Improve critical-state bounds for the fixed-threshold method

Determine whether tighter critical-state sets can improve the practical range of the fixed-global-threshold method and whether a smaller worst-case complexity bound is possible for the one-threshold arbitrary-price lot-sizing problem.

Background

The paper establishes an exact critical-state dynamic program for one common all-units discount threshold with arbitrary time-varying prices and variable end-of-period inventory bounds. The resulting algorithm has cubic arithmetic complexity and uses a quadratic-size critical-state grid. The authors leave unresolved whether the state set can be tightened in a way that materially expands the method’s practical applicability, and whether the worst-case bound for the one-threshold arbitrary-price case can be reduced.