Synthetic and Runs-Rules Charts Combined With an &xmacr; Chart: Theoretical Discussion

A synthetic and a runs‐rules &xmacr; charts that are combined with a basic &xmacr; chart are called a Synthetic‐ &xmacr; and an improved runs‐rules &xmacr; charts, respectively. This paper gives the zero‐state and steady‐state theoretical results of the Synthetic‐ &xmacr; and improved runs‐rules &xmacr; monitoring schemes. The Synthetic‐ &xmacr; and improved runs‐rules schemes can each be classified into four different categories, that is, (i) non‐side‐sensitive, (ii) standard side‐sensitive, (iii) revised side‐sensitive, and (iv) modified side‐sensitive. In this paper, we first give the operation and, secondly, the general form of the transition probability matrices for each of the categories. Thirdly, in steady‐state, we show that for each of the categories, the three methods that are widely used in the literature to compute the initial probability vectors result in different probability expressions (or values). Fourthly, we derive the closed‐form expressions of the average run‐length (ARL) vectors for each of the categories, so that, by multiplying each of these ARL vectors with the zero‐state and steady‐state initial probability vectors, yield the zero‐state and steady‐state ARL expressions. Finally, we formulate the closed‐form expressions of the extra quadratic loss function for each of the categories.