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.

翻译：本文研究单品种经济批量问题，其生产成本函数为分段线性形式。该批量问题作为批量问题领域的基石，根据文献\cite{ou}可适用于多种重要的生产规划问题（其特殊情形）。当生产成本的断点数量$m$可变时，该问题因\cite{Florian1980}证明为NP-hard而难以处理。对于固定$m$，文献\cite{Koca2014}给出了$O(T^{2m+3})$时间复杂度的算法，随后\cite{ou}将其改进为$O(T^{m+2}\log(T))$，其中$T$为规划周期的期数。\newline 我们提出一种更高效的$O(T^{m+2})$时间复杂度算法，该算法优于Ou提出的最新算法，并通过若干可能具有独立价值的新颖算法技术推导得出。