Optimising Shewhart charts in parallel production lines

I describe a methodology for optimising n Shewhart -charts x operating on parallel production lines in a factory. The goal is to maximise the factory-wide probability of detecting an out-of-control condition subject to a constraint on the expected number of false signals. I use non-linear programming to appropriately set the -charts’ x control limits incorporating information about the probability of each production line going out-of-control. Using this approach, factories can set their quality control systems to optimally detect out-of-control conditions. Given some distributional assumptions, I also present a one-dimensional optimisation methodology that allows for the efficient optimisation of very large factories.