Optimization of sample sizes, sampling intervals, and control limits of the X¯&S$$ \overline{X}\&S $$ chart system monitoring independent quality characteristics

This article designs an X¯&S$$ \overline{X}\&S $$ control chart system for monitoring process shifts in mean and standard deviation in a multistage manufacturing system. Each of the X¯&S$$ \overline{X}\&S $$ chart combination in a chart system monitors one of the critical quality characteristics (dimensions) of a product. The design algorithm optimally allocates the detection power of the chart system among different stages as well as between the X¯$$ \overline{X} $$ chart and S chart within each stage based on the values of certain parameters (e.g., process capability, magnitude of the process shift) that would affect the performance of the chart system. Meanwhile, the sample sizes, sampling intervals, and control limits of the X¯&S$$ \overline{X}\&S $$ charts are also optimized. The optimization design is carried out using false alarm rate and inspection capacity as constraints. Consequently, the performance of the system as a whole is improved without requiring additional cost and effort for inspection. The results of the comparative studies show that from an overall viewpoint, the optimal X¯&S$$ \overline{X}\&S $$ chart system is more effective (in terms of reduction in detection time) than the traditional 3-sigma X¯&S$$ \overline{X}\&S $$ chart system as well as a suboptimal X¯&S$$ \overline{X}\&S $$ chart system by about 53 and 26 %, respectively. Some useful guidelines have been brought forth to aid the users to adjust the sample sizes, sampling intervals, and control limits of the charts in a system.