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We consider a periodic review inventory system with multiclass demands and fixed setup cost. Demand arrivals of each class are assumed to be a Poisson process, and a lost-sales setting is adopted. The demand classes are classified by the cost of their unsatisfied demands. We consider two cases: the leftover inventory at the end of a restocking interval is entirely discarded or entirely carried over to the next period. We obtain the optimal rationing policy, the optimal ordering policy and the optimal duration of the periodic review interval that minimize the average cost per unit time. We derive the differential equations satisfied by the value function characterized by the on-hand inventory level and the residual restocking time. This value function does not have the traditional modularity and convexity properties. Hence, the optimal policy is derived directly based on the ordinary differential equations satisfied by the value function. Moreover, some structural properties of the optimal policy such as the monotone property of the optimal rationing policy are obtained.

Periodic review inventory models with multiclass demands and fixed order costs