Repositioning ridesplitting vehicles through pricing: Atwo-region simulated study
C. V. Beojonea , N. Geroliminisa,∗ and Y. Yinba École Polytechnique Fédérale de Lausanne (EPFL), Lausanne CH-1015, Switzerlandb University of Michigan, Ann Arbor, Michigan 48109-2125, USA

caio.beojone@epfl.ch, nikolas.geroliminis@epfl.ch, yafeng@umich.edu∗ Corresponding author
Extended abstract submitted for presentation at the 11th Triennial Symposium on
Transportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius IslandApril 1, 2022
Keywords: Shared mobility, Repositioning, Model Predictive Control, Traffic Flow Models

1INTRODUCTION

Every day ride-sourcing services grow and promote new service options at the same time that
regulators try to identify and mitigate the negative externalities of these operations to the society(Rayle et al., 2016).
Geographical variations on demand can create an unbalance between service demand anddriver supply. In this direction, keeping drivers well positioned is of vital importance to maintaina satisfactory service quality. Most strategies in the literature are reactive in the sense that they
rely on past events such as losses of requests. However, if customers face recurrent losses, they
are likely to change to a mode that is always ready to serve them.
MFD-based models can describe the dynamics of state evolution in urban networks partitioned in a number of homogeneously congested regions to predict near future conditions.
Examples of its use include taxis and ride-hailing (solo rides) (Nourinejad & Ramezani, 2020).
Note that, the studies still lack the effects of ridesplitting on repositioning strategies and detailed
analysis on driver decision process. The reader is acknowledged that this does not configure asystematic, nor comprehensive literature review, which will come in the final version of the paper.
Herein, we develop relocation strategies for ride-sourcing vehicles using a fare optimization
controller subject to a dynamic MFD-based model as Beojone & Geroliminis (2021), which also
provides drivers with an estimate for their earnings according to their repositioning decisions. The
fleet operator uses a continuous-time Markov chain to estimate earnings for a given decision. To
the best of our knowledge, this is the first attempt to integrate fare optimization and repositioning
in a scenario with ridesplitting.

2MODEL DESCRIPTION

The model assumes that the TNC provides two service options that idle drivers can serve. The
first option, ride-hailing, aims customers that prefer to pay higher fares to ride alone. The
second one, ridesplitting, aims customers that accept some detour and longer travel times inexchange for cheaper fares. Another important assumption is that ridesplitting is limited toserve two passengers simultaneously per vehicle, which can comply with most of the current data
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C. V. Beojone and N. Geroliminis

2on ridesplitting. Table 1 is derived from the list of activities. Pick-up and drop-off activities
were aggregated to form a single state. Vehicles in states RH and S2 are assumed completely
busy and cannot receive new assignments. At the same time, vehicles in states I and S1 areconsidered available for new assignments (S1 only for ridesplitting assignments). For simplicity,
we assume that private vehicles actions besides driving (e.g. cruising for parking) are negligible.
Table 1 – List of states based on the activities vehicles perform.
Activity
Idle
Repositioning
Ride-hailingStateIRP
RH

AccumulationnIo (t)nRPod (t)nRH
od (t)

Activity
Single ridesplitting
Shared ridesplitting
Private vehicle

StateS1
S2
PVAccumulation
nS1
od (t)nS2
od (t)nPodV (t)To summarize the description states and activities of ride-sourcing and private vehicles, Figure
1A presents the state space with their respective transitions. The urban area is composed of aset R = {1, 2} of regions. Mass conservation equations [1] keep track of the number of vehicles
in each state and their remaining distance to be traveled in an M-Model (Sirmatel et al., 2021).
The only exception is for idle ride-sourcing vehicles, which have no trips to complete. Instead,they cruise waiting for the assignment of a new passenger (similarly to cruising for parking),
thus, there is no remaining distance, only the number of vehicles.
R1

A

B R1

Legend:: Private vehicles: Idle ride-sourcing: Repositioning: Ride-hailing: Single ridesplitting: Shared ridesplitting

R2

R2

Figure 1 – (A) Two-region state-space for ride-sourcing vehicles and private vehicles with theirschematic state transitions in the MFD-based model (system level model). (B) Ride-sourcing
driver state-space and transitions in the Markov chain model (individual level model).
KKṅK
od (t) =Iod (t) − Ood (t)

(1a)

KK
K
Ṁod(t) =Iod(t)LK
od (t) − nod (t)vo (t)(1b)

KWhere nK
od (t) and Mod (t) are the number of drivers and the total remaining distance to beK (t) andtravelled for drivers at an activity K at area o to area d, respectively. Furthermore, Iod
K (t) represent the total inflows and outflows for these drivers, respectively. LK (t), in the otherOododhand, is their average trip length and vo (t) is the current average speed at that area.
Given the large frequency of events (passenger arrivals, deliveries, transfer flows) one canapproximate the revenue generation for the company by means of a continuous rate (instead ofa service by service basis). Furthermore, if the regions are reasonable uniform, we can assume
that drivers split earnings equally. Therefore, drivers’ earning generation depends on the activity
they are performing and booking and distance fares.
We assume that drivers try to maximize their earnings but they are unable to accurately
estimate earnings themselves (due to limited information and rationality). To help drivers,
the service provider uses a continuous-time Markov chain to depict near-future activities andTRISTAN XI Symposium

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C. V. Beojone and N. Geroliminis

3earnings of individual drivers. Figure 1B depicts the transitions on the Markov chain. Notethat this Markov chain is depicted on an individual level, i.e., arrival rates, trip-completion andtransfer flows are individualized based on current information of the MFD model. Moreover, it
assumes that drivers will only make this decision the moment they are about to become idle,
therefore, there is no transition from idle to repositioning state because the probability making
this decision again in the near-future is negligible.
KKKπ̇od(t) = −Õod(t)πod(t) +XkkI˜od(t)πod(t)(2)

kK
KWhereP ˜k πkod (t) is the instantaneous probability of a driver being in state Kod ; and Õod andk Iod πod (t) are the individualized outflows and sum of inflows towards state Kod . The starting
solutions of the Markov chain represent drivers’ possible decisions, and their resulting probabilities are used to estimate their earnings. Earnings from derive from bookings and traveling,which are computed from to the number of new assignments and the production of busy drivers,
respectively. A logit model depicts drivers’ final decisions back into the MFD-model.

3MODEL PREDICTIVE CONTROLLER

In this prototype application, we consider that the objective is to decrease the number of lostride requests for ride-sourcing services (called abandonments). Moreover, the responsible forsetting fares and supplying drivers with estimates on their near-future earnings is the service
provider, which can control booking and traveling fares independently within a bounded rangessss([fB,min, fB,max] and [fT,min, fT,max]). The design of the fare optimization problem of the ridesourcing service is formulated in Equation [3].

minssfB (t),fT (t)J=Z tf Xt0Xsplod(vo (t), nsav (t)) λsod (t) dts∈S (o,d)∈R2s.t.: Equation [1]ssfB,min≤ fBs (t) ≤ fB,maxs
sfT,min≤ fTs (t) ≤ fT,maxv(t), nK
od (t), ... ≥ 0

Xnkods (t)(3a)

t ∈ [t0 , tf ](3b)t ∈ [t0 , tf ] and s ∈ S(3c)t ∈ [t0 , tf ] and s ∈ S(3d)t ∈ [t0 , tf ](3e)nsav (t) =ks ∈Ks

In the objective function (Equation [3a]) the total number of lost passengers is describeds (t) (probability of losing an arriving passenger), and λs (t) (passenger arrival rate forusing plod
oda service s ∈ S = {H, S} – trisride-hailing or ridesplitting). Equation [3b] summarizes the massconservation equations of the MFD-model in Equation [1]. Equations [3c] and [3d] bound thevalues for fares. Finally, Equation [3e] guarantees that the states of the model remain nonnegative. In this prototype implementation, we neglect the market effects on customers anddrivers decisions for joining or leaving the system. This is a direction for further research.
We prepared a rolling time horizon MPC controller. At each time step of 3 minutes, thecontroller tries to minimize abandonments for the next 10 time-steps. The feedback loop provides
an estimation of system states. The optimization output is the set of fares. Fares affect indirectly
loss probabilities and outflows, therefore we solve the problem with a pattern search algorithm
at each step. We assume that the demand prediction module is based on historical data.

4

COMPUTATION RESULTS

The observation of abandonment rates shows how repositioning has potential to decrease thenumber of lost requests. Fixing fares and allowing drivers to move between regions according toTRISTAN XI Symposium

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C. V. Beojone and N. Geroliminis

4

their expectations on revenues was capable to decrease the number of lost requests by more than
70%. Optimizing fares dynamically, brought abandonments to a near-zero situation, decreasing
by 98% compared to the non-repositioning scenario. However, the impacts of these decisions
on traffic are secondary to the problem as shown in Figure 2, where the optimization of fares
decreased the travelling speeds, in general. Furthermore, the greatest decreases in speed were in
the most congested periods of each region. We must point out that, in all cases, travelling speeds
remained above the critical speeds (13 km/h), i.e., the system never entered a hyper-congested
state (accounting for the background traffic as in Figure 1A).

1.5

Ridesplitting
8Region 1

40.5

00212Time (h)

30030

30

61Region 2102Speed (km/h)

Abandonments (pas/h)Ride-hailing
103

1232020150Time (h)No-repositioning

252515123100Time (h)Fixed prices

123Time (h)Optimized prices

Figure 2 – Comparison of realized abandonments and average travelling speed among three scenarios for (left) ride-hailing and (right) ridesplitting). Measurements extracted from the simulation/plant results for each time-step of 3 minutes.

5FINAL CONSIDERATIONSIn this paper we developed relocation strategies for ride-sourcing vehicles using a fare optimization controller which also provides drivers with an estimate for their earnings for their repositioning decisions. Our main results show that repositioning could brought passenger abandonments
to a near-zero scenario at the expense of decreasing travelling speeds, especially at the momentsof higher demand.
Such findings are expected in a problem with a single objective (minimize abandonments).
We must highlight that, in this case, the decrease in abandonments is a lot more pronounced thanthe decrease in speeds, which did not enter a hyper-congested state. More evidence is neededbut these findings provide a likely path for testing the impacts of different regulatory strategies
under a transient situation. Nevertheless, this work contributes by expanding the test of fare
optimization to ride-sourcing services with the presence of ridesplitting option and providing
a powerful tool to estimate near-future earnings through a Markov chain, given the additional
options drivers and passengers have in these travel modes.

References
Beojone, Caio Vitor, & Geroliminis, Nikolas. 2021. A dynamic multi-region MFD model for ride-sourcing
systems with ridesplitting. Working paper (submitted to Transportation Science).
Nourinejad, Mehdi, & Ramezani, Mohsen. 2020. Ride-Sourcing modeling and pricing in non-equilibrium
two-sided markets. Transportation Research Part B: Methodological, 132, 340–357. 23rd International
Symposium on Transportation and Traffic Theory (ISTTT 23).
Rayle, Lisa, Dai, Danielle, Chan, Nelson, Cervero, Robert, & Shaheen, Susan. 2016. Just a better taxi?
A survey-based comparison of taxis, transit, and ridesourcing services in San Francisco. Transport
Policy, 45, 168–178.
Sirmatel, Isik Ilber, Tsitsokas, Dimitrios, Kouvelas, Anastasios, & Geroliminis, Nikolas. 2021. Modeling, Estimation, and Control in Large-Scale Urban Road Networks with Remaining Travel DistanceDynamics. Transportation Research Part C: Emerging Technologies, 128, 103157.

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