During loading and unloading steps, energy is consumed when cranes lift containers, while energy is often wasted when cranes drop containers. By optimizing the scheduling of cranes, it is possible to reduce energy consumption, thereby lowering operational costs and environmental impacts. In this paper, we introduce a single-crane scheduling problem with energy savings, focusing on reusing the energy from containers that have already been lifted and reducing the total energy consumption of the entire scheduling plan. We establish a basic model considering a one-dimensional storage area and provide a systematic complexity analysis of the problem. First, we investigate the connection between our problem and the semi-Eulerization problem and propose an additive approximation algorithm. Then, we present a polynomial-time Dynamic Programming (DP) algorithm for the case of bounded energy buffer and processing lengths. Next, adopting a Hamiltonian perspective, we address the general case with arbitrary energy buffer and processing lengths. We propose an exact DP algorithm and show that the variation of the problem is polynomially solvable when it can be transformed into a path cover problem on acyclic interval digraphs. We introduce a paradigm that integrates both the Eulerian and Hamiltonian perspectives, providing a robust framework for addressing the problem.

翻译：在集装箱装卸作业过程中，起重机提升集装箱时消耗能量，而下降集装箱时能量往往被浪费。通过优化起重机调度方案，可以降低能耗，从而减少运营成本与环境影响。本文提出一种考虑节能的单起重机调度问题，重点关注已提升集装箱能量的再利用，以降低整体调度方案的总能耗。我们建立了考虑一维存储区域的基础模型，并对该问题进行了系统性的复杂度分析。首先，我们探究了该问题与半欧拉化问题之间的关联，并提出了一种加法近似算法。随后，针对能量缓冲区与处理长度受限的情形，我们提出了一种多项式时间动态规划算法。接着，采用哈密顿视角，我们研究了能量缓冲区与处理长度任意的普遍情况。我们提出了一种精确动态规划算法，并证明当该问题可转化为无环区间有向图的路径覆盖问题时，其变体具有多项式时间解法。最后，我们提出了一种融合欧拉与哈密顿双重视角的范式，为该问题提供了稳健的求解框架。