This paper deals with an optimization problem in which a set of inspection operations have to be processed so that a fixed production schedule is as reliable as possible. The problem is formalized as a single machine scheduling problem whose objective is to minimize a regular step total cost function. Mixed Integer Linear Programming models and Lagragean relaxation are described for the case without release dates. Experimental results are reported and discussed to evaluate the performance of a standard mathematical programming solver applied to different special cases and to the general case.