Optimal design of a synthetic chart for monitoring process dispersion with unknown in-control variance

For a stable manufacturing process, quality problems are often caused by changes in process dispersion. Although there have been plenty of research on the monitoring of process dispersion, the existing studies of synthetic charts for monitoring process dispersion only focus on the upward shift monitoring. However, the decrease shift monitoring is also necessary and important. In this paper, a synthetic $S^2$ chart is proposed to simultaneously monitor both upward and downward shifts and it consists of a Shewhart‐type two‐sided $S^2$ sub‐chart and a conforming run length sub‐chart. In the known in‐control variance case, the conforming run length sub‐chart only needs a lower control limit and the proposed synthetic $S^2$ chart is shown to be average run length (ARL) unbiased. The effect of parameter estimation on the proposed synthetic $S^2$ chart is also investigated as it is an important issue especially in the real manufacturing processes. Considering that the in‐control variance is usually unknown and needs to be estimated by Phase I samples in practice, a new synthetic $S^2$ chart in which conforming run length sub‐chart also only needs a lower control limit is developed when the in‐control variance is estimated. Furthermore, optimal designs for both known and unknown parameter cases are studied. The advantage of the proposed chart in performance is shown in the results of the comparison with the ARL‐unbiased $S^2$ chart. Also, an example illustrates the construction and application procedure of this proposed chart.