## 3 Multi-Modal Multi-Objective Journey-Planning Optimisation Model
\[\forall y^{k}\in P\
exists\Big{(}\left(y^{l}\in P\wedge i\in F\right)\Big{|} \left(f_{i}\left(y^{k}\right)>f_{i}\left(y^{l}\right)\right)\Big{)}, \tag{2}\]  
\[\forall y^{k}\in P\exists\Big{(}\left(y^{l}\in P\wedge i\in F\right)\Big{|} \left(f_{i}\left(y^{k}\right)>f_{i}\left(y^{l}\right)\right)\Big{)}. \tag{3}\]  
Formally, we can characterise the validity of any given multi-modal journey in terms of a time-expanded graph model, where the non-public transport networks have copies of each edge corresponding to departures at each possible time interval \(t\) (e.g., \(E^{pt}\) for the set of pedestrian edges departing at time interval \(t\)). On the other hand, the public transport network has edges corresponding to each scheduled trip between each pair of consecutive stops in all scheduled trips. Let \(x^{k}_{etij}\) denote a binary variable indicating whether or not transport mode \(v\) is used to travel from node/stop \(i\) to node/stop \(j\) departing at time \(t\) in journey \(k\). A feasible journey must be spatially contiguous (Constraint 4). Specifically, Constraint 4 states that each non-origin and non-destination node entered must also be exited. Note that, for the case of connections between transport modes, the edge set \(E^{vt}\) includes zero length edges to nodes in the same position but in different transport networks. Journeys begin at the origin (Constraint 5) and end at the destination (Constraint 6). \begin{table}
\begin{tabular}{c c c c c} \hline  & **Pedestrian** & **E-Scooter** & **Taxi** & **Public Transport** \\ \hline Fixed cost (GBP) & 0 & 1 & 2.5 & 0 \\ Daily cost (GBP) & 0 & 0 & 0 & 4.5 \\ Distance cost (GBP/m) & 0 & 0 & 0.00125 & 0 \\ Time cost (GBP/s) & 0 & 0.0025 & 0 & 0 \\ Response time (s) & 0 & 120 & 300 & N/A \\ Maximum speed (m/s) & 1.111 & 3.89 & 31.29 & N/A \\ CO\({}_{2}\) (g/m) & 0.00011 & 0.007 & 0.12 & 0.0411 \\ Calories (kCal/m) & 0.06 & 0 & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Vehicle parameters. \[\sum_{v\in V}\sum_{t\in T}\sum_{(i,j)\in E^{vt}|i\
eq O}x^{k}_{vtij}=\sum_{v\in V} \sum_{t\in T}\sum_{(j,l)\in E^{vt}|l\
eq D}x^{k}_{vtij},\ \forall j\in N\setminus O\cup D,\ \forall y^{k}\in P, \tag{4}\]  
\[\sum_{v\in V}\sum_{t\in T}\sum_{j\in N^{vt}}x^{k}_{vtOj}=1,\ \forall y^{k}\in P, \tag{5}\]  
\[\sum_{v\in V}\sum_{t\in T}\sum_{j\in N^{vt}}x^{k}_{vtjD}=1,\ \forall y^{k}\in P, \tag{6}\]  
Constraint 7 characterises time-feasible paths within this time-space network-graph-based formulation; specifically, nodes can only be exited at a time after they have been entered. \(\tau_{vtij}\) denotes the travel time between nodes \(i\) and \(j\) while using transport network \(v\) departing at time interval \(t\). \(d_{t}\) is the time associated with time interval \(t\). Additionally, the journey may not begin before the earliest departure time \(\alpha\) (Constraint 8). \[\sum_{v\in V}\sum_{t\in T}\sum_{(i,j)\in E^{vt}|i\
eq O}d_{t}x^{k}_{vtij}+\tau_ {vtij}\leq\sum_{v\in V}\sum_{t\in T}\sum_{(j,l)\in E^{vt}|i\
eq D}d_{t}x^{k}_{ vtij},\ \forall j\in L\setminus O\cup D,\ \forall y^{k}\in P, \tag{7}\]  
\[\sum_{j\in L}\sum_{v\in V}\sum_{t\in T}d_{t}x^{k}_{vtOj}\geq\alpha,\ \forall y^{k}\in P, \tag{8}\]  
Commuter queries might also include budget, latest arrival time, maximum transfers and unacceptable transport mode constraints. In this work, such constraints are to be treated as commut preferences, which they apply to filter journey options output from the proposed approach. Note that they could also be trivially addressed during the optimisation stage by pruning infeasible solutions as soon as they emerge.