On dynamic lot sizing with bounded inventory for a perishable product

It is well-known in the single-item dynamic lot-sizing (DLS) literature that the DLS problem for a perish-able product (DLS-P) can be solved in polynomial time (see, e.g., Hsu [1]), and that the DLS problem with bounded inventory (DLS-BI) is also polynomially solvable (see, e.g., Love [2]). However, the computational complexity of the DLS problem with bounded inventory for a perishable product (DLS-BI-P) remains to be open. In this note, we answer this open problem and show that DLS-BI-P is NP-hard. Recently, Jing and Mu [20] presented an exact algorithm for an important extension of DLS-BI-P and claimed that the pro-posed algorithm is polynomial, while Jing and Chao [19] also developed an exact algorithm for another important extension of DLS-BI-P. In this note, we also point out that both of these two exact algorithms are exponential, and that the algorithm by Jing and Chao [19] may not generate an optimal solution in general.& COPY; 2023 Elsevier Ltd. All rights reserved.