Optimal design of a distribution-free quality control scheme for cost-efficient monitoring of unknown location

Traditionally, a cost-efficient control chart for monitoring product quality characteristic is designed using prior knowledge regarding the process distribution. In practice, however, the functional form of the underlying process distribution is rarely known a priori. Therefore, the nonparametric (distribution-free) charts have gained more attention in the recent years. These nonparametric schemes are statistically designed either with a fixed in-control average run length or a fixed false alarm rate. Robust and cost-efficient designs of nonparametric control charts especially when the true process location parameter is unknown are not adequately addressed in literature. For this purpose, we develop an economically designed nonparametric control chart for monitoring unknown location parameter. This work is based on the Wilcoxon rank sum (hereafter WRS) statistic. Some exact and approximate procedures for evaluation of the optimal design parameters are extensively discussed. Simulation results show that overall performance of the exact procedure based on bootstrapping is highly encouraging and robust for various continuous distributions. An approximate and simplified procedure may be used in some situations. We offer some illustration and concluding remarks.     

Design and implementation of CUSUM exceedance control charts for unknown location

Nonparametric control charts provide a robust alternative in practice when the form of the underlying distribution is unknown. Nonparametric CUSUM (NPCUSUM) charts blend the advantages of a CUSUM with that of a nonparametric chart in detecting small to moderate shifts. In this paper, we examine efficient design and implementation of Phase II NPCUSUM charts based on exceedance (EX) statistics, called the NPCUSUM-EX chart. We investigate the choice of the order statistic from the reference (Phase I) sample that defines the exceedance statistic. We see that choices other than the median, such as the 75th percentile, can yield improved performance of the chart in certain situations. Furthermore, observing certain shortcomings of the average run-length, we use the median run-length as the performance metric. The NPCUSUM-EX chart is compared with the NPCUSUM-Rank chart based on the popular Wilcoxon rank-sum statistic. We also study the choice of the reference value, k, of the CUSUM charts. An illustration with real data is provided.     

A Distribution‐free Control Chart for the Joint Monitoring of Location and Scale

Traditional statistical process control for variables data often involves the use of a separate mean and a standard deviation chart. Several proposals have been published recently, where a single (combination) chart that is simpler and may have performance advantages, is used. The assumption of normality is crucial for the validity of these charts. In this article, a single distribution‐free Shewhart‐type chart is proposed for monitoring the location and the scale parameters of a continuous distribution when both of these parameters are unknown. The plotting statistic combines two popular nonparametric test statistics: the Wilcoxon rank sum test for location and the Ansari–Bradley test for scale. Being nonparametric, all in‐control properties of the proposed chart remain the same and known for all continuous distributions. Control limits are tabulated for implementation in practice. The in‐control and the out‐of‐control performance properties of the chart are investigated in simulation studies in terms of the mean, the standard deviation, the median, and some percentiles of the run length distribution. The influence of the reference sample size is examined. A numerical example is given for illustration. Summary and conclusions are offered. Copyright © 2011 John Wiley & Sons, Ltd.     

Distribution-free triple EWMA control chart for monitoring the process location using the Wilcoxon rank-sum statistic with fast initial response feature

The exponentially weighted moving average (EWMA) control chart is a memory-type chart known to be more efficient in detecting small and moderate shifts in the process parameter. The double EWMA (DEWMA) chart is an extension of the EWMA chart that is more effective than the latter in the detection of small-to-moderate shifts. This paper proposes a new distribution-free (or nonparametric) triple EWMA (TEWMA) control chart based on the Wilcoxon rank-sum (W) statistic to improve the detection ability in the process location parameter. Moreover, a new fast initial response (FIR) feature is added to further improve the sensitivity of the new TEWMA chart. The performance of the proposed TEWMA chart with and without FIR features is compared to those of the existing EWMA and DEWMA W charts. It is observed that the TEWMA chart with and without FIR features is superior to the competing charts in most situations. A real-life illustration is provided to show the application and implementation of the new chart.     

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.     

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 Distribution-Free Shewhart Quality Control Chart Based on Signed-Ranks

Since their inception by Walter Shewhart in the late 1920s, most control chart developments have been distribution-based procedures in the sense that the process output is assumed to follow a specified probability distribution (normal for continuous measurements and binomial or Poisson for attribute data). Due to Deming's influence and their widespread adoption as one of the seven basic tools of total quality management (TQM), control charts have been applied to processes where data may be markedly nonnormal. In this article, we propose a distribution-free (or nonparametric) statistical quality control chart for monitoring a process center. The proposed chart is of the Shewhart type and is based on the signed-ranks of grouped observations. The exact false alarm rate and the in-control average run length of the proposed chart are computed by using the null distribution of the well-known Wilcoxon signed-rank statistic. The out-of-control run lengths are computed exactly for normal underlying distributions and by simulation for uniform, double exponential, and Cauchy shift alternatives. Efficiency studies show that the proposed chart is more efficient than the traditional Shewhart X-bar chart under heavy-tailed distributions (the double exponential and the Cauchy) but is less efficient under light-tailed distributions (the uniform and the normal).     

A new nonparametric double exponentially weighted moving average control chart

There are many practical situations where the underlying distribution of the quality characteristic either deviates from normality or it is unknown. In such cases, practitioners often make use of the nonparametric control charts. In this paper, a new nonparametric double exponentially weighted moving average control chart on the basis of the signed-rank statistic is proposed for monitoring the process location. Monte Carlo simulations are carried out to obtain the run length characteristics of the proposed chart. The performance comparison of the proposed chart with the existing parametric and nonparametric control charts is made by using various performance metrics of the run length distribution. The comparison showed the superiority of the suggested chart over its existing parametric and nonparametric counterparts. An illustrative example for the practical implementation of the proposed chart is also provided by using the industrial data set.     

Design and implementation issues for a class of distribution-free Phase II EWMA exceedance control charts

Distribution-free (nonparametric) control charts can play an essential role in process monitoring when there is dearth of information about the underlying distribution. In this paper, we study various aspects related to an efficient design and execution of a class of nonparametric Phase II exponentially weighted moving average (denoted by NPEWMA) charts based on exceedance statistics. The choice of the Phase I (reference) sample order statistic used in the design of the control chart is investigated. We use the exact time-varying control limits and the median run-length as the metric in an in-depth performance study. Based on the performance of the chart, we outline implementation strategies and make recommendations for selecting this order statistic from a practical point of view and provide illustrations with a data-set. We conclude with a summary and some remarks.     

An extended nonparametric homogeneously weighted moving average sign control chart

Parametric (or traditional) control charts are based on the assumption that the quality characteristic of interest follows a specific distribution. However, in many applications, there is a lack of knowledge about the underlying distribution. To this end, nonparametric (or distribution-free) control charts have been developed in recent years. In this article, a nonparametric double homogeneously weighted moving average (DHWMA) control chart based on the sign statistic is proposed for monitoring the location parameter of an unknown and continuous distribution. The performance of the proposed chart is measured through the run-length distribution and its associated characteristics by performing Monte Carlo simulations. The DHWMA sign chart is compared with other nonparametric sign charts, such as the homogeneously weighted moving average, generally weighted moving average (GWMA), double GWMA, and triple exponentially weighted moving average sign charts, as well as the traditional DHWMA chart. The results indicate that the proposed chart performs just as well as and in some cases better than its competitors, especially for small shifts. Finally, two examples are provided to show the application and implementation of the proposed chart.     

A double generally weighted moving average exceedance control chart

Since the inception of control charts by W. A. Shewhart in the 1920s, they have been increasingly applied in various fields. The recent literature witnessed the development of a number of nonparametric (distribution‐free) charts as they provide a robust and efficient alternative when there is a lack of knowledge about the underlying process distribution. In order to monitor the process location, information regarding the in‐control (IC) process median is typically required. However, in practice, this information might not be available due to various reasons. To this end, a generalized type of nonparametric time‐weighted control chart labeled as the double generally weighted moving average (DGWMA) based on the exceedance statistic (EX) is proposed. The DGWMA‐EX chart includes many of the well‐known existing time‐weighted control charts as special or limiting cases for detecting a shift in the unknown location parameter of a continuous distribution. The DGWMA‐EX chart combines the better shift detection properties of a DGWMA chart with the robust IC performance of a nonparametric chart, by using all the information from the start until the most recent sample to decide if a process is IC or out‐of‐control. An extensive simulation study reveals that the proposed DGWMA‐EX chart, in many cases, outperforms its counterparts.     

On developing sensitive nonparametric mixed control charts with application to manufacturing industry

Control charts are designed under the normality assumption of the quality characteristic of the process. However, the normality assumption rarely holds in practice. In non-normal conditions, parametric charts tend to display more false alarm rates and invalid out-of-control comparisons. The exponentially weighted moving average chart is a frequently used memory-type control chart for monitoring the process target that only performs effectively under the smoothing parameter's small choices. This study proposes a nonparametric mixed exponentially weighted moving average-progressive mean chart based on sign statistic (NPMEP_SN) under simple and ranked set sampling schemes to address this said drawback. Normal and non-normal distributions are included in this study to observe the proposed chart's in-control behavior and out-of-control efficacy. The prominent feature of the proposed schemes is that it works efficiently in detecting small and persistent shifts in the process location corresponding to the given values of the smoothing parameter. The proposed scheme is also tested under the ranked set sampling scheme to enhance the NPMEP_SN chart's performance (hereafter named “NPMEP_RSN”). The performance of the proposed charts is investigated through simulations using run-length profiles. The proposed schemes were seen to outperform other alternatives, specifically under the ranked set sampling scheme. A real data-set related to the diameter of a piston ring is included as a demonstration of the proposal.     

A Bivariate Semiparametric Control Chart Based on Order Statistics

In this article, a new bivariate semiparametric Shewhart‐type control chart is presented. The proposed chart is based on the bivariate statistic (X_(r), Y_(s)), where X_(r) and Y_(s) are the order statistics of the respective X and Y test samples. It is created by considering a straightforward generalization of the well‐known univariate median control chart and can be easily applied because it calls for the computation of two single order statistics. The false alarm rate and the in‐control run length are not affected by the marginal distributions of the monitored characteristics. However, its performance is typically affected by the dependence structure of the bivariate observations under study; therefore, the suggested chart may be characterized as a semiparametric control chart. An explicit expression for the operating characteristic function of the new control chart is obtained. Moreover, exact formulae are provided for the calculation of the alarm rate given that the characteristics under study follow specific bivariate distributions. In addition, tables and graphs are given for the implementation of the chart for some typical average run length values and false alarm rates. The performance of the suggested chart is compared with that of the traditional χ² chart as well as to the nonparametric SN² and SR² charts that are based on the multivariate form of the sign test and the Wilcoxon signed‐rank test, respectively. Finally, in order to demonstrate the applicability of our chart, a case study regarding a real‐world problem related to winery production is presented. Copyright © 2016 John Wiley & Sons, Ltd.     

An EWMA signed ranks control chart with reliable run length performances

During the design phase of a control chart, the determination of its exact run length properties plays a vital role for its optimal operation. Markov chain or integral equation methods have been extensively applied in the design phase of conventional control charts. However, for distribution-free schemes, due to the discrete nature of the statistics being used (such as the sign or the Wilcoxon signed rank statistics, for instance), it is impossible to accurately compute their run length properties. In this work, a modified distribution-free phase II exponentially weighted moving average (EWMA)-type chart based on the Wilcoxon signed rank statistic is considered and its exact run length properties are discussed. A continuous transformation of the Wilcoxon signed rank statistic, combined with the classical Markov chain method, is used for the determination of the average run length in the in- and out-of control cases. Moreover, its exact performance is derived without any knowledge of the distribution of sample observations. Finally, an illustrative example is provided showing the practical implementation of our proposed chart.     

Bivariate semiparametric control charts for simultaneous monitoring of process mean and variance

In the present article, two semiparametric bivariate control charts are presented, which use order statistics and are effective in jointly monitoring of possible shifts in the process mean and/or variance. To achieve that both the median location (or more generally the location of a specific order statistic) and the number of specific observations of the test sample lying between the control limits are taken into account. The false alarm rate and the in-control average run length are not affected by the marginal distributions, while the effect of the dependence structure on them is negligible; therefore, they can be used as fully nonparametric charts. A performance-comparison study is carried out, and an illustrative example is provided using a real-world data set.     

A Distribution-Free Multivariate Control Chart

Monitoring multivariate quality variables or data streams remains an important and challenging problem in statistical process control (SPC). Although the multivariate SPC has been extensively studied in the literature, designing distribution-free control schemes are still challenging and yet to be addressed well. This article develops a new nonparametric methodology for monitoring location parameters when only a small reference dataset is available. The key idea is to construct a series of conditionally distribution-free test statistics in the sense that their distributions are free of the underlying distribution given the empirical distribution functions. The conditional probability that the charting statistic exceeds the control limit at present given that there is no alarm before the current time point can be guaranteed to attain a specified false alarm rate. The success of the proposed method lies in the use of data-dependent control limits, which are determined based on the observations online rather than decided before monitoring. Our theoretical and numerical studies show that the proposed control chart is able to deliver satisfactory in-control run-length performance for any distributions with any dimension. It is also very efficient in detecting multivariate process shifts when the process distribution is heavy-tailed or skewed. Supplementary materials for this article are available online.     

Nonparametric signed‐rank control charts with variable sampling intervals

Variable sampling interval (VSI) charts have been proposed in the literature for normal theory (parametric) control charts and are known to provide performance enhancements. In the VSI setting, the time between monitored samples is allowed to vary depending on what is observed in the current sample. Nonparametric (distribution‐free) control charts have recently come to play an important role in statistical process control and monitoring. In this paper a nonparametric Shewhart‐type VSI control chart is considered for detecting changes in a specified location parameter. The proposed chart is based on the Wilcoxon signed‐rank statistic and is called the VSI signed‐rank chart. The VSI signed‐rank chart is compared with an existing fixed sampling interval signed‐rank chart, the parametric VSI ‐chart, and the nonparametric VSI sign chart. Results show that the VSI signed‐rank chart often performs favourably and should be used.     

An Efficient Nonparametric EWMA Wilcoxon Signed‐Rank Chart for Monitoring Location

The statistical performance of traditional control charts for monitoring the process shifts is doubtful if the underlying process will not follow a normal distribution. So, in this situation, the use of a nonparametric control charts is considered to be an efficient alternative. In this paper, a nonparametric exponentially weighted moving average (EWMA) control chart is developed based on Wilcoxon signed‐rank statistic using ranked set sampling. The average run length and some other associated characteristics were used as the performance evaluation of the proposed chart. A major advantage of the proposed nonparametric EWMA signed‐rank chart is the robustness of its in‐control run length distribution. Moreover, it has been observed that the proposed version of the EWMA signed‐rank chart using ranked set sampling shows better detection ability than some of the competing counterparts including EWMA sign chart, EWMA signed‐rank chart, and the usual EWMA control chart using simple random sampling scheme. An illustrative example is also provided for practical consideration. 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.     

Nonparametric Profile Monitoring using Dynamic Probability Control Limits

This article focuses on monitoring nonparametric profile with time‐varying sample sizes and random predictors. Traditional profile monitoring schemes, whose control limits are often determined before the monitoring initiates, are constructed based on perfect knowledge of profile sample sizes and predictors. In practice, however, our foreknowledge about future random sample sizes and predictors is seldom available. An inappropriate assumption or estimation of the sample sizes model and/or predictors distribution function may lead to unexpected performance of traditional control charts. To overcome this problem, we propose a kernel‐based nonparametric profile monitoring scheme which integrates the multivariate exponentially weighted moving average procedure with the probability control limits. The success of the proposed chart lies in the use of dynamic control limits which are determined online, essentially aiming at guaranteeing the conditional probability that the charting statistic exceeds the control limit at present given that there is no alarm before the current time point to meet a pre‐specified false alarm rate. The simulation studies show that the proposed control scheme has good in‐control and out‐of‐control performances under various scenarios of time‐varying sample sizes and random predictors. Copyright © 2016 John Wiley & Sons, Ltd.