Inland waterway transport is a low CO2 emission alternative to road transport. A shift towards more inland waterway transport could also help reduce road congestion and noise pollution. Infrastructure bottlenecking, particularly at locks, is part of the reasons preventing this shift. Congestion is leading to delays. Locks can be physically improved, or the passage of the vessels through the locks can be optimized through scheduling. Recent work introduced a novel switching-max-plus-linear system approach to scheduling vessels passing through networks of waterways and locks, also introducing a novel routing component to the scheduling problem. Switching max-plus-linear systems are a convenient way to model scheduling systems using max-plus-linear algebra. The switching-max-plus-linear model only considered locks with a single chamber that can only process one vessel at a time. Additionally, it only considered four specific waterway network configurations, rather than any arbitrary network configuration. Real locks can have multiple chambers, and they can process multiple vessels at the same time if they are placed according to regulations in the two-dimensional space of the chamber. The scope of this report was then to build upon this switching max-plus-linear model by adding support for arbitrary network configurations, and multi-chamber, multi-vessel locks with proper twodimensional ship placement, to answer the main research question: How can multi-vessel, multichamber locks with ship placement be integrated into the SMPL IWT scheduling model? Three mathematical scheduling models formulated as SMPL systems were introduced, each subsequent model building on the previous one. The first introduced support for arbitrary network configurations. The second introduced support for multi-chamber locks and allowed vessels to pass through lock chambers at the same time, provided that their assigned one-dimensional sizes fit into the assigned one-dimensional capacity of the chamber. The third introduced support for twodimensional ship placement through a Tetris-like placement sequence also modeled as a switching max-plus-linear system. The models were translated to mixed integer linear programming models, and arrival time and arrival time offset objectives were added, so they could be used as the rules by which a scheduler would build and solve offline scheduling optimization models on a case-by-case basis. Auxiliary objectives to promote vessels slowing down, rather than waiting stationary in the waiting areas, were also added. The models were all found to be working as intended and implemented correctly through the use of a number of verification cases and tests. Complexity tests showed that the solution times for all three models already became larger than a practical limit of 15-20 minutes for simple scenarios with 10-15 vessels. A heuristic model that mimics how vessels are assigned to lockages in practice was built. Comparisons to the scheduling models on the verification cases showed negligible differences for the models in single-lock cases, but it indicated that the multi-vessel, multi-chamber models may outperform practice in multi-lock cases. Future research is recommended to focus on online optimization to account for disturbances, distributed optimization to reduce calculation times, and validation of the model’s performance with real data.

Inland waterway transport is a low CO2 emission alternative to road transport. A shift towards more inland waterway transport could also help reduce road congestion and noise pollution. Infrastructure bottlenecking, particularly at locks, is part of the reasons preventing this shift. Congestion is leading to delays. Locks can be physically improved, or the passage of the vessels through the locks can be optimized through scheduling. Recent work introduced a novel switching-max-plus-linear system approach to scheduling vessels passing through networks of waterways and locks, also introducing a novel routing component to the scheduling problem. Switching max-plus-linear systems are a convenient way to model scheduling systems using max-plus-linear algebra. The switching-max-plus-linear model only considered locks with a single chamber that can only process one vessel at a time. Additionally, it only considered four specific waterway network configurations, rather than any arbitrary network configuration. Real locks can have multiple chambers, and they can process multiple vessels at the same time if they are placed according to regulations in the two-dimensional space of the chamber. The scope of this report was then to build upon this switching max-plus-linear model by adding support for arbitrary network configurations, and multi-chamber, multi-vessel locks with proper twodimensional ship placement, to answer the main research question: How can multi-vessel, multichamber locks with ship placement be integrated into the SMPL IWT scheduling model? Three mathematical scheduling models formulated as SMPL systems were introduced, each subsequent model building on the previous one. The first introduced support for arbitrary network configurations. The second introduced support for multi-chamber locks and allowed vessels to pass through lock chambers at the same time, provided that their assigned one-dimensional sizes fit into the assigned one-dimensional capacity of the chamber. The third introduced support for twodimensional ship placement through a Tetris-like placement sequence also modeled as a switching max-plus-linear system. The models were translated to mixed integer linear programming models, and arrival time and arrival time offset objectives were added, so they could be used as the rules by which a scheduler would build and solve offline scheduling optimization models on a case-by-case basis. Auxiliary objectives to promote vessels slowing down, rather than waiting stationary in the waiting areas, were also added. The models were all found to be working as intended and implemented correctly through the use of a number of verification cases and tests. Complexity tests showed that the solution times for all three models already became larger than a practical limit of 15-20 minutes for simple scenarios with 10-15 vessels. A heuristic model that mimics how vessels are assigned to lockages in practice was built. Comparisons to the scheduling models on the verification cases showed negligible differences for the models in single-lock cases, but it indicated that the multi-vessel, multi-chamber models may outperform practice in multi-lock cases. Future research is recommended to focus on online optimization to account for disturbances, distributed optimization to reduce calculation times, and validation of the model’s performance with real data.

Inland waterways form a natural network infrastructure with the capacity for waterborne transport of people and goods for moving freight from seaports to the hinterland. Recently, Inland Waterway Transport (IWT) has been promoted more extensively by the European Union and various governments as it plays a crucial role in reducing road congestion and CO2 emissions from transport. However, the advantages of IWT are not fully exploited due to inefficiencies in the logistics system, such as long waiting times at locks and sub-optimal navigation on waterways. Currently, no scheduling at infrastructures or routing optimisation of the overall waterway network is happening. The scheduling of vessels through a lock is usually performed on a First In First Out basis, providing an opportunity for improvement. Hence, this thesis aims to design a scheduling strategy for generating an optimal plan for sending inland vessels through a waterway network with minimal delays, yielding a significant positive impact on the modal shift towards IWT. 
A promising approach to scheduling problems is by using Switching Max-Plus-Linear (SMPL) systems. SMPL systems have proven to be effective in various Discrete-Event Systems and transportation networks. Using SMPL models is convenient since non-linear scheduling problems can be described linearly using Max-Plus operators without compromising on the system dynamics. Moreover, as the SMPL systems can be transformed into Mixed-Integer-Linear-Programming (MILP) problems, it is also possible to use fast optimisers for solving the scheduling problems.
This thesis will show how one can describe IWT systems, consisting of; waterways, vessels and locks, as SMPL systems. The optimal schedule for the inland vessels is determined based on multiple input parameters, including waterway network lay-out, the sailing speeds of vessels and arrival deadlines of the vessels. The scheduler will return the individual vessel routing and overall vessel order in the waterway network. This routing and order selection is defined using binary control variables, turning the IWT scheduling problem into a MILP problem, which will allow finding the solution to large scale IWT scheduling problems in a reasonable computation time. Furthermore, this thesis will show how the goal of minimising the cumulative arrival times of all vessels in a network can be achieved. This is done for different types of waterway network cases, for which the results are shown and analysed.