Frontiers in Statistical Quality Control 11

Shewhart’s idea of predictability and modern statistics
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A random sample of n items is inspected and the number of marginallyconforming items dl and nonconforming items d2 in the sample are counted. The probability mass function for the trinomial distribution is: 1 and the function for the sampling plan's probability of acceptance is: Bray et al. Sampling plans based on Bray et al.

It should be noted that these sampling plans are one-sided, based on two upper specification limits conventionally represented by the symbols m and M, whereas in this paper, these two specification limits are defined as U , and U2respectively. In [26], the authors first explored the case of semi-curtailed sampling where inspection would cease as soon as the number of nonconforming items or nonconforming plus marginally-conforming items discovered in the sample were sufficient to cause lot rejection.

This was followed by consideration of the case of fully-curtailed inspection where, in addition to ceasing inspection due to early lot rejection, sampling inspection would cease as soon as lot acceptance was evident. The authors provide equations for the average sample number ASN as well as the maximum likelihood estimators MLE of the proportions marginallyconforming and nonconforming in the lot and their associated asymptotic vari- ances under these forms of curtailed inspection.

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Visit store. The papers presented here were carefully selected and reviewed by the scientific program committee, before being revised and adapted for this volume. We discuss necessary and sufficient conditions to achieve values of probabilities of misleading signals PMS smaller than or equal to 0. Another principal application of acceptance sampling has been in the quality control of the net contents of packaged products sold in the marketplace. No ratings or reviews yet. Beoordeel zelf slecht matig voldoende goed zeer goed. These changes pose new opportunities for researchers in statistical methodology, including those interested in surveillance and statistical process control methods.

The relationship between the percent saving in inspection and the efficiency of the estimators is also provided. In their second paper [27], the authors extended their research to address threeclass attribute sampling plans in a multiple sampling context, with particular focus on double sampling. Expressions for the ASN, the MLEs of the proportions marginally-conforming and nonconforming after a specified number of lots have been inspected, and the relations between the estimators' asymptotic variances and the ASN are obtained under uncurtailed, semi-curtailed, and fully-curtailed inspection in the double sampling context, then generalized to multiple sampling.

As the full details of Shah and Phatak's two papers are beyond the intended scope of this one, the interested reader is advised to consult their work.

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The main focus of this edited volume is on three major areas of statistical quality control: statistical process control (SPC), acceptance sampling and design of. Editorial Reviews. From the Back Cover. The main focus of this edited volume is on three major areas of statistical quality control: statistical process control.

As his papers were largely of an evolving nature, this section will focus primarily on Clements [9]. His approach differs slightly from that of Bray et al. If both of the following inequalities are satisfied: d, lc, and d2 lc, then the lot is accepted; otherwise, it is rejected. In this case, the function for the sampling plan's probability of acceptance is simply the cumulative distribution function for the trinomial distribution: Clements [9] used Table A of [30] as a launching pad. This table provides code letters, sample sizes, and associated acceptance and rejection numbers based on various acceptance quality limits AQL under normal inspection.

As much as possible, Clements limited himself to using the preferred sample sizes and AQL values to construct sets of possible trinomial-based alternatives to the binomialbased plans in the table. He pointed out that the system he developed worked best for AQL values less than 4. For code letters from F to K, adjustments to sample sizes were necessary. However, he was able to succeed in using a single set of acceptance numbers along each diagonal in his version of the master table. Allen University of Guelph , Brown [5] investigated extending the three-class attributes sampling plan approach to a sampling-byvariables framework.

Newcombe and Allen [22] later published an abbreviated version of this thesis. The primary focus of this work was the application where the variable of interest is distributed according to a normal distribution with unknown mean and standard deviation and the marginally-conforming and nonconforming specification limits are one-sided. However, the thesis [5] does include a chapter dealing with techniques for applying the method under situations of nonnormality. Somewhat paralleling the methodology of Bray et al. Such a sampling plan is specified by a sample size, an acceptability constant with respect to the sum of the proportions marginally-conforming and nonconforming, and another acceptability constant with respect to the proportion nonconforming, i.

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A random sample of n items is inspected and the sample mean x,- and standard deviation s are calculated. The probability of acceptance of a lot of quality pl, p2 involves the bivariate noncentral t distribution as indicated below: Pr? The special case of the bivariate noncentral t distribution needed to solve this probability was originally given by Owen [23]. Brown gives details of both the exact and an approximate method in her thesis [5]. Newcombe and Allen [22] also give the necessary details. This method permits the usual two-class procedure to be used to determine the acceptability constants. Brown also provides guidance for matching His method was to use the binomial-based plan's AQL value as the process nonconforming value p2 and then choose AQL values that were odd numbers of steps higher than p2 for the process marginally-conforming values pi.

For each higher p, value selected, the sample size corresponding to the next lower code letter was used. For example, for an AQL of 1. With p2 fixed at 1. Clements also proposed using a narrow hmit technique see [25] for establishing the value of the specification limit separating conforming from marginally-conforming items when the standard deviation is known and the distribution of the quality characteristic is normal. For a one-sided sampling plan with upper specification limits, the additional specification limit is determined as follows: Samples of this additional information are included in Table 1 for the example earlier discussed.

Clements also includes approximate formulae for determining generic trinomial sampling plans based on the assumption of quality characteristics distributed according to a normal distribution with known standard deviation, as well as graphical operating characteristic curves and contours for example sampling plans.

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The thesis includes APL programming code to aid in developing three-class sampling plans. At the time of their review, the current practice involved a samplingby-variables approach. The method was found to suffer from a number of deficiencies including using a sample mean as an indicator of individual apple quality as well as ignoring the variability and non-normality of the measured variables.

Furthermore, the approach was determined to be biased in the producer's favour. The authors concluded that a more appropriate sampling plan solution for grading this produce would involve classification by attributes.

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The practice was to grade lots of apples into one of four grades A, B, C, or R and criteria existed to grade individual apples according to these same grades. Lacking, however, was a standard in terms of quality levels and fraction nonconforming. To address this deficiency, the authors used simulation methods for establishing critical values for grading the individual lots while maintaining approximately the same expectations with respect to different grades from past history of the various growers' lines under the sampling-by-variables plan.

As four classes of acceptability or grades were involved, the applicable probability models identified were the quadrivariate hypergeometric distribution for isolated lots and the quadrinomial distribution for a continuing series of lots. The authors developed their formulae for the four classification probability functions based on the quadrinomial distribution and dominance logic with respect to individual sample item classifications, then applied the rules in the order from R to A rather than from A to R to reduce producer-oriented bias.

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The authors conclude the paper with several recommendations to enhance the implementation of the insightful solution they developed. The approach used by the paper's authors is interesting in that it takes a rather obscure grading approach and translates it into terms that are more transparent and amenable to critical evaluation by purchasers and consumers. Their approach is no doubt applicable in principle to many practical applications beyond the grading of produce. They reviewed two valuation schemes. The first QVF, assigned a value of 0 to conforming items, a variable value v between 0 and to marginally-conforming items, and a value of 1 to nonconforming items.

The second scheme QVF2 assigned a value of 1 to conforming items, a variable value greater than 1 to marginally-conforming values, and another variable but greater yet value to nonconform- ing items. The authors reported that their experiments and analyses did not reveal any significant advantage of one scheme over the other so this section will focus only on QVF,.

Their method involves specifying a sample size and a single critical value n, V,. A random sample of n items is inspected and the number of marginallyconforming items d l and nonconforming items d2 in the sample are counted.

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If the following inequality is satisfied: then the lot is accepted; otherwise, it is rejected. The hnction for the sampling plan's probability of acceptance for valid values of V, is: To aid in the implementation of such sampling plans, the authors provide approximate formulae for calculating n and V, to give the desired probabilities of acceptance, given p, vectors for both acceptable and rejectable quality and specified producer's and consumer's risks. They also suggest considering several different values of v and verifying the actual resulting risks against those specified before deciding upon a sampling plan.

This section provides them for purposes of completeness. The probability mass function for three-class or trivariate hypergeometric distribution is as follows: The function for such a sampling plan's probability of acceptance is: For the subset of such sampling plans where c2 is fixed at 0, the simplified function for the sampling plan's probability of acceptance is: Finally, the function for a finite-lot sampling plan's probability of acceptance under the QVFl approach is: 4.

However, their work may be extended to the two-sided scenario where combined control of the proportions outside both the upper and lower specification limits is required. Brown had suggested an approximate method that is frequently used for two-class sampling plans in the "further research" section of her thesis [5]. Once the values of k12and k2 are determined for the one-sided, unknown mean and standard deviation case, these values may be converted into critical values for the estimates o f p 1 2and p2respectively using the "M method" described in, for example, Schilling [25, Ch.

Uniform minimum variance unbiased estimates UMVUE of the values of p12and p2 can then be computed from the sample statistics, with the estimates due to the upper specification limits being added to their lower specification limit counterparts and then compared to the previously determined critical values to decide lot acceptance. As pointed out in Brown [5], the method is approximate as the actual probability of acceptance of such a lot also depends on how much of the proportion of interest is outside each specification limit; the approximation is good, however, because the operating characteristic band is very narrow see Baillie [2].

Where the quality characteristic of interest may only be classified into one of two states, this is not an option. However, where the quality characteristic is measurable, a potentially useful approach that may be employed to create a third class of acceptability is narrow-limit gauging also referred to as compressed-limit or reduced-limit gauging as discussed by Schilling [25]. The method was originally created to permit a form of sampling-by-attributes inspection to be used as an alternative to samplingby-variables inspection with approximately equivalent control.

As already men- tioned in 3. Although the method is traditionally associated with characteristics distributed according to a normal distribution with known standard deviation, it would seem reasonable to alter these conditions in some applications. In particular, with respect to sampling inspection applications where consumer protection is emphasized, a normal distribution centered at the desired lot target mean could be used with a maximum standard deviation calculated to provide the required proportion nonconforming p2 beyond the upper or lower specification limit or both.

Once the mean and standard deviation are established for the model normal distribution, narrow limit s can be readily determined such that the total lot proportion outside these new limits Ca12 becomes associated with the usual low probability of acceptance assigned to p2 alone. Values of n, c12,and c2 may then be determined to complete the sampling plan. It should also be noted that model distributions other than the normal distribution could be justified in such narrow limit determinations. More generally, sampling plans are usually designed with consideration to both the quality level that should be accepted with a specified high probability producer's risk quality or PRQ and the quality level that should be accepted with a specified low probability consumer's risk quality or CRQ.

In the case of threeclass sampling plan design, values for both the proportion marginally-conforming and proportion nonconforming need to be specified as a vector or pair pl, p2 for both the PRQ and CRQ scenarios.