In-control ARL Research Articles

Asymmetric Control Limits for Weighted-Variance Mean Control Chart with Different Scale Estimators under Weibull Distributed Process

Shewhart charts are the most commonly utilised control charts for process monitoring in industries with the assumption that the underlying distribution of the quality characteristic is normal. However, this assumption may not always hold true in practice. In this paper, the weighted-variance mean charts are developed and their population standard deviation is estimated using the three subgroup scale estimators, namely the standard deviation, median absolute deviation and standard deviation of trimmed mean for monitoring Weibull distributed data with different coefficients of skewness. This study aims to compare the out-of-control average run length of these charts with the pre-determined fixed value of the in-control ARL in terms of different scale estimators, coefficients of skewness and sample sizes via extensive simulation studies. The results indicate that as the coefficients of skewness increase, the charts tend to detect the out-of-control signal more rapidly under identical magnitude of shift. Meanwhile, as the size of the shift increases under the same coefficient of skewness, the proposed charts are able to locate the shifts quicker and the similar scenarios arise as a sample size raised from 5 to 10. A real data set from survival analysis domain which, possessing Weibull distribution, was to demonstrate the usefulness and applicability of the proposed chart in practice.

CUSUM design for detection of event-rate increases for a Poisson process

In quantifying the performance of a CUSUM chart for detecting upward shifts in event rate, it has been recommended that steady-state evaluation of performance measures such as ARL be used. In this article, the methodology for making such evaluations using the Markov-chain approach is presented, for the case of an exponential CUSUM. This is much more efficient than the alternative of simulation, which is still in use. It is also shown that if one is using steady-state ARL as a measure of detection performance, one can find better choices for the CUSUM parameter k than that provided by the SPRT-based formula. Two types of shift in event rate are considered, and corresponding tables of recommended choices of CUSUM parameters (k, h) are presented for ten levels of in-control ARL, and for nine sizes of shift. These tables can assist quality engineers in the design of CUSUMs for monitoring inter-event times in steady-state operation. It is also shown that these exponential CUSUM tables may be used to find values for the parameters of a geometric CUSUM or a Bernoulli CUSUM chart for monitoring a proportion, provided the in-control value of the proportion is no more than approximately 0.5%.

QREI's Best Paper Awards 2020

QREI's Best Paper Awards 2020

Designing an Effective Combined Shewhart-CUSUM Control Scheme with Exponentially Distributed Data

In this paper, the Combined Shewhart-CUSUM control scheme has been proposed to monitor the production process when the quality characteristic follows exponential distribution to quickly detect the shift in the process. The simulated values of ARL are determined after the transformation of the data into approximate normal distribution by Nelson transformation method and adding Shewhart control limits to existing CUSUM Control Chart. Scheme parameters (value of k and h) and out of control ARL are calculated at various shift and at various in-control ARL. Parameters are also calculated to detect δ standard deviation shifts, which may be helpful to the quality control practitioners in designing the Combined Shewhart-CUSUM scheme when data is highly skewed.

Conditional design of the CUSUM median chart for the process position when process parameters are unknown

ABSTRACTIn practice, different practitioners will use different Phase I samples to estimate the process parameters, which will lead to different Phase II control chart's performance. Researches refer to this variability as between-practitioners-variability of control charts. Since between-practitioners-variability is important in the design of the CUSUM median chart with estimated process parameters, the standard deviation of average run length (SDARL) will be used to study its properties. It is shown that the CUSUM median chart requires a larger amount of Phase I samples to sufficiently reduce the variation in the in-control ARL of the CUSUM median chart. Considering the limitation of the amount of the Phase I samples, a bootstrap approach is also used here to adjust the control limits of the CUSUM median chart. Comparisons are made for the CUSUM and Shewhart median charts with estimated parameters when using the adjusted- and unadjusted control limits and some conclusions are made.

Improved control charts for fraction non-conforming based on hypergeometric distribution

In this paper we introduce the control chart for fraction non-conforming based on the Hypergeometric distribution, and propose the exact Hypergeometric test for a population proportion to detect significant changes in the process quality level. In contrast to the common Binomial p-chart, which is especially suitable for sampling from continuous production processes, the Hypergeometric p-chart can be established also for periodical production processes, where sampled items successively produced during one period are not stochastically independent and the probability of a sampled item being non-conforming depends on the previous sampled item. Further, while the Binomial p-chart does not take the fluctuation of the production process into account, the Hypergeometric p-chart enables consideration of the population size of periodical production appropriately. The Hypergeometric p-chart leads in general to lower probabilities of the type I error, i.e. false rejection of the hypothesis of statistical control, to a considerably higher in-control ARL, and thus to a significantly lower frequency of false alarms. Moreover, it provides monitoring of a manufacturing process with a considerably lower inspection effort than its Binomial counterpart. Finally, the exact Hypergeometric test leads to more accurate test decisions in detecting process shifts compared with the common used approximate Binomial test.

A fuzzy nonparametric Shewhart chart based on the bootstrap approach

Abstract In this paper, we consider a nonparametric Shewhart chart for fuzzy data. We utilize the fuzzy data without transforming them into a real-valued scalar (a representative value). Usually fuzzy data (described by fuzzy random variables) do not have a distributional model available, and also the size of the fuzzy sample data is small. Based on the bootstrap methodology, we design a nonparametric Shewhart control chart in the space of fuzzy random variables equipped with some L2 metric, in which a novel approach for generating the control limits is proposed. The control limits are determined by the necessity index of strict dominance combined with the bootstrap quantile of the test statistic. An in-control bootstrap ARL of the proposed chart is also considered.

Phase II Shewhart-type Control Charts for Monitoring Times Between Events and Effects of Parameter Estimation

Monitoring times between events (TBE) is an important aspect of process monitoring in many areas of applications. This is especially true in the context of high-quality processes, where the defect rate is very low, and in this context, control charts to monitor the TBE have been recommended in the literature other than the attribute charts that monitor the proportion of defective items produced. The Shewhart-type t-chart assuming an exponential distribution is one chart available for monitoring the TBE. The t-chart was then generalized to the tr-chart to improve its performance, which is based on the times between the occurrences of r (≥1) events. In these charts, the in-control (IC) parameter of the distribution is assumed known. This is often not the case in practice, and the parameter has to be estimated before process monitoring and control can begin. We propose estimating the parameter from a phase I (reference) sample and study the effects of estimation on the design and performance of the charts. To this end, we focus on the conditional run length distribution so as to incorporate the ‘practitioner-to-practitioner’ variability (inherent in the estimates), which arises from different reference samples, that leads to different control limits (and hence to different IC average run length [ARL] values) and false alarm rates, which are seen to be far different from their nominal values. It is shown that the required phase I sample size needs to be considerably larger than what has been typically recommended in the literature to expect known parameter performance in phase II. We also find the minimum number of phase I observations that guarantee, with a specified high probability, that the conditional IC ARL will be at least equal to a given small percentage of a nominal IC ARL. Along the same line, a lower prediction bound on the conditional IC ARL is also obtained to ensure that for a given phase I sample, the smallest IC ARL can be attained with a certain (high) probability. Summary and recommendations are given. Copyright © 2014 John Wiley & Sons, Ltd.

Optimal EWMA of linear combination of Poisson variables for multivariate statistical process control

In this paper, we propose a new process control chart for monitoring correlated Poisson variables, the EWMA LCP chart. This chart is the exponentially weighted moving average (EWMA) version of the recently proposed LCP chart. The latter is a Shewhart-type control chart whose control statistic is a linear combination of the values of the different Poisson variables (elements of the Poisson vector) at each sampling time. As a Shewhart chart, it is effective at signalling large process shifts but is slow to signal smaller shifts. EWMA charts are known to be more sensitive to small and moderate shifts than their Shewhart-type counterparts, so the motivation of the present development is to enhance the performance of the LCP chart by the incorporation of the EWMA procedure to it. To ease the design of the EWMA LCP chart for the end user, we developed a user-friendly programme that runs on Windows© and finds the optimal design of the chart, that is, the coefficients of the linear combination as well as the EWMA smoothing constant and chart control limits that together minimise the out-of-control ARL under a constraint on the in-control ARL. The optimization is carried out by genetic algorithms where the ARLs are calculated through a Markov chain model. We used this programme to evaluate the performance of the new chart. As expected, the incorporation of the EWMA scheme greatly improves the performance of the LCP chart.

Optimal linear combination of Poisson variables for multivariate statistical process control

In this paper we analyze the monitoring of p Poisson quality characteristics simultaneously, developing a new multivariate control chart based on the linear combination of the Poisson variables, the LCP control chart. The optimization of the coefficients of this linear combination (and control limit) for minimizing the out-of-control ARL is constrained by the desired in-control ARL. In order to facilitate the use of this new control chart the optimization is carried out employing user-friendly Windows© software, which also makes a comparison of performance between this chart and other schemes based on monitoring a set of Poisson variables; namely a control chart on the sum of the variables (MP chart), a control chart on their maximum (MX chart) and an optimized set of univariate Poisson charts (Multiple scheme). The LCP control chart shows very good performance. First, the desired in-control ARL (ARL0) is perfectly matched because the linear combination of Poisson variables is not constrained to integer values, which is an advantage over the rest of charts, which cannot in general match the required ARL0 value. Second, in the vast majority of cases this scheme signals process shifts faster than the rest of the charts.

DESIGN OF THE c AND np CHARTS WHEN THE PARAMETERS ARE ESTIMATED

The performance of attribute control charts is usually evaluated under the assumption of known process parameters (i.e., the nominal proportion of nonconforming units or the nominal number of nonconformities). However, in practice, these process parameters are rarely known and have to be estimated from an in-control historical Phase I data set. In this paper, we investigate the number m of Phase I samples that would be necessary for the c and p charts with estimated parameters in order to obtain similar in-control ARL's as in the known parameters case. We also propose new specific chart parameters that allow the c and np charts with estimated parameters to have approximately the same in-control ARL's as the ones obtained in the known parameter case.

The Design and Performance of the Multivariate Synthetic-T 2 Control Chart

One of the objectives of research in statistical process control is to obtain control charts that show few false alarms but, at the same time, are able to detect quickly the shifts in the distribution of the quality variables employed to monitor a productive process. In this article, the synthetic-T 2 control chart is developed, which consists of the simultaneous use of a CRL chart and a Hotelling's T 2 control chart. The ARL is calculated employing Markov chains for steady and zero-state scenarios. A procedure of optimization has been developed to obtain the optimum parameters of the synthetic-T 2, for zero and steady cases, given the values of in-control ARL and magnitude of shift which needs to be detected rapidly. A comparison between (standard T 2, MEWMA, T 2 with variable sample size, and T 2 with double sampling) charts reveals that the synthetic-T 2 chart always performs better than the standard T 2 chart. The comparison with the remaining charts demonstrate in which cases the performance of this new chart makes it interesting to employ in real applications.

The Revised m-of-k Runs Rule

ABSTRACT Sensitizing rules based on runs and scans are widely used to increase the sensitivity of the classical Shewhart control chart in detecting small process shifts. The use of these rules frequently results in a substantial reduction of the in-control average run length ARL and/or in a poor performance of the out-of-control average run length for moderate to large process shifts. To overcome these weaknesses, we introduce the revised m-of-k runs rule and we study the average run length performance of the Shewhart control chart supplemented with the revised runs rule. Our extensive numerical experimentation and the comparisons made with other competitive control charts reveal that the revised runs rule demonstrates an improved performance in the detection of small to moderate shifts while maintaining the same superiority for detecting large shifts.

Use of independent runs for the improvement of the response of Shewhart control charts

The paper deals with an extension of standard Shewhart charts witch provide independent runs of given length, or IR-charts, with the purpose of improving the performance of the former by allowing a faster detection of small and medium shifts in a process mean. The author purposely chooses to remain in the field of indipendent sample observations, which are much more frequently used in SPC compared to other types of charts due to their simplicity, and shows how careful choice of independent runs, formed by the same sample means used in a standard X-chart, can be adopted in order to reduce ARL values and, therefore, the expected time of detection of out-of-control situations. The result is obtained without the main drawback of dependent runs procedures as summarised in the Western Electric Company runs rules and others recalled in the paper, with lies in the fact that their in-control ARL is too low, thus providing a too short expected time before a false out-of control alarm occurs.

An NP Control Chart Using Double Inspections

The np control chart is used widely in Statistical Process Control (SPC) for attributes. It is difficult to design an np chart that simultaneously satisfies a requirement on false alarm rate and has high detection effectiveness. This is mainly because one is often unable to make the in-control Average Run Length ARL 0 of an np chart close to a specified or desired value. This article proposes a new np control chart which is able to overcome the problems suffered by the conventional np chart. It is called the Double Inspection (DI) np chart, because it uses a double inspection scheme to decide the process status (in control or out of control). The first inspection decides the process status according to the number of non-conforming units found in a sample; and the second inspection makes a decision based on the location of a particular non-conforming unit in the sample. The double inspection scheme makes the in-control ARL 0 very close to a specified value and the out-of-control Average Run Length ARL 1 quite small. As a result, the requirement on a false alarm rate is satisfied and the detection effectiveness also achieves a high level. Moreover, the DI np chart retains the operational simplicity of the np chart to a large degree and achieves the performance improvement without requiring extra inspection (testing whether a unit is conforming or not).

An Optimal Filter Design Approach to Statistical Process Control

Many control charts can be viewed as charting the output of a linear filter applied to process data, with an alarm sounded when the filter output falls outside a set of control limits. We generalize this concept by considering a linear filter in its most general time-invariant form. We provide a strategy for optimizing the filter coefficients in order to minimize the out-of-control ARL, while constraining the in-control ARL to some desired value. The optimal linear filters exhibit a number of interesting characteristics, in particular when the process data are autocorrelated. In many situations, they also substantially outperform an optimally designed exponentially weighted moving average (EWMA) control chart.

On the Statistical Design of Geometric Control Charts

We address several theoretic issues involved in the design of geometric control charts. Primary among these is the question of setting probability limits, based on notions of “unbiasedness” and “conditional power”. A nearly ARL-unbiased design is defined by setting the in-control ARL as near as possible, given the discrete nature of the geometric distribution, to the peak of the ARL curve. An optimal design criterion, based on minimizing the sum of out-of-control ARLs for upward and downward shift is also suggested. These are shown to produce improved conditional power compared with the usual design.

False Alarm Probability Function of Cusum Charts

In this paper, standard Cusum charts with head start value zero are compared with fast initial response (FIR) Cusum charts proposed by Lucas and Crosier (Technometrics 24: 199–205, 1982) from the viewpoint of the false alarm probability function. The focus in this comparison is conservativeness. The concept of conservativeness is based on the relationship between the in-control ARL and the false alarm probability function by regarding the Cusum statistic as a Markov process. The comparison shows that standard Cusum charts are conservative in contrast to FIR Cusum charts. Moreover, the concept of false alarm probability function provides an easy probabilistic interpretation of the in-control ARL.

Design of change detection algorithms based on the generalized likelihood ratio test

AbstractA design procedure for detecting additive changes in a state‐space model is proposed. Since the mean of the observations after the change is unknown, detection algorithms based on the generalized likelihood ratio test, GLR, and on window‐limited type GLR, are considered. As Lai (1995) pointed out, it is very difficult to find a satisfactory choice of both window size and threshold for these change detection algorithms. The basic idea of this article is to estimate, through the stochastic approximation of Robbins and Monro, the threshold value which satisfies a constraint on the mean between false alarms, for a specified window size. A convenient stopping rule, based on the first passage time of an F‐statistic below a fixed boundary, is used to terminate the iterative approximation. Then, the window size which produces the most desirable out‐of‐control ARL, for a fixed value of the in‐control ARL, can be selected. These change detection algorithms are applied to detect biases on the measurements of ozone, recorded from one monitoring site of Bologna (Italy). Comparisons of the ARL profiles reveal that the full‐GLR scheme provides much more protection than the window‐limited GLR schemes against small shifts in the process, but the modified window‐limited GLR provides more protection against large shifts. Copyright © 2001 John Wiley & Sons, Ltd.

Control Charts Applicable When the Fraction Nonconforming is Small

It has been proposed that when the proportion of nonconforming product is extremely small that one base 3σ control charts on the 0.2777th power of X, the number of items sampled until a nonconforming item is found. This choice of exponent renders the transformed variable approximately normal in distribution. In this paper, that proposal is compared to the alternative of using a logarithmic transformation of X as the basis for 3σ control charts. Expressions are derived for the OC curves of both procedures. It is shown further that if probability limits are used in lieu of 3σ limits, the OC curves of the two procedures are identical. If probability limits are used, the power transformation is preferred to the log transform, since of the two, its distribution is more nearly normal and, thus, more closely justifies the use of the supplementary tests commonly used to enhance the power of control charts. Computation of the average run lengths shows that the power transformation gives poor protection against worsening quality and that the log transformation results in an unacceptably short in-control ARL value. It is shown that either transformed value gives greatly improved performance when used in an EWMA scheme. The power transformation is recommended over the log transformation as the basis for an EWMA chart because it will detect an increase in the fraction nonconforming of up to a factor of three with a smaller resultant ARL. Moreover, it will detect any decrease in the fraction nonconforming more quickly.