Abstract
The size of the solution space associated with the tripmatching problem has made the search for highorder ridepooling prohibitive. We introduce hyperpooled rides along with a method to identify them within urban demand patterns. Travellers of hyperpooled rides walk to common pickup points, travel with a shared vehicle along a sequence of stops and are dropped off at stops from which they walk to their destinations. While closely resembling classical mass transit, hyperpooled rides are purely demanddriven, with itineraries (stop locations, sequences, timings) optimised for all cotravellers. For 2000 trips in Amsterdam the algorithm generated 40 hyperpooled rides transporting 225 travellers. They would require 52.5 vehicle hours to travel solo, whereas in the hyperpooled multistop rides, it is reduced sixfold to 9 vehicle hours only. This efficiency gain is made possible by achieving an average occupancy of 5.8 (and a maximum of 14) while remaining attractive for all cotravellers.
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Introduction
Ridepooling offers a promising alternative in the context of urban mobility, being an intermediate mode between private rides and mass transit. However, its full potential has arguably not been yet realised, neither in practice nor in theory. Consequently, the scalability limits of pooled services in urban mobility remain unknown. In this study, we explore ways to increase the efficiency of ridepooling systems, namely by means of increasing service occupancy and reducing vehicle mileage of ridepooling systems while remaining attractive for travellers.
Ridepooling companies primarily offer doortodoor services, picking up and dropping off travellers along partially shared routes. Unlike public transit services, common pickup and dropoff points, i.e. (virtual) stops, are rarely introduced. Furthermore, the majority of ridepooling algorithms seek to maximise operational efficiency by focusing on realtime operations while disregarding the attractiveness of the yielded services, thereby implicitly assuming that users are captive to the service. Passengeroriented solutions which focus on maximising users’ utility are scarce. Similarly, offline planning tools which could potentially unlock the full potential of bundling requests are limited.
We introduce a novel kind of pooled rides which is hereby denominated as hyperpool rides. Travellers of hyperpooled rides walk to common pickup points, travel by means of a shared vehicle along a sequence of stops and are dropped off at stops from which they walk to their destinations. To identify attractive hyperpooled rides, a sequence of bundling operations is conducted, resulting in moving successively from ‘private rides’ through ‘doortodoor pooled rides’ to arrive at ‘compact stoptostop rides’, and demonstrates the benefits of such rides in improving ridepooling efficiency. In particular, we are interested in rides which can be attractively composed using common pickup and dropoff points. These compact stoptostop pooled rides are instrumental in introducing the ultimate level of pooling. Subsequently, multiple such stoptostop pooled rides may be bundled so as to compose a hyperpooled ride, where travellers are picked up at common stop points, and the vehicle serves intermediate stops enroute. Our findings demonstrate that for such rides, occupancy increases substantially, improving thus the efficiency of pooled services and in some instances even starts resembling public transport operations while offering an ondemand service. Figure 1 illustrates ten selected hyperpooled rides in Amsterdam which were yielded by employing the proposed algorithm.
The proposed bottomup, hierarchical approach allows addressing the combinatorial increase of the associated search space consisting of all possible permutations of travel requests, which makes exact methods impractical. The central challenge to the ridepooling problem pertains to its computational complexity  the number of feasible groups of travellers grows combinatorially, and even for relatively small problems, the search space quickly exceeds reasonable processing capabilities. An exact matching algorithm for solving the ridesharing problem presented by Kucharski and Cats (2020) in^{1} is able to attain a sufficiently confined search space. This was achieved by (i) leveraging on user utility formulations so as to explicitly limit the consideration set to attractive pooling combinations only, and; (ii) exploiting properties of the socalled shareability graphs to effectively reduce the search space in the hierarchical searches. This study leverages the same principles to further explore higher degrees of pooled rides, and thereby devise ridepooling solutions which contribute to improved efficiency and sustainability objectives, i.e. by increasing the occupancy (reducing the vehicle mileage needed for transporting passengers). In the experiments reported by^{1}, doortodoor pooled rides of degree (number of cotravellers) greater than five, which already yielded a search space of size 10^{20}, were seldom obtained. Conversely, with the two new levels of compact pooling proposed here, rides composed of up to 14 travellers, which would have otherwise required handling a search space of 10^{57} size, are identified. Hence, an exhaustive search is rendered impossible (calculated as a number of 14element subsets of a 2000element set with every possible sequence of 14 origins and destinations: \((\begin{array}{c}2000\\ 14\end{array})14!14!\)).
The potential of ridepooling to improve urban mobility has triggered a bursting stream of research^{2,3,4,5,6}, which led to an algorithmic breakthrough in the seminal works of Santi et al. (2014)^{7} and AlonsoMora et al. (2017)^{8}. First, the socalled shareability network was introduced along with a methodology to match travellers in pairs^{7}. This network representation was then further exploited with a comprehensive algorithm to efficiently match incoming requests with available vehicles by sequentially adding new cotravellers into vehicles with empty seats and adjusting their routes^{8}. By applying cutoffs on maximal detours and delays, the pooled rides are acceptable for travellers and by minimising the service costs (vehicle hours) when solving the matching problem, the solution becomes optimal for the supply side (platform and/or drivers).
Recent developments in ridepooling research constitute a series of improvements to the original solutions, including: algorithmic—to reduce computational complexity^{9,10,11}; heuristic—to narrow problem search space^{12,13}; spatial decomposition—to reduce problem size^{14}; or demand prediction—to better anticipate future requests^{15,16}.
In parallel, behavioural research has made advancements in gaining a better understanding of users’ socalled willingness to share^{17,18,19,20}. The discomfort associated with sharing has been quantified and can be now included in classical utility formulas within the random utility modelling framework. Either added as an alternative specific constant (fixed penalty^{21}) or traveltime multiplier^{22}. Furthermore, the extent of actual detours and delays of pooling were estimated from empirical data^{23,24,25}, and the inherent variability of pooling travel and wait time has been incorporated into users’ choice models to capture the impacts of ridepooling reliability^{26}.
The ridepooling problem suffers from combinatorial growth of search space, yet when rides are restricted to mutually attractive ones only, the search space implodes for practical problems and can be searched exhaustively without having to use heuristics. This is made possible by reformulating the fixed timewindows of a maximal acceptable detour and delay into utilitybased compensation formulas as proposed and demonstrated in our previous study ‘Exact Matching of Attractive Shared Rides—ExMAS’^{1}, on which we build here. Such, utilitydriven approaches, not only put the service user in the centre but also significantly reduce the search space. Other related usercentric works include accounting for individual customers’ pooling benefits and comparing it to existing providercentred pooling mechanisms^{27}, focus on the sensitivity of the results to user preferences in terms of the level of service (time to be served and excess trip time)^{28}, or introduce willingness to share as a compensatory cost function at the individual passenger level to explicitly include the utility of pooled travellers^{29}.
Ridepooling can be considered as an intermediate mode of travel positioned between private ridehailing and public transport, yet in practice—for example, in terms of its occupancy level—it still remains much closer to the former. There is considerable evidence to suggest that ridehailing contributes to congestion^{30,31,32}. In contrast, a number of studies argue that ridepooling can potentially become complementary to public transport^{33,34,35}. The critical differences between currently offered pooled services and public transport pertain to fares and travel times. Pooled services are typically more convenient: doortodoor, hence involving no transfers; yet remain significantly more expensive.
One way to bridge this gap is the introduction of socalled pickup and dropoff points (PUDOs)  which introduce access/egress walk into ridepooling^{36}. Such services have been offered, for example, by Uber (UberExpresPool), MOIA and ViaVan (see a relevant market review by Foljanty (2021)^{37}). From the operator’s perspective, stops are introduced to minimise operating costs (5% vehiclekilometres reduction in Munich^{38}) within acceptable access/egress walking distances^{39}. Experiments with various PUDO spacing and density^{40} revealed an average ride occupancy increase by 20% and mileage decrease by 27%. Solutions to the stoplocation problem are inspired by pickup delivery locations^{41} or NPhard multimeetingpoint route search^{42}. Notably, in this study, we formulate the problem so that it can be searched exhaustively.
Research on ridepooling shares several key similarities with ondemand public transport (for a recent review of the latter, see ref. ^{43}) that are worth highlighting here. The mathematical complexity is similarly challenging, with similar supplyside and demandside constraints^{44}. In both public transport and ridepooling a critical mass is needed to take off^{45} and economies of scale share similarities^{46}. A virtuous cycle results from a positive feedback loop between service improvements, reduced travel times and increased demand, equivalent to the Mohring Effect^{47}, In the case of ridepooling this process is governed by the improved matching of users^{48}. At the same time, in the shortrun, increased demand leads to higher vehicle loads and longer detours (similar to crowding in PT). The key difference is the bottomup approach, while ondemand public transport typically departs from predefined fixed lines and timetables and dynamically adjusts them to the demand^{49}, here we start from the disaggregated demand and hierarchically build rides of increasing aggregation until we reach higher level occupancies.
Several recent studies have made initial steps in contributing toward bridging between these two approaches. Integrated public transport and flexible ondemand transport passenger assignment models^{50,51,52} developed using agentbased simulation models. Schluter et al. (2021)^{53} demonstrated that DRT systems are able to reduce the economic and environmental costs of transportation between rural and urban areas. For medium cities, demandresponsive services minimise detours, waiting times and walking distances experienced by passengers while allowing for a higher shareability and efficiency of the service^{54}. The potential efficiency gains that can be made by bundling travel demand in the wide spectrum, which goes beyond forming shared rides on the one hand and without yet having to rely on introducing flexible variants to predefined service lines remains largely unknown.
Hyperpool is an analytic, offline, utilitybased algorithm. For a given disaggregated travel demand provided in the form of individual trip requests known in advance, attractive pooled rides of increasing levels of aggregation (as illustrated in Fig. 2) are successively identified. The main objective is to increase service occupancy levels while remaining attractive to users. The algorithm relies on explicit utility formulas that tradeoff any detour, delay and discomfort induced by successive kinds of pooling against the discount offered for sharing the ride. The utility formulas are userspecific and assess the attractiveness of four pooling levels. Utility is calculated for each travellerride pair taking based on userspecific behavioural parameters (e.g. value of time, willingness to share), ridespecific attributes (ride, waiting and walking times) and service settings (pricing).
First, all feasible doortodoor ridepooling rides, which can be composed for the input demand pattern provided in the form of individual trip requests are identified using ExMAS. In the next stage, these doortodoor ridepooling rides are searched to identify candidates for compact stoptostop rides. In particular, the optimal pickup point, departure time and dropoff point for each ride are identified to then assess whether the additional discounts compensate for the discomfort of walking for all cotravellers. Importantly, we note that those mutually attractive compact stoptostop rides have a structure similar to a single trip request. Thanks to this resemblance, a set of such rides can be subject to a new bundling operation. These rides are then provided as an input to ExMAS and with a few minor modifications can be pooled into the desired hyperpool rides. In the last, crucial step, ExMAS finds pairs, triples and quadruples of stoptostop pooled rides and bundles them together. This seconddegree pooling (pooling rides which have already been pooled) is critical to leap beyond the current limits of ridepooling systems. The hierarchical, utilitybased approach guarantees that all identified rides are attractive for all cotravellers (i.e. their perceived generalised cost does not exceed that of private ridehailing). Notwithstanding, we do not claim for completeness, i.e. not all feasible rides (stoptostop and hyperpooled) are identified using our method. The proposed method can be, therefore, considered a form of heuristic search in the very large solution space of highdegree pooling.
By following the abovementioned stepwise approach, one obtains the complete set of rides from which, by solving the bipartite matching (classical for ridepooling), the final solution is obtained. In the solution, each of the travellers is assigned to a ride, which may be private, pooled or hyperpooled, depending now not only on individual preferences but also on global optimality considerations (mileage reductions). Key performance indicators (KPIs) of interest at the system, user and ride levels can be then analysed. In the results, we report: travel times, walk times, fares, passenger utilities (decomposed into trip fare, invehicle travel time and walk times to access and egress to/from stop points) and their relative changes across pooling options, ride occupancy and ride fareefficiency (vehicle travel time divided by the total fare).
The proposed hyperpool method is replicable and reproducible using a publicly available source code and parameterised through a set of design variables (e.g. prices and discounts) and behavioural parameters (value of time, willingness to share, etc.). The hyperpool algorithm was tested for demand levels of up to 8000 trip requests per hour and maximal batches of 2000 travellers. Such levels are large enough for many realworld problems, and in the following, we demonstrate that they induce a critical mass of demand level, beyond which attractive hyperpooled rides emerge.
Results
The hyperpool method is applied to pool a batch of 2000 trips requested within half an hour in Amsterdam, The Netherlands. In the following, we first showcase how 10 travellers are hyperpooled together into a single multistop shared ride, followed by a systemwide impact analysis of hyperpooled rides for the Amsterdam case study.
Experimental setting
We sample 2000 Amsterdam PM peak trips from an updated version of the Albatross dataset^{55}. This demand is considered inelastic (no induced demand), i.e. travellers can only choose from the ridehailing and ridepooling alternatives offered. The actual Amsterdam fare for private ridehailing rides of 1.5€/km is used and the following discounts are assumed: 25% for doortodoor ridepooling, 66% for stoptostop and 75% for multistop pooling, i.e. the perkilometre fares of 1.5, 1.11, 0.5 and 0.375 €/km, respectively (corresponding to respective λ’s in the Methods section). Public transport fare in Amsterdam is ca. 0.3 €/km, hence only 25% cheaper than the proposed hyperpool rides. The invehicle time multiplier for pooling is set to \({\beta }_{d2d}^{t}=1.3\), and multipliers of 1.5 and 1.2 are specified for walking (β^{d}) and waiting (β^{d}) times, respectively. Travellers’ valueoftime is normally distributed around 12€/h with a standard deviation of σ = 1.5€/km to represent travellers’ heterogeneity. Stop locations are searched over all nodes of the road network (we did not limit searches to PT stops) within a 7.5min walk time radius centred around origins and destinations (\({t}_{w}^{max}\)). Fixed and networkwide flat speeds of 1.5 m/s for walking and 8 m/s (28.8 km/h) for vehicle movements (which can be easily refined with space and timevarying speeds) are specified. The detailed OSM graph of Amsterdam is used to obtain travel times and distances^{56}. For public transport connection attributes, the actual GTFS public transport timetable and Open Trip Planner query engine are used (following the method described in ref. ^{34}).
Hyperpool showcase
To showcase the potential of hyperpooling, we select ten trip requests from the Amsterdam PM peak travel demand, which the proposed algorithm managed to bundle into a single hyperpooled ride. Figure 3 shows pooling levels resulting from consecutive bundling steps starting from private rides (a) through doortodoor pooled rides (b) and stoptostop rides (c) to ultimately yield a single hyperpooled ride (d). The latter results in ten requests with a total length of 65 km being served by a single 9.6 km multistop ride. If these requests were served by private rides, the latter would have resulted in 2.25 vehiclehours, compared to only 0.44 with the generated hyperpool ride (as detailed in Table 1). The total passengerwalking time is 78 min and the total passengerinvehicle time is 157 min. Remarkably, the total disutility for each of the travellers bundled in this ride is lower than for the alternative private ride (as our method guarantees that only travellers who willingly join a shared ride based on their utility considerations are mutually compatible). Moreover, for some travellers, the hyperpooled ride is more attractive than the public transport (PT) alternative (depending also on their valueoftime). While offering an efficient bundling of travel requests, the total invehicle travel times are still 50% of those associated with PT and the total walking time is 30% shorter. In monetary terms, the total disutility drops from 127 with private rides to 95€for the hyperpool, which is also considerably lower than for PT (112€). Notably, the total collected fare drops from €97 for private rides to only 24. This reduction in revenues is partially compensated by the cost reductions resulting from fewer vehicle hours. In addition, service attractiveness may result in induced demand which in turn would enlarge the overall revenues. Furthermore, service contribution to reducing externalities may justify subsidies. Note that the hyperpool solution obtained by the proposed algorithm is identified from the gigantic search space of 3.86 × 10^{40} combinations, otherwise practically impossible without employing the proposed hierarchical method.
Half an hour hyperpooling in Amsterdam
To demonstrate the scalability of the proposed approach and its systemwide effects, it is next applied and investigated for a travel demand consisting of 2000 passenger trips requested within a 30min batch (4000 trips/hour) in Amsterdam. The impact of consecutive levels of pooling is analysed starting with private rides, then adding doortodoor rides with ExMAS, to thereafter identify stoptostop rides and finally generate hyperpooled rides. At each of these steps, the matching problem (detailed in the Methods section) is solved and the impact of the newly added rides on system performance is analysed.
The resulting rides’ compositions obtained at respective pooling levels are shown in Table 2. Each newly introduced level of pooling attracts a significant share of travellers. Doortodoor pooling is attractive for over 1300 travellers, and 171 of them can be successfully bundled into stoptostop rides. Eventually, over 225 travellers are successfully matched into attractive hyperpooled rides. Notably, introducing new levels of pooling does not affect the number of private rides in the solution, which remains stable at around 30% of the travellers who are not matched with any other cotraveller. The impact of the newly introduced pooled rides is thus mainly on bundling more efficiently travellers who are already pooling. The number of feasible pooled rides (last column) identified by ExMAS exceeds 150,000. This number increases only slightly for stoptostop rides but then grows substantially into over a million feasible hyperpooled rides.
Importantly, the demand of 225 travellers is successfully supplied using only 40 hyperpooled rides composed, on average, of 5.6 travellers. If each one of the 225 travellers had travelled alone, they would require 52.5 vehiclehours. This figure is reduced by the hyperpooled multistop rides sixfold to 9 vehicle hours only, allowing to attain an average occupancy of 5.8. If we treat vehiclehours reduction as a proxy for cost reductions (emission, congestion and other externalities), this may allow discounting of the fare by the assumed rate.
In general, introducing new levels of pooling improves all the KPIs of interest (Table 3). Vehicle hours are reduced, traveller’s costs (disutility) are lower, invehicle hours are reduced, and occupancy rates rise. Vehicle hours are reduced from 274 h for private rides to 171 for the solution obtained when offering all pooling services. The generalised total travel times of passengers, i.e. increase in disutility, is reduced from 7.56€to 6.65€ per passenger. Invehicle hours are lowest for private rides, then increase for doortodoor pooling, drop when stoptostop rides are introduced and drop again when hyperpooling is introduced. This reduction is achieved by inducing modest average walking times. The mean fare collected by the service provider drops from 5.92€to 4.02€per passenger. The average occupancy of pooling increases only slightly when new pooling services are introduced (from 1.53 for doortodoor pooling to 1.60 for hyperpooled). The greatest efficiency gain (fares per vehicle hours) is yielded for doortodoor pooling, that is, introducing hyperpooling is not directly attractive for the service provider, whose efficiency drops from 50.8 to 47 €per vehicle hour, yet remains at a significantly higher level when compared to the private ridehailing solution (47 versus 43€/vehkm).
A more detailed analysis of individual trips is presented in Figs. 4–6. This series of panels (distinguishing between rides/travellers levels) shows how introducing increasing levels of pooling impacts the solution. The different services within each solution are depicted using distinct colours.
First, we investigate how ride occupancy varies as a function of trip length in Fig. 4. One can see in the last panel the massive impact of hyperpooled rides, reaching, to the best of our knowledge uncommon levels. The stoptostop rides (introduced in panel c) are not the main contributor to system efficiency, but rather serve as a mean to an end, i.e. being instrumental for composing hyperpooled rides in the next step. The compact stoptostop rides have occupancy comparable to doortodoor rides, but are typically shorter. The following trend can be observed twice: first: longer private trips (blue) are more often pooled into doortodoor rides (green)—compare panel a with b; second: longer doortodoor rides (orange) are more often pooled into hyperpooled rides (red)—compare panel c with d. Attractive stoptostop rides are typically short (presumably due to constrains of finding the optimal stop locations). However, this is not the case for hypepooled rides, which can be both short and long. In particular, most private rides (blue) are for short distances, where it is hard to identify the grouping that compensates for discomfort.
Figure 5 shows how attractive are pooled rides of various types for travellers. It can be observed how travellers’ utility improves relative to a private ride. Here, each dot denotes one of our 2000 travellers and the xaxis distinguishes the travellers’ based on their respective valueoftime (β^{t}). The grey background denotes the public transport alternative, which spans from nonattractive to highly competitive versus private rides. At first a clear pattern can be observed with a lower bound of attractiveness for doortodoor pooling (b)—the relative gains of pooling are greater for travellers with lower valueoftime. The impact of stoptostop pooling (c) on travellers’ utility is negligible. Stoptostop rides are bounded within the relative attractiveness of doortodoor ridepooling. Hyperpooling, however, breaks this trend and sharply reduces (dis)utilities, which can now be halved. While the main trend is still visible, it is not as evident anymore— effective hyperpooled rides are generated and selected also by travellers with a relatively high valueoftime as well. Similarly to the impact on occupancy, the stoptostop rides are not attractive per se, despite being offered for a significant discount, i.e. 1.11€/km for a doortodoor ride is reduced in this experiment to 0.5€/km). This discount is apparently not sufficient to compensate for walking induced (which has a high perceived discomfort), yet the true benefits materialise once hyperpooled services are offered. Notably, hyperpooled rides are inclusive for both high and low valueoftime travellers. This is a surprising finding, since such services were expected to be attractive mainly to travellers with low valueoftime.
Next, Fig. 6 focuses on the perspective of the service provider. Obviously, under fixed demand, offering discounts always yields lower total revenues. Discounts are offered on the premise that ridepooling allows a potential cost reduction. To further investigate it, the total fares paid are divided by vehicle hours, which can be treated as a proxy of operating costs. Due to flat perkilometre fares and fixed network speeds, offering private rides (panel a) always yields 44 €/vehhour (xaxis). On the contrary, this varies significantly for doortodoor rides (panel b). While for a small number of rides, doortodoor pooled rides are less efficient, for the majority, effectiveness increases and reaches up to 200 €/vehhour (e.g. when five highly compatible requests are pooled together). The introduction of stoptostop rides (panel c) proves to be ineffective and the discount is not compensated for by the reduced mileage. Again, similarly to the two previous figures, this trend is strongly reversed with hyperpooled rides, with the majority of the mutually attractive rides generated being also more beneficial for the service provider than private rides. Although some of the identified hyperpooled rides may be less profitable for operators than the private rides, the majority are more effective. This suggests that under such settings, subsidies for the service provider may not be necessary. These disaggregated results can be compared with the fare per vehicle hour reported in Table 3. Introducing hyperpooling reduces effectiveness compared to doortodoor rides (from 50 to 47), yet it remains more costeffective when compared to private rides only (47 to 43).
To understand how both the service design variables and the exogenous behaviour parameters can impact the results, we perform the following sensitivity analysis. We investigate the hyperpooling results under various price settings and travellers’ value of time. For the 225 travellers in the hyperpooled rides generated for the 2000 trips in Amsterdam, we test whether the proposed alternative remains attractive. The results are presented in Fig. 7, where attractiveness is compared to private rides (blue) and PT (green). We decrease and increase the perkilometre fare of hyperpooling (\({\lambda }_{{\mathbb{h}}}\), in the first two panels) and the value of time (β^{t}, in the last two panels). First, we show how it impacts individual travellers (first and third panels) and then if all members comprising hyperpooled rides still find it attractive (second and fourth panels).
If we increase the hyperpooling price from 0.375 to 0.5€/km, only 30% of the 225 travellers who found hyperpooling attractive will optout. However, for 70% of the hyperpooled rides, at least one traveller becomes unsatisfied following this price increase, resulting in its disintegration. When a default value of time (β^{t} = 12€/h) is used, hyperpooling is attractive in comparison to PT for 80% of the travellers. This figure increases to almost 100% when the value of time drops to 8€/h. On the other hand, with a value of time of 16€/h only 40% of hyperpooled rides are attractive when compared to private alternative and 20% visavis public transport.
Discussion
The proposed hyperpool method yields insofar unattainable high occupancy levels while ensuring that those are mutually attractive ridepooling trips. Up to 14 travellers are pooled together in rides where the discomfort induced by pooling is compensated by reduced fares. The proposed approach hence offers great potential for travellers (whose disutility of travelling can be reduced), for policymakers (contributing towards sustainability goals by means of reducing vehiclekilometres) and for service providers (improving service efficiency levels). The main contribution pertains to the generation of hyperpooled rides as they break the current limits of classical ridepooling. The intermediate service of stoptostop rides is instrumental in attaining attractive hyperpooled at the subsequent step, yet their own attraction value is found to be limited.
Despite only 11% of the travellers choosing to travel by means of hyperpooled rides in the Amsterdambased experiment, the impact of such rides on overall pooling levels is considerable. Obviously, the critical mass in the demand levels is greater than for classical pooling, i.e. more travellers are needed to identify attractive rides of a high degree. Yet a demand level of 4000 trip requests per hour already allows for the identification of hyperpool rides for which 10% of the travellers willingly opt. The number of travellers making use of hyperpool rides may rise due to induced demand and modal shifts, thereby potentially leading to the emergence of a virtuous cycle. It should be noted that for some of the new trip requests considered in the case study, hyperpooled rides were even more attractive than the public transport alternative. This is particularly notable since the quality of PT in Amsterdam is considered high^{34}, thereby making it difficult for ondemand services to compete.
In our experiments, we set the fare levels of hyperpool services to be similar to those of public transit (0.3€/km for PT and 0.375€/km for hyperpooling), expecting such services to be highly attractive for travellers, yet not costefficient for the service provider. Surprisingly, the significant mileage reductions and high occupancy levels that were attained by the hyperpooling solutions effectively compensated for the reduced fees. The costeffectiveness of hyperpooled rides exceeds the one of private ridehailing; however, the cost of larger vehicles needs to be considered. The hyperpooled rides were found to be selected by travellers with a variety of valueoftime, demonstrating that the obtained hyperpooled solutions managed to cater for diverse travellers and needs by offering an affordable alternative while also effectively compensating travellers with greater values of time.
By employing behavioural constraints and capitalising on the hierarchical nature of the problem, the proposed hyperpool algorithm is able to drastically reduce the size of the search space. However, this also implies that the algorithm is not exhaustive since we gradually increase the discount with the newly introduced pooling levels. Furthermore, the solution is generated offline. Transferring the proposed method and findings to a realtime setting remains an open challenge. In developing the proposed hyperpool approach, the travellers’ perspective is deliberately adopted, treating the fleet operations as exogenous. This is deemed reasonable, since the focus here is on the potential of a given demand to be pooled while assuming that the available fleet is sufficiently large. In other words, it is assumed that the fleet size is such that there are vehicles available to serve the pooled rides without additional delays beyond those caused by the compromise among the cotravellers to minimise joint departure delays. Another consequence of the adopted perspective is that stops are set so as to minimise the walking time for the respective group of cotravellers. Even a minor shift in stop location and departure time may significantly shorten the overall vehicle route and avoid detours^{39} (or, in other words, shift the user equilibrium solution towards the system optimum one). This can improve the solution and further reduce mileage. On a related note, vehiclehours as reported in this study (mileage) only constitute a rough approximation of the actual costs. First, deadheading and repositioning are not considered, focusing strictly on the distance travelled with a passenger onboard only. Second, no considerations are made regarding the potential deployment of different vehicle types. Rides composed of more than three travellers require special kinds of vehicles, resembling minibuses—presumably more expensive to operate, which may be taken into consideration in future research.
We believe that the proposed approach is not only beneficial for improving the efficiency of ridepooling but may also inspire the development of novel solutions to the transit network design problem, which is commonly solved by means of (meta)heuristics (see ref. ^{57}). While the resulting hyperpool rides do not correspond to public transport lines (i.e. recurrent service configuration patterns), they may be used as spatial patterns driving the network design solutions towards a new optima. Broader experiments with larger travel demand sets may reveal recurrent patterns, and clustering methods may be used to spatially and temporally aggregate hyperpooled rides into potential service lines.
An additional direction for further research pertains to the development of personalised pricing strategies and related revenue management techniques where discounts are determined for each travellerride pair. Furthermore, whether discounts offered for hyperpool services can also be made attractive for the system provider remains an open question. Traditionally, ridepooling services are considered a commercial product. However, arguably, once they reach occupancy levels comparable to transit, one may consider treating them as public services, e.g. subject to tendering and subsidies. Related policy decisions depend on the extent to which these services contribute to sustainability goals and their added value to urban accessibility, which largely depends on whether they compete with or complement public transport^{34} and active modes of transport.
Methods
The proposed method revolves around the Exact Matching of Attractive Shared rides (ExMAS) algorithm^{1}. Doortodoor attractive pooled rides computed with ExMAS are compressed into attractive stoptostop rides. Subsequently, identified attractive stoptostop pooled rides are themselves pooled once again using a modified version of the ExMAS algorithm, as detailed below.
Travel demand data
The proposed method computes the hyperpooling for a given demand pattern known in advance Q = {Q_{1}, Q_{2}, …, Q_{n}}, with a single trip i defined in terms of its origin, destination and departure time:
The origins and destinations are the nodes N in the road network graph (o_{i}, d_{i} ∈ N). We propose an offline method, suitable for processing travel demand data that are readily and publicly available for multiple cities, e.g. New York^{58} or Chicago^{59}.
Doortodoor ridepooling with ExMAS
We start by computing attractive pooled doortodoor rides using our ExMAS algorithm. (Full methodological details of the ExMAS algorithm are available in ref. ^{1} while its software implementation is publicly available at https://github.com/RafalKucharskiPK/ExMAS. Here we provide only a brief introduction). ExMAS first identifies the pairwise shareable trip requests and gradually exploits the search space of pooled rides of increasing degree (number of travellers). First, the algorithm aims at extending identified pairwise shareable trip request pairs into triplets, then extending triplets into quadruples, and so forth. ExMAS exhaustively exploits the search space of attractive pooled rides and terminates when no ride can be extended further. The feasible pooled rides resulting from ExMAS serve as input for the new pooling services proposed here.
In ExMAS, pooled rides are considered attractive if the pooledride utility is greater than the utility associated with the alternative private ride for all respective cotravellers. In a doortodoor private (nonshared) ride, the utility (U_{p}) is composed of a direct travel time (t_{p}) weighted by β^{t} (corresponding to the valueoftime) and a distancebased fare (λ_{p}l):
To identify attractive pooling, the above utility term is compared with the doortodoor pooled ride utility (U_{d2d}):
For a traveller to select the pooled alternative, the lower distancebased fare (λ_{d2d} < λ_{p}) needs to at least compensate for the longer travel time (t_{d2d} ⩾ t_{p}), delay induced by pooling (d_{d2d} > 0) and additional discomfort (expressed as \({\beta }_{d2d}^{t}\, > 1\) to represent the socalled willingnesstoshare). Mind that the utility here is, in fact, a disutility and is negative, due to the negative sign in λ’s and negative values of β^{t}. Thus, the computed utility values correspond to generalised travel costs. The delay in offline ridepooling is understood as the absolute difference between the desired and actual departure time (as detailed in ref. ^{1}). The experiments performed in this study examine the deterministic part of the utility function (ε = 0). To account for heterogeneity among travellers, i.e. taste variations, travellerspecific valueoftime \({\beta }_{i}^{t}\) drawn from a normal distribution are incorporated into the utility calculation.
ExMAS outputs the set of attractive stoptostop rides R_{d2d} = {r_{d2d}}, with a ride defined generically as:
where Q_{r} is a set of served trips, O_{r} and D_{r} are the ordered sequences of served trips’ origins and destinations, respectively, and \({t}_{r}^{p}\) denotes traveller’s desired departure time.
Note that ExMAS generates socalled FIFO trips only, i.e. where the sequence of pickups precedes the sequence of dropoffs.
Optimal pickup point for stoptostop pooled rides
Next, we identify which doortodoor pooled rides generated by ExMAS can be served using common pickup and dropoff points. For each pooled ride, the algorithm examines whether travellers can walk to a single common pickup point and arrive at a common dropoff point while yielding a superior travel alternative, using the following method.
The following utility formula is used for assessing the attractiveness of a stoptostop pooled ride, where travellers walk to a pickup point, travel directly to the dropoff point, and walk to their destination:
Similarly to ExMAS, only attractive stoptostop pooled rides are considered, i.e. such that U_{s2s} ⩾ U_{d2d}, for each cotraveller i ∈ Q_{r}. This means that the reduced price (λ_{s2s} < λ_{d2d}) and possibly shorter travel time (t_{s2s,i} ⩽ t_{d2d,i}) must at least compensate for the induced walking discomfort (β^{w}w_{s2s,i}). Mind that no additional invehicle discomfort (\({\beta }_{s2s}^{t}\,\approx\, {\beta }_{d2d}^{t}\)) is applied, since the ride is anyhow pooled. The discomfort associated with walking time (β^{w} > 1) is explicitly accounted for. Moreover, invehicle travel time is likely to become shorter since the number of pickup and dropoff locations is reduced, and their locations are determined so as to reduce detours. This effect gives an additional slack, which makes it easier to compensate for the negative impact of walking.
The yielded stoptostop pooled rides are denoted by r_{s2s} and are defined by means of a set of served travellers (Q_{r}) and a triplet of origin (pickup), destination (dropoff) and departure time:
The following procedure identifies the optimal pickup and dropoff locations and respective departure times for a given stoptostop ride. Instead of finding the optimal triplet (in threedimensional search space) decomposes the problem into three independent steps solved sequentially. The algorithm first loops over common pickup points, then determines the optimal departure time, and finally loops over common dropoff points. This exhaustive search did not result in a computational bottleneck since the search space solutions are sufficiently small and can be easily performed in parallel. This approach was undertaken rather than applying an optimisation programme for a nonlinear objective function on a multidimensional discrete search space, which is considered out of scope for this study:

1.
Identifying access points (i.e. virtual stops) accessible from all the origins within a fixed walking time threshold \({t}_{w}^{max}\), either based on the full network graph or only among a predetermined set of locations (e.g. existing public transport stops).

2.
For each candidate pickup point, determine the optimal departure time. To this end, the delay experienced by each traveller is calculated as the absolute difference between the desired departure time t_{i} and the departure time to walk to the pickup point (t_{w}(o_{i}, o)) and reach it at the time of common departure t:
$${d}_{s2s,i,o,t}= {t}_{i}t{t}_{w}({o}_{i},o)$$(7)Subsequently, the optimal departure time t from stop point o is determined as the minimum over the squared sum of delays:
$$t= \mathop{\arg\min}\limits_{t\in {\mathbb{R}}}\sum\limits_{i\in r}{({d}_{s2s,i,o,t})}^{2}$$(8)This step is independent of the dropoff point d and it is thus possible to execute it once for each origin to thereafter use it for all candidate destinations in the next step.

3.
Finally, all potential dropoff locations from which destinations are reachable within a fixed walking time threshold \({t}_{w}^{max}\) are explored.
We can now identify the optimal ride by tracing how the total utility changes for a given triplet (pickup location, departure time, and dropoff location). Note that most of the utility components change as a function of the triplet specifications: the pickup location and departure time determine access walk time and delays, respectively, and the dropoff location determines the egress walk time. In addition, the combination of pickup and dropoff locations determines invehicle time changes. Similarly to eq. (8), to avoid utility imbalances between cotravellers, the socalled log sum formula is applied as follows:
Since the goal is to identify attractive rides, we further consider only stoptostop pooled rides, which are found to be more attractive than their doortodoor pooled service counterparts for all respective cotravellers:
This can be further relaxed to identify stoptostop rides which are more attractive than private rides: U_{s2s} ⩾ U_{p}  yet we did not exploit this direction in the illustrative experiment. The outcome of this step is used as input to the subsequent hyperpool algorithm.
Pooling stoptostop rides to the hyperpooled rides
Stoptostop pooled rides are, just like private rides, defined using a triplet: origin, destination and departure time (compare eqs. (1) and (9)). They can, therefore, be treated as input to the ExMAS algorithm and thereby bundle pooled rides again into socalled hyperpooled rides. In short, we aim to pool stoptostop rides, which are already in themselves composed of several rides bundled by means of single pickup and dropoff locations).
To this end, the utility of a stoptostop pooled ride is expressed, in a fashion similar to a private ride in eq. (2), using the following formula:
Here, λ_{s2s}l is the fare paid by the travellers (fares are equal among the cotravellers in the stoptostop pooled ride, since the travel distance is identical for all fellow travellers) and t_{s2s} is the direct invehicle time (between the pickup location and the dropoff location). The previously identified pickup and dropoff locations are hereby treated as the origin o and destination d of such a pooled ride, respectively. Similarly, the optimal departure time computed earlier is used as the desired departure t for all the respective pooled travellers. The maximal valueoftime β^{t} among cotravellers is employed in subsequent calculations since it serves as an upper bound in reducing the available slack for pooling while ensuring that the hyperpooled ride is attractive for the most demanding cotraveller included in the pool.
Next, the algorithm identifies which of such rides may be pooled again so as to form hyperpooled rides \({\mathbb{h}}\). A formula similar to eq. (3) for doortodoor pooled rides is used:
An additional discount is applied (\({\lambda }_{{\mathbb{h}}} \,<\, {\lambda }_{s2s}\)) as well as a penalty (\({\beta }_{{\mathbb{h}}}^{t}\)) which is lower than (\({\beta }_{d2d}^{t}\)), since travellers are already pooled.
Similarly to ExMAS, the hyperpool algorithm first identifies pairwise shareable rides and matches them together. From such pairs, a new shareability graph is constructed, in which we explore pooled rides of a gradually increasing degree. Note that each stoptostop ride is composed of at least two individual trip requests; thus, any pair of shareable stoptostop rides pool together at least four individual trip requests.
Several adjustments to the original ExMAS algorithm have to be made when applied to the stoptostop pooled rides. First, while single trips are pairwise independent, stoptostop rides are composed of individual trips which may be overlapping. For instance, a stoptostop ride composed of trips {1, 2} cannot be pooled with another stoptostop ride composed of trips {2, 3}  since traveller 2 cannot be pooled with him or herself. We, therefore, add a constraint: pooling candidates need to be mutually exclusive (\({{\bf{Q}}}_{r}\cap {\bf{Q}}^{\prime} =\varnothing\)). Second, it is essential to ensure that each ride remains attractive for all cotravellers included in the bundle, i.e. \({U}_{{\mathbb{h}},i} \,>\, {U}_{p,i}\), for each i ∈ Q_{r}. It should be noted that this condition is tested at the stoptostop ride level only, which does not guarantee attractiveness at the individual level (for each single trip request). While for some cotravellers, stoptostop pooling is very attractive and close to both origin and destination, for others, it may lie at the limit of attractiveness. Similarly, ExMAS outputs the delay relative to the desired departure time, which for the hyperpooled rides becomes an optimal departure time computed with (eq. (8)), thus the individual delay evaluated with eq. (7) needs to be updated. For hyperpooling travellers, similar to stoptostop rides, the delay is the absolute difference between the actual and desired departure from the origin to arrive at the pickup point in time to enter the shared vehicle. For some travellers, shifting the departure due to hyperpooling is positive (moving closer to the desired departure), while for others, it is negative. Based on a series of experiments, it can be concluded that using aggregated utility formulas of eq. (11) and eq. (12) well approximates the individual utilities encountered by cotravellers, and only a small share of identified hyperpool rides does not meet the attractiveness criteria (ca. 5% travellers), which can be easily postprocessed (as we do in the experiments).
Employing the above adjustments, the original ExMAS algorithm successfully pools stoptostop rides into hyperpool rides. The output of the original ExMAS \({{\bf{R}}}_{{\mathbb{h}}}\) is now enriched with information concerning the access and egress times of individual travellers:
Q_{s2s} denotes a set of stoptostop rides pooled together, O_{r} and O_{r} are the sequences of pickup and dropoff locations, respectively. Access and egress walk times of individual travellers can be obtained from their stoptostop rides, while the departure from the first stop \({t}_{r}^{p}\) and the route along the intermediate stops O_{r} and O_{r} allows calculating their invehicle travel times and delays.
Solution
Applying the previous steps results in a rich set of feasible rides of various kinds, within which it is now possible to search for the optimal solution. Since typically, each traveller has multiple travel (at the very least a private ride is available and typically multiple pooled rides of various kinds are feasible), it is necessary to find the ride that will be performed by each traveller. The global choice set is now composed of rides of four kinds:
To address this, a socalled tripride coverage problem is formulated as an assignment problem. Hereby, each trip i is unilaterally matched with a ride r ∈ R. The typical formulation^{1,8} where the assignment is formulated as the problem of identifying a binary vector x_{r} of length equal to the number of rides ∥R∥ is adopted, i.e. an assignment variable which takes the value one if a ride is selected and zero otherwise (second constrain of eq. (15)). The costs C_{R} are then expressed as the cost of each ride c_{r}, multiplied by the assignment variable x_{r} and aggregated for all rides (eq. (15)).
The assignment must assign each trip request to exactly one ride, obtained through the rowwise sum of assignment variable x_{r} in the tripride incident matrix I_{i,r}. The latter is a binary matrix, in which each entry is one if ride r serves trip i and zero otherwise (first constrain of eq. (15)). Eventually, the solution to the problem (eq. (15)) is the set of rides R^{*} ⊆ R such that x_{r} = 1 ∀ r ∈ R^{*}. The matching problem is formulated using the following binary programme:
Typically, two perspectives on cost are considered: the one taken by the service provider (minimising the sum of vehicle hours) and the one taken by travellers (maximising the sum of utilities U_{r,i}). Despite taking a strictly demandorientated stance in this study, operator’s costs, i.e. vehicle hours, are minimised in solving this matching problem. This choice is made since the algorithm has already ensured that all the pooled rides in the search space are attractive for all travellers, thus, when optimising for the service operator, one does not run the risk of including unattractive rides in the solution. Moreover, we are explicitly interested in identifying the potential for highoccupation pooled rides, i.e. those for which vehicle hours are minimal.
Optionally, one may filter the solution R in eq. (15) to consist only of a subset of services. For instance, in the experiments performed in this study, the impact of successively introducing new kinds of services on systemwide KPIs is investigated.
Implementation
The hyperpool algorithm solves a service planning problem in an acceptable time. It is parameterised via a custom configuration file and can be easily deployed on external computational servers. The opensource Python code is publicly available, allowing for further developments and experiments. The pseudocode algorithm is presented in Algorithm 1 below. It starts with trip requests as input data to travel demand and gradually searches for attractive rides of four kinds, finally matching travellers with rides by solving the coverage problem.
The hyperpool algorithm outputs contain information concerning the ride serving each trip request (traveller). Rides in the solution can be of four kinds: private doortodoorride (for those who couldn’t be successfully matched with anyone), pooled doortodoor rides, pooled stoptostop rides or multistop rides. The fare paid by each traveller depends on the distance travelled and the type of service. Other components of travellers’ utility: detour, delay, and walk time are stored in the solution for further analysis.
Algorithm 1
Hyperpool  pseudocode for the algorithm
Despite the radical reduction in the search space size, it may (similarly to the ExMAS algorithm) still become prohibitive under certain settings (beyond some demand levels and discount rates). Nevertheless, 10minute batches with up to 2000 trip requests are computed within less than two hours on a standard laptop (MacBook Pro 2021 M1 chip, 16GB RAM), sufficient for offline strategic analyses of most practical realworld ridepooling problems. Furthermore, the stoptostop rides algorithm can be solved in parallel. The bottleneck of the current implementation seems to be in the second run of ExMAS when stoptostop rides are bundled into hyperpooled rides. Since at this step, sharing is not penalised by applying yet another multiplier (\({\beta }_{{\mathbb{h}}}^{t}\)) and with a considerable discount being offered, one often ends up with a very large number of identified hyperpooled rides (over a million in our halfhour batch, which includes only 250 out of 2000 individual trip requests).
Key performance indicators
The following set of key performance indicators is generated to measure the performance of the obtained hyperpool rides:

vehicle hours, i.e. total time spent by vehicles to serve the demand (empty kilometres, like deadheading and repositioning, are not included here)

passenger utility, i.e. total (dis)utility of travellers, both total as well as decomposed into:

trip fare (direct monetary costs)

invehicle travel time

walk time (to access and egress to/from stop points)


fareefficiency, i.e. vehicle travel time divided by the total fare. Expressed in €per vehiclekilometre and used as a proxy of service efficiency for the provider.

and ride occupancy:
$$\left(\sum\limits_{i\in {\bf{Q}}}{t}_{p}\right)\Bigg/\left(\sum\limits_{r\in {{\bf{R}}}^{* }}{t}_{r}\right)$$(16)The above formulation of the ’timeaveraged effective occupancy’ is arguably the most meaningful KPI. While in public transit, the occupancy is calculated for the actual passengerhours, the shortestpath passengerhours are specified here, i.e. the total passengerhours for the private rides \(\left({\sum }_{i\in {\bf{Q}}}{t}_{p}\right)\) as the nominator. In the denominator, the proposed metric takes the actual vehicle hours of the selected rides \(\left({\sum }_{r\in {{\bf{R}}}^{* }}{t}_{r}\right)\). We propose this formulation for the rideoccupancy metric, since it is sensitive to detours (which are included in the denominator only) and directly reflects the compactness of pooling.
Data Availability
The input data to reproduce the experiment is available on the public repository https://doi.org/10.5281/zenodo.12806397. The road graph data used in experiments was downloaded from OpenStreeMap public dataset on July 2024. The raw data with the results of the conducted experiments is available on the public data repository: https://doi.org/10.5281/zenodo.12721529.
Code availability
Hyperpool Python code, along with scripts to reproduce the experiment is available on a public repository: https://github.com/RafalKucharskiPK/ExMAS/releases/tag/hyperpool and under https://doi.org/10.5281/zenodo.12806397 without use restrictions.
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Acknowledgements
This research was supported by the CriticalMaaS project (Grant Number 804469), which is financed by the European Research Council and Amsterdam Institute for Advanced Metropolitan Solutions, as well as by the National Science Centre in Poland programme OPUS 19 (Grant Number 2020/37/B/HS4/01847).
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R.K.: conceptualisation; methodology; data curation; formal analysis; investigation; visualisation; writing—original; writing—editing and review; funding acquisition. O.C.: conceptualisation; methodology; resources; validation; writing—original; writing—editing and review; supervision, funding acquisition.
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Kucharski, R., Cats, O. Hyper pooling private trips into high occupancy transit like attractive shared rides. npj. Sustain. Mobil. Transp. 1, 6 (2024). https://doi.org/10.1038/s44333024000064
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DOI: https://doi.org/10.1038/s44333024000064