In this paper, we propose a variable sampling interval Shewhart control chart to monitor the coefficient of variation (CV) squared, denoted by VSI SH-
γ
2
. The new model overcomes the ARL-biased ...(average run length) property of the control chart monitoring the CV in a previous study by designing two one-sided charts rather than one two-sided chart. Moreover, the effect of measurement error on the performance of the VSI SH-
γ
2
control chart is investigated. The incorrect formula for the distribution of the CV in the presence of measurement error in a former study is fixed. Numerical simulations show that the precision errors and accuracy errors do have negative influences on the VSI SH-
γ
2
chart. An appropriate strategy based on the obtained results is suggested to reduce these negative effects.
•Classify the asymptotic behavior SIR models with noise perturbing to linear terms.•Construct a threshold value based on coefficients.•The negativity of threshold implies the extinction of the model ...at exponential rate.•If it is positive, the solution converges to unique stationary measure in total variation.•We can use this technique to classify models with noise perturbing to non linear terms.
In this paper, we classify the asymptotic behavior for a class of stochastic SIR epidemic models represented by stochastic differential systems where the Brownian motions and Lévy jumps perturb to the linear terms of each equation. We construct a threshold value between permanence and extinction and develop the ergodicity of the underlying system. It is shown that the transition probabilities converge in total variation norm to the invariant measure. Our results can be considered as a significant contribution in studying the long term behavior of stochastic differential models because there are no restrictions imposed to the parameters of models. Techniques used in proving results of this paper are new and suitable to deal with other stochastic models in biology where the noises may perturb to nonlinear terms of equations or with delay equations.
In many industrial manufacturing processes, the quality of products can depend on the relative amount between two quality characteristics X and Y. Often, this calls for the on-line monitoring of the ...ratio Z = X/Y as a quality characteristic itself by means of a control chart. A large number of control charts monitoring the ratio have been investigated in the literature under the assumption of independent normal observations of the two quality characteristics. In practice, due to the high frequency in sensor data collection, both autocorrelation and cross-correlation between consecutive observations can exist for X and Y and should be modelled to protect against the false alarm rate inflation when implementing a control chart for monitoring the ratio Z = X/Y. In this paper, we tackle this problem by investigating the performance of the Phase II Shewhart-type RZ control chart monitoring the ratio of two normal variables whose relationship is captured by a bivariate time series autoregressive model VAR(1), which can also account for the cross-correlation between the two quality characteristics. With the numerical study, we discuss how the design and the statistical performance of the Shewhart-type RZ control chart change with the VAR(1) model's parameters. We also provide an example to illustrate the use of the Shewhart-type RZ control chart with bivariate time series of observations in a furnace process.
In this paper, we investigate the effect of the measurement error on the performance of the cumulative sum (CUSUM) control charts monitoring the coefficient of variation. The measurement errors are ...supposed to follow a linear covariate error model. The obtained results show that the precision error ratio and the accuracy error have negative impact on the chart performance. Moreover, in order to overcome the difficulty in predetermining a specific value for the process shift size, we suggest to optimize the parameters of the charts according to the random shift size in a given interval. The robustness of the proposed method is studied. An example is given to illustrate the use of the CUSUM charts on a real quality control problem from sintering process.
Monitoring the ratio between two random normal variables plays an important role in many industrial manufacturing processes. In this paper, we suggest designing two one‐sided Shewhart control charts ...monitoring this ratio. The numerical results show that the one‐sided charts have more advantages compared with the two‐sided Shewhart chart proposed previously in the literature. Moreover, we investigate the effect of measurement error on the performance of these control charts where the measurement error is supposed to follow a linear covariate error model. The change of model parameters from an in‐control condition to an out‐of‐control is presented without using a strict assumption about the independence of the shift size from measurement errors. A valuable finding from this study is that taking multiple measurements per item is not an effective way to reduce the negative effect of measurement error on the Shewhart charts' performance.
We investigate in this paper a new type of control chart called VSI EWMA‐RZ by integrating the variable sampling interval feature (VSI) with the exponentially weighted moving average (EWMA) scheme to ...monitor the ratio of two normal random variables. Because the distribution of the ratio is skewed, we suggest designing two separated one‐sided charts instead of one two‐sided chart. A new coefficient is introduced allowing us to be free to choose a sampling interval provided that it optimizes the performance of the control chart. We also make a direct comparison between the VSI EWMA‐RZ charts and standard EWMA‐RZ control charts. The numerical simulations show that the proposed charts outperform the standard EWMA charts in detecting process shifts. An application is illustrated for the implementation of the VSI EWMA‐RZ control charts in the food industry.
In many industrial manufacturing processes, the ratio between two normal random variables plays a key role in ensuring quality of products. Thus, monitoring this ratio is an important task that is ...well worth considering. In this paper, we combine a variable sampling interval (VSI) strategy with a cumulative sum (CUSUM) scheme to create a new type of control chart for purpose of tracking the ratio between two normal variables. The average time to signal (ATS) and the expected average time to signal (EATS) criteria are used to evaluate the performance of the new VSI CUSUM RZ control chart. The numerical results show that the proposed control chart has much more attractive performance in comparison with the standard CUSUM‐RZ control chart and the VSI EWMA‐RZ control chart.
In the industrial practice, control charts are frequently implemented assuming that the quality characteristic of interest can be accurately measured without errors. In general, this assumption is ...not realistic: measurement error always exists in quality control applications and may considerably affect the performance of control charts in detecting the occurrence of an out-of-control condition. In this paper, the effect of measurement error on the statistical performance of Shewhart
t
and EWMA
t
control charts is investigated. Several tables are provided to show how the statistical performance of these control charts changes with different sources of the measurement error. The obtained results show that the measurement errors have a significant influence on the performance of both the Shewhart
t
and EWMA
t
control charts. Two examples in the analytical chemistry and food industry are presented to illustrate the use of the proposed charts.
This paper focuses on a stochastic SIR model with Beddington–DeAngelis functional response, perturbed by a complex type of noises. By introducing a threshold, sufficient and almost necessary ...conditions for permanence and extinction of the disease are given.
PDF
