Sample size required for convergence

Table 3-5 displays the sample sizes required for the convolution approximation to converge to Spk within a designated accuracy ε= 0.12(0.01)0.03.

{

}

Pr S_pk′′ −S_pk ≤ε ≥ − 1 α

For example, for Spk = 1.33 with risk α= 0.025, a sample size of n ≥ 3831 ensures that the difference between sample estimate and actual parameter would be no greater than 0.03 with 97.5% confidence. Thus, if ˆS = 1.33, then we may conclude that the actual S_pk pk is greater than 1.3, actually in the interval of (1.30, 1.36), with 97.5% confidence. Note that the investigation is not for practical purpose. But, the computations illustrate the rate of convergence for the convolution approximation to converge to actual Spk.

Table 3-5. Sample size required for the convolution approximation to converge.

Designated Accuracy, ε Spk α

0.12 0.11 0.10 0.09 0.08 0.07 0.06 0.05 0.04 0.03 1.00 0.1 88 106 129 159 203 266 363 523 819 1459 0.05 127 151 184 228 289 378 518 744 1164 2072 0.025 167 200 242 299 380 496 676 975 1524 2288 1.33 0.1 159 189 230 284 361 472 644 929 1210 1789 0.05 226 270 327 405 513 672 875 1319 1543 2706 0.025 298 355 430 531 673 880 1199 1727 2183 3831 1.50 0.1 203 242 294 364 461 603 822 1184 1852 3296 0.05 289 345 418 517 655 840 1168 1683 2631 4680 0.025 380 453 549 678 859 1122 1529 2203 3443 5225 1.67 0.1 253 302 366 453 574 750 1022 1473 2303 4097 0.05 361 430 521 644 811 1066 1452 2092 3271 5818 0.025 473 564 683 767 1068 1396 1901 2738 4280 7455 2.00 0.1 366 437 529 654 828 1082 1474 2124 3321 5444 0.05 521 621 752 929 1177 1538 2094 3017 4716 8387 0.025 683 731 985 1217 1540 2013 2741 3946 6170 10970

Chapter 4

Product Acceptance Determination Based on the S

Index

4.1. Introduction

Acceptance sampling plans are practical tools for quality assurance applications, which provide the supplier and the customer a general decision rule for lot sentencing that meets both of their requirements for product quality. Acceptance sampling plans, however, cannot avoid the risk of accepting undesired poor product lots, nor can it avoid the risk of rejecting good product lots unless 100% inspection is implemented. A well-designed sampling plan can effectively reduce the difference between the required and the actual supplied product quantity. An acceptance sampling plan is a statement regarding the required sample size for product inspection and the associated acceptance criteria for sentencing each individual product lot. The criteria used for measuring the performance of an acceptance sampling plan is based the operating characteristic (OC) curve which quantifies the risks for vendors and buyers. The OC curve plots the probability of accepting an individual lot versus the actual product fraction of defectives, which displays the discriminatory power of the sampling plan.

For product quality protection and company’s profit, both the vendor and the buyer would focus on certain points on the OC curve to reflect their benchmark risk. The vendor (supplier) would usually focus on a specific product quality level, called acceptable quality level (AQL), which would yield a high probability, 1− , of accepting a lot. On the other hand, the buyer (consumer) α would focus on a point at the other end of the OC curve, called lot tolerance percent defective (LTPD), which would result in a low probability, β, of accepting a lot. The LTPD is a quality level specified by the consumer, setting a specified low probability of accepting a lot for product with defective level as high as LTPD, and is the poorest quality level that the consumer is willing to accept. The α probability, also called the producer’s risk, occurs when an acceptable product lot is rejected. The β probability, also called the consumer’s risk, occurs when the product with unacceptable quality is accepted. Thus, a well-designed sampling plan must provide a probability

of at least 1− of accepting a lot if the product quality level is at the contracted AQL level and a α probability of no more than β if the level of the product quality is at the LTPD level, the designated undesired quality level preset by the customer. That is, the OC curve of the acceptance sampling plan must pass through the two designated points (AQL, 1− ) and (LTPD, α β ).

There are a number of different ways of classifying the acceptance sampling plans. One major classification is by attributes and variables. The primary advantage of variables sampling plans is that the same operating characteristic curve can be obtained with a smaller sample size than that required by an attributes sampling plan. The precise measurements required by a variables plan would probably cost more than the simple classification of items required by an attributes plan, but the reduction in sample size may more than offset this exact expense. Such saving may be especially marked if inspection is destructive and the item is expensive (see Schilling [41], Duncan [9] and Montgomery [31]). The basic concepts and models of statistically based on variables sampling plans were introduced by Jennett and Welch[19]. Lieberman and Resnikoff [28] developed extensive tables and OC curves for various AQLs for MIL-STD-414 sampling plan. Owen [32] considered variables sampling plans based on the normal distribution, and developed sampling plans for various levels of probabilities of Type I error when the standard deviation is unknown. Das and Mitra [8] have investigated the effect of non-normality on the performance of the sampling plans. Guenther [11] developed a systematic search procedure, which can be used with published tables of binomial, hyper-geometric, and Poisson distributions to obtain the desired acceptance sampling plans. Stephens [43] provided a closed form solution for single sample acceptance sampling plans using a normal approximation to the binomial distribution. Hailey [12] presented a computer program to obtain single sampling plans with minimum sample size based on either the Poisson or binomial distribution. Hald [13] gave a systematic exposition of the existing statistical theory of lot-by-lot sampling inspection by attributes and provided some tables for the sampling plans. Comparisons between variables sampling plans and attributes sampling plans were investigated by Hamaker [14], who concluded that the expected sample size required by variable sampling is smaller than those for comparable attributes sampling plans. Govindaraju and Soundararajan [10] developed variables sampling

plans that match the OC curves of MIL-STD-105D. Suresh and Ramanathan [46] developed a sampling plan based on a more general symmetric family of distributions.

Due to the sampling cannot guarantee that every defective item in a lot will be inspected, the sampling plan involves risks of not adequately reflecting the quality conditions of the lot. Such risk is even more significant as the rapid advancement of the manufacturing technology and stringent customers demand is enforced. Particularly, when the required product fraction of defectives is very low, often measured in parts per million (PPM). The required number of inspection items must be enormously large in order to adequately reflecting the actual product fraction of defectives or process yield. The yield index Spk establishes the relationship between the manufacturing specifications and the actual process performance, and provides an exact yield measure on the normal processes. In this paper, we consider a variables sampling plan based on the Spk index as a quality benchmark to deal with lot sentencing problem for processes with very low fraction of defectives.

4.2. Process Capability Indices

4.2.1. Process capability indices Ca, Cp, Cpk, Cpm, Cpmk

Process capability indices have been proposed to the manufacturing industry, to provide numerical measures on process performance. Those indices establish the relationship between the actual process performance and the manufacturing specifications, which have been the focus of the recent research in statistical and quality assurance literatures. The explicit forms of the indices are defined as follows:

where USL and LSL are the upper and lower specification limits, respectively, m = (USL + LSL)/2 is the midpoint of the specification limits, d = (USL − LSL)/2 is the half length of the specification interval, T is the target value, μ is the process mean, and σ is the process

standard deviation.

The index Ca measures only the process centering (process accuracy), and ignores the process variation (process precision). The index Cp measures the overall process variation relative to the specification tolerance, therefore can not reflect the tendency of process centering (see Juran [20], Sullivan [44], [45] and Kane [21]). In order to reflect the deviations of process mean from the target value, several indices similar in nature to Cp, such as Cpk, Cpm, Cpmk, have been proposed.

Those indices take into consideration the magnitude of process variance as well as process location (see Hsiang and Taguchi [16], Chan et al [4], Boyles [2], Ruczinski [40] and Pearn et al [35]). The index Cpk have a relationship to the actual process yield, which can be expressed as

2 (3Φ C_pk) 1− ≤Yield≤ Φ(3C_pk),

(Boyles [2]). To emphasize the loss in a product’s worth when one of its characteristics departs from the target value T, the two indices Cpm and Cpmk are defined by being related to the idea of squared error loss loss X( ) (= X T− )².

4.2.2. The yield index Spk

For a long time, process yield has been a standard criterion used in the manufacturing industry as a common measure on process performance, and defined as the percentage of processed product unit that falls within the manufacturing specification limits. For product units falling out of the manufacturing tolerance, additional cost would be incurred to the factory for scrapping or repairing the product. All passed product units, which incur no additional cost to the factory, are equally accepted by the producer. The above indices Cpk, Cpm, Cpmk can provide only a lower bound estimation on the process yield Yield≥ Φ ×2 (3 index value) 1− . Note that the highest value that a process yield might be would not be concerned. For example, if the index value is C, then the yield of the process would be equal to or greater than 2 (3 ) 1Φ C − .

On the other hand, the yield index Spk, proposed by Boyles [3], can provide an exact measure on the process yield, which can be expressed as

= 2 (3 _pk) 1 Yield Φ S − .

The Spk index is defined as

1 -1 1 1

The Spk index outperforms other indices in providing a one-to one relationship to the process yield.

We remark that the indices presented above are designed to monitor the performance for stable normal or near-normal processes with symmetric tolerances. In practice, the process mean μ and the process variance σ² are unknown. To calculate the index value, sample data must be collected, and a great degree of uncertainty may be introduced into the assessments due to sampling errors. As the use of the capability indices grows more widespread, users are becoming educated and sensitive to the impact of the estimators and their distributions, learning that capability measures must be reported in confidence intervals or via capability testing. Statistical properties of the estimators of those indices under various process conditions have been investigated extensively, including Chan et al. [4], Pearn et al. [35], Kotz and Johnson [23], Vännman and Kotz [49], Ruczinski [40], Vännman [47], Kotz and Lovelace [25], Borges and Ho [1], Hoffman [15], Zimmer et al. [55], Kotz and Johnson [24], Lee et al. [26], Spiring et al. [42], Pearn et al. [36], Montgomery [31], Hubele et al. [17], Lin and Sheen [29], Wang [50], Wu [52], Mathew et al. [30].

4.2.3. Sampling distribution of the estimated Spk

To estimate the yield measurement index Spk, we consider the following natural estimator ˆpk

The exact distribution of ˆS is analytically intractable. However, a useful approximate _pk

distribution of ˆS can be furnished by considering an expansion of ˆ_pk S . Lee et al [26] _pk considered a normal approximation to the distribution of ˆS by using an expansion technique. _pk The expansion of ˆS , which is denoted ˆ_pk S′ in this paper, is normally distributed, and can be _pk expressed as follows

2 2

Thus, the approximate probability density function of the approximate distribution can be expressed as:

4.3. Designing Spk Variables Sampling Plan

Consider a variables sampling plan for controlling the lot fraction of defectives (nonconformities). Since the quality characteristic is a variables, there will exist either a USL or an LSL, or both, that defined the acceptable values of this parameter. A well-designed sampling plan must provide a probability of at least 1− of accepting a lot if the lot fraction of defectives α is at the contracted AQL. The sampling plan must also provide a probability of acceptance no more than β if the lot fraction of defectives is at the LTPD level, the designated undesired level preset by the buyer. Thus, the acceptance sampling plan must have its OC curve passing through those two designated points (AQL, 1− ) and (LTPD, α β ). To determine whether a given process is capable, we can first consider the following testing hypothesis:

H0: p = AQL (process is capable), H1: p = LTPD (process is incapable).

For normally distributed processes with single characteristic, the index Spk is used to establish the relationship between the manufacturing specification and the actual process performance, which provides an exact measure on the process yield. That is, the null hypothesis with proportion defective, H0: p = AQL is equivalent to test process capability index with H0: Spk ≥ SAQL, where SAQL is the level of acceptable quality for Spk index. For instance, if the fraction of defectives p = AQL of vendor’s product is about 66 PPM, then the probability of consumer accept the lots will larger than 100(1−α)% . On the other hand, if the fraction of defectives of vendor’s product, p = LTPD, is about 2700 PPM, then the probability of consumer would accept no more than 100 % . Then, from the relationship between the index value and fraction of defectives, we β could obtain the equivalent SAQL = 1.33 and SLTPD = 1.00 based on the process index Spk.

Therefore, the required inspection sample size n and critical acceptance value c0 for the sampling plans are the solution to the following two nonlinear simultaneous equations.

Pr{Accepting the lot| p = AQL}≥ −1 α , Pr{Accepting the lot| p = LTPD}≤ . β

As described earlier, the asymptotic sampling distribution of ˆS is normally distributed with a _pk mean Spk and a variance (a²+b²)/36nφ²(3S_pk). The probability of accepting the lot then can be

Therefore, the required inspection sample size n and critical acceptance value c0 of ˆS for the _pk sampling plans can be obtained by solving the following two nonlinear simultaneous equations.

(^φ )

(

)

β φ

π

∞ ⎡ ⎤

≥ ⎢− − ⎥

⎢ + ⎥

+ ⎣ ⎦

∫₀

2

2

2 2

2 2

18 (3 )

(3 )

18n ^LTPD exp ^LTPD ( _LTPD)

c

n S

S x S dx

a b

a b ,

where SAQL > SLTPD. We note that the required sample size n is the smallest possible value of n satisfying the above two equations, and determining the [n] as sample size, where [n] means the least integer greater than or equal to n. Since the calculation of critical values involve two parameters a and b, functions of the two indices Cp and Ca, we have to consider the effect of a²+b² from various (Cp, Ca) for given a fixed performance requirement Spk.

Figure 4-1(a). Surface plot of S1. Figure 4-1(b). Surface plot of S2.

Figure 4-2. Surface plot of S1 and S2.

Pearn et al [36] discussed that for given a fixed performance requirement Spk, the differences among those calculated critical values corresponding to various values of Cp and Ca, are sufficiently small and can be neglected. Also, they showed that the factor (a²+b²)/36 (3φ² S )

is insensitive to the value changes of Cp and Ca in all cases, except for Ca = 1. Consequently, the critical values c0 may be considered as a constant, which is independent of the process characteristics Cp and Ca for a fixed performance requirement Spk. n

In order to illustrate how we solve the above two nonlinear simultaneous equations, we let

(^φ )

Figure 4-3(a). The required sample size n as surface plot with α= 0.01 (0.01) 0.10 and β = 0.01(0.01)0.10 under (SAQL, SLTPD) = (1.33, 1.00).

Figure 4-3(b). The critical acceptance value c0 as surface plot with α= 0.01 (0.01) 0.10 and β = 0.01(0.01)0.10 under (SAQL, SLTPD) = (1.33, 1.00).

Figure 4-3(c). The required sample size n as surface plot with α= 0.01 (0.01) 0.10 and β = 0.01(0.01)0.10 under (SAQL, SLTPD) = (1.50, 1.33).

Figure 4-3(d). The critical acceptance value c0 as surface plot with α= 0.01 (0.01) 0.10 and β = 0.01(0.01)0.10 under (SAQL, SLTPD) = (1.50, 1.33).

For SAQL = 1.33 and SLTPD = 1.00, Figures 4-1(a) and 4-1(b) display the surface plots of equations S1(n, c0) and S2(n, c0) with -riskα = 0.05 and -riskβ = 0.05, respectively. Figure 4-2 displays the surface plots of equations S1(n, c0) and S2(n, c0) simultaneously with -riskα = 0.05 and

-risk

β = 0.05 under SAQL = 1.33 and SLTPD = 1.00, respectively. In Figure 4-2, interaction of S1(n, c0) and S2(n, c0) is (n, c0) = (66, 1.1412), which is the solution to the two nonlinear simultaneous equations. That is, in this case, the minimum required sample size n = 66 and critical acceptance value c0 = 1.1412 of the sampling plan based on the capability index Spk.

To investigate the behavior of the critical acceptance values and required sample sizes, we perform extensive calculations to obtain the solution of the two nonlinear equations with various parameters. Figure 4-3(a) displays the required sample size n as surface plot with the probabilities α= 0.01(0.01)0.10 and β = 0.01(0.01)0.10 under (SAQL, SLTPD) = (1.33, 1.00). Figure 4-3(b) displays the critical acceptance value c0 as surface plot with the probabilities α= 0.01(0.01)0.10 and β = 0.01(0.01)0.10 under (SAQL, SLTPD) = (1.33, 1.00). Figures 4-3(c) and 4-3(d) show the required sample size n and the critical acceptance value c0 as surface plot with α= 0.01(0.01)0.10 and β = 0.01(0.01)0.10 under (SAQL, SLTPD) = (1.50, 1.33), respectively.

From Figures 4-3(a) and 4-3(d), we observe that the larger of the risk (α or β ) which producer or customer could suffer, the smaller is the required sample size n. This phenomenon can be explained intuitively, as if we hope that the chance of wrongly concluding a bad process as good or good lots as bad ones is small, then more sample information is needed to judge the lot quality. Further, for fixed SAQL, SLTPD and -riskα , the corresponding critical acceptance values become smaller when the -riskβ becomes larger. On the other hand, for fixed SAQL, SLTPD, and

-risk

β , the corresponding critical acceptance values become larger when the -riskα becomes larger. This can also be explained by the same reasoning as above. Consequently, the required sample size is smaller when the difference between SAQL and SLTPD is significant since the judgment will then be relatively easier to reach correct decision.

For practical applications purpose, we calculate and tabulate the critical acceptance values and required sample sizes for the sampling plans, with commonly used -riskα , -riskβ , SAQL and SLTPD. Table 4-1 display (n, c0) values for -riskα = 0.01, 0.025(0.025)0.10 and -riskβ = 0.01,

0.025(0.025)0.10, with various benchmarking quality levels, (SAQL, SLTPD) = (1.33, 1.00), (1.50, 1.33), (1.67, 1.50), (2.00, 1.67). Based on the designed sampling plan, the practitioners can determine the number of production items to be sampled for inspection and the corresponding critical acceptance value. For example, if the benchmarking quality level (SAQL, SLTPD) is set to (1.33, 1.00) with producer’s -riskα = 0.01 and customer’s -riskβ = 0.05, then the corresponding sample size and critical acceptance value can be obtained as (n, c0) = (101, 1.1142). The lot will be accepted if the 101 inspected production items yield measurements with ˆS ≥ 1.1142. _pk

Table 4-1. (n, c0) values for -riskα = 0.01, 0.025(0.025)0.10,

For the proposed sampling plan to be practical and convenience to use, a step-by-step procedure is provided below.

Step 1: Decide the process capability requirements (i.e. set the values of SAQL and SLTPD), set the -risk

α , the chance of wrongly concluding a capable process as incapable, and the -riskβ , the chance of wrongly concluding a bad lot as good one.

Step 2: Check the Table 4-1 to find the critical value c0 and the required sample size n for inspection based on given values of α-risk, -riskβ , SAQL and SLTPD.

Step 3: Calculate the value of ˆS from these _pk n inspected samples.

Step 4: Make decisions that accept the entire lot if the estimated ˆS value is greater than the _pk critical value c0. Otherwise, we reject the entire lot.

4.4. Accuracy of Spk Variables Sampling Plans

To assess the accuracy of the Spk sampling plan, we simulate N = 10000 lots in each combination shown in Table 4-1 to calculate the probability of accepting lots to compared with the probability of accepting the lot from the corresponding (α β, ) and (SAQL, SLTPD). Table 4-2 displays the probability of accepting the lot from simulation according to the rule in Table 4-1.

For the case of (SAQL, SLTPD) = (1.33, 1.00) , the simulation results indicate for each combination of α-risk (the producer’s risk) = 0.01, 0.025(0.025)0.10 and -riskβ (the customer’s risk) = 0.01, 0.025(0.025)0.10, the probabilities of accepting the lots are all greater than the corresponding 1− under Sα AQL = 1.33, and the probabilities of accepting the lots are all slightly greater than the corresponding β under SLTPD = 1.00, but the magnitude of LP− are β no greater than 0.0205.

For the case of (SAQL, SLTPD) = (1.50, 1.33) , the simulation results indicate for each combination of α- risk = 0.01, 0.025(0.025)0.10 and -riskβ = 0.01, 0.025(0.025)0.10, the probabilities of accepting the lots are almost greater than the corresponding 1− under Sα AQL = 1.50 (except for α= 0.025 and β = 0.01, the probability of accepting the lot is 0.9749), and the probabilities of accepting the lots are all slightly greater than the corresponding β under SLTPD = 1.33, but with LP− no greater than 0.0206. β

Table 4-2. Probabilities of accepting the lot for -riskα = 0.01, 0.025(0.025)0.10, -risk

β = 0.01,0.025(0.025)0.10 with various (SAQL, SLTPD) by simulation with N=10000.

SAQL = 1.33 SAQL = 1.50 SAQL = 1.67 SAQL = 2.00

AP: the probability of accepting the lot under SAQL

LP: the probability of accepting the lot under SLTPD

For the case of (SAQL, SLTPD) = (1.67, 1.50) , the simulation results indicate for each combination of -riskα and -riskβ = 0.01, 0.025(0.025)0.10, the probabilities of accepting the lots are all close to the corresponding 1− under Sα AQL = 1.67 with AP− −(1 α) no greater than 0.0094, and the probabilities of accepting the lots are all slightly greater than the corresponding -riskβ under SLTPD = 1.50, but with LP− no greater than 0.0089. β

For the case of (SAQL, SLTPD) = (2.00, 1.67) , the simulation results indicate for each

combination of ( , )α β = 0.01, 0.025(0.025)0.10, the probabilities of accepting the lots are all greater than the corresponding 1− under Sα AQL = 2.00, and the probabilities of accepting the lots are mostly greater than the corresponding -riskβ under SLTPD = 1.67 (except for ( , )α β = (0.05, 0.05) and (0.05, 0.1), the probability of accepting the lot is 0.0495 and 0.0977, respectively), but with LP− no greater than 0.0054. The simulation results clearly indicate that the β probabilities of accepting lots are all very close to the preset value of 1− or α β in all the cases we investigated. Thus, the sampling plan based on the normal approximation to the distribution of the estimated Spk is adequately reliable to the engineers for their in-plant applications.

combination of ( , )α β = 0.01, 0.025(0.025)0.10, the probabilities of accepting the lots are all greater than the corresponding 1− under Sα AQL = 2.00, and the probabilities of accepting the lots are mostly greater than the corresponding -riskβ under SLTPD = 1.67 (except for ( , )α β = (0.05, 0.05) and (0.05, 0.1), the probability of accepting the lot is 0.0495 and 0.0977, respectively), but with LP− no greater than 0.0054. The simulation results clearly indicate that the β probabilities of accepting lots are all very close to the preset value of 1− or α β in all the cases we investigated. Thus, the sampling plan based on the normal approximation to the distribution of the estimated Spk is adequately reliable to the engineers for their in-plant applications.