Average Time To Signal Research Articles

Comparison of accumulate-combine and combine-accumulate methods in multivariate CUSUM charts for mean vector

We compared two basic methods, combine-accumulate method and accumulate-combine method, using the past quality information in multivariate quality control procedure for monitoring mean vector of multivariate normal process. When small or moderate shifts have occurred, accumulate-combine method yields smaller average run length (ARL) and average time to signal (ATS) than combine-accumulate method. On the other hand, we have found from our numerical results that combine-accumulate method has better performances in terms of switching behavior than accumulate-combine method. In industry, a quality engineer could select one of the two method under the comprehensive consideration about the required time to signal, switching behavior, and other physical factors in the production process.

A Combined Control Chart to Monitor the Process Mean in a Two‐Stage Sampling

This paper proposes a new combined control chart for monitoring the mean of a process. A sample of size n is split into two sub‐samples of sizes n1 and n2 = n ‐ n1, determined by an optimization search. The units of the first sub‐sample are evaluated by attributes and plotted on an npx control chart. If this chart signals an out‐of‐control condition, then values of the quality characteristic of interest are collected from the units of the second sub‐sample, and the sample mean is calculated and plotted on an control chart. If both control charts signal, then the process is halted for adjustment. The possibility that all n items will not be inspected may lead to a reduction in both the cost and time spent on examining the sampled items. The performance of the proposed procedure is compared to that of two separate and npx control charts. The proposed procedure exhibits superior performance to the control chart for a variety of sample sizes, n, and shifts, δ, of the target mean. The average time to signal (ATS) for the combined control chart was lower than that calculated for a single or npx control chart, indicating that the combined control chart is an efficient tool for monitoring the process mean. Copyright © 2013 John Wiley & Sons, Ltd.

Multi-objective genetic algorithm for economic statistical design of [formula omitted] control chart

Control charts are widely used for monitoring the quality of a product or a process. Their implementation cost motivates researchers to design them with the lowest cost and most desirable statistical properties. Usually, the cost function is optimized subject to statistical properties. However, the cost function also depends on statistical properties, and minimizing it as the only objective is not an efficient method of economic statistical design of control charts. In this paper, cost function, as well as statistical properties, including probability of Type I error, power of X¯ control chart, and Average Time to Signal (ATS), are considered as objectives; the corresponding constraints are also used. Then, a Multi-Objective Genetic Algorithm for Economic Statistical Design (MOGAESD) is proposed for identifying the Pareto optimal solutions of control chart design. The preferred solution is selected by the designer. The performance of the proposed method is compared through some numerical examples reported in the literature. The results show that the proposed approach is effective.

Optimal Variable Sample Size and Sampling Interval MSE Chart Based on Improved Square Root Transformation

In the field of statistical process control (SPC), control charts for attributes are widely used to detect the out-of-control condition by checking the number of nondefective units or nondefective in a sample. In this article, we use the average time to signal (ATS) and the average number of observations to signal (ANOS) to evaluate the performance of the optimal variable sample size and sampling interval (VSSI) improved square root transformation (ISRT) mean square error (MSE) (VSSI_ ISRT_ MSE) control chart for attribute data. In addition, this control chart will be used to monitor: (1) the difference between the process mean and the target value, and (2) the process variance shifts. We found that the optimal VSSI_ ISRT_ MSE chart performs better than the specific VSSI, the optimal variable sampling interval (VSI), and the fixed parameters (FP) ISRT_MSE charts. An example is given to illustrate this new proposed approach.

The computation of average run length and average time to signal: an overview

Control charts are widely used in industries to monitor a process for quality improvement. Evaluation of the average run length (ARL) or average time to signal (ATS) plays an important role in the design of control charts and performance comparison. In this paper, we review several basic and popular procedures, including the Markov chain and integral equation methods for computing ARL, ATS and associated run length distributions for cumulative sum charts, exponentially weighted moving average charts and combined control charts, respectively. Some important references and key formulations are provided for practitioners.

On monitoring process variability under double sampling scheme

Abstract The presence of variation in all manufacturing and measurement processes is a natural phenomenon and is the key factor which affects the performance of all types of processes. A better understanding of the causes of variability in any processes is necessary to improve the process. For an efficient monitoring of process variability, we have suggested a set of variance type control charts based on auxiliary characteristics and evaluated their performances in terms of Average Time to Signal (ATS) (the performance measure at every point of variability shift) and Average Extra Quadratic Loss (AEQL) (the performance measure over the whole process shift range) under normal and gamma process environments. We have also examined the effects of contaminated environments on the ATS performance of different variance based charting structures. Illustrative examples on some selective variance type control structures are also provided for procedural details. Finally we have closed with concluding remarks about this study.

Cumulative Sum Control Charts for Simultaneously Monitoring Means and Variances of Multiple Quality Variables

Multivariate cumulative sum (CUSUM) control charts for simultaneously monitoring both means and variances under multivariate normal process are investigated. Performances of multivariate CUSUM schemes are evaluated for matched fixed sampling interval (FSI) and variable sampling interval (VSI) features in terms of average time to signal (ATS), average number of samples to signal (ANSS). Multivariate Shewhart charts are also considered to compare the properties of multivariate CUSUM charts. Numerical results show that presented CUSUM charts are more efficient than the corresponding Shewhart chart for small or moderate shifts and VSI feature with two sampling intervals is more efficient than FSI feature. When small changes in the production process have occurred, CUSUM chart with small reference values will be recommended in terms of the time to signal.

Switching properties of CUSUM charts for controlling mean vector

Abstract Some switching properties of multivariate control charts are investigated when theinterval between two consecutive sample selections is not xed but changes accordingto the result of the previous sample observation. Many articles showed that the perfor-mances of variable sampling interval control charts are more ecient than those of xedsampling interval control charts in terms of average run length (ARL) and average timeto signal (ATS). Unfortunately, the ARL and the ATS do not provide any informationon how frequent a switch is being made. We evaluate several switching properties oftwo sampling interval Shewhart and CUSUM procedures for controlling mean vectorof correlated quality variables.Keywords: Average number of switches to signal, average sampling interval, probabilityof switch. 1. Introduction The purpose of a control chart is to detect assignable causes of variation so that thesecauses can be found and eliminated. During the control process, one wishes to detect anydeparture from a satisfactory state as quickly as possible and identify which attributes areresponsible for the deviation. Thus, a good control chart detects shifts quickly in the processwhen the process is out-of-control state, and produces few false alarms when the process isin-control state.Many situations in industrial quality control involve a vector of measurements of twoor more related quality variables rather than a single quality variable. When the qualityvariables are correlated, one could obtain better sensitivity by using multivariate controlprocedures than separate control variables for each of the quality variables.Traditional sampling practice in using a control chart is to take samples from the processat xed sampling interval (FSI) and performances of control charts have been developedwhen the sampling interval between samples is xed. In recent years, application of variable

An EWMA t chart with variable sampling intervals for monitoring the process mean

This paper proposes a variable sampling interval (VSI) version of the fixed sampling interval (FSI) exponentially weighted moving average (EWMA) t chart developed by Zhang et al. for monitoring the changes in the process mean. An optimal design strategy based on the average time to signal (ATS) is presented. We determine the optimal parameters for the VSI EWMA t chart using a Markov chain approach so that the chart has the desired robustness property against errors in estimating the process standard deviation or changing standard deviation. Also, we explain how the various parameters of this VSI EWMA t chart can be computed and how the use of the VSI feature improves the statistical efficiency of the FSI EWMA t and FSI EWMA X-bar charts in terms of out-of-control ATS performances. Comparisons with the FSI EWMA t and FSI EWMA X-bar charts are performed.

Cumulative Conformance Count Charts with VariableSample Sizes

Cumulative count of conforming (CCC) chart has been shown effective for high-yield processes with very low fraction of nonconforming items. Traditional CCC chart assumes the products from a process are inspected item by item. Recently, the CCC chart was generalized to conform to the industrial practice where items are inspected sample by sample without according to the production order. Different from the traditional chart, the generalized CCC chart monitors the cumulative count of samples until a nonconforming sample is encountered. In order to improve the power of the generalized CCC chart to detect process changes, this study applied the control scheme of variable sample size (VSS) to the generalized CCC chart. The average time to signal (ATS) process change of the proposed chart was derived by Markov chain approach and taken as performance measure to evaluate its statistical efficiency. Performance comparisons between the proposed chart and generalized CCC chart are made to evaluate its usefulness.

What are the best sample sizes for the Xbar and CUSUM charts?

The X ¯ and CUSUM control charts are used most widely for monitoring the mean of a quality characteristic x. This article studies the sample sizes n X ¯ and n CUSUM of these two charts in the domain of statistical design. However, the study takes the important sampling inspection cost (including the variable and fixed cost components) into consideration. The chart performance will be measured by a weighted average ATS (Average Time to Signal). There are two interesting findings based on the results of this study: (1) The X ¯ chart becomes more statistically effective for detecting mean shifts when the fixed sampling cost cannot be neglected and/or when the mean shift range is small. If n X ¯ and n CUSUM are set as 4 and 1, respectively, based on some conventional wisdom, the simple X ¯ chart often outperforms the more complicated CUSUM chart from an overall viewpoint. (2) Under all circumstances, the overall statistical performance of both charts can be improved, or significantly improved, by the optimization design. The optimal values of n X ¯ and n CUSUM depend on the ratio between the fixed and variable sampling costs, the range of the mean shift, and the in-control Average Time to Signal. For the general cases, the best sample sizes are n X ¯ = 3 or 4, and n CUSUM =2 or 3.

Method and Application of Multistage Machining Quality Control Based on Minimum Overall Mean Quality Loss

To improve the quality control effect and decrease the manufacturing cost in the product’s multistage machining processes, based on the quality loss of single stage machining process, an overall mean quality loss model of multistage machining processes is established by the integration of Duncan cost model, Taguchi quality loss model, mean shift index and average time to signal (ATS). By using this quality loss model, the overall mean quality loss optimal function of multistage machining processes is constructed and the control parameters of this function are optimally solved. Besides, The Shewhart control chart and Cause-selecting control chart can be optimally designed with the control parameters of this optimal function. Finally, on the basis of an application, the multistage machining quality control based on minimum overall mean quality loss is verified to improve the control chart’s performance under the condition of the minimum overall mean quality loss.

A side sensitive modified group runs control chart to detect shifts in the process mean

Gadre and Rattihalli [5] have introduced the Modified Group Runs (MGR) control chart to identify the increases in fraction non-conforming and to detect shifts in the process mean. The MGR chart reduces the out-of-control average time-to-signal (ATS), as compared with most of the well-known control charts. In this article, we develop the Side Sensitive Modified Group Runs (SSMGR) chart to detect shifts in the process mean. With the help of numerical examples, it is illustrated that the SSMGR chart performs better than the Shewhart's X¯ chart, the synthetic chart [12], the Group Runs chart [4], the Side Sensitive Group Runs chart [6], as well as the MGR chart [5]. In some situations it is also superior to the Cumulative Sum chart p9] and the exponentially weighed moving average chart [10]. In the steady state also, its performance is better than the above charts.

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.

Design of control charts with m‐of‐m runs rules

AbstractThis paper investigates economic–statistical properties of the X̄ charts supplemented with m‐of‐m runs rules. An out‐of‐control condition for the chart is either a point beyond a control limit or a run of m‐of‐m successive points beyond a warning limit. The sampling process is modeled by a Markov chain with 2m states. The steady‐state probability for each state and the average run length (ARL) from each state of the Markov chain are derived in explicit formulas. Then the stationary average run length (SALR) is derived so as to develop an economic–statistical model. Using this model, the design parameters are optimized by minimizing the cost function with constraints on the average time to signal (ATS). The X̄ chart supplemented with m‐of‐m runs rules is compared with the Shewhart X̄ chart in terms of the SARL and the cost function. Sensitivity of the design parameters with respect to the cost function is also analyzed. General guidelines for implementing the X̄ chart with m‐of‐m runs rules are presented from those observations. It should be emphasized that supplementing run rules may provide feasible and efficient solutions even if the sample size is limited, while the Shewhart X̄ chart may not. Copyright © 2009 John Wiley & Sons, Ltd.

Economic-Statistical Design of X-bar Control Charts Using Taguchi Loss Functions

Abstract Duncan is the pioneer to provide a control chart design from economic-view point in 1956 and ensure that the economic design of control charts is really lower the cost compared with a conventional Shewhart charts. An economic design does not consider the statistics properties, such as type I or type II error and average time to signal (ATS). In order to improve these issues, an economic-statistical design of control charts was proposed under the statistical consideration. Usually, a product is accepted if the product measurements meet specification requirements and there are no quality losses in the most of economic or economic-statistics design of control charts. Taguchi's Quadratic loss function was employed in this research to construct the quality losses for economic-statistical design of control charts with multiple assignable causes. A numerical example is also used to illustrate its working of the proposed model and to compare the lost cost among the pure economic, economic-statistical control charts design and economic-statistical control charts design using Taguchi's loss function. The example shows that the social loss from the Taguchi's Quadratic loss function is the most important cost compared with the original cost items.

A new statistical process control method to monitor and diagnose bivariate normal mean vectors and covariance matrices simultaneously

In this paper, in order to find an adequate method of monitoring the mean vector and covariance matrix of a production process simultaneously, first, some available univariate control methods were reviewed and evaluated. Then, the maximum exponentially weighted moving average method with a better potential application and good performances in terms of average time to signal (ATS) criterion was selected to be extended to the bivariate case. In the extended procedure, by proper transformation of the control parameters, the primary control space is transformed such that all control elements have the same probability distributions. In this case, only the maximum absolute value of the transformed elements is monitored to control both the mean vector and the covariance matrix simultaneously. Moreover, a heuristic procedure is proposed to interpret and diagnose the out-of-control parameter(s). Finally, not only the performances of the proposed method are evaluated and compared to the ones from the two recent methods in terms of ATS but also the percentage of correct classification of the proposed method was estimated by extensive simulation experiments.

Performance Comparisons for the Synthetic Control Chart for Detecting Increases in Fraction Nonconforming

The performance of the recently proposed synthetic control chart for detecting increases in fraction nonconforming is re-evaluated and comparisons with a wider range of alternative charts are presented. It is shown that the previously reported strong performance of the synthetic chart was due to the implicit inclusion of a ‘head-start’ feature when computing the reported values of average time to signal (ATS). The comparisons considered here, using both initial-state and steady-state evaluations of ATS, show that, without the head-start feature, the performance of the synthetic chart is not much better than that of the np chart and also that there are better alternatives, in particular, the use of a CUSUM of the run lengths of conforming items (termed the run-length CUSUM). Consideration may also be given to the use of the RL2 chart, which is based on the moving sum of two successive conforming run lengths. While the superior performance of the run-length CUSUM is clearly demonstrated, the RL2 chart also performs quite well, except for specified circumstances, when it should not be used.

Control charts applied to simulated sow herd datasets

Statistical control charts were used to detect process change in pig production. Two charts were tested to detect small deviations in production processes: the cumulative sum (CUSUM) control chart and the exponentially weighted moving average (EWMA) control chart. A Monte-Carlo simulation was used for developing an optimal design of the EWMA and the CUSUM charts. The traits piglets born in total and the return to oestrus rate were considered. Over a given time period, small shifts were purposely implemented to test the performance of the charts. The average time to signal (ATS) and false positive rate (FPR) were taken as classification parameters to evaluate the performance of the charts. All shifts in the number of piglets born in total were detected with CUSUM and EWMA control charts. The trait piglets born in total showed an ATS ranging from 1.3 (FPR = 33.5%) to 6.8 weeks (FPR = 1.2%) using the CUSUM chart. The EWMA chart presented an ATS which ranged between 2.0 (FPR = 14.9%) and 6.3 (FPR = 1.9%) weeks. The application of the CUSUM to the return to oestrus rate resulted in an ATS of 2.6 (FPR = 38.3%) to 15.6 weeks (FPR = 3.0%) and the EWMA chart produced a signal between 4.1 (FPR = 14.5%) and 16.4 weeks (FPR = 1.4%). Both charts appear to be useful tools for tracking commercial swine farm processes and detecting emerging change in process performance.

A VARIABLE SAMPLE SIZE S2-EWMA CONTROL CHART FOR MONITORING THE PROCESS VARIANCE

Monitoring the stability of measures dispersion from a process quality parameter is an important aspect of Statistical Process Control which should be carefully planned by practitioners. To perform this task, this paper proposes an adaptive EWMA chart as a practical and efficient tool. The developed EWMA chart is the Variable Sample Size (VSS) version of a static S2-EWMA control chart previously developed by one of the authors to monitor the sample variance. The way to compute the design parameters of this VSS S2-EWMA control chart is discussed and an optimal design strategy based on the Average Time to Signal (ATS) after a shift in process dispersion is presented. The statistical performance of the VSS S2-EWMA has been evaluated by means of a comparison with two other EWMA charts: the static S2-EWMA and the adaptive (VSI) S2-EWMA allowing to vary the sampling intervals. The obtained results show how the possibility of varying the sample size significantly improves the statistical performance over the static S2-EWMA; furthermore, some interesting findings suggest to implement the VSS S2-EWMA with respect to the VSI S2-EWMA when some particular process operating conditions occur.