On the Estimation Error in Zero-Inflated Poisson Model for Process Control

The control chart based on the geometric distribution (geometric chart) has been shown to be competitive with p- or np-charts for monitoring the proportion nonconforming, especially for applications in high quality manufacturing environments. However, implementing a geometric chart is often based on the assumption that the in-control proportion nonconforming is known or accurately estimated. For a high quality process, an accurate parameter estimate may require a very large sample size that is seldom available. In this paper we investigate the sample size effect when the proportion nonconforming is estimated. An analytical approximation is derived to compute shift detection probabilities and run length distributions. It is found that the effect on the alarm probability can be significant even with sample sizes as large as 10,000. However, the average run length is only affected mildly unless the sample size is small and there is a large process improvement. In practice, the quantitative results of the paper can be used to determine the minimum number of items required for estimating the control limits of a geometric chart so that certain average run length requirements are met.

Since the run length distribution is generally highly skewed, a significant concern about focusing too much on the average run length (ARL) criterion is that we may miss some crucial information about a control chart’s performance. Thus it is important to investigate the entire run length distribution of a control chart for an in-depth understanding before implementing the chart in process monitoring. In this paper, the percentiles of the run length distribution for the double sampling (DS) X chart with estimated process parameters are computed. Knowledge of the percentiles of the run length distribution provides a more comprehensive understanding of the expected behaviour of the run length. This additional information includes the early false alarm, the skewness of the run length distribution, and the median run length (MRL). A comparison of the run length distribution between the optimal ARL-based and MRL-based DS X chart with estimated process parameters is presented in this paper. Examples of applications are given to aid practitioners to select the best design scheme of the DS X chart with estimated process parameters, based on their specific purpose.

The nonparametric (distribution-free) control charts are robust alternatives to the conventional parametric control charts when the form of underlying process distribution is unknown or complicated. In this paper, we consider two new nonparametric control charts based on the Hogg–Fisher–Randle (HFR) statistic and the Savage rank statistic. These are popular statistics for testing location shifts, especially in right-skewed densities. Nevertheless, the control charts based on these statistics are not studied in quality control literature. In the current context, we study phase-II Shewhart-type charts based on the HFR and Savage statistics. We compare these charts with the Wilcoxon rank-sum chart in terms of false alarm rate, out-of-control average run-length and other run length properties. Implementation procedures and some illustrations of these charts are also provided. Numerical results based on Monte Carlo analysis show that the new charts are superior to the Wilcoxon rank-sum chart for a class of non-normal distributions in detecting location shift. New charts also provide better control over false alarm when reference sample size is small.

A study of the run length distribution of a Shewhart control chart is important before the chart is implemented in process monitoring. Interpretations based on the average run length (ARL) alone can be misleading since the in-control run length distribution of a Shewhart chart is highly skewed. This article highlights the importance of computing the percentage points of the run length distribution, which enables practitioners to determine the qth- (0 < q < 100) percentage points of the time to signal (PPTS (q)) and the qth- percentage points of the number of individual units sampled, say, I (q), to detect a shift for a Shewhart chart. Examples will be given to illustrate the benefits and usefulness of computing the two performance measures, PPTS (q) and I (q). Formulas to aid practitioners in computing the percentage points of the run length distribution and to determine the two performance measures will also be provided.

On the average run lengths of quality control schemes using a Markov chain approach

We propose a method for obtaining run-length distribution for Shewhart control charts with supplementary runs rules. We use Markov chain imbedding for obtaining the run length distribution. The method has several advantages. It is easy and fast to calculate and can be applied to other areas such as reliability with these advantages. In addition, accurate and clear distribution can be obtained. In this paper, we calculate probabilities of the run length according to the changes of the variances through S² control charts with supplementary runs rules. When the variance increases, the average run length (ARL) decreases rapidly. Also, when the degree of freedom increases, the ARL decreases.

Exponentially Weighted Moving Average (EWMA) control charts yield insights into data in a way more comprehensible to the practitioners and researchers because of its capability in discovering small to moderate process mean shifts. EWMA control chart is incorporated with conforming run length (CRL) chart, named synthetic EWMA chart, to enhance the performance of the chart in detecting the out-of-control signal. Synthetic EWMA chart based on ranked set sampling (RSS) for monitoring process mean has been proposed as it advanced the detection of chart over a series of mean shifts. With the situation that normality assumption is scarcely attain in practice, we proposed synthetic EWMA median chart based on RSS. Rather than select average run length (ARL) as sole performance evaluating tool, the median and percentiles of run-length distribution are used to examine the performance of the proposed chart as it provides more information on the entire run-length distribution. Near-optimal parameters of the proposed chart will be acquired by setting the incontrol ARL at a designated value. The run length performances of the proposed chart are then compared with the existing charts such as EWMA median chart based on RSS.

The c‐chart or the control chart for nonconformities is designed for the case where one deals with the number of defects or nonconformities observed. A control chart can be developed for the total or average number of nonconformities per unit, which is well modeled by the Poisson distribution. In this paper the c‐chart will be studied, where the usual operation of the c‐chart will be extended by introducing a Bayesian approach for the c‐chart. Control chart limits, average run lengths, and false alarm rates will be determined by using a Bayesian method. These results will be compared with the results obtained when using the classical (frequentist) method. Copyright © 2014 John Wiley &amp; Sons, Ltd.

ABSTRACTThe issue of the effects of parameter estimation on the in-control performance of control charts has motivated researchers for several decades. In this context, recently, acknowledging what has been called by some the practitioner-to-practitioner variability, a new perspective has been advocated, namely, the study of the conditional distribution of the in-control average run length (or the conditional false-alarm rate), which is more meaningful in practice. Adopting this new perspective, some authors have analyzed the conditional distribution of the false-alarm rate (or of the in-control average run length) of and of S charts separately. However, since the and S charts are not typically used separately but together or jointly in many applications, here we study the effects of parameter estimation on the performance of the two charts applied jointly (called the joint charts). For the joint charts, defining the joint false-alarm rate as the probability that at least one of the two charts ( and S) issues a false alarm, we obtain its conditional distribution, some quantiles of interest (upper prediction bounds for it) and the number of Phase I samples required to guarantee that with a high probability the conditional joint false alarm rate will not exceed a maximum tolerated value. We assume normality, consider Sp (the square root of the pooled variance) as the Phase I estimator of the process standard deviation, and consider two possibilities regarding the chart: (1) centered at and (2) centered at a specified target value. The results show (and we formally prove) that, whereas the required number of Phase I samples may be very large for the joint charts, interestingly, it lies between the corresponding numbers of samples required by the chart and by the S chart individually; so, considering the performance of the charts from the perspective of their joint use may slightly alleviate the required number of Phase I samples.

The geometric chart has proven more effective than Shewhart p or np charts to monitor the proportion nonconforming in high-quality processes. Implementing a geometric chart commonly requires the assumption that the in-control proportion nonconforming is known or accurately estimated. However, accurate parameter estimation is very difficult and may require a larger sample size than that available in practice in high-quality process where the proportion of nonconforming items is very small. Thus, the error in the parameter estimation increases and may lead to deterioration in the performance of the control chart if a sample size is inadequate. We suggest adjusting the control limits in order to improve the performance when a sample size is insufficient to estimate the parameter. We propose a linear function for the adjustment constant, which is a function of the sample size, the number of nonconforming items in a sample, and the false alarm rate. We also compare the performance of the geometric charts without and with adjustment using the expected value of the average run length (ARL) and the standard deviation of the ARL (SDARL).

In recent years cumulative sum (CUSUM) control charts have become increasingly popular as an alternative to Shewhart's control charts. These charts use sequentially accumulated information in order to detect out-of-control conditions. They are philosophically related to procedures of sequential hypothesis testing (the relation being similar to that existing between Shewhart's charts and classical procedures for hypothesis testing). In the present paper we present a new approach to design of CUSUM-Shewhart control schemes and analysis of the associated run length distributions (under the assumption that the observations correspond to a sequence of independent and identically distributed random variables). This approach is based on the theory of Markov chains and it enables one to analyze the ARL (Average Run Length), the distribution function of the run length, and other quantities associated with a CUSUM-Shewhart scheme. In addition, it enables one to analyze situations in which out-of-target conditions are not present initially, but rather appear after a substantial period of time during which the process has operated in on-target mode (steady state analysis). The paper also introduces an APL package, DARCS, for design, analysis, and running of both one- and two-sided CUSUM-Shewhart control schemes and gives several examples of its application.

Acceptance Sampling Plans are employed to safeguard against the inconsistent degradation of quality standards, especially when lots are of higher quality due to sophisticated technology, leading to a greater quantity of defect-free items. In this paper, a new sampling plan based on the Zero-Inflated Poisson Distribution (ZIP_F) with fuzzy parameters is proposed. As the production process improves due to technological advancements, the occurrence of defect-free items becomes more prevalent. The Zero-Inflated Poisson (ZIP_F) distribution is identified as the appropriate probability distribution for determining the number of defects in these situations. The primary aim is to develop a procedure for sampling with fuzzy parameters as a reference plan under the Zero-Inflated Poisson distribution (ZIP_F) framework. The fuzzy operating characteristic (fOC-ZIP) function of the sampling plan is assessed, and a comparative analysis is conducted with the existing sampling plan based on the Poisson distribution.

The effects of parameter estimation are examined for the well-known c-chart for attributes data. The exact run length distribution is obtained for Phase II applications, when the true average number of non-conformities, c, is unknown, by conditioning on the observed number of non-conformities in a set of reference data (from Phase I). Expressions for various chart performance characteristics, such as the average run length (ARL), the standard deviation of the run length (SDRL) and the median run length (MDRL) are also obtained. Examples show that the actual performance of the chart, both in terms of the false alarm rate (FAR) and the in-control ARL, can be substantially different from what might be expected when c is known, in that an exceedingly large number of false alarms are observed, unless the number of inspection units (the size of the reference dataset) used to estimate c is very large, much larger than is commonly used or recommended in practice. In addition, the actual FAR and the in-control ARL values can be very different from the nominally expected values such as 0.0027 (or ARL0=370), particularly when c is small, even with large amounts of reference data. A summary and conclusions are offered.

Runs rules are used to increase the sensitivity of the Shewhart control chart in detecting small and moderate process mean shifts. Most of the charts incorporating runs rules are designed based on the average run length (ARL). It is known that the shape of the run length distribution changes according to the magnitude of the shift in the process mean, ranging from highly skewed when the process is in-control to approximately symmetric when the shift is large. Since the shape of the run length distribution changes with the magnitude of the shift in the mean, the median run length (MRL) provides a more meaningful explanation about the in-control and out-of-control performances of a control chart. In this article, we propose the design of the revised m-of-k runs rule based on MRL. In addition, the standard deviation of the run length (SDRL) of the revised m-of-k rule will also be studied. The revised m-of-k runs rule, suggested by Antzoulakos and Rakitzis (2008), was originally designed based on ARL. The Markov chain technique is employed to obtain the MRLs. Compared with the standard chart, the MRL results show that the revised rules give better performances for small and moderate mean shifts, while maintaining the same sensitivity towards large mean shifts. The MRL results are in accordance with the results obtained by Antzoulakos and Rakitzis (2008), where the rules are designed based on ARL.

The average run length (ARL), the average number of in-control signals before an out-of-control signal occurs, is a critical measure of the performance of a control chart. Various methods have been explored for estimating the ARL, including Markov chain-based approximation, integration method, and the Monte Carlo method, especially for time-dependent charting statistics. Although computationally expensive, the Monte Carlo method is generic and applicable to all control charts. Typically, it involves choosing the maximum run length in advance and discarding iterations that fail to produce out-of-control signals to accommodate some unusually lengthy runs. However, these discarded runs provide information for estimating the run length. This paper introduces a new Monte Carlo approximation that retains failed iterations as Type I censored observations for ARL estimation. Our approach relies on the memoryless expectation of the run length distribution to have a simple and efficient ARL estimator. This assumption is necessary for mean estimation in Type I censored data and aligns with existing ARL approximations used in popular control charts like Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) charts. Numerical simulations demonstrate our method’s computational and statistical efficiency across the mean, median, EWMA, and CUSUM charts.