Crane Scheduling Problem with Energy Saving

TL;DR

This work addresses energy-efficient crane scheduling in a 1D storage area by reusing energy from lowering operations within an energy buffer $e$. It develops a two-pronged approach: (i) a graph-theoretic, Eulerian-driven formulation with zero and positive buffers, including additive-approximation methods and dynamic programming under bounded parameters; and (ii) a Hamiltonian perspective that reduces the problem to a path cover on interval digraphs, with exact DP solutions and a tractable acyclic special case via maximum matching. Key contributions include a unified framework linking semi-Eulerization and path covers, several polynomial-time algorithms under bounded parameters, and a principled extension to arbitrary energy buffers with both theoretical and practical implications for green crane scheduling. The results offer rigorous foundations for energy-aware scheduling in automated container terminals and pave the way for more realistic energy models and real-world validation.

Abstract

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.

Figures (7)

• Figure 1: Layout at Container Terminal
• Figure 2: Example of job information
• Figure 3: Illustration of the two-level graph with auxiliary edges and penalty edges corresponding to the optimal solution for the example in Figure \ref{['job_info_fig']} with energy buffer $e = 1$
• Figure 4: Illustration of the two-level graph with auxiliary edges and penalty edges corresponding to the optimal solution for the example in Figure \ref{['job_info_fig']} with energy buffer $e = 2$
• Figure 5: Illustration of two possible transitions for the example in Figure \ref{['job_info_fig']}.

Theorems & Definitions (13)

• Lemma 1
• Lemma 2
• Definition 3
• Definition 4
• Lemma 5
• Theorem 6
• Theorem 7
• Definition 8
• Definition 9
• Definition 10