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Abstract and Figures

With the popularization of transportation network companies (TNCs) (e.g., Uber, Lyft) and the rise of autonomous vehicles (AVs), even major car manufacturers are increasingly considering themselves as autonomous mobility-on-demand (AMoD) providers rather than individual vehicle sellers. However, matching the convenience of owning a vehicle requires providing consistent service quality, taking into account individual expectations. Typically, different classes of users have different service quality (SQ) expectations in terms of responsiveness, reliability, and privacy. Nonetheless, AMoD systems presented in the literature do not enable active control of service quality in the short term, especially in light of unusual demand patterns, sometimes allowing extensive delays and user rejections. This study proposes a method to control the daily operations of an AMoD system that uses the SQ expectations of heterogeneous user classes to dynamically distribute service quality among riders. Additionally, we consider an elastic vehicle supply, that is, privately-owned freelance AVs (FAVs) can be hired on short notice to help providers meeting user service-level expectations. We formalize the problem as the dial-a-ride problem with service quality contracts (DARP-SQC) and propose a multi-objective matheuristic to address real-world requests from Manhattan, New York City. Applying the proposed service-level constraints, we improve user satisfaction (in terms of reached service-level expectations) by 53% on average compared to conventional ridesharing systems, even without hiring additional vehicles. By deploying service-quality-oriented on-demand hiring, our hierarchical optimization approach allows providers to adequately cater to each segment of the customer base without necessarily owning large fleets.
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A business class for autonomous mobility-on-demand: Modeling service
quality contracts in dynamic ridesharing systems
Breno A. Beirigoa, Rudy R. Negenborna, Javier Alonso-Moraa, Frederik Schultea
aDelft University of Technology, Delft, The Netherlands
With the popularization of transportation network companies (TNCs) (e.g., Uber, Lyft) and the rise of
autonomous vehicles (AVs), even major car manufacturers are increasingly considering themselves as au-
tonomous mobility-on-demand (AMoD) providers rather than individual vehicle sellers. However, matching
the convenience of owning a vehicle requires providing consistent service quality, taking into account in-
dividual expectations. Typically, different classes of users have different service quality (SQ) expectations
in terms of responsiveness, reliability, and privacy. Nonetheless, AMoD systems presented in the literature
do not enable active control of service quality in the short term, especially in light of unusual demand
patterns, sometimes allowing extensive delays and user rejections. This study proposes a method to control
the daily operations of an AMoD system that uses the SQ expectations of heterogeneous user classes to
dynamically distribute service quality among riders. Additionally, we consider an elastic vehicle supply,
that is, privately-owned freelance AVs (FAVs) can be hired on short notice to help providers meeting user
service-level expectations. We formalize the problem as the dial-a-ride problem with service quality contracts
(DARP-SQC) and propose a multi-objective matheuristic to address real-world requests from Manhattan,
New York City. Applying the proposed service-level constraints, we improve user satisfaction (in terms of
reached service-level expectations) by 53% on average compared to conventional ridesharing systems, even
without hiring additional vehicles. By deploying service-quality-oriented on-demand hiring, our hierarchical
optimization approach allows providers to adequately cater to each segment of the customer base without
necessarily owning large fleets.
Keywords: Autonomous vehicle, Mobilty-on-demand, Ridesharing, Service quality, On-demand hiring,
Multi-objective optimization.
1. Introduction
The upcoming autonomous vehicle (AV) transition is expected to re-shape urban transportation in the
next decades. The current personal mobility paradigm, mainly based on vehicle ownership, is likely to be
phased out as autonomous mobility-on-demand (AMoD) systems develop (Litman, 2017). All major car
manufacturers have already anticipated this shift, announcing their plans to use shared autonomous vehicle5
(SAV) fleets to provide seamless transportation solutions to travelers rather than selling individual vehicles
(Hyland and Mahmassani, 2017).
However, most AMoD systems are unable to actively control service quality in the short-term, at the
operational level. Once a particular service level is defined (e.g., in terms of maximum waiting times), they
determine a fleet size that maintains such level at a reasonable rate (see, e.g., Alonso-Mora et al., 2017;10
Fagnant et al., 2015; Boesch et al., 2016; Vazifeh et al., 2018). As high this rate might be, even a small lack
of reliability can undermine the acceptance of mobility-on-demand services, especially for customers who
Email addresses: (Breno A. Beirigo), (Rudy R. Negenborn), (Javier Alonso-Mora), (Frederik Schulte)
© 2021. This manuscript version is made available under the CC-BY-NC-ND 4.0
cannot afford service rejection or excess delays. User acceptance, however, is a crucial factor in ridesharing
systems: peak performance can only be achieved when a sufficiently large number of riders are willing to
participate (Krueger et al., 2016).15
Moreover, ridesharing research generally does not consider that different customer groups widely vary in
their behavior and desires (Zeithaml et al., 2001). Companies from various sectors acknowledge that profits
and perceived service quality (SQ) can be improved by directly catering to segments of their customer base.
For example, airline passengers can generally choose from first, business, or economy class, whereas current
on-demand urban transportation network companies (TNCs), such as Uber and Lyft, offer a range of service20
tiers, spanning from pooled riders up to executive limousine services. Similarly, future AMoD systems are
also likely to consider customer preferences when matching overlapping itineraries to construct viable routes.
This study proposes a hierarchical multi-objective approach to dynamically control service quality —
defined in terms of responsiveness, reliability, and privacy — to meet user needs. To this end, we take the
perspective of a ridesharing AMoD company servicing a diversified user base where riders have both different25
sharing preferences and service level (SL) requirements, which are expressed in terms of maximum tolerance
delays. By varying such parameters, we define a strict service quality contract (SQC) that specifies the
minimum SQ-settings for three user classes, namely, business (B), standard (S), and low-cost (L). Whenever
the operating fleet cannot sustain the service quality demanded by incoming requests, third-party vehicles
are hired online to guarantee the expected user experience required by each SQ class.30
Although fleet size elasticity is already a core element of current TNCs, drivers’ availability remains a
bottleneck for reliable operations. An AV-based solution tends to be more robust: autonomous ridesharing
platforms do not need to overcome the opportunity cost of drivers but the opportunity cost of idle AVs.
These could include, for instance, staying parked (possibly incurring parking costs) or fulfilling other tasks to
their owners. Similar to Campbell (2018), in this study, we consider that many AV owners would find value35
in sharing their vehicles when not in use. Once vehicle availability is disassociated from driver availability,
independent AV owners could profit from otherwise unproductive idle times (e.g., parked at work), making
their vehicles available to AMoD systems in exchange for compensation on a contract basis. On the other
hand, by leveraging the underutilized capacity of a freelance autonomous vehicles (FAV) fleet, providers can
consider a substantially larger pool of hireable vehicles to resolve supply and demand mismatches, adjusting40
the fleet size in the short term to meet users’ service level expectations.
For example, if 1% of the cars daily entering Manhattan were autonomous, a pool of about 7,000 vehicles
could be activated (Beaton, 2019). The proposed model excels to the extent that it can take advantage of
unutilized, abundant vehicle supply to react in the short term to demand fluctuations. Although this scenario
is more likely to unfold in an AV-predominant world, it is worth noting that our method is agnostic to vehicle45
automation. The freelance fleet could comprise both AV and non-AV owners (a mix prone to appear in a
transition phase to widespread automation), and operations could also rely entirely on a dynamically hired
fleet like current TNCs.
Figure 1 illustrates how a batch of requests from different SQ classes are serviced under this scenario,
using a set of working and hireable vehicles. It can be seen from the output sub-figure 1-(b) that the high-SQ50
class request B5 prompts a hiring operation (H3W3) and enjoys a private ride. In contrast, the low-SQ
class request L4 waits longer to be picked up and does not travel directly to its destination (the ride is
shared with user S2 momentarily). The working vehicle 2 stops rebalancing to service ODs 1, 2, and 4,
whereas candidate vehicle 3 is hired to service ODs 5 and 3. Since working vehicle 1 cannot access any users
in time, it repositions to the origin of the hireable vehicle 3 (presumably, an undersupplied region), whereas55
the idle FAV 4 is dismissed. Therefore, hireable FAVs join the working fleet to help providers honoring
service quality contracts by preventing delays and rejections from happening.
1.1. Contributions
Our contribution is fourfold. First, we propose a new approach to dynamically control service quality in
AMoD systems, allowing providers to significantly better meet user service-level expectations. Second, we60
acknowledge the inherent differences between user profiles by segmenting the demand into service quality
classes and formalize the requirements of these profiles using detailed SQCs, which are upheld fully on
Route OD
Class service quality:
W1 = Rebalancing
W2 = {1,1’,2,4,2’,4}
H3 W3 ={5,5’,3,3’}
H4 = Dismissed
Fleet status: Rebalancing
(a) (b)
Figure 1: (a) The user base is segmented into three classes, namely, business (B), standard (S), and low-cost (L), which
have different expectations towards service quality. The fleet comprises working vehicles (third-party- or company-owned) and
hireable vehicles (third-party-owned). (b) To adequately fulfill user expectations across classes, we rely on on-demand vehicle
hiring, inflating the fleet size to avoid service-level violations.
an operational level using online hiring (i.e., short-term fleet size elasticity). This setup allows providers
greater leeway since they can trade off class-specific user delay tolerance and on-demand hiring. Third, we
contribute to the DARP literature by modeling such SQCs through service quality constraints and show65
how a typical AMoD provider can benefit from enforcing service quality in a real-world scenario. Finally, to
deal with large-scale real-world instances, we propose a matheuristic that creates a graph of feasible visiting
plans and optimally assigns these plans to active vehicles, hierarchically minimizing vehicle hiring and user
1.2. Organization of the paper70
The subsequent sections are divided as follows. Section 2 reviews the concepts encompassed by our prob-
lem formulation. Section 3 presents a multi-objective mixed integer linear programming (MILP) formulation
for the dial-a-ride problem with service quality contracts (DARP-SQC) as well as a dynamic formulation
for it, and describe how objectives optimized hierarchically in order of importance. Section 4 introduces
a multi-objective matheuristic for the dynamic version of the problem. Then, Section 5 describes the ex-75
perimental settings we use to create a series of case studies. These case studies comprise different user
base compositions, service-level enforcement rates, and SQC settings for each SQ class. Next, Section 6
reports the results for both static and dynamic formulations, highlighting the tradeoffs between fulfilling
user expectations and hiring new vehicles. Finally, Section 7 concludes our work and presents an outlook
for future research on service quality and fleet size elasticity.80
2. Literature review
The ridesharing problem of pooling multiple users in a single ride has its roots in the dynamic variant of
the classic dial-a-ride problem (DARP), introduced by Cordeau (2006). DARPs relate to the transportation
of passengers or goods between specific origins and destinations at the request of users (Cordeau et al.,
2007), explicitly controlling user inconvenience through service quality constraints (Berbeglia et al., 2010).85
Subsequently, we review the concept of service quality across several vehicle routing studies, highlighting
the characteristics that enable us to develop the AMoD system we propose.
2.1. Service levels, service rate, and fleet size
Service level,quality of service, and service quality are terms used interchangeably in DARP formulations
to denote the degree to which a system can avoid excessive delays or comply with prespecified time window90
constraints (see Paquette et al., 2009). However, compliance with these service levels may differ. Studies
occasionally allow for violations from the preferred service levels and consider that providers can make
selective visits, deciding which requests to accommodate (Ho et al., 2018). This is the case for most recent
AMoD studies, in which the quality of proposed matching and schedule operations is also measured regarding
the service rate (i.e., the percentage of requests serviced).95
Service levels, service rates, and fleet sizes have been consistently shown to be inextricably linked in
both static- and dynamic-demand contexts. Under a static demand assumption, where users do not react
to the service quality offered, studies focus on determining fleet sizes required to achieve acceptable service
rates assuming target service levels. For example, Molenbruch et al. (2017) describe an approximately
linear relationship between service level settings and various performance indicators and demonstrate that100
operational efficiency can be improved even further if heterogeneous user expectations regarding service
levels are considered. Accordingly, in an analysis of the minimum fleet sizing problem, Vazifeh et al. (2018)
have shown that the shorter the maximum delays between two consecutive trips, the bigger is the fleet
required to keep service at a reasonable rate.
On the other hand, under a dynamic-demand assumption, fleet sizes ultimately affect the demand model105
once subpar service quality outcomes may cause users to opt for competing transportation modes in the
long run. For example, Liu et al. (2019) show that over consecutive interactions with a multimodal system,
users eventually learn to make consistent choices based on the experienced historical service quality of travel
modes (viz., mobility-on-demand and public transit). Similarly, H¨orl et al. (2021) computes for different
fleet sizes, states where demand, cost, and waiting times are in equilibrium. Since the authors assume a110
fleet cost-covering model, increasing fleet size may eventually decrease the demand since higher operational
costs lead to higher fares, ultimately pricing out users. In such models, therefore, the concept of service rate
is more nuanced since customers ultimately refrain from using the service upon learning that their expected
service levels cannot be fulfilled.
Ultimately, regardless of the type of demand context assumed, upon determining a compromise between115
service quality and fleet productivity, an initial fleet size is chosen to launch operations. Typically, fleet sizing
is regarded as a tactical decision once adding extra vehicles is a long-term investment (H¨orl et al., 2019).
Consequently, as the operational environment changes, operators must periodically re-evaluate the system
performance to adjust the fleet size accordingly. Failing to (promptly) do so, may ultimately inconvenience
users (due to vehicle under-supply) or increase operational costs (due to vehicle over-supply).120
In contrast, this study considers AMoD operators can increase and decrease fleet size at the operational
level to quickly react to supply and demand imbalances. This way, as long as there are sufficient hire-
able vehicles, our method enables operators to consistently meet target heterogeneous user service quality
2.2. Heterogeneous users125
The literature on classic routing problems provides many variants regarding user heterogeneity formu-
lations. On heterogeneous dial-a-ride (HDARP) formulations, for example, a heterogeneous fleet has to
service multiple user categories with different transportation requirements (Parragh, 2011). However, most
categories focus on vehicle layout or equipment availability (Ho et al., 2018); service levels generally do not
vary among users.130
As argued by Molenbruch et al. (2017), it is plausible to assume service level requirements depend on
why a trip is undertaken (e.g., scheduled appointment, leisure activity, commuting). Moreover, as shared
autonomous transportation start to compete with current modes, users may adopt AMoD services that best
reflect or improve their current ride experience.
By explicitly addressing heterogeneous user preferences, transportation providers can segment their user135
base and improve their operational efficiency and profitability by catering to each class accordingly. For
instance, Wong et al. (2008) extend a model of urban taxi services in congested networks to the case
of multiple user classes (low- and high-income), multiple taxi modes (normal and luxury) with distinct
combinations of service area restrictions (urban or rural), and fare levels (mileage- and congestion-based).
Chen et al. (2016) show that AMoD providers can find a compromise between profitability and service140
coverage by offering high value of travel time (VOTT) users, refined, work-enhancing vehicle environments
at higher fares. Lokhandwala and Cai (2018) allow users to choose either private or shared rides and the
extent to which they can tolerate deviations from the original trip distance. Zhang et al. (2015) also model
the user’s willingness to share rides and adapt their waiting tolerances from random hourly incomes. Beirigo
et al. (2020), on the other hand, consider equity aspects of service quality for AMoD in areas with limited145
public transit access.
Addressing heterogeneous user requirements is also common outside transportation-on-demand formu-
lations. For example, Smith et al. (2010) investigate a generalization of the dynamic traveling repairperson
problem (DTRP) that heavily penalizes service delays of high-priority demand classes. Bulh˜oes et al. (2018)
consider a vehicle routing problem with service levels (VRP-SL), where a logistics provider has to balance150
service quality and operational costs, prioritizing different groups of requests according to their relative
Still, throughout all these studies, users either leave the system unserved or endure longer delays when
these requirements cannot be fulfilled. In contrast, we consider rejections as an unacceptable breach of
service quality contract, such that we seek to inflate fleet size to at least guarantee full coverage.155
2.3. Heterogeneous vehicles & fleet size elasticity
Although most studies on urban mobility assume that a single service provider takes care of all requests,
this centralized setting is unlikely to happen in reality since multiple providers can exist in one operation
area (Ho et al., 2018). Despite optimal from a fleet operational perspective, centralized operations entail a
transition from a diversified mobility market to a monopolistic market, possibly leading to less competition160
and, consequently, higher prices for passengers (Vazifeh et al., 2018), and different services in different zones
(Beirigo et al., 2018).
Vehicle automation further diversifies the mobility market since idle privately-owned AVs may also
become micro-operators. Once vehicles remain parked about 95% of the time (Shoup, 2017), incentives for
individuals to simultaneously own and share their AVs when they are not in use will likely be high (Campbell,165
2018). These idle hireable vehicles, which we refer to as FAVs, may be readily available during predefined
time windows to join larger, centrally controlled fleets in exchange for compensation. As suggested by
Hyland and Mahmassani (2017) and shown by Beirigo et al. (2021), AV fleet managers can significantly
benefit from such short-term fleet size elasticity, increasing or decreasing the fleet to adequately meet the
demand, by either hiring privately-owned AVs or drivers with non-AVs.170
2.4. Shared autonomous vehicles systems
Following the taxonomy of Narayanan et al. (2020) concerning the operation of SAV systems, we model
an on-demand booking service with dynamic vehicle assignment and mixed sharing system (both ridesharing
and carsharing setups are possible).
Table 1 shows how our work fits into related literature concerning service quality, fleet characteristics,175
and objective function. First, the “Service quality” header comprises three columns where we identify
whether users (i) can demand shared and/or private rides, (ii) can choose from heterogeneous service levels,
and (iii) can have priority over others. The “Fleet” header groups four columns indicating the SAV fleet
characteristics, namely vehicle homogeneity (concerning capacity), vehicle ownership (company or private),
availability (permanent or time windowed), and fleet size inflation enabled by dynamic hiring. Finally, the180
“Objective” column summarizes the goal of each approach. Typically, works that do not aim to minimize
rejections assume travelers are willing to wait indefinitely but penalize this waiting accordingly. For instance,
Hyland and Mahmassani (2018) prevent travelers from going from assigned to unassigned as well as being
reassigned more than once, besides penalizing the assignment to busy vehicles picking up or dropping off.
We refer to these penalties in the objective function as “Min. assignment to busy” since they aim to improve185
the service quality of users previously assigned. In contrast, when waiting is bounded, a sufficiently large
fleet size is considered. In Fagnant and Kockelman (2018), for example, the minimum fleet size is defined
using a “seed” day simulation in order to guarantee travelers are never rejected and do not wait for more
than ten minutes. Regarding the objective function labels, when a method aims to minimize costs that
depend solely on the distance traveled, we label it with “D” (i.e., min. travel distance). Conversely, if other190
elements are used to calculate the cost (e.g., hourly salaries, VOTT, travel times), we use the label “C”.
In contrast to typical formulations, we analyze a highly diversified market scenario in which private AV
owners provide freelance rides to a mobility firm in exchange for compensation. The firm is assumed to own
a homogeneous fleet but occasionally hire third-party vehicles to avoid user service-level violations. To make
it more realistic, we assume such a backup fleet comprises heterogeneous vehicles with different capacities.195
However, opposed to most HDARP formulations in which vehicle/user compatibility depends on physical
characteristics, we treat seats as commodities. Any customer can be transported by any vehicle as long as
service quality can be maintained.
Table 1: On-demand booking and dynamic vehicle assignment ridesharing systems classified according to service quality, fleet characteristics, and objective function.
Objectives separated by “-” are linearly combined and objectives separated by “>” are optimized hierarchically.
Service quality Fleet
availability Dynamic
hiring Objective
Fagnant and Kockelman (2018) S - - - C P - W
Alonso-Mora et al. (2017) S - - - C P - R - W
Fiedler et al. (2018) S - - - C P - D
Gueriau and Dusparic (2018) S - - - C P - P
Simonetto et al. (2019) S - - - C P - W
Santos and Xavier (2015) S - - - C P - R - C
Wallar et al. (2019) S - - C P - H - U
Hyland and Mahmassani (2018) P - - - C P - W - D - B
Gurumurthy and Kockelman (2018) P - - - C P - W
Fagnant and Kockelman (2014) P - - - C P - W
Chen et al. (2016) P - - C P - C
Beirigo et al. (2021) P - - C, P P, W F
Lokhandwala and Cai (2018) S, P - - C P - D - S
Zhang et al. (2015) S, P - - C P - C
This study S, P ✓ ✓ C, P P, W R>H>S>W
Objectives: W: Min. waiting, R: Min. rejections, H: Min. fleet size, S: Min. service level violation, D: Min. travel distance,
C: Min. cost, F: Max. profit, P: Max. vehicle pickups, U: Max. utilization vehicle capacity, B: Min. assignment to busy.
3. Problem formulations
In this section, we introduce the mathematical model for the problem at hand considering a static context200
(Section 3.1) and present the changes necessary to implement the dynamic version (Section 3.2). The static
formulation allows us to assess the impact of organizing operations (i.e., fleet size and mix) to meet the
service-level expectations prescribed in SQCs. On the other hand, the dynamic formulation can be seen
as an operational tool that allows providers to control service quality on short notice, constantly routing,
re-routing, rebalancing, and hiring AVs to keep a consistent user experience even in the light of unexpected205
demand patterns. Since both formulations aim to minimize a multi-objective function hierarchically, Section
3.3 further explains how this process is carried out using a lexicographic method.
3.1. Static DARP-SQC
The DARP-SQC is modeled on a digraph G= (N, E ). The node set Nis partitioned into {P, D, O}
where P={1, ..., n}is both the set of pickup nodes and request indices, D={n+ 1, ..., 2n}is the set of210
destination nodes, and Ois the set of origins okof vehicles kK. We consider a free-floating fleet where
vehicles do not need to depart from or come back to a central station. They can start from different origins
and park nearby the delivery location of the last dropped request upon finishing the service. We let Zbe
the set of discretized locations on a street network, such that all nodes in Nmap to a single location in Z,
using the function g:NZ. The shortest path between nodes iand jin Nis given by Zg(i),g(j)Z215
which we denote as Zi,j for simplicity. In turn, the minimum travel time to traverse path Zi,j is ti,j.
Each vehicle khas capacity Qkand is available to work on a contract during a time window [ek, l k]. For
company vehicles, this window spans the entire planning horizon, whereas, for third-party vehicles, various
time windows may be set. Users are segmented into service quality classes cCsuch that the set of
request indices Pcan be further partitioned as P=S
Pc. The arc set Eis defined as E={(i, j)|i220
O, j Por i, j PD , i ̸=jand i̸=n+j}. To each node iNis associated a load qi, corresponding
to the number of passengers, such that qi= 0 iO. Regarding the pickup and destination nodes,
qi0iPand qi=qiniD, that is, the load acquired upon picking up a request has to be equally
consumed in the request’s destination. Nodes iNare also associated with a service duration di, which
may correspond to, for instance, embarking and disembarking delays, for iPand iD, respectively.225
Service levels for SQ classes cCare represented by constants αcand βc. The former corresponds to the
expected maximum pickup delay over the revealing time eiof request iPc. The latter is the maximum
total delay tolerated by user i. Hence, users from class care satisfied if picked up within αcbut can tolerate
up to a βcdelay, which comprises both pickup and in-vehicle delays. The binary parameter ρcis 1, if users
from SQ class cCdemand private rides (i.e., ridesharing is disabled).230
The decision variable xk
i,j is 1 when vehicle kKtraverses arc (i, j)Eand the load of a vehicle kupon
leaving node iNis ωk
i. On the other hand, the decision variable yidetermines whether the service level
of user iPis achieved, setting a viable range for i’s pickup delay variable δi. If we can meet user iservice
levels, that is, variable yiis 1, then 0 δiαci. On the contrary, if yiis 0, the minimum service-level rate
σciof SQ class ciCdoes not cover user i. In this case, user ineeds to wait longer, that is, αci< δiβc.235
Regardless of the case, the variable ∆k
i, representing the delay of request iPinside vehicle k, can take up
all the remaining tolerated delay budget not spent while picking up i, that is, ∆k
iβciδi. Our service
quality formulation associates user satisfaction with complying with expected maximum pickup delays αc
because users are typically more inconvenienced while waiting for a vehicle (e.g., due to weather conditions,
safety concerns) than while waiting inside it. Still, worse than waiting for a vehicle is having the request240
rejected. Hence, adopting more flexible pickup delays, controlled by service-level rates, gives the provider
more leeway to fulfill upcoming requests. Finally, variable τk
icorresponds to the time at which vehicle k
arrives at node iN. All elements of the DARP-SQC problem are summarized in Table 2.
3.1.1. Multi-objective function
Let Xbe the solution space, such that s∈ X is given by s={xk
i,j , yi, δi,k
i, τ k
i, ωk
i}. We consider a
multi-objective function with non-commensurate objectives, ranked according to their respective priority.
Table 2: Sets, parameters, and variables of the DARP-SQC.
CSorted SQ classes (from highest to lowest priority)
PPickup nodes and request indices
PcPickup nodes and request indices of SQ class cC
DDelivery nodes
OOrigin nodes okof vehicles kK
ZDiscrete street network locations
Zi,j Shortest path between locations iand j
okOrigin node of vehicle kK
ciSQ class of request iP
QkCapacity of vehicle kK
ti,j Travel time from node ito node j
diService duration at node iN
qiNumber of passengers of request i
eiEarliest pickup time of request i
ekContract start time of vehicle k
lkContract end time of vehicle k
σcService-level enforcement rate of SQ class cC
αcExpected max. pickup delay of users in SQ class cC
βcTotal delay tolerance of users in SQ class cC
ρc(Binary) 1 if SQ class cCdoes not allow ridesharing, 0 otherwise
i,j (Binary) 1 if vehicle ktraverses arc (i,j ), 0 otherwise
yi(Binary) 1 if user iservice level is achieved, 0 otherwise
δiPickup delay of user i
iIn-vehicle delay of user iin vehicle k
iArrival time of vehicle kat node i
iLoad of vehicle kafter visiting node i
First, since using smaller vehicles may help to decrease external costs (e.g., by occupying less parking and
road space, consuming less fuel/power, and lowering emissions), we prioritize solutions that minimize by
minimizing the total number of seats across all vehicles with different capacities
fcapacity(s) = X
ok,j .(1)
Second, to prioritize solutions featuring fewer vehicles and, ultimately, increased sharing, we aim to minimize
the fleet size
ffleet(s) = X
ok,j .(2)
Next, assuming classes in Care sorted in decreasing priority order (e.g., business, standard, low-cost), we
aim to minimize, the sum of service level violations
c(s) = X
(yi1) | ∀cC},(3)
and then the total waiting time for each class
c(s) = X
i| ∀cC}.(4)
Finally, the multi-objective function is given by
f(s) = {fcapacity(s), ffleet(s), f violate
c(s)cC, f wait
3.1.2. MILP model245
The formulation of the DARP-SQC is as follows:
i,j , yi, δi,k
i, τ k
i, ωk
i) (6)
Subject to:
i,j = 1 iP(7)
i,j X
i,n+j= 0 jP, k K(8)
ok,i 1kK(9)
i,j X
j,i 0kK, j PD(10)
ilkkK, iN(11)
j(ti,j +di+τk
i,j i, j N, k K(12)
i+ti,n+i+di)iP, k K(13)
i,j i, j N, k K(14)
max{0, qi} ≤ ωk
imin{Qk, Qk+qi} ∀iN, k K(15)
yilσc· |Pc|mcC(16)
αci(1 yi)δiαciyi+βci(1 yi)iP(17)
iei+δiiP, k K(18)
i+δiβciiP, k K(19)
i=qiiP:ρci= 1 (20)
i,n+i= 1 iP:ρci= 1 (21)
i,j ∈ {0,1} ∀kK, i, j N(22)
yi∈ {0,1}, δiNiP(23)
iNkK, iP(24)
i, ωk
iNkK, iN(25)
The aim of the hierarchical multi-objective function (6) is first to determine the minimum fleet size and
vehicle mix that satisfies the demand and then to improve class service levels. Constraints (7) and (8) ensure
that all users are serviced exactly once and that the same vehicle visits their origin and destination nodes.
Constraints (9) and (10) guarantee that every scheduled vehicle kdeparts from its origin okand stops at the250
delivery node of its last request. Constraints (11) impose that vehicles can only pick up and deliver users
within their working time window. The consistency of arrival and ride times is ensured by constraints (12)
and (13), whereas the consistency of load variables is ensured by inequalities (14) and (15). Constraints
(16)-(21) implement the user SQCs. First, constraints (16) enforce that the service-level expectations of a
minimum share of users from each class are met. Next, Constraints (17), (18), and (19) ensure that service255
levels are consistent with maximum pickup and in-vehicle delays. Then, equalities (20) and (21) ensure
that users whose class entails a private ride are picked up by empty vehicles and travel directly to their
destination. Finally, constraints (22)-(25) declare the variables.
The quadratic constraints (12) and (14) are linearized as in Cordeau et al. (2007):
iti,j +diMk
i,j (1 xk
i,j )i, j N, k K(26)
i,j (1 xk
i,j )i, j N, k K(27)
The validity of the Big-M constants Wand Min (26) and (27) is ensured by setting Wk
i,j min{2Qk,2Qk+
qi}and Mk
i,j max{0, li+ti,j +diej} ∀kKand i, j N.260
3.2. Dynamic DARP-SQC
In order to deal with real-world instances, we break apart the time horizon into κ-second rounds t
{1,2, . . . , T }. At each round t, we process a request batch indexed by Pt, which is comprised of requests
accumulated throughout the time slot [κ·(t1), κ ·t). Hence, passengers wait at most κseconds to have
their requests handled by the system. If rejected, we assume they walk away or may opt to re-enter the265
system in the next round, possibly choosing a different service quality class. Additionally, we search for
available freelance vehicles parked throughout locations in Zto construct the set of backup FAVs KFAV
which can privately service all requests in Pt.
In the following, we further detail the elements we use to expand the static DARP-SQC problem definition
presented in Section 3.1 to accommodate dynamic execution. These elements are summarized in Table 3.270
Requests. A request riis defined by a tuple {i, i, τk
i, τ k
i, ei, ei, qi, ci}, which consists of origin iP,
destination iD, with respective expected arrival times τk
iand τk
i, placement time ei, minimum delivery
time ei=ei+ti,i, load qi, and service level class ci.
Vehicles. We assume all vehicles kare associated to visiting plans vk={Pk,Rk,Sk}. Sets Pkand
Rkare comprised of passengers (i.e., picked up requests) and assigned requests, respectively. In turn,
m}is a sequence of nodes representing the vehicle’s itinerary, such that SkN. While
0is vehicle k’s last visited node, Sk
mare the subsequent mnodes to visit. Hence, vehicle k’s current
load after visiting Sk
0in the interval [τk
, τ k
0) is given by
that is, the sum of the loads qiof each request riin the passenger set Pk.
Visiting plan feasibility. A valid visiting plan vkfrom vehicle kconsists of a sequence of trip pairs275
(i, j)∈ {(Sk
m)}that represent a feasible solution with respect to the following
C1) ejτk
i+di+ti,j τk
j(arrival consistency)
C2) ωk
i+qjQk(load consistency)
C3) iPρci= 1 j=iand jPρcj= 1 ωk
i= 0 (privacy requirement)280
C4) τk
ilk(vehicle contract deadline)
It is worth noting that when a request is assigned to a vehicle, we assume that it can still be further delayed
or even picked up by different vehicles but never rejected. Such flexibility allows a higher number of feasible
rides to arise at each period, which ultimately favors the inclusion of high-priority users by disrupting
previous visiting plans.285
Vehicle states and visiting plan types. At each round, a vehicle kcan execute three actions, namely,
(i) move to pick up or deliver requests, (ii) stay idle at its current position, and (iii) move empty to another
location. Accordingly, we consider that vehicles kexecuting (i), (ii), or (iii), are in states servicing,idle, or
rebalancing, respectively. The best plan is chosen from a set of feasible candidates Vk, which is partitioned
based on the possible vehicle states using290
Table 3: Sets, parameters, and variables of the dynamic formulation.
ZRegional centers of street network locations Z
KPAV Company vehicles
tHireable vehicles at time t
tVehicles hired at time t
tParked vehicles at time t
PkPassengers (picked up requests) of vehicle k
RkAssigned requests (non picked up) of vehicle k
SkNode itinerary of vehicle k
vkVisiting plan {Pk,Rk,Sk}of vehicle k
RCandidate visiting plans where kis in the rebalancing state
SCandidate visiting plans where kis in the servicing state
idle Candidate visiting plan where kis in the idle state
VkFeasible visiting plans Vk
S∪ Vk
R∪ {vk
idle}for vehicle k
VVisiting plans S
Vkin ERTV graph
ViVisiting plans in Vincluding request i
iVisiting plans in Vthat meet request itarget SL
PARequests previously assigned to vehicles
BtRequest batch placed in period t
tRequests in Ptof class cC
tOrigins okZof hired vehicles KH
tPickup nodes of service-level violated requests at time t
JtRebalancing targets O
γhire Hiring penalty
γsl Service-level violation penalty
γreject Rejection penalty
κRound duration
lkContract deadline of vehicle k
TTotal time horizon
δmax Maximal hiring delay
hk(Binary) 1 if vehicle kKFAV
tis hired, 0 otherwise
xv(Binary) 1 if visiting plan vis chosen, 0 otherwise
yi(Binary) 1 if service level of request iis achieved, 0 otherwise
zi(Binary) 1 if request iis rejected, 0 otherwise
δiv Pickup delay of user iin visiting plan v
vDelay sum of all requests in visiting plan v
vDelay sum of class crequests in visiting plan v
acNumber of service-level violations in class c
SSet of candidate visiting plans where vehicle is servicing users, that is, |Pk|>0 or |Rk|>0,
idle Candidate visiting plan where vehicle is idle, that is, Rk=Pk=and |Sk|= 1, such that Sk
a parking place,
RSet of candidate visiting plans where vehicle is rebalancing, that is, Rk=Pk=and |Sk|= 2,
such that Sk
0is a parking place and Sk
1is a rebalance target.
Updating vehicle progress. Visiting plans are updated using the function updateProgress(k,t ), where
the time period tis used to compute the progress of vehicle kthroughout the itinerary Skand update sets
Rkand Pk. First, regardless of node type, i∈ Sk, if τk
iei,Sk=Sk\ {i}. If iis a pickup node, then
Rk=Rk\ {ri}and Pk=Pk∪ {ri}, whereas if iis a delivery node, then Pk=Pk\ {ri}.295
Optimal itineraries. Based on the passengers Pkof vehicle kand a candidate request set R, there can
be several feasible Sitineraries created using nodes from Pk∪ R that fulfill constraints C1, C2, C3, and C4.
These itineraries compose candidate visiting plans v={Pk,R,S} in the set Vk
S. The total delay associated
with an itinerary Sof visiting plan vconsists of the sum of the arrival delays at destination nodes, which is
given by
i∈ S∩D
Using delays ∆v, we can determine the visiting plan featuring the global minimum total delay
g= arg min
From a SQ class perspective, the total delay of requests from class cconsists of the sum of delays to reach
these requests’ destinations, which is given by
Likewise, using delays ∆c
v, we can determine the visiting plan featuring the minimum total delay for class c
c= arg min
Hiring. We outsource requests to privately-owned vehicles (FAVs) whenever the current fleet can no
longer meet the minimum service level requirements of SQ classes. For each period t, we search regional
centers for available FAVs that can back up the PAV fleet if it cannot service a request batch Ptadequately.
We assume that such freelance vehicles are readily available to join the fleet in exchange for compensation,
and their supply is sufficient to match the overall excess demand. We collect these vehicles in set KFAV
{closestFAVToRequest(i, Z* )| ∀iPt}, where closestFAVToRequest(i, Z* )is a function that returns
the vehicle kparked at regional center location in ZZthat can pick up request i.
To determine regional centers in Z, we implement a variant of the facility-location problem proposed
by Toregas et al. (1971). Our formulation aims to determine the minimum set of facilities that together can
cover all other locations in Zwithin δmax time units. For instance, a long δmax results in fewer regional305
centers, increasing the pickup time of requests and, consequently, service-level violations. Conversely, a
short δmax leads to many regional centers, allowing the hireable fleet to access users faster. In this study,
we refer to δmax as the maximal hiring delay and adjust it such that hired vehicles can pick requests within
their expected service level, regardless of class.
All vehicles kKFAV
thave contract deadlines min{lk, T }, such that kis dismissed at a later time tas310
long as t>=lk. Finally, to guarantee PAV availability, we set lk=Tfor all vehicles kKPAV. Additionally,
we consider hired vehicles to have precisely the capacity of the number of passengers determined by the
request they are supposed to service. Therefore, throughout the simulation, the working fleet dynamically
changes from homogeneous to heterogeneous, especially at high-demand times.
Decision variables. Decision variables are grouped in solution tuples s={xv, yi, zi, hk}with xv, yi, zi, hk
{0,1}such that the solution space is defined using
X={xv, yi, zi, hk| ∀v∈ Vk,kKand iPt}.
Variable xv= 1 when the visiting plan v∈ Vkfrom vehicle kKis chosen, and variables yiand zihelp315
to distinguish how user iis serviced. If yi= 1, the system could meet request iclass service levels, whereas
if zi= 1 request iis rejected. Finally, for vehicles kKFAV
t,hk= 1 signals that kwas hired to carry out a
visiting plan at the current period t.
3.2.1. Multi-objective function
For the dynamic version of our problem, we also propose a multi-objective function. First, we aim to
minimize, in turn, the sum of rejection penalties γreject given by the ordered list of objectives
c(s) = X
γreject ·zi| ∀cC}.(30)
After preventing rejections, we aim to hire the least number of vehicles necessary to meet user service
levels, such that we aim to minimize
fhire(s) = X
Qkhk. . (31)
Next, we seek to minimize the sum of violation penalties γsl in the ordered list of objectives
c(s) = X
γsl ·(yi1) | ∀cC}.(32)
Then, we consider the minimization of waiting times across classes:
c(s) = X
v·xv| ∀cC}.(33)
Finally, we define the multi-objective function of our dynamic formulation using
f(s) = {freject
c(s)cC, f hire (s), f violate
c(s)cC, f wait
According to each class priority, the order of the objectives guarantees that visiting plans are set up first320
to avoid rejections. In the next steps, vehicles are hired only to prevent service level violations, which are
minimized subsequently also following the order of C. Once optimal service level distribution is guaranteed,
the solution featuring the minimum waiting time is chosen.
3.3. Lexicographic method for multi-objective optimization
We use the lexicographic method to determine the optimal solution for the multi-objective function
f(s) (for both static and dynamic models). This method consists of solving, in decreasing priority order, a
sequence of single-objective problems
Minimize {fr(s)|fl(s)ψl, l = 1, . . . , r 1, s X },
where r∈ {1,2,...,|f|} is the index of the current objective. A solution sis feasible only if it does not325
degrade the optimal values ψlfound for previous objectives flwith higher priority.
Although we adopt a hierarchical optimization approach, it is worth noting that service quality objectives
such as fviolate and fwait are good candidates for linear combination. By deliberately weighing the tradeoffs,
providers could establish a utility function that best represents their efforts to cater to users across classes.
4. A matheuristic for the dynamic DARP-SQC330
The following sections describe how we build on the state-of-the-art method proposed by Alonso-Mora
et al. (2017) to solve the problem’s dynamic version. The method was chosen because (i) its conventional
formulation offers optimality guarantees and (ii) it is flexible to accommodate the contributions put forward
in this study, namely, service level constraints and hiring capabilities.
First, request demand and vehicle supply information are combined into a pairwise-shareability graph335
(Section 4.1), which is subsequently used to iteratively compute a graph of feasible visiting plans (Section
4.2). Next, visiting plans are optimally assigned to vehicles using a multi-objective MILP formulation whose
goal is to minimize user dissatisfaction hierarchically (according to the importance of each class) while hiring
the least number of vehicles (Section 4.3). Then, a MILP formulation is used to optimally rebalance idle
vehicles to underserved areas, where either service level violations (i.e., extra delays or rejections) or vehicle340
hirings occurred (Section 4.4).
The necessary steps to calculate a complete solution are described in Algorithm 1 and summarized in
Figure 2, which provides an overview of our method using as an example the input presented in Figure 1
(a). Since the construction steps I and II (where feasible visiting plans are determined) are followed by
assignment steps III and IV (where vehicles are optimally assigned to visiting plans), the complete strategy345
consists of a matheuristic, that is, a heuristic that incorporates phases where mathematical programming
models are solved (Archetti and Speranza, 2014).
Algorithm 1: Dynamic DARP-SQC matheuristic
Input: Request index batches P={B1, B2, B3,...,BT}for total horizon T, service level classes C, initial fleet
KPAV randomly distributed throughout street network locations in Z, and regional centers Z.
2for t= 1,2,...,T do
4K=K\ {k}if tlkkK//Remove vehicles whose contracts expired
5PA={i| ∀riS
Rk}//Build set of assigned requests
6Pt=BtPA//Build set of requests in progress
t={closestFAVToRequest(i, Z* )| ∀iPt}
9Build RV graph using vehicles Kand requests Pt(Section 4.1)
10 From RV graph, build the ERTV graph with feasible visiting plans V(Section 4.2)
11 (MILP) Match a visiting plan in Vto each vehicle in K(Section 4.3)
12 K=K\ {k}if hk= 0 kKFAV
t//Remove unhired FAVs
13 O
t={ok|hk= 1 kKFAV
t}//Origins of hired vehicles
14 PU
t={i|yi= 0, riPt}//Origins of SL violated requests
15 Jt={i| ∀ iO
t}//Rebalancing targets
16 KP
t={k|if vk=vk
idlekK}//Idle vehicles
17 Create rebalancing visiting plans Vk
Rusing KP
tand Jt
18 (MILP) Match rebalancing visiting plans to vehicles (Section 4.4)
19 Implement all visiting plans v∈ Vkif xv= 1 kK
4.1. Pairwise request-vehicle graph
The pairwise request-vehicle (RV) graph is based on the idea of shareability graphs proposed by Santi
et al. (2014). In the RV graph, two requests are connected only if they can be serviced by an empty virtual350
vehicle starting at the origin of one of them. Likewise, a vehicle is connected to a request if the request can
be serviced by the vehicle.
Since we consider previously assigned requests can still be picked up by different vehicles, these requests
also integrate the RV graph construction. At each time period t, after updating all vehicle tuples according
to the current period tas well as dismissing hired vehicles with expired contract deadlines, we define the355
S={1, 2, 3} | B={5} | L={4}
W={1, 2} | H={3, 4} III III IV
1: 0,  3Rebalancing
2: 011’242’4’ Servicing
3: 055’33’ Servicing
4: 0Idle
Visiting plan
Idle vehicle
Figure 2: Method overview for the input example presented in Figure 1 (a). Input is processed into a RV graph (Figure 3)
where initial ridesharing opportunities are identified. Then, the RV graph is used to build an ERTV -graph (Figure 4), which
connects all vehicles and requests through feasible visiting plans. Next, these plans are optimally assigned to available vehicles
(Figure 5). Finally, remaining idle vehicles are optimally rebalanced to underserved areas (Figure 6). The output consists of
updated vehicles plans, which are illustrated in Figure 1 (b).
Pairwise shareability graph
S={1, 2, 3}, B={5}, L={4}
W={1, 2}, H={3, 4} Request-request
Figure 3: RV graph resulting from the problem input presented in Figure 1(a).
set of assigned requests PA={i| ∀riS
Rk}, and the set of all requests Pt=BtPA. Then, we use the
total fleet Kand requests Ptto build the RV graph.
Figure 3 presents the RV graph created from the problem input described in Figure 1(a). The graph
features six request-request (RR) pairs and seven request-vehicle (RV) pairs.
4.2. Extended trip-request-vehicle graph360
Similarly to Alonso-Mora et al. (2017), we use the RV graph to compute the request-trip-vehicle (RTV)
graph, where request and vehicle nodes are connected to trip nodes, which are comprised of request sets.
Feasible trips featuring q∈ {1,2}request derive directly from the RV graph, whereas trips involving q3
requests are feasible only if all possible subtrips with q1 requests are also feasible. By carrying out this
sub-feasibility assessment process iteratively, increasingly higher q-trip combinations can be evaluated up to365
the capacity of vehicle k, that is, q∈ {1,2, . . . , Qk}.
In contrast to the original RTV graph formulation, where trip nodes consist of a request set that refers
to a unique minimum waiting visiting plan, we propose visiting plan nodes (v-nodes). We consider a single
request set Rcan lead to up to |C|+ 1 visiting plans. Besides the minimum total delay plan vk
g, we also
II Extended Request-Trip-Vehicle graph
 022’
 044’
S2 142
 011’242’4’
 0121’2’44
 044’121’2’
 05
 055’
 04
 044’
Figure 4: ERTV graph created from the RV graph showed in Figure 3. For brevity, we omit the elements concerning vehicle
H3. Edges connecting requests ito visiting plans vweight requests’ pickup delays δiv, whereas edges connecting vehicles to
visiting plans weight plans’ total delays ∆v(i.e., sum of all users’ pickup and in-vehicle delays).
consider the minimum total delay plans vk
cfor each class c∈ {ci|ri∈ R}. Adding extra visiting plans370
focusing on SQ classes’ total delay minimization adds flexibility to the assignment algorithm once it can
consider a larger pool of itineraries to balance service level provision between classes.
We refer to this expanded version, where request sets generate several visiting plans, as the extended
trip-request-vehicle (ERTV) graph. Hence, whenever a vehicle kcan successfully service a request set R,
instead of adding edges e(ri,R)ri∈ R and e(R, k), we add edges where Ris replaced by visiting plans v,375
which are guaranteed to be valid with respect to feasibility constraints C1, C2, C3, and C4.
Once these visiting plans entail a one-to-one relationship to both vehicles and requests, we can attribute
weights to edges. As a result, a feasible visiting plan vleads to edges e(ri, v, δiv )ri∈ R and e(v, k, v),
where δiv is the delay to pick up request rithrough sequence Sand ∆vis the total waiting time to realize
We employ an exhaustive search over all feasible itineraries created from Rto find the visiting plans vk
and vk
c. However, it is worth noting that we are able to do so because we consider low-capacity vehicles with
at most four seats. In order to consider high-capacitated vehicles, local improvement methods or exploration
heuristics might be necessary to curb computation time.
Additionally, to ensure every vehicle will be assigned to a visiting plan, we add two extra edges. For385
parked vehicles, we add edge e(k, vk
idle,0) to guarantee they can stay parked. Consequently, if vk
idle is assigned
to kKFAV
t,kdid not need to be hired. For unloaded vehicles cruising to pick up users (i.e., Pk=and
|Rk| ≥ 1), we add edge e(k, vk
stop,0), where vk
stop ∈ Vk
Ris a special rebalance case, where Rkis emptied and
the vehicle parks at a nearby location in Z. Both vk
idle and vk
stop have zero-valued weights because we aim
to avoid unnecessary trips, such that vehicles remain still whenever possible. Vehicles can also continue to390
carry out their incumbent visiting plan, which is naturally added to the ERTV graph during the construction
phase. Algorithm 2 presents all the construction steps of the ERTV graph.
Finally, upon adding edges that link requests ito visiting plans v, we populate the set Viof plans where
ican be serviced. Then, we define set VSL
i⊆ Viwhere δiv αci, which consists of the set of plans where
request idesired service levels are satisfied.395
Figure 4 displays a portion of the ERTV graph based on the pairwise RV graph. It shows how requests
Algorithm 2: ERTV graph construction
Input: RV graph from period twith request-request (RR) edges e(ri, rj) with i, j Ptand vehicle-request (RV)
edges e(ri, k) with iPtand kK.
Output: ERTV graph with request and v-node edges e(ri, v, δiv) and v-node and vehicle edges e(v, k, v).
1Function getVisitingPlans(k, R)
g//Min. total delay (28)
4for c∈ {ci|ri∈ R} do
c//Min. total delay for class c(29)
6return V
7Function addVisitingPlansToERTV(V)
8Add request and v-node edge e(ri, v, δiv )ri∈ R,vV
9Add v-node and vehicle edge e(v, k, v)vV
10 Function addFeasiblePlansFromRequests(Rk
q, k, R)
11 for R ∈ Rdo
12 V=getVisitingPlans(k, R)
13 if V̸=then
14 Rk
q← R
15 addVisitingPlansToERTV(V)
16 begin
17 for kKdo
18 addVisitingPlansToERTV({vk
stop, v k
19 Rk
q=∅ ∀q∈ {1,2,...,Qk}
20 R={{ri}|∀e(ri, k)RV graph}//Candidate one-request trips
21 addFeasiblePlansFromRequests(Rk
1, k, R)
22 R={{ri, rj}|∀ri, rjRk
1}//Candidate two-request trips
23 R={R | ∀ R ∈ Re(R1,R2)RV graph}//Filter unfeasible
24 addFeasiblePlansFromRequests(Rk
2, k, R)
25 for q∈ {3,...,Qk}do
26 R={Ri∪ Rj| ∀ Ri,RjRk
27 R={R | ∀ R ∈ R∧ |R| =q}//Select candidate q-request trips
28 R={R | ∀ R ∈ R∧ ∀ri∈ R,R \ riRk
q1}//Filter candidate unfeasible trips
29 addFeasiblePlansFromRequests(Rk
q, k, R)
Table 4: Feasible request combinations for each vehicle throughout the iterations of Algorithm 2. Valid combinations are created
based on the RV graph showed in Figure 3 considering four-seat vehicles, totaling 135 pickup and delivery combinations.
N. of requests (N. of possible visiting plans)
Vehicle 1(1) 2(6) 3(90) 4(2,520) Total
W1 B5 - - 1
W2 S1 S1,S2 - - 7
S2 S2,L4 - - 7
L4 S1,L4 S1,S2,L4 - 97
H3 S2 S2,S3 - - 7
S3 S3,B5 - - 4*
L4 S2,L4 - - 7
B5 L4,B5 - - 4*
H4 L4 L4 - - 1
*No ridesharing in itineraries featuring business users
Pt={1,2,4,5}from period tare associated to vehicles 1, 2, and 4 through visiting plans V1={v17, v18 },
V2={v0, v1, . . . , v14}, and V4={v15 , v16}, respectively. It is worth noting that vehicles 1, 2, and 4 include
their incumbent visiting plans, namely, v18,v7, and v16. This way, whenever appropriate, vehicles can
continue carrying out these plans. Moreover, for vehicle 2, the plans v10,v11, and v12 stem from the same400
request set R={r1, r4, r2}. Since Rfeatures two different SQ classes, v10 is the visiting plan achieved
seeking to minimize the total delay, whereas v11 and v12 are the visiting plans achieved when the goal is
to minimize the delays of standard and low-cost classes. A similar outcome can be found for plans v4,
v5, and v6, which all stem from request set R={r2, r4}. Table 4 presents the number of pickup and
delivery permutations evaluated at each iteration of the Algorithm 2, considering vehicles can carry up to405
four passengers. Since some permutations might violate requests’ service-level requirements, only a subset
of them ends up becoming valid visiting plans.
4.3. Visiting plan assignment formulation
Upon creating period t’s ERTV graph embeding feasible visiting plans V=S
Vk, we formulate an
assignment problem, formalized using the following MILP model:410
f(xv, yi, zi, hk)
Subject to:
xv= 1 kK(35)
zi= 1 X
xv= 1 iPA(37)
yilσc· |Pc
The optimization problem aims at minimizing the multi-objective function (34) in order to find the least
number of hirings necessary to meet user service level expectations throughout SQ classes. Constraints
(35) guarantee that each vehicle realizes a single visiting plan. In turn, constraints (36) ensure that every
unassigned user is either assigned or rejected, whereas (37) guarantee that previously assigned requests
continue so but allows the original plan to be changed (other vehicles can carry out the service). Constraints415
(38), guarantee the consistency of the hiring process, such that hk= 1 whenever a visiting plan from Vk
of vehicle kKFAV
tis chosen. Constraints (39) guarantee the consistency of each variable yi, which is 1
Visiting plan assignment
(B SL)
-Working = 1, 2, 3 / Hired = 3/ Idle = 1
-SL met = 1, 2, 3, 4, 5 / Delayed: / Rejected =
-W1 = 0/ W2 = 011’242’4’ / W3 = 055’33’
Vehicles: 1, 2, 3, 4 Candidate plans: {,1,…,}
Requests: {1, 2, 3, 4, 5}
Optimal assignment (lexicographic method)
Total waiting
(B SL)
SL violations
(B SL)
Figure 5: At each period, feasible visiting plans are optimally assigned to available vehicles such that rejections, hirings, service
level violations, and waiting times are hierarchically minimized. The order objectives are addressed following the lexicographic
method is indicated by the arrows.
only when the expected service level of request iis fulfilled. Finally, analogously to the static formulation,
constraints (40) enforce that the service-level expectations of a minimum share of users from each class are
Figure 5 illustrates the assignment step for the input example presented in Figure 1(a). The feasible
visiting plans featured in Figure 4 are optimally assigned to associated vehicles according with the multi-
objective function (34). From the output, it can be seen that all user expectations were met (i.e., no delays
or rejections). Additionally, previously rebalancing vehicle 2 was assigned to pick up users 2 and 4, whereas
vehicle 3 was hired to pick up requests 3 and 5.425
Feasibility. Due to constraints (40), the MILP model is infeasible whenever the number of vehicles is
insufficient to honor SQCs. Still, we can determine the best possible outcome by minimizing violations to a
relaxed version:
yi+aclσc· |Pc
t|mcC, (41)
where acrepresents the number of service level violations in class c, such that 0 aclσc· |Pc
Then, we can minimize acacross classes using the ordered list of objectives
c(s) = ac| ∀cC}.(42)
4.4. Idle vehicle rebalancing formulation
Due to the spatiotemporal characteristics of transportation requests, vehicles can end up stranded in low-
demand areas of a city, operating at subpar productivity rates. Hence, to address ongoing supply-demand
imbalances, relocation strategies are commonly employed to route idle vehicles to regions where requests
are more prone to appear. Following Alonso-Mora et al. (2017), we fix supply and demand imbalances430
by optimally rebalancing parked vehicles KP
t={k|if vk=vk
idlekK}to locations where the system
failed to meet user needs. In the original formulation, vehicle rebalancing targets were the pickup points of
rejected users. Since our hiring setup can avoid rejections altogether, we adopt two new rebalancing stimuli
to move vehicles throughout the street network. First, since hiring indicates a lack of proper vehicle supply,
we consider that the origins of hired vehicles KH
t={k|hk= 1 kKFAV
t}are also rebalancing targets.435
We refer to these origins as O
t={ok| ∀kKH
t}, since they are a subset of regional centers Z. Second, we
consider that service-level violations in a particular area also indicate insufficient vehicle supply. This way,
besides the origins of rejected users, the origins of assigned but dissatisfied users also integrate the list of
rebalancing targets. We refer to the origins of service-level violated requests as PU
t={i|yi= 0, riPt},
such that the list of rebalancing targets is Jt={i| ∀ iO
t}. Finally, we minimize the total travel440
times to reach the rebalancing targets Jt.
For each vehicle kKP
twe define a set of candidate visits Vk={vk
R, where visiting plans
vk∈ Vk
Rare based on sequences {Sk|Sk={Sk
0, i} ∀iJt}. Then, we use variables xvto optimally assign
vehicles to targets as follows:
Subject to:
xv= 1 kKP
xv= min |Jt|,|KP
The objective function (43) aims at minimizing the total rebalancing delay. Equations (44) guarantee445
that all available idle vehicles KP
tare used to reach targets in Jt(if any), and equations (45) ensure that
each idle vehicle ends up either rebalancing or staying parked. Figure 6 illustrates how the rebalancing
process takes place for the idle vehicles resulting from the assignment step illustrated in Figure 5.
5. Experimental study
In the following sections, we present the SQ parameters (Section 5.1), describe how simulation settings450
such as trip and travel data are set up (Section 5.2), and design twelve operational scenarios (Section 5.3)
to study the influence of different user-base and FAV-supply configurations.
5.1. Service quality settings
The primary objective of our experiments is to assess the operational impact of actively controlling service
quality for a heterogeneous user base. Since the interplay between service levels, fleet size, and service denial455
has already been thoroughly examined in the literature (see, e.g., Molenbruch et al., 2017; H¨orl et al., 2021),
we focus our analysis on the operational adjustments required to service the request demand entirely, such
that formerly inconvenienced users can also be successfully serviced under strict SQCs.
In order to design a heterogeneous user base, we first define in Section 5.1.1 the class SQC configurations
adopted in this study. Next, to evaluate the outcome of meeting SQCs, in 5.1.2, we present three service-level460
enforcement rate scenarios, where we vary provider’s leeway to violate the SQCs. Finally, to evaluate how
class SQC configurations can influence fleet operations, in Section 5.1.3, we propose three user segmentation
scenarios, where we vary the predominance of a certain class within the transportation demand.
Idle vehicle rebalancing
-Rebalancing = 1/ Idle = 1
-W1 = 0, 3
Optimal assignment
Total distance traveled to reach rebalancing targets
Idle vehicles: 1
Rebalancing targets:
-Origins of requests with unmet service levels =
-Hired vehicle origins = 3
Figure 6: Idle working vehicles are optimally rebalanced to underserved areas.
Table 5: Service quality contract (SQC) for each SQ class.
class Ride
Service level (min)
Priority Expected max. pickup (α) Max. total delay (β)
Business 1st private 3 7
Standard 2nd shared 5 7
Low-cost 3rd shared 7 7
5.1.1. Service quality contracts
Table 5 summarizes the SQC settings we adopt for each user class. Besides priority order and sharing465
preferences, it shows the expected maximum pickup and total delays of each SQ class. The class-specific
settings can represent, for instance, high-, intermediate-, and low-SQ requirements, possibly arising from
high-, middle-, and low-income users, respectively. Besides income, the criteria to choose a suitable class
can also be related to the nature of the appointment of a user. For example, trips to scheduled events
(e.g., meetings, concerts, doctor appointments) are usually stricter, requiring higher service rates and levels.470
Alternatively, to make a parallel with current urban mobility options (regarding the expected quality of
service), the low-cost class best represents former transit passengers, whereas standard and business classes
best represent former ridesharing and carsharing/taxi passengers, respectively.
5.1.2. Service-level enforcement rate
We assess the outcome of different service-level enforcement rates σcthrough four scenarios where we475
assume the system addresses all SQ classes cCusing a common rate σ∈ {0,0.8,0.9,1}. The scenario where
σ= 0 simulates the typical AMoD system in which service-level preferences are not enforced. The remainder
scenarios allow us to investigate how increasingly enforcing the fulfillment of service-level expectations in
10% steps across user classes affects hiring, until the point at which service-level expectations are fully
upheld, for σ= 1. Hence, scenarios σ= 0 and σ= 1 represent lower and upper service-level bounds,480
whereas σ= 0.8 and σ= 0.9 allow the system to violate service-level expectations of 20% and 10% of the
requests, respectively. Although we do not consider costs in our model, the service-level enforcement rates
are tunable parameters that allow providers to trade off vehicle hiring and user dissatisfaction costs.
Table 6: User base segmentation scenarios define different distributions of SQ classes across the same transportation demand
by varying, in turn, the proportion of requests in each class.
SQ class
Business Standard Low-cost
B+ 68% 16% 16%
S+ 16% 68% 16%
L+ 16% 16% 68%
Figure 7: Optimal distribution of regional centers with corresponding reachable locations throughout the street map of Man-
hattan, New York City, considering a maximal hiring delay of 150s at 30km/h.
5.1.3. User base segmentation
To investigate the interplay between requests of distinct SQ classes, we create three user base segmen-485
tation scenarios, namely, B+, S+, and L+, in which we vary the proportion of users belonging to each
SQ class in the transportation demand (Table 6). We assume that the service quality demanded by each
user base follows a normal distribution in which the predominant class of such base covers the average SQ
requirements of most users (68%) whereas the other two classes can adequately service the rest.
The proposed scenarios aim to represent service quality requirements arising in different regions, on490
different occasions, or at different times. For example, user base L+ may better represent the vicinity of
a campus area, where most users (presumably students) are willing to wait longer in exchange for cheaper
rides. Conversely, the user base B+ fits affluent areas where privacy and high responsiveness are prone to
play a more significant role.
Throughout the simulation, we use the user base segmentation scenarios to determine the share of495
requests from the Manhattan, New York City taxi dataset that belongs to each SQ class.
5.2. Simulation settings
Street network. We use the Osmnx package (see Boeing, 2017) to construct a multidigraph from the
street network of Manhattan comprised of 4,548 locations and 9,701 links. Travel times are drawn from the
shortest distances between the street network locations, considering an average travel speed of 30km/h. For500
simplicity, we assume that congestion is an exogenous effect; neither the time of the day nor the number of
vehicles traveling throughout the street network affects travel times.
Region center distribution. Figure 7 shows the distribution of 68 regional centers, optimally determined
considering a maximal hiring delay s= 150s. This value allows that available hireable vehicles can fulfill
the expectations of even business users who have placed their request at the beginning of a thirty-second505
Transportation demand settings. We run our simulation on 42,702 real-world taxi requests from
Manhattan, New York City, occurring in the evening peak, from 18h to 19h, on the first day of February
2011. The raw dataset, containing detailed taxi information for the whole city, is processed in three phases.
First, we filter out the demands occurring outside Manhattan. To do so, we remove all requests whose510
origin/destination GPS coordinates cannot be matched (within a 50 meters range) to an intersection of such
graph or whose travel distances are small (below 50 meters). After the matching process, we replace origin
and destination GPS locations with their correspondent node IDs in the graph. Second, we select only the
relevant trip data fields required to carry out our simulation, namely, pickup times (which are adapted to
request times), origin/destination IDs, and number of passengers. Additionally, we filter out records whose515
number of passengers is higher than four (i.e., the maximum vehicle capacity we consider). After this step,
the final shares of requests requiring one, two, three, and four seats are 77.00%, 16.54%, 4.53%, and 1.93%,
respectively. Finally, we use the roulette wheel selection to randomly attribute SQ classes to users, following
the proportions specified in the user base segmentation scenarios.
5.3. Case study configuration520
We combine both service-level enforcement rates {0, 0.8, 0.9, 1}(Section 5.1.2) and user base segmen-
tation scenarios {B+, S+, L+}(Section 5.1.3) to create a series of case studies for the static and dynamic
formulations. Regardless of the case study considered, users expect to be serviced according to the SQC
configuration presented in Table 5. Table 7 summarizes the settings used to create the instances for both
5.3.1. Static DARP-SQC
Each instance is a combination of a request batch of size n∈ {15,20}and a user base segmentation
u∈ {B+,S+,L+}. To create such instances, we collect nrequests from thirty-second batches of our
trip data and randomly distribute SQ classes according to the proportions defined in the user base u. We
maintain requests’ original placement times as well as pickup and delivery locations but set all passenger530
counts to one to enable more ridesharing opportunities.
Since we assume a no-rejection policy, we adjust the fleet size and the disposition of the vehicles according
to the settings of each instance. First, to guarantee all requests can be reached by a vehicle on time, we
determine the minimum set of regional centers from which vehicles can access all nodes in less than three
minutes (i.e., the shortest expected maximum pickup delay of all SQ classes). Then, for each request, we535
assume there will be a four-seat vehicle, available throughout the entire planning horizon, stationed at a
regional center that can adequately reach it, such that |K|=n. Although our model allows for vehicle
heterogeneity, we do not investigate such a feature in this formulation to curb computational complexity.
Creating a viable instance for a heterogeneous fleet would require stationing at least Qmax ×nvehicles at
each regional center, assuming vehicle capacities can range from one to Qmax = max{Qk| ∀kK}.540
5.3.2. Dynamic DARP-SQC
We assume that the provider relies on an initial working fleet of 1,000 four-seat vehicles to service the
transportation demand described in 5.2. If the policy allows, the provider can inflate the fleet size by
occasionally hiring privately-owned vehicles (on a single-ride basis) to meet minimum class service level
requirements. By default, we assume the capacity of the hired vehicles is equal to the number of passengers545
of the request that first prompted the hiring.
5.4. Dynamic formulation benchmarking
We implement three policies by varying the objective function of our matching algorithm:
Min. waiting (MW): Standard formulation, with no service level constraints and no hiring, where the
goal is to hierarchically minimize the rejection penalties and waiting times across classes fMW ={freject
C, f wait
Min. fMW (xv, zi) s.t. (35), (36), (37).
SL constraints (SL): No hiring formulation, enforcing service-level constraints, and aiming to hierar-
chically minimize fSL ={ffeasible
ccC, f reject
ccC, f violate
ccC, f wait
Min. fSL (xv, yi, zi, ac) s.t. (35), (36), (37), (39), (40), (41).
Table 7: Instance settings for both static and dynamic problem formulations.
Problem formulation
Characteristic Static Dynamic
#Requests {15, 20}42,702
#Instances 5 request samples -
Round duration (κ) 30s 30s
#Rounds 1 120
Company fleet size - 1,000
Capacity of company vehicle - 4 seats
Hireable fleet size Equal to n. of requests Unlimited
Capacity of hireable vehicle 4 seats 1, 2, 3, or 4 seats
SL constraints + Hire (SLH): Proposed formulation, enforcing service-level constraints, allowing
hiring, and aiming to hierarchically minimize fSLH ={ffeasible
ccC, f reject
ccC, f hire, f violate
C, f wait
Min. fSLH (xv, yi, zi, hk, ac) s.t. (35), (36), (37), (38), (39), (40), (41).
Although we assume providers are allowed to break service level expectations of up to (1 σc) percent
of users for each class cC, we consider rejections a more critical service-level violation than delays.550
Following this assumption, throughout all policies, we first seek to minimize rejections. Subsequently, policy
MW focuses on minimizing waiting times, whereas policies SL and SLH focus on minimizing service-level
violations, which cover both rejections and pickup delays. Particularly in SLH, we use the hiring capabilities
to prevent rejections from happening even further, once avoiding user rejection is more important than fleet
size minimization. Ultimately, by contrasting MW, SL, and SLH, we can assess the effect of enforcing555
service-level constraints alone as well as how they influence vehicle hiring.
6. Results
The static and dynamic formulations were developed in Python and Java, respectively, and all MILPs
were implemented using Gurobi 8.1.0. The experiments were performed on an AMD Opteron central pro-
cessing unit (CPU) running at 2.10 GHz and 128 GB of random access memory (RAM).560
The static formulation results (Section 6.1) help us to understand how fleet size varies to completely
accommodate the service quality requirements of different user bases throughout the whole optimization
horizon. In contrast, the dynamic formulation results (Section 6.2) highlight the tradeoffs involved in
controlling service quality online by periodically organizing operations to account for the needs of incoming
request batches.565
6.1. Static DARP-SQC
Table 8 presents the results for the static DARP-SQC model. Each instance is run for a maximum of
five hours. The figures for each number of requests, user base, and service rate combination are obtained
by averaging the results of the five different request distributions. In column “N. of hired,” we present the
average number of hired vehicles, considering that the fleet size equals the number of requests. Next, for570
each SQ class, we show the pickup and ride delays. Finally, column “N. of solved inst.” presents the number
of instances that could be solved optimally. We use these instances to calculate the averages.
The fleet sizes achieved under the no service level enforcement scenario (i.e., σ= 0) represent the
minimum number of vehicles required to pick up all users. In contrast, the slightly higher fleet sizes achieved
under the 80%, 90%, and 100% service rates highlight the compromise between the number of vehicles and the575
class delay tolerances enforced by the SQC constraints. From an AMoD platform’s perspective, these results
illustrate how exploring users’ tolerance to extra delays effectively decreases fleet size while maintaining the
strict service quality imposed by previously laid out agreements. By setting up balanced SQCs, platforms
Table 8: Number of vehicles hired, pickup delay, and ride delay for the static DARP-SQC instances across all case studies.
Figures correspond to the average of the results achieved for instances that could be solved to optimality.
N. of
req. User
base Service
rate N. of
Pickup delay (s) Ride delay (s) N. of
15 B+ 0% 10.4 138.3 175.6 203.5 0.0 22.9 12.9 5
80% 11.8 90.9 134.2 221.6 0.0 48.4 12.9 5
90% 12.2 90.5 109.1 220.5 0.0 48.4 12.9 5
100% 12.2 90.5 109.1 220.5 0.0 48.4 12.9 5
S+ 0% 9.0 57.6 143.3 176.3 0.0 46.8 57.0 4
80% 9.2 57.6 126.0 175.0 0.0 60.7 56.6 4
90% 9.2 57.6 126.0 175.0 0.0 60.7 56.6 4
100% 9.8 57.6 115.0 175.0 0.0 49.8 56.6 4
L+ 0% 9.0 68.8 152.3 165.1 0.0 0.0 67.3 4
80% 9.2 59.2 147.9 163.6 0.0 0.0 67.3 4
90% 9.5 59.2 124.6 163.6 0.0 0.0 60.8 4
100% 9.5 59.2 124.6 162.4 0.0 0.0 61.9 4
20 B+ 0% 14.0 124.2 227.8 205.6 0.0 28.5 8.1 5
80% 16.8 73.1 162.0 223.5 0.0 28.5 12.3 5
90% 17.2 73.1 134.4 199.2 0.0 6.0 11.5 5
100% 17.2 73.1 134.4 199.2 0.0 6.0 11.5 5
S+ 0% 11.0 170.0 199.7 159.1 0.0 25.9 0.2 2
80% 12.5 94.6 193.0 184.7 0.0 23.9 30.8 2
90% 13.5 76.0 166.7 174.8 0.0 14.5 7.2 2
100% 14.5 74.8 141.0 173.8 0.0 14.5 7.2 2
L+ 0% 10.0 116.0 201.7 181.7 0.0 42.3 49.7 1
80% 10.0 116.0 103.3 197.7 0.0 151.3 39.4 1
90% 10.0 116.0 103.3 197.7 0.0 151.3 39.4 1
100% 11.0 116.0 69.2 197.7 0.0 133.2 35.2 1
can avoid excessive hiring by exploiting the delay tolerance of specific classes while guaranteeing maximum
performance for high-demanding user classes.580
From Table 8, we can also see that, even for a small twenty-request batch, optimal solutions could be
found only for a few instances from user bases S+ and L+. Conversely, the predominance of business-class
requests in the user base B+ allows reaching optimal solutions in every case. Having more standard and
low-cost class users, who are willing to share a ride and wait longer, leads to a larger number of possible
routes, which translates to additional complexity. Ultimately, the results indicate that addressing realistic585
instances for the DARP-SQC requires more computationally efficient algorithms.
6.2. Dynamic DARP-SQC
Throughout the following sections, we provide an exploratory analysis of the aggregate results regarding
all user bases and service-level enforcement rates for all policies. We use the case study consisting of user
base S+ and service-level enforcement rate σ= 0.9 to illustrate our results further and refer to it as the590
reference case study. In this case study, the most frequent SQ class (i.e., the standard class) imposes a
balanced SQC when compared to its counterparts, requiring reasonably fast service levels and also allowing
for ridesharing. On the other hand, the reference service-level enforcement rate allows us to thoroughly
assess the flexibility achieved by violating the expectations of 10% of the users.
6.2.1. Service quality distribution595
Table 9 summarizes the average service quality outcome across classes achieved by each policy for all
case studies. The column “Serviced” shows the percentage of users picked up by the AMoD system. In turn,
the column “Met SL” presents the percentage of satisfied users, serviced according to their expected service
levels. Subsumed under the header “Violated SL”, we present the two service level violations we consider in
our approach. The column “Delayed” indicates the percentage of users whose desired service levels could not600
be fulfilled, whereas the column “Rejected” indicates the percentage of requests that could not be serviced.
It is worth noting that each percentage under the “Serviced” column is the sum of corresponding percentages
under “Met SL” and “Delayed” columns.
The user base segmentation scenarios play a significant role in the number of users serviced when vehicle
hire is not enabled. Once user base scenario B+ comprises more business users, which expect shorter delays605
and private rides, less sharing occurs, leading to about 10% fewer pickups than the other bases. However,
by exploiting the user delay tolerances, our SL policy can achieve higher service rates than the MW policy
across case studies.
By delaying users who can wait longer, the SL-policy matching process can enable additional possibili-
ties to combine overlapping routes. Although policy MW minimizes user class rejections and waiting times610
hierarchically, the lack of service-level enforcement constraints prevents the optimization process from ade-
quately harnessing class delay tolerances. For example, for the reference case study, enforcing service-level
constraints leads to a 29.81% percentage point increase over the 55.92% number of satisfied users (i.e., met
service levels) from policy MW. At the same time, from Table 9, we can see that the number of serviced
users increases moderately (from 96.14% to 98.16%). Ultimately, these results suggest that our service-level615
enforcement constraints were useful to reorganize pickups, resulting in a 53.3% increase in service quality
across classes in the reference case study.
Figures 8 and 9, detail the service-level distribution for all requests, considering user base S+ and service-
level enforcement rate σ= 0.9. In Figure 8, we can see that because policy MW does not enforce users’
class service-level expectations, pickup waiting times of business users end up being the highest among the620
classes. Once we aim to first minimize rejections across classes in MW, this result suggests that by delaying
the business users, the optimization process could pick up more users from the remaining classes. Figure 8
also shows that policy SL rejects thirty four business users (0.4% of the 6,831 business requests) but is able
to achieve the expected three-minute maximum waiting time for around 75% of them.
Service-level enforcement rate versus vehicle hiring. Figure 9 offers an alternative perspective on625
how each policy makes use of the 10% service level violation allowed by adopting the service-level enforcement
rate σ= 0.9. For SL, which features the relaxed version of the service-level enforcement constraints (41),
Table 9: Service quality achieved using each policy for all case studies.
base Service
Violated SL
Policy Serviced Met SL Delayed Rejected
B+ 0% Min. waiting 84.99% 37.35% 47.64% 15.01%
80% SL constrs. 85.09% 55.32% 29.77% 14.91%
SL constrs. + Hire 100.00% 91.98% 8.02% -
90% SL constrs. 85.18% 56.01% 29.17% 14.82%
SL constrs. + Hire 100.00% 95.91% 4.09% -
100% SL constrs. 84.92% 56.84% 28.08% 15.08%
SL constrs. + Hire 100.00% 100.00% - -
S+ 0% Min. waiting 96.14% 55.92% 40.22% 3.86%
80% SL constrs. 98.19% 85.58% 12.61% 1.81%
SL constrs. + Hire 100.00% 91.05% 8.95% -
90% SL constrs. 98.16% 85.73% 12.43% 1.84%
SL constrs. + Hire 100.00% 95.58% 4.42% -
100% SL constrs. 98.37% 86.69% 11.68% 1.63%
SL constrs. + Hire 100.00% 100.00% - -
L+ 0% Min. waiting 96.53% 78.53% 18.00% 3.47%
80% SL constrs. 98.00% 93.52% 4.48% 2.00%
SL constrs. + Hire 99.34% 96.14% 3.20% 0.66%
90% SL constrs. 97.79% 94.07% 3.72% 2.21%
SL constrs. + Hire 99.63% 98.05% 1.59% 0.37%
100% SL constrs. 97.76% 94.41% 3.35% 2.24%
SL constrs. + Hire 100.00% 100.00% - -
Figure 8: Pickup waiting time distribution and number of rejected users for each SQ class across all policies for user base S+
and service-level enforcement rate σ= 0.9 for policies SL and SLH. The dashed lines mark the expected max. pickup delays of
user classes.
Figure 9: Service-level violation breakdown into percentages of rejected and delayed users for each SQ class across all policies
for user base S+ and service-level enforcement rate σ= 0.9 for policies SL and SLH.
we can verify how this relaxation took place across classes. For business and standard users, the share of SL
violated requests are 11.24% and 17.73%, which correspond to 1.24 and 7.73 percentage points higher than
the 10% violation limit. In contrast, for the SLH policy, the violations for business and standard users total630
3.73% and 5.60%, figures significantly lower than the SL violation limit. This result could indicate that the
optimization process is overhiring vehicles since there was still extra room for violating user expectations
(up to 10%). However, by analyzing which user class hired vehicles are addressing the most, we can further
understand why the service violation limit was underutilized.
Table 10 expands the summary statistics presented in Table 9. For each user base segmentation scenario,635
it shows for every user SQ class the percentage of users picked up by company or freelance vehicles as well
as the percentage of users whose service-level expectations are violated. From Table 10, we can see that
vehicle hiring occurs predominantly to fulfill business user requests. Therefore, these hirings end up freeing
the four-seat company vehicles that would have been used to transport these requests privately. As a result,
the ridesharing requests from standard and low-cost classes can enjoy a larger vehicle supply, which leads640
to fewer service-level violations. This extra vehicle supply, in turn, is reflected in the underutilization of the
service-level violation limits.
Long-term fairness. Ultimately, it is worth noting that, although we enforce class service levels, individual
users may experience service-level violations repeatedly, over a longer time horizon. To further minimize
user dissatisfaction, a real-world service provider could strive to make up for repeating users who have had645
their service levels violated in previous rides. Provided that users iPare associated with unique ids in
the AMoD system, the variable of yican be set up ad-hoc to grant users the best service level. This way, σc
rates could also be applied individually by keeping track of each traveler’s service level over multiple rides.
The effect of rebalancing. Regarding the rebalancing approach, the inclusion of service-level violations
as an additional stimulus to prompt rebalancing (besides service rejection) has been shown to increase fleet650
productivity. It can be seen from Figure 10 that in MW, idle vehicles can still be found until about 18:30,
whereas in SL there are no idle vehicles from about 18:10 and onward. The number of idle vehicles drops
sharply as soon as the system fails to fulfill user expectations across classes. As a result, by rebalancing
vehicles earlier, policy SL contributes to achieving a two percentage point increase in the number of users
picked up compared to MW. Moreover, when users cannot be rejected (i.e., policy SLH), the hiring operations655
have successfully replaced the user rejection stimuli to indicate where vehicles are needed.
Table 10: Share of requests whose service-level expectations were met (through company or freelance vehicles) or violated, according to SQ class for each user base
distribution, service-level enforcement rate, and policy.
base Service
Business Standard Low-cost
Policy Company Freelance Violated Company Freelance Violated Company Freelance Violated
B+ 0% Min. waiting 14.6% - 85.4% 73.3% - 26.7% 98.3% - 1.7%
80% SL constrs. 38.2% - 61.8% 87.1% - 12.9% 96.0% - 4.0%
SL constrs. + Hire 60.4% 29.7% 9.9% 92.1% - 7.9% 100.0% - -
90% SL constrs. 38.7% - 61.3% 87.5% - 12.5% 98.2% - 1.8%
SL constrs. + Hire 64.1% 31.2% 4.8% 94.8% * 5.2% 100.0% - -
100% SL constrs. 39.3% - 60.7% 89.4% - 10.6% 98.7% - 1.3%
SL constrs. + Hire 66.8% 33.2% - 99.4% * - 100.0% - -
S+ 0% Min. waiting 25.9% - 74.1% 56.0% - 44.0% 85.8% - 14.2%
80% SL constrs. 86.0% - 14.0% 82.9% - 17.1% 96.7% - 3.3%
SL constrs. + Hire 75.4% 16.8% 7.8% 88.4% * 11.3% 100.0% - -
90% SL constrs. 88.8% - 11.2% 82.3% - 17.7% 97.7% - 2.3%
SL constrs. + Hire 78.2% 18.1% 3.7% 92.8% 1.6% 5.6% 100.0% - -
100% SL constrs. 90.3% - 9.7% 83.1% - 16.9% 98.4% - 1.6%
SL constrs. + Hire 76.6% 23.4% - 97.9% 2.1% - 99.9% * -
L+ 0% Min. waiting 25.4% - 74.6% 61.8% - 38.2% 94.9% - 5.1%
80% SL constrs. 83.8% - 16.2% 87.3% - 12.7% 97.2% - 2.8%
SL constrs. + Hire 82.6% 9.0% 8.4% 88.4% - 11.6% 99.0% - *
90% SL constrs. 84.4% - 15.6% 89.0% - 11.0% 97.5% - 2.5%
SL constrs. + Hire 80.3% 15.9% 3.7% 92.6% 1.2% 6.2% 99.5% - *
100% SL constrs. 85.5% - 14.5% 90.1% - 9.9% 97.5% - 2.5%
SL constrs. + Hire 77.2% 22.8% - 95.8% 4.2% - 99.3% * -
Values lower than 0.01% are marked with “*” and absent data are marked with “-”
Table 11: Average number of hired vehicles per capacity and summary statistics for the hired fleet in each round.
base Service
Avg. hired/Vehicle capacity Hired vehicles
1 2 3 4 Avg. Median Max.
B+ 80% 76.39% 19.41% 2.74% 1.46% 223.88 309 493
90% 79.67% 15.79% 3.21% 1.32% 243.75 336 497
100% 79.92% 14.75% 4.16% 1.17% 254.74 355 520
S+ 80% 76.41% 15.48% 4.82% 3.29% 32.79 22 102
90% 73.45% 17.40% 5.45% 3.70% 47.41 43 168
100% 74.29% 18.57% 4.94% 2.20% 60.32 65 167
L+ 80% 66.49% 26.89% 3.36% 3.26% 15.64 8 72
90% 77.33% 17.55% 3.50% 1.62% 30.85 22 104
100% 76.71% 18.01% 2.61% 2.67% 56.07 51 191
6.2.2. Fleet size inflation
From the overall fleet statistics presented in Table 11, we can see that the average, maximum, and median
numbers of hired seats throughout all rounds are proportional to the service quality of each user base and
service-level enforcement rate. For instance, we can see that the overrepresentation of business-class users660
is translated accordingly in the fleet of vehicles hired to service formerly dissatisfied users in policies MW
and SL. Moreover, most hired vehicles end up being of low capacity. Once we assume that a hired vehicle’s
capacity equals the number of seats required by the request that prompted the hiring, this outcome is
congruent with the characteristics of the user demand (77% of the requests demand only one seat). For the
reference case study, Table 11 shows that at the demand peak, the total fleet capacity grows by 168 vehicles665
(see the hiring peak in Figure 10 between 18:15 and 18:30).
6.2.3. Whole day operation
In Figure 11, we show the fleet status for a full-day instance considering 210,269 requests configured
according to the reference case study. Hirings occur predominantly in the evening, and the surplus number
of vehicles over the 1000-vehicle fleet is lower than 200. From the figure, we can conclude that the fixed670
working fleet can meet the entire demand most of the time and still be sub-utilized at some periods (e.g.,
before 6:00). Our results stress how much vehicle downtime a provider can avoid by partially relying on the
FAV fleet to completely fulfill the demand.
6.2.4. Computational complexity
The same shortcomings found for the static version regarding the more flexible user-segmentation sce-675
narios are also present in the dynamic version. In these scenarios, the lexicographical method cannot be
completed on some rounds due to the maximum computation time we have imposed on the optimization
process. Consequently, some assignments are sub-optimal, covering a subset of the objectives (high-priority
first). This outcome is particularly present for scenario L+. From Table 11, we can see that the maximum
number of hired vehicles for this user segmentation for a 100% service rate is higher than the S+, where users680
have a lower waiting tolerance. Additionally, Table 9 shows that 0.37% of low-cost users are rejected while
using the hiring policy SLH, where hired vehicles are made available to pick up all users. This indicates
that, for some rounds, the hierarchical optimization stops prematurely, failing to minimize class rejections
and vehicle hire.
6.3. Managerial insights685
With widespread AV adoption, cities may implement policies that prioritize high occupancy vehicles or
even charge empty-seat taxis. Such policies serve two main purposes. First, they discourage excessive low-
occupancy AV rides, counterbalancing potential rises in overall vehicle kilometers traveled (VKT). Second,
they contribute as alternative sources of income for cities, especially in the shadow of an expected revenue
drop due to less parking space requirements (Nourinejad et al., 2018). Hence, keeping high occupancy rates690
Min. waiting (MW) SL constraints (SL) SL constraints + Hire (SLH)
Carrying Idle Rebalancing Cruising to pick up
1 2 3 4 passenger(s)Vehicle status:
Figure 10: Size of the working fleet every thirty seconds over the course of one hour for each policy, considering user base S+
and a 90% SL enforcement for SL and SLH. Colors help to identify the part-to-whole ratio of vehicles per status at each period.
Carrying Idle Rebalancing Cruising to pick up
1 2 3 4 passenger(s)Vehicle status:
Figure 11: Size of the working fleet every thirty seconds over the course of an entire day, considering user base S+ and a 90%
SL enforcement for SLH, policy. Colors help to identify the part-to-whole ratio of vehicles per status at each period.
and reducing total VKT are critical factors for future AMoD providers. Our multi-objective function covers
both goals once it determines the minimum fleet size and mix to meet the demand expectations.
However, despite advantageous for the operator, single-ride contracts are likely to be unpopular among
owners. Under such an agreement, there might be occasions where vehicles are made available for a whole
day but make only a couple of trips. In this low-profit scenario, owners could, for instance, link their695
participation to a minimum compensation, such that their vehicles are used more frequently.
Alternatively, to guarantee frequent use, providers could establish long-duration contracts to actively
manage and rebalance vehicles as if they belonged to the fixed working fleet. Under such contracts, however,
operators may be required to bear all costs within the leasing period. These may include not only the costs
to rebalance and service users but also the opportunity costs of hireable vehicles working for other competing700
operators. Therefore, once costs are considered, the rider’s satisfaction must be adequately translated into
user class fares to cover the provider’s outlay concerning vehicle hire.
Moreover, when hiring, we assume there is an infinite number of FAVs readily available in each request’s
surroundings. This assumption entails that there exists an efficient hiring strategy in place, which hires
vehicles of adequate capacities in advance and rebalances them to locations where the own fleet historically705
cannot fulfill the demand. Such an anticipatory hiring strategy could be the key to reaching a more reasonable
compromise between FAV owners and providers’ objectives in a real-world scenario. In this case, independent
owners have an extra incentive to make their vehicles available to join the provider’s fleet once they are
compensated even before they are assigned to a request.
7. Conclusions710
In this study, we have introduced a new operational approach to actively control service quality in AMoD
systems, increasing and decreasing the number of used vehicles in the short term to meet diversified user
expectations. We have used these expectations to establish service quality contracts, allowing heterogeneous
users to choose ride experiences that best match their preferences, especially regarding maximum waiting
times and willingness to share a ride. Based on an experimental study using New York City taxi data, we715
have found that the developed approach allows us to significantly improve the service quality of all considered
user categories. Enforcing the proposed service-level constraints, we can meet the expectations of 85.7% of
the users across classes, a 53% average increase in comparison to conventional ridesharing systems. When
hiring is enabled, we can meet the expectations of 95.6% of the user requests, at the expense of a mild fleet
inflation (a maximum surplus of 168 FAVs is observed at the evening demand peak).720
The proposed method allows providers to make a compromise between avoiding service-level violations
and hiring extra vehicles, steering the vehicle supply to service the highest priority or most profitable
customer segments, according to the particular conditions of an operational environment (e.g., availability
of idle vehicles, hiring costs, user dissatisfaction costs). Nevertheless, hiring occurs only to the extent that
it preserves minimum service level requirements. Hence, we add to recent literature by providing an active725
means to control service quality on an operational level and distinguishing services classes in terms of user
Moreover, we have formalized the problem using a MILP formulation and proposed a matheuristic
to deal with large-scale real-world instances. This matheuristic encompasses the construction of feasible
ridesharing visiting plans and the optimal assignment of plans to available vehicles. Besides ride-matching,730
the assignment phase also includes a reactive rebalancing strategy that uses both service level violations and
vehicle hire as stimuli to reposition vehicles. Using a lexicographic method, we have solved a multi-objective
function where the primary goal is to minimize rejections (i.e., the most significant source of inconvenience)
to subsequently minimize fleet size and service-level violations. We have also evaluated to what extent hiring
extra vehicles affects overall service quality and fleet usage, by designing twelve scenarios where we vary735
the rate at which providers commit to fulfilling user service level expectations. In this way, our results help
understand the operational impact of meeting user service quality expectations and the tradeoffs entailed
by occasionally violating service quality expectations.
Although we have restricted the analysis to three user segments, the service quality classes we propose are
general enough to serve as the basis for more diversified customer-centric services. For the classes considered,740
our results show that, regardless of the user base, most hired vehicles join the initial fleet to service one-
passenger requests from the high-quality user segment (i.e., the business class). These findings indicate
that providers may benefit from leveraging the strengths of two AMoD paradigms, namely, ridesharing and
single-passenger personal mobility, on a common platform, without the necessity of owning the entire fleet.
Future work will focus on setting the relation between service quality and total costs. This way, operators745
can weigh the advantage of 1) investing in PAVs or hiring FAVs online to avoid service quality contract
breaches and 2) compensating users in the event of poor service. Ultimately, service fares must reflect PAV
operational costs (e.g., maintenance, management, cleaning, investment, parking, insurance, city fees), FAV
owner’s custom preferences and hiring costs, and the platforms’ reputation as a function of the consistent
fulfillment of service quality expectations. Additionally, a promising future direction is considering AMoD750
providers integrate a broader transportation ecosystem. In this setting, different modes are orchestrated to
achieve overarching mobility goals, such as mitigating congestion and distributing accessibility over different
regions and demographics. From the providers’ perspective, such an integration would require weighing a
series of alternative factors (e.g., congestion pricing, empty-vehicle fees, parking costs, ride subsidization
schemes) into the fleet management strategy. Finally, one could also focus on addressing the inherent755
uncertainty of the proposed scenario, in which both vehicle supply and demand unfold throughout time.
This research is supported by the project “Dynamic Fleet Management (P14-18 – project 3)” (project
14894) of the research programme i-CAVE, partly financed by the Netherlands Organization for Scientific
Research (NWO), domain Applied and Engineering Sciences (TTW).760
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Min. waiting (MW) SL constraints (SL) SL constraints + Hire (SLH)
Carrying Idle Rebalancing Cruising to pick up
1 2 3 4 passenger(s)Vehicle status:
Figure 12: Test case depicted in Figure 10 considering less computation time and a smaller graph of feasible solutions. The
number of users picked up drops about 10% for policies MW and SL. Due to the premature termination of the lexicographic
method, policy SLH is unable to realize all the hires necessary to pick up all requests.
8. Execution time vs. solution quality855
We decrease execution time by parallelizing the construction of both RV and ERTV graphs. As in
Alonso-Mora et al. (2017), to build the RV graph, we determine suitable pairs for requests in parallel,
whereas to build ERTV, we create vehicle visiting plans in parallel, exploring increasingly longer visiting
sequences for each vehicle separately. Also, we tradeoff solution quality with faster computation by limiting
the sizes of RV and ERTV graphs and imposing maximum execution timeouts.860
In this study, for the RV graph, we assume requests can be connected to up to sixty vehicles and as
many requests as possible. If hiring is enabled, we guarantee that the request-vehicle edge set will always
include the edge featuring the closest backup hireable vehicle, even if better options (i.e., closer vehicles) in
the working fleet are available. Doing so allows us to achieve our objective of minimizing rejections once
the hireable edge will certainly integrate the pool of visiting plans. As for the ERTV graph, we consider the865
vehicle visiting plan’s exploration timeout to 0.4 seconds and let the optimal assignment method execute
for at most 240 seconds in the Gurobi solver. We refer to this configuration as C60 4 240.
To show how limiting the execution times and graph size affects solution quality, we rerun our experiments
considering that for the RV graph, each request is connected to the closest fifteen vehicles (including the
hireable, if applicable) and fifteen requests, therefore totaling at most thirty pairs. Regarding the ERTV870
graph, we determine a timeout of 0.2 seconds per vehicle to interrupt the exploration of candidate visiting
plans and set up a thirty-second maximum execution time for the matching formulation. We refer to this
configuration as C30 2 30.
Table 12 presents the execution times for all test cases assuming the limitations described for both
configurations. Similar to the static formulation, the higher the flexibility of the predominant user class,875
the more complex the problem, resulting, therefore, in longer execution times. In Figure 12 and 13, we
replicate Figures 10 and 11 considering configuration C30 2 30. Under stricter timeouts, the optimization
process has fewer opportunities to find high-quality solutions, leading to decreased sharing (see the increase
in vehicles carrying one passenger) and substantially more hirings at the demand peaks. Additionally, less
computational time may harm user experience even when hiring is enabled. In Figure 10, for example, 0.01%880
of users are rejected because the maximum execution time expires before a no-rejection solution could be