Article: Joint scheduling of laden and empty containers with turnover and port-level availability constraints Journal: European Journal of Industrial Engineering (EJIE) 2026 Vol.22 No.5 pp.1 - 22 Abstract: Maritime logistics plays a vital role in global trade, handling over 80% of the world's cargo by volume. However, significant imbalances in trade flows across regions often result in costly inefficiencies due to the accumulation or shortage of empty containers at ports. This paper considers an integrated container scheduling problem in maritime logistics, which jointly manages the delivery of laden containers and the repositioning of empty containers across a port network. The problem incorporates realistic operational constraints, including vessel capacities, port-to-port travel times, container turnover delays, and limited initial container availability at each port. In particular, the explicit consideration of port-level container availability and turnover delays provides a more realistic representation of container circulation and reuse throughout the transportation network. We first establish that the problem is NP-hard in the strong sense and formulate it as a mixed-integer programming (MIP) model. To efficiently solve large-scale instances, we develop a Lagrangian relaxation-based heuristic that decomposes the original problem into tractable subproblems and iteratively generates lower bounds and feasible solutions. Extensive computational experiments demonstrate that the proposed heuristic consistently produces high-quality solutions across different network sizes and operating conditions. The results further show that the method remains effective under variations in initial container availability, cargo demand, and inter-port travel times, demonstrating its robustness and practical applicability to maritime container scheduling. [Submitted: 11 March 2026; Accepted: 11 September 2026] Inderscience Publishers - linking academia, business and industry through research

Title: Joint scheduling of laden and empty containers with turnover and port-level availability constraints

Authors: Eun-Seok Kim; Sangmin Lee; Ik Sun Lee

Addresses: School of Business and Management, Queen Mary University of London, London E1 4NS, UK ' School of Information Convergence, Kwangwoon University, Seoul, 01897, South Korea ' School of Business, Dong-A University, Seo-gu, Busan 602-760, South Korea

Abstract: Maritime logistics plays a vital role in global trade, handling over 80% of the world's cargo by volume. However, significant imbalances in trade flows across regions often result in costly inefficiencies due to the accumulation or shortage of empty containers at ports. This paper considers an integrated container scheduling problem in maritime logistics, which jointly manages the delivery of laden containers and the repositioning of empty containers across a port network. The problem incorporates realistic operational constraints, including vessel capacities, port-to-port travel times, container turnover delays, and limited initial container availability at each port. In particular, the explicit consideration of port-level container availability and turnover delays provides a more realistic representation of container circulation and reuse throughout the transportation network. We first establish that the problem is NP-hard in the strong sense and formulate it as a mixed-integer programming (MIP) model. To efficiently solve large-scale instances, we develop a Lagrangian relaxation-based heuristic that decomposes the original problem into tractable subproblems and iteratively generates lower bounds and feasible solutions. Extensive computational experiments demonstrate that the proposed heuristic consistently produces high-quality solutions across different network sizes and operating conditions. The results further show that the method remains effective under variations in initial container availability, cargo demand, and inter-port travel times, demonstrating its robustness and practical applicability to maritime container scheduling. [Submitted: 11 March 2026; Accepted: 11 September 2026]

Keywords: maritime logistics; container scheduling; empty container repositioning; ECR; mixed integer programming; MIP; Lagrangian heuristic.

DOI: 10.1504/EJIE.2026.157628

European Journal of Industrial Engineering, 2026 Vol.22 No.5, pp.1 - 22

Received: 11 Mar 2026
Accepted: 11 Sep 2026
Published online: 12 Nov 2026 *

Title: Joint scheduling of laden and empty containers with turnover and port-level availability constraints

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