Optimal T2 Control Chart with a Double Sampling Scheme – An Alternative to the MEWMA Chart

Recent studies have shown that a double sampling (DS) scheme yields improvements in detection times of process shifts over variable ratio sampling (VRS) methods that have been extensively studied in the literature. Additionally, a DS scheme is more practical than some of the VRS methods since the sampling interval is fixed. In this paper, we investigate the effect of double sampling on cost, a criterion as important as detection rate. We study economic statistical design of the DS T² chart (ESD DS T²) so that designs are found that are economically optimal but yet meet desired statistical properties such as having low probabilities of false searches and high probabilities of rapid detection of process shifts. Through an illustrative example, we show that relatively large benefits can be achieved in a comparison with the classical T² chart and the statistical DS T² charts with our ESD DS T² approach. Furthermore, the economic performance of the ESD DS T² charts is favorably compared to the MEWMA and other VRS T² control charts in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

New Synthetic EWMA and Synthetic CUSUM Control Charts for Monitoring the Process Mean

Exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts are potentially powerful statistical process monitoring tools because of their excellent speed in detecting small to moderate persistent process shifts. Recently, synthetic EWMA (SynEWMA) and synthetic CUSUM (SynCUSUM) control charts have been proposed based on simple random sampling (SRS) by integrating the EWMA and CUSUM control charts with the conforming run length control chart, respectively. These synthetic control charts provide overall superior detection over a range of mean shift sizes. In this article, we propose new SynEWMA and SynCUSUM control charts based on ranked set sampling (RSS) and median RSS (MRSS) schemes, named SynEWMA‐RSS and SynEWMA‐MRSS charts, respectively, for monitoring the process mean. Extensive Monte Carlo simulations are used to estimate the run length characteristics of the proposed control charts. The run length performances of these control charts are compared with their existing powerful counterparts based on SRS, RSS and MRSS schemes. It turns out that the proposed charts perform uniformly better than the Shewhart, optimal synthetic, optimal EWMA, optimal CUSUM, near‐optimal SynEWMA, near‐optimal SynCUSUM control charts based on SRS, and combined Shewhart‐EWMA control charts based on RSS and MRSS schemes. A similar trend is observed when constructing the proposed control charts based on imperfect RSS schemes. An application to a real data is also provided to demonstrate the implementations of the proposed SynEWMA and SynCUSUM control charts. Copyright © 2014 John Wiley & Sons, Ltd.     

Control Charts monitoring Product's Loss to Society

Taguchi introduced a new philosophy in quality control that accounts for the economic loss associated to process variation measured by deviations from the target value of a product quality characteristic. The Taguchi loss function has been considered in the design of control charts only for the computation of costs associated with nonconformities. This paper considers sample statistics based on the Taguchi loss function as a means to implement Shewhart control charts monitoring both the deviation from the target and dispersion of normally distributed quality characteristics. The aim of this proposed control chart is to perform on‐line quality control of a process by monitoring its quality loss cost performance over time. To compute the quality loss performance, we consider a nominal‐the‐best quality characteristic. The statistical performance of the proposed control charts has been evaluated and compared with that of widely used control charts. Implementing target costing philosophy by means of one of the proposed charts is also discussed. An example illustrates the Taguchi control chart in a practical implementation. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust economic control chart design

Designing a control chart involves making fundamental decisions about the control chart parameters. Traditionally, practitioners select design parameters using ad hoc procedures. Academic research has been challenging this tradition by introducing a more rigorous criterion for selecting the design parameters based on economic criteria. However, there has been limited success in implementing economic designs. This research develops the concept of Robust Economic Design of control charts where multiple economic and process scenarios are considered in control chart design. By developing a robust optimization technique for economic control chart design, we hope to promote a better understanding of industrial implementation of economic designs of control charts. The effectiveness of this technique is illustrated through examples and a sensitivity analysis.     

Using Control Charts to Monitor the Performance of Large-Scale Systems

ABSTRACT

The initial goal of this project was to provide Alcan Rolled Products Company of Kingston, Ontario, with control charts to monitor its new Paint Line Three (PL3) aluminium coil coating facility. As the project progressed, however, it became apparent that the following factors made PL3 ill-suited to traditional statistical quality control methods: the inexact nature of the coil coating science, the level of process interdependence and complexity of the operation, and the economic and technical infeasibility of installing sophisticated measuring devices at all stages of the coating line.

Consequently, an effort was made to model the entire system on a macroscopic, rather than a microscopic, level and to use the results of qualitative lab tests to make inferences about the paint line's process capability. To accomplish this, the length of scrap produced by every lot processed in a 1-year period was used to create a statistical model of the of the frequency and severity of the two most costly quality shortcomings: surface defects and coating problems.

This article outlines the methodology used and demonstrates how a complex phenomenon such as scrap creation can be modeled by using a combination of well-known statistical distributions. The statistical model can be used to transform raw data into meaningful graphical representations of large-scale system performance.     

Relationships Among Control Charts Used with Feedback Control

Feedback control is common in modern manufacturing processes and there is a need to combine statistical process control in such systems. Typical types of assignable causes are described and fault signatures are calculated. A fault signature can be attenuated by the controller and an implicit confounding among faults of different types is discussed. Furthermore, the relationships between various control statistics are developed. Control charts have been proposed previously for deviations from target and for control adjustments. We describe why one or the other can be effective in some cases, but that neither directly incorporates the magnitude (or signature) of an assignable cause. Various disturbance models and control schemes, both optimal and non‐optimal, are included in a mathematically simple model that obtains results through properties of linear filters. We provide analytical results for a widely‐used model of feedback control. Copyright © 2006 John Wiley & Sons, Ltd.     

The Economic Design of ―x Charts: A Proactive Approach

It has long been recognized that poor quality can only result in higher costs. Yet, the idea of reducing cost through better quality is not fully realized. Current models for the economic design of control charts provide strategies to maintain existing quality levels. In this research, a comprehensive cost model is developed to incorporate two cost functions. A reactive function, which accounts for all quality related costs incurred while maintaining a stable level of the process, and a proactive function, which accounts for the cost of process improvement. Using incremental economics, the two cost functions are assembled to allow an evaluation of process improvement alternatives based on their economic worth. Procedures for obtaining economically optimum designs for controlling the process mean are developed and designed experiments are utilized to investigate model performance over a wide range of input parameters. The results indicate that the model is sensitive to changes in 13 parameters, especially when the magnitude of the process shift is small. Copyright © 2004 John Wiley & Sons, Ltd.     

CUSUM Control Charts for the Monitoring of Zero‐inflated Binomial Processes

Zero‐inflated probability models are used to model count data that have an excessive number of zeros. These models are mostly useful in modeling high‐yield or health‐related processes. The zero‐inflated binomial distribution is an extension of the ordinary binomial distribution that takes into account the excess of zeros. In this paper, one‐sided cumulative sum (CUSUM)‐type control charts are proposed for monitoring increases or decreases in the parameter p of a zero‐inflated binomial process. The results of an extensive numerical study concerning the statistical design of the proposed schemes as well as their practical implementation are provided. Copyright © 2015 John Wiley & Sons, Ltd.     

New Synthetic CUSUM and Synthetic EWMA Control Charts for Monitoring the Process Mean using Auxiliary Information

The cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts are potentially powerful process monitoring tool because of their excellent speed in detecting small to moderate shifts in the process parameters. These control charts can be further improved by integrating them with the conforming run length control chart, resulting in the synthetic CUSUM (SynCUSUM) and synthetic EWMA (SynEWMA) charts. In this paper, we enhance the detection abilities of the SynCUSUM and SynEWMA charts using the auxiliary information. With suitable assumptions, the proposed control charts encompass the existing SynCUSUM, SynEWMA, CUSUM, and EWMA charts. Extensive Monte Carlo simulations are used to study the run length profiles of the proposed control charts. It turns out that the proposed near‐optimal control charts with the auxiliary information perform uniformly and substantially better than the existing near‐optimal SynCUSUM, SynEWMA, CUSUM, and EWMA charts. The proposed and existing control charts are also illustrated with the help of an example. Copyright © 2017 John Wiley & Sons, Ltd.     

Effects of Model Accuracy on Residual Control Charts

Residual control charts are acknowledged to be effective tools for statistical process control of multistage processes. In these monitoring procedures, the models on the stage‐wise correlation should be first derived before the control charts are implemented. Therefore, the monitoring performance is inevitably affected by the model fitting scheme. Most of the previous works are under the assumption that the derived models represent the process behavior perfectly. Far less is known about the effects of the model inaccuracy on the monitoring performance. To investigate the effects of the underlying models on the monitoring performance, residual control charts based on two different modeling schemes are compared in this paper. The results indicate that the charting performance is correlated with the model fitting schemes. That is, a more accurate model will significantly increase the detection power and decrease the false alarm rate as well. Copyright © 2015 John Wiley & Sons, Ltd.     

An Analysis of Four Spectral Control Charts

Control charts are used in manufacturing settings to monitor and detect changes in a process. Spectral control charts, based on the periodogram evaluated at the Fourier frequencies, have been designed to detect the onset of periodic behavior. In this article, two additional spectral charts formed using established periodogram tests are proposed. Comparisons of the average run lengths of the spectral charts based on these tests are made using Monte Carlo techniques. These comparisons evaluate how well the methods detect both simple and compound periodic behavior corresponding to both Fourier and non-Fourier frequencies. The charts are also compared using an industrial data set involving the widths of coating on silicon wafers. The analysis will illustrate both the strengths and weaknesses of each procedure and give suggestions as to appropriate usage.     

Economic and economic-statistical designs of the side sensitive group runs chart with auxiliary information

This article studies the economic and economic-statistical designs of the auxiliary information (AI) based side sensitive group runs (SSGR-AI) chart. The regression estimator that consists of information not only from the primary variable but also from the auxiliary variable is integrated into the control charting statistic. Optimal designs of the SSGR-AI chart, for the minimization of the expected cost function with and without statistical constraints, are developed based on (i) average run length (ARL) and (ii) expected average run length (EARL). Furthermore, sensitivity analyses are conducted, that is, the impact of various input parameters on the optimal parameters and costs for different values of correlation coefficients (ρ) between the primary and auxiliary variables are investigated. In addition, the effects of incorrect specification of the size of the shift on the optimal cost of the SSGR-AI chart are studied. The comparative study reveals that the SSGR-AI chart is superior to the exponentially weighted moving average-AI (EWMA-AI) and synthetic-AI (Syn-AI) charts, for both designs, by giving the smallest costs.     

Optimal Design of VSI ―X Control Charts for Monitoring Correlated Samples

This paper develops an economic design of variable sampling interval (VSI)―X control charts in which the next sample is taken sooner than usual if there is an indication that the process is off‐target. When designing VSI―X control charts, the underlying assumption is that the measurements within a sample are independent. However, there are many practical situations that violate this hypothesis. Accordingly, a cost model combining the multivariate normal distribution model given by Yang and Hancock with Bai and Lee's cost model is proposed to develop the design of VSI charts for correlated data. An evolutionary search method to find the optimal design parameters for this model is presented. Also, we compare VSI and traditional ―X charts with respect to expected cost per unit time, utilizing hypothetical cost and process parameters as well as various correlation coefficients. The results indicate that VSI control charts outperform the traditional control charts for larger mean shift when correlation is present. In addition, there is a difference between the design parameters of VSI charts when correlation is present or absent. Copyright © 2005 John Wiley & Sons, Ltd.     

Directionally Sensitive Multivariate Control Charts in Practice: Application to Biosurveillance

Multivariate control charts are used for monitoring multiple series simultaneously, for the purpose of detecting shifts in the mean vector in any direction. In the context of disease outbreak detection, interest is in detecting only an increase in the process means. Two practical approaches for deriving directional Hotelling charts are Follmann's correction and Testik and Runger's quadratic programming. However, there has not been an extensive comparison of their practical performance. Moreover, in practice, many of the underlying method assumptions are often violated, and the theoretically guaranteed performance might not hold. In this work, we compare the two directionally sensitive approaches: a statistically based approach and an operations research solution. We evaluate Hotelling charts as well as two extensions to multivariate exponentially weighted moving average charts. We examine practical performance aspects such as robustness to often‐impractical assumptions, the amount of data required for proper performance, and computational aspects. We perform a large simulation study and examine performance on authentic biosurveillance data. Copyright © 2013 John Wiley & Sons, Ltd.     

Economic–statistical Design of Maximum Exponentially Weighted Moving Average Control Charts

The maximum exponentially weighted moving average (MaxEWMA) control chart effectively combines the two EWMA charts into one chart and monitors both increases and decreases in the mean and/or variability. In this paper, we develop the economic–statistical design of the MaxEWMA control chart in which the Taguchi's quadratic loss function is incorporated into the Duncan's economical model. Numerical simulations are executed to minimize the expected total cost model and determine the optimal decision variables, including the sample size, sampling interval, control limit width, and the smoothing constant of the MaxEWMA control chart. It is shown that the optimal control limit width and smoothing constant increase as the optimal cost value increases and that both the optimal sample size and sampling interval always decrease as the magnitudes of mean and/or variance shifts increase. Copyright © 2012 John Wiley & Sons, Ltd.     

Constrained optimization model for the design of an adaptive X chart

An economic-statistical model is developed for variable parameters (VP) X charts in which all design parameters vary adaptively, that is, each of the design parameters (sample size, sampling interval and control-limit width) vary as a function of the most recent process information. The cost function due to controlling the process quality through a VP X chart is derived. During the optimization of the cost function, constraints are imposed on the expected times to signal when the process is in and out of control. In this way, required statistical properties can be assured. Through a numerical example, the proposed economicstatistical design approach for VP X charts is compared to the economic design for VP X charts and to the economic-statistical and economic designs for fixed parameters (FP) X charts in terms of the operating cost and the expected times to signal. From this example, it is possible to assess the benefits provided by the proposed model. Varying some input parameters, their effect on the optimal cost and on the optimal values of the design parameters was analysed.     

Control Charts for the Lognormal Mean

Among a set of tools that form the core of statistical process control, statistical control charts are most commonly used for controlling, monitoring, and improving processes. The conventional control charts are based on the assumption that the distribution of the quality characteristic to be monitored follows the normal distribution. However, in real applications, many process distributions may follow a positively skewed distribution such as the lognormal distribution. In this study, we discuss the construction of several control charts for monitoring the mean of the lognormal distribution. A real example is used to demonstrate how these charts can be applied in practice. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Economic design of exponential charts for time between events monitoring

The occurrence of defects or non-conformities in manufacturing processes can usually be modeled by a homogeneous Poisson process. However, the process parameter may change over time and it can be monitored with statistical process control techniques. Control charts based on an exponential distribution, called exponential charts in this paper, can be developed to monitor the occurrence rate of such events. For manufacturers, the economic objective of production is very important and has to be optimized. An economic approach is developed in this paper for the design of exponential charts. We compare and contrast the performances of statistical design, economic design and economic–statistical design. The usefulness of the proposed economic design approach is justified. The relationships among these designs are illustrated through numerical examples. In particular, the economic–statistical design approach is interpreted from a multi-objective optimization viewpoint. The limitations of the approach as well as future research are also discussed.