Monitoring Bernoulli processes considering measurement errors and learning effect

This paper presents the effect of measurement errors and learning on monitoring processes with individual Bernoulli observations. A cumulative sum control chart is considered to evaluate the possible impacts of measurement errors and learning. We propose a time‐dependent learning effect model along with measurement errors and incorporate them into the Bernoulli CUSUM control chart statistic. The performance of the Bernoulli CUSUM control chart is then merely assessed by comparing the average number of observations to signal (ANOS) under two proposed conditions with the condition of no possible errors. Thus, the ANOS values are obtained under different proportions of non‐conforming items, once considering errors due to measurement by inspectors, and once considering both errors and learning effect together. The experimental results show that the efficiency of the control chart to detect assignable causes deteriorates in the presence of measurement errors and enhances when learning affects operators' performance. The proposed approach has a potential to be used in monitoring high‐quality Bernoulli processes as well as disease diagnosis, and other health care applications with Bernoulli observations.     

An np‐control chart for inspection errors and repeated classifications

The np‐control chart has been used to monitor the conforming fraction in process production, and it is assumed that no classification errors occur during the inspection process. Increases in the sample size and/or the number of repeated classifications of the inspected items can reduce the impact of the classification errors. In this paper, an np ‐control chart is proposed, and the monitored statistics are based on the results of independent repeated classifications with classification errors during the inspection process. The aim of the proposed control chart is to have the same performance as a control chart without classification errors. Numerical examples illustrate the proposal. Copyright © 2011 John Wiley & Sons, Ltd.     

On the performance of coefficient of variation charts in the presence of measurement errors

In the literature, coefficient of variation control charts have been introduced under the assumption of no measurement errors. However, measurement errors always exist in practice, and they do affect the performance of control charts in the detection of an out‐of‐control situation. In this paper, we therefore study the performance of a coefficient of variation Shewhart‐type control chart (Shewhart‐CV chart) and also one‐sided coefficient of variation exponentially weighted moving average–type control charts (EWMA‐γ² charts) using a model with linear covariates. Moreover, we propose and study the performance of a two‐sided EWMA‐γ² chart using a model with linear covariates. Several figures and tables are provided and analyzed to evaluate the statistical performance of these control charts for different sources of measurement errors. The obtained results show that the precision and accuracy errors significantly affect the performance of both the Shewhart‐CV and EWMA‐γ² control charts. An example illustrating the use of this study is finally presented.     

Acceptance control and guardbanding for error‐prone individual measurements

The concept of fractional nonconformance was recently proposed to assess the probability of conformance when measurements are error‐prone. Applications of fractional nonconformance assessment include acceptance sampling inspection and short‐run process control. In this study, we introduce a fractional nonconformance‐based one‐sided acceptance control chart to monitor a short‐run process as well as decide the acceptability of products manufactured from the process. The guardband technique is also incorporated in the proposed approach to reduce the impact of measurement errors. Guardband selection is investigated for both independent and autocorrelated processes. Our analysis shows that guardbanding is beneficial for short‐run production environments. The optimum guardbands obtained under risk and cost models are also found to be consistent.     

On the Performance of Shewhart median Chart in the Presence of Measurement Errors

In the literature, median control charts have been introduced under the assumption of no measurement error. However, measurement errors always exist in practice and may considerably affect the ability of control charts to detect out‐of‐control situations. In this paper, we investigate the performance of Shewhart median chart by using a linear covariate error model. Several figures and tables are presented and commented to show the statistical performance of Shewhart median control chart in the presence of measurement errors. We also investigate the positive effect of taking multiple measurements for each item in a subgroup on the performance of Shewhart median chart. An example illustrates the use of Shewhart median chart in the presence of measurement errors. Copyright © 2016 John Wiley & Sons, Ltd.     

Design of control charts with m‐of‐m runs rules

This paper investigates economic–statistical properties of the X? charts supplemented with m‐of‐m runs rules. An out‐of‐control condition for the chart is either a point beyond a control limit or a run of m‐of‐m successive points beyond a warning limit. The sampling process is modeled by a Markov chain with 2m states. The steady‐state probability for each state and the average run length (ARL) from each state of the Markov chain are derived in explicit formulas. Then the stationary average run length (SALR) is derived so as to develop an economic–statistical model. Using this model, the design parameters are optimized by minimizing the cost function with constraints on the average time to signal (ATS). The X? chart supplemented with m‐of‐m runs rules is compared with the Shewhart X? chart in terms of the SARL and the cost function. Sensitivity of the design parameters with respect to the cost function is also analyzed. General guidelines for implementing the X? chart with m‐of‐m runs rules are presented from those observations. It should be emphasized that supplementing run rules may provide feasible and efficient solutions even if the sample size is limited, while the Shewhart X? chart may not. Copyright © 2009 John Wiley & Sons, Ltd.     

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

The 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 t and EWMA t charts for monitoring changes in the process mean

The performance of an X‐bar chart is usually studied under the assumption that the process standard deviation is well estimated and does not change. This is, of course, not always the case in practice. We find that X‐bar charts are not robust against errors in estimating the process standard deviation or changing standard deviation. In this paper we discuss the use of a t chart and an exponentially weighted moving average (EWMA) t chart to monitor the process mean. We determine the optimal control limits for the EWMA t chart and show that this chart has the desired robustness property. Copyright © 2009 John Wiley & Sons, Ltd.     

M‐ATTRIVAR: An attribute‐variable chart to monitor multivariate process means

In this paper, an attribute‐variable control chart, namely, M‐ATTRIVAR, is introduced to monitor possible shifts in a vector of means. The monitoring starts using an attribute chart (classifying the units as approved or not using a gauge) and continues in such a way until a warning signal is given, shifting the control to a variable chart for the next sampling. If the variable chart does not confirm the warning, the monitoring returns to an attribute control. Otherwise, the monitoring remains with the variable chart. Whenever any of the charts (attribute or variable) signals an alarm, the control scheme triggers an alarm. The main advantage of this new proposal is the possibility of judging the state of the process only by the attribute chart most of the time (normally more economical and faster). The performance of the M‐ATTRIVAR control chart is compared versus the main competitor (T² control chart) in terms of performance detection (out‐of‐control average run length) but also economically (average sampling cost). The M‐ATTRIVAR is always cheaper than T², and in many scenarios, it detects quicker process shifts than the T² control chart. A numerical example illustrates a practical situation.     

A Synthetic median control chart for monitoring the process mean with measurement errors

Recent studies show that the Shewhart median chart is widely used for detecting shifts in a process, but it is often rather inefficient in detecting small or moderate process shifts. In order to overcome this problem, a Synthetic chart can be used. This chart outperforms the Shewhart‐type chart because it uses the information about the time interval between two consecutive nonconforming samples. In this paper, we propose and study the Phase II Synthetic median control chart. A Markov chain methodology is used to evaluate the statistical performance of the proposed chart. Moreover, its performance is investigated in the presence of measurement errors, which are modelled by a linear covariate error model. We provide the results of an extensive numerical analysis with several tables and figures in order to show the statistical performance of the investigated chart, for both cases of measurement errors and no measurement errors. Finally, an example illustrates the use of the Synthetic median chart.     

A more accurate formulation for a class of modelsfor the economic design of control charts

Some of the published models for the economic design of control charts use the expected cost per unit of output as the objective function to be minimized. In these models, the computation of steady-state probabilities that the process is in each possible state does not take into account the effect of the corrective action that may follow a signal from the control chart. As a result, the expected cost per unit is overestimated and the selection of the chart parameters is not optimal. The purpose of this paper is to eliminate this inaccuracy by proposing the exact formulation and to estimate the magnitude of errors resulting from the inaccurate formulation of the objective function. By solving numerical examples of joint design of X and R charts, it is shown that these errors are typically very large and consequently it is imperative to use the exact formulation, proposed in this paper, to avoid inefficient and costly control chart designs. Finally, an additional opportunity for further cost improvements in process control is identified and discussed.     

A CCC‐r chart for high‐yield processes

The cumulative count of a conforming (CCC) chart is used to monitor high‐quality processes and is based on the number of items inspected until observing r non‐conforming ones. This charting technique is known as a CCC‐r chart. The function of the CCC‐r chart is the sensitive detection of an upward shift in the fraction defectives of the process, p. As r gets larger, the CCC‐r chart becomes more sensitive to small changes of upward shift in p. However, since many observations are required to obtain a plotting point on the chart, the cost is fairly high. For this trade‐off problem it is necessary to determine the optimal number of non‐conforming items observed before a point is plotted, the sampling (inspection) interval, and the lower control limit for the chart. In this paper a simplified optimal design method is proposed. For illustrative purposes, some numerical results for the optimal design parameter values are provided. The expected profits per cycle obtained using the proposed optimal design method are compared with those obtained using other misspecified parameter values. The effects of changing these parameters on the profit function are shown graphically. Copyright © 2001 John Wiley & Sons, Ltd.     

Linear profiles monitoring in the presence of nonnormal random errors

Profile monitoring is a technique to test the stability of the relationship between a response variable and explanatory variables over time. The most relevant linear profile monitoring methods have been constructed using the normality assumption. However, the normality assumption could be violated in many quality control applications. In this study, we consider a situation in which the random errors in a linear profile model follow a skew‐normal distribution. The skew‐normal distribution is a generalized version of the normal distribution. A new Shewhart‐type chart and exponentially weighted moving average (EWMA) chart, named the Shewhart^R and EWMA^R charts, respectively, are constructed based on residuals to monitor the parameters of linear profile model. The simulation results show that the multivariate EWMA chart is sensitive to the normality assumption and that the proposed Shewhart^R and EWMA^R charts have good ability to detect big and small‐to‐moderate process shifts, respectively. An example using photo mask techniques in semiconductor manufacturing is provided to illustrate the applications of the Shewhart^R and EWMA^R charts.     

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.     

On Modified Successive Sampling Based Control Charting Schemes

The presence of variation cannot be avoided in different kinds of manufacturing and non‐manufacturing processes. A better understanding of the causes of variability in any process is necessary to improve the process. Control chart is a very frequently used tool for checking whether the process parameters are stable or not. The current study devises a sampling technique, named as modified successive sampling scheme, that is not only cost‐effective but also efficient as compared with the simple random sampling scheme. A number of Shewhart‐type control charts are proposed based on the said sampling scheme, and average run length is used as a performance indicator. Based on the average run length values, all the proposed charts are compared with existing Shewhart control chart for both positive and negative shifts in the process. Finally, the new proposals are applied to a real dataset where the variable of interest is an inner diameter of automobile engine piston rings made of steel. Copyright © 2015 John Wiley & Sons, Ltd.     

A New Maximum EWMA Control Chart for Simultaneously Monitoring Process Mean and Dispersion Using Auxiliary Information

The maximum exponentially weighted moving average (MaxEWMA) control charts have gained considerable attention for simultaneously detecting both increases and decreases in the mean and/or dispersion of a process. In this paper, we propose a new auxiliary information‐based (AIB) MaxEWMA control chart, called the AIB‐MaxEWMA chart. The AIB‐MaxEWMA chart encompasses the existing MaxEWMA chart. Extensive Monte Carlo simulations are performed to evaluate the average run length, standard deviation of the run length, and diagnostic abilities of the AIB‐MaxEWMA chart. An extensive comparison reveals that the AIB‐MaxEWMA chart performs uniformly better than the MaxEWMA chart. An example is also used to explain the implementation and working of the AIB‐MaxEWMA chart. Copyright © 2017 John Wiley & Sons, Ltd.     

Design of an X̅ control chart with guaranteed joint in‐control and out‐of‐control performances when the mean and variance are estimated

During phase I analysis, the mean and variability of the process are estimated, in order to obtain the control limits for the chart. The effect of the estimation errors in the performance of the chart (phase II) has been widely studied in the literature. However, the papers have been mainly orientated to the in‐control performance of the chart. In this paper, the in‐control and out‐of‐control performances of the control chart are considered at the same time. Therefore, the number of samples to be taken in the phase I analysis is calculated, taking into account the possible deterioration of both metrics. In addition, an adjustment in the control limit is proposed, to reduce this number of samples.     

A New Synthetic Exponentially Weighted Moving Average Control Chart for Monitoring Process Dispersion

Exponentially weighted moving average (EWMA) control charts have been widely recognized as a potentially powerful process monitoring tool of the statistical process control because of their excellent speed in detecting small to moderate shifts in the process parameters. Recently, new EWMA and synthetic control charts have been proposed based on the best linear unbiased estimator of the scale parameter using ordered ranked set sampling (ORSS) scheme, named EWMA‐ORSS and synthetic‐ORSS charts, respectively. In this paper, we extend the work and propose a new synthetic EWMA (SynEWMA) control chart for monitoring the process dispersion using ORSS, named SynEWMA‐ORSS chart. The SynEWMA‐ORSS chart is an integration of the EWMA‐ORSS chart and the conforming run length chart. Extensive Monte Carlo simulations are used to estimate the run length performances of the proposed control chart. A comprehensive comparison of the run length performances of the proposed and the existing powerful control charts reveals that the SynEWMA‐ORSS chart outperforms the synthetic‐R, synthetic‐S, synthetic‐D, synthetic‐ORSS, CUSUM‐R, CUSUM‐S, CUSUM‐ln S², EWMA‐ln S² and EWMA‐ORSS charts when detecting small shifts in the process dispersion. A similar trend is observed when the proposed control chart is constructed under imperfect rankings. An application to a real data is also provided to demonstrate the implementation and application of the proposed control chart. Copyright © 2014 John Wiley & Sons, Ltd.     

A Bayesian model for the joint optimization of quality and maintenance decisions

We develop a model for the economic design of a Bayesian control chart for monitoring a process mean. The process may randomly suffer failures that result in a non‐operating state, and thus call for an immediate corrective maintenance action, as well as assignable causes that shift the process mean to an undesirable level. Quality shifts, apart from poorer quality outcome and higher operational cost, also result in higher failure rate. Consequently, their removal, besides improving the outcome quality and reducing the quality‐related cost, is also a preventive maintenance action since it reduces the probability of a failure. The proposed Bayesian model allows the determination of the design parameters that minimize the total expected quality and maintenance cost per time unit. The effectiveness of the proposed model is evaluated through the comparison of its expected cost against the optimum expected cost of the simpler variable‐parameter Shewhart chart. Copyright © 2010 John Wiley & Sons, Ltd.     

A Robust Control Chart

This article studies alternative standard deviation estimators that serve as a basis to determine the control chart limits used for real‐time process monitoring (phase II). Several existing (robust) estimation methods are considered. In addition, we propose a new estimation method based on a phase I analysis, that is, the use of a control chart to identify disturbances in a data set retrospectively. The method constructs a phase I control chart derived from the trimmed mean of the sample interquartile ranges, which is used to identify out‐of‐control data. An efficient estimator, namely the mean of the sample standard deviations, is used to obtain the final standard deviation estimate from the remaining data. The estimation methods are evaluated in terms of their mean squared errors and their effects on the performance of the phase II control chart. It is shown that the newly proposed estimation method is efficient under normality and performs substantially better than standard methods when disturbances are present in phase I. Copyright © 2012 John Wiley & Sons, Ltd.