About Shewhart control charts to monitor the Weibull mean

In this paper, a new reparametrization expressed in terms of the process mean for Weibull distribution is studied; thus, the monitoring of the process mean can be made directly. Additionally, we call attention that the asymptotic control limits for control chart by central limit theorem (CLT) may lead to a serious erroneous decision. Definitively, they can only be used to signal small/medium shifts in the process mean but with a very very large sample size. We present guidelines for practitioners about the minimum sample size needed to match out‐of‐control average run length (ARL₁) with the exact and asymptotic control limits in function of the shape parameter after an extensive simulation study. The proposed schemes are applied to monitoring the Weibull mean parameter of the strength distribution of a carbon fibber used in composite materials.     

Evaluation of Shewhart time-between-events-and-amplitude control charts for correlated data

In recent years, several techniques based on control charts have been developed for the simultaneous monitoring of the time interval T and the amplitude X of events, known as time-between-events-and-amplitude (TBEA) charts. However, the vast majority of the existing works have some limitations. First, they usually focus on statistics based on the ratio $\frac{X}{T}$, and second, they only investigate a reduced number of potential distributions, that is, the exponential distribution for T and the normal distribution for X. Moreover, until now, very few research papers have considered the potential dependence between T and X. In this paper, we investigate three different statistics, denoted as Z₁ , Z₂ , and Z₃ , for monitoring TBEA data in the case of three potential distributions (gamma, normal, and Weibull), for both T and X, using copulas as a mechanism to model the dependence. An illustrative example considering times between machine breakdowns and associated maintenance illustrates the use of TBEA control charts.     

Shewhart and EWMA t control charts for short production runs

Short‐run productions are common in manufacturing environments like job shops, which are characterized by a high degree of flexibility and production variety. Owing to the limited number of possible inspections during a short run, often the Phase I control chart cannot be performed and correct estimates for the population mean and standard deviation are not available. Thus, the hypothesis of known in‐control population parameters cannot be assumed and the usual control chart statistics to monitor the sample mean are not applicable. t‐charts have been recently proposed in the literature to protect against errors in population standard deviation estimation due to the limitation of available sampling measures. In this paper the t‐charts are tested for implementation in short production runs to monitor the process mean and their statistical properties are evaluated. Statistical performance measures properly designed to test the chart sensitivity during short runs have been considered to compare the performance of Shewhart and EWMA t‐charts. Two initial setup conditions for the short run fixing the population mean exactly equal to the process target or, alternatively, introducing an initial setup error influencing the statistic distribution have been modelled. The numerical study considers several out‐of‐control process operating conditions including one‐step shifts for the population mean and/or standard deviation. The obtained results show that the t‐charts can be successfully implemented to monitor a short run. Finally, an illustrative example is presented to show the use of the investigated t charts. Copyright © 2010 John Wiley & Sons, Ltd.     

Monitoring count data with Shewhart control charts based on the Touchard model

The most common control chart used to monitor count data is based on Poisson distribution, which presents a strong restriction: The mean is equal to the variance. To deal with under- or overdispersion, control charts considering other count distributions as Negative Binomial (NB) distribution, hyper-Poisson, generalized Poisson distribution (GPD), Conway–Maxwell–Poisson (COM-Poisson), Poisson–Lindley, new generalized Poisson–Lindley (NGPL) have been developed and can be found in the literature. In this paper we also present a Shewhart control chart to monitor count data developed on Touchard distribution, which is a three-parameter extension of the Poisson distribution (Poisson distribution is a particular case) and in the family of weighted Poisson models. Its normalizing constant is related to the Touchard polynomials, hence the name of this model. It is a flexible distribution that can account for both under- or overdispersion and concentration of zeros that are frequently found in non-Poisson count data. Consequences in terms of speed to signal departures of stability of the parameters are obtained when incorrect control limits based on non-Touchard distribution (like Poisson, NB or COM-Poisson) are used to monitor count data generated by a Touchard distribution. Numerical examples illustrate the current proposal.     

Correction factors for Shewhart and control charts to achieve desired unconditional ARL

In this paper we derive correction factors for Shewhart control charts that monitor individual observations as well as subgroup averages. In practice, the distribution parameters of the process characteristic of interest are unknown and, therefore, have to be estimated. A well-known performance measure within Statistical Process Monitoring is the expectation of the average run length (ARL), defined as the unconditional ARL. A practitioner may want to design a control chart such that, in the in-control situation, it has a certain expected ARL. However, accurate correction factors that lead to such an unconditional ARL are not yet available. We derive correction factors that guarantee a certain unconditional in-control ARL. We use approximations to derive the factors and show their accuracy and the performance of the control charts – based on the new factors – in out-of-control situations. We also evaluate the variation between the ARLs of the individually estimated control charts.     

Simple adjustments to improve control limits on attribute charts

Many SPC software packages determine control chart limits for attribute data using normal approximations. For some sample sizes and/or process parameter values these approximations are far from adequate, mainly due to skewness in the exact distribution. Significant improvements can be made to the probabilistic accuracy of control limits by simple adjustments derived from Cornish and Fisher expansions for percentage points. The adjustments are preferable to normalizing transformations in that the original scale of the data is retained and most SPC packages allow self-determined control limits to be inserted. The overall objective is to bring operational meaning and practice for attribute charts more into line with charts for variables.     

A clustering approach to identify the time of a step change in Shewhart control charts

Control charts are the most popular statistical process control tools used to monitor process changes. When a control chart indicates an out‐of‐control signal it means that the process has changed. However, control chart signals do not indicate the real time of process changes, which is essential for identifying and removing assignable causes and ultimately improving the process. Identifying the real time of the change is known as the change‐point estimation problem. Most of the traditional methods of estimating the process change point are developed based on the assumption that the process follows a normal distribution with known parameters, which is seldom true. In this paper, we propose clustering techniques to estimate Shewhart control chart change points. The proposed approach does not depend on the true values of the parameters and even the distribution of the process variables. Accordingly, it is applicable to both phase‐I and phase‐II of normal and non‐normal processes. At the end, we discuss the performance of the proposed method in comparison with the traditional procedures through extensive simulation studies. Copyright © 2008 John Wiley & Sons, Ltd.     

Visualizing when prior information can reduce the necessary size of a reference sample when estimating in‐control parameters for Shewhart and CUSUM charts for a normal process

We consider the construction of Shewhart and cumulative sum (CUSUM) charts for a normal process when the in‐control mean and standard deviation must be estimated from a reference sample. Unless the reference sample size is extremely large, substituting the unknown mean and standard deviation with the sample estimates from the reference sample will introduce alarming variability in the conditional (on the reference sample estimates) in‐control average run length. In most applications, some prior information for the mean and standard deviation is available. We investigate how effective Bayes estimators can be in terms of reducing the necessary size of the reference sample. We consider both uniform and conjugate priors, and we show how to construct a graphic that depicts the relationship between the precision in the prior and the relative error in the conditional in‐control average run length.     

Shewhart control charts to detect mean and standard deviation shifts based on grouped data

A Shewhart control chart is proposed based on gauging theoretically continuous observations into multiple groups. This chart is designed to monitor the process mean and standard deviation for deviations from stability. By assuming an underlying normal distribution, we derive the optimal grouping criterion that maximizes the expected statistical information available in a sample. Control charts based on grouped observations are superior to standard control charts based on variables, such as X and R charts, when the quality characteristic is difficult or expensive to measure precisely, but economical to gauge.     

An exact method for designing Shewhart and S2 control charts to guarantee in-control performance

The in-control performance of Shewhart and S² control charts with estimated in-control parameters has been evaluated by a number of authors. Results indicate that an unrealistically large amount of Phase I data is needed to have the desired in-control average run length (ARL) value in Phase II. To overcome this problem, it has been recommended that the control limits be adjusted based on a bootstrap method to guarantee that the in-control ARL is at least a specified value with a certain specified probability. In this article we present simple formulas using the assumption of normality to compute the control limits and therefore, users do not have to use the bootstrap method. The advantage of our proposed method is in its simplicity for users; additionally, the control chart constants do not depend on the Phase I sample data.     

The effects of violations of the multivariate normality assumption in multivariate Shewhart and MEWMA control charts

A multivariate Shewhart and a multivariate exponentially weighted moving average control charts are types of multivariate control charts for monitoring the mean vector. For those control charts, a multivariate normal distribution is an important assumption that is used to describe a behavior of a set of quality characteristics of interest. This research explores the sensitivity of average run lengths and standard deviation of run lengths for the multivariate Shewhart and the multivariate exponentially weighted moving average control charts when the normality assumption is incorrect.     

Detecting changes in location using distribution‐free control charts with big data

This paper proposes a simple distribution‐free control chart for monitoring shifts in location when the process distribution is continuous but unknown. In particular, we are concerned with big data applications where there are sufficient in‐control data that can be used to specify certain quantiles of interest which, in turn, are used to assess whether the new, incoming data to be monitored are in control. The distribution‐free chart is shown to lose very little power against the Shewhart charts designed for normally distributed data. The proposed charts offer a practical and robust alternative to the classical Shewhart charts which assume normality, particularly when monitoring quantiles and the data distribution is skewed. The effect of the size of the reference sample is examined on the assumption that the quantiles are known. Conclusions and recommendations are offered.     

Effect of the measurement errors on two one-sided Shewhart control charts for monitoring the ratio of two normal variables

Monitoring the ratio between two random normal variables plays an important role in many industrial manufacturing processes. In this paper, we suggest designing two one-sided Shewhart control charts monitoring this ratio. The numerical results show that the one-sided charts have more advantages compared with the two-sided Shewhart chart proposed previously in the literature. Moreover, we investigate the effect of measurement error on the performance of these control charts where the measurement error is supposed to follow a linear covariate error model. The change of model parameters from an in-control condition to an out-of-control is presented without using a strict assumption about the independence of the shift size from measurement errors. A valuable finding from this study is that taking multiple measurements per item is not an effective way to reduce the negative effect of measurement error on the Shewhart charts' performance.     

Percentile‐based control chart design with an application to Shewhart X̅ and S2 control charts

We present a method to design control charts such that in‐control and out‐of‐control run lengths are guaranteed with prespecified probabilities. We call this method the percentile‐based approach to control chart design. This method is an improvement over the classical and popular statistical design approach employing constraints on in‐control and out‐of‐control average run lengths since we can ensure with prespecified probability that the actual in‐control run length exceeds a desired magnitude. Similarly, we can ensure that the out‐of‐control run length is less than a desired magnitude with prespecified probability. Some numerical examples illustrate the efficacy of this design method.     

Using statistical process control charts for the continual improvement of asthma care

BACKGROUND: Home monitoring of lung function using simple, inexpensive tools to measure peak expiratory flow rate (PEFR) has been possible since the 1970s. Yet although current national and international guidelines recommend monitoring of PEFRs via traditional run charts, their use by both patients and physicians remains low. The role of statistical process control (SPC) theory and charts in the serial monitoring of lung function at home were explored and applied to the direct care of patients with asthma. The method represents an integration of collective professional and improvement knowledge with the related disciplines of continual improvement, SPC, system thinking/system dynamics, paradigms, and the learning community/organization. CASE STUDIES: Use of PEFR control charts for four patients cared for at the Asthma-Allergy Clinic and Research Center (Shreveport, La) is described. The key to good asthma control is the ability to optimize lung function by reducing the variation between serial lung function measurements and thereby generate a safe range of function. Knowledge of the type of variation (special cause or common cause) in the system helps in focusing clinical decision making. Case 4, an 11-year-old boy, for example, shows how control charts were used to learn the effects of a new inhaled corticosteroid. Comparison of the last 14 days of baseline and the last 14 days of open label use of the inhaled corticosteroid showed an obvious improvement in actual PEFR values--which a run chart or comparison of means would have easily demonstrated. The control chart showed that this child's care process at baseline was functionally at risk for severe asthma (46% personal best) and that the effect of the new medication not only elevated the mean function but shifted the range of function from 46%-72% personal best to 78%-102% personal best. At this new range of function the patient's system of care was not capable of delivering values that are at risk for severe asthma. Unless the range of function the change in care is capable of producing is specifically quantitated, misinterpretation of improvement data can occur. DISCUSSION: Developing the concept of the PEFR control chart involved examining and challenging traditional mental models for monitoring PEFR at home in the care of asthma, acquiring a better understanding of the workings of dynamic systems and with system thinking, and sharing what was learned with patients and seeking their input. CONCLUSIONS: The PEFR control chart employs an interesting statistical platform that enables the integration of knowledge of serial measurements and knowledge of the variation between those measurements into a tool with which to better assess the asthma care process being followed. This tool provides clinical insights, practical knowledge, and opportunities unavailable to patients and physicians via traditional PEFR charting.     

Phase II control charts for monitoring dispersion when parameters are estimated

Shewhart control charts are among the most popular control charts used to monitor process dispersion. To base these control charts on the assumption of known in-control process parameters is often unrealistic. In practice, estimates are used to construct the control charts and this has substantial consequences for the in-control and out-of-control chart performance. The effects are especially severe when the number of Phase I subgroups used to estimate the unknown process dispersion is small. Typically, recommendations are to use around 30 subgroups of size 5 each.

?We derive and tabulate new corrected charting constants that should be used to construct the estimated probability limits of the Phase II Shewhart dispersion (e.g., range and standard deviation) control charts for a given number of Phase I subgroups, subgroup size and nominal in-control average run-length (ICARL). These control limits account for the effects of parameter estimation. Two approaches are used to find the new charting constants, a numerical and an analytic approach, which give similar results. It is seen that the corrected probability limits based charts achieve the desired nominal ICARL performance, but the out-of-control average run-length performance deteriorate when both the size of the shift and the number of Phase I subgroups are small. This is the price one must pay while accounting for the effects of parameter estimation so that the in-control performance is as advertised. An illustration using real-life data is provided along with a summary and recommendations.     

Optimisation design of attribute control charts for multi-station manufacturing system subjected to quality shifts

The qualities of products are a major concern in any production system; thus implementing efficient inspection policies is of great importance to reduce quality-related costs. This article addresses the problem of finding optimal inspection policies for the multi-station manufacturing system (MMS) subjected to quality shifts to minimise total quality-related cost. Each station of the MMS may stay at either in-control condition or out-of-control condition, which may lead to different nonconforming product rates. Markov chain method is used to calculate the steady-state probability distribution (SSPD). Based on the SSPD, the cost structure of this MMS is analysed. The economical optimisation model of attribute control charts (ACCs) is then established, in which the decision variables are the control chart parameters: sampling interval, sample size and control limit. The ACCs optimisation model is resolved by the proposed integrated algorithm combining heuristic rule and tabu search. This approach is verified through an application case taken from a mobile phone shell production company. The results of comparative analysis show that the proposed model is much more economical than both the current outgoing inspection strategy and the regular np control chart. The sensitivity analysis of four input parameters is also conducted.     

Monitoring automatically controlled processes using statistical control charts

Statistical process control (SPC) and automatic process control (APC) are integrated under a hierarchical scheme. The integrated scheme is called statistical and automatic process control (SAPC). SAPC is based on the comparison of the outputs of the actual system and parallel systems that model the actual system under assumed deterministic disturbances. While APC provides a continuous controller intervention, SPC acts as a supervisory controller to detect process disturbances. A cross-correlation method is used to estimate the time when the disturbance starts to affect the system as well as the type of the disturbance. After the magnitude of the disturbance is determined, a counter action is applied on the system to minimize its effect.     

Monitoring multivariate coefficient of variation with upward Shewhart and EWMA charts in the presence of measurement errors using the linear covariate error model

In practice, measurement errors exist and ignoring their presence may lead to erroneous conclusions in the actual performance of control charts. The implementation of the existing multivariate coefficient of variation (MCV) charts ignores the presence of measurement errors. To address this concern, the performances of the upward Shewhart-MCV and exponentially weighted moving average MCV charts for detecting increasing MCV shifts, using a linear covariate error model, are investigated. Explicit mathematical expressions are derived to compute the limits and average run lengths of the charts in the presence of measurement errors. Finally, an illustrative example using a real-life dataset is presented to demonstrate the charts’ implementation.