## 2 Materials and Methods
### First Step: Representation of the State and Simulation of Weather Forecasts  
The state comprises general information about the installation process and the current problem instance. The latter particularly subsumes information about the current date, the number, position, and loading state of vessels, the number of turbines to build, and the number of finished turbines. Additionally, the state contains general information, e.g., about costs or capacities required for the optimizer as given in Table 2 and information on the duration and restrictions of the considered operations. As described in Section 1.1, the installation of a wind turbine occurs sequentially from bottom to top. This sequentiality allows simplifying the planning problem by applying aggregate operations, effectively reducing the number of considered operations from 12 to 4. While the basic operations given in Table 1 directly refer to tasks given in the process description, the aggregate operations, given in Table 3, subsume sequences of basic operations. Therefore, the operation _Load OWT_ is comprised of the loading of one tower, one nacelle, three blades, and one hub. The operation _Install OWT_ subsumes the operations: _position at foundation_, _jack-up_, _install tower_, _install nacelle_, \(3\times\)_install blade_, _install hub_, and finally _jack-down_.  
Finally, the state comprises the current weather forecast given as an interval of probable values for the wave height and wind speed for each hour within the prediction horizon \(N\). As for this article, no real-world forecasts are available, forecasts are simulated from historical weather records as proposed in [9]. Compared to the original proposal, the forecast uncertainty has been adapted to provide more realistic assumptions. Therefore it is assumed that the uncertainty \(\delta(t)\) starts at 0.0 for the current time instance, reaches 0.25 at 168 h into the future, and from there quickly increases to 0.65 at 336 h, 0.95 at 504 h, and from there increases even faster. The simulation first determines the average wind speed \(\mu_{s}^{*}\) and wave height \(\mu_{h}^{*}\) over the prediction horizon \(N\). Afterward, for each time instance, the simulation obtains the measured values \(\mu_{s}\) and \(\mu_{h}\) from the historical recordings. Finally, the simulation acquires the forecast interval by multiplying the average values with the uncertainty and then adding/subtracting this value from the measurement: \(forecast_{s}(t)=\mu_{s}\pm(\mu_{s}^{*}\cdot\delta(t))\) and \(forecast_{h}(t)=\mu_{h}\pm(\mu_{h}^{*}\cdot\delta(t))\). Figure 4 shows the measurements as a black, dotted line and the generated intervals as gray areas for each hour of a forecasting horizon of three weeks. Additionally, the figure denotes the limits for a jack-up operation as a red line. The figure shows that the intervals strictly conform to the measured values in the beginning but begin to broaden quite quickly. As a result, larger proportions of the forecast intervals exceed the limits. This increase effectively reduces the probability of meeting these limits further into the forecast.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Operation** & **Base Duration (h)** & **Max. Wind (m/s)** & **Max. Wave (m)** \\ \hline Travel to Base Port & 4 & 21 & 2.5 \\ Travel to Construction Site & 4 & 21 & 2.5 \\ Load OWT & 12 & - & - \\ Install OWT & 19 & Depending on progress \\ \hline \hline \end{tabular}
\end{table}
Table 3: Aggregate operations used by the MILP model.