Economic and economic statistical design of T 2 control chart with two adaptive sample sizes

The Hotelling's T ² control chart, a direct analogue of the univariate Shewhart X¯ chart, is perhaps the most commonly used tool in industry for simultaneous monitoring of several quality characteristics. Recent studies have shown that using variable sampling size (VSS) schemes results in charts with more statistical power when detecting small to moderate shifts in the process mean vector. In this paper, we build a cost model of a VSS T ² control chart for the economic and economic statistical design using the general model of Lorenzen and Vance [The economic design of control charts: A unified approach, Technometrics 28 (1986), pp. 3–11]. We optimize this model using a genetic algorithm approach. We also study the effects of the costs and operating parameters on the VSS T ² parameters, and show, through an example, the advantage of economic design over statistical design for VSS T ² charts, and measure the economic advantage of VSS sampling versus fixed sample size sampling.     

The robust economic statistical design of the Hotelling's T2 chart

ABSTRACT

Economic statistical designs aim at minimizing the cost of process monitoring when a specific scenario or a set of estimated process and cost parameters is given. But, in practice the process may be affected by more than one scenario which may lead to severe cost penalties if the wrong design is used. Here, we investigate the robust economic statistical design (RESD) of the T² chart in an attempt to reduce these cost penalties when there are multiple scenarios. Our method is to employ the genetic algorithm (GA) optimization method to minimize the total expected monitoring cost across all distinct scenarios. We illustrate the effectiveness of the method using two numerical examples. Simulation studies indicate that robust economic statistical designs should be encouraged in practice.     

Evaluation of the economic statistical design of the multivariate T2 control chart with multiple variable sampling intervals scheme: NSGA-II approach

The economic and statistical merits of a multiple variable sampling intervals scheme are studied. The problem is formulated as a double-objective optimization problem with the adjusted average time to signal as the statistical objective and the expected cost per hour as the economic objective. Bai and Lee's [An economic design of variable sampling interval ¯X control charts. Int J Prod Econ. 1998;54:57–64] economic model is considered. Then we find the Pareto-optimal designs in which the two objectives are minimized simultaneously by using the non-dominated sorting genetic algorithm. Through an illustrative example, the advantages of the proposed approach are shown by providing a list of viable optimal solutions and graphical representations, which indicate the advantage of flexibility and adaptability of our approach.     

A statistically adaptive sampling policy to the Hotelling's T2 control chart: Markov chain approach

ABSTRACT

We present an alternative sampling scheme for the Hotelling's T² control chart with variable parameters (VP T²) which allows the sampling interval h, the sample size n, and control limit k to vary between minimum and maximum values while keeping the warning line fixed over time. Our method uses only one measurement scale to overcome the difficulties of using two scales in practice. Later, we demonstrate the merits of the method in terms of its performance in detecting small-to-moderate shifts and its ease of application.     

A new Markov chain approach for the economic statistical design of the VSS T2 control chart

An economic statistical design model for a T² chart which uses a variable sample size (VSS) feature is developed in this article. This study mainly differs from the others conducted in the field. In that a new approach is offered to achieve closed form of some statistical criteria. In other words, the proposed formulas can be considered as a better alternative approach in designing the VSS control charts in terms of simplicity and yet providing the users with better optimal solutions.     

Economic and economic statistical designs for three-level control charts

In some situations, an appropriate quality measure uses three or more discrete levels to classify a product characteristic. For these situations, some control charts have been developed based on statistical criteria regardless of economic considerations. In this paper, we develop economic and economic statistical designs (ESD) for 3-level control charts. We apply the cost model proposed by Costa and Rahim.[Economic design of X charts with variable parameters: the Markov chain approach, J Appl Stat 28 (2001), 875–885] Furthermore, we assume that the length of time that the process remains in control is exponentially distributed which allows us to apply the Markov chain approach for developing the cost model. We apply a genetic algorithm to determine the optimal values of model parameters by minimizing the cost function. A numerical example is provided to illustrate the performance of the proposed models and to compare the cost of the pure economic and ESD for three-level control charts. A sensitivity analysis is also conducted in this numerical example.     

A comparison study of effectiveness and robustness of robust economic designs of T2 chart using genetic algorithm

ABSTRACT

Recently, researchers have tried to design the T² chart economically to achieve the minimum possible quality cost; however, when T² chart is designed, it is important to consider multiple scenarios. This research presents the robust economic designs of the T² chart where there is more than one scenario. An illustrative example is used to demonstrate the effect of the model parameters on the optimal designs. The genetic algorithm optimization method is employed to obtain the optimal designs. Simulation studies show that the robust economic designs of T² chart are more effective than traditional economic design in practice.     

Economic and economic-statistical designs of the T2 control charts with SVSSI sampling scheme

Recent studies have shown that the adaptive T² chart with two different sampling interval and three sample sizes (SVSSI) shows a good performance in detecting small to large shifts in the process mean. This paper investigates the economic and economic statistical designs of the SVSSI T² charts. We use the Markov chain approach to developing the cost model proposed by Costa and Rahim (Journal of applied statistics 2001; 28: 875–885). A genetic algorithm approach is used to find the optimal solutions. Using numerical examples, we illustrate the performance of the proposed model and compare the statistical, economic, and economic statistical designs of the SVSSI T² chart with respect to the economic and statistical criteria. Furthermore, we compare the performance of the SVSSI T² chart with the other T² control schemes.     

A modified economic-statistical design of the T2 control chart with variable sample sizes and control limits

Recent studies have shown that using variable sampling size and control limits (VSSC) schemes result in charts with more statistical power than variable sampling size (VSS) when detecting small to moderate shifts in the process mean vector. This paper presents an economic-statistical design (ESD) of the VSSC T² control chart using the general model of Lorenzen and Vance [22]. The genetic algorithm approach is then employed to search for the optimal values of the six test parameters of the chart. We then compare the expected cost per unit of time of the optimally designed VSSC chart with optimally designed VSS and FRS (fixed ratio sampling) T² charts as well as MEWMA charts.     

Bivariate copulas on the Hotelling's T2 control chart

In this paper, we propose five types of copulas on the Hotelling's T² control chart when observations are from exponential distribution and use the Monte Carlo simulation to compare the performance of the control chart, which is based on the Average Run Length (ARL) for each copula. Five types of copulas function for specifying dependence between random variables are used and measured by Kendall's tau. The results show that the copula approach can be fitted the observation and we can use copula as an option for application on Hotelling's T² control chart.     

Economic statistical design of the T2 control chart under the Weibull shock model with multiple assignable causes

A proper monitoring of stochastic systems is the control charts of statistical process control and drift in characteristics of output may be due to one or several assignable causes. Although much research has been done on the design of control charts, the economic statistical design of the T2 control chart under the Weibull shock model with multiple assignable causes has not yet been addressed. Therefore, we tried to deal with it in this paper and thus we developed a cost model based on the variable sampling interval. We also give an example to support the practical use of T2 chart under the Weibull shock model with multiple assignable causes. Based on the optimization of the average cost per unit of time and taking into account the different combination values of Weibull distribution parameters, optimal values of design parameters were derived and calculated. Then, the cost models under the influence of single assignable cause and multiple assignable causes under the same cost and time parameters were compared. Also, a sensitivity analysis was conducted in which the variability of loss cost and design parameters due to change of cost and time and Weibull distribution parameters were evaluated.     

Economic-statistical design of synthetic double sampling T2 chart

Abstract

An economic-statistical design of the synthetic double sampling (synDS) T² chart is presented in this study. The cost function is minimized to obtain the optimal design parameters of the synDS T² chart by incorporating the statistical constraints or the constraints on the average number of samples. An example is provided and a sensitivity analysis is conducted to study the effect of model parameters on the optimal solution of the design. The numerical comparison shows that the synDS T² chart performs better than the synthetic T² chart and the multivariate exponentially weighted moving average chart, in terms of the cost.     

New robust estimators for detecting non-random patterns in multivariate control charts: a simulation approach

In the past decade, different robust estimators have been proposed by several researchers to improve the ability to detect non-random patterns such as trend, process mean shift, and outliers in multivariate control charts. However, the use of the sample mean vector and the mean square successive difference matrix in the T ² control chart is sensitive in detecting process mean shift or trend but less sensitive in detecting outliers. On the other hand, the minimum volume ellipsoid (MVE) estimators in the T ² control chart are sensitive in detecting multiple outliers but less sensitive in detecting trend or process mean shift. Therefore, new robust estimators using both merits of the mean square successive difference matrix and the MVE estimators are developed to modify Hotelling's T ² control chart. To compare the detection performance among various control charts, a simulation approach for establishing control limits and calculating signal probabilities is provided as well. Our simulation results show that a multivariate control chart using the new robust estimators can achieve a well-balanced sensitivity in detecting the above-mentioned non-random patterns. Finally, three numerical examples further demonstrate the usefulness of our new robust estimators.     

Three-level control charts with variable sample size,sampling interval,and control limits

Recently, three-level control chart has been developed for monitoring situations in which an appropriate quality measure uses three discrete levels to classify a product characteristic. In this paper, the variable parameters control charts for multinomial data are developed with a three-level classification scheme. We compare it with various adaptive charts to show that this modified scheme can improve the performance. In order to evaluate and compare the performance of this scheme, adjusted average time to signal is used as the performance measure. Results indicate that the proposed chart has improved performance and is relatively sensitive to small shifts.     

The economic design of multivariate binomial EWMA VSSI control charts

Since multi-attribute control charts have received little attention compared with multivariate variable control charts, this research is concerned with developing a new methodology to employ the multivariate exponentially weighted moving average (MEWMA) charts for m-attribute binomial processes; the attributes being the number of nonconforming items. Moreover, since the variable sample size and sampling interval (VSSI) MEWMA charts detect small process mean shifts faster than the traditional MEWMA, an economic design of the VSSI MEWMA chart is proposed to obtain the optimum design parameters of the chart. The sample size, the sampling interval, and the warning/action limit coefficients are obtained using a genetic algorithm such that the expected total cost per hour is minimized. At the end, a sensitivity analysis has been carried out to investigate the effects of the cost and the model parameters on the solution of the economic design of the VSSI MEWMA chart.     

Integration of classification algorithms and control chart techniques for monitoring multivariate processes

We propose new multivariate control charts that can effectively deal with massive amounts of complex data through their integration with classification algorithms. We call the proposed control chart the ‘Probability of Class (PoC) chart’ because the values of PoC, obtained from classification algorithms, are used as monitoring statistics. The control limits of PoC charts are established and adjusted by the bootstrap method. Experimental results with simulated and real data showed that PoC charts outperform Hotelling's T ² control charts. Further, a simulation study revealed that a small proportion of out-of-control observations are sufficient for PoC charts to achieve the desired performance.     

Identifying the time of a step change with multivariate single control charts

Change point estimation procedures simplify the efforts to search for and identify special causes in multivariate statistical process monitoring. After a signal is generated by the simultaneously used control charts or a single control chart, add-on change point procedure estimates the time of the change. In this study, multivariate joint change point estimation performance for simultaneous monitoring of both location and dispersion is compared under the assumption that various single charts are used to monitor the process. The change detection performance for several structural changes for the mean vector and covariance matrix is also discussed. It is concluded that choice of the control chart to obtain a signal may affect the change point detection performance.     

An extension of Banerjee and Rahim model in economic and economic statistical designs for multivariate quality characteristics under Burr XII distribution

The design parameters of the economic and economic statistical designs of control charts depend on the distribution of process failure mechanism or shock model. So far, only a small number of failure distributions, such as exponential, gamma, and Weibull with fixed or increasing hazard rates, have been used as a shock model in the economic and economic statistical designs of the Hotelling T² control charts. Due to both theoretical and practical aspects, the lifetime of the process under study may not follow a distribution with fixed or increasing hazard rate. A proper alternative for this situation may be the Burr distribution, in which the hazard rate can be fixed, increasing, decreasing, single mode, or even U-shaped. In this research article, economic and economic statistical designs of the Hotelling T² control charts under the Burr XII shock models under two uniform and non uniform sampling schemes were proposed, constructed, and compared. The obtained design models were implemented by a numerical example, and a sensitivity analysis was conducted to evaluate the effect of changing parameters of shock model distribution on the optimum values of the proposed design models. The results showed that first the proposed designs under non uniform sampling scheme perform better and second the optimum values of the designs are not significantly sensitive to changing of the Burr XII distribution parameters. We showed that the obtained design models are also true for the beta Burr XII shock model.     

C434. Local bounds on power of T2 in scale families

In this paper, we have developed a likelihood ratio factorization technique for variable selection for multiple observations model. The asymptotic distribution of the selection criterion is also given. This technique has been applied to a marketing data set for illustration of the technique developed.