Extra Quadratic Loss Research Articles

Performance of the Shiryaev–Roberts‐Type Scheme in Monitoring Weibull Shape Parameter Based on Type II Censored Data

ABSTRACTThe Weibull distribution is very effective in modelling many different failure mechanisms given its intrinsic flexibility to represent different hazard functions via suitable selections of the shape and scale parameters. Given that the shape parameter of the distribution directly controls the hazard rate, we discuss a Shiryaev–Roberts (SR)–type scheme for monitoring the shape parameter with a fixed scale parameter when using Type II right‐censored Weibull lifetime data. A comparative analysis with existing schemes from the literature is provided by the average run length (ARL), along with the standard deviation of run length (SDRL) and some run‐length percentiles. The average extra quadratic loss (AEQL), the relative mean index (RMI) and the performance comparison index (PCI) are also considered to assess the overall performance. Our simulation results display certain advantages of the SR‐type scheme compared to those of traditional charts in detecting changes in censored lifetime data. An example based on the breaking strengths of the carbon fibre is also presented to demonstrate the proposed method's effectiveness in real situations.

A time-varying EWMA-CV control chart based on Upper Triangular Minimax ranked set sampling method

When the process mean fluctuates with time and process standard deviation is a linear function of process mean, the coefficient of variation is usually used to measure the process variation in industry. On the other hand, the use of Ranked Set Sampling as a smart sampling technique enables the ranking of samples before measurement. The purpose of this research is to introduce a new sampling method referred to as Upper Triangular Minimax Ranked Set Sampling and design a time-varying Exponentially Weighted Moving Average control chart to monitor the coefficient of variation. Also, in this research, average run length, standard deviation of run length, median of run length, extra quadratic loss, relative average run length and performance comparison index criteria under two states including zero-state and steady-state have been used to compare the performance of the designed control chart with the ones provided in the literature. The results show the superiority of the proposed control chart based on the novel sampling method rather than the control charts available in the literature. The application of the proposed control chart is also shown through a real case and compared with available competitors in the literature.

A novel adaptive CUSUM system for efficient process mean monitoring: An application in piston ring manufacturing process

The adaptive versions of charting tools are well-established statistical monitoring techniques for detecting unknown changes in the process over a range of shifts. A re-weighted adaptive CUSUM mean (RACUSUM) charting scheme has been proposed in this study for monitoring the range of shifts. The proposed RACUSUM statistic estimates the unknown changes in the ongoing process using an EWMA statistic by plugging in an unbiased shift estimator and choosing a suitable smoothing constant according to the estimated shift to update the slack value of the CUSUM statistic. By employing Monte Carlo simulation using the zero- and steady-state processes, the proposal’s run-length (RL) profiles have been calculated. The proposed RACUSUM chart shows more sensitive behavior under the shift delay process than in the case of zero-state. A comparative analysis between the proposal and competitors has been presented using average RL, extra quadratic loss (EQL), and the relative mean index (RMI) as metrics for 1-sided and 2-sided charting schemes. The robustness of the proposed 2-sided RACUSUM chart against non-normality has also been reported. The proposal’s applicability has been demonstrated using two cases, one from the artificial and the other from a dataset of piston ring manufacturing industrial process.

Efficient and distribution-free charts for monitoring the process location for individual observations

Sudden and sequential variations are crucial in industrial and production processes. To track these consistent changes in process parameters, effective charting methods are needed. The generally weighted moving average (GWMA) chart outperforms the exponentially weighted moving average (EWMA) chart in detecting small changes under various design parameters. However, its application relies on process distribution normality assumptions. This study presents new distribution-free GWMA control charts for individual measurements when the central limit theorem doesn't apply under simple random sampling. The charts' robustness and performance are evaluated under symmetric, skewed, and contaminated process environments, using run length properties, relative mean index (RMI), and extra quadratic loss (EQL) for overall assessment. The proposed chart outperforms existing charts in detecting specific and over-the-range shifts with appropriate design parameter choices. It's been applied to an electronics dataset where voltage on constant capacitance serves as a key quality characteristic, validating the theoretical findings.

An enhanced nonparametric quality control chart with application related to industrial process

In various practical situations, the information about the process distribution is sometimes partially or completely unavailable. In these instances, practitioners prefer to use nonparametric charts as they don’t restrict the assumption of normality or specific distribution. In this current article, a nonparametric double homogeneously weighted moving average control chart based on the Wilcoxon signed-rank statistic is developed for monitoring the location parameter of the process. The run-length profiles of the newly developed chart are obtained by using Monte Carlo simulations. Comparisons are made based on various performance metrics of run-length distribution among proposed and existing nonparametric counterparts charts. The extra quadratic loss is used to evaluate the overall performance of the proposed and existing charts. The newly developed scheme showed comparatively better results than its existing counterparts. For practical implementation of the suggested scheme, the real-world dataset related to the inside diameter of the automobile piston rings is also used.

Ameliorating the diagnostic power of combined shewhart-memory-type control chart strategies for mean

A great deal of research in statistical process control (SPC) involves the integration of different ideas to achieve the optimal detection of anomalies in the location parameter of a process. Likewise, the prime rationale of the combined (Shewhart-Memory-type) strategies is to promptly detect the shift of any size: small, moderate or large. An effort, along the same lines, is made to detect shifts in the mean of a normally distributed process. The usual difference estimator of the process mean instead of the sample mean is employed in the Mixed EWMA-Crosier CUSUM Combined Shewhart Mixed EWMA-CUSUM and Combined Shewhart Mixed EWMA-Crosier CUSUM charts to make their upgraded versions. This enhances the sensitivity of these charts against any shift size. The performance of the charts is appraised and compared with some notable charts of the same ilk, with reference to the average run length (ARL), relative average run length, extra quadratic loss and performance comparison index. An illustrative example is also provided.

Novel combined Shewhart-CUmulative EWMA-SUM mean charts without- and with measurement error

The precision of process monitoring often encounters challenges in determining the exact shift size. Therefore, combined control charts have gained considerable attention because of their excellent speed to detect simultaneously small-to-moderate and large-size shifts. The effectiveness of the applied quality control methods strongly depends on the performance of the measurement system. Measurement error presence contributes significantly negatively toward the performance of the usual control charting schemes. This article proposes novel two-sided combined Shewhart-Cumulative EWMA-sum (Shewhart-CUESUM) control charts designed to efficiently monitor the mean of normally distributed processes. In addition, to address measurement errors, the M-Shewhart-CUESUM chart is proposed, incorporating an additive measurement error model. Evaluation of the charts through Monte-Carlo simulations, considering metrics such as average run length (ARL), extra quadratic loss, relative ARL, and performance comparison index. It is found that the combined Shewhart-CUESUM outperforms than CUESUM chart. The results show that the presence of measurement errors can significantly diminish the charts’ performance, which can be mitigated by utilizing a multiple measurements scheme. Among the different well-established combined charts examined, the M-Shewhart-CUESUM chart shows considerably more sensitive to detecting simultaneously detect small and large size shifts. To employ simulated datasets to illustrate the impact of measurement errors and demonstrate the implications of the proposed charts on process mean shifts.

The Development and Evaluation of Homogenously Weighted Moving Average Control Chart based on an Autoregressive Process

This research aims to investigate a Homogenously Weighted Moving Average (HWMA) control chart for detecting minor and moderate shifts in the process mean. A mathematical model for the explicit formulae of the average run length (ARL) of the HWMA control chart based on the autoregressive (AR) process is presented. The efficacy of the HWMA control chart is evaluated based on the average run length, the standard deviation of run length (SDRL), and the median run length (MRL). As illustrations of the design and implementation of the HWMA control chart, numerical examples are provided. In numerous instances, a comparative analysis of the HWMA control chart relative to the Extended Exponentially Weighted Moving Average (Extended EWMA) and cumulative sum (CUSUM) control charts with mean process shifts is performed in detail. Additionally, the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI) are utilized to evaluate the performance of control charts. For various shift sizes, the HWMA control chart is superior to the Extended EWMA and CUSUM control charts. This study applies empirical data from the area of economics to validate the explicit formula of ARL values for the HWMA control chart. Doi: 10.28991/HIJ-2024-05-01-02 Full Text: PDF

Generalized multivariate EWMA control chart supplemented with bivariate ranked set techniques

ABSTRACT Existing univariate exponential weighted moving average (EWMA) control charts with ranked set schemes are used for monitoring related characteristics independently. Also, existing multivariate EWMA control charts are generally applicable for simple random sampling (SRS) based processes. In this work, a generalized multivariate EWMA control chart is proposed with bivariate ranked set techniques (BVRST) for simultaneously monitoring. Furthermore, an extensive comparative analysis between proposed generalized multivariate EWMA and classical control chart is conducted. In this regard, we have used various performance measures, which cover average run length, median of run length, relative average run length, performance comparison index, and extra quadratic loss. The performance measures are obtained using the Monte Carlo simulation (MCS) approach. Results revealed that the proposed generalized multivariate EWMA control chart has an outstanding detection ability relative to the existing multivariate control charts. Additionally, existing multivariate control charts are found special cases of the proposed control chart. A real-life example from the agriculture field is included in which chemical properties (residual sodium and calcium-magnesium contents) of irrigation water are monitored through the control charts. The results revealed that the proposed control chart is proficient in diagnosing small variations in the chemical characteristics relative to the existing control charts.

Performance Evaluation of HWMA Control Chart based on AR(p) with Trend Model to Detect Shift Process Mean

The main goal of this study is to establish explicit solutions for the average run length (ARL) of the Homogenously Weighted Moving Average control chart when subjected to autoregressive with trend process. The accuracy of the explicit formula for the ARL is evaluated in comparison to the numerical integral equation method. To evaluate the two approaches, the accuracy percentage was employed. A determination is carried out of the HWMA control chart’s effectiveness using the median run length (MRL), the standard deviation of run length (SDRL), and the average run length (ARL). A comprehensive comparison is performed between the HWMA control chart, the Extended Exponentially Weighted Moving Average (EEWMA), and the cumulative sum (CUSUM) control charts with mean process shifts to illustrate the design and implementation of the HWMA control chart. As criteria for various values of design parameters, the performance of these control charts can also be evaluated using the relative mean index (RMI), the average extra quadratic loss (AEQL), and the performance comparison index (PCI). To evaluate the effectiveness of our explicit formula approach, we employ this formula on copper price data.

A nonparametric mixed exponentially weighted moving average-moving average control chart with an application to gas turbines.

This study aims to develop a nonparametric mixed exponentially weighted moving average-moving average (NPEWMA-MA) sign control chart for monitoring shifts in process location, particularly when the distribution of a critical quality characteristic is either unknown or non-normal. In literature, the variance expression of the mixed exponentially weighted moving average-moving average (EWMA-MA) statistic is calculated by allowing sequential moving averages to be independent, and thus the exclusion of covariance terms results in an inaccurate variance expression. Furthermore, the effectiveness of the EWMA-MA control chart deteriorates when the distribution of a critical quality characteristic deviates from normality. The proposed NPEWMA-MA sign control chart addresses these by utilizing the corrected variance of the EWMA-MA statistic and incorporating the nonparametric sign test into the EWMA-MA charting structure. The chart integrates the moving average (MA) statistic into the exponentially weighted moving average (EWMA) statistic. The EWMA-MA charting statistic assigns more weight to recent w samples, with weights for previous observations decling exponentially. Monte Carlo simulations assess the chart's performance using various run length (RL) characteristics such as average run length (ARL), standard deviation of run length (SDRL), and median run length (MRL). Additional measures for overall performance include the average extra quadratic loss (AEQL) and relative mean index (RMI). The proposed NPEWMA-MA sign control chart demonstrates superior performance compared to existing nonparametric control charts across different symmetrical and asymmetric distributions. It efficiently detects process shifts, as validated through both a simulated study and a real-life example from a combined cycle power plant.

A new adaptive CUSUM control chart based on Hampel function to monitor location parameter

AbstractTo identify small‐to‐moderate shifts (also known as outliers and anomalies) in the process parameters (location and/or dispersion), the classical cumulative sum (CUSUM) and exponential weighted moving average (EWMA) control charts are famous. The traditional CUSUM control chart is less capable to detect out‐of‐control signals of 1.0 standard deviation shifts. The advanced versions of the traditional CUSUM and EWMA control charts are ACUSUM (adaptive CUSUM) and AEWMA (adaptive EWMA) control charts, respectively. The aforementioned issue of the traditional CUSUM control chart can be addressed through the ACSUUM control chart. It also identifies various sizes of shifts in the process parameters. In order to efficiently identify and remove outliers and anomalies (i.e., shifts) from datasets, the score functions are typically used to establish ACUSUM control charts. The goal of the study is to propose a new ACUSUM control chart based on the Hampel (score) function and it is symbolized as control chart. Different sizes of shift in the process location parameter can be distinguished using the control chart. The primary novelty is the substitution of a proposed time‐varying parameter for the typical CUSUM control chart's prioritization of the process characteristic by appropriate weights. The proposed control chart's run length feature is obtained through the Monte Carlo simulation technique for zero‐state and steady‐states. For evaluating a single shift, performance measures like average run length (ARL), standard deviation of run length (RL), and different percentiles of run length are preferred. However, for evaluating a range of shifts, extra quadratic loss, relative average run length, and performance comparison index are used. Comparison based on performance measures shows the superiority of the proposed control chart against the classical CUSUM and EWMA, ACUSUM, AEWMA, mixed EWMA‐CUSUM, and mixed CUSUM‐EWMA control charts. Furthermore, the proposed control chart is also implemented with real‐life data which is taken from cost of medical insurance to demonstrate its competency in statistical process control field for quality engineers, practitioners, experts, and researchers.

Optimal designs of the omnibus SPRT control chart for joint monitoring of process mean and dispersion

The vast majority of control schemes related to the sequential probability ratio test (SPRT) are designed for the purpose of monitoring only the process mean. Nonetheless, most manufacturing processes are vulnerable to external factors that cause the process mean and variability to change simultaneously. It is, therefore, crucial to consider a joint scheme for monitoring both the location and scale parameters of a production process. In this article, we develop a scheme that combines both mean and variance information in a single SPRT, known as the omnibus SPRT (OSPRT) chart. Expressions for the run-length properties of the OSPRT chart are derived by means of the Markov chain approach. We also propose optimal designs for the OSPRT chart based on two different metrics, i.e. by minimising the average time to signal and the average extra quadratic loss. Through a comprehensive analysis, this article reveals that the optimal OSPRT chart outperforms the classical -S, weighted-loss cumulative sum, absolute-value SPRT, and two maximum weighted-moving-average-type charts. The optimal OSPRT chart also has the advantage of collecting a small number of samples on average before producing a decision. Finally, the implementation of the OSPRT chart is presented with a wire bonding industrial dataset.

Integration of hybrid score functions to design new adaptive EWMA control chart to monitor the process mean shift

Small-to-moderate shifts in the process parameters can be detected using the traditional Exponentially Weighted Moving Average (EWMA) control chart. The classical EWMA control chart's parameter helps only to identify the specific size of the shift. However, various shifts occur in real-world processes, so the conventional EWMA control chart performs inefficiently for off-target shifts. This problem has been confronted by applying an adaptive EWMA (AEWMA) control chart, an advanced version of the classical EWMA control chart. The score functions are utilized to design the AEWMA control chart. Generally, the Huber and Bi-square (score) functions have unique characteristics, such as monotone and redescended, respectively.There has not been any investigation into the simultaneous and/or joint deployment of score functions. So, by taking this point as motivation, this study aims to propose a new score function by blending the Huber and Bi-square characteristics into one hybrid formulation and then utilize it to design a new AEWMA control chart to monitor different sizes of the shift in the process location parameter. To evaluate the proposed control chart performance against other control charts, Monte Carlo simulation is applied to obtain the run length properties under zero-state and steady-state environments. Performance measures, such as average run length, standard deviation of run length, and median of run length, are considered to evaluate the performance at a single shift. In contrast, extra quadratic loss, relative average run length, and performance comparison index are utilized to evaluate effectiveness for a specific range of shifts. The comparison based on performance measures and visual presentation reveals the superiority of the proposed control chart. Besides, the proposed control chart is employed in the real-world chemical engineering process industry problem, demonstrating excellent detection ability of anomalies in air pollutant data. The comparison based on performance measures and visual presentation reveals the good performance of the proposed control chart.

Mixed Tukey Exponentially Weighted Moving Average-Modified Exponentially Weighted Moving Average Control Chart for Process Monitoring

The goal of this study is to present the mixed Tukey exponentially weighted moving average-modified exponentially weighted moving average control chart (MEME-TCC) for monitoring process location with symmetric and skewed distributions in an attempt to significantly improve detection ability. With the benefits of nonparametric assumption robustness. The average and median run lengths are supporting measurements for assessing the performance of a monitoring scheme using Monte Carlo simulation. Furthermore, the average extra quadratic loss (AEQL), relative mean index (RMI), and performance comparison index (PCI) can all be used to evaluate overall performance criteria. The proposed chart is compared with existing charts such as; EWMA, MEWMA, TCC, MEME, MMEE, and MMEE-TCC. The comparison result shows that the proposed chart is the best control chart for detecting small to moderate shifts among all distributional settings. Nevertheless, the EWMA chart detects large shifts more effectively than other charts, except in the case of the gamma distribution, where MEWMA performs best. The results of adapting the proposed control chart to two sets of real data corresponded to the research findings. Doi: 10.28991/ESJ-2023-07-03-014 Full Text: PDF

Tukey-EWMA control chart with variable sampling intervals for process monitoring

In this study, the Tukey-Exponential Weighted Moving Average control chart with variable sampling interval (VSI Tukey-EWMA) is proposed. The performance of the proposed control chart is investigated for several distributions and compared with Tukey-EWMA and VSI EWMA control charts. We show that the VSI Tukey-EWMA control chart outperforms the Tukey-EWMA control chart and it performs similarly to the VSI EWMA control chart. The average extra quadratic loss is calculated to examine the overall performance over the whole process shift range by using the Markov chain approach. Finally, the VSI Tukey-EWMA and Tukey-EWMA control charts are applied on real data.

An unbiased function‐based Poisson adaptive EWMA control chart for monitoring range of shifts

AbstractFor monitoring the number of nonconformities per unit in industrial processes during the inspection, Poisson control charts are most widely deployed. These charting structures are referred to as attribute control charts as the quality characteristic under study is based on a nominal scale other than a quantitative or measured scale. In this study, Poisson adaptive exponentially weighted moving average (PAEWMA) has been designed and the performance of the said chart under a steady‐state situation along with the zero‐state condition has been studied for detecting changes over an unknown range of shifts. The enactment of the proposed PAEWMA design with existing schemes has been evaluated and compared using run‐length (RL) profiles such as; average RL (ARL), the standard deviation of RL (SDRL), and some percentile points of the RL distribution. The extra quadratic loss and relative mean index measures have been used for comparing the range of shifts. The comparison reveals that the proposed PAEWMA chart under the steady‐state situations performs far better than the competitors. The effect of estimation of the process parameter is also a part of the study. The three applications of the proposal have been included; which are taken from the artificial dataset, from past study monitoring dataset, and the other from the aircraft accident monitoring data for the implementation of the proposal.

An improved adaptive EWMA control chart for monitoring time between events with application in health sector

AbstractNonconforming events rarely occur in high‐quality processes, and the time between events () is likely to be followed by a skewed distribution, such as an exponential distribution. This paper proposes the upper and lower‐sided improved adaptive exponentially weighted moving control charts for monitoring data modeled by the exponential distribution. The proposed control charts are labeled as control charts, that is, the and control chart, respectively. The control chart detects the upward shifts, while the control chart identifies the downward shifts in the process. The Monte Carlo simulations are used to approximate the run length distribution of the proposed control charts. Numerical results associated with various performance measures, such as average run length (), standard run length (), median run length (), extra quadratic loss (), relative average run length (), and performance comparison index () are computed. The proposed control charts compared to the respective existing control charts, such as , , , and control charts, at a single shift, as well as over a specified range of shifts. The comparison reveals that the proposed control charts outperform the existing control charts for both single specified shifts and over a certain range of shifts. Finally, real‐life data of hospital stay time for male patients with traumatic brain injury are analyzed for the applicability of the control chart in the health sector.

An adaptive EWMA control chart based on Hampel function to monitor the process location parameter

AbstractThe classical memory Exponential Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) control charts (CCs) are important to detect small‐to‐moderate sizes of shift in the process location and/or dispersion parameters. To make the memory control chart (CC) more efficient for broad range of shift, the score functions in particular Huber and Bi‐square functions are incorporated in the classical memory CCs. These score functions help to formulate the parameters of memory CCs and are called Adaptive EWMA (AEWMA) and Adaptive CUSUM (ACUSUM). The AEWMA and ACUSUM are capable to handle various sizes (i.e., small, moderate, and large) of shift. The aim of this study is to utilize Hampel function as the alternative of the mentioned score functions, and propose a new AEWMA CC, symbolized as AEWMAHample. To develop the proposed AEWMAHampleCC, the standard EWMA statistic and Hampel function are incorporated in the classical EWMA CC structure. The proposed AEWMAHampleCC is also efficient to handle certain sizes of shift in the process location parameter. Performance evaluation procedures, like average run length is considered for a particular shift. Similarly, the performance comparison index, relative average run length, and extra quadratic loss function are estimated over a range of shift. All these performance measures are calculated from run length (RL) characteristic of the proposed control chart. An algorithm in MATLAB is developed based on Monte Carlo simulation procedure to generate the RLs. Some existing control charts including EWMA, CUSUM, MEC, MCE, , , IACCUSUM, , and are taken for comparison reason. For specific sizes of shift, the proposed AEWMAHampleCC reveals the superiority. Besides, to address the practical significance for practitioners and quality engineers, the real‐life and simulated data examples are considered.

Enhanced fault detection models with real-life applications

<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>