On the Performance of Shewhart-type Synthetic and Runs-rules Charts Combined with an &Xmacr; Chart

A synthetic &Xmacr; chart is a combination of a conforming run‐length chart and an &Xmacr; chart, or equivalently, a 2‐of‐(H + 1) runs‐rules (RR) chart with a head‐start feature. However, a synthetic &Xmacr; chart combined with an &Xmacr; chart is called a Synthetic‐ &Xmacr; chart. In this article, we build a framework for Shewhart Synthetic‐ &Xmacr; 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‐ &Xmacr; 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 &Xmacr; and RR charts) is conducted.