Adjusted Average Time To Signal Research Articles

Triple exponentially weighted moving average control charts without or with variable sampling interval for monitoring the coefficient of variation

The coefficient of variation (CV) chart is widely used in the process where the standard deviation is proportional to the mean. In this work, two separate one-sided triple exponentially weighted moving average control charts are proposed to monitor the CV squared (denoted as TEWMA- γ 2 chart). By adding the variable sampling interval (VSI) feature to the proposed - γ 2 charts, the VSI TEWMA- γ 2 charts are further proposed to improve the performance of the TEWMA- γ 2 charts. Through extensive Monte Carlo simulations, the zero-state and steady-state run length (RL) performances are presented. The comparisons with the existing CV charts, based on several run length metrics show the superiority of the proposed charts, especially for small shifts. Moreover, the overall measure, expected ARL (EARL), also shows the outperformance of the γ 2 charts. In addition, based on the average time to signal (ATS) and adjusted ATS (AATS), it is shown that the VSI strategy makes the TEWMA- γ 2 charts detect the shifts more quickly. An illustrative example from the sintering process is finally provided to display the application of the proposed charts.

Economic design of max charts using Taguchi’s loss function

Recent researches have shown that a single control chart, Max chart, simplifies and performs as effectively as joint and S charts for monitoring both the process mean and standard deviation. This paper develops an economic design of Max chart based on a unified model and embellishes it with Taguchi’s quality loss function. A simplified algorithm is applied to determine the optimal settings of three control chart parameters: the sample size, the sampling interval and the control limit. A numerical example is presented to illustrate its application. A fractional factorial experiment is conducted to study the sensitivity of the input parameters on the optimal designs of Max chart and joint and S charts. Statistical and economical properties including in-control average time to signal (ATS), out-of-control ATS, adjusted average time to signal (AATS) and minimum expected cost per hour of the two charts are compared. The results show that the two charts operate with approximately equal cost but different false alarm rate and detecting speed for the assignable cause without and with consideration of process failure mechanism. Critical input parameters that cause these differences are then investigated. This paper concludes that Max chart can be a good alternative to joint and S charts from not only statistical and simplified properties but also economical considerations.

Development of U control chart by variable sample size and sampling interval to improve the statistical properties

AbstractIt has been proved that the quality control charts with variable sampling schemes are more effective than the classical ones in improving statistical measures. The Average Number of False Alarms (ANF), Average Number of Samples (ANS), Average Number of Inspected Items (ANI), and Adjusted Average Time to Signal (AATS) have been the most important statistical measures always being attending in the evaluation of control charts. Thus, this study was conducted aiming at illustrating a comprehensive analytical review on the U control chart via the statistical measures. To this end, different levels of the possible factors were determined, and the results of calculating the statistical measures and the obtained parameters on the sampling schemes of the U control chart were presented. The results indicated that the variable U control charts were capable of improving the effectiveness of statistical measures, especially for detecting shifts and number of false alarms.

Applying SVSSI sampling scheme on np-chart to Decrease the Time of Detecting Shifts - Markov chain approach, and Monte Carlo simulation

One of the main criteria for judging about the power of control charts is their ability in fast detection of deviations and shifts in the process. Average time to signal (ATS) or adjusted average time to signal (AATS) are among such criteria calculated under a certain state and assumption. Several studies have shown that using the idea of ​​variable design for control charts, by separating their limits to the safe and the warning regions, can allow quick discovery of shifts and increase sensitivities to small changes. In this paper, a new variable sampling scheme with three sample sizes and two different sampling intervals, called SVSSI, is developed to increase the efficiency of the np control chart. Through various numerical examples, the performance of this scheme is evaluated by calculating ATS and AATS values using Markov chain method. Monte Carlo simulation method is used to validate the results of Markov chain method of SVSSI sampling scheme. In comparison with other schemes, better performance of applying SVSSI is proved in all conditions.

Statistically constrained economic design of control chart with Weibull in-control time for processes subject to multiple assignable causes

PurposeThe purpose of this paper is to develop an economic statistical design based on the concepts of adjusted average time to signal (AATS) and ANF for control chart under a Weibull shock model with multiple assignable causes.Design/methodology/approachThe design used in this study is based on a multiple assignable causes cost model. The new proposed cost model is compared with the same cost and time parameters and optimal design parameters under uniform and non-uniform sampling schemes.FindingsNumerical results indicate that the cost model with non-uniform sampling cost has a lower cost than that with uniform sampling. By using sensitivity analysis, the effect of changing fixed and variable parameters of time, cost and Weibull distribution parameters on the optimum values of design parameters and loss cost is examined and discussed.Practical implicationsThis research adds to the body of knowledge relating to the quality control of process monitoring systems. This paper may be of particular interest to practitioners of quality systems in factories where multiple assignable causes affect the production process.Originality/valueThe cost functions for uniform and non-uniform sampling schemes are presented based on multiple assignable causes with AATS and ANF concepts for the first time.

Weighted adaptive multivariate CUSUM charts with variable sampling intervals

ABSTRACTThe adaptive multivariate CUSUM (AMCUSUM) chart has received considerable attention because of its superior sensitivity against a range of mean shift sizes than that of the conventional non-adaptive multivariate CUSUM (MCUSUM) chart. Recently, weighted AMCUSUM (WAMCUSUM) charts with a fixed sampling interval (FSI) have been proposed, called the WAMCUSUM-FSI charts, which provide more sensitivity than the AMCUSUM-FSI charts. In this paper, we extend this work and propose WAMCUSUM charts with variable sampling interval (VSI), named the WAMCUSUM-VSI charts, for efficiently monitoring the mean of a multivariate normally distributed process. The Monte Carlo simulation method is used to compute the average time to signal (ATS) and the adjusted ATS (AATS) profiles of the existing and proposed charts. It is found that the WAMCUSUM-VSI charts perform substantially and nearly uniformly better than the WAMCUSUM-FSI charts in terms of the ATS and AATS performance criterion. An example is given to explain the implementation of the WAMCUSUM charts with fixed and VSIs.

Multi-Objective Genetic Algorithm for Economic Statistical Design of the T2 Control Chart with Variable Sample Size: The Updated Markov Chain Approach

Abstract T 2 control charts are used to primarily monitor the mean vector of quality characteristics of a process. Recent studies have shown that using variable sample size (VSS) schemes results in charts with more statistical power for detecting small to moderate shifts in the process mean vector. In this study, we have presented a multiple-objective economic statistical design of VSS T2 control chart with the adjusted average time to signal (AATS) as the statistical objective and the expected cost per hour as the economic objective. Then, a multi-objective genetic algorithm for economic statistical design is proposed for identifying the Pareto optimal solutions of control chart design. Through an illustrative example, the advantages of the proposed approach are shown by providing a list of viable optimal solutions and graphical representations, which indicate the advantage of flexibility and adaptability of our approach.

The Run Sum Hotelling's χ2 Control Chart with Variable Sampling Intervals

Control charts are widely used for process monitoring and quality control in manufacturing industries. Implementing variable sampling interval (VSI) control schemes on control charts rather than traditional fixed sampling interval procedure can significantly improve the control chart's efficiency. In this paper, the VSI run sum (RS) Hotelling's χ2 chart is proposed. The optimal scores and parameters of the proposed chart are determined using an optimization technique to minimize the following: (i) out‐of‐control average time to signal (ATS); (ii) adjusted ATS (AATS), when the exact shift size can be specified; (iii) expected ATS; or (iv) expected AATS, when the exact shift size cannot be specified. The Markov chain method is used to evaluate the zero‐state ATS and expected ATS, and steady‐state AATS and expected AATS of the proposed chart. The results show that the VSI RS Hotelling's χ2 chart significantly outperforms the standard RS Hotelling's χ2 chart and the former also performs well compared with other competing charts. By adding more scoring regions, the efficiency of the VSI RS Hotelling's χ2 chart can be further enhanced. An illustrative example using data from a manufacturing process is presented to demonstrate the application of the VSI RS Hotelling's χ2 chart. The application of the proposed chart in a quality improvement program can be extended to management and service industries. Copyright © 2016 John Wiley & Sons, Ltd.

The variable sampling interval run sum [formula omitted] control chart

Traditional control charts for process monitoring are based on taking samples from the process at fixed length sampling intervals. More recently, research works focused on the use of variable sampling intervals (VSIs), where the lengths of the sampling intervals are varied according to the process quality. A short sampling interval is considered when the process quality indicates a possible out-of-control situation while a long sampling interval is considered, otherwise. In this paper, the VSI run sum (RS) X‾ chart is proposed with its optimal scores and parameters determined using an optimization technique to minimize the out-of-control average time to signal (ATS) or the adjusted average time to signal (AATS). A Markov-chain method is used to evaluate both the ATS and AATS of the proposed chart, for the zero and steady state cases, respectively. Results show that the VSI RS X‾ chart is considerably more efficient than the basic RS X‾ chart. The VSI RS X‾ chart performs generally well compared with other competing charts, such as the standard X‾, synthetic X‾, exponentially weighted moving average (EWMA) X‾, VSI X‾ and VSI EWMA X‾ charts. The sensitivity of the VSI RS X‾ chart can be enhanced further by adding more scoring regions or a head-start feature. An illustrative example is presented to explain the implementation of the proposed VSI RS X‾ chart.

A different and simple approach for comparing sampling methods in quality control

Purpose – The purpose of this paper is to measure the performance of a sampling method through the average number of samples drawn in control. Design/methodology/approach – Matching the adjusted average time to signal (AATS) of sampling methods, using as a reference the AATS of one of them the paper obtains the design parameters of the others. Thus, it will be possible to obtain, in control, the average number of samples required, so that the AATS of the mentioned sampling methods may be equal to the AATS of the method that the paper uses as the reference. Findings – A more robust performance measure to compare sampling methods because in many cases the period of time where the process is in control is greater than the out of control period. With this performance measure the paper compares different sampling methods through the average total cost per cycle, in systems with Weibull lifetime distributions: three systems with an increasing hazard rate (shape parameter β=2, 4 and 7) and one system with a decreasing failure rate (β=0, 8). Practical implications – In a usual production cycle where the in control period is much larger than the out of control period, particularly if the sampling costs and false alarms costs are high in relation to malfunction costs, the paper thinks that this methodology allows us a more careful choice of the appropriate sampling method. Originality/value – To compare the statistical performance between different sampling methods using the average number of samples need to be inspected when the process is in control. Particularly, the paper compares the statistical and economic performance between different sampling methods in contexts not previously considered in literature. The paper presents an approximation for the average time between the instant that failure occurs and the first sample with the process out of control, as well.

Economic Design of Max Charts

A single control chart, Max chart, proposed by Chen and Cheng (1998) and slightly modified by Chen and Huang (2006), was shown to simplify and perform as effectively as joint and S charts for monitoring both the process mean and standard deviation. This article studies the economic design of Max charts, and a simplified algorithm is applied to determine the optimal settings of three control chart parameters: the sample size, sampling interval, and control limit. A numerical example is presented to illustrate its application. A fractional factorial experiment is conducted to study the sensitivity of the input parameters on the optimal designs of Max chart and joint and S charts. Statistical and economical properties including in-control average time to signal (ATS), out-of-control ATS, adjusted average time to signal (AATS), and minimum expected cost per hour of the two charts are compared. The results show that the two charts operate with approximately equal cost but different false alarm rate and detecting speed for the assignable cause without and with consideration of process failure mechanism. Critical input parameters that cause these differences are then investigated. We conclude that Max chart can be a good alternative to joint and S charts from not only statistical and simplified properties but also economical considerations.

Controlling over-adjusted process means and variances using VSI cause selecting control charts

Process adjusted unnecessarily is a common problem in statistical process control. Incorrect adjustment of a process may result in shifts in process mean, process variance, or both, ultimately affecting the quality of products. The article considers the variable process control scheme for two dependent process steps with incorrect adjustment. We construct the variable sampling interval (VSI) Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts in order to effectively monitor the quality variable produced by the first process step with incorrect adjustment and the quality variable produced by the second process step with incorrect adjustment, respectively. The performance of the proposed VSI control charts is measured by the adjusted average time to signal (AATS) derived using a Markov chain approach. An example of process control for automobile braking system shows the application and performance of the proposed joint VSI Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts in detecting small and median shifts in mean and variance for the two dependent process steps with incorrect adjustment. The performance of the VSI Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts and the fixed sampling interval (FSI) Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts are compared via the numerical analysis results. These demonstrate that the former is much faster in detecting shifts in mean and variance. Whenever quality engineers cannot specify the values of variable sampling intervals, the optimal VSI Z X ¯ - Z S 2 and Z e ¯ - Z S e 2 control charts are suggested. Furthermore, the impacts of misusing Z Y ¯ - Z S y 2 charts to monitoring the process mean and variance in the second step are also investigated.

Control charts with predetermined sampling intervals

PurposeThe purpose of this paper is to investigate a new sampling methodology previously proposed for systems with a known lifetime distribution: the Predetermined Sampling Intervals (PSI) method.Design/methodology/approachThe methodology is defined on basis of system hazard cumulative rate, and is compared with other approaches, particularly those whose parameters may change in real time, taking into account current sample information.FindingsFor different lifetime distributions, the results obtained for adjusted average time to signal (AATS) using a control chart for the sample mean are presented and analysed. They demonstrate the high degree of statistical performance of this sampling procedure, particularly when used in systems with an increasing failure rate distribution.Practical implicationsThis PSI method is important from a quality and reliability management point of view.Originality/valueThis methodology involves a process by which sampling instants are obtained at the beginning of the process to be controlled. Also this new approach allows for statistical comparison with other sampling schemes, which is a novel feature.

Using VSI EWMA charts to monitor dependent process steps with incorrect adjustment

The article considers the variable process control scheme for two dependent process steps with incorrect adjustment. Incorrect adjustment of a process may result in shifts in process mean, ultimately affecting the quality of products. We construct variable sampling interval (VSI) EWMAZX and EWMAZe control charts to effectively monitor the quality variable produced by the first process step with incorrect adjustment and the quality variable produced by the second process step with incorrect adjustment. The performance of the proposed VSI control charts is measured by the adjusted average time to signal (AATS) derived using a Markov chain approach. An example of the automobile braking system with incorrect adjustment shows the application and performance of the proposed VSI EWMAZX and EWMAZe control charts in detecting shifts in process mean. Furthermore, the performance of the proposed VSI EWMAZX and EWMAZe control charts and the fixed sampling interval (FSI) EWMAZX and EWMAZe control charts are compared by numerical analysis results. These demonstrate that the former is much faster in detecting small and median shifts in mean. The optimum VSI EWMAZX and EWMAZe control charts are also proposed using optimization technique to minimize AATS when quality engineers cannot specify the values of variable sampling intervals. It has been found that the optimum VSI EWMAZX and EWMAZe control charts always work better than the VSI EWMAZX and EWMAZe control charts.

Adaptive sampling enhancement for Hotelling’s T2 charts

This paper makes a study of an adaptive sampling scheme useful to increase the power of the fixed sampling rate (FSR) T 2 control chart. In our study, the three parameters of T 2 control chart: the sample size, the sampling interval, and the upper percentage factor that is used for determining the action limit, vary between two values for a relaxed or tightened control, depending on the most recent T 2 value. The average time to signal (ATS) and adjusted average time to signal (AATS) a shift in the process mean vector for the new chart are derived and regarded as an objective function respectively to optimize its design parameters. With some minor changes, the new chart can be reduced to the variable sampling interval (VSI) T 2 chart, the sample size (VSS) T 2 chart, the variable sample size and sampling interval (VSSI) T 2 chart, or the FSR T 2 chart. Numerical comparisons among them are made and discussed. Furthermore, the effects of the initial sample number (use for estimating the in-control process parameters) upon the chart’s performance and adaptive design parameters are presented.

Adaptive control schemes for two dependent process steps

The article considers the statistical adaptive process control for two dependent process steps. We construct adaptive sample size and sampling interval (ASSI) Z X ¯ and Z e ¯ control charts to monitor the quality variables produced by the first process step and the second process step at the end of the second process step and effectively distinguish which process step is out-of-control. The performance of the proposed ASSI control charts is measured by adjusted average time to signal (AATS) using Markov chain approach. An example of automobile braking system shows the application and performance of the proposed ASSI Z X ¯ and Z e ¯ control charts in detecting shifts in process mean. Furthermore, the performance of the ASSI Z X ¯ and Z e ¯ control charts and the nonadaptive sample size and sampling interval Z X ¯ and Z e ¯ control charts are compared by some numerical analysis results. It demonstrates that the former is much faster than the latter in detecting small shifts in mean.

Controlling‐dependent process steps using variable sample size control charts

AbstractThe article considers the variable process control scheme for two dependent process steps. We construct variable sample size (VSS) ZX̄ and Zē control charts to effectively monitor the quality variable produced by the first process step and the quality variable produced by the second process step 2, respectively. The performance of the proposed VSS control charts is measured by the adjusted average time to signal (AATS) using a Markov chain approach. An example of producing cotton yarn system shows the application and performance of the proposed VSS ZX̄ and Zē control charts in detecting shifts in process mean. Furthermore, the performance of the VSS ZX̄ and Zē control charts and the fixed sample size (FSS) ZX̄ and Zē control charts are compared by numerical analysis results. These demonstrate that the former is much faster in detecting small shifts in mean. Copyright © 2006 John Wiley & Sons, Ltd.

Adaptive sampling interval cause-selecting control charts

This article considers the statistical adaptive process control for two dependent process steps. We construct an adaptive sampling interval Z X control chart to monitor the quality variable produced by the first process step, and use the adaptive sampling interval Z e control chart to monitor the specific quality variable produced by the second process step. By using the proposed adaptive sampling interval control charts, we can quickly detect and distinguish which process step is out of control. The performance of the proposed adaptive sampling interval control charts is measured by the adjusted average time to signal (AATS), which was derived by a Markov chain approach, for an out-of-control process. An empirical automobile braking system example shows the application and the performance of the proposed adaptive sampling control charts in detecting shifts in process means. Some numerical results obtained demonstrated that the performance of the proposed adaptive sampling cause-selecting control charts outperforms the fixed sampling interval cause-selecting control charts.