A Novel exact algorithm for economic lot-sizing with piecewise linear production costs
Kleitos Papadopoulos
TL;DR
This work tackles the single-item economic lot-sizing problem with piecewise-linear production costs (ELS-PL). It develops an exact dynamic-programming-based algorithm that achieves $O(T^{m+2})$ time for fixed $m$, by extending Ou's ELS-PC framework with new state representations ($\Pi$, $\nu$, $\omega$) and a novel monotonic rebordering technique to efficiently manage candidate production patterns. The key contribution is removing the RMQ-based acceleration used in prior work and introducing elementary data structures and a sorting lemma to support fast state transitions, yielding a theoretical speedup from $O(T^{m+2}\log T)$ to $O(T^{m+2})$. This advancement broadens the practicality of exact solutions for ELS-PL and is likely transferable to related ELS variants, potentially speeding production planning in long-horizon contexts.
Abstract
In this paper, we study the single-item economic lot-sizing problem with production cost functions that are piecewise linear. The lot-sizing problem stands as a foundational cornerstone within the domain of lot-sizing problems. It is also applicable to a variety of important production planning problems which are special cases to it according to \cite{ou}. The problem becomes intractable when $m$, the number of different breakpoints of the production-cost function is variable as the problem was proven NP-hard by \cite{Florian1980}. For a fixed $m$ an $O(T^{2m+3})$ time algorithm was given by \cite{Koca2014} which was subsequently improved to $O(T^{m+2}\log(T))$ time by \cite{ou} where $T$ is the number of periods in the planning horizon.\newline We introduce a more efficient $O(T^{m+2})$ time algorithm for this problem which improves upon the previous state-of-the-art algorithm by Ou and which is derived using several novel algorithmic techniques that may be of independent interest.