# Combining Optimization and Simulation for Designing a Robust Short-Sea Feeder Network
## 3 Solution Approach  
Our iterative solution method is based on the hybrid optimization simulation framework proposed by [PERSON] et al. [34]. The solution method combines an optimization model (also referred to as the  
Figure 2: An example to show duration and synchronization violations due to harsh weather. The mother vessel has to wait for 10 h in ocean hub 1s, causing a synchronization violation and a duration violation of five hours.  
master problem) and a simulation model. The role of the optimization model is to select routes for mother and daughter vessels from a set of a priori generated routes, while the simulation is a discrete event simulation model that evaluates the robustness of solutions (a solution is a combination of routes) with respect to uncertain weather conditions. The simulation model results are used to update the costs of the simulated solutions and then faded back to the optimization model. Alternating between optimization and simulation models is repeated until no new improved solution is found. The proposed solution method is illustrated in Figure 3. According to the classification of [PERSON] et al. [35], the proposed approach is within the alternate simulation-optimization category.  
A static set of routes for the mother and daughter vessels is generated using the provided input data. For each of the daughter vessel types, the master problem then selects the cost-minimizing routes from this set by solving an integer programming model (described in more detail in Section 4.1). Afterwards, the combination of routes chosen by the master problem is simulated using historical weather data to estimate the solution's real-world performance (described in more detail in Section 4.2).  
As explained in Section 2.2, a duration violation is caused by harsh weather conditions that lead to a delay long enough to prevent the vessel from completing its round trip within its scheduled duration. Synchronization violations can amplify delays as they may transfer the delay of one vessel to another. A duration violation will automatically lead to an initial disruption when starting the next round trip. The magnitude of a duration violation is defined as the number of hours by which a vessel is late compared to the allowed duration of one round trip.  
We add a penalty cost based on the magnitude of duration violations from the simulation to the costs of the selected combination of routes. This additional penalty cost makes solutions (or route combinations) prone to duration violations less attractive in subsequent iterations of solving the master problem. When the master problem is solved again with updated route costs, it might choose a new solution with lower costs. The iterations between the master problem and the simulation model continue until all of the selected solutions have been simulated, and no new improving solutions are found. This feedback approach between the master problem and the simulation model allows generating good solutions based on the trade-off between operational costs and robustness (i.e., penalty costs).  
Different strategies to provide robustness against disruptions due to harsh weather and/or synchronization violations can be included in the optimization and/or simulation model (discussed more in Section 4.3). Thus, the framework can also be used to evaluate the potential benefit of introducing performance-improving strategies, for example, permitting speed-ups in case of a delay (see, e.g., [PERSON] et al. [30], [PERSON] et al. [36]).  
Figure 3: An overview of the optimization-simulation framework. From input data, the routes are generated. The master problem and the simulation model are iteratively solved until no new solution is found.