Process Monitoring Using Truncated Gamma Distribution

Exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) charts are used to detect smaller shifts in process parameters. The usual EWMA and CUSUM charts depend on the normality assumption for a better detection ability. This study proposes an efficient EWMA control chart based on the spirit of Tukey control chart, especially designed for skewed distributions. The performance of the proposed and the competing charts is measured using different length properties such as average run length (ARL), standard deviation of run length (SDRL), and median run length (MDRL). We have observed that the proposed chart is quite efficient at detecting process shifts of smaller magnitude, especially for skewed distributions. For practical considerations, the proposed chart is implemented at aerospace manufacturing data on industrial production index.

<abstract> <p>Nonconforming events are rare in high-quality processes, and the time between events (TBE) may follow a skewed distribution, such as the gamma distribution. This study proposes one- and two-sided triple homogeneously weighted moving average charts for monitoring TBE data modeled by the gamma distribution. These charts are labeled as the THWMA TBE charts. Monte Carlo simulations are performed to approximate the run length distribution of the one- and two-sided THWMA TBE charts. The THWMA TBE charts are compared to competing charts like the DHWMA TBE, HWMA TBE, THWMA TBE, DEWMA TBE, and EWMA TBE charts at a single shift and over a range of shifts. For the single shift comparison, the average run length (ARL) and standard deviation run length (SDRL) measures are used, whereas the extra quadratic loss (EQL), relative average run length (RARL) and performance comparison index (PCI) measures are employed for a range of shifts comparison. The comparison reveals that the THWMA TBE charts outperform the competing charts at a single shift as well as at a certain range of shifts. Finally, two real-life data applications are presented to evaluate the applicability of the THWMA TBE charts in practical situations, one with boring machine failure data and the other with hospital stay time for traumatic brain injury patients.</p> </abstract>

Zero-inflated models are frequently used to deal with data having many zeros. A commonly used model for over-dispersed data containing zeros is known as the zero-inflated Poisson model. However, to account for the heterogeneity of counts that leads to excess variance besides inflation of zeros in the data using a more flexible model than the zero-inflated Poisson model, a zero-inflated negative binomial (ZINB) is suggested. In the present study, Shewhart and exponentially weighted moving average (EWMA) control charts are suggested to monitor the ZINB data. The charts are compared using the average run length and standard deviation of run length by using extensive Monte Carlo simulations. Besides a comprehensive simulation study assuming different settings of parameters of ZINB, a real data set is used to show the practicality of the proposed charts. The results indicate that the EWMA chart is better than the Shewhart chart.

PurposeThis work aims to focuse on improving the performance of the new exponentially weighted moving average (NEWMA) scheme for monitoring process dispersion. The authors use the generalized time-varying fast initial response (GFIR) to further enhance the detection ability of variability NEWMA control charts at the process startup. The performance of the proposed chart and other schemes discussed in this article are evaluated; and compared using the average run length (ARL) and standard deviation run length (SDRL) measures. It is observed that the ARL of the proposed scheme is quicker in detecting small and moderate shifts in the process dispersion than its counterparts. The real-life application of the proposed scheme is presented.Design/methodology/approachThe dynamic parameter of GFIR is used to enhance the detection ability of variability NEWMA control charts. The authors apply GFIR to the control limit of variability NEWMA scheme. This further narrows the control limit, hence enabling it to swiftly detect small and moderate changes in process dispersion.FindingsThe authors present the performance comparisons by examining the ARL properties of the proposed chart and its counterparts. The performance comparison shows that the proposed chart is highly sensitive in detecting small and intermediate process shifts. The real-life application presented also supports the study’s conclusion from the simulation studies. The performance comparison of the proposed chart and its counterparts shows that the proposed scheme is efficient in detecting process abnormalities, especially at the startup.Originality/valueIn terms of the control limits, the proposed chart is the generalized variability NEWMA control chart in which all the previously proposed NEWMA variant schemes can be obtained. Also, the newly proposed control scheme is more efficient in detecting small or moderate persistent shifts in the process dispersion.

ABSTRACTRuns rules are usually used with Shewhart-type charts to enhance the charts' sensitivities toward small and moderate shifts. Abbas et al. in 2011 took it a step further by proposing two runs rules schemes, applied to the exponentially weighted moving average (EWMA) chart and evaluated their average run length (ARL) performances using simulation. They showed that the proposed schemes are superior to the classical EWMA chart and other schemes being investigated. Besides pointing out some erroneous ARL and standard deviation of the run length (SDRL) computations in Abbas et al., this paper presents a Markov chain approach for computing the ARL, percentiles of the run length (RL) distribution and SDRL, for the two runs rules schemes of Abbas et al. Using Markov chain, we also propose two combined runs rules EWMA schemes to quicken the two schemes of Abbas et al. in responding to large shifts. The runs rules (basic and combined rules) EWMA schemes will be compared with some existing control charting methods, where the former charts are shown to prevail.

Measurement error is a usually met distortion factor in real-world applications that influences the outcome of a process. The performance of the control charts can be affected in the presence of measurement error which also leads to erroneous conclusions regarding the average run length. In this paper, we proposed regression estimator-based mixed exponentially weighted moving average-cumulative sum (MxEMCUS) control chart to monitor the process mean and investigate the effect of measurement error on shift detection ability of the proposed control chart. The performance of MxEMCUS control chart is evaluated in the cases of mean shift with and without measurement error. The linear covariate model is used to evaluate the control chart in the presence of measurement error. The case of multiple measurements and linear increasing variance is also investigated in the presence of measurement error. The average run length and standard deviation of run length are used as the performance measuring tool. Based on simulation results, it is concluded that in the presence of measurement error, proposed chart has better performance as compared to other existing charts. An example with real data set is given to demonstrate the implementation of MxEMCUS control chart.

On the average run lengths of quality control schemes using a Markov chain approach

Bayesian Control charts are emerging as the most efficient statistical tools for monitoring manufacturing processes and providing effective control over process variability. The Bayesian approach is particularly suitable for addressing parametric uncertainty in the manufacturing industry. In this study, we determine the monitoring threshold for the shape parameter of the Inverse Gaussian distribution (IGD) and design different exponentially-weighted-moving-average (EWMA) control charts based on different loss functions (LFs). The impact of hyperparameters is investigated on Bayes estimates (BEs) and posterior risks (PRs). The performance measures such as average run length (ARL), standard deviation of run length (SDRL), and median of run length (MRL) are employed to evaluate the suggested approach. The designed Bayesian charts are evaluated for different settings of smoothing constant of the EWMA chart, different sample sizes, and pre-specified false alarm rates. The simulative study demonstrates the effectiveness of the suggested Bayesian method-based EWMA charts as compared to the conventional classical setup-based EWMA charts. The proposed techniques of EWMA charts are highly efficient in detecting shifts in the shape parameter and outperform their classical counterpart in detecting faults quickly. The proposed technique is also applied to real-data case studies from the aerospace manufacturing industry. The quality characteristic of interest was selected as the monthly industrial production index of aircraft from January 1980 to December 2022. The real-data-based findings also validate the conclusions based on the simulative results.

The article introduces a novel Bayesian AEWMA Control Chart that integrates different loss functions (LFs) like the square error loss function and Linex loss function under an informative prior for posterior and posterior predictive distributions, implemented across diverse ranked set sampling (RSS) designs. The main objective is to detect small to moderate shifts in the process mean, with the average run length and standard deviation of run length serving as performance measures. The study employs a hard bake process in semiconductor production to demonstrate the effectiveness of the proposed chart, comparing it with existing control charts through Monte Carlo simulations. The results underscore the superiority of the proposed approach, particularly under RSS designs compared to simple random sampling (SRS), in identifying out-of-control signals. Overall, this study contributes a comprehensive method integrating various LFs and RSS schemes, offering a more precise and efficient approach for detecting shifts in the process mean. Real-world applications highlight the heightened sensitivity of the suggested chart in identifying out-of-control signals compared to existing Bayesian charts using SRS.

The nonparametric (distribution-free) control charts are robust alternatives to the conventional parametric control charts when the form of underlying process distribution is unknown or complicated. In this paper, we consider two new nonparametric control charts based on the Hogg–Fisher–Randle (HFR) statistic and the Savage rank statistic. These are popular statistics for testing location shifts, especially in right-skewed densities. Nevertheless, the control charts based on these statistics are not studied in quality control literature. In the current context, we study phase-II Shewhart-type charts based on the HFR and Savage statistics. We compare these charts with the Wilcoxon rank-sum chart in terms of false alarm rate, out-of-control average run-length and other run length properties. Implementation procedures and some illustrations of these charts are also provided. Numerical results based on Monte Carlo analysis show that the new charts are superior to the Wilcoxon rank-sum chart for a class of non-normal distributions in detecting location shift. New charts also provide better control over false alarm when reference sample size is small.

Control charts for monitoring the coefficient of variation (γ) are useful for processes with an inconsistent mean (μ) and a standard deviation (σ) which changes with μ, by monitoring the consistency in the ratio σ over μ. The synthetic‐γ chart is one of the charts proposed to monitor γ, and its attractiveness lie in waiting until a second point to fall outside the control limits before a decision is made. However, existing synthetic‐γ charts do not differentiate between the points falling outside the upper control limit (UCL) and lower control limit (LCL). Hence, this paper proposes a side‐sensitive synthetic‐γ chart, where successive nonconforming samples must either fall above the UCL or below the LCL. Formulae to compute the average run length (ARL), the standard deviation of the run length (SDRL) and expected average run length (EARL) are derived using the Markov chain approach, and the algorithms to obtain the optimal charting parameters are proposed. Subsequently, the optimal charting parameters, ARL, SDRL and EARL values for various numerical examples are shown. Comparisons show that the side‐sensitive synthetic‐γ chart consistently outperforms the existing synthetic‐γ chart, especially for small shifts. The proposed chart also consistently outperforms the Shewhart‐γ chart, while showing comparable or better performance than the Exponentially Weighted Moving Average (EWMA) chart for most shift sizes, except for very small shifts. Finally, this paper shows the implementation of the proposed chart on an industrial example.

Control charts are prominent tools in monitoring various variables of manufacturing processes. In this paper, the control chart based on sample median is proposed for process location parameter using Shewhart?s approach. The proposed control chart is developed under some distributions including normal distribution. Its performance is studied in terms of power, average run length (ARL) and standard deviation of run length (SDRL). Also, the effect of non normality on this control chart is discussed.. KEYWORDS :ARL, Control limits, Location parameter, Median, Non-normality, Power.

Control charts are used to detect assignable causes in different manufacturing and nonmanufacturing processes. This study presents a new maximum exponentially weighted moving average (Max-EWMA) chart for joint unit interval time and magnitude monitoring. To this end, beta distribution is considered for time whereas simplex distribution is used for magnitude. Average run length (ARL), standard deviation of run length (SDRL), and different quantiles are used to evaluate the performance of the Max-EWMA chart. A real data example is also included in the study to show the application of the proposed chart. The results provide evidences that the Max-EWMA chart is efficient in detecting small- to medium-sized shifts.

In this study, we propose a design for a control chart for the normal distribution using a modified double sampling scheme. The probability of in control, out of control, the average run length (ARL), and the standard deviation of run length (SDRL) of the proposed control chart is derived. The proposed control chart using the modified double sampling scheme is found to be more efficient in detecting small shifts of a process in comparison with existing, single, and double sampling control charts. The efficiency of the proposed control chart is investigated in terms of ARL and SDRL via simulation and application to a real data set.

Exponentially Weighted Moving Average (EWMA) control charts yield insights into data in a way more comprehensible to the practitioners and researchers because of its capability in discovering small to moderate process mean shifts. EWMA control chart is incorporated with conforming run length (CRL) chart, named synthetic EWMA chart, to enhance the performance of the chart in detecting the out-of-control signal. Synthetic EWMA chart based on ranked set sampling (RSS) for monitoring process mean has been proposed as it advanced the detection of chart over a series of mean shifts. With the situation that normality assumption is scarcely attain in practice, we proposed synthetic EWMA median chart based on RSS. Rather than select average run length (ARL) as sole performance evaluating tool, the median and percentiles of run-length distribution are used to examine the performance of the proposed chart as it provides more information on the entire run-length distribution. Near-optimal parameters of the proposed chart will be acquired by setting the incontrol ARL at a designated value. The run length performances of the proposed chart are then compared with the existing charts such as EWMA median chart based on RSS.