# A Multi-Agent Optimization Approach for Multimodal Collaboration in Marine Terminals
## 2 Problem Description and Modeling Approach
### Modeling Approach  
To the best of the authors' knowledge, the defined problem setting has not been previously studied. An important difference with existing literature is that vessels may be delayed to reduce truck congestion at the landslide. Therefore, to select an appropriate solution methodology, inspiration was drawn from problems in the literature that share similar features. The MAS modeling paradigm is considered highly suitable for this problem because of the involvement of multiple autonomous yet interdependent stakeholders, each with distinct goals and incomplete information about others. Previous approaches to modeling container terminal operations using MAS are well-documented in the literature. For example, ref. [19] introduces a micro-level MAS framework for container terminals, while ref. [20] examines MAS-based coordination of barge and terminals to improve hinterland transport planning. Likewise, ref. [18] focuses on the coordination of internal trucks to manage cargo unloading operations. However, the previous models have a more limited scope, as agents only represent internal terminal functions without addressing stakeholder coordination and multimodal collaboration. A closely related problem from MAS that inspires our modeling approach is the Multi-Agent Pickup and Delivery Problem (MAPD), which shares several design similarities. Multi-agent pickup and delivery is the problem of allocating tasks for agents and finding the shortest paths without collisions. Both involve multiple interrelated agents, each with its own objectives and limited knowledge of others' plans, creating potential conflicts in decision-making. Additionally, both problems take into account dynamic scheduling and time constraints, which can disrupt planned operations. Finally, resource sharing and coordination are critical in both cases, as agents must efficiently allocate and manage shared resources to optimize overall system performance while balancing individual priorities. Solving MAPD problems often involves prioritization techniques and heuristics to enhance optimization. For instance, ref. [21] applies prioritized planning to develop efficient paths, combined with Large Neighborhood Search (LNS) for iterative solution improvement. Similarly, ref. [22] integrates task deadlines into a priority-based framework, using bounding and pruning techniques to maximize task assignments. Ref. [23] also leverages prioritization to ensure collision-free routes when assigning multiple goals to agents, striking a balance between managing conflicting objectives and task completion, with LNS aiding the search process. A key difference between the examined problem and MAPD problems is the less apparent presence of a unified optimization goal in the latter case. While MAPDstudies typically involve independent agents working toward system-wide optimization, the examined problem lacks a clear, single optimization goal due to the varying, sometimes conflicting, objectives of different stakeholders. This complexity necessitates balancing these competing interests rather than pursuing a single objective, often addressed in the literature through multi-objective representations. Multi-objective optimization is a common approach in scheduling operations with multiple, possibly conflicting goals, which is a key characteristic of the examined problem. Examples include job shop scheduling [24], timetabling [25], and machine scheduling [26]. In supply chain-related problems with multiple objectives, examples include truck scheduling in cross-docking centers, where the focus is on maximizing reliability against breakdowns while minimizing outbound truck tardiness [27], ship scheduling in congested marine terminals, balancing environmental, economic, and social objectives [28], and integrated terminal management, minimizing ship service times and yard crane operational costs [29]. The standard methodology for solving these problems are multi-objective evolutionary algorithms (MOEAs), like NSGA-II [30] and SPEA2 [31].