# A Multi-Agent Optimization Approach for Multimodal Collaboration in Marine Terminals
## 3 Model Formulation  
The problem statement involves three distinct actor groups, each with specific requirements and objectives, which are modeled as decision-making agents within an MAS. The first group, consisting of vessels, is represented by the **berth allocation agent**, which seeks to minimize both waiting times and processing times for the vessels, by ensuring a berth allocation for timely mooring and cargo handling. The second group, comprising logistics companies, is represented by the **truck arrival management agent**, aiming to incur reduced deviation costs by aligning truck arrivals with their preferred time periods and optimizing pickup and delivery schedules. Lastly, we define the **terminal processes agent** that acts as an independent coordinating agent, responsible for overseeing the system processes and verifying the feasibility of all solutions by balancing the needs of the vessels and logistics companies, while maintaining smooth terminal operations. While the framework features a terminal coordinator, its function is not to dictate actions but to facilitate information exchange and conflict resolution between agents. The coordinator acts as a neutral intermediary, identifying conflicts over shared resources and enabling agents to adjust their plans in order to identify mutual beneficial solutions. This design ensures that the decentralized nature of the MAS is preserved, with coordination achieved through adaptation rather than central control.  
The proposed approach assumes that agents collaborate by exchanging solutions, represented by their associated costs, in order to converge on a mutually acceptable outcome. While this assumption facilitates the exploration of coordinated decision-making, it may not fully reflect the current business priorities or operational constraints faced by terminal stakeholders. In particular, the model does not explicitly account for the asymmetry of power among actors, such as the disproportionate influence of shipping lines compared to smaller logistics providers or trucking companies. Although this imbalance is partially captured by the sequential structure of the multi-agent system, where vessels act first, the broader institutional and governance issues surrounding stakeholder inequality and trust remain beyond the scope of this study. Instead, the primary objective is to assess the viability and effects of different coordination and prioritization mechanisms in cargo handling. Demonstrating the value of such mechanisms is a necessary first step toward future work that may address practical barriers to collaboration, including data sharing, trust, and power imbalances.  
The overall goal of the MAS is to determine Pareto optimal solutions by deploying search algorithms that take into account all requirements set by the involved agents. In Section 3.1, the berth allocation agent is described, followed by the truck arrival management agent in Section 3.2, and by the terminal processes agent in Section 3.3. Given the involved agents are interrelated, many parameters and variables are shared across them. For conciseness, these parameters and variables will be introduced only when they first appear in the tables within this section. In Figure 2, a schematic representation illustrates the interactions among variables, parameters and processes (to be introduced in the following sections) within the proposed MAS.