 1.180 1.172 1.164 1.158 1.152 1.147 1.142 1.138 1.134 1.130
1.381 1.320 1.280 1.252 1.231 1.214 1.200 1.189 1.179 1.171 1.164 1.157 1.151 1.146 1.141 1.137 1.133 1.130
1.379 1.318 1.279 1.251 1.230 1.213 1.199 1.188 1.178 1.170 1.163 1.156 1.151 1.145 1.141 1.136 1.133 1.129
1.378 1.317 1.278 1.250 1.229 1.212 1.199 1.187 1.178 1.169 1.162 1.156 1.150 1.145 1.140 1.136 1.132 1.128
1.376 1.316 1.277 1.249 1.228 1.211 1.198 1.187 1.177 1.169 1.161 1.155 1.149 1.144 1.140 1.135 1.132 1.128
1.374 1.314 1.275 1.247 1.227 1.210 1.197 1.186 1.176 1.168 1.161 1.154 1.149 1.143 1.139 1.135 1.131 1.127
5.2. 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 CPU values without investigating further the magnitude of the difference between the two suppliers. In practice, the customer would consider choosing the new supplier only if the new supplier’s capability is significantly better than the existing supplier’s by a magnitude of h > 0, due to the high cost of the supplier replacement. Our exact approach, in this case, can be used to test the corresponding hypothesis: H0 : C PU 2 6 C PU 1 þ h; H1 : C PU 2 > C PU 1 þ h: b PU 2 = C b PU 1 is larger than The decision rule is similar to Phase I. We would reject H0, accept that CPU2 > CPU1 + h, if R ¼ C or equal to the critical value c0, where c0 satisfies PrfR P c0 jH0 : C PU 2 6 C PU 1 þ h; n1 ; n2 ; and C PU 1 P Cg 6 a: For all combinations of (CPU1, CPU2) under H0, the maximal critical value occurs at CPU1 = C and CPU2 = C + h, and the larger the a, the smaller the critical value. Thus, we calculate the critical value c0 with the probability PrfR P c0 jC PU 1 ¼ C; C PU 2 ¼ C þ h; n1 ; n2 g ¼ a: If the test rejects the null hypothesis H0, then there is sufficient information to conclude that supplier II is significantly better than supplier I by a magnitude of h, and the replacement of Supplier I by Supplier II would be suggested. Table 6 shows some critical values for some minimum quality requirements C of CPU suggested by Montgomery (2001), the magnitude h = 0.1(0.1)0.5 of the difference between the two suppliers and n1 = n2 = n = 30(10)200 with a = 0.05. 5.3. Required sample size In Phases I and II, the supplier selection procedure solely depends on the a risk, incorrectly judging H0 as H1, and not takes into account the b risk, incorrectly judging H1 as H0, which is comparatively unfavorable to H1 and the new supplier. This is due to the additional cost of replacing the existing supplier with the new one. Once the sample sizes and the a risk are defined, the power of test, also known as 1 b, will be fixed. Fig. 2 plots some particular power values versus CPU2 for CPU1 = 1.0, 1.5, 2.0, n1 = n2 = n = 30, 50, 100, 150, 200, and a = 0.05. It can be seen that the larger the sample size, the larger the power of test, consequently, and the smaller the b risk. To decrease the b risk and at the same time maintain the a risk in a small level, we could increase the sample sizes. By calculating the power for a specific value of CPU2, we could obtain the minimal sample size required for designated power and a risk. The required sample size could be calculated by recursive search with the following two constraints:
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Table 6 Critical values for rejecting CPU2 6 CPU1 + h with a = 0.05 n
(CPU1, CPU2) (1.25, 1.35)
(1.25, 1.45)
(1.25, 1.55)
(1.25, 1.65)
(1.25, 1.75)
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
1.501 1.433 1.390 1.358 1.335 1.316 1.301 1.289 1.278 1.269 1.261 1.253 1.247 1.241 1.236 1.231 1.227 1.223
1.611 1.538 1.491 1.458 1.433 1.413 1.397 1.383 1.372 1.362 1.353 1.346 1.339 1.333 1.327 1.322 1.317 1.313
1.721 1.643 1.593 1.558 1.531 1.510 1.492 1.478 1.466 1.455 1.446 1.438 1.430 1.424 1.418 1.413 1.408 1.403
1.831 1.748 1.695 1.657 1.629 1.606 1.588 1.573 1.560 1.549 1.539 1.530 1.522 1.515 1.509 1.503 1.498 1.493
1.941 1.853 1.797 1.757 1.727 1.703 1.684 1.668 1.654 1.642 1.631 1.622 1.614 1.607 1.600 1.594 1.588 1.583
n
(CPU1, CPU2) (1.45, 1.55)
(1.45, 1.65)
(1.45, 1.75)
(1.45, 1.85)
(1.45, 1.95)
1.479 1.413 1.370 1.340 1.317 1.299 1.284 1.272 1.262 1.253 1.245 1.238 1.232 1.226 1.221 1.216 1.212 1.208
1.574 1.503 1.458 1.426 1.402 1.382 1.367 1.354 1.343 1.333 1.325 1.317 1.311 1.305 1.299 1.294 1.290 1.286
1.668 1.594 1.546 1.512 1.486 1.466 1.449 1.435 1.424 1.413 1.405 1.397 1.390 1.383 1.378 1.373 1.368 1.363
1.762 1.684 1.633 1.598 1.570 1.549 1.531 1.517 1.505 1.494 1.484 1.476 1.469 1.462 1.456 1.451 1.446 1.441
1.857 1.774 1.721 1.683 1.655 1.632 1.614 1.599 1.585 1.574 1.564 1.556 1.548 1.541 1.535 1.529 1.524 1.519
(1.60, 1.70)
(1.60, 1.80)
(1.60, 1.90)
(1.60, 2.00)
(1.60, 2.10)
1.467 1.402 1.360 1.330 1.307 1.289 1.275 1.263 1.252 1.244 1.236 1.229 1.223 1.217 1.212 1.208 1.204 1.200
1.552 1.483 1.439 1.407 1.384 1.365 1.349 1.337 1.326 1.316 1.308 1.301 1.294 1.289 1.283 1.279 1.274 1.270
1.638 1.565 1.518 1.485 1.460 1.440 1.424 1.411 1.399 1.389 1.381 1.373 1.366 1.360 1.354 1.349 1.345 1.340
1.723 1.647 1.598 1.563 1.536 1.516 1.499 1.484 1.472 1.462 1.453 1.445 1.438 1.431 1.425 1.420 1.415 1.411
1.809 1.729 1.677 1.641 1.613 1.591 1.573 1.558 1.546 1.535 1.525 1.517 1.509 1.503 1.496 1.491 1.486 1.481
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 n
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
(CPU1, CPU2)
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Fig. 2. Power curve for CPU1 = 1.0, 1.5, and 2.0.
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PrfR P c0 jH0 : C PU 2 6 C PU 1 ; n1 ; n2 ; and C PU 1 P Cg 6 a; and; PrfR P c0 jH1 : C PU 2 > C PU 1 ; n1 ; n2 ; and C PU 1 P Cg P 1 b: For users’ convenience in applying the supplier selection procedure, we tabulate the sample sizes required in Table 7 for various designated selection power with the minimal capability requirement CPU = 1.00, 1.25, 1.45, 1.60, the magnitude of difference CPU2 CPU1 = 0.15(0.05)1.00, and designated power=0.90, 0.95, 0.975, 0.99. Note that if the process of Supplier I is investigated first, that is n1 is fixed, we can increase n2 to satisfy the requirement of both risks. In practice, both suppliers would be asked to sample with equal sample size. In Table 7, we show the sample size required for circumstances with n1 = n2. For example, if the minimal capability requirement CPU is 1.25, the designated a and b risks are 0.05 (i.e. designated power of test = 0.95), and the expected magnitude of difference CPU2 CPU1 is 0.3, then the required sample size for both suppliers to sample is 267. 5.4. Comparison with Chou’s approximation We compared our exact approach with Chou’s approximation using simulations. Table 8 displays the simulation results for actual Type I error a each with 107 simulated samples for CPU1 = CPU2 = 1.00, 1.33, 1.50, 1.67, 2.00, n = 30(10)100, and a = 0.05. The results indicate that our exact approach is indeed more accurate than Chou’s approximation. While Chou’s approximation method requires equal sample sizes, n1 = n2, the proposed exact approach is applicable for unequal sample sizes n1 – n2, which is more accurate and more general.
Table 7 Sample size required for testing H0: CPU2 6 CPU1 versus H1: CPU2 > CPU1 CPU1
CPU2
Power
CPU1
0.90
0.95
0.975
0.99
CPU2
Power 0.90
0.95
0.975
0.99
1.00
1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00
535 314 210 152 116 93 76 64 55 48 43 38 35 31 29 27 25 23
675 396 264 191 146 116 96 81 69 60 53 48 43 39 36 33 31 29
810 474 316 229 175 139 114 96 82 72 63 57 51 47 43 39 36 34
980 574 383 277 211 168 138 116 99 87 77 68 62 56 51 47 44 41
1.25
1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25
763 445 295 212 161 128 104 87 75 65 57 51 46 41 38 35 32 30
965 561 372 267 203 161 131 110 94 81 71 64 57 52 47 43 40 37
1157 673 447 320 243 192 157 131 112 97 85 76 68 62 56 52 48 44
1403 816 540 388 295 233 190 159 135 117 103 92 82 74 68 62 57 53
1.45
1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45
983 570 376 269 204 161 131 109 93 80 71 63 56 51 46 43 40 37
1241 719 474 339 257 203 165 137 117 101 88 78 70 63 58 53 49 45
1488 862 569 407 308 243 197 164 140 121 106 94 84 76 69 63 58 54
1805 1046 690 493 373 294 239 199 169 146 128 113 101 91 83 76 70 65
1.60
1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60
1167 674 444 317 239 189 153 127 108 93 82 72 65 58 53 48 45 41
1475 852 560 400 302 238 193 160 136 117 103 91 81 73 66 61 56 52
1768 1021 672 479 362 285 231 192 163 140 123 109 97 87 79 72 67 62
2145 1238 815 581 438 345 280 232 197 170 148 131 117 106 96 87 80 74
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Table 8 The actual Type I error a of Chou’s approximation and our exact approach CPU
1.00
1.33
1.50
1.67
2.00
n
Chou’s
Exact
Chou’s
Exact
Chou’s
Exact
Chou’s
Exact
Chou’s
Exact
30 40 50 60 70 80 90 100
0.0544 0.0531 0.0526 0.0521 0.0520 0.0515 0.0516 0.0516
0.0500 0.0500 0.0498 0.0499 0.0501 0.0499 0.0504 0.0504
0.0543 0.0532 0.0525 0.0520 0.0520 0.0518 0.0517 0.0516
0.0498 0.0499 0.0502 0.0497 0.0500 0.0500 0.0500 0.0503
0.0543 0.0532 0.0526 0.0521 0.0519 0.0518 0.0518 0.0516
0.0500 0.0497 0.0499 0.0497 0.0502 0.0504 0.0499 0.0504
0.0544 0.0531 0.0526 0.0521 0.0521 0.0517 0.0519 0.0516
0.0500 0.0499 0.0499 0.0499 0.0499 0.0501 0.0507 0.0504
0.0543 0.0532 0.0526 0.0522 0.0521 0.0518 0.0519 0.0516
0.0503 0.0502 0.0504 0.0507 0.0503 0.0507 0.0505 0.0503
6. Supplier selection implementations 6.1. WDM data analysis The company importing WDM products has a minimal requirement on the polarization dependent loss (PDL). The minimal requirement of the PDL characteristic is CPU = 1.25. Kolmogorov–Smirnov normality tests for data of both suppliers are performed first to confirm if the data of both suppliers are normally distributed. Both tests result in p-value >0.15, which means that data of both suppliers are normally distributed. Histograms of the data are shown in Fig. 3. 6.2. Phase I: Selecting a better process capability To determine if Supplier II provides a better process capability of the WDM products than Supplier I, we perform the hypothesis testing: H0: CPU2 6 CPU1 versus H1: CPU2 > CPU1. For the PDL data displayed in Tables 4 and 5, we calculate the sample means, sample standard deviations and the sample estimators for both suppliers, and obtain that x1 ¼ 0:06079, b PU 1 ¼ 1:2936, C b PU 2 ¼ 2:04527 and thus R = 1.58107. To compute the critical x2 ¼ 0:05018, s1 = 0.00495, s2 = 0.00486, C value, we develop a ‘Visual C++’ computer program to handle the complicated computations. (The program can be obtained by request.) The program reads the input parameters CPU1, CPU2, the corresponding sample sizes n1, n2, and a, the risk for wrongly rejecting CPU2 6 CPU1 while actually CPU2 > CPU1 is true, and outputs with the critical value. We run the C++ program with n1 = 105, n2 = 100, CPU1 = CPU2 = 1.25 (the minimal capability requirement of CPU) and a = 0.05 to obtain the critical value as 1.1924. Since the testing statistic R = 1.58107 > 1.1924, we therefore conclude that the Supplier II is superior than Supplier I with 95% confidence level. 6.3. Phase II: Magnitude outperformed measurement To investigate further the magnitude of the capability difference between the two suppliers, we perform the hypothesis testing: H0: CPU2 6 CPU1 + h versus H1: CPU2 > CPU1 + h. We run the C++ program with n1 = 105, n2 = 100, CPU1 = 1.25
Fig. 3. Histograms of the two PDL data.
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Table 9 Critical values and decisions of testing the two WDM suppliers CPU1 CPU2 h c0 Decision
1.25 1.45 0.20 1.3813 Reject H0
1.25 1.55 0.30 1.4758 Reject H0
1.25 1.65 0.40 1.5704 Reject H0
1.25 1.66 0.41 1.5799 Reject H0
1.25 1.67 0.42 1.5893 Do not Reject H0
(the minimal capability requirement of CPU), CPU2 = 1.25 + h, h = 0.2(0.1)0.4, 0.41, 0.42 and a = 0.05 to obtain the critical values for various h. The decisions of the hypotheses are shown in Table 9. Based on the testing results, we can conclude that the Supplier II has a manufacturing capability that is significantly better than the Supplier I by a magnitude of 0.41, i.e. CPU2 > CPU1 + 0.41. 6.4. Sample size required for a designated power Assume that the minimal requirement of CPU values for all candidate processes is 1.25, and the customer would replace the existing supplier with a new one if the capability of the new supplier is significantly better than the existing supplier by a magnitude of 0.3. Then, to make a hypothesis testing with a designated selection power of 0.90, the sample size required to sample is 212 as in Table 7. In the WDM supplier selection problem, since the sample sizes of two suppliers are smaller than 212, the selection power for testing H0: CPU2 6 CPU1 versus H1: CPU2 > CPU1 would be less than 0.90. In fact, the power of test for CPU2 = 1.66 is 0.8489, that is the b risk of incorrectly accepting CPU2 6 CPU1 while actually CPU2 > CPU1 is true is up to 0.1511. In order to reduce the b risk, we would suggest the customer to sample for a designated power with as large sample size as in Table 7. 7. Conclusions For stable normal processes with one-sided specification limits, capability indices CPU and CPL have been widely used in the manufacturing industry to provide numerical measures on process capability. In this paper, we considered the supplier selection problem for two one-sided processes, and presented an exact analytical approach to solve the problem. The proposed approach which generalizes the existing method is applicable to test the hypothesis H0: CPU2 6 CPU1 versus H1: CPU2 > CPU1 with unequal sample sizes, and the decisions made are more reliable. In practice, often the replacement of suppliers would be made only if the new supplier’s capability is significantly better than the existing supplier’s by a magnitude of h > 0. The proposed exact approach, in this case, can be used to test the corresponding hypothesis H0: CPU2 6 CPU1 + h versus H1: CPU2 > CPU1 + h. Sample size required for a specific value of CPU2 is also investigated under a designated selection power and confidence level. The results are useful to practitioners or engineers. A real-world application comparing two WDM manufacturing processes is presented to illustrate the practicality of the exact approach to actual data collected from the factories. Acknowledgements The authors would like to thank the Editor and the anonymous referees for their valuable comments and suggestions, which significantly improved the presentation of this paper. This paper was partially supported by National Science Council of Taiwan Grant NSC-95-2118-M-009-004-MY2. References Bothe, D.R., 1997. A capability study for an entire product. In: ASQC Quality Congress Transactions, Nashville, vol. 46, pp. 172–178. Bothe, D.R., 1999. A capability index for multiple process streams. Quality Engineering 11 (4), 613–618. Bothe, D.R., 2002. Statistical reason for the 1.5r shift. Quality Engineering 14 (3), 479–487. Boyles, R.A., 1991. The Taguchi capability index. Journal of Quality Technology 23 (1), 17–26. Chou, Y.M., 1994. Selecting a better supplier by testing process capability indices. Quality Engineering 6 (3), 427–438. Huang, D.Y., Lee, R.F., 1995. Selecting the largest capability index from several quality control processes. Journal of Statistical Planning and Inference 46 (3), 335–346. Kane, V.E., 1986. Process capability indices. Journal of Quality Technology 18 (1), 41–52. Kotz, S., Johnson, N.L., 2002. Process capability indices – A review, 1992–2000. Journal of Quality Technology 34 (1), 1–19. Montgomery, D.C., 2001. Introduction to Statistical Quality Control, 4th ed. John Wiley & Sons, Inc., New York, NY. Pearn, W.L., Chen, K.S., 1997. Multi-process performance analysis: A case study. Quality Engineering 10 (1), 1–8. Pearn, W.L., Chen, K.S., 2002. One-sided capability indices CPU and CPL: Decision making with sample information. International Journal of Quality & Reliability Management 19 (3), 221–245.
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Pearn, W.L., Shu, M.H., 2003a. Manufacturing capability control for multiple power distribution switch processes based on modified Cpk MPPAC. Microelectronics Reliability 43 (6), 963–975. Pearn, W.L., Shu, M.H., 2003b. Lower confidence bounds with sample size information for Cpm with application to production yield assurance. International Journal of Production Research 41 (15), 3581–3599. Pearn, W.L., Wu, C.W., 2006. Variables sampling plans with PPM fraction of defectives and process loss consideration. Journal of Operational Research Society 57 (4), 450–459. Pearn, W.L., Kotz, S., Johnson, N.L., 1992. Distributional and inferential properties of process capability indices. Journal of Quality Technology 24 (4), 216–231. Pearn, W.L., Lin, G.H., Chen, K.S., 1998. Distributional and inferential properties of the process accuracy and process precision indices. Communications in Statistics: Theory & Methods 27 (4), 985–1000. Wu, C.W., Pearn, W.L., 2005. Measuring manufacturing capability for couplers and wavelength division multiplexers. International Journal of Advanced Manufacturing Technology 25 (5–6), 533–541.
European Journal of Operational Research 195 (2009) 381–393 www.elsevier.com/locate/ejor
Stochastics and Statistics
Supplier selection for one-sided processes with unequal sample sizes W.L. Pearn a, H.N. Hung b, Ya Ching Cheng a,* a
Department of Industrial Engineering & Management, National Chiao Tung University, Taiwan b Institute of Statistics, National Chiao Tung University, Taiwan Received 13 September 2007; accepted 31 January 2008 Available online 9 February 2008
Abstract In this paper, we consider the supplier selection problem which deals with comparing two one-sided processes and selecting the one that has a significantly higher capability value. We present an exact approach to tackle the supplier selection problem. Testing hypotheses with two phases for comparing two processes are considered. Critical values of the tests are calculated to determine the selection decisions. Sample size required for a designated selection power and confidence level is also investigated. The results provide useful information to practitioners. The proposed approach generalizes the existing method by not requiring equal sample sizes for the two processes. An application example on comparing two wavelength division multiplexer (WDM) manufacturing processes is presented to illustrate the practicality of our approach to actual data collected from the factories. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Supplier selection; Process capability index; Nonconformities
1. Introduction Process capability indices have been widely used in the manufacturing industry, to measure the process capability of meeting the product quality requirement preset by the product designers or the process engineers. Several capability indices, including Cp, CPU, CPL, Cpk, Cpm, and Cpmk, are developed for this purpose (Kane, 1986; Pearn et al., 1992; Pearn et al., 1998). Those indices essentially compare the predefined manufacturing specifications with the actual production performance of the investigated quality characteristics and are defined as follows: USL LSL USL l l LSL USL l l LSL ; C PU ¼ ; C PL ¼ ; C pk ¼ min ; ; Cp ¼ 6r 3r 3r 3r 3r 9 8 > > = < USL LSL USL l l LSL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; C pm ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; C pmk ¼ min > ; :3 r2 þ ðl T Þ2 3 r2 þ ðl T Þ2 > 6 r2 þ ðl T Þ2 where USL is the upper specification limit, LSL is the lower specification limit, l is the process mean, r is the process standard deviation (overall process variation), and T is the target value. While the indices Cp, Cpk, Cpm, and Cpmk are appropriate for normal processes with two-sided specification limits, the indices CPU and CPL are designed for processes with one-sided specification limit, and only one of USL and LSL is required in this case.
*
Corresponding author. Tel.: +886 3 5712121x57357. E-mail address: [email protected] (Y.C. Cheng).
0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.01.050
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The index Cpk measures actual process performance based on the process yield (fraction of product items conforming to the manufacturing specifications). With a fixed value of Cpk, the bounds of process yield are given by 2U(3Cpk) 1 6 yield 6 U(3Cpk), where U() is the cumulative distribution function of the standard normal distribution N(0, 1) (Boyles, 1991). For instance, if Cpk = 1.00, then it guarantees that the yield will be no less than 99.73%, or equivalently, the fraction of nonconformities is no greater than 2700 parts per million. Those indices have been used to establish the MPPAC (multi-process performance analysis chart) for factory defective control applications (Pearn and Chen, 1997; Pearn and Shu, 2003a,b), process performance analysis for multiple quality characteristics (Bothe, 1997; Wu and Pearn, 2005), better supplier selection (Chou, 1994; Huang and Lee, 1995), capability measures for multiple manufacturing streams (Bothe, 1999), variables acceptance sampling plans for lot sentencing (Pearn and Wu, 2006), and many others. Kotz and Johnson (2002) presented a thorough review on the development of process capability indices in the past 10 years. 2. Indices CPU, CPL and fraction of nonconformities For normally distributed processes with one-sided specification limit USL or LSL, formula of the process yield, Pr(X < USL), or Pr(X > LSL) in terms of CPU or CPL can be established as the following: X l USL l < ¼ PrðZ < 3C PU Þ ¼ Uð3C PU Þ; PrðX < USLÞ ¼ Pr r r l X l LSL < ¼ PrðZ > 3C PL Þ ¼ Uð3C PL Þ; PrðX > LSLÞ ¼ Pr r r where Z is the variable of standard normal N(0, 1). For convenience of presentation, we define CI = CPU or CPL. Therefore, the corresponding nonconformities in parts per million (NCPPM) for a well controlled normally distributed process can be calculated exactly, as NCPPM = 106 [1 U(3CI)]. Table 1 displays the corresponding NCPPM for CPU, CPL = 1.00, 1.25, 1.33, 1.45, 1.50, 1.60, 1.67, and 2.00. In current practice, a process is called inadequate if CI < 1.00; it indicates that the process is not adequate with respect to the production tolerances, either the process variation r2 needs to be reduced or the process mean l needs to be shifted closer to the target value. A process is called capable if 1.00 6 CI < 1.33; it indicates that caution needs to be taken regarding the process distribution and some process control is required. A process is called satisfactory if 1.33 6 CI < 1.50; it indicates that process quality is satisfactory, material substitution may be allowed, and no stringent quality control is required. A process is called good if 1.50 6 CI < 1.67. A process is called excellent if 1.67 6 CI < 2.00; it indicates that process quality exceeds satisfactory. Finally, a process is called super if CI P 2.00 (Pearn and Chen, 2002). Table 2 summarizes the above five conditions and the corresponding CI values.
Table 1 Corresponding NCPPM for some specific values of CI CI
NCPPM
1.00 1.25 1.33 1.45 1.50 1.60 1.67 2.00
1349.898 88.417 33.037 6.807 3.398 0.793 0.272 0.001
Table 2 Some commonly used capability requirement quality conditions Quality condition
CI values
Inadequate Capable Satisfactory Good Excellent Super
CI < 1.00 1.00 6 CI < 1.33 1.33 6 CI < 1.50 1.50 6 CI < 1.67 1.67 6 CI < 2.00 2.00 6 CI
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Table 3 Some minimal capability requirements of CPU and CPL for new and special processes Index value
Production process types
1.25 1.45 1.60
Existing processes New processes, or existing processes on safety, strength, or critical parameters New processes on safety, strength, or critical parameters
Montgomery (2001) recommended some minimum quality requirements on CPU and CPL, as presented in Table 3, for specific process types which must run under some designated capability conditions. It would be desirable to determine a bound so that practitioners could expect to find the true value of the process capability no less than the bound with certain level of confidence. Noting that for newly setup processes on safety, strength, or with critical parameters, the fraction of nonconformities should target no more than 0.8 PPM, a more stringent requirement set for possible 1.5r mean shift (Bothe (2002)). 3. WDM supplier selection Consider a company importing the couplers, splitters and wavelength division multiplexers (WDM). These devices are important passive components to the optical fiber communications, which divide route or combine multiple optical signals. Fiber-optic couplers either split optical signals into multiple paths or combine multiple signals on one path. Like electrical currents, a flow of signal carriers, in this case photons, comprise the optical signal. However, an optical signal does not flow through the receiver to the ground. Rather, at the receiver a detector absorbs the signal flow. Multiple receivers, connected in a series, would receive no signal passing the first receiver, which would absorb the entire signal. Multiple parallel optical output ports must be connected to divide the signal between the ports, which reduces the signal magnitude. The number of input and output ports characterizes a coupler. The most interesting type of fiber coupler is the wavelength division multiplexer (WDM). 3.1. Optical characteristic Polarization dependent loss (PDL) is one of the most important characteristics of those optical passive components. Polarization dependent loss (PDL) is a measure of the peak-to-peak difference during the signal transmission in an optical component or system with respect to all possible states of polarization. Components in fiber-optic telecommunications networks must be insensitive to changes in the polarization state of the signal, a requirement that is particularly challenging for surface-relief diffraction gratings since those with metallic coatings tend to polarize the light incident on them. The polarization of the constant, and fully polarized, input signal is varied. As the polarization of the incident light varies, the output signal shows a corresponding change in power. The measure of polarization dependent loss (PDL) is expressed in units of decibels (dB), with upper specification limit, USL, set to 0.08 dB. The company importing the couplers, splitters and WDM has two suppliers manufacturing the WDM and competing for the company’s orders. The company would like to determine which supplier provides better WDM products. The characteristic PDL is the one of the key measurements for the comparison. Table 4 displays the PDL data collected from Table 4 PDL data of suppliers I and II (in dB) Supplier I PDL data 0.058 0.051 0.065 0.058 0.064 0.060 0.059 0.059 0.058 0.068 0.061 0.056 0.060 0.061 0.069
0.064 0.059 0.058 0.054 0.064 0.061 0.056 0.051 0.059 0.059 0.063 0.054 0.072 0.068 0.062
Supplier II PDL data 0.066 0.060 0.055 0.061 0.060 0.064 0.067 0.061 0.063 0.067 0.064 0.064 0.061 0.055 0.060
0.062 0.051 0.067 0.070 0.058 0.061 0.066 0.058 0.054 0.064 0.067 0.062 0.058 0.058 0.070
0.057 0.060 0.058 0.056 0.057 0.068 0.064 0.060 0.065 0.061 0.067 0.070 0.056 0.049 0.062
0.057 0.064 0.054 0.068 0.051 0.054 0.064 0.054 0.064 0.061 0.064 0.052 0.062 0.055 0.068
0.062 0.057 0.064 0.058 0.062 0.061 0.058 0.061 0.055 0.062 0.062 0.057 0.062 0.071 0.064
0.052 0.049 0.041 0.052 0.052 0.057 0.056 0.049 0.051 0.045 0.051 0.042 0.048 0.052 0.053
0.045 0.060 0.055 0.041 0.045 0.049 0.048 0.048 0.045 0.048 0.055 0.055 0.049 0.055 0.048
0.058 0.042 0.054 0.042 0.053 0.052 0.048 0.054 0.050 0.048 0.038 0.047 0.046 0.055
0.049 0.041 0.053 0.062 0.048 0.059 0.050 0.052 0.049 0.061 0.046 0.052 0.045 0.056
0.054 0.047 0.057 0.052 0.049 0.054 0.057 0.040 0.046 0.060 0.049 0.045 0.047 0.047
0.054 0.046 0.048 0.050 0.057 0.052 0.047 0.051 0.046 0.053 0.044 0.050 0.045 0.054
0.051 0.053 0.055 0.048 0.055 0.052 0.044 0.053 0.050 0.053 0.049 0.050 0.051 0.048
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Supplier I with sample size n1 = 105, and Supplier II with n2 = 100. In current factory applications, most practitioners would take the intuitive and simple approach to just compare the sample estimates of CPU1 and CPU2 for the two suppliers, then make conclusion. Such approach ignores the sampling errors, which is considered highly unreliable. 3.2. Existing solution method Chou (1994) developed an approximation method for selecting better suppliers based on one-sided capability index CPU and CPL. A rule for testing H0: CPU1 P CPU2 versus H1: CPU1 < CPU2 is proposed. The rule, requiring equal sample sizes b PU 1 < C b PU 2 and satisfaction of the condition A < expfv2 ð1 2aÞ=2g, where n1 = n2 from the two suppliers, rejects H0 if C 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n b 2 þ 2 aC b PU 1 C b PU 2 b 2 þ 2 aC ; A ¼ 2= aC PU 1 PU 1 a = 9n/(n 1) and v21 ð1 2aÞ is the (1 2a)th quantile of a v2 distribution with one degree of freedom. The test statistic A is the log-likelihood ratio statistic approximating the exponential of the v2 distribution with one degree of freedom under the large-sample theory. The approximation method requires equal sample sizes, n1 = n2 = n, in the applications. Therefore, it cannot be applied to the above WDM example. In the following, we develop an exact method for selecting better suppliers, which is also applicable for cases with n1 – n2. 4. Development of the exact method To compare the two suppliers, we consider the hypothesis testing for comparing the two CPU values, H0: CPU1 P CPU2 versus H1: CPU1 < CPU2 (or equally, H0: CPU2/CPU1 6 1 versus H1: CPU2/CPU1 > 1). Under the assumption that the quality characteristic of the two processes, X1, and X2 are independent normally distributed, X 1 N ðl1 ; r21 Þ and X 2 N ðl2 ; r22 Þ, we have X 1 N ðl1 ; r21 =n1 Þ, X 2 N ðl2 ; r22 =n2 Þ. Therefore, b PU 1 C
USL X 1 USL X 1 ¼ ¼ = 3S 1 3r1
sffiffiffiffiffi S 21 ; r21
b PU 2 C
USL X 2 USL X 2 ¼ ¼ = 3S 2 3r2
sffiffiffiffiffi S 22 ; r22
and the testing statistic R, ratio of the two estimators, can be expressed as follows:
1 2 N USLl ; 9n12 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N C PU 2 ; 9n2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3r2 b PU 2 = C b PU 1
F n1 1;n2 1 ¼
F n1 1;n2 1 : R¼C 1 ; 9n11 N USLl N C PU 1 ; 9n11 3r1 Thus, the distribution of the testing statistic R, is the convolution of the ratio of two normal distributions, and the square root of the F distribution, where the probability density
functions of the two distributions can be obtained below. 1 1 Probability density function of U N ðC PU 2 ; 9n2 Þ=N C PU 1 ; 9n1
Let X 1 N C PU 1 ; 9n11 , X 2 N C PU 2 ; 9n12 and U = X2/X1. Applying the well-known convolution formula fU ðuÞ ¼
Z
1
fX 1 ðx1 ÞfX 2 ðux1 Þjx1 j dx1 1
we can obtain the probability density function (PDF) of U as follows: " # pffiffiffiffiffiffiffiffiffi 2 9 n1 n2 9 9 ðn1 C PU 1 þ n2 C PU 2 uÞ 2 2 exp ðn1 C PU 1 þ n2 C PU 2 Þ exp fU ðuÞ ¼ 2p n1 þ n2 u2 2 2 ( " # 2 2 9 ðn1 C PU 1 þ n2 C PU 2 uÞ n1 C PU 1 þ n2 C PU 2 u : exp þ n1 þ n2 u2 9ðn1 þ n2 u2 Þ 2 n1 þ n2 u2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 2p 3ðn1 C PU 1 þ n2 C PU 2 uÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 1 2U 9ðn1 þ n2 u2 Þ n1 þ n2 u2 where 1 < u < 1.
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Probability density function of V F n1 1;n2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Assume that V F n1 1;n2 1 . By using the technique of change-of-variable, the probability density function (PDF) of V can be derived as follows:
n1 1 fV ðvÞ ¼ 2 n2 1
n121
2 C n1 þn 1 2 C n121 C n221
ðvÞn1 2 ;
n1 þn 2 2 1 n1 1 2 1 þ n2 1 v
where 0 < v < 1. b PU 1 b PU 2 = C Probability density function of C b PU 2 = C b PU 1 then can be expressed as R = U V, and by applying again the convolution The testing statistic R ¼ C formula Z 1
1
r r fU ðuÞfV fR ðrÞ ¼ I > 0
du; u u u 1
Fig. 1. PDF plots of R for sample sizes n1 = n2 = 30, 50, 100, 150, 200 (from bottom to top in plots).
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the probability density function of R then is as follows: " # r n1 2 Z 1 2
1 r 9 ðn C þ n C uÞ 1 PU 1 2 PU 2 u
I > 0 exp fR ðrÞ ¼ A
u 2 n1 þn n1 þ n2 u2 2 2 1 1 u n1 1 r 2 1 þ n2 1 u " # ( 2 2 9 ðn1 C PU 1 þ n2 C PU 2 uÞ n1 C PU 1 þ n2 C PU 2 u : exp þ n1 þ n2 u2 9ðn1 þ n2 u2 Þ 2 n1 þ n2 u2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) 2p 3ðn1 C PU 1 þ n2 C PU 2 uÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi du; 1 2U 9ðn1 þ n2 u2 Þ n1 þ n2 u2 where
n1 1 A¼2 n2 1
n121
pffiffiffiffiffiffiffiffiffi 2 C n1 þn 1 9 n1 n2 9 2 2 2 exp ðn1 C PU 1 þ n2 C PU 2 Þ ; 2p 2 C n121 C n221
U() is the cumulative density function of the standard normal distribution, I is the indicator function, and 1 < r < 1. Fig. 1 plots the probability density function of R for CPU1 = 1.0, 1.5, 2.0, CPU2 = 1.0, 1.5, 2.0, and n1 = n2 = 30, 50, 100, 150, 200 (from bottom to top in plots). The PDF plots of R reveal the following features: b PU 2 = C b PU 1 . 1. The larger the value of CPU2/CPU1, the larger the variance of R ¼ C 2. Distribution of R is single peaked and is almost symmetric to CPU2/CPU1. 3. The larger the sample sizes n1 and n2, the smaller the variance of R, which is certain for all sample estimators. 5. Supplier selection procedure 5.1. Phase I: Selecting a better process capability Assume that the minimum requirement of CPU values for all candidate processes is C, and the existing supplier, Supplier I, has achieved the requirement, i.e. CPU1 P C. If a new supplier, 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. To test whether the new supplier, Supplier II, has a significantly better capability than the existing Supplier I, we consider the hypothesis testing: H0 : C PU 2 6 C PU 1 ; H1 : C PU 2 > C PU 1 ; which is equivalent to H0 : C PU 2 =C PU 1 6 1; H1 : C PU 2 =C PU 1 > 1: b PU 2 = C b PU 1 and a given significance level a, the chance of incorrectly judging Based on the sampling distribution of R ¼ C b PU 2 = C b PU 1 P c0 , where c0 is CPU2 6 CPU1 as CPU2 > CPU1, the decision rule is to reject H0 if the testing statistic R ¼ C the critical value that satisfies PrfR P c0 jH0 : C PU 2 6 C PU 1 ; n1 ; n2 ; and C PU 1 P Cg 6 a: Since the larger the CPU2/CPU1 value, the larger the critical value, we calculate the critical value c0 with the conditions CPU1 = CPU2 = C, i.e. PrfR P c0 jC PU 1 ¼ C PU 2 ¼ C; n1 ; n2 g ¼ a: Table 5 shows the critical values for CPU1 = CPU2 = 1.0(0.1)2.0, n1 = n2 = n = 30(10)200 and a = 0.05. Note that our supplier selection procedure can be applied for cases with unequal sample sizes, n1 – n2. It is also noted that replacing CPU by CPL, the above procedure can also be used to test the hypothesis: H0 : C PL2 6 C PL1 ; H1 : C PL2 > C PL1 :
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Table 5 Critical values for rejecting CPU2 6 CPU1 with n1 = n2 = 30(10)200 and a = 0.05 n
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
CPU1 = CPU2 = C 1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
1.407 1.341 1.299 1.269 1.246 1.228 1.213 1.201 1.191 1.182 1.174 1.167 1.161 1.155 1.150 1.146 1.142 1.138
1.400 1.335 1.294 1.264 1.242 1.224 1.210 1.198 1.188 1.179 1.171 1.164 1.158 1.153 1.148 1.143 1.139 1.136
1.394 1.331 1.290 1.260 1.238 1.221 1.207 1.195 1.185 1.176 1.169 1.162 1.156 1.151 1.146 1.142 1.137 1.134
1.390 1.327 1.286 1.258 1.236 1.219 1.205 1.193 1.183 1.175 1.167 1.160 1.155 1.149 1.144 1.140 1.136 1.132
1.386 1.324 1.284 1.255 1.234 1.217 1.203 1.191 1.182 1.173 1.166 1.159 1.153 1.148 1.143 1.139 1.135 1.131
1.383 1.322 1.282 1.254 1.232 1.215 1.202 1.190 1.180 1.172 1.164 1.158 1.152 1.147 1.142 1.138 1.134 1.130
1.381 1.320 1.280 1.252 1.231 1.214 1.200 1.189 1.179 1.171 1.164 1.157 1.151 1.146 1.141 1.137 1.133 1.130
1.379 1.318 1.279 1.251 1.230 1.213 1.199 1.188 1.178 1.170 1.163 1.156 1.151 1.145 1.141 1.136 1.133 1.129
1.378 1.317 1.278 1.250 1.229 1.212 1.199 1.187 1.178 1.169 1.162 1.156 1.150 1.145 1.140 1.136 1.132 1.128
1.376 1.316 1.277 1.249 1.228 1.211 1.198 1.187 1.177 1.169 1.161 1.155 1.149 1.144 1.140 1.135 1.132 1.128
1.374 1.314 1.275 1.247 1.227 1.210 1.197 1.186 1.176 1.168 1.161 1.154 1.149 1.143 1.139 1.135 1.131 1.127
5.2. 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 CPU values without investigating further the magnitude of the difference between the two suppliers. In practice, the customer would consider choosing the new supplier only if the new supplier’s capability is significantly better than the existing supplier’s by a magnitude of h > 0, due to the high cost of the supplier replacement. Our exact approach, in this case, can be used to test the corresponding hypothesis: H0 : C PU 2 6 C PU 1 þ h; H1 : C PU 2 > C PU 1 þ h: b PU 2 = C b PU 1 is larger than The decision rule is similar to Phase I. We would reject H0, accept that CPU2 > CPU1 + h, if R ¼ C or equal to the critical value c0, where c0 satisfies PrfR P c0 jH0 : C PU 2 6 C PU 1 þ h; n1 ; n2 ; and C PU 1 P Cg 6 a: For all combinations of (CPU1, CPU2) under H0, the maximal critical value occurs at CPU1 = C and CPU2 = C + h, and the larger the a, the smaller the critical value. Thus, we calculate the critical value c0 with the probability PrfR P c0 jC PU 1 ¼ C; C PU 2 ¼ C þ h; n1 ; n2 g ¼ a: If the test rejects the null hypothesis H0, then there is sufficient information to conclude that supplier II is significantly better than supplier I by a magnitude of h, and the replacement of Supplier I by Supplier II would be suggested. Table 6 shows some critical values for some minimum quality requirements C of CPU suggested by Montgomery (2001), the magnitude h = 0.1(0.1)0.5 of the difference between the two suppliers and n1 = n2 = n = 30(10)200 with a = 0.05. 5.3. Required sample size In Phases I and II, the supplier selection procedure solely depends on the a risk, incorrectly judging H0 as H1, and not takes into account the b risk, incorrectly judging H1 as H0, which is comparatively unfavorable to H1 and the new supplier. This is due to the additional cost of replacing the existing supplier with the new one. Once the sample sizes and the a risk are defined, the power of test, also known as 1 b, will be fixed. Fig. 2 plots some particular power values versus CPU2 for CPU1 = 1.0, 1.5, 2.0, n1 = n2 = n = 30, 50, 100, 150, 200, and a = 0.05. It can be seen that the larger the sample size, the larger the power of test, consequently, and the smaller the b risk. To decrease the b risk and at the same time maintain the a risk in a small level, we could increase the sample sizes. By calculating the power for a specific value of CPU2, we could obtain the minimal sample size required for designated power and a risk. The required sample size could be calculated by recursive search with the following two constraints:
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Table 6 Critical values for rejecting CPU2 6 CPU1 + h with a = 0.05 n
(CPU1, CPU2) (1.25, 1.35)
(1.25, 1.45)
(1.25, 1.55)
(1.25, 1.65)
(1.25, 1.75)
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
1.501 1.433 1.390 1.358 1.335 1.316 1.301 1.289 1.278 1.269 1.261 1.253 1.247 1.241 1.236 1.231 1.227 1.223
1.611 1.538 1.491 1.458 1.433 1.413 1.397 1.383 1.372 1.362 1.353 1.346 1.339 1.333 1.327 1.322 1.317 1.313
1.721 1.643 1.593 1.558 1.531 1.510 1.492 1.478 1.466 1.455 1.446 1.438 1.430 1.424 1.418 1.413 1.408 1.403
1.831 1.748 1.695 1.657 1.629 1.606 1.588 1.573 1.560 1.549 1.539 1.530 1.522 1.515 1.509 1.503 1.498 1.493
1.941 1.853 1.797 1.757 1.727 1.703 1.684 1.668 1.654 1.642 1.631 1.622 1.614 1.607 1.600 1.594 1.588 1.583
n
(CPU1, CPU2) (1.45, 1.55)
(1.45, 1.65)
(1.45, 1.75)
(1.45, 1.85)
(1.45, 1.95)
1.479 1.413 1.370 1.340 1.317 1.299 1.284 1.272 1.262 1.253 1.245 1.238 1.232 1.226 1.221 1.216 1.212 1.208
1.574 1.503 1.458 1.426 1.402 1.382 1.367 1.354 1.343 1.333 1.325 1.317 1.311 1.305 1.299 1.294 1.290 1.286
1.668 1.594 1.546 1.512 1.486 1.466 1.449 1.435 1.424 1.413 1.405 1.397 1.390 1.383 1.378 1.373 1.368 1.363
1.762 1.684 1.633 1.598 1.570 1.549 1.531 1.517 1.505 1.494 1.484 1.476 1.469 1.462 1.456 1.451 1.446 1.441
1.857 1.774 1.721 1.683 1.655 1.632 1.614 1.599 1.585 1.574 1.564 1.556 1.548 1.541 1.535 1.529 1.524 1.519
(1.60, 1.70)
(1.60, 1.80)
(1.60, 1.90)
(1.60, 2.00)
(1.60, 2.10)
1.467 1.402 1.360 1.330 1.307 1.289 1.275 1.263 1.252 1.244 1.236 1.229 1.223 1.217 1.212 1.208 1.204 1.200
1.552 1.483 1.439 1.407 1.384 1.365 1.349 1.337 1.326 1.316 1.308 1.301 1.294 1.289 1.283 1.279 1.274 1.270
1.638 1.565 1.518 1.485 1.460 1.440 1.424 1.411 1.399 1.389 1.381 1.373 1.366 1.360 1.354 1.349 1.345 1.340
1.723 1.647 1.598 1.563 1.536 1.516 1.499 1.484 1.472 1.462 1.453 1.445 1.438 1.431 1.425 1.420 1.415 1.411
1.809 1.729 1.677 1.641 1.613 1.591 1.573 1.558 1.546 1.535 1.525 1.517 1.509 1.503 1.496 1.491 1.486 1.481
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 n
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
(CPU1, CPU2)
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Fig. 2. Power curve for CPU1 = 1.0, 1.5, and 2.0.
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PrfR P c0 jH0 : C PU 2 6 C PU 1 ; n1 ; n2 ; and C PU 1 P Cg 6 a; and; PrfR P c0 jH1 : C PU 2 > C PU 1 ; n1 ; n2 ; and C PU 1 P Cg P 1 b: For users’ convenience in applying the supplier selection procedure, we tabulate the sample sizes required in Table 7 for various designated selection power with the minimal capability requirement CPU = 1.00, 1.25, 1.45, 1.60, the magnitude of difference CPU2 CPU1 = 0.15(0.05)1.00, and designated power=0.90, 0.95, 0.975, 0.99. Note that if the process of Supplier I is investigated first, that is n1 is fixed, we can increase n2 to satisfy the requirement of both risks. In practice, both suppliers would be asked to sample with equal sample size. In Table 7, we show the sample size required for circumstances with n1 = n2. For example, if the minimal capability requirement CPU is 1.25, the designated a and b risks are 0.05 (i.e. designated power of test = 0.95), and the expected magnitude of difference CPU2 CPU1 is 0.3, then the required sample size for both suppliers to sample is 267. 5.4. Comparison with Chou’s approximation We compared our exact approach with Chou’s approximation using simulations. Table 8 displays the simulation results for actual Type I error a each with 107 simulated samples for CPU1 = CPU2 = 1.00, 1.33, 1.50, 1.67, 2.00, n = 30(10)100, and a = 0.05. The results indicate that our exact approach is indeed more accurate than Chou’s approximation. While Chou’s approximation method requires equal sample sizes, n1 = n2, the proposed exact approach is applicable for unequal sample sizes n1 – n2, which is more accurate and more general.
Table 7 Sample size required for testing H0: CPU2 6 CPU1 versus H1: CPU2 > CPU1 CPU1
CPU2
Power
CPU1
0.90
0.95
0.975
0.99
CPU2
Power 0.90
0.95
0.975
0.99
1.00
1.15 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00
535 314 210 152 116 93 76 64 55 48 43 38 35 31 29 27 25 23
675 396 264 191 146 116 96 81 69 60 53 48 43 39 36 33 31 29
810 474 316 229 175 139 114 96 82 72 63 57 51 47 43 39 36 34
980 574 383 277 211 168 138 116 99 87 77 68 62 56 51 47 44 41
1.25
1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25
763 445 295 212 161 128 104 87 75 65 57 51 46 41 38 35 32 30
965 561 372 267 203 161 131 110 94 81 71 64 57 52 47 43 40 37
1157 673 447 320 243 192 157 131 112 97 85 76 68 62 56 52 48 44
1403 816 540 388 295 233 190 159 135 117 103 92 82 74 68 62 57 53
1.45
1.60 1.65 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45
983 570 376 269 204 161 131 109 93 80 71 63 56 51 46 43 40 37
1241 719 474 339 257 203 165 137 117 101 88 78 70 63 58 53 49 45
1488 862 569 407 308 243 197 164 140 121 106 94 84 76 69 63 58 54
1805 1046 690 493 373 294 239 199 169 146 128 113 101 91 83 76 70 65
1.60
1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 2.15 2.20 2.25 2.30 2.35 2.40 2.45 2.50 2.55 2.60
1167 674 444 317 239 189 153 127 108 93 82 72 65 58 53 48 45 41
1475 852 560 400 302 238 193 160 136 117 103 91 81 73 66 61 56 52
1768 1021 672 479 362 285 231 192 163 140 123 109 97 87 79 72 67 62
2145 1238 815 581 438 345 280 232 197 170 148 131 117 106 96 87 80 74
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Table 8 The actual Type I error a of Chou’s approximation and our exact approach CPU
1.00
1.33
1.50
1.67
2.00
n
Chou’s
Exact
Chou’s
Exact
Chou’s
Exact
Chou’s
Exact
Chou’s
Exact
30 40 50 60 70 80 90 100
0.0544 0.0531 0.0526 0.0521 0.0520 0.0515 0.0516 0.0516
0.0500 0.0500 0.0498 0.0499 0.0501 0.0499 0.0504 0.0504
0.0543 0.0532 0.0525 0.0520 0.0520 0.0518 0.0517 0.0516
0.0498 0.0499 0.0502 0.0497 0.0500 0.0500 0.0500 0.0503
0.0543 0.0532 0.0526 0.0521 0.0519 0.0518 0.0518 0.0516
0.0500 0.0497 0.0499 0.0497 0.0502 0.0504 0.0499 0.0504
0.0544 0.0531 0.0526 0.0521 0.0521 0.0517 0.0519 0.0516
0.0500 0.0499 0.0499 0.0499 0.0499 0.0501 0.0507 0.0504
0.0543 0.0532 0.0526 0.0522 0.0521 0.0518 0.0519 0.0516
0.0503 0.0502 0.0504 0.0507 0.0503 0.0507 0.0505 0.0503
6. Supplier selection implementations 6.1. WDM data analysis The company importing WDM products has a minimal requirement on the polarization dependent loss (PDL). The minimal requirement of the PDL characteristic is CPU = 1.25. Kolmogorov–Smirnov normality tests for data of both suppliers are performed first to confirm if the data of both suppliers are normally distributed. Both tests result in p-value >0.15, which means that data of both suppliers are normally distributed. Histograms of the data are shown in Fig. 3. 6.2. Phase I: Selecting a better process capability To determine if Supplier II provides a better process capability of the WDM products than Supplier I, we perform the hypothesis testing: H0: CPU2 6 CPU1 versus H1: CPU2 > CPU1. For the PDL data displayed in Tables 4 and 5, we calculate the sample means, sample standard deviations and the sample estimators for both suppliers, and obtain that x1 ¼ 0:06079, b PU 1 ¼ 1:2936, C b PU 2 ¼ 2:04527 and thus R = 1.58107. To compute the critical x2 ¼ 0:05018, s1 = 0.00495, s2 = 0.00486, C value, we develop a ‘Visual C++’ computer program to handle the complicated computations. (The program can be obtained by request.) The program reads the input parameters CPU1, CPU2, the corresponding sample sizes n1, n2, and a, the risk for wrongly rejecting CPU2 6 CPU1 while actually CPU2 > CPU1 is true, and outputs with the critical value. We run the C++ program with n1 = 105, n2 = 100, CPU1 = CPU2 = 1.25 (the minimal capability requirement of CPU) and a = 0.05 to obtain the critical value as 1.1924. Since the testing statistic R = 1.58107 > 1.1924, we therefore conclude that the Supplier II is superior than Supplier I with 95% confidence level. 6.3. Phase II: Magnitude outperformed measurement To investigate further the magnitude of the capability difference between the two suppliers, we perform the hypothesis testing: H0: CPU2 6 CPU1 + h versus H1: CPU2 > CPU1 + h. We run the C++ program with n1 = 105, n2 = 100, CPU1 = 1.25
Fig. 3. Histograms of the two PDL data.
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Table 9 Critical values and decisions of testing the two WDM suppliers CPU1 CPU2 h c0 Decision
1.25 1.45 0.20 1.3813 Reject H0
1.25 1.55 0.30 1.4758 Reject H0
1.25 1.65 0.40 1.5704 Reject H0
1.25 1.66 0.41 1.5799 Reject H0
1.25 1.67 0.42 1.5893 Do not Reject H0
(the minimal capability requirement of CPU), CPU2 = 1.25 + h, h = 0.2(0.1)0.4, 0.41, 0.42 and a = 0.05 to obtain the critical values for various h. The decisions of the hypotheses are shown in Table 9. Based on the testing results, we can conclude that the Supplier II has a manufacturing capability that is significantly better than the Supplier I by a magnitude of 0.41, i.e. CPU2 > CPU1 + 0.41. 6.4. Sample size required for a designated power Assume that the minimal requirement of CPU values for all candidate processes is 1.25, and the customer would replace the existing supplier with a new one if the capability of the new supplier is significantly better than the existing supplier by a magnitude of 0.3. Then, to make a hypothesis testing with a designated selection power of 0.90, the sample size required to sample is 212 as in Table 7. In the WDM supplier selection problem, since the sample sizes of two suppliers are smaller than 212, the selection power for testing H0: CPU2 6 CPU1 versus H1: CPU2 > CPU1 would be less than 0.90. In fact, the power of test for CPU2 = 1.66 is 0.8489, that is the b risk of incorrectly accepting CPU2 6 CPU1 while actually CPU2 > CPU1 is true is up to 0.1511. In order to reduce the b risk, we would suggest the customer to sample for a designated power with as large sample size as in Table 7. 7. Conclusions For stable normal processes with one-sided specification limits, capability indices CPU and CPL have been widely used in the manufacturing industry to provide numerical measures on process capability. In this paper, we considered the supplier selection problem for two one-sided processes, and presented an exact analytical approach to solve the problem. The proposed approach which generalizes the existing method is applicable to test the hypothesis H0: CPU2 6 CPU1 versus H1: CPU2 > CPU1 with unequal sample sizes, and the decisions made are more reliable. In practice, often the replacement of suppliers would be made only if the new supplier’s capability is significantly better than the existing supplier’s by a magnitude of h > 0. The proposed exact approach, in this case, can be used to test the corresponding hypothesis H0: CPU2 6 CPU1 + h versus H1: CPU2 > CPU1 + h. Sample size required for a specific value of CPU2 is also investigated under a designated selection power and confidence level. The results are useful to practitioners or engineers. A real-world application comparing two WDM manufacturing processes is presented to illustrate the practicality of the exact approach to actual data collected from the factories. Acknowledgements The authors would like to thank the Editor and the anonymous referees for their valuable comments and suggestions, which significantly improved the presentation of this paper. This paper was partially supported by National Science Council of Taiwan Grant NSC-95-2118-M-009-004-MY2. References Bothe, D.R., 1997. A capability study for an entire product. In: ASQC Quality Congress Transactions, Nashville, vol. 46, pp. 172–178. Bothe, D.R., 1999. A capability index for multiple process streams. Quality Engineering 11 (4), 613–618. Bothe, D.R., 2002. Statistical reason for the 1.5r shift. Quality Engineering 14 (3), 479–487. Boyles, R.A., 1991. The Taguchi capability index. Journal of Quality Technology 23 (1), 17–26. Chou, Y.M., 1994. Selecting a better supplier by testing process capability indices. Quality Engineering 6 (3), 427–438. Huang, D.Y., Lee, R.F., 1995. Selecting the largest capability index from several quality control processes. Journal of Statistical Planning and Inference 46 (3), 335–346. Kane, V.E., 1986. Process capability indices. Journal of Quality Technology 18 (1), 41–52. Kotz, S., Johnson, N.L., 2002. Process capability indices – A review, 1992–2000. Journal of Quality Technology 34 (1), 1–19. Montgomery, D.C., 2001. Introduction to Statistical Quality Control, 4th ed. John Wiley & Sons, Inc., New York, NY. Pearn, W.L., Chen, K.S., 1997. Multi-process performance analysis: A case study. Quality Engineering 10 (1), 1–8. Pearn, W.L., Chen, K.S., 2002. One-sided capability indices CPU and CPL: Decision making with sample information. International Journal of Quality & Reliability Management 19 (3), 221–245.
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393
Pearn, W.L., Shu, M.H., 2003a. Manufacturing capability control for multiple power distribution switch processes based on modified Cpk MPPAC. Microelectronics Reliability 43 (6), 963–975. Pearn, W.L., Shu, M.H., 2003b. Lower confidence bounds with sample size information for Cpm with application to production yield assurance. International Journal of Production Research 41 (15), 3581–3599. Pearn, W.L., Wu, C.W., 2006. Variables sampling plans with PPM fraction of defectives and process loss consideration. Journal of Operational Research Society 57 (4), 450–459. Pearn, W.L., Kotz, S., Johnson, N.L., 1992. Distributional and inferential properties of process capability indices. Journal of Quality Technology 24 (4), 216–231. Pearn, W.L., Lin, G.H., Chen, K.S., 1998. Distributional and inferential properties of the process accuracy and process precision indices. Communications in Statistics: Theory & Methods 27 (4), 985–1000. Wu, C.W., Pearn, W.L., 2005. Measuring manufacturing capability for couplers and wavelength division multiplexers. International Journal of Advanced Manufacturing Technology 25 (5–6), 533–541.