On the Performance of Shewhart-type Synthetic and Runs-rules Charts Combined with an (X)over-bar Chart

A synthetic (X) over bar chart is a combination of a conforming run-length chart and an (X) over bar chart, or equivalently, a 2-of-(H+1) runs-rules (RR) chart with a head-start feature. However, a synthetic (X) over bar chart combined with an (X) over bar chart is called a Synthetic-(X) over bar chart. In this article, we build a framework for Shewhart Synthetic-(X) over bar 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) over bar 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) over bar and RR charts) is conducted. Copyright (C) 2015 John Wiley & Sons, Ltd.