A Markov chain approach for double-objective economic statistical design of the variable sampling interval X‾$\bm{\bar{X}}$ control chart

An X‾$\bar{X}$ chart with variable sampling interval (VSI) has been shown superior to the traditional X‾$\bar{X}$ chart with fixed ratio sampling (FRS). A constraint approach is not an efficient method for Economic Statistical Design (ESD) of VSI control charts because statistical properties are of the same importance as economic properties and should be optimized simultaneously. Then, a Multi-Objective Genetic Algorithm for ESD is proposed for identifying the Pareto optimal solutions of control chart design. The proposed method allows the practitioner to be provided with a set of optimal designs rather than a single solution, and they can select the locally optimal solution according to the process information. 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. Performance measures of these adaptive control charts are obtained through a Markov chain approach.