Average Number Of Observations To Signal Research Articles

Optimal triple sampling [formula omitted] charts for autocorrelated processes

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

Optimal design of the side sensitive group runs double sampling (SSGRDS) X¯ scheme with estimated process parameters

The side sensitive group runs double sampling (SSGRDS) [Formula: see text] scheme is previously studied under the known process parameters (Case-K) assumption. As the process parameters are hardly known in practical situations, they must be estimated using a suitably chosen in-control (IC) Phase-I sample. However, various studies demonstrated that a substantial amount of Phase-I sample is necessary so that the scheme with unknown process parameters (Case-U) attains a performance almost similar to that of the Case-K counterpart. Since it is challenging to acquire a vast number of IC samples, we examine the optimal designs of the Case-U SSGRDS [Formula: see text] scheme such that the average number of observations to signal (ANOS) and expected ANOS (EANOS) are minimised. The optimal parameters obtained for the Case-U SSGRDS [Formula: see text] scheme enable it to have a performance comparable to that of the Case-K counterpart, without the need for a large number of Phase-I samples. We show that the Case-U SSGRDS [Formula: see text] scheme is effective in identifying the process mean shift of a silicon epitaxial process.

Triple sampling X̄ control chart for gamma process

ABSTRACT Monitoring production processes to provide potential customers with quality products while utilising fewer resources and reducing the cost of inspection is a challenge to quality and industrial engineers. Aiming at developing an improved design using reduced sample sizes, this study proposes a triple sampling (TS) X ¯ chart to monitor shifts in the process mean when the quality characteristic of interest follows the gamma distribution. Optimal parameters are determined by employing three optimisation procedures. The detection ability of the TS X ¯ chart is evaluated and contrasted with competing charts via the average number of observations to signal (ANOS), average run length (ARL) and expected ANOS (EANOS) criteria. When the TS X ¯ chart is designed for detecting small and large shifts, it is noticed that the sample sizes at the first, second and third sampling stages are larger for detecting small shifts. Compared to its competitors, the TS X ¯ chart requires fewer observations on the average to signal a shift; hence, it uses resources more efficiently and incurs lower inspection costs. To aid in the implementation of the TS X ¯ chart, an illustrative example of monitoring the wafer process is provided. Overwhelming evidence based on reported results supports the TS X ¯ chart.

An integrated optimal-GICP design for the SPRT control chart with estimated process parameters based on the average number of observations to signal

ABSTRACT As a variable-sample-size control scheme, the sequential probability ratio test (SPRT) chart is favoured due to its sensitivity and high sampling efficiency. It is frequently documented that the SPRT chart inspects only a small number of observations at each sampling stage, making the said chart extremely appealing to quality engineers. In the current literature, most developments of the SPRT chart are established based on the average run length and average time to signal metrics. However, these metrics do not consider sampling efficiency. In this paper, we develop two optimal designs of the SPRT chart, based on (i) the average value of the average number of observations to signal (AANOS) and (ii) the expected value of the AANOS, with consideration of Phase-I process parameter estimation. We develop a model that integrates the concept of optimization and the guaranteed in-control performance (GICP) framework to neutralize the adverse effects of parameter estimation. Results show that the proposed optimal-GICP design preserves satisfactory out-of-control performances for moderate and large mean shifts, while keeping the false alarm rates at reasonably low levels. Finally, we illustrate an example of the SPRT chart with estimated process parameters for monitoring loop height measurements from a wire bonding dataset.

An Optimal Adaptive Variable Sample Size Scheme for the Multivariate Coefficient of Variation

Development of an efficient process monitoring system has always received great attention. Previous studies revealed that the coefficient of variation (CV) is important in ensuring process quality, especially for monitoring a process where its process mean and variance are highly correlated. The fact that almost all industrial process monitoring involves a minimum of two or more related quality characteristics being monitored simultaneously, this paper incorporates the salient feature of the adaptive sample size VSS scheme into the standard multivariate CV (MCV) chart, called the VSS MCV chart. A Markov chain model is developed for the derivation of the chart’s performance measures, i.e the average run length (ARL), the standard deviation of the run length (SDRL), the average sample size (ASS), the average number of observations to signal (ANOS) and the expected average run length (EARL). The numerical comparison shows that the proposed chart prevails over the existing standard MCV chart for detecting small and moderate upward and downward MCV shifts.

Revised triple sampling control charts for the mean with known and estimated process parameters

ABSTRACT The primary aim of this research is to propose a revised triple sampling (TS) chart, where the derivations of new formulae for computing the average run length of the triple sampling (TS) chart correctly are provided. The secondary aim is to develop the revised TS chart with estimated process parameters. The revised TS charts are compared with the double sampling (DS) , two stage adaptive sample size (AS2) and three stage adaptive sample size (AS3) charts when process parameters are known and estimated using the average run length (ARL), average number of observations to signal (ANOS), average of the average run lengths (AARL), standard deviation of the average run lengths (SDARL), average of the average number of observations to signal (AANOS) and standard deviation of the average number of observations to signal (SDANOS) criteria, where the revised TS charts are found to be superior. Additionally, a table giving the minimum number of Phase-I samples for estimating the process mean so that the revised TS chart with estimated process parameters has the desired in-control AARL and AANOS performances is provided.

A side-sensitive double sampling monitoring scheme with estimated process parameters

In statistical process monitoring (SPM), most of the monitoring schemes are designed assuming that the process parameters for the underlying distribution are known (i.e., Case K). In a variety of contexts, it has been shown that when the parameters used to design the control limits are unknown (i.e., Case U), this greatly affects the monitoring schemes properties. Hence, in this paper, we study the parameter estimation effect of the side-sensitive double sampling (SSDS) monitoring scheme for detecting changes in the process mean when distribution design parameters are estimated from an in-control retrospective sample. A thorough investigation is conducted using the unconditional run-length properties (i.e., average, standard deviation and percentiles), average sample size (ASS) and average number of observations to signal (ANOS) through exact integral formulas and simulations. In addition, the average extra quadratic loss (AEQL), average ratio of the average run-length (ARARL) and performance comparison index (PCI) are used to quantify the run-length of the SSDS scheme from an overall performance perspective. Comparisons with other established monitoring schemes when parameters are unknown indicate that the SSDS scheme has a better overall performance. An illustrative example is also given to facilitate the design and implementation of the new scheme. An additional section briefly discussing the synthetic version of the SSDS scheme is also provided.

Comparative Statistical Analysis of Selected Control Charts for Highly Capable Processes

When a high-quality process is to be controlled by 100% inspection and yes-no decision is employed, several types of charts come into account, e.g. CCC, CCC-r or geometric CUSUM (CCC-CUSUM). The aim of the paper is to examine performance of these charts so that a suitable one can be chosen for a given process. The charts are compared according to the quickness with which the upward shift in the fraction of nonconforming items is detected. The average number of observations to signal (ANOS) instead of the usual average run length (ARL) is determined. While ANOS for CCC or CCC-r charts can be easily calculated based on a geometric or a negative binomial distribution, its computation is quite diffi cult in the case of CCC-CUSUM chart. The corrected diffusion (CD) approximation was used to determine ANOS and the results were verifi ed by Monte Carlo simulation. Zero-state and steady-state (both fi xed-shift and random-shift model) analyses were performed to take different scenarios of the process shift occurrence into account. CCC-3 or CCC-2 and CCC-CUSUM charts were compared. The order r for CCC-r chart was chosen as an optimal value for the given process based on the semi-economic model suggested in Brodecka (2013). Our study revealed that for in-control p0 equal to 0.0002 the CCC-CUSUM chart performs best especially for shifts around the pre-specifi ed out-of-control fraction nonconforming. The CCC-r chart may be comparable or even better in detecting larger shifts. The results of the comparative study were utilized for the choice of the most suitable and best performing control chart to control the high-yield process producing ERG (Exhaust Gas Recirculation) sensors. Comparisons of CCC-r and CCC-CUSUM charts can be found elsewhere in literature, but conclusions seem to be rather inconsistent. To our best knowledge no study dealing with such small in-control fraction nonconforming together with the low risk of false alarm has been published yet. The choice of CUSUM’s parameters and consequent values of ANOS can help practitioners who need to control high-quality processes.

Monitoring Bernoulli processes considering measurement errors and learning effect

AbstractThis paper presents the effect of measurement errors and learning on monitoring processes with individual Bernoulli observations. A cumulative sum control chart is considered to evaluate the possible impacts of measurement errors and learning. We propose a time‐dependent learning effect model along with measurement errors and incorporate them into the Bernoulli CUSUM control chart statistic. The performance of the Bernoulli CUSUM control chart is then merely assessed by comparing the average number of observations to signal (ANOS) under two proposed conditions with the condition of no possible errors. Thus, the ANOS values are obtained under different proportions of non‐conforming items, once considering errors due to measurement by inspectors, and once considering both errors and learning effect together. The experimental results show that the efficiency of the control chart to detect assignable causes deteriorates in the presence of measurement errors and enhances when learning affects operators' performance. The proposed approach has a potential to be used in monitoring high‐quality Bernoulli processes as well as disease diagnosis, and other health care applications with Bernoulli observations.

Optimal design of the side‐sensitive modified group runs (SSMGR) chart when process parameters are estimated

AbstractA side‐sensitive modified group runs (SSMGR) chart is commonly examined under the assumption of known process parameters (denoted as Case‐K). However, in practical situations, the process parameters are seldom known, and it is essential to estimate them from in‐control (IC) phase I samples. From the study of the minimum amount of phase I samples, m that is required by the SSMGR chart when the process parameters are estimated (represented as Case‐U), to achieve practically the same performance as that of the Case‐K, it was found that a huge number of IC phase I samples are needed. Due to time and cost factors, accumulating a substantial amount of IC phase I samples is difficult. To circumvent this problem, we study the optimal designs of the Case‐U SSMGR chart, based on both the zero and steady states' average number of observations to signal (ANOS) and expected ANOS (EANOS). Tables containing optimal charting parameters are provided to ease the implementation of the Case‐U SSMGR chart. By using these optimal parameters specifically designed for the Case‐U SSMGR chart, we can obtain similar zero and steady states' ANOS and EANOS performances to that of the Case‐K. The use of the optimal Case‐U SSMGR chart is demonstrated with real data taken from a hard‐bake process in a manufacturing company.

A linear‐time method for computing the ANOS of Bernoulli and Poisson CUSUM charts

AbstractThe average number of observations to signal (ANOS) is an important measure of the effectiveness of Bernoulli or Poisson cumulative sum (CUSUM) control charts. Being able to quickly and accurately calculate an ANOS vector facilitates effective control chart design. We present a linear‐time method for computing the ANOS of Bernoulli CUSUM charts and generalize to Poisson CUSUM charts. This method overcomes computation challenges associated with previously existing methods.

A study on the run length properties of the side sensitive group runs double sampling (SSGRDS) control chart

The side sensitive group runs double sampling (SSGRDS) chart incorporates the control charting concepts of the side sensitive group runs (SSGR) and double sampling (DS) charts. The SSGRDS chart which combines the efficiency of its basic charts is an effective approach to increase the speed of mean shift detection. The performance of the SSGRDS chart, based on the average number of observations to signal (ANOS), median number of observations to signal (MNOS) and percentiles of the number of observations to signal (PNOS) is investigated in this paper. Based on the results obtained, it is found that the SSGRDS chart becomes more sensitive in detecting mean shifts with an increase in the size of the process mean shift. With the use of MNOS and PNOS to measure the performance of the SSGRDS chart, the entire run length distribution is considered and this leads to a more complete understanding of the performance of the chart. The findings in this paper will provide a clearer picture on the run length properties of the SSGRDS chart which will facilitate practitioners in using the chart.

A side-sensitive modified group runs double sampling (SSMGRDS) control chart for detecting mean shifts

ABSTRACTA side-sensitive modified group runs double sampling (SSMGRDS) chart for detecting mean shifts is proposed. The average number of observations to signal (ANOS) and expected average number of observations to signal (EANOS) performances of the SSMGRDS, , double sampling (DS), side-sensitive group runs double sampling (SSGRDS), side-sensitive modified group runs (SSMGR) and exponentially weighted moving average (EWMA) charts are compared. For the zero-state case, the SSMGRDS chart outperforms all charts under comparison, for all sizes of mean shifts. For the steady-state case, the EWMA chart prevails for small and moderate mean shifts but the SSMGRDS chart prevails for large shifts.

Performance of adaptive np-chart with estimated parameter

Purpose – The purpose of this paper is to compare the performances of np-VP control chart with estimated parameter to the np-VP control chart with known parameter using average time-to-signal (ATS), standard deviation of the time-to-signal (SDTS), and average number of observations to signal (ANOS) as performance measures. Design/methodology/approach – The approach used in this study is probabilistic in which the expected values of performance measures are calculated using probabilities of different estimators used to estimate process parameter. Findings – Numerical results indicate different performances for the np-VP control chart in known and estimated parameter cases. It is obvious that when process parameter is not known and is estimated using Phase I data, the chart does not perform as user expects. To tackle this issue, optimal Phase I estimation scenarios are recommended to obtain the best performance from the chart in the parameter estimation case in terms of performance measures. Practical implications – This research adds to the body of knowledge in quality control of process monitoring systems. This paper may be of particular interest to practitioners of quality systems in factories where products are monitored to reduce the number of defectives and np chart parameter needs to be estimated. Originality/value – The originality of this paper lies within the context in which an adaptive np control chart is studied and the process parameter unlike previous studies is assumed unknown. Although other types of control charts have been studied when process parameter is unknown but this is the first time that adaptive np chart performance with estimated process parameter is studied.

A modified variable sample size and sampling interval controlchart

In this article, a modified version of variable sample size and sampling interval chart using three sampling intervals and two sample sizes, called the VSSIt control chart, is proposed to overcome the weakness of the standard Shewhart (SS) control chart in detecting small and moderate shifts. In order to evaluate the performance of the proposed scheme, average time to signal (ATS), average number of observations to signal (ANOS), and average number of samples to signal (ANSS) are used. Using this control chart can reduce the ATS value around 68, 27, 7, and 1.83 time unit (sampling interval of the SS chart) in detecting shift of size 0.25, 0.5, 0.75, and 1 standard deviation, respectively. The performance of the constructed adaptive chart is compared with various adaptive charts such as variable sample size (VSS), variable sampling interval (VSI), variable sample size and sampling interval (VSSI), and special variable sample size and sampling interval (SVSSI) \( \overline{x} \) control chart. It is shown that the proposed chart always outperforms SS, VSI, VSS, and VSSI schemes. It also can detect shift of size 0.25 standard deviation quicker than SVSSI scheme. Finally, the optimal parameters of the proposed chart are recommended.

Side-sensitive group runs double sampling (SSGRDS) chart for detecting mean shifts

This paper proposes a side-sensitive group runs double sampling (SSGRDS) chart to detect shifts in the process mean. It improves the side-sensitive group runs chart proposed by Gadre and Rattihalli. The implementation of the SSGRDS chart is explained. The newly developed SSGRDS chart is compared with the synthetic, double sampling, synthetic double sampling, side-sensitive group runs and exponentially weighted moving average charts, in terms of the zero-state and steady-state average number of observations to signal (ANOS) and expected average number of observations to signal (EANOS). The zero-state and steady-state ANOS (ssANOS) and EANOS results reveal that the optimal SSGRDS chart generally performs well for detecting small and large mean shifts, from an overall perspective, compared with the optimal versions of other competing charts. This article provides tables of optimal charting parameters to facilitate the design of the SSGRDS chart. From these tables, the user can directly determine the optimal charting parameters for (i) minimising the standardised mean shift , based on different combinations of in-control ANOS and in-control average sample size , or (ii) minimising an overall range of shifts , based on different in-control EANOS and ASS0 combinations.

Synthetic Double Sampling X̄ Chart with Estimated Process Parameters

The synthetic double sampling (SDS) X̄ chart comprises the double sampling (DS) X̄ and conforming run length (CRL) sub-charts. The SDS X̄ chart has been studied under the assumption of known process parameters in the literature. Nevertheless, in practice, process parameters are usually unknown and are estimated from an in-control Phase I dataset. This paper investigates the performance of the SDS X̄ chart with estimated process parameters, in terms of the average run length (ARL), average number of observations to signal (ANOS) and standard deviation of the run length (SDRL). The performance of the SDS X̄ chart with estimated process parameters by minimizing the out-of-control ARL and the out-of-control ANOS is compared with the corresponding chart’s performance with known process parameters. In addition, the minimum number of Phase I samples required by the SDS X̄ chart with estimated process parameters so that it has approximately the same in-control ARL and ANOS performances as the chart with known process parameters is studied. The ARL, ANOS and SDRL properties of the SDS X̄ chart with estimated process parameters differ significantly from that of the chart with known process parameters. Therefore, suitable optimal charting parameters are introduced so that the SDS X̄ chart with estimated process parameters has an adequate performance as its known process parameters counterpart without having to use large number of Phase I samples and sample size.

Synthetic-Type Control Charts for Time-Between-Events Monitoring

This paper proposes three synthetic-type control charts to monitor the mean time-between-events of a homogenous Poisson process. The first proposed chart combines an Erlang (cumulative time between events, Tr) chart and a conforming run length (CRL) chart, denoted as Synth-Tr chart. The second proposed chart combines an exponential (or T) chart and a group conforming run length (GCRL) chart, denoted as GR-T chart. The third proposed chart combines an Erlang chart and a GCRL chart, denoted as GR-Tr chart. By using a Markov chain approach, the zero- and steady-state average number of observations to signal (ANOS) of the proposed charts are obtained, in order to evaluate the performance of the three charts. The optimal design of the proposed charts is shown in this paper. The proposed charts are superior to the existing T chart, Tr chart, and Synth-T chart. As compared to the EWMA-T chart, the GR-T chart performs better in detecting large shifts, in terms of the zero- and steady-state performances. The zero-state Synth-T4 and GR-Tr (r = 3 or 4) charts outperform the EWMA-T chart for all shifts, whereas the Synth-Tr (r = 2 or 3) and GR-T 2 charts perform better for moderate to large shifts. For the steady-state process, the Synth-Tr and GR-Tr charts are more efficient than the EWMA-T chart in detecting small to moderate shifts.

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.

A Novel Adaptive Control Chart with Variable Sample Size, Sampling Interval and Control Limits: A three Stage Variable Parameters (TSVP)

In recent decades, various adaptive X control charts with various variable parameters such as sampling interval,sample size, and control limits have been proposed to improve the efficiency of detecting the out-of-controlconditions for small, medium, and large shifts. The variable parameters (VP) adaptive chart is one of theseproposed charts that have an acceptable performance compared to other schemes especially for small shifts. Inthis article we proposed a modified version of VP chart with three stage variable chart parameters and havecompared it with some other adaptive charts. Here we used three mostly used indicators to evaluate theperformance of these charts which are Average Time to Signal (ATS), Average Number of Observations to Signal(ANOS) and Average Number of Samples to Signal (ANSS). Also we have calculated the optimal points fordifferent adaptive charts and compared them in these points according to the indicators mentioned above.