Hotelling T2 Control Chart Research Articles

Multivariate SPC for recipe preservation of batch processes

AbstractThe performances of the Hotelling's T2 control chart and the squared prediction error control chart based on the multi‐way principal component analysis are evaluated for monitoring within batch process variation for the purpose of recipe preservation. A nonlinear model for simulated batch process data is provided. The model allows for cross correlation of error terms at a given time period and serial correlation of error terms across time periods. The performance characterizations of the two monitoring schemes are provided for a variety of levels of cross correlation and serial correlation. The impact of the time period at which process shifts occur is also investigated for the monitoring schemes. The T2 control chart is recommended for the cases considered. Copyright © 2009 John Wiley & Sons, Ltd.

A Multivariate Robust Control Chart for Individual Observations

To monitor a multivariate process, a classical Hotelling's T2 control chart is often used. However, it is well known that such control charts are very sensitive to the presence of outlying observations in the historical Phase I data used to set the control limit. In this paper, we propose a robust Hotelling's T2-type control chart for individual observations based on highly robust and efficient estimators of the mean vector and covariance matrix known as reweighted minimum covariance determinant (RMCD) estimators. We illustrate how to set the control limit for the proposed control chart, study its performance using simulations, and illustrate implementation in a real-world example.

A New Multivariate Non-Parametric Control Chart Based on Sign Test

Multivariate statistical process control deserves particular attention in the recent scenario. Though, Hotelling T2 control chart is quite popular and widely used technique in this field but its performance is deteriorated when the underlying distribution of the quality characteristics is not following multivariate normal distribution. Hence the need of developing a non-parametric multivariate control chart arises which does not require any underlying distribution. In this work a multivariate non-parametric control chart based on bivariate sign test is proposed. Its performance in both in control and out of control state was evaluated by simulating data from multivariate normal and multivariate t distribution and compared with those of existing multivariate parametric control chart.

A robust alternative to Hotelling's T2 control chart using trimmed estimators

AbstractA control chart is one of the primary techniques used in statistical process control. In phase I, historical observations are analysed in order to construct a control chart with which to determine whether the process has been in control over the period of time in which the data were collected. The presence of multiple outliers may go undetected by the usual control charts, such as Hotelling's T2 due to the masking effect. In this paper we propose a robust alternative to Hotelling's T2 control chart with estimators defined using trimming. Simulation studies show that the proposed control chart is more effective than T2 in detecting outliers. Copyright © 2008 John Wiley & Sons, Ltd.

Causation-Based T2 Decomposition for Multivariate Process Monitoring and Diagnosis

Multivariate process monitoring and diagnosis is an important and challenging issue. The widely adopted Hotelling T2 control chart can effectively detect a change in a system but is not capable of diagnosing the root causes of the change. The MTY approach makes efforts to improve the diagnosability by decomposing the T2 statistic. However, this approach is computationally intensive and has a limited capability in root-cause diagnosis for a large dimension of variables. This paper proposes a causation-based T2 decomposition method that integrates the causal relationships revealed by a Bayesian network with the traditional MTY approach. Theoretical analysis and simulation studies demonstrate that the proposed method substantially reduces the computational complexity and enhances the diagnosability, compared with the MTY approach.

Economic design of an adaptive T2 control chart

Varying three parameters of the Hotelling's T2 control chart, namely, the sample size, the sampling interval length, and the action limit, results in a great improvement over the traditional T2 control chart particularly for the small process shift. This paper presents the economic model for designing the variable parameters T2 chart. In the economic design, a cost function is constructed, involving the cost of sampling and testing, the cost of false alarm, the cost to detect and remove the assignable cause, and the cost when the process is operating out-of-control. Also, a heuristic method of finding optimal values of adaptive design parameters for minimizing the cost function is presented. Variable and fixed parameters T2 control charts are compared with respect to the expected loss per unit time. Furthermore, the effects of the initial sample number (used for estimating the parameters of F-distribution) upon the operating cost and adaptive design parameters are examined.

An approach to controlling two dependent process steps with autocorrelated observations

The observations from the process output are always assumed independent when using a control chart to monitor a process. However, for many processes the process observations are autocorrelated. This autocorrelation can have a significant effect on the performance of the control chart. This paper considers the problem of monitoring the mean of a quality characteristic X on the first process step and the mean of a quality characteristic Y on the second process step, in which the observations X can be modeled as an AR(1) model and observations Y can be modeled as a transfer function of X since the state of the second process step is dependent on the state of the first process step. To effectively distinguish and maintain the state of the two dependent process steps, the Shewhart control chart of residual and the cause-selecting control chart are proposed. The proposed control charts’ performance is measured by the rate of alarm on the proposed charts. From numerical analysis, it shows that the performance of the proposed control charts is much better than the misused Hotelling T2 control chart and the individual Shewhart X and Y control charts.

SHORT RUNS MULTIVARIATE CONTROL CHART FOR PROCESS DISPERSION

The multivariate Hotelling's T2 control chart is designed to be used in a mass production for processes where data to estimate the mean vector and covariance matrix as well as the computation of control limits are available before a production run. Recent years have seen a trend in manufacturing industries to produce smaller lot sizes, a.k.a., low volume production which is a result of increased importance given to just-in-time (JIT) manufacturing techniques, synchronous manufacturing and the reduction of in-process inventory and costs. This new manufacturing environment is also referred to as short runs production or short runs. In a short runs environment, it is difficult or perhaps impossible to establish a reliable historical data set in setting valid control limits and in estimating process parameters due to the availability of insufficient data for a particular process because production runs are usually short and change frequently from one process to another. There is also a need to start charting at or very near the beginning of the run in such a case. Another problem encountered in a short runs production such as in job shops is that there are many different types of measurements so that many different control charts are needed. Standardized control charts that allow different statistics to be plotted on the same chart are extremely useful in short runs. Control charts with standard scale simplify the control charting process in a short runs environment. In this paper, we address the multivariate short runs problems for process dispersion based on individual measurements by presenting the required formulas so that the chart can be used from the start of production, whether or not prior information for estimating the chart's limit and its parameter is available. The proposed chart plots standardized statistics for multiple parts on the same chart. This paper extends the work of the authors in Ref. 13.

Individual Station Monitoring Using Press Tonnage Sensors for Multiple Operation Stamping Processes

In multiple operation stamping processes, a press tonnage signal measured by press tonnage sensors installed on press linkages/uprights, is the summation of die forces at all stations. Different from the current practice of using a whole cycle of press tonnage signals to monitor the compound condition of all stations, this paper proposes a new method to use the partitioned monitoring segments of press tonnage signals to monitor individual station conditions. For this purpose, a generic signal segmentation principle is proposed, and the Hotelling T2 control charts are developed with consideration of the interactions among stations. A real case study of a doorknob stamping process is provided to demonstrate analysis procedures and effectiveness of the developed methodology.

Statistically constrained economic design of the multivariate exponentially weighted moving average control chart

AbstractThis paper shows that economic statistical design can provide for better statistical properties without significantly increasing optimal total costs. Cost comparisons between optimal economic statistical designs and optimal economic designs show no significant cost increases. The average run length (ARL) constraints added by economic statistical design significantly improve the statistical properties of the control chart scheme. False alarm frequency is limited while keeping good shift detection characteristics. In addition, the Multivariate Exponentially Weighted Moving Average (MEWMA) control schemes performed better from the cost standpoint than the benchmark pure statistical design—the Hotelling T2 control chart. This improvement held for unconstrained and constrained designs. Finally, cost comparisons at small values of n showed significant advantage for the MEWMA schemes. Copyright © 2001 John Wiley & Sons, Ltd.

STATISTICAL PROCESS CONTROL FOR TWO GRADE PRODUCTS UNDER TWO DEPENDENT PROCESSES

ABSTRACT In this paper, we propose an efficient approach to distinguish which process step is out of control and examine which kind of special cause occurs the shift of process means for two-grade products produced from two dependent processes. Two EWMA control charts are proposed to detect if the first kind of special causes, which affect one of the two-grade products, occur; and a combined EWMA control chart is designed to examine if another kind of special causes, which affect both the two-grade products, occur in the previous process. Two EWMA-type cause-selecting control charts are proposed to examine if the first kind of special causes, which affect the interested quality characteristic for each grade products, occur in the previous process. A combined cause-selecting control chart is designed to detect if another kind of special causes, which affect both the interested quality characteristics for two-grade products, occur in the current process. A simulated data analysis shows our proposed control charts are superior to Shewhart control charts and Hotelling T2 control charts in distinguishing which process is out of control and detecting which kind of special causes occur for two-grade products.

A Comparison of Multivariate Control Charts for Individual Observations

For multivariate statistical process control with individual observations the usually recommended procedure in the retrospective phase is the Hotelling's T2 control chart. All the observations are pooled to estimate the mean vector and covariance matrix..

Economic Design of T2 Control Charts to Maintain Current Control of a Process

An approximate cost model for a quality control procedure for two or more related variables is investigated. A method is presented to determine the optimal sample size, interval between samples, and critical region parameter for the Hotelling T2 control chart. This model is a multivariate analog of several well-known models for the univariate X̄-chart. It is assumed that only one assignable cause of variation exists and the time between occurrences is exponentially distributed. Numerical results are provided in a particular bivariate case for several values of the cost coefficients. The behavior of the model to variation of the model parameters is discussed.

Minimum Cost Multivariate Quality Control Tests

Abstract An approximate expected cost model for the Hotelling T2 control chart is presented. This control chart is a multivariate analog of the X control chart. A method is presented to determine the optimal sample size, interval between samples, and critical region parameter for the T2 chart. Numerical results are provided for several bivariate problems for various values of the cost coefficients. The model sensitivity to the cost coefficients, the shift parameter, the sample correlation coefficient, and the sample covariance matrix is discussed.