## 3 Multi-Modal Multi-Objective Journey-Planning Optimisation Model  
The multi-modal multi-objective journey-planning problem has two main sets of inputs, a graph \(G=(N,E)\), which defines the nodes \(N\) (junctions) and edges \(E\) (connecting paths) of the non-public transport networks, and general transit feed specification (GTFS) data, which defines the public-transport network structure, including its routes and timetables. The graph \(G\) can be broken down further for each non-public transport mode, where \(G^{p}=(N^{p},E^{p})\) denotes the pedestrian transport network, \(G^{b}=\left(N^{b},E^{b}\right)\) denotes the bike transport network, \(G^{s}=(N^{s},E^{s})\) denotes the e-scooter network (usually the same as the bike network) and \(G^{c}=(N^{c},E^{c})\) denotes the road-vehicle transport network. All nodes, and public transport stops, have a longitude and latitude, which are used to identify the nodes/stops directly connected to the nodes/stops in every other transport network, which define the set of possible transport mode transfer points. Regarding the GTFS data framework, the public transport network is composed of a set of _routes_, which define the stop sequences of each route. Each route has a set of _trips_, which correspond to the set of times each route is repeated. Each trip has a set of _stop times_, which define the stop timetable of that trip. GTFS data includes _calendar_ data, which specifies the dates on which each route operates, _fare_ data, specifying the costs associated with possible sub-trips, and _shape_ data, specifying the exact paths of each route. Our objective is to return, for any given commuter query (origin \(O\), destination \(D\), departure time \(\alpha\)), the set of non-dominated journeys with respect to simultaneous minimisation of the objectives of (i) cost, (ii) travel time, (iii) CO\({}_{2}\) emissions, (iv) inconvenience (measured as the sum of walking and waiting time) and (v) calories used. Table 1 specifies the characteristics and cost structures associated with each transport mode. Travel times and waiting times of journeys are calculated from public transport timetable data and edge traversal times. CO\({}_{2}\) is calculated based on the distance travelled via each transport mode, while calories used are calculated based on the distances walked and cycled. While the number of transfers is, in principle, an element of inconvenience, in this work it is treated as a commuter input constraint and not an objective. The main reason for this is twofold: (i) a general investigation of multi-modal journey planning requires the consideration of transfers; (ii) transfers are the primary source of the problem's complexity that we are trying to address. More generally speaking, we avoid suppressing transfers directly for analysis purposes. We note that our objective function and corresponding solution techniques extend trivially to the case where transfers are an additional minimisation objective. For the purpose of explaining our objective function, let \(y^{k}\) denote a vector defining a single (\(k^{th}\)) multi-modal journey from the origin to the destination and let \(f_{i}\left(y^{k}\right)\) denote the objective value of journey \(y^{k}\) on objective criterion \(i\). We are interested in finding the optimal set of mutually non-dominated journeys with respect to criteria (i) to (v). Objective (1) specifies that we want to find a Pareto set of journeys \(P\) which simultaneously minimises all objective criteria. Constraints (2) and (3) define the mutually non-dominated solutions requirement of a Pareto set. Specifically, Constraint (2) specifies that the Pareto set must not contain any solutions dominated on all criteria by any other solution in the set. Constraint (3) specifies that each solution in the Pareto set must outperform at least one other non-dominated solution on at least one objective criteria. \[\min:f_{i}\left(y^{k}\right),\;\forall i\in F,\;\forall y^{k}\in P, \tag{1}\]  
s.t.