## 1 Introduction  
Mobility-as-a-service (MaaS) is concerned with personal transportation needs in metropolitan areas [1, 2]. It aims to make alternatives to private car journeys more easily accessible [3, 4, 5]. To do this, the offerings of mobility service providers (MSPs) are brought together within a single convenient platform, usually a mobile phone application, to provide integrated journey planning, booking, payment and e-ticket functionality. As outlined by [6], a multiple MSP journey planner provides the most basic functionality of any MaaS app, since booking, payment, ticketing, subscriptions, bundles and behaviour-nudging incentives can all be built on top of it. This work tackles a multi-modal multi-objective journey-planning problem involving four distinct non-private car classes of transport modes: (i) _public transport_, such as bus, train and ferry services; (ii) _taxi services_; (iii) _micro-mobility_ modes, such as hire bikes and e-scooters; (iv) _personal transport_, such as walking and cycling. The minimisation objectives considered (and their reasons for inclusion) include the following: (i) _cost_ (people prefer to pay less); (ii) _arrival time_ (people typically prefer faster journeys); (iii) _CO\({}_{2}\) emissions_ (environmentally conscious people prefer green journeys); (iv) _inconvience_ (measured as the total amount of time spent walking and waiting in a journey, which people typically prefer to avoid if possible); (v) _calorie expenditure_ (some people may like to minimise the required physical exertion of their journey); (vi) _transfers_ (people tend to prefer making fewer connections for convenience and reduced risk of missing connections). For any given commuter query, consisting of a departure time, origin and destination, this objective function characterises a Pareto set of journeys that are mutually non-dominated with respect to these objectives. Such a set of journeys includes the cheapest, fastest, greenest, most convenient and least strenuous. In addition, the Pareto set includes those which balance conflicting objectives. Such a set of journeys may include those attractive to private car users and prompt travel behaviour change. The problem we face is that of being able to generate a Pareto set of multi-modal journeys rapidly in response to commuter queries received through a MaaS app in real-time. A mobile MaaS application provides the means by which commuter queries are made and the query results displayed to commuters. The calculation of multi-modal routes takes place server-side, which permits the use of algorithms, such as the one presented in this work, that rely on pre-processing and on-going background update steps in order to aid a rapid response to commuter queries. In previous work [7], we introduced the multi-modal multi-objective journey-planning problem considered in this work. In contrast to other streams of research related to multi-modal journey planning, the problem considered here is more general. It considers a wider set of transport modes, including public transport, walking, taxis, bikes and also new shared micro-mobility modes such as rentable e-scooters. Previous works have tended to focus only on cars, walking and public transport [8; 9; 10; 11], while also placing constraints on the orders in which modes can be used in journeys, which is a solution-space-reduction approach that is avoided in this work. Ref. [8] proposed an access-node approach where the main parts of journeys are assigned to the public transport network, thereby reducing the size of the path-planning problems to and from the public transport access-nodes. Ref. [9] proposed the Round-Based Public Transit Routing (RAPTOR) algorithm, which is a dynamic programming-based approach for minimising both travel time and number of transfers via walking and public transport networks. Each iteration attempts to find faster journeys to each reachable stop involving one additional transfer. Ref.