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

We consider optimal process control for a production process where items ar e produced in finite lot sizes and are subject to intra-lot and inter-lot p rocess 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 picki ng sample measurements from a lot and tracking them on a process control ch art 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 measurement capacity . For this setting, we develop a model, present an explicit search algorith m, and illustrate that the application of traditional policies, which are t ypically based on i.i.d assumption, could lead to suboptimal results by as much as 17%. Next, we develop a simple interpolation correction to extend t he traditional policies to include interlot and intra-lot variations, and i llustrate that this procedure is close to the optimum. Significance: There are many processes that are subject to intra-lot and in ter-lot variations. The problem is further constrained by acceptable fracti on of false alarms and measurement capacity. The increase in the number of lots at risk of control of such processes, i.e., the lots that are exposed to the process shift but undetected due to beta error, could be as high as 17% if the traditional policies, which are based on i.i.d. assumption, are used in situations with different inter-lot and intra-lot variances. We pro vide a method to obtain the optimal solution in such cases. We then present an approximation procedure which simplified the computation of sampling pl an in terms of the ration of intra-lot to inter-lot variances and show that this approximation is close to the optimum. Moreover this can be used as a sample tool to extend the traditional sampling policies, to include inter- lot and intra-lot variations.