Inventory control with modulated demand and a partially observed modulation process

We consider a periodic review inventory control problem having an underlying modulation process that affects demand and that is partially observed by the uncensored demand process and a novel additional observation data (AOD) process. We present an attainability condition, AC, that guarantees the existence of an optimal myopic base stock policy if the reorder cost $$K=0$$ K = 0 and the existence of an optimal (s, S) policy if $$K>0$$ K > 0 , where both policies depend on the belief function of the modulation process. Assuming AC holds, we show that (i) when $$K=0$$ K = 0 , the value of the optimal base stock level is constant within regions of the belief space and that each region can be described by two linear inequalities and (ii) when $$K>0$$ K > 0 , the values of s and S and upper and lower bounds on these values are constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities. A heuristic and bounds for the $$K=0$$ K = 0 case are presented when AC does not hold. Special cases of this inventory control problem include problems considered in the Markov-modulated demand and Bayesian updating literatures.