Frontiers in Statistical Quality Control 8
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However, the control charts without memory, like the MEWMA a combination of the T 2 and an EWMA procedure Montgomery, for Gaussian observations or nonparametric approaches Qiu, ; Zou and Tsung, will be preferable to deal with a small change for example a small variation of the process mean.
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Despite the details mentioned in previous papers, differences in distribution can be found in control charts. In a manufacturing process, time represents an attribute or variable measure observed as consecutive events of concern. The distribution of the time for an event is known as exponential distribution when the probability of an event in the next small time interval does not vary though time. It is a continuous distribution and widely used to monitor time successive occurrences of events.
Frontiers in Statistical Quality Control 12
Simulations are conducted to compare the performance of the explicit formulas with numerical approximations. The computing time for the explicit formulas is found to be approximately 10 seconds, which is significantly less than the computational time required for numerical approximations.
The accuracy of these explicit expressions has been checked by a comparison with numerical solutions of integral equations based on using Gauss-Legendre integration rules to approximate integrals. The numerical results and the values from the explicit formulas were in excellent agreement.
The computation times required for the numerical computations were approximately 40—50 minutes compared with less than 1 second for the explicit formulas as show in Table 3. Petcharat et al. They compared the computational time between our analytical explicit expressions for the ARL performance with the one obtained via the Gauss-Legendre numerical scheme for integral equations.
They found that those methods are in excellent agreement; however, the computational time of the former takes approximately 1 second while the latter method consumes an approximate computational time of 11 minutes. The accuracy and computational speed for analytical formula was compared with values obtained from the numerical integration method by Fredholm integral equation Rao et al.
They have shown that analytical expressions are very accurate in the same manners of the numerical integration method; however, the latter method is very time consuming in terms of computational time. Dokouhaki and Noorossana presented a copula Markov CUSUM chart, based on the Markov CUSUM chart that is good at monitoring the auto-correlation statistical process control and for monitoring two correlated binomial data sets with an auto-correlation.
There is numerous research related to the multivariate variables control charting focusing on the variable quality characteristics in the statistical process control SPC literature; however, fewer investigations could be found regarding the multivariate attributes control charts relying on the attribute quality characteristics.
Multivariate attribute control charting makes it interesting to monitor the auto-correlated data, since the real-world processes usually include the data based on an auto-correlation structure. Ignoring the auto-correlation structure in developing a multivariate control chart increases the type I and type II errors simultaneously and consequently reduces the performance of the chart.
The most important difficulty in developing a multivariate attributes control charts is the absence of the joint distribution for quality characteristics. This deficiency can be dispelled through the use of a copula approach for developing joint distribution. In that paper, Dokouhaki and Noorossana use the Markov approach for modelling the auto-correlated data. Then, the copula approach is used to make the joint distribution of two auto-correlated binary data series.
Suppose X and Y as two correlated binary data with an auto-correlated structure for each. Thus, the two correlated sequences of X 1 , X 2 , X t , … and Y 1 , Y 2 , …, Y t , …, each with the value of 0 or 1, denote the conforming or non-conforming observations of two characteristics of a process. Consider p as the probability that an item is defective and p which is achieved by the process through the lack of any assignable causes of the process variation. The transition probability of the two-state Markov model regarding the variable X or Y is a conditional probability and would be as:.
Hence, for a typical variable like X whose value are 0 or 1, four transition probabilities can be derived as:. In addition, the estimation of p as the proportion and p as the correlation between two successive observations are. They used the above equations to develop a univariate control chart entitled Markov-based CUSUM which shows a better performance than other related univariate charts.
However, the conditional joint distribution of the variables X and Y needs to be known in order to use the Markov approach in a bivariate control chart. They use the copula function to develop such a distribution since the conditional joint distribution of the two variables cannot be achieved straightforwardly. In order to derive a bivariate copula function, denote the two cdfs F x 1 and F x 2 by the two uniform random variables U and V , respectively.
There are considerable number of multivariate variables control charts focusing on the variable quality characteristics , but fewer control charts could be found regarding the multivariate attribute characteristics relying on attribute quality characteristics. Newcombe and Allen  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  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. 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.
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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. 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. 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. 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 . 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 , 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 .
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 .
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. The establishment of these quality levels is a matter for the producer and consumer to agree upon and ideally should involve consideration of relevant cost information.
Determining a sampling plan to provide the desired performance will usually require a computer program that steps through various combinations of n, c12,and c2 until the required set of values is found.
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However, it should be noted that Cassady and Nachlas  and Clements  provided more efficient algorithms in this regard for their particular versions of these sampling plans. In addition, judging by its title, a paper by Singh et al.