# Combining Optimization and Simulation for Designing a Robust Short-Sea Feeder Network
## 4 Optimization and Simulation Model
### Optimization Model  
The optimization model is based on the approach developed by [PERSON] et al. [10] and consists of a route generation procedure and a master problem. The difference is that, in this study, the optimization model incorporates updated costs from the simulation model. For an efficient route generation, a label-setting algorithm (see, e.g., [PERSON] [37]) is used to generate a priori the routes for mother and daughter vessels. After the routes are generated, the daughter routes are grouped in subsets according to which main route they can be synchronized with. The label setting algorithm is used to limit the set of routes introduced to the optimization model. Only non-dominated routes are retained, which means that similar routes composed of the same ports but in a different order and with higher operational costs are eliminated. The route generation assumes that vessels sail at their design sailing speed, i.e., no speed-up is allowed. Here, it should be pointed out that some of the deterministically dominated routes might perform better in our setting with uncertain travel time, e.g., due to a sequence of port visits that allows avoiding harsh weather conditions. However, identifying these routes during the route generation procedure is, in general, too computationally expensive. Before formulating the optimization model, let us first introduce the following notation: \begin{tabular}{l l} \(\mathcal{P}^{OH}\) & set of ocean hubs, \\ \(\mathcal{P}^{CD}\) & set of Norwegian small ports, \\ \(\mathcal{P}^{CM}\) & set of Norwegian main ports, \\ \(\mathcal{R}^{M}\) & set of all non-dominated main routes, \\ \(\mathcal{R}^{D}\) & set of all non-dominated daughter routes, \\ \(\mathcal{R}^{M}_{p}\) & subset of main routes, for which port \(p\) is served, \(\mathcal{R}^{M}_{p}\subseteq\mathcal{R}^{M}\), \\ \(\mathcal{R}^{D}_{p}\) & subset of daughter routes, for which port \(p\) is served, \(\mathcal{R}^{D}_{p}\subseteq\mathcal{R}^{D}\), \\ \(\mathcal{R}^{D}_{pm}\) & subset of daughter routes that can be synchronized to a main route \(m\) at ocean hub \(p\), \(\mathcal{R}^{D}_{pm}\subseteq\mathcal{R}^{D}_{p}\subseteq\mathcal{R}^{D}\), \\ \(\mathcal{S}\) & set of simulated solutions. If the set is empty, no solutions have been simulated. The size of the set increases for every new solution that is simulated. \\ \end{tabular}   
Parameters  
\begin{tabular}{l l} \(C^{M}_{m}\) & total operational cost of main route \(m\), which includes the weekly time charter cost of mother vessels sailing \(m\), burker costs and port costs, \\ \(C^{D}_{d}\) & operational cost of daughter route \(d\), which includes the weekly time charter cost of daughter vessel sailing \(d\), burker costs and port costs, \\ \(C^{S}_{s}\) & penalty cost of simulated solution \(s\), \\ \(M^{OH}\) & upper limit on the number of daughter vessels that can perform transshipment at an ocean hub, \\ \(M^{S}\) & a value that is marginally larger than \(\varepsilon\), \\ \(S^{D}_{ds}\) & is equal to 1 if daughter route \(d\) belongs to simulated solution \(s\), and 0 otherwise, \\ \(S^{M}_{ms}\) & is equal to 1 if main route \(m\) belongs to simulated solution \(s\), and 0 otherwise, \\ \(S_{s}\) & number of routes which belong to solution \(s\). This can be expressed as follows: \(S_{s}=\sum_{m\in\mathcal{R}^{M}}S^{M}_{ms}+\sum_{d\in\mathcal{R}^{D}}S^{D}_{ds}\), for each simulated solution \(s\), \\ \(\varepsilon\) & auxiliary parameter used to express a less than relation as a less than or equal relation. The value can be as\(x_{m}\) equals to \(1\) if main route \(m\) is selected, and \(0\) otherwise, \(z_{d}\) equals to \(1\) if daughter route \(d\) is selected, and \(0\) otherwise, \(y_{s}\) takes value \(1\) if a simulated solution is included in the optimal solution, and \(0\) otherwise.