In order to monitor univariate auto-correlated processes, many kinds of Control Charts have been proposed in the literature. However, for Multivariate auto-correlated processes, despite of their many applications, Control Charts have been seldom proposed. In this article, based on a method to reduce auto-correlation in the observed data, a Control Chart, called Multivariate Grouped Batch Means (MGBM), is proposed to monitor the mean vector of Multivariate auto-correlated processes. The parameters of this Chart, which is a model-free Chart that does not rely on the modeling structure of the data at hand, are optimized based upon a vector auto-regressive of order-one process. Moreover, the performance of the proposed Control Chart in terms of in-Control and out of- Control average run lengths are investigated by a simulation study of a 2-variate process. The result of the simulation study is encouraging.

We introduce a method for the statistical design of a depth-based Control Chart, using the percentile-based approach. The proposed Control Chart is affine invariant and is asymptotically distribution-free. Generally, the performance of a Control Chart is evaluated with the average run length metric. The average run length metric has a geometric distribution skewed to the right with a large standard deviation and may not be a proper measure for evaluating the Control Chart. Therefore, we use the statistical design method of Control Charts with the PL approach, which is an improvement and development on classical statistical design. By employing constraints on average run length, the length of in-Control and out-of-Control performances are guaranteed with predetermined probabilities and we can ensure that the in-Control run length exceeds the desired value and the out-of-Control run length is less than the desired value. Simulation studies show that the proposed Control Chart is more efficient than the average run length approach.

Water pH and active ingredient concentration are two of the most important variables to consider in the manufacturing process of fungicides. If these variables do not meet the required standards, the quality of the product may be compromised and lead to poor fungicide performance when water is used as the application carrier, which is in most cases. Given the correlation between the variables, these kinds of manufacturing processes must be analyzed in Multivariate settings. Thus, this paper analyzes the variables involved in the process using the Multivariate Control Chart  S introduced by J. A. Vargas. In the original Chart, the arithmetic mean is used as the mean vector estimator. However, in this investigation the arithmetic mean was replaced by the Winsorized Mean for the purpose of evaluating the Chart performance with a robust estimator. The results show that using the new estimator, the Control Chart is able to detect shifts in the variation of the mean vector that the traditional estimator did not. Furthermore, different subgroup sizes for the data were studied in order to examine the performance of the Chart in each case. It was found that the proposed Control Chart is more sensible to changes when the subgroups consist of less observations, since it is able to better identify the outliers in the sample.

Normality is a common assumption for many quality Control Charts. One should expect misleading results once this assumption is violated. In order to avoid this pitfall, we need to evaluate this assumption prior to the use of Control Charts which require normality assumption. However, in certain cases either this assumption is overlooked or it is hard to check. Robust Control Charts and bootstrap Control Charts are two remedial measures that we could use to overcome this issue. In this paper, a new bootstrap algorithm is proposed to construct Hotelling’ s T2 Control Chart. The performance of proposed Chart is evaluated through a simulation study. Our results are compared to the traditional Hotelling’ s T2 Control Chart results and the bootstrap results reported by Phaladiganon et al. [13] using in-Control and out-of-Control average run lengths denoted by ARL0 and ARL1, respectively. The latter case is obtained when the process mean is subject to sustained shifts. Numerical results indicate that the proposed algorithm performs better than the above mentioned methods. The new bootstrap algorithm is also applied to a real data set.

The usual procedure when employing a T2 Control Chart for Multivariate process monitoring is to take samples of fixed size n0 every h0 hours from the process. Recent studies have shown that using variable parameters (VP) schemes results in Charts with more statistical power when detecting small to moderate shifts in the process mean vector. In this paper, the VPT2 Control Chart for monitoring the process mean vector is economically designed. The cost model proposed by Lorenzen and Vance is used here and is minimized through a genetic algorithm (GA) approach.