The Ship Unloading Path Problem (SUPP) refers to the task of determining the optimal unloading path for a vessel that must call at multiple destination ports, under the constraints of cargo demand and operational requirements. The goal is to minimize transportation costs while maximizing operational efficiency [5] .

Suppose a vessel needs to unload cargo at n ports, denoted as the set:

$P=\left\{p_1,p_2,\cdots ,p_n\right\}$ (1)

The vessel’s total cargo volume is Q, and the amount unloaded at port $i$ is $q_i$, subject to the constraint:

${\displaystyle {\sum }_{i=1}^n{q}_i}=Q$ (2)

The objective of the unloading path problem is to identify a route $\pi =\left(p_{{\pi }_1},p_{{\pi }_2},\cdots ,p_{{\pi }_n}\right)$ that visits all ports such that the sailing distance function：

$D\left(\pi \right)={\displaystyle {\sum }_{i=1}^{n-1}d}\left(p_{{\pi }_i},p_{{\pi }_{i+1}}\right)$ (3)

is minimized, where $d\left(p_i,p_j\right)$ denotes the sailing distance or cost between port i and port j.

Therefore, the SUPP is essentially a path optimization problem with cargo volume constraints, whose computational complexity is similar to that of the Traveling Salesman Problem (TSP), but with more complex constraints [32] .

The fundamental idea of the Enumeration Algorithm is to list all possible path schemes, examine their feasibility and cost one by one, and finally select the path with the minimum cost.

In the Ship Unloading Path Problem, if a vessel needs to visit n ports, the total number of possible paths is:

$N=n!$ (4)

For example, when n = 5, the total number of paths is 5! = 120; when n = 10, the total number of paths is approximately 3.6 × 10⁶. As n increases, the number of paths grows exponentially, a phenomenon known as combinatorial explosion [33] .

Therefore, a pure enumeration algorithm is inefficient for large-scale unloading path problems. However, when combined with constraint-based pruning, it can effectively guarantee the global optimal solution for small- and medium-scale problems.

A constrained optimization problem can be formalized as:

$\min f\left(x\right)\text{s}\text{.t}\text{.}{g}_i\left(x\right)\le 0,h_j\left(x\right)=0$ (5)

where $f\left(x\right)$ represents the objective function, and $g_i\left(x\right)$, $h_j\left(x\right)$ denote the inequality and equality constraints, respectively.

In the context of the Ship Unloading Path Problem, a typical cargo capacity constraint can be expressed as follows:

1) Unloading constraint condition

${\displaystyle {\sum }_{k\in \pi }{x}_k}=Q$ (6)

where $x_{ik}$ denotes the quantity of cargo unloaded at port $p_i$ during the k-th step of the path, and Q represents the total cargo carried by the vessel.

2) Remaining Cargo Constraint

$Q-{\displaystyle {\sum }_{j=1}^k{x}_{{\pi }_j}}\ge 0,\forall k$ (7)

which requires that after any unloading step mm, the remaining cargo on the vessel must be non-negative.

3) Feasible Path Constraint

If a given port does not meet its demand, the corresponding path is immediately deemed infeasible and pruned.

By introducing such constraints, the search space in the enumeration process can be significantly reduced, thereby improving computational efficiency [34] .

4) Intelligent Search and Pruning Strategies

To address the inefficiency of enumeration algorithms, researchers commonly adopt pruning strategies. The core idea is that during the enumeration process, if a partial path is found unlikely to yield a better solution, the search along that branch is terminated early.

Common pruning methods include:

Bound-based pruning

If the sum of the current partial path cost C_partial and the theoretical minimum remaining cost LB already exceeds the known optimal solution C*, the branch can be discarded.

Constraint-based pruning

If the current partial path violates cargo capacity or operational sequence constraints, the branch is not expanded further.

Heuristic-based pruning

Paths lacking priority can be filtered early using heuristic functions (e.g., shortest-distance estimates) [35] .

In international shipping scheduling, obtaining the optimal unloading path under port loading and unloading constraints is a key factor for improving transportation efficiency and reducing operating costs. Given that the number of unloading ports is limited and the constraints are clearly defined, this study introduces the Enumeration Method. By exhaustively generating all possible unloading schemes and progressively filtering them based on business rules, the algorithm ultimately identifies the optimal unloading path that satisfies all constraints.

The overall design of the algorithm follows the principle of “generate feasible solutions first, then gradually filter, and finally output the optimal solution”. It consists of five core steps:

1) Information collection and data acquisition;

2) Unloading scheme generation;

3) Voyage path construction and distance calculation;

4) Cargo capacity constraint validation;

5) Optimal path search and manual adjustment.

The overall framework is illustrated in Figure 1 .

Through a human-computer interaction interface, the scheduling plan and the target vessel’s Maritime Mobile Service Identity (MMSI) are collected. Port-related data (loading port, unloading ports, and the unloading volume for each port) is retrieved from the database. The raw data then undergo a cleaning and preprocessing stage, which includes removing missing values, discarding invalid ports, filtering out identical loading and unloading ports, and verifying whether ports belong to the same navigation region. To accelerate subsequent computations, Redis in-memory caching is adopted for data storage and quick access.

All unloading ports of the target vessel are first subject to a regional consistency check. If inconsistencies are detected, the input data must be reselected; if consistent, duplicate ports are removed, and permutations of the remaining ports are enumerated to produce all feasible unloading schemes.

For example, when the vessel with MMSI = 636,022,324 has two unloading ports, NLVSG and DEBRA (as shown in Table 1 ), two feasible schemes can be generated:

Scheme A: NLVSG → DEBRA

Scheme B: DEBRA → NLVSG

1) Voyage Path Construction and Distance Calculation

Based on the last loading port and the unloading ports specified in each unloading scheme, the voyage path is constructed step by step. The inter-port distance table is queried to obtain the distance between any two ports, from which the total voyage distance is calculated. For example, if UYPTP is the last loading port, then:

Scheme A: UYPTP → NLVSG → DEBRA, with a total distance of 6663 nautical miles.

Scheme B: UYPTP → DEBRA → NLVSG, with a total distance of 6909 nautical miles.

According to the “shortest path priority” principle, the scheme with the shorter total distance is selected as the current unloading plan.

2) Cargo Capacity Constraint Validation

For the current unloading scheme, the vessel’s cargo load before and after each unloading port is calculated and compared against the maximum permissible load at that port to verify compliance with draft restrictions. If a port’s constraint is violated, the scheme with the next-shortest voyage distance is selected as the new candidate plan, and the validation process is repeated until all constraints are satisfied.

3) Optimal Path Search and Manual Adjustment

Once cargo capacity constraints are satisfied, the vessel’s estimated arrival, unloading completion, and departure times are further computed, and the results are written into the scheduling plan to generate the optimal unloading path (see Table 2 ). If scheduling personnel deem the result unreasonable due to factors such as policy changes or weather, they may manually input a preferred port sequence. The algorithm then recalculates the arrival and unloading completion times based on the user-defined order, revalidates the new scheme under the defined constraints, and updates the results, thereby producing the final optimal unloading path.

Compared with traditional experience-based manual scheduling methods, this algorithm leverages real business data and systematic computational rules to achieve efficient and accurate unloading path optimization, demonstrating strong practical application value.

The experimental data cover vessel schedules from January to May 2023, including information on target vessels, loading and unloading ports, cargo volumes, and relevant scheduling constraints. Under the principles of maintaining reasonable unloading sequences and satisfying cargo volume restrictions, the algorithm automatically generates and optimizes unloading paths. The resulting numbers of unloading voyages are shown in Figure 2 .

The experimental results demonstrate that the proposed algorithm achieved an overall accuracy exceeding 75% when applied to real scheduling plans from January to May 2023. Accuracy was evaluated by comparing the optimal unloading paths generated by the algorithm with those manually designed by professional schedulers. A solution was considered correct if it was judged by schedulers to be logically consistent with practical unloading operations.

The findings reveal several key observations:

1) Stability across different scenarios. The algorithm consistently maintained high stability across different months and task scales. Even in months with a large number of voyages—such as January and April, with 67 voyages each—it was still able to efficiently generate feasible solutions.

2) Practicality ensured by capacity constraints. By integrating the cargo capacity constraint principle, the algorithm effectively prevented infeasible solutions caused by ship overloading, thereby ensuring that the output was operationally viable.

3) Efficiency compared with manual scheduling. In contrast to traditional experience-based planning, the algorithm substantially reduced repetitive tasks and significantly shortened the time required for schedulers to design paths. This efficiency gain provides strong support for shipping enterprises undergoing digital transformation.

Overall, the numerical experiments validate the effectiveness and applicability of the proposed algorithm in real-world business contexts, demonstrating both robustness and practical utility.

The enumeration-based optimal unloading path search algorithm proposed in this study offers several innovations by integrating path planning with real-world operational constraints. First, by generating all permutations of unloading ports, the algorithm ensures comprehensive path exploration and guarantees the identification of globally optimal solutions. In both the experimental dataset and common practice, the number of unloading ports per voyage does not exceed five, making this exhaustive approach computationally efficient and tractable. Unlike commonly used heuristic methods, this exhaustive approach is not only feasible in shipping scenarios with a limited number of ports but also provides strong interpretability, which is essential for practical decision-making.

Second, the incorporation of cargo capacity constraints into path selection represents a key advancement. The algorithm rigorously verifies whether ship loading and unloading activities at each port satisfy draft restrictions, thereby producing results that are more consistent with actual operational requirements.

Third, the design of a manual adjustment mechanism enhances practical usability. Scheduling personnel can refine system-generated solutions, which then trigger recalculation and validation. This design achieves a synergistic integration of intelligent computation and human expertise.

Finally, the use of databases and caching technologies significantly improves data processing and query efficiency, aligning with the optimization trends of modern logistics information systems.

Collectively, these innovations ensure that the algorithm not only guarantees the theoretical global optimality but also provides flexibility for handling business constraints and manual interventions. The proposed algorithm can be integrated into Terminal Operating Systems (TOS) or vessel management platforms to automatically generate optimal unloading sequences and estimated operation times, while allowing manual adjustments for practical contingencies. This integration can improve berth utilization, reduce vessel idle time, and enhance overall operational efficiency, offering a practical perspective for intelligent maritime logistics management.

Despite its advantages, the algorithm still has notable limitations. First, the computational complexity of enumeration increases factorially with the number of ports, which undermines efficiency in large-scale scenarios. Existing studies often address this challenge by adopting heuristic or branch-and-bound methods to improve scalability. Second, this study focuses solely on cargo capacity constraints, without accounting for additional critical dimensions such as time windows, fuel consumption, carbon emissions, and meteorological conditions. These factors are becoming increasingly important in the context of green and sustainable shipping. Third, the current algorithm relies on static data and lacks the ability to dynamically re-optimize solutions in response to disruptions such as adverse weather conditions or port congestion, thereby limiting its adaptability to real-time uncertainties. Finally, while manual adjustments provide flexibility, they remain heavily dependent on human experience and lack intelligent recommendation support, introducing potential risks of subjective bias.

Future work may proceed in the following directions: 1) integrating enumerat ion with heuristic methods to balance exactness in small- and medium-scale problems with efficiency in large-scale cases; 2) expanding the range of constraints to construct multi-objective models that simultaneously address economic and environmental goals; 3) leveraging real-time data and predictive analytics to enhance adaptability and robustness; and 4) incorporating intelligent recommendation mechanisms into the manual adjustment process to reduce dependence on subjective experience.

In conclusion, while the algorithm demonstrates strong interpretability and alignment with business requirements, further research is needed to improve its scalability, adaptability, and ability to handle multi-dimensional constraints in complex shipping environments.