Exact computational methods for univariate and multivariate control charts under runs rules

This article proposes efficient computational methods for designing and evaluating phases of Shewhart type control charts under runs rules. The efficient computational methods include exact equations or formulas for computing the probability of single-point and run-length properties of control charts. The run-length properties include average, variance, standard deviation, coefficient of variation, and moments. The study implements the proposed computational methods in the design and evaluation phases of well known univariate and multivariate control charts. In this regard, the research considers mean, variance, standard deviation, multivariate Hotelling and generalized variance control charts. Also, various procedures and a code in R language are provided to exemplify the applications of proposed computational methods in control charts. A comprehensive analysis of the behaviour of run-length properties of control charts under runs rules is conducted by considering various choices of factors such as sample size, amount of shift, and choice of runs rules. The proposed efficient computational methods provide the desired results with additional features such as reducing the computational burden, time efficient and being practitioner friendly. Finally, the article presents some real-life examples of univariate and multivariate control charts from manufacturing and winds turbine processes.