A new approach for monitoring process variance

ABSTRACT

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having non-normal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used in such circumstances. In this paper, we propose a new variance chart based on a simple statistic to monitor process variance shifts. We explore the sampling properties of the new monitoring statistic and calculate the average run lengths (ARLs) of the proposed variance chart. Furthermore, an arcsine transformed exponentially weighted moving average (EWMA) chart is proposed because the ARLs of this modified chart are more intuitive and reasonable than those of the variance chart. We compare the out-of-control variance detection performance of the proposed variance chart with that of the non-parametric Mood variance (NP-M) chart with runs rules, developed by Zombade and Ghute [Nonparametric control chart for variability using runs rules. Experiment. 2014;24(4):1683–1691], and the nonparametric likelihood ratio-based distribution-free exponential weighted moving average (NLE) chart and the combination of traditional exponential weighted moving average (EWMA) mean and EWMA variance (CEW) control chart proposed by Zou and Tsung [Likelihood ratio-based distribution-free EWMA control charts. J Qual Technol. 2010;42(2):174–196] by considering cases in which the critical quality characteristic has a normal, a double exponential or a uniform distribution. Comparison results showed that the proposed chart performs better than the NP-M with runs rules, and the NLE and CEW control charts. A numerical example of service times with a right-skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the application of the proposed variance chart and of the arcsine transformed EWMA chart and to compare them with three existing variance (or standard deviation) charts. The proposed charts show better detection performance than those three existing variance charts in monitoring and detecting shifts in the process variance.     

Control Charts for the Mean and Variance based on Changepoint Methodology

This article develops a control chart for the mean and variance of a normal distribution based on changepoint methodology. A Bayesian approach is used to incorporate parameter uncertainty. The resulting control chart plots the probabilities of “no change” as samples become available at the monitoring stage. Average run length considerations are used to set the control limits. Simulations are used to compare the proposed chart with a more traditional Shewhart-type combined control chart for the mean and variance.     

Control Charts for the Generalized Variance Based on Its Predictive Distribution

This article develops a control chart for the generalized variance. A Bayesian approach is used to incorporate parameter uncertainty. Our approach has two stages, (i) construction of the control chart where we use a predictive distribution based on a Bayesian approach to derive the rejection region, and (ii) evaluation of the control chart where we use a sampling theory approach to examine the performance of the control chart under various hypothetical specifications for the data generation model.     

Combined Exponentially Weighted Moving Average Charts for the Mean and Variance Based on the Predictive Distribution

This article develops combined exponentially weighted moving average (EWMA) charts for the mean and variance of a normal distribution. A Bayesian approach is used to incorporate parameter uncertainty. We first use a Bayesian predictive distribution to construct the control chart, and we then use a sampling theory approach to evaluate it under various hypothetical specifications for the data generation model. Simulations are used to compare the proposed charts for different values of both the weighing constant for the exponentially weighted moving averages and for the size of the calibration sample that is used to estimate the in-statistical-control process parameters. We also examine the separate performance of the EWMA chart for the variance.     

A median loss control chart for monitoring quality loss under skewed distributions

The quality and loss of products are crucial factors separating competitive companies in global market. Firms widely employ a loss function to measure the loss caused by a deviation of the quality variable from the target value. Monitoring this deviation from the process target value is important from the view of Taguchi’s philosophy. In reality, there are many situations where the distribution of the quality variable may not be normal but skewed. This paper aims at developing a median loss (ML) control chart for monitoring quality loss under skewed distributions. Both the cases with fixed and variable sampling intervals are considered. Numerical results show that the ML chart with (optimal) variable sampling intervals performs better than the ML chart in detecting small to moderate shifts in the process loss centre or in the difference of mean and target and/or variance of a process variable. The ML chart and the ML chart with variable sampling intervals also illustrate the best performance in detection out-of-control process for a process quality variable with a left-skewed distribution. A numerical example illustrates the application of the proposed control chart.     

A weighted likelihood ratio test-based chart for monitoring process mean and variability

In this paper, a new single exponentially weighted moving average (EWMA) control chart based on the weighted likelihood ratio test, referred to as the WLRT chart, is proposed for the problem of monitoring the mean and variance of a normally distributed process variable. It is easy to design, fast to compute, and quite effective for diverse cases including the detection of the decrease in variability and individual observation case. The optimal parameters that can be used as a design aid in selecting specific parameter values based on the average run length (ARL) and the sample size are provided. The in-control (IC) and out-of-control (OC) performance properties of the new chart are compared with some other existing EWMA-type charts. Our simulation results show that the IC run length distribution of the proposed chart is similar to that of a geometric distribution, and it provides quite a robust and satisfactory overall performance for detecting a wide range of shifts in the process mean and/or variability.     

New V control chart for the Maxwell distribution

Control charts have been popularly used as a user-friendly yet technically sophisticated tool to monitor whether a process is in statistical control or not. These charts are basically constructed under the normality assumption. But in many practical situations in real life this normality assumption may be violated. One such non-normal situation is to monitor the process variability from a skewed parent distribution where we propose the use of a Maxwell control chart. We introduce a pivotal quantity for the scale parameter of the Maxwell distribution which follows a gamma distribution. Probability limits and L-sigma limits are studied along with performance measure based on average run length and power curve. To avoid the complexity of future calculations for practitioners, factors for constructing control chart for monitoring the Maxwell parameter are given for different sample sizes and for different false alarm rate. We also provide simulated data to illustrate the Maxwell control chart. Finally, a real life example has been given to show the importance of such a control chart.     

A HEWMA-CUSUM control chart for the Weibull distribution

The Weibull distribution is one of the most popular distributions for lifetime modeling. However, there has not been much research on control charts for a Weibull distribution. Shewhart control is known to be inefficient to detect a small shift in the process, while exponentially weighted moving average (EWMA) and cumulative sum control chart (CUSUM) charts have the ability to detect small changes in the process. To enhance the performance of a control chart for a Weibull distribution, we introduce a new control chart based on hybrid EWMA and CUSUM statistic, called the HEWMA-CUSUM chart. The performance of the proposed chart is compared with the existing chart in terms of the average run length (ARL). The proposed chart is found to be more sensitive than the existing chart in ARL. A simulation study is provided for illustration purposes. A real data is also applied to the proposed chart for practical use.     

On the improvement of one-sided S control charts

The most common charting procedure used for monitoring the variance of the distribution of a quality characteristic is the S control chart. As a Shewhart-type control chart, it is relatively insensitive in the quick detection of small and moderate shifts in process variance. The performance of the S chart can be improved by supplementing it with runs rules or by varying the sample size and the sampling interval. In this work, we introduce and study one-sided adaptive S control charts, supplemented or not with one powerful runs rule, for detecting increases or decreases in process variation. The properties of the proposed control schemes are obtained by using a Markov chain approach. Furthermore, a practical guidance for the choice of the most suitable control scheme is also provided.     

A control chart for multivariate Poisson distribution using repetitive sampling

Control charts using repetitive group sampling have attracted a great deal of attention during the last few years. In the present article, we attempt to develop a control chart for the multivariate Poisson distribution using the repetitive group sampling scheme. In the proposed control chart, the monitoring statistic from the multivariate Poisson distribution has been used for the quick detection of the deteriorated process to avoid losses. The control coefficients have been estimated using the specified in-control average run lengths. The procedure of the proposed control chart has been explained by using the real-world example and a simulated data set. It has been observed that the proposed control chart is an efficient development for the quick detection of the nonrandom change in the manufacturing process.     

Time truncated attribute control chart for the Weibull distribution using multiple dependent state sampling

In this article, an attribute control chart is proposed for time truncated tests using the Weibull distribution. The design of proposed control chart is presented using the multiple dependent state (MDS) sampling. The control chart coefficients are determined for various specified average run length. The efficiency of the proposed control chart is elaborated with the help of a simulation data and a real data. The proposed control chart perform better than the existing control chart in terms of average run length.     

The Generally Weighted Moving Average Variance Chart

This article extends the generally weighted moving average (GWMA) technique for detecting changes in process variance. The proposed chart is called the generally weighted moving average variance (GWMAV) chart. Simulation is employed to evaluate the average run length (ARL) characteristics of the GWMAV and EWMA control charts. An extensive comparison of these control charts reveals that the GWMAV chart is more sensitive than the EWMA control charts for detecting small shifts in the variance of a process when the shifts are below 1.35 standard deviations. Additionally, the GWMAV control chart performs little better when the variance shifts are between 1.35 and 1.5 standard deviation, and the 2 charts performs similar when the variance shifts are above 1.5 standard deviation. The design of the GWMAV chart is also discussed.     

A new t-chart using process capability index

Control charts play a vital role to enhance the efficiency of the manufacturing process. In many situations, the quality characteristic of interest to be monitored follows a non-normal distribution. In this article, we propose a new control chart using the process capability index when the quality characteristic follows the exponential distribution. The performance of the proposed chart is evaluated using the Monte Carlo simulation. Tables are presented for various values of specified average run length and sample size. The use of the proposed control chart is discussed with the help of an example.     

Designing of a control chart using belief statistic for exponential distribution

A new control chart is proposed by using the belief statistic for the exponential distribution. The structure of the proposed control chart is given to measure the average run length for the shifted process. The comparison of the proposed chart is given with the existing charts in terms of the average run lengths, which shows the outperformance of the proposed chart. The performance of the proposed control chart is also discussed with the help of simulated data.     

A Sequential Rank-Based Nonparametric Adaptive EWMA Control Chart

Nonparametric control chart is useful when the underlying distribution is unknown, or is not likely to be normal. In this article, we provide a sequential rank-based nonparametric adaptive EWMA (NAE) control chart for detecting the persistent shift in the location parameter. This NAE chart is a self-starting scheme and thus can be used to monitor processes at the start-up stages rather than waiting for the accumulation of sufficiently large calibration samples. Moreover, we do not require any prior knowledge of the underlying distribution, and to prespecify any tuning parameter either. A Markov chain model is suggested to calibrate the run-length distribution of NAE, which is shown to have approximate tail probability as a geometric distribution. A simulation study demonstrates that the proposed control chart not only performs robustly for different distributions, but also is efficient in detecting various magnitude of shifts. A real-data example from manufacturing shows that it performs quite well in practical applications.     

Copula-Based Bivariate ZIP Control Chart for Monitoring Rare Events

One difficulty with developing multivariate attribute control charts is the lack of the related joint distribution. So, if it would be possible to generate the joint distribution of two (or more) attribute characteristics, then a bivaraite (or multivariate) attribute control chart can be developed based on Types I and II errors. Copula function is a solution to the matter. In this article, applying the copula function approach, we achieve the joint distribution of two correlated zero inflated Poisson (ZIP) distributions. Then, using this joint distribution, we develop a bivaraite control chart which can be used for monitoring correlated rare events. This copula-based bivariate ZIP control chart is compared with the simultaneous use of two separate univariate ZIP control charts. Based on the average run length (ARL) measure, it is shown that the proposed control chart is much better than the simultaneous use of two separate univariate charts. In addition, a real case study related to the environmental air in a sterilization process is investigated to show the applicability of the developed control chart.     

ON THE EVALUATION OF CONTROL CHART LIMITS BASED ON PREDICTIVE DISTRIBUTIONS

ABSTRACT

We propose a Bayesian approach to obtaining control charts when there is parameter uncertainty. Our approach consists of two stages, (i) construction of the control chart where we use a predictive distribution based on a Bayesian approach to derive the rejection region, and (ii) evaluation of the control chart where we use a sampling theory approach to examine the performance of the control chart under various hypothetical specifications for the data generation model.     

A CUSUM control chart for monitoring the variance when parameters are estimated

CUSUM control chart has been widely used for monitoring the process variance. It is usually used assuming that the nominal process variance is known. However, several researchers have shown that the ability of control charts to signal when a process is out of control is seriously affected unless process parameters are estimated from a large in-control Phase I data set. In this paper we derive the run length properties of a CUSUM chart for monitoring dispersion with estimated process variance and we evaluate the performance of this chart by comparing it with the same chart but with assumed known process parameters.     

Attribute control chart for some popular distributions

The binomial distribution is often used to display attribute control data. In this paper, a statistical model is settled for attribute control chart under truncated life test. By Burr X & XII, inverse Gaussian (IG), and exponential lifetime-truncated distributions, a Shewhart-type attribute control chart is built to display the data. The performance of attributed control chart constructed on truncated life test is evaluated by average run length, which compares the performance of all distributions. Our study arranges that IG is better distribution among all.