# Combining Optimization and Simulation for Designing a Robust Short-Sea Feeder Network
## 4 Optimization and Simulation Model
### Simulation Model  
The optimal solution of the master problem assumes that the shipping system runs under perfect weather conditions, and both mother and daughter vessels sail at design speed. The simulation model evaluates the proposed solution under real-world operating conditions. Harsh weather conditions may force vessels to slow down, affecting the sailing times. In such a situation, synchronization and duration violations can occur. The simulation model adjusts the sailing speed based on wave height. To this end, we divide a route into legs connecting two consecutive ports. For a given leg, waypoints are defined and mark a change in the vessel's travel direction. The segment between two successive waypoints is defined as a sub-leg. The simulation model updates the sailing speed of a vessel at specific points of a leg. Such a point is called an observation point and is defined (i) at a departure port and (ii) after sailing a certain distance, called step distance. Figure 4 shows an example of a route between two ports. There are three sub-legs defined by the ports and waypoints. The observation points are equally separated by the step distance, which is set to be five nautical miles in this example. The background of Figure 4 shows the gridded weather data points. To determine the speed of a vessel, we use historical weather data composed of a grid of data points that contain significant wave height. The parameter'significant wave height' is often used to describe the weather in maritime navigation and is defined as the average height of the highest one-third waves [38]. At each observation point, we extract the significant wave height from the closest data point. Estimating the vessel speed in open water requires detailed information about the vessel, e.g., regarding hull design and loading conditions, to carry out accurate hydrodynamic calculations. Such information is often not available at an early stage of designing feeder networks. Therefore, we use a simplified approximation of the hydrodynamic relationship between wave height and vessel speed in our simulation model. We assume that vessels are often designed to be operated at a particular propulsion power for which the engines run efficiently, and they sail with constant design power that is maintained throughout the route. Since the vessel speed is a function of propulsion power and total resistance, and given that the propulsion power is assumed to be constant, higher waves and winds due to harsh weather cause an increase in total resistance, which results in a decrease of vessel speed. In our model, resistance from waves is calculated according to the STAwave-1 method [39]. The flow chart in Figure 5 shows the overall simulation process. The input to the simulation model is one (or more) feasible solution(s) from the master problem. In Step 1, the simulation model chooses a solution from the master problem. The week to simulate is chosen in Step 2. In Steps 3-5, all routes in the solution are simulated. Once all routes in the solution are simulated for the given week, synchronization and duration violations are calculated in Step 6. Note that these violations are system-specific, i.e., dependent on the set of routes in the given solution and not only on each route individually. If any violations occur, penalty costs are added to the solution in Step 7. Step 8 checks whether the solution has been simulated for all weeks. Finally, Step 9 ensures that all solutions provided by the optimization model are simulated. Figure 4: Illustration of a leg. The vessel changes direction at the waypoints and the speed is updated at every observation point. An important element of the simulation model is to penalize duration and synchronization violations. The total penalty cost for a simulated solution is the accumulated penalty costs for all vessels for each week of the simulation period.