Critical-state algorithms for lot sizing with all-units discounts and variable inventory bounds

Abstract: We study deterministic single-item lot sizing with all-units quantity discounts, linear holding costs, and time-varying end-of-period inventory bounds. Unit prices may vary arbitrarily over time. For one common positive discount threshold, we give an exact algorithm requiring $O(n<sup>3)$ arithmetic operations and $O(n<sup>2)$ working space, including recovery of an optimal purchase plan. The method combines a critical-state theorem in cumulative-purchase coordinates with two prefix-minimum scans per dynamic-programming layer. The structural argument accommodates downward price discontinuities and distinguishes prescribed from free terminal inventory. With a fixed global set of $m$ positive threshold values, the bounds become $O_m(n<sup>{m+2})$ time and $O_m(n<sup>{m+1})$ space. Period-specific thresholds admit the same exact construction, but its state set can be exponential. For commensurate flow data, an inventory-lattice implementation takes $O(PW)$ time, where $P$ is the total number of price tiers and $W$ is the largest inventory-lattice width. We also derive a conditional optimality certificate from a monotone one-threshold relaxation; this certificate does not imply a near-linear worst-case algorithm for arbitrary prices. Exact-arithmetic implementations are checked against exhaustive inventory dynamic programs and tier-enumeration linear programs. An illustrative timing study identifies settings in which the lattice implementation is preferable.