## 4 Proposed Model
### Model Structure  
Figure 5 graphically illustrates the optimization model structure. The fitness value is computed for the specific route plan based on the routes developed by a candidate solution and the constraints and preferences adopted. In particular, the total carrier cost, resulting from traveling distances of individual routes and unit costs, is calculated as the economic component of the problem. Further, the total number of passenger hours in all routes is accumulated to develop the second part of the objective function. These two parameters are appropriately tuned (based on the maximum value found running the algorithm as a single objective) depending on the desirable impact of each one on the objective function. All the examined solutions undergo constraint and preference satisfaction checks before they are compared to each other to develop the optimal routing plan. Indicative (hard) constraints and (soft) preferences include the island service priority, upper and lower bounds for vessel traveling distance or number of served islands, maximum allowable journey time for passengers, avoiding long- or short-lasting trips and giving service priority to more popular destinations. Model economic and non-economic parameters are fine-tuned following normalization based on their mean values before applying the weighting process.  
The H&S network design problem is a complex combinatorial optimization problem and belongs to the NP-hard class of problems, which means that the problem complexity and the computational resource requirements grow significantly as the numbers of nodes and constraints increase. Genetic algorithms (GAs) have been extensively used in recent years to solve large and complex combinatorial optimization problems due to their capability to provide acceptable (near-optimal) solutions within a reasonable computational time for typical problem cases. Figure 6 schematically depicts the optimization structure and methodology. The system is initially fed with all necessary inputs, such as island distances, vessel speeds and unit costs, the number of passengers per destination, etc. Following the typical GA evolutionary process, the algorithm develops a population of initial solutions using random values to optimize the three fundamental parameters, i.e., hub node selection, island assignment to vessels, and island service priority. Individual solutions undergo reproduction through crossover and mutation operations within the population, and new solutions are generated. The solutions are initially checked for validity (constraint and preference fulfillment) and then evaluated in terms of their fitness value. The process terminates when a predefined number of iterations are made without considerably improving the fitness score.  
More specifically, the genetic algorithm is parameterized with a 50-chromosome initial population with the crossover and mutation rates being 0.5 and 0.1, respectively, in all test cases and following a relevant analysis. The termination criterion involves the performance of 20,000 trials without substantial improvement (less than 0.01%) of the fitness value. The problem has been built in the MS Excel environment for easi  
Figure 5: Optimization model structure for short-sea passenger transport.  
Figure 6: Optimization methodology flowchart.  
model implementation. Optimization is performed using the commercial GA software Evolver v8.4.0 of the Palisade Company (now named Lumivero), which runs as an Excel add-in component.