This paper considers a dynamic lot-sizing problem with storage capacity limitation in which backlogging is allowed. For general concave production and inventory costs, we present an O(T2) dynamic programming algorithm where T is the length of the planning horizon. Furthermore, for fixed-charge and nonspeculative costs, we provide O(Tlog T) and O(T) algorithms, respectively. This paper therefore concludes that the time complexity to solve the bounded inventory lot-sizing problem with backlogging is the same as the complexity to solve the uncapacitated lot-sizing problem for the commonly used cost structures