In-control Average Run Length Research Articles

Design of a new attribute chart based on inspection run length

The conventional np chart detects shifts in fraction nonconforming of the process depending on the total number of nonconforming units, which is obtained until all of the units within a sample are completely inspected. To improve the effectiveness of inspection capacity, this paper presents a new attribute control chart, namely the Inspection Run Length (IRL) control chart, which detects shifts in fraction nonconforming of the process depending on a novel inspection strategy. This strategy employs information about the number of conforming and nonconforming units and inspection run length during the inspection process to curtail the number of units that need to be inspected. The optimal design parameters of the proposed control chart are derived based on optimising average run length (ARL) properties. In addition, the performance of the proposed IRL chart is evaluated and compared with the np chart and CRL-type charts. Comparison studies show that the proposed chart achieves a significant reduction in inspection time and cost and an improvement in performance (a lower out-of-control ARL) while holding the false alarm rate ( in-control ARL) at a specified level. Finally, two industrial examples are given to illustrate how to simply apply the proposed chart to the manufacturing process.

Optimal triple sampling [formula omitted] charts for autocorrelated processes

Autocorrelation is a phenomenon in which the characteristics of interest being monitored are dependent. In the presence of autocorrelation, the detection ability of classical control charts deteriorates significantly, and this poses great concerns to practitioners as some processes in real-life settings produce autocorrelated observations. This paper deals with the problem of autocorrelation in Shewhart charts by incorporating the AR(1) and MA(1) models into the classical TS X¯ chart, which is an extension of the Shewhart chart. The performance of the introduced TS X¯ charts for AR(1) and MA(1) is improved by optimizing the parameters of the charts when the in-control average run length (ARL) of the charts is 370. The TS X¯chart for AR(1) is found to outperform the DS X¯chart for AR(1) in the presence of autocorrelation. The performance of the proposed charts is assessed using ARL, standard deviation of the run length (SDRL), average number of observations to signal (ANOS), and standard deviation of the number of observations to signal (SDNOS) criteria. Based on these criteria, the TS X¯chart for MA(1) is found to surpass the TS X¯chart for AR(1) as the autocorrelation level increases.

A Robust Design for the Omnibus SPRT Control Chart Under Skewed Data Distributions

Control charts are widely used in manufacturing industries to ensure that production levels are stable and satisfactory. Recently, the omnibus sequential probability ratio test (OSPRT) control chart was developed for the purpose of monitoring the mean and variability of a process simultaneously. As the OSPRT chart was proposed for the first time in literature, its development relied entirely on the assumption that data follow the Normal distribution. Nonetheless, researchers are frequently reminded that the quality characteristics of manufacturing processes do not necessarily follow the Normal distribution, e.g., strengths of glass fibres, and lifetimes of products. In this paper, we investigate the extent to which the performances of the OSPRT chart designed for the Normal model deteriorate, in situations where the data distributions are Gamma and Lognormal. Results show that the in-control average run length (ARL) and standard deviation of the run length of the OSPRT chart designed for the Normal distribution deteriorate rapidly as skewness increases. To address this issue, we propose a robust design for the OSPRT chart by adjusting its control limits, known as the skewness correction method. It is shown that the skewness-corrected OSPRT chart enjoys a guaranteed in-control ARL, with a justifiable degradation in its out-of-control performances. Besides, we also show some insights into selecting the charting parameters for the skewness-corrected OSPRT chart in order to achieve an optimum out-of-control ARL performance over various shift sizes. The paper wraps up with an illustrative example of the skewness-corrected OSPRT chart for monitoring the weights of radial tyres.

Joint monitoring of location and scale for modern univariate processes

Autocorrelated sequences of individual observations arise in many modern-day statistical process monitoring (SPM) applications. Often times, interest involves jointly monitoring both process location and scale. To jointly monitor autocorrelated individuals data, it is common to first fit a time series model to the in-control process and subsequently use this model to de-correlate the observations so that traditional individuals and moving-range (I-MR) charts can be applied. If the time series model is correctly specified such that the resulting residuals are normal and independently distributed, then applying I-MR control charts to the residual process should work well. However, if the residual process deviates from normality and/or, due to time series model misspecification, contains levels of autocorrelation, the false alarm rate of such a strategy can dramatically rise. In this paper we propose improvements to a recently published distribution-free joint monitoring strategy that permits tighter control of the in-control average run length when model misspecification is a concern. We evaluate its average run length performance and conclude that the improved joint monitoring strategy proposed in this work is a very useful tool for today’s modern SPM practitioner. The proposed joint monitoring scheme is then applied to a real additive manufacturing process to illustrate its implementation in modern practice.

A New Statistical Approach Based on the Access of Electricity Application with Some Modified Control Charts

This article introduces a new probability model based on reflected parameter called the reflected Pareto (RP) distribution. The key properties of the RP model are investigated. A simulation study of the RP model is conducted to evaluate the performances of its estimators. A real-life application is considered to examine the performance of proposed model. The different criteria are discussed numerically as well as graphically to show the flexibility of the RP model. The exponential weighted moving average control charts based on the maximum likelihood and modified maximum likelihood estimators for the shape parameter of the RP distribution are obtained. Detailed simulation results of proposed charts are performed to examine and analyze the performance of these charts with three in-control average run length values and two sample sizes. Finally, the application of the proposed control charts is shown by considering a real-life data set.

Control charts for high-dimensional time series with estimated in-control parameters

In this article, we study the effect of misspecification caused by fitting the target process in the Phase I analysis of the monitoring procedure on the behavior of several types of multivariate exponentially weighted moving average (MEWMA) control charts in the high-dimensional setting. In particular, the classical MEWMA control charts, whose control statistics are based on the exact and asymptotic Mahalanobis distance, are considered together with the novel approaches where the Euclidean distance and the diagonalized Euclidean distance are employed in the construction of control statistics. The high-dimensional distributions of the control statistics are deduced at each time. These results are later used to assess the performance of the considered control charts under misspecification. Both theoretical and empirical findings lead to the conclusion that the control charts based on the Euclidean distance and the diagonalized Euclidean distance are robust to misspecification for moderate dimensions of the data-generating model, whereas they tend to overestimate the in-control average run lengths (ARLs) in the case of larger dimensions. On the other hand, the control schemes based on the Mahalanobis distance are considerably affected by the estimation of the parameters of the target process, and their application results in drastically smaller values of the ARLs, especially when the dimension of the data-generating model is large.

Enhancing Control: Unveiling the Performance of Poisson EWMA Charts through Simulation with Poisson Mixture DATA

Poisson-Exponentially Weighted Moving Average (PEWMA) charts are one of the most frequently used control charts for monitoring count data. But as real-world data often shows overdispersion—prevalent in manufacturing, health care, economics, and marketing—the standard Poisson distribution falls short. One of the ways to tackle overdispersion is to use Poisson mixture distributions. Our study examines Average Run Length (ARL) performance in the presence of Poisson mixture distribution in the PEWMA control charts. Through meticulously designed experiments, we explore different control parameter combinations and employ simulation to evaluate the process. Our graphs illustrate the performance of the PEWMA control chart, offering desired in-control ARL across parameter combinations. Finally, the performance of the PEWMA control chart is presented for the real process data of fastener production.

An optimal economic design of control chart for correlated Poisson variables: The double-dimension LCP chart

This work proposes a control chart that uses a statistic that is a linear combination of Poisson variables considering the sampling methodology called Double Dimension. This proposal aims to reduce costs when the Poisson variables used in process monitoring are expensive or difficult to measure. Therefore, they are not measured every time the control is performed but only when there are indications that the process is out of control. A user-friendly program was developed to obtain the performance of this chart and estimate the parameters that optimize the out-of-control ARL (average run length) for a shift that the user wants to detect as quickly as possible, restricted to a fixed value of in-control ARL. In addition, numerical results are presented in different scenarios, which allows verifying the performance and sensitivity of the proposed methodology. The results show that implementing the double dimension methodology offers similar performance as always measuring the cheap and expensive variables but with significant sampling savings.

An ARL-unbiased modified chart for monitoring autoregressive counts with geometric marginal distributions

Geometrically distributed counts arise in the industry. Ideally, they should be monitored using a control chart whose average run length (ARL) function achieves a maximum when the process is in control; that is, the chart is ARL-unbiased. Moreover, its in-control ARL should coincide with a reasonably large and prespecified value. Because dependence among successive geometric counts is occasionally a more sensible assumption than independence, we assess the impact of using an ARL-unbiased chart specifically designed for monitoring independent geometric counts when, in fact, these counts are autocorrelated. We derive an ARL-unbiased modified chart for monitoring geometric first-order integer-valued autoregressive or GINAR(1) counts. We provide compelling illustrations of this chart and discuss its use to monitor other autoregressive counts with a geometric marginal distribution.

An ARL-Unbiased Modified np-Chart for Autoregressive Binomial Counts

Abstract Independence between successive counts is not a sensible premise while dealing, for instance, with very high sampling rates. After assessing the impact of falsely assuming independent binomial counts in the performance of np-charts, such as the one with 3-σ control limits, we propose a modified np-chart for monitoring first-order autoregressive counts with binomial marginals. This simple chart has an in-control average run length (ARL) larger than any out-of-control ARL, i.e., it is ARL-unbiased. Moreover, the ARL-unbiased modified np-chart triggers a signal at sample t with probability one if the observed value of the control statistic is beyond the lower and upper control limits L and U. In addition to this, the chart emits a signal with probability γ L {\gamma_{L}} (resp. γ U {\gamma_{U}} ) if that observed value coincides with L (resp. U). This randomization allows us to set the control limits in such a way that the in-control ARL takes the desired value ARL 0 {\operatorname{ARL}_{0}} , in contrast to traditional charts with discrete control statistics. Several illustrations of the ARL-unbiased modified np-chart are provided, using the statistical software R and resorting to real and simulated data.

Multivariate semiparametric control charts for mixed-type data.

A useful tool that has gained popularity in the Quality Control area is the control chart which monitors a process over time, identifies potential changes, understands variations, and eventually improves the quality and performance of the process. This article introduces a new class of multivariate semiparametric control charts for monitoring multivariate mixed-type data, which comprise both continuous and discrete random variables (rvs). Our methodology leverages ideas from clustering and Statistical Process Control to develop control charts for MIxed-type data. We propose four control chart schemes based on modified versions of the KAy-means for MIxed LArge KAMILA data clustering algorithm, where we assume that the two existing clusters represent the reference and the test sample. The charts are semiparametric, the continuous rvs follow a distribution that belongs in the class of elliptical distributions. Categorical scale rvs follow a multinomial distribution. We present the algorithmic procedures and study the characteristics of the new control charts. The performance of the proposed schemes is evaluated on the basis of the False Alarm Rate and in-control Average Run Length. Finally, we demonstrate the effectiveness and applicability of our proposed methods utilizing real-world data.

The Performance of S Control Charts for the Lognormal Distribution with Estimated Parameters

Control charts, one of the powerful tools in statistical process control (SPC), are widely used to monitor and detect out-of-control processes in the manufacturing industry. Many researchers have pointed out the effects of using estimated parameters on the average run length (ARL) performance metric. Most of the previous literature has studied the expected value of the average run length (AARL) and the standard deviation of the average run length (SDARL) to evaluate the performance of control charts. In this article, we study the performance of three S control charts, the Shewhart S-chart, the median absolute deviation (MAD) control chart, and the lognormal S control chart, for a lognormal distribution in terms of the AARL and SDARL. Simulation results indicate the sample size to reach the specified in-control ARL value is very large. The lognormal S control chart has a smaller SDARL value than the other two S-charts.

On designing efficient multivariate exponentially weighted moving average control chart for compositional data using variable sample size

ABSTRACT Conventionally, the standard process monitoring control charts (s) focused on fixed sample size (). An optimal statistical scheme is proposed in this study using a variable sample size to enhance the performance of a multivariate exponentially weighted moving average () control chart () for compositional data () (i.e. - ) based on a coordinate representation using isometric log-ratio transformation (). A methodology is proposed to obtain the optimal parameters by considering the zero-state () average run length () and the steady-state () average run length () conditions of the process. The statistical performance of the proposed is evaluated based on a continuous-time Markov chain () method for both cases (i.e. the and the ) using a fixed value of in-control (IC) average run length . For benchmarking reasons, the out-of-control () performance of the - is compared against the traditional - with in terms of ; the proposed shows better performance than the - . The and of the - are always less than that of the - at some certain level of shifts. The proposed - performs, on average, () and (for ) more effectively than - . Moreover, it is found that the number of variables (d) has a negative impact on the run length characteristics of the - . When the value of d increases, the and of the - also increase. The of the - is less than the of the - for all the combinations of sample size (n) and d, i.e. under , the proposed performs on average (for d = 3) and (for d = 5) better than the situation. An example of an industrial problem of grid production for a European plant is also given to study the statistical significance and implementation of the - over the existing - .

Distribution-free multivariate time-series monitoring with analytically determined control limits

We consider the problem of detecting a shift in the mean of a multivariate time-series process with general marginal distributions and general cross- and auto-correlation structures. We propose a distribution-free monitoring procedure that does not need model fitting or trial-and-error calibration for control limits, which makes the procedure convenient to be implemented when a facility consists of many processes to be monitored. The main idea is to convert each observation vector into a one-dimensional T 2 quantity that captures cross-correlation. The T 2 quantities form a univariate auto-correlated process, and CUSUM statistics are constructed on the T 2 quantities. Then using the fact that the CUSUM statistics on the auto-correlated process behave as a reflected Brownian motion asymptotically under some conditions, the control limits of the CUSUM procedure are analytically determined by setting the first-passage time of the Brownian motion equal to a target in-control average run length. We compare the performance of our procedure with three competing procedures on simulated data with various cross- and auto-correlation and real data from a wafer etching process. The proposed procedure delivers actual in-control average run lengths close to the target and shows comparable or better performance in detecting a shift in mean than the competitors.

The 𝑛𝑝-Chart with 3-𝜎 Limits and the ARL-Unbiased 𝑛𝑝-Chart Revisited

Abstract In the statistical process control literature, counts of nonconforming items are frequently assumed to be independent and have a binomial distribution with parameters ( n , p ) (n,p) , where 𝑛 and 𝑝 represent the fixed sample size and the fraction nonconforming. In this paper, the traditional n ⁢ p np -chart with 3-𝜎 control limits is reexamined. We show that, even if its lower control limit is positive and we are dealing with a small target value p 0 p_{0} of the fraction nonconforming ( p ) (p) , this chart average run length (ARL) function achieves a maximum to the left of p 0 p_{0} . Moreover, the in-control ARL of this popular chart is also shown to vary considerably with the fixed sample size 𝑛. We also look closely at the ARL function of the ARL-unbiased n ⁢ p np -chart proposed by Morais [An ARL-unbiased n ⁢ p np -chart, Econ. Qual. Control 31 (2016), 1, 11–21], which attains a pre-specified maximum value in the in-control situation. This chart triggers a signal at sample 𝑡 with probability one if the observed number of nonconforming items, x t x_{t} , is beyond the lower and upper control limits (𝐿 and 𝑈), probability γ L \gamma_{L} (resp. γ U \gamma_{U} ) if x t x_{t} coincides with 𝐿 (resp. 𝑈). A graphical display for the ARL-unbiased n ⁢ p np -chart is proposed, taking advantage of the qcc package for the statistical software R. Furthermore, as far as we have investigated, its control limits can be obtained using three different search algorithms; their computation times are thoroughly compared.

Power guidelines for process monitoring

We present a process for designing monitoring procedures that includes practical power guidelines. These guidelines are based upon the average run length (ARL). The specific guideline metric is the ratio of the in-control ARL (ARLic) to the out-of-control ARL (ARLoc). Our recommended design process uses that ARL Ratio, in combination with the ARLic, ARLoc, and ARL curve, to design effective process monitoring procedures. For adequate power, we generally recommend an ARL Ratio of 20 or more and argue that monitoring procedures with ARL Ratios less than 10 should usually be avoided. An area of caution lies between ARL Ratios of 10 and 20, allowing us to propose a stoplight type model for use in monitoring procedure design. We also discuss exceptions to these guidelines as well as a methodology to incorporate power considerations in other approaches to control chart design.

Attribute Control Chart for Rayleigh Distribution Using Repetitive Sampling under Truncated Life Test

A control chart is an important tool in statistical process monitoring that is useful to monitor and improve production process quality. In this article, an attribute control chart using repetitive sampling under a truncated life test is proposed for monitoring the mean life of the product where the lifetime follows the Rayleigh distribution. The repetitive sampling parameters and the control limit coefficients of the chart are determined so that the in-control average run length (ARL) is very close to the target ARL. Tables of ARL values for various shift sizes in the scale parameter were presented, and the performance of the proposed chart is compared with the existing attribute control charts using the out-of-control ARL. The proposed control chart is shown to outperform the existing control charts in terms of ARL. An illustrative example is given to demonstrate the application of the proposed chart.

A design parameter-free geometric Kullback-Leibler information control chart for monitoring Bernoulli processes

This paper considers a design parameter-free Kullback-Leibler information (KLI) control chart for monitoring the nonconforming proportion p of Bernoulli processes in Phase II. The chart statistic is derived from the Kullback-Leibler information based on a geometric distribution. Unlike conventional charts for Phase II, the proposed chart does not require design parameters. The users only need to determine a desired in-control average run length. The performance of conventional control charts deteriorates if the prespecified optimal design parameter is not appropriate for the actual size of the shift in the monitored quality characteristic. A design parameter-free approach prevents this situation, and the time needed to find the optimal design parameter can be saved as well. The performance of charts is evaluated by the average number of observations to signal under a specific shift in p, and the penalized relative mean index assesses the overall performance. The proposed chart has the best overall performance for detecting an upward shift in p compared with the cumulative count of conforming (CCC-r), Bernoulli cumulative sum (CUSUM), Bernoulli exponentially weighted moving average (EWMA), Bernoulli generalized likelihood ratio (GLR), geometric CUSUM, and binomial GLR charts. The KLI chart has decent performance for detecting a downward shift in p if the size of the shift is small. For detecting a possible shift in p to both sides, the overall performance of the KLI chart is better than those of the charts.

Effect of Parameter Estimation on the Performance of Shewhart <img src=image/13425522_01.gif>-joint Chart Looked at in Terms of the Run Length Distribution

Using spread-charts to monitor process variation and thereafter using the <img src=image/13425522_02.gif>-chart to monitor the process mean after is a common practice. To apply these charts independently using estimated 3-sigma limits is common. Recently, some authors considered the application of <img src=image/13425522_02.gif> and R-charts together as a charting scheme, <img src=image/13425522_01.gif>-chart when the standards are known, Case KK, only the mean standard is known, Case KU and both standards unknown, Case UU. The average run length (ARL) performance criterion was used. However, because of the skewed nature of the run length (RL) distribution, many authors have frowned at the use of ARL as a sole performance measure and encouraged the percentiles of the RL distribution instead. Therefore, the cdfs of the RLs of the chart under the cases mentioned will be derived in this work, and the percentiles are used to look at the chart for Case KU and the yet to be considered case of the chart, Case UK where only the process variance is known is included for comparison. These are the contribution to the existing literature. <img src=image/13425522_01.gif>-chart performed better in Case KU than in Case UK and the unconditional in-control median run length described the behavior of the chart better than the in-control ARL.

A VARIABLE SAMPLE SIZE SYNTHETIC CHART FOR THE COEFFICIENT OF VARIATION

A variable sample size (VSS) synthetic chart to monitor the coefficient of variation is proposed in this paper to improve the performance of the existing synthetic chart. A description of how the chart operates, as well as the formulae for various performance measures (i.e., the average run length (ARL), standard deviation of the run length (SDRL), average sample size (ASS), and expected average run length (EARL)) are proposed. The algorithms that optimise the out-of-control ARL (ARL1) and EARL (EARL1), subject to the constraints in the in-control ARL (ARL0) and ASS (ASS0), are also proposed. Subsequently, optimal charting parameters for various numerical examples are obtained. The proposed chart shows a significant improvement over the existing synthetic -chart. Comparisons with other -charts also show that the proposed chart performs better than the Shewhart- and VSS- charts under all cases, while showing better performance than the exponentially weighted moving average (EWMA) and VSS EWMA- charts for moderate and large shift sizes. Finally, this paper shows the implementation of the proposed chart on an actual industrial example.