TU Delft Repository

This thesis proposes a new solution to minimize the number of batteries needed to schedule ships depending on the presented harbour. The proposed solution is an extension to the bin packing problem. Item Fragmentation (BPP-IF) and Conflicts (BPPC) were added to allow ships to carry more batteries and to prevent batteries from being scheduled for multiple ships, respectively. Negative Items (BPP-NI) were added to allow for the batteries to be charged when they were not scheduled to be used. The combination of the three bin packing extensions (BPPC-IF-NI) formed the basis for the proposed solution. By a polynomial time reduction we showed that BPPC-IF-NI is NP-hard and therefore cannot be analytically solved in polynomial time. The solution can be approximated with an updated version of the heuristic presented by Ekici (2022). The main idea in this heuristic approach is to generate a set of independent sets of the conflict graph and determine the packing of the bins using the independent sets with the help of the integer linear program ISB-IP. An iterative framework is used where 𝐿 sets of 𝑇 independent sets are generated at each iteration and considered together with the independent sets used in the best solution to obtain a better solution. After testing many possible combinations of 𝐿 and 𝑇, it was found that 𝐿 = 50 and 𝑇 = 10 gives the best results, meaning that half of the number of the initial independent sets (𝐿-value) containing a maximum of half the number of items in each set (𝑇-value) where added in each iteration. When using these parameters, it was found that the solution of the heuristic was the same as the true optimal value. This was checked by inserting the power set without conflicts into the ISB-IP. The performance of the proposed solution was tested on generated instances. It was seen that there is no correlation between the computation time and the number of conflicts. It was also found that there was a strong correlation between the number of conflicts and the number of batteries used in the solution, without any clear outliers. The proposed solution is able to very quickly approximate the minimum number of batteries needed to schedule the ships for the given harbour. Since most instances are approximated in a couple of seconds, it can also be used to update schedules when complication arise on a smaller scale. This would allow harbours to switch to electric ships, while reducing the cost by minimizing the number of expensive ship batteries.

Tipping occurs when a critical point is reached, beyond which a perturbation leads to persistent system change. Here, we present observational indications demonstrating presently ongoing noise-tipping of a real-world system. Noise in a river system is associated with the changing flow rate. In particular, we consider the upper Rhine River delta, where flow and sediment fluxes are partitioned over the two downstream branches (bifurcates) of an important river bifurcation. Field observations show that a sequence of peak flows in the 1990s resulted in sudden sediment deposition in one bifurcate, triggering a persistent and ongoing change in the flow partitioning. This has caused the system to move toward an alternative equilibrium state or attractor. An idealized model confirms that a river bifurcation system under such conditions is prone to tipping, and provides insight on the onset of tipping.