Motivated by distribution problems arising in the supply chain of Haleon, we investigate a discrete optimization problem that we call the "container delivery scheduling problem". The problem models a supplier dispatching ordered products with shipping containers from manufacturing sites to distribution centers, where orders are collected by the buyers at agreed due times. The supplier may expedite or delay item deliveries to reduce transshipment costs at the price of increasing inventory costs, as measured by the number of containers and distribution center storage/backlog costs, respectively. The goal is to compute a delivery schedule attaining good trade-offs between the two. This container delivery scheduling problem is a temporal variant of classic bin packing problems, where the item sizes are not fixed, but depend on the item due times and delivery times. An approach for solving the problem should specify a batching policy for container consolidation and a scheduling policy for deciding when each container should be delivered. Based on the available item due times, we develop algorithms with sequential and nested batching policies as well as on-time and delay-tolerant scheduling policies. We elaborate on the problem's hardness and substantiate the proposed algorithms with positive and negative approximation bounds, including the derivation of an algorithm achieving an asymptotically tight 2-approximation ratio.