Determining control points of a production system under a chance constraint

A production system operates at a speed which is a random variable, with a known distribution function. Given the routine control point, the actual accumulated production observed at that point, and the rate of demand, the decision-maker determines the next control point. Consequently the interval between two control points will be maximum or minimum under a probabilistic constraint that insures that at any point the actual production will not fall below the planned production at a given confidence level 1-α. The problem is applied to semiautomated production processes where the advancement of the process cannot be measured or viewed continuously, but the process has to be controlled in discrete points by the decision-maker. A formula for determining the next control point for a general case distribution function was developed. Furthermore, examples for the normal, uniform and beta distributions were examined. In general, the solution differs for shortage and surplus and whether the demand rate is smaller or greater than the α-th quantile of the distribution function of the speed.