# Multi-Objective Routing and Categorization of Urban Network Segments for Cyclists
## 2 Materials and Methods
### Formulation of the Objective Functions
#### 2.1.2 Comfort-/Speed-/Accessibility-Related Criteria  
Comfort is an important criterion for cyclists when choosing their route on the city network. There are several factors that affect the comfort of cycling, including the slope of the segment, pedestrian flows, and total route travel times.  
Travel time: Travel time is a critical factor that significantly influences cyclists' route choices. Cyclists, like any other traveler, seek to reach their destination efficiently within a reasonable time [28]. Let \(t_{\text{i,j,k}}\) be the travel time of each link \(N_{\text{i,j,k}}\) in the network. Then, the following objective function:  
\[F_{6}(y)=\min\sum_{ij\in R}\sum_{k\in K}t_{ijk}y_{ijk} \tag{6}\]  
is introduced to minimize the total route travel time.  
Gradient (uphill-downhill): The slope of the route is a factor that affects the comfort of cycling. Cyclists usually prefer routes with a slight slope, as uphill and downhill routes may require more effort and may affect the comfort of travel, especially over longer distances [31]. It is also quite common for a cyclist to change their route to \"avoid\" a steep part of the route. If parameter \(g_{\text{i,j,k}}\) denotes the gradient of each link \(N_{\text{i,j,k}}\) (%), then one can introduce the following objective function:  
\[F_{7}(y)=\min\sum_{ij\in R}\sum_{k\in K}g_{ijk}l_{ijk}y_{ijk} \tag{7}\]to minimize the overall slope of the route.  
Width of pavement/pedestrian crossing: The results of [PERSON]'s research [32] show that riding on pedestrian pavement is common among cyclists. Three main components of their choice are avoiding vehicles, increasing the smoothness of the route, and unclear infrastructure markings. Recall that sidewalks measured below the 1.5 m threshold are excluded from the study. If \(w_{\text{i,j,k}}\) is the width of the pedestrian pavement or sidewalk of each link N\({}_{\text{i,j,k}}\), then the following objective function:  
\[F_{8}(y)=\max\sum_{ij\in R}\sum_{k\in K}w_{ijk}l_{ijk}y_{ijk} \tag{8}\]  
is needed to be maximized.  
Pedestrian load: Pedestrian loads on a sidewalk or pedestrian crossing can have a significant impact on cyclists. In general, cyclists are more likely to avoid routes with high pedestrian loads as they perceive these routes to be more dangerous. This is because high pedestrian loads can pose several risks to cyclists, such as the risk of colliding with pedestrians and the risk of being forced off the road [33]. Cyclists respond to pedestrian density by either modifying their speed or getting off their bikes and taking other avoidance measures whenever necessary [34]. Then, we seek to minimize the objective function of  
\[F_{9}(y)=\min\sum_{ij\in R}\sum_{k\in K}p_{ijk}l_{ijk}y_{ijk} \tag{9}\]  
where \(p_{\text{i,j,k}}\) is defined as the pedestrian load per network link in pedestrians/15 min/m.  
Note that we have defined nine objective functions, some of which need to be maximized, while others need to be minimized. This forms a multi-objective optimization problem in which improving an objective (i.e., reducing the overall travel time of the trip) might result in deteriorating another (i.e., forcing the cyclist to take a steep route), and vice versa. This is why determining the set of Pareto-optimal solutions is important.