Tugboat Scheduling Optimization

• Tugboat Scheduling Problem is a combinatorial optimization challenge in maritime logistics that assigns tugboats to complex barge shipments.
• It employs a mixed-integer programming model paired with an adaptive large neighborhood search to handle constraints like capacity, time windows, and flow conservation.
• This integrated approach enhances operational cost efficiency, resource utilization, and real-time scheduling for inland waterway transport.

The tugboats scheduling problem is a specialized combinatorial optimization challenge in maritime logistics, characterized by the assignment of tugboats to transport multiple barges across a network of origins, destinations, and idle barge sites. The scheduling must honor operational constraints such as tugboat capacity, work time limits, time windows at nodes, and complex flow conservation requirements within a drop-and-pull operational mode. In modern formulations, the problem is addressed through rigorous mathematical models integrated with advanced metaheuristics, enabling effective real-world solution for large‐scale instances.

1. Mathematical Foundations

The core of the tugboats scheduling problem is the “Multi-trip Multi-barge Drop-and-Pull Barges Shipment Problem.” The mixed-integer programming (MIP) formalization models a network $G = (N, A)$, where tugboats $P$ execute two types of orders:

• TypeF: transport of full barges (pickup and delivery pairing)
• TypeE: repositioning of empty barges, possibly requiring multiple detachment actions

The high-level MIP objective function is:

$$\min \sum_{p \in P} \sum_{(i, j) \in N \times N} \left( C_p^T t_{i,j} + C_p^D d_{i,j} \right)x_{i,j,p}$$

where $x_{i, j, p}=1$ if tugboat $p$ sails from node $i$ to node $j$, $t_{i, j}$ and $d_{i, j}$ are travel time and distance, and $C_p^T$, $C_p^D$ are time- and distance-based cost coefficients. Principal constraints include:

• Each tugboat begins and concludes at virtual origin $s$ and destination $s'$.
• Flow conservation at all nodes:

$$\forall p \in P, \forall j \in N\setminus\{s, s'\}: \sum_{i \in N\setminus\{s'\}} x_{i,j,p} - \sum_{k \in N\setminus\{s\}} x_{j,k,p} = 0$$

• Capacity restrictions per tugboat: $y_{j,p}^E + y_{j,p}^F \leq K$, $K=5$.
• Time consistency for arrivals and services at nodes:

$$z_{j,p} \geq z_{i,p} + s_{i,p} + t_{i,j} - M(1-x_{i,j,p})$$

The MIP model supports exact solution approaches for small problem instances and functions as a decoding tool for candidate schedules generated by metaheuristic frameworks.

2. Adaptive Large Neighborhood Search (ALNS) Algorithm

To resolve scheduling instances of realistic scale, which are computationally prohibitive for direct MIP solution, the adaptive large neighborhood search (ALNS) algorithm is employed. The principal features are:

• Initial Solution Construction: Utilizes an adaptive greedy insertion heuristic. In a two-phase process, full barge orders (TypeF) are inserted first, followed by empty barge orders (TypeE), each sequenced to minimize incremental cost (“loss”).
• Destroy and Repair Operators: Six destroy operators (e.g., random removals, greedy removals, route removals) and eleven repair operators (e.g., greedy insertion, regret greedy insertion, noisy and adaptive insertion) manipulate candidate solutions by removing and optimally reinserting orders.
• Adaptive Operator Selection: Each operator maintains a weight that updates dynamically by performance. Probabilities of selection and weight updates are governed by:

$\rho_j = \frac{w_j}{\sum_{i=1}^k w_i}$

$$w_j^{\text{updated}} = b w_j + (1-b) \frac{\pi_j}{\theta_j}$$

with $b$ as a decay factor, $\pi_j$ the accumulated score, and $\theta_j$ the usage count.

• Decoding Model Integration: The candidate solution is validated and evaluated by the MIP decoder, ensuring adherence to operational constraints and accurately computing the objective function.
• Acceptance Mechanism: Improvement solutions are accepted directly, and non-improving solutions may be accepted with probability

$$\nu = \exp\left[\frac{\text{loss}(x) - \text{loss}(x')}{T}\right]$$

simulating the effect of annealing to escape local optima.

3. Performance Profile

Extensive computational testing across both oceanic and inland settings indicates:

• For small/medium instances, ALNS reliably achieves optimal or near-optimal solutions, with improvement rates approaching 100% over naïve initial solutions.
• For large instances, solution stability slightly decreases but high-quality outcomes persist with practical computation times.
• ALNS demonstrably outperforms “strengthened” mathematical models: while MIP solvers manage only small instances, ALNS comfortably scales to dozens of orders and multiple tugboats.

A plausible implication is that ALNS, in conjunction with a strict decoding phase, represents an effective and generalizable approach to large-scale maritime scheduling challenges.

4. Operational Relevance and Application

The scheduling solution targets improved tugboat utilization in inland waterway transportation. Key operational aspects:

• Accommodates drop-and-pull mode, enabling simultaneous barge detachment and independent tug movement.
• Explicitly models both full barge shipments and complex repositioning of empty barges, including cooperative multi-tugboat scenarios.
• Integrates the combinatorial flexibility of ALNS with the exactness of MIP decoding, resulting in robust satisfaction of working time, time window, and capacity restrictions.
• Delivers reductions in operational cost, increased resource utilization, and scalability to real-time daily scheduling.
• Noted limitations are sensitivity to algorithmic parameters (operator weights, cooling schedule), inherent stochastic behavior, and potentially high computation times in extreme problem sizes.

5. Future Avenues for Research

Ongoing and future work priorities include:

• Uncertainty Modeling: The introduction of stochastic programming and robust optimization to capture fluctuating environmental and operational variables (e.g., weather, congestion, delay).
• Multi-objective Expansion: Augmenting the model to optimize additional criteria such as emissions, fuel efficiency, and service reliability for broader environmental and social considerations.
• Dynamic and Real-Time Scheduling: Developing methods for continual reoptimization as orders and conditions evolve during a work shift.
• Network Scale-Up: Extending current models to encompass complex, intermodal, and multi-port networks with richer interaction dynamics.
• Comparative and Hybrid Methods: Comprehensive benchmarking against alternative metaheuristics (genetic algorithms, particle swarm optimization) and potential integration of machine learning to further refine adaptive operator selection mechanisms.

6. Significance in Maritime Logistics Optimization

The integration of exact mixed-integer programming with a flexible ALNS metaheuristic has advanced the state-of-the-art in barge and tugboat scheduling. This strategy resolves the scaling limitations of pure mathematical programming and achieves notable real-world applicability through its operational precision and adaptability. The approach is suited for inland waterways with complex demand structures and heterogeneous fleets, contributing directly to logistical efficiency and resource minimization in modern freight networks.

The framework elaborated in "ALNS for Tugboat Scheduling in Inland Waterway" (Ma, 24 Sep 2025) encapsulates current best practice for this scheduling problem. Its empirical results and proposed future directions establish a structured roadmap for continued innovation in maritime transport optimization.