## 2 Materials and Methods
### Third Step: MILP Formulation for the Incremental Scheduling of Offshore Operations
\[X_{v,k}^{cap} =X_{v,k-1}^{cap}-Y_{v,k}^{out}+Y_{v,k}^{load} \forall k\in\{2\ldots N\};\forall v\in\{1\ldots V\} \tag{2}\] \[X_{v,k}^{cap} \leq CAP \forall k\in\{1\ldots N\};\forall v\in\{1\ldots V\}\] (3) \[Y_{v,k}^{loc} =Y_{v,k-1}^{loc}-Y_{v,k}^{toPort}+Y_{v,k}^{toSite} \forall k\in\{2\ldots N\};\forall v\in\{1\ldots V\} \tag{4}\]  
The support variables \(Y_{v,k}^{busy}\), and \(Y_{v,k}^{atBay}\) track if a specific vessel is generally busy and, additionally, if it is currently performing a loading operation at the base port. Constraints (5)-(9) record these durations. Constraints (5)-(8) thereby mark the corresponding entries in \(Y_{v,t}^{busy}\) for \(t=k+1\) to \((k+D_{o,k}-1)\) for the corresponding vessel \(v\) and the scheduled operation \(o\). This indicates that no other operation can be scheduled during this period (c.f. Constraint (10)). As can be seen from this formulation, the constraints do not mark the time instance \(k\) itself, as this is already marked in the corresponding decision variables. Consequently, these constraints also subtract one hour from the duration. In contrast, \(Y_{v,k}^{atBay}\) denotes the full duration from \(k\) to \(k+D_{o,k}\). \[\sum_{t=k+1}^{k-1+D_{1,k}}Y_{v,t}^{busy} \geq Y_{v,k}^{out}\cdot(D_{1,k}-1) \forall k\in\{1\dots N\};\forall v\in\{1\dots V\} \tag{5}\] \[\sum_{t=k+1}^{k-1+D_{2,k}}Y_{v,t}^{busy} \geq Y_{v,k}^{totPort}\cdot(D_{2,k}-1) \forall k\in\{1\dots N\};\forall v\in\{1\dots V\}\] (6) \[\sum_{t=k+1}^{k-1+D_{3,k}}Y_{v,t}^{busy} \geq Y_{v,k}^{toSite}\cdot(D_{3,k}-1) \forall k\in\{1\dots N\};\forall v\in\{1\dots V\}\] (7) \[\sum_{t=k+1}^{k-1+D_{4,k}}Y_{v,t}^{busy} \geq Y_{v,k}^{load}\cdot(D_{4,k}-1) \forall k\in\{1\dots N\};\forall v\in\{1\dots V\}\] (8) \[J =\sum_{v=1}^{V}\Bigg{(}\sum_{k=1}^{N}\left(Y_{v,k}^{loc}\right) \cdot C_{v}^{o}+\sum_{k=1}^{N}\left(Y_{v,k}^{atBay}\right)\cdot C_{v}^{p}+\sum _{k=1}^{N}\left(Y_{v,k}^{toSite}\right)\cdot C_{v}^{m}+\sum_{k=1}^{N}\left(Y_{ v,k}^{toPort}\right)\cdot C_{v}^{m}\Bigg{)} \tag{1}\]0  
Constraint (10) exploits this behavior and uses the resulting support variable \(Y^{busy}\) and the decision variables to ensure that each vessel is only involved in a single operation at a time. \[J =\sum_{v=1}^{V}\Bigg{(}\sum_{k=1}^{N}\left(Y_{v,k}^{loc}\right) \cdot C_{v}^{o}+\sum_{k=1}^{N}\left(Y_{v,k}^{atBay}\right)\cdot C_{v}^{p}+\sum _{k=1}^{N}\left(Y_{v,k}^{toSite}\right)\cdot C_{v}^{m}+\sum_{k=1}^{N}\left(Y_{ v,k}^{toPort}\right)\cdot C_{v}^{m}\Bigg{)} \tag{1}\]1  
Constraint (11) limits the number of available loading bays to a specified value \(BAYS\). This constraint uses the parameter \(INUISE_{k}^{busy}\), which denotes how many loading bays are in use by previous schedules at each time instance \(k\). \[J =\sum_{v=1}^{V}\Bigg{(}\sum_{k=1}^{N}\left(Y_{v,k}^{loc}\right) \cdot C_{v}^{o}+\sum_{k=1}^{N}\left(Y_{v,k}^{atBay}\right)\cdot C_{v}^{p}+\sum _{k=1}^{N}\left(Y_{v,k}^{toSite}\right)\cdot C_{v}^{m}+\sum_{k=1}^{N}\left(Y_{ v,k}^{toPort}\right)\cdot C_{v}^{m}\Bigg{)} \tag{1}\]2  
Constraint (12) ensures that the optimizer only schedules installation activities if they can be finished within the prediction horizon \(N\). To record the duration, the model uses a provided matrix of expected durations \(D_{o,k}\). This matrix denotes the duration of an operation \(o\) if it is started at a time instance \(k\). Therefore, the index \(o\) refers to the aggregate operations _install_\(ovt=1\), _move to port = 2, _move to site = 3_ and _load components = 4_. This matrix is calculated outside of the optimizer, as described before, in Section 2.3. \[J =\sum_{v=1}^{V}\Bigg{(}\sum_{k=1}^{N}\left(Y_{v,k}^{loc}\right) \cdot C_{v}^{o}+\sum_{k=1}^{N}\left(Y_{v,k}^{atBay}\right)\cdot C_{v}^{p}+\sum _{k=1}^{N}\left(Y_{v,k}^{toSite}\right)\cdot C_{v}^{m}+\sum_{k=1}^{N}\left(Y_{ v,k}^{toPort}\right)\cdot C_{v}^{m}\Bigg{)} \tag{1}\]3  
The last two support variables reduce the symmetry of the formulation by recording the finishing time of operations.