# Multicriteria Route Planning for In-Operation Mass Transit under Urban Data
## 4 Multicriteria Route Planning for Mass Transit  
### Problem Formulation  
We adopt and modify the grid-like graph as a weighted directed graph \(G=(V,E)\), in which \(V\) is the set of \(n\) nodes (grids) and \(E\) refers to the set of edges (trajectories). We set \(F=(f_{ij})\) as the cost matrix associated with \(E\), \(F\) corresponds to length (fuel consumption, or construction cost), and a route \(R\) in consist of \(V\) and \(E\) in the targeted problem. Another matrix associated with the selected trajectories \(E^{\prime}\) and stations \(V^{\prime}\) is route \(R^{\prime}=(\{E^{\prime}\},\{V^{\prime}\})\) with \(PF(R)\), which refers to the expected PF of a temporary route \(R^{\prime}\) derived from the PF inference module. The PF is neither superimposable nor monotonic between routes; specifically, both PF and relative input features for the inference model are not monotonic. That is, a reinference is mandatory to retrieve an accurate PF of a route. Precisely, our targeted problem is defined as follows. Given constraints consisting of (1) range for planning \(A\) (2) a set of must-visit stations \(S_{M}=\{S_{M0},\,\ldots\,\,,\,S_{M1}\}\) (3) number of recommended stations \(r\) (4) maximum length \(L\), our goal is to recommend a trajectory in a designated area along with a set of stations \(S=S_{M}+S_{R}\) so as to retain the Pareto-optimal between (1) **profit** (_the PF value to be maximized, referring to the social welfare for the public_) and (2) **cost** (_the cost for construction and maintenance of the route to be minimized. The cost is positively correlated to the length of the route, referring to the sustainability for the government_), in which \(S_{M}\) refers to the set of must-visit stations that shall be settled as the stations and included in the final route, \(S_{R}\) is the recommended stations \(\{S_{R0},\,\ldots\,\,,\,S_{Rr}\}\) that shall be derived by the given algorithm and included in the final route. Meanwhile, the Pareto-optimal here indicates that one method is able to retain the solutions under certain constraints (i.e., area size, or number of stations) if no other method has better performance in both objectives ('profit' and 'cost'). For instance, given certain constraint (scenario), if two methods have the better performance in either of the two objectives against each other, and no other method has better performance (or could be called as the \"improved satisfaction\") over these two methods in both objectives, then these two methods are both defined to reach the Pareto-optimal under this given scenario. Precisely, the targeted problem can be formulated in the following equations:  
\[\left\{ \begin{array}{c}\text{Maximize PF}\Big{(}\Big{\{}\sum_{e_{ij}\in R }e_{ij}\Big{\}},\big{\{}\sum_{v_{i}\in R}v_{i}\Big{\}}\Big{)}.\\ \text{Minimize }\sum_{e_{ij}\in R}\text{F}(e_{ij})\end{array} \right. \tag{1}\]  
s.t. \[\sum_{e_{ij}\in R}\text{F}(e_{ij})\leq L \tag{2}\]  
\[E^{\prime}\subset E_{A},\ V^{\prime}\subset V_{A},\ R\subset A\ with\ R=\big{(}\{E^{\prime}\},\{V^{\prime}\}\big{)} \tag{3}\]  
\[S\subset V\prime \tag{4}\]  
\[S=S_{M}+S_{R} \tag{5}\]  
\[S_{M_{i}}\subset\ V^{\prime}\qquad\forall\ S_{M} \tag{6}\]