## 2 Materials and Methods
### Third Step: MILP Formulation for the Incremental Scheduling of Offshore Operations  
After preprocessing the weather forecasts to obtain an estimation of the duration for each aggregate operation at each time instance, the open loop calculates a plan for the prediction horizon \(N\). As can be seen from the state of the art, several possible formulations and approaches exist for this task. Following the assumptions given in Section 2.1, i.e., that plans cover about a week or even more and that deviations can be anticipated comparably early due to the long duration of operations, this article proposes to exploit these extended time frames and to strive for exact mathematical solutions. Consequently, this section proposes a Mixed-Integer Linear Program for the scheduling. As this article considers multiple vessels with equal capabilities, a Flexible JSSP formulation is selected. According to the classification in Section 1.3, there exist three basic formulations: a precedence-based formulation, a sequence-based formulation, or a time-indexed formulation. Whereas time-indexed formulations are the slowest to compute, they provide several advantages over the other formulations for this application. On the one hand, it is possible to directly make use of varying processing times, i.e., to directly include the estimated duration for aggregate operations. For sequence-based formulations, this is hard to achieve due to their abstract representation of time. On the other hand, a time-indexed formulation can easily be parameterized to obtain plans for a predetermined time frame, i.e., the prediction horizon \(N\). This is very hard to formulate for precedence-based models. Such models index the scheduling problem over the number of operations, and thus, always must schedule all operations. For the proposed incremental approach, this will be impossible in most of the cases, as forecasts only remain reliable for comparably short forecast horizons. Therefore, the optimizer might not be able to schedule all operations within the prediction horizon, rendering the problem infeasible. For time-indexed models, the problem can be reformulated to instruct the optimizer to schedule as many operations as possible within the prediction horizon, favoring incremental approaches. As stated before, the Mixed-Integer formulation used in this article is an extension to the one used in [9]. This article extends the formulation to handle multiple installation vessels and by constraints, which limit the number of available loading bays in the base port. Moreover, it presents a modification to the overall formulation, which includes so-called penalty terms. These terms, e.g., provide benefits for finishing an operation early, to reduce the model's symmetry, and to speed up the optimization. Table 2 in Section 1.6 provides an overview of the relevant model parameters and variables. The model aims to minimize the cost function \(J\) given in Equation (1) under consideration of the constraints given in Equations (2)-(15). The MILP optimizes a plan for the prediction horizon \(N\) determined by the sampling step size \(T\) and the previously defined number of planning periods \(P\) as \(N=P*T\) (c.f. Section 2.1. Cost function \(J\) sums the cost for vessels being offshore, fuel cost for traveling between the site and base port, and hourly costs for port operations. Therefore, it sums each occurrence of such an operation and, respectively, every hour a vessel is offshore and multiplies these sums with the associated costs (first line of Equation (1)). It is advised to choose different costs for  
Figure 5: Example transition matrix \(P^{n}\) with three operations of duration 2, 3 and 1, which results in 7 states: The first operation with one helper state, the second operation with two helper states, the third operation with no helper state and the final state. each vessel to reduce model symmetry.