Lot sizing problem with multi-mode replenishment and batch delivery

•Single-item lot sizing with multi-mode replenishment & batch deliveries is studied.•Replenishment cost is stepwise (called also full-truck-load or multiple setup cost).•We show the NP-hardness, give a 2-approximation algorithm, show it admits an FPTAS.•Existing results are improved after the transformation to the single-mode problem.•Polynomial time algorithm for the case with 2 modes & divisible batches is proposed.