On the performance of Shewhart-type synthetic and runs-rules charts combined with an X chart

Abstract

A synthetic X chart is a combination of a conforming run-length chart and an X chart, or equivalently, a 2-of-(H + 1) runs-rules (RR) chart with a head-start feature. However, a synthetic X chart combined with an X chart is called a Synthetic- X chart. In this article, we build a framework for Shewhart Synthetic- X and improved RR (i.e., 1-of-1 or 2-of-(H + 1) without head-start) charts by conducting an in-depth zero-state and steady-state study to gain insight into the design of different classes of these schemes and their average run-length performance using the Markov chain imbedding technique. More importantly, we propose a modified side-sensitive Synthetic- X chart, and then using overall performance measures (i.e., the extra quadratic loss, average ratio of average run-length, and performance comparison index), we show that this new chart has a uniformly better performance than its Shewhart competitors. We also provide easy-to-use tables for each of the chart's design parameters to aid practical implementation. Moreover, a performance comparison with their corresponding counterparts (i.e., synthetic X and RR charts) is conducted.