The variable sampling interval run sum X‾ control chart

Traditional control charts for process monitoring are based on taking samples from the process at fixed length sampling intervals. More recently, research works focused on the use of variable sampling intervals (VSIs), where the lengths of the sampling intervals are varied according to the process quality. A short sampling interval is considered when the process quality indicates a possible out-of-control situation while a long sampling interval is considered, otherwise. In this paper, the VSI run sum (RS) $\bar{X}$ chart is proposed with its optimal scores and parameters determined using an optimization technique to minimize the out-of-control average time to signal (ATS) or the adjusted average time to signal (AATS). A Markov-chain method is used to evaluate both the ATS and AATS of the proposed chart, for the zero and steady state cases, respectively. Results show that the VSI RS $\bar{X}$ chart is considerably more efficient than the basic RS $\bar{X}$ chart. The VSI RS $\bar{X}$ chart performs generally well compared with other competing charts, such as the standard $\bar{X}$, synthetic $\bar{X}$, exponentially weighted moving average (EWMA) $\bar{X}$, VSI $\bar{X}$ and VSI EWMA $\bar{X}$ charts. The sensitivity of the VSI RS $\bar{X}$ chart can be enhanced further by adding more scoring regions or a head-start feature. An illustrative example is presented to explain the implementation of the proposed VSI RS $\bar{X}$ chart.     

A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem

The Generalized Fermat Problem (in the plane) is: given $n\ge 3$ destination points find the point ${\stackrel{?}{x}}^1$ which minimizes the sum of Euclidean distances from ${\stackrel{?}{x}}^1$ to each of the destination points.The Weiszfeld iterative algorithm for this problem is globally convergent, independent of the initial guess. Also, a test is available, $\stackrel{?}{a}$ priori, to determine when ${\stackrel{?}{x}}^1$ a destination point. This paper generalizes earlier work by the first author by introducing an asymmetric Euclidean distance in which, at each destination, the $x$-component is weighted differently from the $y$-component. A Weiszfeld algorithm is studied to compute ${\stackrel{?}{x}}^1$ and is shown to be a descent method which is globally convergent (except possibly for a denumerable number of starting points). Local convergence properties are characterized. When ${\stackrel{?}{x}}^1$ is not a destination point the iteration matrix at ${\stackrel{?}{x}}^1$ is shown to be convergent and local convergence is always linear. When ${\stackrel{?}{x}}^1$ is a destination point, local convergence can be linear, sub-linear or super-linear, depending upon a computable criterion. A test, which does not require iteration, for ${\stackrel{?}{x}}^1$ to be a destination, is derived. Comparisons are made between the symmetric and asymmetric problems. Numerical examples are given.     

Optimal design of the double sampling X¯ chart with estimated parameters based on median run length

The double sampling (DS) $\overline{X}$ chart when the process parameters are unknown and have to be estimated from a reference Phase-I dataset is studied. An expression for the run length distribution of the DS $\overline{X}$ chart is derived, by conditioning and taking parameter estimation into account. Since the shape and the skewness of the run length distribution change with the magnitude of the mean shift, the number of Phase-I samples and sample sizes, it is shown that the traditional chart’s performance measure, i.e. the average run length, is confusing and not a good representation of a typical chart’s performance. To this end, because the run length distribution is highly right-skewed, especially when the shift is small, it is argued that the median run length (MRL) provides a more intuitive and credible interpretation. From this point of view, a new optimal design procedure for the DS $\overline{X}$ chart with known and estimated parameters is developed to compute the chart’s optimal parameters for minimizing the out-of-control MRL, given that the values of the in-control MRL and average sample size are fixed. The optimal chart which provides the quickest out-of-control detection speed for a specified shift of interest is designed according to the number of Phase-I samples commonly used in practice. Tables are provided for the optimal chart parameters along with some empirical guidelines for practitioners to construct the optimal DS $\overline{X}$ charts with estimated parameters. The optimal charts with estimated parameters are illustrated with a real application from a manufacturing company.