Parallelism of continuous- and discrete-time production planning problems

We consider a capacitated production-inventory problem in both discrete- and continuous-time, stationary settings. In the discrete-time setting we analyze the infinite horizon capacitated dynamic lot-sizing problem, find the optimal solution and characterize its properties. For the continuous-time setting we formulate a new problem, which we claim to be an appropriate counterpart of the above discrete-time problem. No other Counterpart model was found in the literature, including the vast literature on optimal control. which presumably deals with similar problems but in a continuous-time framework. The new problem formulation is the basis for a new class of models, which forms an alternative way of analyzing certain dynamic lot-sizing problems. This new alternative could sometimes be simpler than the analysis in the discrete case.