A New Distribution‐free Control Chart for Joint Monitoring of Unknown Location and Scale Parameters of Continuous Distributions

While the assumption of normality is required for the validity of most of the available control charts for joint monitoring of unknown location and scale parameters, we propose and study a distribution‐free Shewhart‐type chart based on the Cucconi 1 statistic, called the Shewhart‐Cucconi (SC) chart. We also propose a follow‐up diagnostic procedure useful to determine the type of shift the process may have undergone when the chart signals an out‐of‐control process. Control limits for the SC chart are tabulated for some typical nominal in‐control (IC) average run length (ARL) values; a large sample approximation to the control limit is provided which can be useful in practice. Performance of the SC chart is examined in a simulation study on the basis of the ARL, the standard deviation, the median and some percentiles of the run length distribution. Detailed comparisons with a competing distribution‐free chart, known as the Shewhart‐Lepage chart (see Mukherjee and Chakraborti 2 ) show that the SC chart performs just as well or better. The effect of estimation of parameters on the IC performance of the SC chart is studied by examining the influence of the size of the reference (Phase‐I) sample. A numerical example is given for illustration. Summary and conclusions are offered. Copyright © 2013 John Wiley & Sons, Ltd.     

Distribution‐free Lepage Type Circular‐grid Charts for Joint Monitoring of Location and Scale Parameters of a Process

In the last 5 years, research works on distribution‐free (nonparametric) process monitoring have registered a phenomenal growth. A Google Scholar database search on early September 2015 reveals 246 articles on distribution‐free control charts during 2000–2009 and 466 articles in the following years. These figures are about 1400 and 2860 respectively if the word ‘nonparametric’ is used in place of ‘distribution‐free’. Distribution‐free charts do not require any prior knowledge about the process parameters. Consequently, they are very effective in monitoring various non‐normal and complex processes. Traditional process monitoring schemes use two separate charts, one for monitoring process location and the other for process scale. Recently, various schemes have been introduced to monitor the process location and process scale simultaneously using a single chart. Performance advantages of such charts have been clearly established. In this paper, we introduce a new graphical device, namely, circular‐grid charts, for simultaneous monitoring of process location and process scale based on Lepage‐type statistics. We also discuss general form of Lepage statistics and show that a new modified Lepage statistic is often better than the traditional of Lepage statistic. We offer a new and attractive post‐signal follow‐up analysis. A detailed numerical study based on Monte‐Carlo simulations is performed, and some illustrations are provided. A clear guideline for practitioners is offered to facilitate the best selection of charts among various alternatives for simultaneous monitoring of location‐scale. The practical application of the charts is illustrated. Copyright © 2016 John Wiley & Sons, Ltd.     

A combination of max‐type and distance based schemes for simultaneous monitoring of time between events and event magnitudes

Traditionally, two isolated sequential stopping rules are employed for monitoring the time of occurrence of an event (T) and the magnitude of an event (X) . Recently, several researchers recommend monitoring T and X together using some unified approach. A unified approach based on combinations of two statistics, one for monitoring T and the other for X , is often more efficient. Likewise, a new approach of simultaneous monitoring of location and scale parameters of a process, combining a max and a distance based statistics, is recently introduced in literature. Motivated by such emerging concepts, we design a new scheme combining a Max‐type and a Distance‐type schemes, referred to as the MT scheme, to monitor T  and X simultaneously and efficiently. It retains the advantages of both the Max‐type and the Distance‐type schemes for joint inference. The proposed scheme is very competent in detecting a shift in the process distribution of T  or X or both. Moreover, it is computationally simpler. It has nice exact expressions for design parameters. Therefore, it is easier to implement. It has a distinct advantage over its traditional counterparts in detecting moderate to large shifts. Finally, we illustrate the implementation of the proposed scheme with a real dataset of damage caused by outbreak of fire disaster.     

A distribution-free CUSUM chart for joint monitoring of location and scale based on the combination of Wilcoxon and Mood statistics

We propose a distribution-free cumulative sum (CUSUM) chart for joint monitoring of location and scale based on a Lepage-type statistic that combines the Wilcoxon rank sum and the Mood statistics. Monte Carlo simulations were used to obtain control limits and examine the in-control and out-of-control performance of the new chart. A direct comparison of the new chart was made with the original CUSUM Lepage based on Wilcoxon rank sum and Ansari-Bradley statistics. The result is a more powerful chart in most of the considered scenarios and thus a more useful CUSUM chart. An example using real data illustrates how the proposed control chart can be implemented.     

Two economically optimized nonparametric schemes for monitoring process variability

Controlling and reducing process variability is an essential aspect for maintaining the product or service quality. Even though most practitioners believe that an increasing process variability is often a more severe concern than a shift in location, barely a few research paid attention to the cost-efficient monitoring of process variability. Some of the existing studies addressed the dispersion aspect, assuming that the quality characteristic is Gaussian. Non-normal and complex distributions are not uncommon in modern production processes, time to event processes, or processes involving service quality. Unfortunately, we find no literature on economically designed nonparametric (distribution-free) schemes for monitoring process variability. This article introduces two Shewhart-type cost-optimized nonparametric schemes for monitoring the variability of any unknown but continuous processes to fill the research gap. The proposed monitoring schemes are based on two popular two-sample rank statistics for differences in scale parameters, known as the Ansari–Bradley statistic and the Mood statistic. We assess their actual performance for a set of process scenarios and illustrate the design along with the implementation steps. We discuss a practical problem related to product quality management. It is expected that the proposed schemes will be beneficial in various industrial operations.     

CRPS chart: Simultaneously monitoring location and scale under data‐rich environment

The detection performance of a conventional control chart is usually degraded by a large sample size as in Wang and Tsung. This paper proposes a new control chart under data‐rich environment. The proposed chart is based on the continuous ranked probability score and aims to simultaneously monitor the location and the scale parameters of any continuous process. We simulate different monitoring schemes with various shift patterns to examine the chart performance. Both in‐control and out‐of‐control performances are studied through simulation studies in terms of the mean, the standard deviation, the median, and some percentiles of the average run length distribution. Simulation results show that the proposed chart keeps a high sensitivity to shifts in location and/or scale without any distributional assumptions, and the outperformance improves, as the sample size becomes larger. Examples are given for illustration.     

Misleading signals in joint schemes for the mean vector and covariance matrix

In multivariate statistical process control, it is recommendable to run two individual charts: one for the process mean vector and another one for the covariance matrix. The resulting joint scheme provides a way to satisfy Shewhart's dictum that proper process control implies monitoring both process location and spread. The multivariate quality characteristic is deemed to be out of control whenever a signal is triggered by either individual chart of the joint scheme. Consequently, a shift in the mean vector can be misinterpreted as a shift in the covariance matrix and vice versa. Compelling results are provided to give the quality control practitioner an idea of how joint schemes for the mean vector and covariance matrix are prone to trigger misleading signals that will likely lead to a incorrect diagnostic of which parameter has changed.     

A global scheme for controlling the mean and standard deviation of a process based on the optimal design of gauges

Control charts for variables are a reference tool for the statistical monitoring of a quantitative variable. However, there are processes in which the exact measurement of the variable is highly complex or costly. In these cases, one option is to base the monitoring on the number of units that are classified by a gauge as being above, below, or between a pair of reference limits. With this approach, the process control achieves the economy and agility of control by attributes. In the literature, one can find different alternatives to such schemes. In this paper, a more general formulation is introduced that subsumes many of the previously proposed schemes into a new control scheme by means of an additional parameter. Through a specially designed optimization procedure, it is possible to obtain the parameters that maximize the scheme's performance against a specified process shift.     

Simultaneous monitoring of origin and scale of a shifted exponential process with unknown and estimated parameters

The distribution of consumer lifetimes, high-voltage of current in semiconductor transistors, and the risk associated with monitoring health care often come with a threshold. A two-parameter (or shifted) exponential distribution is, in general, regarded as a better statistical model in such situations compared with a traditional (one-parameter) exponential model. Research on inferential problems associated with two-parameter exponential distributions, including monitoring schemes for the parameters of this model, is active. Currently, all existing monitoring schemes for origin and scale parameters of a shifted exponential distribution assume that the process parameters are known (Case-K). The actual values of the process parameters are, however, rarely known in practice. The traditional method of estimating parameters from a set of a (Phase-I) reference sample and plug them in for Phase-II monitoring affects the performance of a monitoring scheme. Skewed processes, like the two-parameter exponential process, exacerbate this problem. The present article shows that even a reference sample of size 50,000 cannot guarantee nominal in-control performances of monitoring schemes when the actual process parameters are unknown (Case-U). To address this problem, we develop monitoring schemes based on max and distance statistics for simultaneously monitoring the two parameters of a shifted exponential process in Case-U. We show that the proposed schemes perform well. We illustrate the practical application of the proposed procedures by analyzing data about the production of an electronic component.     

Design and comparison of some Shewhart‐type schemes for simultaneous monitoring of Weibull parameters

Weibull distribution is one of the most important probability models used in modeling time between events, system reliability, and particle sizes, among others. Therefore, efficiently and consequently monitoring certain changes in Weibull process is considered as an important research topic. Various statistical process monitoring schemes have been developed for monitoring different process parameters, including some for Weibull parameters. Most of these schemes are, however, designed to monitor and control a single process parameter, although there are two important model parameters for Weibull distribution. Recently, several researchers studied various schemes for jointly monitoring the mean and variance of a normally distributed process using a single plotting statistic. Nevertheless, there is still dearth of researches in joint monitoring of non‐normal process parameters. In this context, we develop some control schemes for simultaneously monitoring the scale and shape parameters of processes that follow the Weibull distribution. Implementation procedures are developed, and performance properties of various proposed schemes are investigated. We also offer an illustrative example along with a summary and recommendations.     

Two CUSUM schemes for simultaneous monitoring of parameters of a shifted exponential time to events

Two‐parameter (shifted) exponential distribution is widely applied in many areas such as reliability modeling and analysis where time to failure is protected by a guaranty period that induces an origin parameter in the exponential model. Despite a large volume of works on inferential aspects of two‐parameter exponential distribution, only few studies are done from the perspective of process monitoring. In the modern production process, where items come with a warranty, we often encounter shifted‐exponential time between events from consumers' perspective, and therefore, in this paper, we propose two CUSUM schemes for joint monitoring of the origin and scale parameters based on the Maximum Likelihood estimators. We study the in‐control behavior of the proposed procedures via Markov chain approach as well as applying Monte Carlo. We provide detailed implementation strategies of the two schemes along with the follow‐up procedures to identify the source of shifts when an out‐of‐control signal is obtained. We examine the performance properties of CUSUM schemes and find that the two proposed schemes offer performance advantages over the Shewhart‐type schemes especially for monitoring small to moderate shifts. Further, we provide some guidance for choosing the appropriate schemes and study the effect of reference parameter k of the CUSUM schemes. We also investigate the optimal design of reference values both in known and unknown shift cases. Finally, two examples are given to illustrate the implementation of the proposed approach.     

Time‐domain hybrid formulations for wave simulations in three‐dimensional PML‐truncated heterogeneous media

We are concerned with the numerical simulation of wave motion in arbitrarily heterogeneous, elastic, perfectly‐matched‐layer‐(PML)‐truncated media. We extend in three dimensions a recently developed two‐dimensional formulation, by treating the PML via an unsplit‐field, but mixed‐field, displacement‐stress formulation, which is then coupled to a standard displacement‐only formulation for the interior domain, thus leading to a computationally cost‐efficient hybrid scheme. The hybrid treatment leads to, at most, third‐order in time semi‐discrete forms. The formulation is flexible enough to accommodate the standard PML, as well as the multi‐axial PML. We discuss several time‐marching schemes, which can be used à la carte, depending on the application: (a) an extended Newmark scheme for third‐order in time, either unsymmetric or fully symmetric semi‐discrete forms; (b) a standard implicit Newmark for the second‐order, unsymmetric semi‐discrete forms; and (c) an explicit Runge–Kutta scheme for a first‐order in time unsymmetric system. The latter is well‐suited for large‐scale problems on parallel architectures, while the second‐order treatment is particularly attractive for ready incorporation in existing codes written originally for finite domains. We compare the schemes and report numerical results demonstrating stability and efficacy of the proposed formulations. Copyright © 2014 John Wiley & Sons, Ltd.     

On the Performance of Control Charts for Simultaneous Monitoring of Location and Dispersion Parameters

A single control chart is very famous to control assignable causes that shift the process because of variations in parameters (e.g., location and dispersion). Simultaneous monitoring of processes is another popular approach used for the bilateral processes. In this study, we have proposed the mixed control charts for simultaneously monitoring of process location and dispersion parameters. We have used the idea of mixed exponential weighted moving average and cumulative sum charts and designed the charting structures for simultaneous monitoring. The proposals are compared with several existing counterparts. The comparisons reveal numerous advantages of the proposed charts over the other existing scheme. The practical application of the proposed charts is also highlighted using an illustrative example based on a real dataset. Copyright © 2016 John Wiley & Sons, Ltd.     

An improved design of exponentially weighted moving average scheme for monitoring attributes

The Exponentially Weighted Moving Average (EWMA) schemes are a potent tool for monitoring small to moderate variations in the quality characteristics in production lines of manufacturing industries. Practitioners in various sectors widely use the EWMA schemes for scrutinising both the variables and attributes. In the present article, we investigate a modified EWMA scheme based on the power of the difference between the actual number of nonconforming items and its technical specification in an in-control (IC) situation. We abbreviate it as a wEWMA scheme and show that the traditional EWMA scheme is a particular case of the proposed scheme when the power is unity. We establish that the powers lower than unity are more effective for detecting smaller shifts, while for detecting substantial variations in process parameter, one should prefer higher powers greater than unity. Noting that possible magnitude of a shift is often unknown, we propose the optimal design procedure of the scheme, including the determination of its charting parameters to ensure the best overall performance. The results reveal that the optimal wEWMA schemes can be beneficial in detecting a shift very quickly when the sample size is small, particularly for high-precision production processes.     

The Max EWMAMS control chart for joint monitoring of process mean and variance with individual observations

A traditional approach to monitor both the location and the scale parameters of a quality characteristic is to use two separate control charts. These schemes have some difficulties in concurrent tracking and interpretation. To overcome these difficulties, some researchers have proposed schemes consisting of only one chart. However, none of these schemes is designed to work with individual observations. In this research, an exponentially weighted moving average (EWMA)‐based control chart that plots only one statistic at a time is proposed to simultaneously monitor the mean and variability with individual observations. The performance of the proposed scheme is compared with one of the two other existing combination charts by simulation. The results show that in general the proposed chart has a significantly better performance than the other combination charts. Copyright © 2010 John Wiley & Sons, Ltd.     

On the Monitoring of Simple Linear Berkson Profiles

We consider the quality of a process, which can be characterized by a simple linear Berkson profile. One existing approach for monitoring the simple linear profile and two new proposed schemes are studied for charting the simple linear Berkson profile. Simulation studies demonstrate the effectiveness and efficiency of one of the proposed monitoring schemes. In addition, a systematic diagnostic approach is provided to spot the change point location of the process and to identify the parameter of change in the profile. Finally, an example from semiconductor manufacturing is used to illustrate the implementation of the proposed monitoring scheme and diagnostic approach. Copyright © 2012 John Wiley & Sons, Ltd.     

A New Exponentially Weighted Moving Average Control Chart for Monitoring the Process Mean

Exponentially weighted moving average (EWMA) quality control schemes have been recognized as a potentially powerful process monitoring tool because of their superior speed in detecting small to moderate shifts in the underlying process parameters. In quality control literature, there exist several EWMA charts that are based on simple random sampling (SRS) and ranked set sampling (RSS) schemes. Recently, a mixed RSS (MxRSS) scheme has been introduced, which encompasses both SRS and RSS schemes, and is a cost‐effective alternative to the RSS scheme. In this paper, we propose new EWMA control charts for efficiently monitoring the process mean based on MxRSS and imperfect MxRSS (IMxRSS) schemes, named EWMA–MxRSS and EWMA–IMxRSS charts, respectively. Extensive Monte Carlo simulations are used to estimate the run length characteristics of the proposed EWMA charts. The run length performances of the suggested EWMA charts are compared with the classical EWMA chart based on SRS (EWMA–SRS). It turns out that both EWMA–MxRSS and EWMA–IMxRSS charts perform uniformly better than the EWMA–SRS chart when detecting all different shifts in the process mean. An application to a real data set is provided as an illustration of the design and implementation of the proposed EWMA chart. Copyright © 2014 John Wiley & Sons, Ltd.     

An Area‐based Methodology for the Monitoring of General Linear Profiles

Statistical profile monitoring has received much attention during recent years. While numerous contributions and applications have been demonstrated in the literature, the control statistics across many of the proposed methodologies have mainly remained unchanged, which somehow hinders further improvement of the monitoring schemes. In this paper, however, we propose a novel approach to leverage the information in the area formed between the sampled and in‐control profile to improve the monitoring scheme performance. Specifically, we develop a control statistic based on the convolution of the observed and in‐control profiles to monitor the shifts in the slope and intercept parameters. We also extend the mean square statistic to area weighted total sum of squares, to more effectively monitor the shift in the standard deviation. Extensive simulation studies are conducted to demonstrate the performance of the proposed methodology in comparison with some of the existing approaches. Copyright © 2016 John Wiley & Sons, Ltd.     

Nonparametric cost‐minimized Shewhart‐type process monitoring with restricted false alarm probability

Traditional Duncan‐type models for cost‐efficient process monitoring often inflate type I error probability. Nevertheless, controlling the probability of type I error or false alarms is one of the key issues in sequential monitoring of specific process characteristics. To this end, researchers often recommend economic‐statistical designs. Such designs assign an upper bound on type I error probability to avoid excessive false alarms while achieving cost optimality. In the context of process monitoring, there is a plethora of research on parametric approaches of controlling type I error probability along with the cost optimization. In the nonparametric setup, most of the existing works on process monitoring address one of the two issues but not both simultaneously. In this article, we present two distribution‐free cost‐efficient Shewhart‐type schemes for sequentially monitoring process location with restricted false alarm probability, based, respectively, on the sign and Wilcoxon rank‐sum statistics. We consider the one‐sided shift in location parameter in an unknown continuous univariate process. Nevertheless, one can easily extend our proposed schemes to monitor the two‐sided process shifts. We evaluate and compare the actual performance of the two monitoring schemes employing extensive computer simulation based on Monte Carlo. We investigate the effects of the size of the reference sample and the false alarm constraint. Finally, we provide two illustrative examples, each based on a realistic situation in the industry.