This paper studies the optimal policy for a periodic-review inventory system in which the production costs consist of a fixed cost and a piecewise linear convex variable cost. Such a cost function can arise from alternate sources of supply or from the use of overtime production. We fully characterize the structure of the optimal policy for the single-period problem. For the multi-period problem, the optimal policy can have disconnected production regions and complicated optimal produce-up-to levels, which implies that implementation of the optimal policy may not be practical. Fortunately, careful investigation shows that the optimal policy has some interesting properties. The structure of the optimal policy outlined by these properties leads to a practical and close-to-optimal heuristic policy. In an extensive numerical study, the average gap is only 0.02% and the worst gap is 1.37%.