Double-sampling control charts for attributes

In this article, we propose a double-sampling (DS) np control chart. We assume that the time interval between samples is fixed. The choice of the design parameters of the proposed chart and also comparisons between charts are based on statistical properties, such as the average number of samples until a signal. The optimal design parameters of the proposed control chart are obtained. During the optimization procedure, constraints are imposed on the in-control average sample size and on the in-control average run length. In this way, required statistical properties can be assured. Varying some input parameters, the proposed DS np chart is compared with the single-sampling np chart, variable sample size np chart, CUSUM np and EWMA np charts. The comparisons are carried out considering the optimal design for each chart. For the ranges of parameters considered, the DS scheme is the fastest one for the detection of increases of 100% or more in the fraction non-conforming and, moreover, the DS np chart is easy to operate.     

Determining Optimal Number of Samples for Constructing Multivariate Control Charts

Normally, an average run length (ARL) is used as a measure for evaluating the detecting performance of a multivariate control chart. This has a direct impact on the false alarm cost in Phase II. In this article, we first conduct a simulation study to calculate both in-control and out-of-control ARLs under various combinations of process shifts and number of samples. Then, a trade-off analysis between sampling inspection and false alarm costs is performed. Both the simulation results and trade-off analysis suggest that the optimal number of samples for constructing a multivariate control chart in Phase I can be determined.     

Transition probabilities and ARL performance of Six Sigma zone control charts

In this article, Six Sigma zone control charts (SSZCCs) are proposed for world class organizations. The transition probabilities are obtained using the Markov chain approach. The Average Run Length (ARL) values are then presented. The ARL performance of the proposed SSZCCs and the standard Six Sigma control chart (SSCC) without zones or run rules is studied. The ARL performance of these charts is then compared with those of the other standard zone control charts (ZCCs), the modified ZCC and the traditional Shewhart control chart (SCC) with common run rules. As expected, it is shown that the proposed SSZCC outperforms the standard SSCC without zones or run rules for process shifts of any magnitude. When compared to the other standard ZCCs and the Shewhart chart with common run rules, it is observed that the proposed SSZCCs have much higher false alarm rates for smaller shifts and hence they prevent unwanted process disturbances. The application of the proposed SSZCC is illustrated using a real time example.     

Properties and performance of the c-chart for attributes data

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.     

Parameter estimation and design considerations in prospective applications of the � chart

The effects of parameter estimation on the in-control performance of the Shewhart X¯ chart are studied in prospective (phase 2 or stage 2) applications via a thorough examination of the attained false alarm rate (AFAR), the conditional false alarm rate (CFAR), the conditional and the unconditional run-length distributions, some run-length characteristics such as the ARL, the conditional ARL (CARL), some selected percentiles including the median, and cumulative run-length probabilities. The examination involves both numerical evaluations and graphical displays. The effects of parameter estimation need to be accounted for in designing the chart. To this end, as an application of the exact formulations, chart constants are provided for a specified in-control average run-length of 370 and 500 for a number of subgroups and subgroup sizes. These will be useful in the implementation of the X¯ chart in practice.     

Multivariate control chart based on the highest possibility region

The T ² control chart is widely adopted in multivariate statistical process control. However, when dealing with asymmetrical or multimodal distributions using the traditional T ² control chart, some points with relatively high occurrence possibility might be excluded, while some points with relatively low occurrence possibility might be accepted. Motived by the thought of the highest posterior density credible region, we develop a control chart based on the highest possibility region to solve this problem. It is shown that the proposed multivariate control chart will not only meet the false alarm requirement, but also ensure that all the in-control points are with relatively high occurrence possibility. The advantages and effectiveness of the proposed control chart are demonstrated by some numerical examples in the end.     

Wilcoxon-type rank-sum control charts based on progressively censored reference data

Abstract

In this article, we introduce a new distribution???free Shewhart???type control chart implementing a modified Wilcoxon-type rank sum statistic based on progressive Type-II censoring reference data. The proposed chart is also a tool for monitoring the incomplete data, because the censoring scheme applied allows the protection of experimental units at an early stage of the testing procedure. The setup of the new nonparametric control chart is presented in detail, while its operating characteristic function is studied. Explicit formulae for the evaluation of Alarm Rate and Average Run Length values for both in-control and out-of-control situations are established. A numerical study carried out depicts the performance and robustness of the proposed control chart. For illustration purposes, a practical example is also discussed.     

Accounting for phase I sampling variability in the performance of the MEWMA control chart with estimated parameters

In this article, we assess the performance of the multivariate exponentially weighted moving average (MEWMA) control chart with estimated parameters while considering the practitioner-to-practitioner variability. We evaluate the chart performance in terms of the in-control average run length (ARL) distributional properties; mainly the average (AARL), the standard deviation (SDARL), and some percentiles. We show through simulations that using estimates in place of the in-control parameters may result in an in-control ARL distribution that almost completely lies below the desired value. We also show that even with the use of larger amounts of historical data, there is still a problem with the excessive false alarm rates. We recommend the use of a recently proposed bootstrap-based design technique for adjusting the control limits. The technique is quite effective in controlling the percentage of short in-control ARLs resulting from the estimation error.     

Distribution-Free Quality Control Charts Based on Signed-Rank-Like Statistics

In this article we propose three distribution-free (or nonparametric) statistical quality control charts for monitoring a process center when an in-control target center is not specified. These charts are of the Shewhart-type, the exponentially moving average-type, and the cumulative sum-type. The constructions of the proposed charts require the availability of an initial reference sample taken when the process was operating in-control to calculate an estimator for the unknown in-control target process center. This estimated center is then used in the calculation of signed-rank-like statistics based on grouped observations taken periodically from the process output. As long as the in-control process underlying distribution is continuous and symmetric, the proposed charts have a constant in-control average run length and a constant false alarm rate irrespective of the process underlying distribution. Other advantages of the proposed distribution-free charts include their robustness against outliers and their superior efficiency over the traditional normal-based control charts when applied to processes with moderate- or heavy-tailed underlying distributions, such as the double exponential or the Cauchy distributions.     

Performance of the zone control chart

Average run lengths of the zone control chart are presented, The performance of this chart is compared with that of several Shewhart charts with and without runs rules, It is shown that the standard zone control chart has performance similar to some even simpler charts and a much higher false alarm rate than the Shewhart chart with all of the common runs rules. It is also shown that a slightly modified zone control chart outperforms the Shewhart chart with the common runs rules.     

Comments on the Mixture Detection Rule Used in SPC Control Charts

When calculating independently the false alarm rate of the eight usual runs rules used in SPC control chart, it appears that the proposed rule designed to detect mixture patterns corresponds to a Type-I error strongly lower than the seven other rules. This discrepancy is underlined and the mixture rule is showed to be useless both for in-control and out-of-control processes. Thus a modification of the mixture detection rule is proposed and the impact of this new mixture rule is then illustrated and discussed using Monte Carlo calculations.     

The RL2 chart versus the np chart for detecting upward shifts in fraction defective

The use of the np chart for monitoring fraction-defective is well-established, but there are a number of relatively simple alternatives based on run-lengths of conforming items. Here, the RL₂ chart, based on the moving sum of two successive conforming run-lengths, is investigated in order to provide SPC practitioners with clear-cut guidance on the comparative performance of these competing charts. Both sampling inspection and 100% inspection are considered here, and it is shown that the RL₂ chart can often be considerably more efficient than the np chart, but the comparative performance depends on the false-alarm rate used for the comparison. Graphs to aid parameter-choice for the RL₂ chart are also provided.     

Two group inspection-based control charts for dispersion matrix

In this article, we propose the two control charts, i.e. the ‘VMAX Group Runs’ (VMAX-GR) and ‘VMAX Modified Group Runs’ (VMAX-MGR) control charts based on the bivariate normal processes, for monitoring the covariance matrix. The proposed charts give the faster detection of a process change and have better diagnostic feature. It is verified that the VMAX-GR and the VMAX-MGR charts give a significant reduction in the out-of-control ‘Average Run Length’ (ARL) in the zero state, as well as in the steady state, as compared to the synthetic control chart based on the VMAX statistic and the generalized variance |S| chart.     

Bivariate copulas on the Hotelling's T2 control chart

In this paper, we propose five types of copulas on the Hotelling's T² control chart when observations are from exponential distribution and use the Monte Carlo simulation to compare the performance of the control chart, which is based on the Average Run Length (ARL) for each copula. Five types of copulas function for specifying dependence between random variables are used and measured by Kendall's tau. The results show that the copula approach can be fitted the observation and we can use copula as an option for application on Hotelling's T² control chart.     

An optimal control chart for finite matrix sequences at some unknown change point

We present a new measure for evaluating the performance of control charts to detect abrupt changes of finite matrix sequences. The objective is to minimize the probability that the control chart fails to raise the alarm at unknown change point time for a given in-control average run length. We construct and prove the optimal control chart with dynamic control limits in different pre- and post-change distributions. We validate the optimality of the proposed chart by conducting exhaustive experiments on both simulation study and real-world data.     

A binomial CUSUM chart for detecting large shifts in fraction nonconforming

This article studies a unique feature of the binomial CUSUM chart in which the difference (d _t ?d ₀) is replaced by (d _t ?d ₀)² in the formulation of the cumulative sum C _t (where d _t and d ₀ are the actual and in-control numbers of nonconforming units, respectively, in a sample). Performance studies are reported and the results reveal that this new feature is able to increase the detection effectiveness when fraction nonconforming p becomes three to four times as large as the in-control value p ₀. The design of the new binomial CUSUM chart is presented along with the calculation of the in-control and out-of-control Average Run Lengths (ARL₀ and ARL₁).     

A Robust Control Chart for Monitoring the Mean of an Autocorrelated Process

Residual control charts are frequently used for monitoring autocorrelated processes. In the design of a residual control chart, values of the true process parameters are often estimated from a reference sample of in-control observations by using least squares (LS) estimators. We propose a robust control chart for autocorrelated data by using Modified Maximum Likelihood (MML) estimators in constructing a residual control chart. Average run length (ARL) is simulated for the proposed chart when the underlying process is AR(1). The results show the superiority of the new chart under several situations. Moreover, the chart is robust to plausible deviations from assumed distribution of errors.     

A joint-adaptive np control chart with multiple dependent state sampling scheme

Although the classical Shewhart np control chart has been widely used to detect an out-of-control status of manufacturing process, it is static and there is lack of responsiveness to slight process changes. In this paper, an adaptive np control chart with a joint sampling strategy combining double sampling (DS) and variable sampling interval (VSI) is developed. The multiple dependent state sampling scheme is adopted to further improve the performance of the control chart. An economical design model to minimize the general cost of using the proposed chart is established and solved by a genetic algorithm. The numerical results show that comparing to traditional static np control chart, the proposed np chart yields better performance in terms of shorter time to signal an out-of-control process and less expected cost per unit of time. Comparisons are made to show the capability of the proposed chart in yielding average reductions of 5.01% and 8.89%, in the cost of the proposed model compared to situations in which either the DSVSI np chart or the traditional np chart is used.     

An investigation of the effects of inspection errors on the run-length control charts

For continuous inspection schemes in an automated manufacturing environment, a useful alternative to the traditional p or np chart is the Run-Length control chart, which is based on plotting the run lengths (the number of conforming items) between successive nonconforming items. However, its establishment relies on the error-free inspection assumption, which can seldom be met in practice. In this paper, the effects of inspection errors on the Run-Length chart are investigated based on that these errors are assumed known. The actual false alarm probability and the average number inspected (ANI) in the presence of inspection errors are studied. This paper also presents the adjusted control limits for the Run-Length chart, which can provide much closer ANI curves to the ones obtained under error-free inspection.     

A Multivariate Synthetic Control Chart for Monitoring Process Mean Vector

In this article, a multivariate synthetic control chart is developed for monitoring the mean vector of a normally distributed process. The proposed chart is a combination of the Hotelling's T ² chart and Conforming Run Length chart. The operation, design, and performance of the chart are described. Average run length comparisons between some other existing control charts and the synthetic T ² chart are presented. They indicate that the synthetic T ² chart outperforms Hotelling's T ² chart and T ² chart with supplementary runs rules.