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Platelets are short-life blood components used in hospital blood transfusion centers. Excluding time for transportation, testing, and arrangement, clinically transfusable platelets have a mere three-day life-span. This paper analyzes a periodic review inventory system for such perishable products under two replenishment modes. Regular orders are placed at the beginning of a cycle. Within the cycle, the manager has the option of placing an emergency order, characterized by an order-up-to level policy. We prove the existence and uniqueness of an optimal policy that minimizes the expected cost. We then derive the necessary and sufficient conditions for the policy, based on which a heuristic algorithm is developed. A numerical illustration and a sensitivity analysis are provided, along with managerial insights. The numerical results show that, unlike typical inventory problems, the total expected cost is sensitive to the regular order policy. It also shows that the optimal policy is sensitive to changes in the expected demand, suggesting to decision makers the significance of having an accurate demand forecast.