In this paper, we propose a new model of the Inventory Routing Problem (IRP) encountered in the context of reverse logistics. The vehicles here are required not only to deliver forward goods to retailers but also to collect simultaneously returned goods at their locations. The new model will be referred to as the Inventory Routing Problem with Simultaneous Delivery and Pickup (IRPSDP). The IRPSDP is based on the existing Periodic Vehicle Routing Problem (PVRP) model. The objective of the IRPSDP is to minimize the average inventory holding and travelling costs over a given m-day period. The IRPSDP treats the visit frequency as a decision variable instead of a given parameter, to seek the best trade-off between both costs. We present a mathematical formulation and give heuristic procedures to solve the problem. Finally, we test the performance of our approach by conducting some numerical experiments.