Process control for items produced in lots with inter and intra lot variations

We consider optimal process control for a production process where items are produced in finite lot sizes and are subject to intra-lot and inter-lot process variations. The process can randomly shift from the in-control state to the out-of-control state through mean-shift after producing a lot. The framework is to monitor the process at predetermined lot intervals by picking sample measurements from a lot and tracking them on a process control chart to detect the mean-shift. The objective is to obtain the control policy by minimizing the lots exposed to the mean-shift before detection, subject to an acceptable fraction of false alarms and limited measurements capacity. For this setting, we develop a model, present an explicit search algorithm, and illustrate the application of traditional policies, which are typically based on i.i.d assumption, and could lead to suboptimal results by as much as 17%. Next, we develop a simple interpolation correction to extend the traditional policies to include inter-lot and intra-lot variations, and illustrate that this procedure is close to the optimum.