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In this paper, the problem of efficient control of continuous-review perishable inventory systems is addressed from the control-theoretic perspective. In the considered systems, the goods at a distribution center used to fulfill unknown, variable demand are subject to exponential decay. The stock is replenished from multiple supply sources characterized by different latency in procuring orders. A new supply policy which incorporates the Smith predictor to counteract the adverse effects of delay is developed. The proposed policy guarantees that the assigned storage space at the distribution center is never exceeded. Moreover, we show that with appropriately chosen controller parameters all of the demand imposed at the distribution center is realized from the readily available resources. As a result, the maximum service level is ensured.