Cumulative Count Of Conforming Research Articles

A design parameter-free geometric Kullback-Leibler information control chart for monitoring Bernoulli processes

This paper considers a design parameter-free Kullback-Leibler information (KLI) control chart for monitoring the nonconforming proportion p of Bernoulli processes in Phase II. The chart statistic is derived from the Kullback-Leibler information based on a geometric distribution. Unlike conventional charts for Phase II, the proposed chart does not require design parameters. The users only need to determine a desired in-control average run length. The performance of conventional control charts deteriorates if the prespecified optimal design parameter is not appropriate for the actual size of the shift in the monitored quality characteristic. A design parameter-free approach prevents this situation, and the time needed to find the optimal design parameter can be saved as well. The performance of charts is evaluated by the average number of observations to signal under a specific shift in p, and the penalized relative mean index assesses the overall performance. The proposed chart has the best overall performance for detecting an upward shift in p compared with the cumulative count of conforming (CCC-r), Bernoulli cumulative sum (CUSUM), Bernoulli exponentially weighted moving average (EWMA), Bernoulli generalized likelihood ratio (GLR), geometric CUSUM, and binomial GLR charts. The KLI chart has decent performance for detecting a downward shift in p if the size of the shift is small. For detecting a possible shift in p to both sides, the overall performance of the KLI chart is better than those of the charts.

A new exponentially weighted moving average chart with an adaptive control scheme for high yield processes—An application in injection molding process

AbstractAnalytical formulae for computing the average time to signal (ATS) value for the exponentially weighted moving average (EWMA) cumulative count of conforming (CCC) chart using the Markov chain procedure are derived, as the performance of this chart is currently studied using simulation in the literature. Additionally, the variable sampling interval (VSI) EWMA CCC chart for monitoring the cumulative counts of items inspected until a nonconforming item is obtained in a high yield process and is developed to increase the sensitivity of the basic EWMA CCC chart. By using the Markov chain procedure, optimal parameters of the VSI EWMA CCC chart in minimizing the ATS value are obtained and provided to facilitate the chart's implementation in practice. It is found that the VSI EWMA CCC chart shows an impressive improvement over the basic CCC, VSI CCC, and EWMA CCC charts. An example using real data from an injection molding process producing micro‐prism array of an optical element is given to demonstrate the implementation of the VSI EWMA CCC chart in practice. The originality of this manuscript lies in the derivation of new Markov chain‐based analytical formulae for computing the ATS of the EWMA CCC chart and the incorporation of the VSI technique into the EWMA CCC chart to boost the out‐of‐control detection speed of the latter.

Monitoring high-quality processes with one-sided conditional cumulative counts of conforming chart

Due to the development of high-quality manufacturing processes, a cumulative count of conforming (CCC) chart has been demonstrated to be useful in high-quality manufacturing processes for monitoring the nonconforming fraction. When the quality characteristic is asymmetric and the one-directional shifting is of interest, a one-sided control chart is more appropriate than a two-sided chart. We construct a one-sided CCC chart for high-quality processes based on previous runs to improve its sensitivity. The results show that the proposed conditional procedure method of the one-sided CCC chart can detect the shift quickly when the process begins to deteriorate at the beginning. A real case example of medication error is illustrated using the proposed charting technique. It shows that the proposed method is more suitable than traditional p-chart. The proposed conditional one-sided CCC procedure is useful in manufacturing that involves high-quality processes.

Multi-objective Economic-Statistical Design of Cumulative Count of Conforming Control Chart

Cumulative Count of Conforming (CCC) charts are utilized for monitoring the quality characteristics in high-quality processes. Executive cost of control charts is a motivation for researchers to design them with the lowest cost. Usually in most researches, only one objective named cost function is minimized subject to statistical constraints, which is not effective method for economic-statistical design of control charts. In this paper, a multi-objective model for the economic-statistical design of the CCC control chart is developed. Then, multi-objective evolutionary algorithm (NSGA-II) for obtaining the Pareto optimal solution of the model is proposed. A numerical example is applied to illustrate the effectiveness of the proposed model. This model leads to lower cost and smaller probability of Type I and Type II errors, compared with economic model. In addition, a sensitivity analysis is done to investigate the effect of input parameters on the best solutions of the proposed model.

Variable Sampling Interval Cumulative Count of Conforming Chart with Runs Rules

The cumulative count of conforming (CCC) chart is effective in detecting very low fraction of nonconforming items for high yield manufacturing processes. In this study, a combination of runs rules and variable sampling interval feature is proposed to a lower sided CCC chart by inspecting the items one by one. The performance measures of the control chart are derived by using the Markov chain approach. The numerical comparisons show that the performance of the CCC chart can be improved by adding the runs rules and varying the sampling interval.

Performance of cumulative count of conforming chart of variable sampling intervals with estimated control limits

The control chart with cumulative count of conforming (CCC) items is very useful for high-quality processes. The current research of CCC chart performance with parameter estimation is focusing on the fixed sampling interval (FSI) scheme. In this paper, we investigate the effect of performance of CCC chart with variable sampling intervals with estimated control limits. The best design parameters of CCC chart with variable sampling intervals with known process nonconforming rate p0 are verified when such p0 is estimated with Bayesian estimator. The average time to signal (ATS) and its standard deviation (SDATS) are calculated when the high-quality process is in-control and out-of-control. Its efficiency is compared with that of FSI CCC chart, which shows a great cost reduction in practice.

Change point estimation of high-yield processes experiencing monotonic disturbances

In this paper, we first propose a maximum likelihood estimator (MLE) of a change point in high-yield processes, where the only assumption is that the change belongs to a family of monotonic changes. Following a signal from the cumulative count of conforming (CCC) control chart, the performance of the proposed monotonic change-point estimator is next evaluated by comparing its performances to the ones designed for step-changes and linear-trend disturbances through extensive simulation experiments involving different single step-changes, linear-trend disturbances, and multiple-step changes. The results show that when the type of change is not known a priori, using the proposed change-point estimator is useful, because it provides accurate and precise estimates of the change points for almost all of the shift magnitudes and all of the change types considered in this paper. In addition, the applicability of the proposed method is illustrated using a real case.

Properties and performance of one-sided cumulative count of conforming chart with parameter estimation in high-quality processes

The one-sided cumulative count of conforming (CCC) chart is a useful method to monitor nonconforming fraction in high-quality manufacturing processes. The nonconforming fraction parameter is assumed to be known when implementing a one-sided CCC chart. In this study, we investigated the impact of estimated nonconforming fraction, [pcirc] 0, in a one-sided CCC chart. The run length distribution is derived as well as the conditional probability of a false alarm rate (CFAR), conditional average run length (CARL) and its standard deviation (CSDRL). Simulation results are conducted to evaluate the effect of [pcirc] 0 in a one-sided CCC chart. The results show that values of CFAR, CARL and CSDRL are close to the nominal values for a large sample. The impact of estimation errors was also studied. We find that CFAR decreases for large [pcirc] 0. Thus, a large value of [pcirc] 0 is suggested for fewer false alarms.

Identifying time of a monotonic change in the fraction nonconforming of a high-quality process

When a signal is detected by control charts, a search begins to identify and eliminate the sources of this signal. Knowing when a process has changed is very helpful for this purpose. The unknown special point that the process changed for the first time is referred to as change point. In this paper, we propose a maximum-likelihood estimator for the behavior model of the process fraction nonconforming in a high-quality process monitored with a cumulative count of conforming (CCC) control chart. We estimate the time of change without requiring the prior knowledge of the change type rather than we assume the type of change present belongs to a family of monotonic changes. Then, we compare the performance of the proposed change point estimator relative to estimators for the process fraction nonconforming change point derived under a single step and a linear trend change assumption. We do this for a number of monotonic change types following a signal from a CCC control chart. Finally, the application of the proposed change point estimator is shown through a real case.

An improved self-starting cumulative count of conforming chart for monitoring high-quality processes under group inspection

The cumulative count of conforming (CCC) chart as a main statistical process control tool for monitoring high-quality processes has been widely studied. However, its applicability is limited to situations where units of product are inspected sequentially, or item by item. Motivated by real-life problems, this paper proposes an improved control charting technique for high-quality processes under group inspection. It integrates a self-starting feature and an approximately ARL- (average run length) unbiased design. The chart, named the CCCG chart, monitors the cumulative count of conforming samples until a non-conforming one is encountered. The ‘G’ in the subscript stands for ‘Group’. The self-starting feature caters to the need in practice to start monitoring a production process as soon as possible. The approximately ARL-unbiased design is developed to significantly improve the chart's sensitivity to process deterioration compared with the conventional design. The performance of the CCCG chart in phase I when the control limits are sequentially updated has been examined. Simulated data examples are presented to demonstrate the use and efficiency of the proposed technique. The CCCG chart is a natural generalisation of the traditional CCC chart and thus includes the latter as a special case.

An Economic Approach to the Management of High-Quality Processes

Modern manufacturing developments have forced researchers to investigate alternative quality control techniques for high-quality processes. The cumulative count of conforming (CCC) control chart is a powerful alternative approach for monitoring high-quality processes for which traditional control charts are inadequate. This study develops a mathematical model for the economic design of the CCC control chart and presents an application of the proposed model. On the basis of the results of the application, the economic and classical CCC control chart designs of the CCC control chart are compared. The optimal design parameters for different defective fractions are tabulated, and a sensitivity analysis of the model is presented for the CCC control chart user to determine the optimal economic design parameters and minimum hourly costs for one production run according to different defective fractions, cost, time, and process parameters. Copyright © 2012 John Wiley & Sons, Ltd.

Estimating the change point of the cumulative count of a conforming control chart under a drift

In a high quality process, the fraction of nonconforming is very low. In this area, standard Shewhart control charts are no longer useful. The Cumulative Count of Conforming (CCC) control charts, which enumerates the number of conforming items between the occurrences of two nonconforming ones, have been shown to be effective in the monitoring of high quality processes. When the CCC control chart signals an out-of-control condition, the process engineers should search for the source of the assignable causes. Knowing the exact time of the process change would help them to reduce the time for identification of the assignable causes. This paper provides a maximum likelihood estimator for the change point of the nonconforming level of the high quality process with a linear trend. Then, a Monte Carlo simulation is applied to evaluate the performance of the proposed estimator. In addition, the proposed estimator is compared with the MLE of the process fraction nonconforming, derived under a single step change. The results show that the proposed estimator outperforms the MLE designed for step change, when a linear trend disturbance is present in the process.

Cumulative Conformance Count Charts with VariableSample Sizes

Cumulative count of conforming (CCC) chart has been shown effective for high-yield processes with very low fraction of nonconforming items. Traditional CCC chart assumes the products from a process are inspected item by item. Recently, the CCC chart was generalized to conform to the industrial practice where items are inspected sample by sample without according to the production order. Different from the traditional chart, the generalized CCC chart monitors the cumulative count of samples until a nonconforming sample is encountered. In order to improve the power of the generalized CCC chart to detect process changes, this study applied the control scheme of variable sample size (VSS) to the generalized CCC chart. The average time to signal (ATS) process change of the proposed chart was derived by Markov chain approach and taken as performance measure to evaluate its statistical efficiency. Performance comparisons between the proposed chart and generalized CCC chart are made to evaluate its usefulness.

On the conditional decision procedure for high yield processes

Cumulative Count of Conforming (CCC) items chart has shown to be an effective tool for monitoring high yield processes. However, this chart uses a single count value to determine whether a change in a process has occurred. This makes the chart relatively insensitive to process shifts. To improve the performance of the chart, values of the previous runs or observations were incorporated into the decision rule using conditional probability. This paper investigates the performance of the modified decision rule and shows that the results obtained for the average run length are only true for the case of independent observations. An appropriate relationship for the average run length is developed and the results are compared to the modified decision rule mathematically and numerically. The results indicate that the average run length values obtained from the modified decision rule always underestimates the true values.

Economic design of cumulative count of conforming charts under inspection by samples

This paper deals with the economic issues involved in the design of control charts for high quality manufacturing processes. The cumulative count of conforming (CCC) chart has been shown effective for such processes. However, the applicability of the conventional CCC chart is limited to the case that items are inspected one by one in the order of production. When items are inspected lot by lot or sample by sample without preserving or according to the original production ordering, modification is needed. In this paper, it is proposed to monitor the cumulative number of samples inspected until a nonconforming sample is encountered. An economic model is developed for designing such a generalized CCC chart. Numerical examples are given and sensitivity analysis is performed to demonstrate the use of the economic design model.

Cumulative count of conforming chart with variable sampling intervals

The control chart with cumulative count of conforming (CCC) items has shown to be useful for process monitoring in automated and discrete manufacturing. It does not require a fixed sample size and can be useful for one-at-a-time data. The current research relies on data from the fixed sampling interval (FSI) scheme. In this paper, we investigate the use of the CCC chart when the variable sampling interval scheme is used. The average time to signal (ATS) is calculated, and its efficiency is compared with that of FSI CCC chart. The use of variable sampling interval scheme may further enhance the cost effectiveness of control chart implementation from a practical point of view.

Designing Accurate Control Charts Based on the Geometric and Negative Binomial Distributions

Attribute control charts are used effectively to monitor for process change. Their accuracy can be improved by judiciously selecting the sample size. The required sample sizes to achieve accuracy can be quite restrictive, especially when the nominal proportions of non-conforming units are quite small. The usual attribute control chart has a set sample size and the number of non-conforming units in the sample is plotted. If, instead of setting a specific sample size the number of non-conforming units is set, an alternative monitoring process is possible. Specifically, the cumulative count of conforming (CCC-r) control chart is a plot of the number of units that must be tested to find the rth non-conforming unit. These charts, based on the geometric and negative binomial distributions, are often suggested for monitoring very high quality processes. However, they can also be used very efficiently to monitor processes of lesser quality. This procedure has the potential to find process deterioration more quickly and efficiently. Xie et al. (Journal of Quality and Reliability Management 1999; 16(2):148–157) provided tables of control limits for CCC-r charts for but focused mainly on high-quality processes and the tables do not include any assessments of the risk of a false alarm or the reliability of detecting process change. In this paper, these tables are expanded for processes of lesser quality and include such assessments using the number of expected monitoring periods (average run lengths (ARLs)) to detect process change. Also included is an assessment of the risk of a false alarm, that is, a false indication of process deterioration. Such assessments were not included by Xie et al. but are essential for the quality engineer to make sound decisions. Furthermore, a hybrid of the control charts based on the binomial, geometric and negative binomial distributions is proposed to monitor for process change. Copyright © 2005 John Wiley & Sons, Ltd.

Economic Design of A Dynamic CCC – r Chart for High-Yield Processes

Recently, Ohta et al. [Quality and Reliability Engineering International 17: 439-446, 2001] have studied the economic design of CCC(Cumulative Count of Conforming)-r charts for high-yield processes assuming a fixed hazard rate. Generally, however, the hazard rate is varying over time. With the ageing of a process, the hazard rate gets first gradually smaller, while during the last stage of the wear-out process, the hazard rate increases. For such processes, the Weibull distribution is used to model the circumstances. In this paper, we discuss an economic design of a dynamic CCC – r chart with time-varying parameters. Concretely, we propose a process control model for a Weibull distributed-shock model and determine the initial values and dynamic decision rules for the time-varying parameters of the CCC–r chart, that is, the required number of nonconforming observations r, the sampling interval h, and the lower control limit LCL maximizing the expected profit per unit time derived from the process. Finally, we compare the profits obtained applying a dynamic CCC – r chart with those obtained with a traditional static CCC – r chart.

Optimal control limits for CCC charts in the presence of inspection errors

AbstractThe control chart based on cumulative count of conforming (CCC) items between the occurrence of two non‐conforming ones, or the CCC chart, has been shown to be very useful for monitoring high‐quality processes. However, as in the implementation of other Shewhart‐type control charts, it is usually assumed that the inspection is free of error. This assumption may not be valid and this may have a significant impact on the interpretation of the control chart and the setting of control limits. This paper first investigates the effect of inspection errors and discusses the setting of control limits in such cases. Even if inspection errors are considered, the average time to alarm increases in the beginning when the process deteriorates. Since this is undesirable, the control limits in the presence of inspection errors should be set so as to maximize the average run length when the process is at the normal level. A procedure is presented for solving this problem. Copyright © 2003 John Wiley & Sons, Ltd.

On economic design of cumulative count of conforming chart

Control charts based on Cumulative Count of Conforming (CCC) have been shown to be useful in high-quality processes. In the design stage of any chart in manufacturing industry, economic factors should be taken into account. In this paper, an economic model for CCC chart design is developed. A simplified algorithm is used to search the optimal setting of the sampling and control parameters. Sensitivity analysis is presented and numerical examples are used to illustrate the procedure. Compared with related studies, the model presented here could produce lower cost and smaller Type II error than the existing approach and the procedure is easy to implement.