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Effective Heuristics for Distributing Vehicles in
Free-floating Micromobility Systems
Lacy M. Greening, Alan L. Erera
School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332,
lacy.greening@gatech.edu, alan.erera@isye.gatech.edu
The recent proliferation of free-floating, shared micromobility systems has resulted in idle bikes and e-scooters
crowding sidewalks. A key operating challenge for such systems is to determine a best distribution of these
vehicles across a spatial region at any point in time to maximize system performance measured by metrics
like trips served per time or average vehicle utilization. We present simple but effective simulation-based
heuristic approaches for determining such a best distribution using an approach that discretizes both space
and time. To gauge the quality of these heuristics, we propose an integer programming optimization model
that uses a time-space network representation of such systems to determine a best vehicle distribution with
complete foresight of a known demand scenario; solving this model for a large sample of possible demand
scenarios yields an a posteriori upper bound. We additionally develop a two-stage stochastic programming
model that can be used to find initial vehicle distributions for smaller problem instances (with more coarse
discretizations) using a small set of sample days. Computational experiments using a test case study focusing
on the city of Atlanta, Georgia are used to demonstrate the value the proposed approaches offer to both
cities and companies, and provide evidence that simulation-based heuristics outperform alternative solution
methods and generate provably high-quality solutions for many instance types.
1. Introduction
Shared mobility is defined as the shared use of a car, scooter, bicycle, or other travel mode with
the goal of providing short-term access to transportation on an as-needed basis (Shaheen et al.
2016). Since their arrival in 1994, these services have increased accessibility and reduced driving
and personal vehicle ownership. Shared mobility systems can be categorized by the length of trip
taken by the user: short (0-5 miles), mid-length (5-15 miles), and long-distance (greater than 15
miles). Of the three distances, short trips comprise almost 60% of the total number taken per year
in the United States; these are usually served by micromobility transport services (CBI 2020). The
micromobility industry initially began with station-based bike sharing (SBBS) systems, which then
evolved to free-floating bicycle sharing (FFBS) systems, and now includes free-floating e-bike and
e-scooter systems. In SBBS systems, available bikes are docked at stations throughout a city, and
these stations have fixed capacities. In free-floating systems, some types of bikes can be locked to
1
2Greening, Erera: Heuristics for Distributing Vehicles
any fixed structure, such as a bike rack or light pole, while other types of bikes and scooters can
be left wherever the rider ends their trip. Free-floating system operators face lower start-up costs
and a simpler deployment process since very little fixed infrastructure is required; they need not
invest in stations or spend time deciding and negotiating where fixed stations should and can be
placed.
One major difference between station-based systems and free-floating systems is the rider expe-
rience, where a rider now refers to a user of a micromobility service. In station-based systems, a
potential rider travels to a dedicated station with available vehicles; if a vehicle is unavailable at the
nearest station, the next station may not be in a convenient location for the rider. Additionally, the
rider must leave the vehicle at the end of a trip at a station with available capacity. Depending on
the final destination, there may not be a station with capacity nearby, and this may discourage the
potential rider from renting a vehicle. In free-floating systems, a rider may locate a vehicle much
closer to their origin because they are not constrained to stations, and even more convenient is
the ability to leave the vehicle wherever the trip ends. Thus, the transition to free-floating systems
has increased rider flexibility and has led to higher levels of satisfaction by decreasing total travel
times. However, rider flexibility and associated satisfaction depends strongly on the ability to locate
vehicles near trip origins. Instead of locating stations and deciding on their vehicle inventories,
free-floating system operators must decide how to distribute a vehicle fleet across many potential
inventory locations within the service region.
Many free-floating micromobility systems use a simple but novel approach to reposition vehicles.
In station-based systems, companies typically manage all aspects of the operations: repositioning
and repairing, as well as charging, if electric. However, many free-floating system operators employ
independent contractors for charging and repositioning. These contractors receive a fixed fee to
find and collect ready-to-be-charged vehicles, clean, make minimal repairs, charge overnight, and
then release their charged vehicles to several possible locations by a target time the next morning.
This nightly process allows the company to use a simple redistribution scheme to move its vehicles,
just relying on instructions to chargers for replacing charged vehicles. Such a scheme can also be
used for dynamic redistribution by either employing contractors to pick up, charge, and release
vehicles throughout the day or by using crowdsourcing to reposition vehicles when needed. Note
that system operators that use crowd-sourced chargers provide flexibility to them regarding where
to return vehicles, and this may make it more difficult to precisely distribute a fleet geographically.
Regardless of when vehicle repositioning may occur, it is nevertheless important to decide where
the fleet of vehicles should be located if possible at any given point in time to maximize one or
more system performance objectives.
Greening, Erera: Heuristics for Distributing Vehicles 3
In this paper, we consider the problem of determining how to distribute a fleet of electric micro-
mobility vehicles within a geographic region at a specific time. Our objective is to maximize vehicle
utilization (and hence trips served) during a planning horizon, and we examine the sensitivity
of this performance metric to changes in daily starting locations to determine what benefits are
achieved from strategically positioning vehicles. We study various approaches for creating a good
distribution of vehicles, and we conclude that strategically positioning vehicles at the beginning
of an operating day can improve their overall availability to riders and increase their usage. To
summarize the primary contributions of this work, we:
•develop effective simulation-based optimization heuristics for determining high-quality distri-
butions of fleets of micromobility vehicles serving uncertain trip demands over daily planning
horizon, including a simple greedy construction heuristic and multiple improvement heuristics
based on swap neighborhoods;
•propose important implementation strategies for simulation-based heuristics for this problem,
including a so-called Approximate Gradient improvement heuristic that targets and evaluates
via detailed simulation only the likely most-effective region-to-region vehicle swap pairs;
•create an upper bounding approach for this problem that uses a novel mixed integer linear
programming formulation that captures some important features of the probabilistic matching
of available vehicles to potential trips within origin regions;
•build a two-stage stochastic optimization model to approximate the true multi-stage opti-
mization problem, and demonstrate its effectiveness compared to simpler simulation-based
heuristics; and
•conduct a case study using data representative of e-scooter fleet operations in Atlanta, Georgia,
that shows the overall effectiveness of simulation-based heuristics for determining provably
high-quality fleet distributions.
The remainder of this paper is organized as follows. In Section 2, we discuss relevant literature
focusing on station-based and free-floating micromobility sharing systems. In Section 3, we formu-
late a multi-stage optimization problem for determining the best point-in-time vehicle distribution
for free-floating vehicle systems. In Section 4, we present optimization-based approaches for deter-
mining upper bounds to this optimization problem, and both a two-stage stochastic programming
model and also simulation-based construction and improvement heuristics for finding heuristic
solutions. In Section 5, we use a case study based on data from Atlanta, Georgia to demonstrate
the effectiveness of these approaches on practical problems. Finally, Section 6 highlights potential
areas of future work.
4Greening, Erera: Heuristics for Distributing Vehicles
2. Literature Review
An important challenge for micromobility systems is providing riders with adequate vehicle avail-
ability near their origins (and station capacity near their destinations for SBBS systems), given
budget and space constraints. Critical decisions include how many vehicles to include in a fleet and
how and when to distribute vehicles throughout a city. For a comprehensive overview of research
addressing such problems within shared mobility systems see Laporte et al. (2018) and Shui and
Szeto (2020).
Research on fleet-sizing for SBBS systems must consider station locations and capacities and
their effects on satisfying demand. There has been work on this topic using queueing theory (George
and Xia 2011, Fricker and Gast 2016), stochastic network flow models (Shu et al. 2013), and bi-
level programming (Nair and Miller-Hooks 2014). However, work on fleet-dimensioning for FFBS
systems is not as prevalent. One reason for this is that free-floating system fleets are currently
constrained by local governments, and companies tend to choose to deploy the maximum number
allowed by regulation. However, as city governments implement new policies that require companies
to pay a fee for each deployed vehicle, fleet-sizing research for FBBS systems to maximize ridership
profitability may become more relevant.
A key operational problem for micromobility systems is vehicle repositioning for reuse. Repo-
sitioning for SBBS systems can either be conducted statically or dynamically. Two important
problems include the static bicycle repositioning problem (SBRP) and the dynamic repositioning
and routing problem (DRRP). The SBRP is to determine optimal routes for one or many trucks to
reposition bikes, typically at night, whereas the DRRP creates routes to reposition vehicles during
the operating day. Most research addresses static repositioning for station-based systems by solving
mixed integer linear programming (MIP) models. Solution techniques include various heuristics,
e.g., clustering, simulation-based, tabu search, etc., as well as exact algorithms and decomposition
techniques (Benchimol et al. 2011, Raviv et al. 2013, Dell’Amico et al. 2014, Forma et al. 2015,
Erdo˘gan et al. 2015, Szeto et al. 2016, Li et al. 2016, Kloim¨ullner and Raidl 2017, Schuijbroek et al.
2017, Bulh˜oes et al. 2018). Research on DRRP for SBBS systems has increased in recent years;
most of the work in the area has relied on various decomposition and heuristic methods to solve
stochastic and dynamic optimization models (Contardo et al. 2012, Chemla et al. 2013, Ghosh et al.
2017, Zhang et al. 2017, Warrington and Ruchti 2019, Brinkmann et al. 2019, Legros 2019). Repo-
sitioning for free-floating systems has also become more prevalent as free-floating e-scooter and
e-bike systems have continued to grow. Three recent papers model and solve a static-repositioning
problem for a FFBS system using multiple repositioning vehicles. Pal and Zhang (2017) propose a
hybrid nested large neighborhood search metaheuristic with variable neighborhood descent to solve
a MIP for the static complete rebalancing problem. Liu et al. (2018) formulate a similar problem,
Greening, Erera: Heuristics for Distributing Vehicles 5
but address unmet demand and the inconvenience that riders face when trying to find a bike.
Osorio et al. (2021) model the process of charging electric vehicles while completing rebalancing
operations, including modeling the routing of pickup and delivery operations.
Related to vehicle repositioning and most closely related to the work in this paper are research
works that focus on determining the best number of vehicles at each station for SBBS systems or
within specific areas for FFBS systems. Many of the papers above that solve SBRP or DRRP also
determine optimal or target inventory levels for stations, while others use target inventory levels as
inputs. Nair and Miller-Hooks (2011) were one of the first to study station inventory levels. They
developed a stochastic program with an objective of satisfying demand and minimizing the costs of
relocating vehicles across all stations. Raviv and Kolka (2013) studied the dynamics of inventory
at a single station in order to model and decrease rider dissatisfaction. Datner et al. (2017) built off
this work by incorporating station interactions in their mathematical models. Shu et al. (2013) use
a network flow model with Poisson arrivals to determine the distribution of bicycles across a set of
stations, the number of stations needed, and the value of daily redistribution. Vogel et al. (2014)
determined optimal station inventory levels for all stations when minimizing relocation costs using
a MIP that was solved using a matheuristic approach. Thus, there has been a substantial amount
of work on station-based system inventory levels. However, we are not aware of published research
on the ideal number of free-floating micromobility vehicles for specific regions within a service area.
3. Problem Description
We consider the operational-level problem of determining a best point-in-time distribution of a
fleet of free-floating, electric micromobility vehicles across a geographic service area, where the
objective is to maximize the number of potential rider trips served via successful matches to
available vehicles. Each rider wishes to begin their trip immediately, and thus demands become
known without any lead time (no reservations). The rider attempts to find a vehicle nearby to their
current location to begin the trip; if none is found readily, then this demand is abandoned. When
a rider matches with an available vehicle, they begin travel and the trip concludes at a destination
location where the vehicle is released. Released vehicles are available to be immediately used again
to serve new trips. Since electric micromobility vehicles rely on a battery to operate, they lose
charge during operations and may not be available for reuse before a charging operation or battery
swap is conducted. Many micromobility services face naturally low demand during the overnight
hours, and therefore this is a natural time to use for battery charging and vehicle repositioning. It
may also make sense to charge and reposition during normal operating hours.
To determine the best distribution of a fleet at some time, it is critical to understand how that
fleet will be used to serve trips moving forward in time after the positioning decisions. Doing so is
6Greening, Erera: Heuristics for Distributing Vehicles
challenging even if the set of possible future trips, along with their timing, origins, and destinations,
are known in advance and it becomes more difficult when demand is uncertain. In this paper, we
build a model of these operations using a discretized time-space mathematical network G= (N,A).
Suppose that the start of the planning horizon is considered the beginning of discrete time period
t= 1. The horizon T={1,2, ..., T }then consists of Tdiscrete time periods during which vehicles
are potentially used by riders, and suppose this discretization is chosen such that a vehicle cannot
serve more than one rider in any single time period. Also suppose that the service area is discretized
into a set Rof individual regions, i.e., the location and size of each region r∈ R are inputs, where
R≡ |R|. Let the nodes (r, t)∈ N represent the region rat the beginning of time twith a vector
of attributes that describe the vehicle fleet there. Arcs a∈ A connect time-space nodes forward in
time and allow vehicles to transition from a state at (r, t) to one at (r0, t0) where t0> t.
Let Vbe a set containing the vehicles in the micromobility fleet, where Vis used to denote the
fleet size, i.e., V≡ |V|. The best point-in-time distribution specifies the subset V1
r⊆ V to be located
in each region r∈ R at the beginning of the planning horizon, as shown in Figure 1 for a service area
composed of equally-sized regions. To model vehicle battery charge, we assume that each vehicle
v∈ V has a battery life Bvat t= 1 and that charge level decreases over time with vehicle usage; in
this research, we assume charge decreases directly proportional to miles traveled, but other more
sophisticated models might be used. In the models we will propose, a homogeneous fleet of vehicles
is assumed and we refer to the initial distribution of vehicles by the vector v=v1
1, . . . , v 1
R, where
each v1
irepresents the number of fully-charged vehicles in region iat the beginning of the planning
horizon.
Rider demand is also modeled using this time-space discretization. Let Dt
ij be the set of potential
trips to travel from region ito region jbeginning in time interval t∈ T , and Dt
ij ≡ |Dt
ij |; a potential
trip represents an opportunity to serve a rider given the availability of an appropriate vehicle.
An individual trip in this set may also have other attributes, for example its total distance. For
potential trip k∈ Dt
ij to be satisfied, the rider must be able to locate an available vehicle with
sufficient remaining battery life in origin region iat time t.
Demand in any given system is likely to vary with uncertainty over time, so a stochastic model
is appropriate for {Dt
ij }; for now, the particular stochastic model for these quantities is not par-
ticularly important although we do assume that they are exogenous to system state information
like the number of vehicles in regions or the number of prior trips executed. Given a geographic
and time discretization, however, it may be too optimistic to assume that a potential rider in i
at time twill find any vehicle vin the region with sufficient charge to complete their trip, so a
stochastic model for this matching is needed. To model a potential rider matching with a battery-
feasible vehicle, we assume the following simple probabilistic model. Consider the set SjDt
ij (ω) of
Greening, Erera: Heuristics for Distributing Vehicles 7
V1
1V1
2V1
3V1
4
V1
5V1
6V1
7V1
8
V1
9V1
10 V1
11 V1
12
V1
13 V1
14 V1
15 V1
16
Figure 1 Distribution of the set of fleet vehicles Vacross the set of regions R, where R= 16.
all potential trips outbound from region iat time tin some realization ω, and suppose that nit
is the number of potential trips in the set. First, we use Bernoulli trials to determine which of
these trips start near enough to any scooters with a success likelihood pt
ithat increases with the
density of available vehicles in i. In this paper, we assume that this success likelihood is given by
the following expression:
pt
i= min 1,vt
i
100Ai,(1)
where vt
iis the initial inventory of vehicles in region ifor time tand Aiis the size of the region
measured in square miles. Note that the resulting binomial number of successes, nS
it may exceed
the number of vehicles available in i. Thus, we next determine if a trip matches with a feasible
vehicle by drawing a random trip and vehicle separately; if the vehicle is feasible for the trip, then
this trip is executed with this vehicle removed from the pool. If not, the trip is not executed and
another match draw is considered. In this paper, we assume that all executed trips require only a
single time period to complete for simplicity.
Given this operational model, we can now pose a general multi-stage stochastic optimization
problem for determining the set of initial vehicles by region {V1
r}to maximize the expected number
of executed trips during the planning horizon. To reduce excessive notation, we limit this formula-
tion to the case where vehicles used for trips are only occupied for a single time period. Consider
the following multi-stage model:
max
{V1
r}Eω
X
t∈T X
(i,j)∈R×R
dt
ij (ω)
(2a)
s.t.X
r∈R
V1
r=V,(2b)
{(V2
rj (ω), d1
rj (ω))}=fω, r, {D1
rj (ω)},V1
r,∀r∈ R,∀ω, (2c)
{(Vt+1
rj (ω), dt
rj (ω))}=fω, r, {Dt
rj (ω)},Vt
r(ω),∀r∈ R,∀t∈ T \ {1},∀ω, (2d)
Vt
r(ω) = X
i∈R
Vt
ir(ω),∀r∈ R,∀t∈ T \ {1},∀ω. (2e)
8Greening, Erera: Heuristics for Distributing Vehicles
Figure 2 Visualization of vehicle state transition and trip demands executed.
The primary decision variables are the initial sets of vehicles in each region r∈ R. The formulation
additionally relies on accounting variables Vt
rj and Vt
r, defined for each uncertain realization ωand
t > 1, which account for the sets of vehicles moving between regions (or staying within region) from
period t−1 to t. On the input side, the outcome space of scenarios ωdefines not only the uncertain
trip demand over time but also the uncertainty in matching potential trips to vehicles over time.
Constraint (2b) ensures that the subsets V1
rpartition the vehicle fleet, where the summation is
defined on sets.
Expressions (2c) and (2d) model the primary state transitions from one period to the next. Note
that the transition function fis the same for each time period tand for each rbut relies on some
region characteristics, the potential demand, and the initial vehicle fleet; this function additionally
depends directly on ωsince the process by which potential trips are matched with available vehicles
is a source of uncertainty in addition to trip demand uncertainty. The state transition can be
characterized completely by the sets of vehicles Vt+1
rj moving from rto any other region j(including
remaining in j=r) serving trips beginning in period t. Figure 2 graphically depicts state transition
outcomes for region iduring time period t, depicting the new sets of vehicles moving from ito
jand kand the counts of completed trips, along with the set of vehicles remaining in region i
(after a within-region trip or in inventory) and the count of completed within-region trips. Finally,
expression (2e) simply aggregates the outbound vehicle sets from all regions iin the prior period
to form the input set Vt
rfor period t.
While the multi-stage optimization problem (2) is useful for summarizing this decision problem,
it is very difficult to address directly without some simplification. The primary difficulty is that
the outcome space of the uncertainty set is extremely large and, given that individual vehicles are
Greening, Erera: Heuristics for Distributing Vehicles 9
tracked with specific battery levels, the state space is likewise. In the following section, we will
introduce approaches for finding good (but suboptimal) solutions to (2) and a mechanism to assess
the quality of those solutions via comparison to an approximate upper bound.
4. Solution Approaches
In this section, we will develop approaches for finding provably good (but not necessarily optimal)
solutions to (2). Mixed-integer programming (MIP) models will be developed to find upper bounds
on the optimal objective, and then we will extend the simplest variant of these models to develop a
two-stage stochastic MIP model that maximizes a sample average objective as a heuristic for solving
(2). The second set of approaches will use more detailed simulations of operations to evaluate
prospective solutions within construction algorithms and within improvement local search. The
most detailed simulator will also be used with a larger sample size to assess and compare the
quality of solutions produced by all heuristic methods.
4.1. Deterministic Demand Optimization Model
First, we consider the special case of (2) with a single uncertain potential trip demand realization
{Dt
ij (ω)}which is assumed to be known in advance. In this section, we will show that a MIP built
using Gas a vehicle flow network can be used to determine a tight upper bound on the maximum
number of trips that can be served given this demand realization and a single realization of the
uncertainty governing the process of matching potential trips to available vehicles.
Importantly, the model proposed here does not assume complete knowledge of ω; although
the realization of potential demand is known, the model approximates the stochastic process of
matching potential rides with available vehicles in a region. We will show that the approximation
developed leads to an upper bound on the maximum trips served. Another relaxation used in
this model is to assume that each vehicle is initially fully-charged and has an unlimited battery
life. Since each vehicle stays in the same battery state for the duration of the horizon T, vehicle
movements may be simplified to single-commodity flows on arcs in A.
Let integer variables xt
ij represent the number of vehicles that move from region i∈ R to region
j∈ R during time interval t∈ T and yt
irepresent the number of vehicles that remain in region i
during time interval t. Integer variables v1
idenote the initial inventory of vehicles assigned to region
iat the beginning of the planning horizon. Binary variables wt
iindicate if at least one vehicle is
available within region iat the beginning of time tand ut
ijk indicate if rider ksuccessfully finds a
vehicle to travel from region ito region jduring time t. Finally, let continuous auxiliary variables
pt
irepresent the probability of finding an available vehicle in region iduring time t. We can now
formulate the single-realization (SR) optimization problem as follows:
10 Greening, Erera: Heuristics for Distributing Vehicles
max X
t∈T X
i∈R X
j∈R
xt
ij (3a)
s.t.X
i∈R
v1
i=V, (3b)
v1
i=X
j∈R
x1
ij +y1
i,∀i∈ R,(3c)
X
j∈R
xt
ji +yt
i=yt+1
i+X
j∈R
xt+1
ij ,∀i∈ R,∀t∈ T ,(3d)
pt
i≤1,∀i∈ R,∀t∈ T ,(3e)
p1
i≤v1
i
100Ai
,∀i∈ R,(3f)
pt
i≤yt−1
i+Pj∈R xt−1
ji
100Ai
,∀i∈ R,∀t∈ T \ {1},(3g)
pt
i−Qt
ijk ≤ut
ijk ,∀i, j ∈ R,∀t∈ T ,∀k∈ Dt
ij ,(3h)
pt
i−Qt
ijk ≥ut
ijk −1,∀i, j ∈ R,∀t∈ T ,∀k∈ Dt
ij ,(3i)
xt
ij ≤X
k∈Dt
ij
ut
ijk ,∀i, j ∈ R,∀t∈ T ,(3j)
X
j∈R X
k∈Dt
ij
ut
ijk −X
j∈R
xt
ij ≤X
j∈R
Dt
ij wt
i,∀i∈ R,∀t∈ T ,(3k)
yt
i≤V(1 −wt
i),∀i∈ R,∀t∈ T ,(3l)
xt
ij ∈Z+,∀i, j ∈ R,∀t∈ T ,(3m)
yt
i∈Z+,∀i∈ R,∀t∈ T ,(3n)
v1
i∈Z+,∀i∈ R,(3o)
wt
i∈ {0,1},∀i∈ R,∀t∈ T ,(3p)
ut
ijk ∈ {0,1},∀i, j ∈ R,∀t∈ T ,∀k∈ Dt
ij (3q)
The objective is to maximize the number of vehicle trips served over the planning horizon modeled.
Constraints (3b) limit the number of vehicles to be distributed at the beginning of the day to the
fleet size V. Constraints (3c) and (3d) are typical flow balance constraints that capture vehicle
movement between regions across time, where the vehicle inventories in period 1 are given by the
allocation decisions and then the inventories in the remaining periods are determined by the vehicle
movement and holding decisions.
Constraints (3e) to (3i) are used in to create a relaxed model of a realization of the vehicle-to-rider
matching process. Input parameters Qt
ijk are random variable draws for each k∈ ∪t∈T ∪i,j∈R Dt
ij
from the standard Uniform(0,1) and will be used within the approximate simulation. Constraints
(3e) to (3g) determine an upper bound on the likelihood that an individual rider finds an available
Greening, Erera: Heuristics for Distributing Vehicles 11
Figure 3 Network structure for optimizing the starting distribution for a given day with known demand.
vehicle in region ias defined by (1); note, however, that the optimization model may decide to
lower this probability artificially in some region iat time tif doing so leads to a better objective
function, hence this is a relaxation. Given the probabilities selected by the model, constraints (3h)
and (3i) force ut
ijk = 1 if Qt
ijk < pt
iand ut
ijk = 0 if Qt
ijk > pt
isimulating a Bernoulli trial. We also
remark here that if pt
i=Qt
ijk , the model can choose either value for ut
ijk .
The sum of riders able to locate vehicles is an upper bound on the number of trips executed
from region ito region jat time t, specified by constraints (3j). Note that the sum of the simulated
upper bounds could be greater than the number of vehicles in the region; in this case, the flow
balance constraint would limit the total demand served, but the optimization model can pick the
most advantageous demands to serve. Again, this flexibility given to the optimization model is
another relaxation to the actual state transition dynamics. Constraints (3k) and (3l) ensure that
available vehicles are used to serve successfully matched rider requests. If the model chooses to
execute fewer trips than the total number of successful potential matches, the binary variable wt
i
must be set to value one to ensure that no inventory can be held forward to time period t+ 1. Note
that the left-hand side of (3k) is the number of successful rider-to-vehicle matches less the number
of executed trips; thus, this difference is bounded by the total possible outbound trips from region
iat time t. Finally, in constraints (3l), the number of vehicles that might remain in inventory in
region iduring time twill never exceed the fleet size. An illustration of the network structure of
(3) is given in Figure 3.
4.2. Sample Average Upper Bound
To determine a good sample average upper bound on the maximum expected number of trips that
can be served given a fixed fleet of vehicles Vand set of regions Rgiven by (2), model (3) is solved
12 Greening, Erera: Heuristics for Distributing Vehicles
for many instances, each of which represents a single realization sof rider demands with associated
uniform random variables to be used in approximating the rider-to-vehicle matching process. It is
important to note here that the realization scontains a subset of the information ωin a complete
uncertain realization. Let Sbe a reasonably large pool of such scenarios, each equally likely to
occur and each of which is associated with sets {Dt
ij }and uniformly-distributed Qt
ijk values for
each kin the union of these demand sets.
Let zSR (s) be the optimal objective function value for model (3) for a given scenario s∈ S;
note also that the fleet distribution given by v=v1
1, . . . , v 1
Rmay differ for each such scenario.
We know that zSR(s) is an upper bound on the maximum number of trips that can be served for
scenario sgiven that the uncertain realization sis known in advance, the initial fleet distribution
is optimized separately for s, and that the model relaxes the actual state transitions. Thus, we can
specify a sample average a posteriori upper bound UAP on the expected number of trips that can
be served by simply averaging the optimal objective values over the scenario set S:
UAP =Ps∈S zSR (s)
|S| ,
We will use this sample average a posteriori bound to evaluate the quality of any initial distri-
bution of vehicles produced by any of the methods proposed in the following sections.
4.3. Simplified Two-stage Stochastic Integer Programming Model
Building on ideas used in constructing the SR model (3), we now introduce a two-stage stochastic
optimization model (SIP) for determining an initial vehicle distribution v. Again, the optimization
model will use simplifications of the state transition dynamics. As a two-stage model, the optimiza-
tion approach is also able to use complete foresight for each scenario when deciding which trips to
serve and which to reject. For these reasons, vehicle distributions generated by solving a SIP model
will be suboptimal solutions to the decision problem (2). Like the SR model, the SIP model will
also ignore vehicle battery life and assume that all vehicles are homogeneous during the planning
horizon and are availble to complete any potential trip. The trip-to-vehicle matching process is
simplified as follows: all potential rides are matched to a vehicle as long as one is located within
the origin region. As we will see, the SIP model grows in size quickly for realistic-sized instances
and this latter simplification allows for optimization with a larger set of scenarios.
In this two-stage decision problem, the first-stage decision variables are again the initial fleet dis-
tribution inventories given by v1
ifor each region i∈ R. In the second stage, the model optimistically
assumes that all uncertainties are simultaneously revealed (in this case, the potential trip demands
between all region pairs at all times) and the recourse is to determine which of these potential trips
to serve over time. Note then that for a large enough fleet size, all potential trips can be served for
Greening, Erera: Heuristics for Distributing Vehicles 13
all instances given an appropriate initial distribution. When such an initial distribution cannot be
found, the optimization model here chooses which potential trips to serve and which to reject with
complete foresight about how such choices lead to future distributions of vehicles across locations.
Of course, these future distributions then determine which future trips can (and cannot) be served.
Given a finite set S0of potential trip scenarios, we can formulate this two-stage optimization
problem as a MIP with recourse variables for each scenario s; assume that the sample is Monte
Carlo, such that each scenario shas equal probability of occurring (i.e., ps= 1/S, where S=|S0|).
For a small enough set S0, these MIP instances can be solved directly using an integer-programming
solver. Of course, when S0is not representative of the true set of uncertain scenarios this additional
approximation will degrade the quality of the fleet distribution decision further. Since the model
is structurally very similar to the single-scenario SR model, we will use the same potential trip
input parameter and vehicle flow notation for the second-stage decision variables, augmented with
a scenario indicator. Let Ds,t
ij ∀i, j ∈ R,∀t∈ T represent the potential trip demand from region
ito region jduring time tfor scenario s∈ S0. The second-stage decision variables, each indexed
by scenario s, are then xs,t
ij for executed trips between region iand jinitiated in time t,ys,t
ifor
vehicles held in region ifrom tto t+ 1, and ws,t
iindicating whether potential trip demand exceeds
executed trips outbound from region iduring t.
Given this notation, we formulate the SIP model then as follows:
max 1
|S0|X
s∈S0X
t∈T X
i∈R X
j∈R
xs,t
ij (4a)
s.t.X
i∈R
v1
i=V, (4b)
v1
i=X
j∈R
xs,1
ij +ys,1
i,∀i∈ R,∀s∈ S0,(4c)
X
j∈R
xs,t
ji +ys,t
i=X
j∈R
xs,t+1
ij +ys,t+1
i,∀i∈ R,∀t∈ T ,∀s∈ S 0,(4d)
xs,t
ij ≤Ds,t
ij ,∀i, j ∈ R,∀t∈ T ,∀s∈ S0,(4e)
X
j∈R
Ds,t
ij −X
j∈R
xs,t
ij ≤X
j∈R
Ds,t
ij ws,t
i,∀i∈ R,∀t∈ T ,∀s∈ S 0,(4f)
ys,t
i≤ V−X
j∈R
Ds,t
ij !(1 −ws,t
i),∀i∈ R,∀t∈ T ,∀s∈ S 0,(4g)
xs,t
ij ∈Z+,∀i, j ∈ R,∀t∈ T ,∀s∈ S0,(4h)
ys,t
i∈Z+,∀i∈ R,∀t∈ T ,∀s∈ S 0,(4i)
ws,t
i∈ {0,1},∀i∈ R,∀t∈ T ,∀s∈ S 0,(4j)
v1
i∈Z+,∀i∈ R (4k)
14 Greening, Erera: Heuristics for Distributing Vehicles
The constraints that modeled the probability of an arriving rider finding an available vehicle,
(3h) and (3i) above, were removed to simplify the model. Furthermore, (4e) provides a simple upper
bound in this case on the maximum number of potential trips that can be served between regions
in any time period. Constraints (4f) and (4b) again work together to ensure that no inventory
is held in region iif any potential demand remains unserved. In this case, the first term on the
left-hand side of (4f) counts all of the potential trips outbound from iduring tdirectly, and the
right-hand side coefficient in (4g) can be reduced since we cannot hold more inventory than the
fleet size less the potential outbound demand.
Of course, we should also note that the SIP model with a single scenario in Sis another type of
single-scenario MIP that, like SR, can be used to create a sample-average a posteriori upper bound
when solved multiple times each with a single scenario. Since the SIP formulation uses a weaker
relaxation of the trip-to-vehicle matching process, doing will so create a weaker upper bound but
perhaps with less computational effort. We will refer to an upper bound created this way as UAPR
since the SIP formulation is a relaxation of the SR formulation.
4.4. Overview of Simulation-based Heuristics
Solving the SIP model with a large number of scenarios is likely to be challenging, especially when
the number of regions and time periods is fairly large. Thus, other heuristic approaches that do not
rely on solving large MIPs may be effective alternatives especially if they can capture the uncertain
state transition dynamics with more fidelity than is possible with integer variables and linear
constraints. In this section, we develop two types of simulation-based heuristics. The first type is
for greedily constructing an initial distribution of vehicles from an empty initial solution, while the
second type is for improving an initial solution distribution via local search. For these approaches,
we propose two slightly different discrete-event simulations to use when evaluating a potential
change to the current solution. The first, which is only used within the construction approach, is
a deterministic-match (DM) simulation that approximates the vehicle-to-rider matching step for
simplicity, while the second is a more accurate probabilistic-match (PM) simulation.
Both simulation approaches assess the performance of some initial distribution vof homogeneous
fully-charged vehicles by computing the number of trips served over the planning horizon Tfor
each scenario sin a set of scenarios S, and then estimating the expectation with the sample mean
number of trips. To do so, first a complete set of potential trip demands {Dt
ij (s)}is generated using
the specified input stochastic model and the initial vehicle state {V1
i}is generated using the initial
vehicle distribution v. Each scenario salso stores two additional Uniform[0,1] random variable
outcomes associated with each potential trip in the demand sets. These will be used as needed
below in the trip-to-vehicle matching process.
Greening, Erera: Heuristics for Distributing Vehicles 15
The trips executed during period toutbound from each region are determined sequentially by
region. For region i, given the vehicle set Vt
i(s) and potential trips {Dt
ij (s)}, a vehicle-to-trip
matching process is executed. Consider the set of potential trips Dt
ioutbound from region iat time
period t, and suppose they are placed in an ordered list. Each potential trip in this list also has a
destination region. The simulation then determines whether each potential trip can be executed,
one by one. For the DM simulation, we assume that each trip can potentially be served if a battery-
feasible vehicle remains in the region. This match process is simulated by drawing one of the
remaining vehicles in the region using the second stored uniform random variable for the current
potential trip, and if the vehicle has sufficient charge to execute the trip given its destination,
the ride is successful and the vehicle is removed from the currently available pool. For the PM
simulation, we additionally assume that a rider might arrive for a trip in some subarea of region i
without nearby vehicles. Thus, for each potential trip we first determine whether the rider origin
is nearby any vehicles using a Bernoulli trial with the first stored uniform random variable and
probability (1). Successful trips here then use the approach described above to determine whether
they can be matched to a battery-feasible available vehicle. Whenever a trip does not successfully
match with a vehicle in either simulation, it is removed from the system without being served and
classified as lost demand.
Consistent with the problem description, when a trip is executed that vehicle is occupied for a
single time period tand it becomes available for re-use in period t+ 1 in the destination region jof
the trip. Each vehicle’s remaining battery life is tracked over time; after serving a trip, battery life
is reduced by the amount required by duration of the trip in minutes. Additionally, if a vehicle’s
battery charge falls below a specified minimum value, the vehicle is marked as inoperable at the
end of its current trip and is no longer allowed to move. Vehicles that are not used for a trip in
period tremain within their current region. By tracking individual trips and vehicles used for all
regions iin time period t, the new vehicle state {Vt+1
i}is generated for the next time period and
the simulations continue until the end of the final period in the horizon t=T. Furthermore, the
number of executed trips dt
ij (s) are recorded for each time period t, origin i, and destination j.
The final number of executed trips for scenario sis
ρ(s) = X
t∈T X
i,j∈R
dt
ij (s)
and then the sample average number of trips served across all scenarios is
ρ=1
|S| X
s∈S
ρ(s)
16 Greening, Erera: Heuristics for Distributing Vehicles
4.5. Simulation-based Construction Heuristic
The initial simulation-based heuristic is a best-improving construction heuristic. This heuristic
begins with a starting distribution where all regions are empty, i.e., vehicle counts are 0 everywhere
and v=0. It then determines the value of adding a small number nvehicles to each region by
updating the starting distribution vaccordingly and then calling either DM(v) or PM(v) as the
simulation with a fixed set of scenarios Sas described above, and decides to add those nvehicles
where they yield the largest simulated number of trips served. This process repeats until the entire
fleet is deployed across regions.
As a simulation-based heuristic, the construction heuristic relies on simulation results to make
best greedy decisions in any given iteration. Since acquiring the performance metric (ρ, the sample
average number of trips served in this case) by simulation can be computationally expensive, we
employ several ideas to speed-up the heuristic without degrading solution quality. First, the DM
simulation is faster than the PM simulation since it simplifies the matching process; computational
results show that it is typically two to three times faster. Therefore, we use DM initially during
the heuristic as the first vehicles are distributed across regions and then we switch to PM once a
minimum sample average ρmin of rides is served.
As the number of regions within the service area increases, checking the value of adding n
vehicles to each region can also lead to long computation times. To combat this, the heuristic only
evaluates adding additional vehicles to the regions experiencing the most lost demand given the
current vehicle distribution v. A list Lof the Lregions with most lost (unserved) potential trip
demand is returned by each simulation upon completion, where Lmay vary for the DM versus the
PM simulation. During an iteration, the construction heuristic will only evaluate adding vehicles
to the first Lregions in the lost demand list for the current solution v. The values of LDM and LPM
are selected by the operator for the DM and PM simulations respectively to achieve the desired
trade-off between solution quality and computation time.
At the completion of an iteration, the region rmax that results in the largest increase in objective
function is where the nvehicles are greedily placed; ties are broken to the lower-numbered region.
This process is repeated until the entire fleet has been distributed. Both the number of vehicles to
distribute nand the number of regions to evaluate Lcan be defined differently when using the DM
simulation versus the PM simulation. For example, when initially using the simpler DM simulation,
the operator may be willing to distribute more vehicles (i.e., increase nDM) and evaluate fewer
regions (i.e., decrease LDM) to decrease the time needed in this stage of the construction heuristic.
Pseudocode for the construction heuristic is shown in Algorithm 1, where ρmin is the minimum
sample average number of rides served needed to switch to PM.
Greening, Erera: Heuristics for Distributing Vehicles 17
Algorithm 1: Construction Heuristic
Input : V,R,ρmin,nDM ,nPM,LDM ,LPM
Output: Starting distribution v
1Initialize v←− 0,sim ←− DM, L ←− LDM regions with highest total potential trips ;
2while Pi∈R v1
i< V do
3ρmax ←− 0;
4for l∈ L do
5(ρ0,L0) = sim(v+nsimel);
6if ρ0> ρmax then
7ρmax ←− ρ0,rmax ←− l,Lmax ←− L0;
8end
9end
10 v←− v+nermax ,L ←− Lmax;
11 if ρmax ≥ρmin then
12 sim ←− PM;
13 end
14 end
4.6. Simulation-based Improvement Heuristics
We now describe simulation-based heuristics that, given a current distribution vcof vehicles, seek
to find a new distribution vthat yields a higher objective function value. The approaches developed
in this section are in the local search family which seek vby iteratively moving from a current
solution to a neighboring solution with a better objective function value. Since the neighborhood
evaluations here will involve simulation, it is critical again to choose neighborhoods and search
strategies judiciously.
Each of the improvement heuristics we propose is a type of relocation heuristic used to determine
if relocating vehicles locally from region ito region jresults in an improvement in ρ. When
evaluating objective function values, the improvement heuristics will rely on the more expensive
but higher fidelity PM simulation. Each of the improvement heuristics we develop seeks to relocate
uvehicles from region ito region jduring an iteration where u≥1. The first three improvement
heuristics implement a swap neighborhood, where a solution is a neighbor to the incumbent if u
vehicles are moved from region ito region j. The neighborhood, then, is all such solutions that
can be generated from an incumbent by moving uvehicles from any ito any different j. The three
heuristics differ only in how they search this neighborhood and select an improving neighbor each
iteration.
Suppose that a swap neighborhood is defined by a complete list of region pairs (i, j ) where i6=j.
During an iteration, a local search heuristic considers each of these region pairs in sequence. The
first heuristic, First Improvement (FI), completes a neighborhood search iteration given incumbent
18 Greening, Erera: Heuristics for Distributing Vehicles
vcwhen it finds the first neighbor solution defined by region pair (i, j) in the swap neighborhood
such that ρ>ρc; at this point, the incumbent is updated and the search is restarted back at the
beginning of the list of region pairs. The second heuristic, First-Continuous (FC) Improvement,
also completes an iteration when the first improving neighbor is found. However, given the new
incumbent the search for the next improving swap neighbor (i, j) proceeds from the the pair in the
list defining the swap neighborhood. This feature can potentially allow the FC algorithm to more
quickly complete a search of all of the region-to-region swap pairs. Finally, the Best Improvement
(BI) heuristic exhaustively searches the complete neighborhood swap region pair list each iteration
and updates the incumbent, if an improving solution is found, to the improving solution with the
highest value of ρ. Pseudocode for all three heuristics are shown in Appendix A.
The other improvement heuristic uses what we denote as an Approximate Gradient (AG) method
to more quickly identify a swap neighbor with a nearly-best improvement during a single iteration.
While the complete BI heuristic requires O(R2) neighbor evaluations each iteration, the AG method
instead requires O(R) (or more precisely, 2R) simulation evaluations. This heuristic, described in
Algorithm 2, first selects the region from which to remove uvehicles by determining which leads
to the minimum loss in ρc−ρwhen removing uvehicles. It then selects the region to receive the u
vehicles by determining which region leads to the maximum gain in ρ−ρcwhen adding uvehicles.
Similar to the construction heuristic, AG only evaluates adding vehicles in any iteration to the top
Lregions when ordered by decreasing lost demand. Additionally, when removing vehicles, AG as
prioritizes the evaluation of a small number of regions. Here, we order the regions in decreasing
order of a ratio that intends to determine the value of vehicles there. This ratio is the total number
of idle vehicles in inventory at isummed over the planning horizon divided by the total lost demand
at i. Given this ordering, AG only evaluates the first Iregions for removal of vehicles during any
iteration. Using the ratio instead of simply the number of idle vehicles adjusts for regions that use
a large number of idle vehicles during some periods to prepare for upcoming periods with high trip
demand. Note that the benefit of restricting the evaluation of swap neighbors in this way is that
it requires only L+Isimulation calls during each iteration. The pseudocode for Algorithm (2) is
written assuming that each call to the simulation returns when needed a list Iof the Iregions
with largest idle vehicles to lost demand ratio and a list Lof the Lregions with most lost demand.
Finally, note that in the improvement heuristics, when the number of vehicles to be relocated,
u, is greater than 1, all heuristics dynamically alter the search neighborhood by decreasing uafter
some iterations. Specifically, the heuristics decrease uby 1 when they fail to find any remaining
improvements after a complete search of the current neighborhood. When ureaches 0, the heuristic
terminates. If region ihas fewer than uinitial vehicles, the number of vehicles currently assigned
in the initial distribution to the region is temporarily used as u. In other words, if region ihas
Greening, Erera: Heuristics for Distributing Vehicles 19
only u−1 vehicles and the heuristic is determining the value of moving these vehicles to another
region, the heuristic will use u−1 while evaluating region i.
Algorithm 2: Approximate Gradient
Input : vc,u
Output: Improved starting distribution v
1Initialize v←− vc, (ρ, I,L)←− P M (v) ;
2while u > 0do
3dmin ←− ρ,dmax ←− 0;
4for i∈ I do
5n←− min{u, vi};
6if ρ−(ρ0←− PM(v−nei)) < dmin then
7dmin ←− ρ−ρ0;
8Regionmin ←− i;
9end
10 end
11 for l∈ L do
12 n←− u;
13 if (ρ0←− PM(v+nel)) −ρ > dmax then
14 dmax ←− ρ0−ρ;
15 Regionmax ←− l;
16 end
17 end
18 n←− min{u, vRegionmin };
19 if ρ < (ρ0,I0,L0)←− PM(v+neRegionmax −neRegionmin )then
20 v←− v+neRegionmax −neRegionmin ,ρ←− ρ0,I ←− I0,L ←− L0;
21 else
22 u=u−1;
23 end
24 end
5. Computational Study
In this section, we describe the design and results of a case study focused on an electric micromobil-
ity scooter system in Atlanta, Georgia to evaluate the performance of our heuristic approaches for
determining geographic fleet distributions. Specifically, we present results that highlight that: (i)
finding UAP using the SR model (3) leads to much tighter, more realistic upper bounds for assessing
heuristic performance; (ii) a simulation-based construction heuristic can be used alone, if tuned
20 Greening, Erera: Heuristics for Distributing Vehicles
properly, to find very good fleet distributions; (iii) improvement heuristics can be used to improve
weaker initial vehicle distributions to identify equally good, if not better, distributions than fine-
tuned constructed solutions; (iv) the distributions found using the SIP optimization model work
best when the fleet size is large enough to serve most demand, but are still outperformed by distri-
butions resulting from simulation-based heuristics; and (v) incorporating a small number of fleet
redistribution operations during the operating day can lead to large performance improvements.
The discrete-event simulations, heuristics, and MIP models were implemented in Python 3.7.9
and IP models were solved using Gurobi 12.10 with all settings left at their default value. Experi-
ments were run on a Linux computing cluster, which uses HTCondor 8.8.12 for job management.
Each node in the cluster uses multi-core 2.4 GHz processors with 8 GB of RAM each.
5.1. Instance and Simulation Descriptions
The instances used in this case study are built to represent a system operated within a 9 square-mile
service area centered on downtown Atlanta. The planning horizon studied is a 14-hour operating
day from 7 AM to 9 PM discretized into 20-minute time periods. Different instances are generated
which differ in how the service area is partitioned into a set Rof subregions and the size of the
vehicle fleet Voperated. There are 16 total instances, where the numbers of regions range from 13
to 100 and the fleet sizes range from 500 to 2,000 vehicles.
Because the partitioning of a service area into individual regions was not the focus of this study,
we created the region layouts with one of two goals in mind: (i) closely modeling a currently-
deployed system in Atlanta or (ii) stress-testing the heuristic performance by using larger instances.
The layouts of the two smaller sets of size 13 and 20 regions were selected to focus on goal (i)
and are based on Atlanta population density and system observations of a shared scooter system
currently operating there. The layouts of the two larger sets of sizes 50 and 100 regions were selected
for goal (ii) and divide the service area into equally-sized regions organized in a grid-like pattern.
We note that any reasonable geographic discretization will likely require vehicles to be allocated
within each region to specific locations; this intra-region allocation problem is outside the scope of
this study. The region set layouts are illustrated in Figure 4.
For this study, we generate a baseline set Sof S= 100 potential trip demand scenarios; this
baseline set will be used to compute all upper bounds and to simulate the performance and assess
any distribution of vehicles using the DM or PM simulations. Subsets of Swill be used for the SIP
models. Each s∈ S includes potential trip demand sets Dt
ij (s) and two Uniform[0,1] random variable
outcomes for each potential trip in the sets. Potential trip demands from region itraveling to region
jduring time tare generated using independent Poisson random variables with time-dependent
and geographically-dependent rate parameters, where the rates used are the total expected number
Greening, Erera: Heuristics for Distributing Vehicles 21
(a) R= 13 (b) R= 20
(c) R= 50 (d) R= 100
Figure 4 The four region layouts.
of potential trips at region imultiplied by the probability that the trip is destined for region j.
The trip demand rates for each region are estimated based on real-world Atlanta system data
and represent demand typical for a weekday. The expected number of potential trips across the
service area is about 8,500 trips per day for each of the region layouts. The time-dependent origin-
destination probability matrices are generated similarly for the 13- and 20-region layouts, whereas
the 50- and 100-region layouts assume that trip destinations are equally-probable among the current
region and the 8 immediately adjacent. Each scenario s∈ S is equally-probable, and thus Scan be
considered a Monte Carlo sample. Note of course that this is not necessary for any of the models
proposed (under small modifications), and thus it is possible for example to give higher weighting
in an empirical sample to more recent or more representative scenarios.
For this study, we assume that the fleet of vehicles is homogeneous; that is, the battery level of
each vehicle in the fleet is assumed to be 100% at the beginning of the planning horizon and each
vehicle operates in the same manner. Battery life is tracked throughout the planning horizon and is
reduced by the amount required by duration of the trip in minutes. We approximate a 1% decrease
for every 4.5 minutes traveled, assuming that it takes about 15 minutes to travel one mile (Wiggers
22 Greening, Erera: Heuristics for Distributing Vehicles
RAverage Trip
Distance (mi)
13 1.2
20 1.5
50 0.7
100 0.5
Table 1 Average distance per trip taken by successful riders.
2019). If the battery level drops below 3% (i.e., below the level needed to travel one mile), the
vehicle is designated inoperable and can no longer be matched to potential trips. To determine an
estimate of the distance traveled for each trip, we calculate the Manhattan distance between the
origin and destination centroids. For intra-region trips, the distance traveled is assumed to be the
average distance between two points in the region (assuming a square region of the correct area).
In Table 1, the average distance of all trips is given for each of the region layout types. We of course
realize that the average distance of trips is not likely in practice to vary with the decision about
how to partition an area into regions, but we thought it would be useful to have some variation
here to represent systems with different vehicle types. Additionally, the probability that an arriving
rider finds a vehicle in their origin region is defined by (1). Again note that an operator can define
an alternative expression for this probability using system-specific data as needed.
5.2. Analysis of Sample Average Upper Bounds
First, we study the effect on the quality of sample-average upper bounds when using a more detailed
representation of the vehicle-to-trip matching process within the optimization model. To do so, we
present results that demonstrate the value of determining the sample average a posteriori upper
bound UAP on the expected number of trips that can be served when using the SR model (3) using
the methods described in Section 4.2. These results are then compared to the weaker upper bounds
UAP R that result from computing similar sample average bounds using the single-scenario variant
of the SIP model as described in Section 4.3. In Figure 5, a comparison of the average number of
total potential trips across all scenarios (Max), UAP R, and UAP is shown for each region set and
each fleet size.
A first observation is that the bounds behave similarly as we vary fleet size across all region
layouts, as seen by the similar behavior exhibited in each region plot. It is clear that the largest
benefit from the tighter UAP bounds is when the fleet of vehicles is small; in such cases, it is not
possible to serve all potential trips even with an optimal fleet distribution. There are two main
reasons for the large difference between UAP and Max when the fleet is small: (i) the fleet is not
large enough to serve all demand across the service area; and (ii) the rider-to-vehicle matching
problem is more difficult given the smaller number of fleet vehicles. The small fleet size effect
Greening, Erera: Heuristics for Distributing Vehicles 23
Figure 5 Comparison of methods bounding the expected number of trips that can be served for a given instance.
appears also in the UAP R bound, which uses the weaker relaxation of the trip-to-vehicle matching
process by assuming all riders will be matched to an available vehicle if one is located in their
origin region; in this case, the MIP is still unable to satisfy all trip demand for small fleet sizes.
The second reason is also a consequence of the first: the small number of fleet vehicles results in
a small likelihood of successful matches in some regions. As the fleet size grows, more of the trip
demand can be satisfied, and thus the two bounds eventually converge to the average number of
total potential trips when the fleet has 2,000 vehicles.
5.3. Construction Heuristic Computational Results
We next evaluate the effectiveness of using the simulation-based construction heuristic to gener-
ate good initial distributions for each of the instances. In this study, we test different parameter
values to tune the construction heuristic for specific instances, where the focus is to improve the
number of successful trips while maintaining a reasonable computation time. We set a maximum
compute time of 3.5 hours for an execution of the heuristic to ensure that there is sufficient time
to later improve the solution, if desired. We test two values for ρmin, i.e., the amount of trips
24 Greening, Erera: Heuristics for Distributing Vehicles
R V ρmin/Max nDM LDM nPM LPM ρBC %BG %GC %IP Time
(hrs)
13
500 0.4 30 5 10 5 4,026 53.1% 3.7% 4.5% 0.7
1,000 0.4 30 5 30 5 6,607 23.1% 11.9% 4.2% 0.8
1,500 0.5 30 5 30 5 7,839 8.7% 20.2% 2.5% 0.5
2,000 0.5 30 5 20 5 8,438 1.8% 64.5% 3.4% 1.5
20
500 0.4 40 5 10 5 4,312 49.9% 7.3% 8.5% 0.6
1,000 0.4 40 5 30 5 6,573 23.7% 18.4% 7.5% 0.3
1,500 0.4 20 5 20 5 7,798 9.5% 45.5% 9.5% 0.7
2,000 0.5 20 5 10 5 8,432 2.1% 80.3% 9.6% 3.4
50
500 0.5 20 5 10 10 4,239 50.2% 4.9% 5.5% 0.3
1,000 0.5 20 5 10 10 6,830 19.8% 4.3% 1.1% 1.6
1,500 0.5 20 5 10 10 7,849 7.9% 29.3% 3.7% 1.7
2,000 0.5 20 5 10 5 8,370 1.8% 74.4% 5.5% 1.7
100
500 0.4 20 5 20 10 4,872 43.7% 9.8% 9.2% 0.6
1,000 0.5 20 10 10 20 6,958 19.6% 6.8% 1.8% 3.0
1,500 0.4 40 5 10 10 7,897 8.7% 15.6% 1.8% 2.7
2,000 0.5 20 10 10 10 8,406 2.8% 46.8% 2.6% 3.0
Table 2 Performance of initial distributions generated by the tuned construction heuristic.
that must be satisfied before switching from DM to PM, which are 40% and 50% of the maxi-
mum expected number of potential trips across the planning horizon (Max). When distributing
marginal vehicles each iteration, we test three values for nsim and two values for Lsim; for DM, we
use nDM ={20,30,40}and LDM =max{5,d0.05Re},max{5,d0.1Re}, and we alter the parame-
ters for PM to nPM ={10,20,30}and LPM =max{5,d0.1Re},max{5,d0.2Re}in an attempt to
increase precision. Notice that at least 5 regions will be evaluated as greedy distribution choices
each iteration; we do this to ensure a reasonably-sized set of candidates for the instances with fewer
regions.
To assess performance, we calculate both metrics relative to UAP and also the improvement in a
constructed distribution over a simple, but effective initial distribution, namely proportional. The
proportional distribution is created by determining the proportion of total potential trip origins
within each region and distributing the fleet accordingly. That is, if 10% of all trip origins across
the planning horizon occur in region i, 10% of the fleet will be initially distributed to region i. To
compare the best constructed distributions’ expected number of trips ρBC to UAP , we calculate
%BG as the percent gap between the UAP bound and ρBC . To compare ρBC with the proportional
distribution, we calculate %GC as the percent gap closed by the constructed distribution (i.e.,
(ρBC −ρP)/(Max −ρP)) and %IP as the constructed distribution’s percent improvement over the
proportional distribution as baseline. For all three metrics, a higher value is better. Table 2 sum-
marizes the performance results of the vehicle distributions generated by the tuned construction
heuristic for each instance.
From the table, we observe that the constructed distributions are able to significantly improve
the expected number of rides served as compared to the proportional distribution. Interestingly,
Greening, Erera: Heuristics for Distributing Vehicles 25
the results are fairly consistent across the instances with varying number of regions. As the fleet
size increases, the construction heuristic is able to generate distributions that cut the proportional
distribution’s gap by over half in almost every instance using a fleet of 2,000 vehicles. While the
larger improvements in %GC for the larger fleets are due in part to the original proportional gaps
being small, the construction heuristic still manages to find a distribution that outperforms an
already very good distribution. We observe this in %IP, where almost all instances have values of 2%
or more with many improvements in the 8-10% range. Overall, the simulation-based construction
heuristic appears to be quite effective.
5.4. Improvement Heuristic Computational Results
In this section, we evaluate the performance of the proposed improvement heuristics by improving
two initial distributions (ID) for each instance. The first distribution is a heuristic-constructed
(Con) distribution using the parameters ρmin = 0.5Max, nDM =nPM = 30, LDM = max{5,d0.05Re},
and LPM = max{5,d0.1Re}. These parameter settings lead to faster computation times, while
keeping the number of vehicles distributed each iteration small enough such that a good final
distribution results. The second distribution to improve is the proportional (Pro) distribution pre-
viously described in Section 5.3. The computation time limit used for the improvement heuristics
is 5 hours and includes both the construction time and additional improvement time. Thus, if a
heuristic-constructed distribution requires a substantial amount of time, the improvement heuristic
has limited time to improve the solution.
All four improvement heuristics use the parameter uas the initial number of vehicles to be
relocated each iteration. Recall that uis reduced by 1 whenever an improvement is not found
after all swap neighbors are checked. Thus, as uincreases, the computation time of the heuristic
generally increases. The benefit of a larger uvalue, i.e., step size, is that there is a greater chance
of moving away from a local maximum. In our experiments, we tested different uvalues (i.e.,
u∈ {5,10,15}) to determine which value led to the most trips served and found that u= 15
consistently performed well. Thus, the results we present in this section will use this setting. In AG,
the percent of regions we evaluate each iteration is either the top 10% or 20% ranked by largest
total lost demand (i.e., L= max{5,d0.1Re}, when 10%) or the ratio of idle vehicles to lost demand
(i.e., I= max{5,d0.1Re}, when 10%). Similar to the construction heuristic, we ensure that at least
5 regions are included as origins and 5 as destinations when defining the search neighborhood.
In addition to %BG, %GC, and %IP, we also compare the improved solutions to their original
initial distribution using %IID as the percent improvement over the initial distribution (i.e., Con
or Pro) and to the best constructed distribution found in Section 5.3 using %BCI as the percent
improvement over ρBC. The performance results of the best distribution generated for each instance
using the four improvement heuristics are shown in Table 3.
26 Greening, Erera: Heuristics for Distributing Vehicles
R V ID ρID IH L, I ρIH %IID %BG %GC %IP ρBC %BCI Time
(hrs)
13
500 Pro 3,863 FC 0 4,030 4.3% 53.1% 3.8% 4.6% 4,026 0.1% 3.3
1,000 Con 6,604 BI 0 6,644 0.6% 22.7% 13.6% 4.8% 6,605 0.6% 5.0
1,500 Con 7,838 BI 0 7,862 0.3% 8.5% 22.6% 2.8% 7,839 0.3% 5.0
2,000 Pro 8,161 AG 5 8,445 3.5% 1.7% 66.1% 3.5% 8,438 0.1% 1.9
20
500 Con 4,255 BI 0 4,309 1.3% 50.0% 7.2% 8.4% 4,312 -0.1% 5.0
1,000 Con 6,510 FC 0 6,643 2.0% 22.9% 21.2% 8.7% 6,571 1.1% 5.0
1,500 Pro 7,118 FC 0 7,828 10.0% 9.1% 47.5% 10.0% 7,798 0.4% 5.0
2,000 Con 8,412 BI 0 8,437 0.3% 2.0% 80.9% 9.6% 8,432 0.1% 5.0
50
500 Con 4,135 AG 10 4,239 2.5% 50.2% 4.9% 5.5% 4,239 0.0% 1.2
1,000 Pro 6,756 AG 10 6,850 1.4% 19.6% 5.4% 1.4% 6,830 0.3% 2.3
1,500 Pro 7,572 AG 10 7,871 3.9% 7.6% 31.6% 4.0% 7,849 0.3% 3.6
2,000 Pro 7,936 AG 10 8,358 5.3% 1.9% 72.3% 5.3% 8,370 -0.1% 5.0
100
500 Con 4,674 AG 20 4,950 5.9% 42.8% 11.7% 11.0% 4,872 1.6% 3.2
1,000 Pro 6,847 AG 10 6,985 2.0% 19.2% 8.3% 2.2% 6,958 0.4% 3.7
1,500 Pro 7,760 AG 10 7,973 2.7% 7.8% 24.1% 2.8% 7,897 1.0% 3.6
2,000 Pro 8,189 AG 10 8,396 2.5% 2.9% 44.6% 2.5% 8,406 -0.1% 5.0
Table 3 Tuned improvement heuristic results when improving either a pre-specified constructed or proportional
initial distribution.
We first observe that the best-performing improvement heuristic is able to find improved distri-
butions for all instances that outperform their ID solution, as evidenced by %IID. This highlights
the heuristics’ abilities to find improving solutions even when the starting solution is already very
good. We again see that the resulting distributions improve significantly upon those generated
with the proportional distribution approach. When compared to the best constructed solutions, we
see that two instances for fleets of 2,000 vehicles do not reach the same number of expected rides
served as the best constructed solutions. It should be noted, however, that this is likely due to the
improvement heuristic running out of time, as time spent was at the maximum allowed. It is also
true the best constructed solutions are already very good for the largest fleet sizes.
Of the four improvement heuristics, we consistently observe AG yielding the best solutions within
the time limit. We also note that the computation times for BI and FC are typically at the run
time limit, signaling that more improvement may be possible. If the operator has additional time,
they may want to allow a longer run time in an attempt to find better fleet distributions. Of the
four heuristics, FI never leads to the best fleet distribution. This heuristic struggled to find good
solutions within the 5-hour limit and consistently produced distributions with the lowest expected
number of rides served.
Note that in many instances the proportional distribution is improved to serve the same amount
or more rider demand as the improved heuristic-constructed distribution. This demonstrates that
the improvement heuristics are capable of improving weaker starting distributions to solutions
that are comparable to those that began with heuristic-constructed solutions. Therefore, in certain
Greening, Erera: Heuristics for Distributing Vehicles 27
R V |S0|zSIP ρSIP %BG %GC %IP ρIH %IHI Time
(hrs)
13
500 20 7,566 3,919 54.4% 1.4% 1.7% 4,030 -2.8% 0.70
1,000 100 8,163 6,480 24.6% 6.3% 2.2% 6,644 -2.5% 3.15
1,500 100 8,545 7,750 9.8% 10.7% 1.3% 7,862 -1.4% 0.35
2,000 100 8,589 8,243 4.0% 18.9% 1.0% 8,445 -2.4% 0.02
20
500 5 7,404 4,015 53.4% 0.9% 1.0% 4,309 -6.8% 0.09
1,000 50 8,127 6,303 26.8% 7.6% 3.1% 6,643 -5.1% 7.47
1,500 100 8,533 7,731 10.2% 41.0% 8.6% 7,828 -1.3% 4.24
2,000 100 8,607 8,330 3.3% 69.2% 8.2% 8,437 -1.3% 0.45
50
500 4 7,523 4,103 51.8% 1.9% 2.1% 4,239 -3.2% 4.82
1,000 5 8,141 6,411 24.8% -19.4% -5.1% 6,850 -6.4% 0.83
1,500 50 8,426 7,686 9.8% 12.1% 1.5% 7,871 -2.4% 1.67
2,000 100 8,504 8,296 2.6% 61.7% 4.5% 8,358 -0.8% 2.03
100
500 1 7,596 4,630 46.5% 4.0% 3.8% 4,950 -6.5% 0.23
1,000 5 8,216 6,952 19.6% 6.4% 1.7% 6,985 -0.5% 4.59
1,500 25 8,472 7,922 8.4% 18.4% 2.1% 7,973 -0.6% 2.32
2,000 25 8,599 8,322 3.8% 28.4% 1.6% 8,396 -0.9% 0.45
Table 4 SIP-generated distribution performance results.
situations, the operator could solely rely on improving operator-generated distributions and not
suffer any loss of quality in their final solutions. The only issue that could arise is that starting
with a poor distribution does typically require more computation time to improve.
5.5. Computational Results Using the Two-stage SIP Model
Next, we analyze the quality of initial vehicle distributions generated by solving the SIP model
described in Section 4.3. A subset S0⊆ S of sample scenarios is defined for each instance such that
the commercial solver can find a solution within a provable 1% optimality gap within an 8-hour
time limit. To define S0, we first determine the maximum number of sample scenarios that can be
solved and then randomly sample from the set S.
Table 4 summarizes performance results for the vehicle distributions generated by the SIP model.
The table includes two output values for the SIP model: (i) the SIP model objective value zSIP
and (ii) the expected number of trips served ρSIP when the vehicle distribution is evaluated using
the PM simulation. Recall that even if the SIP model is solved with the complete scenario set
(|S0|= 100), these two values will not be equal because the SIP model both relaxes the vehicle-to-
trip matching process and also makes optimal choices about which demands not to serve since, as
a two-stage model, the complete set of potential trips is known when making recourse decisions
for all time periods. We additionally compare ρSIP to performance metrics for vehicle distributions
generated by the improvement heuristics using %IHI as the percent improvement over ρIH.
One primary observation from the results is that the vehicle distributions generated by the
improvement heuristics outperform those generated by the SIP model for every instance. Thus, it is
28 Greening, Erera: Heuristics for Distributing Vehicles
difficult to create a two-stage stochastic optimization approach that works as well as a simulation-
based heuristic for these problem instances. However, the SIP solutions still outperform the pro-
portional vehicle distributions, except for the 50-region, 1,000-vehicle fleet instance. This likely
results from the need to use a very small scenario set of size 5 for that instance. We see that as
the fleet size grows, the difference between zSIP and ρSIP decreases. The primary reason for this is
that more trip demand can be served with a larger fleet of vehicles, mitigating the impact of the
approximation of the vehicle-to-trip matching process. Since the problem of distributing a larger
fleet is generally easier to solve, more scenarios can also be included in these cases improving the
quality of the resulting distribution of vehicles. It is also interesting to note that although three
of the four instances with a 500-vehicle fleet had short computation times, increasing the size of
S0quickly leads to computational intractability and a solution could not be found within the time
limit. Overall, we find that using the SIP model produces good vehicle distributions, but tends to
underperform when compared to distributions generated by the simulation-based construction and
improvement heuristics.
5.6. Incorporating Mid-Day Redistribution Operations
As a final component to this computational study, we now analyze the benefits that an operator
may be able to achieve if they were to include redistribution operations at fixed times during the
planning horizon. We consider the problem where the operator specifies the initial distribution
of vehicles along with two additional midday target distributions, where the two re-distribution
operations are planned for: (i) before noon when the vehicles have been used for morning commute
trips and may need to be moved for afternoon riders; and (ii) for the late afternoon to prepare
for evening commute trips. For these experiments, we use the AG improvement heuristic with
parameters u= 15 and L=I= max{5,d0.1Re}. To determine the first distribution, we initialize AG
with the proportional distribution assuming that a single distribution is to be used for the whole
planning horizon. For the second and third distributions, AG is initialized with the distribution
found by the heuristic for the previous sub-horizon; so, the best distribution for t= 1 is the initial
distribution for the planned distribution before noon. When optimizing each of these distributions,
the heuristic only considers the time periods in the sub-horizon from the distribution time through
to the time of the next reset operation. That is, for the initial distribution, AG will optimize
considering the time periods prior to noon and ignore the remaining. Since the planning horizons
are shorter, the simulation-based heuristics require less computation time overall.
Table 5 shows the improvement in performance when incorporating these two additional redis-
tribution operations. The column ρIHR shows the total sample average number of trips served given
the three distributions during the planning horizon. Note that for the smaller instances with large
Greening, Erera: Heuristics for Distributing Vehicles 29
fleet sizes, the redistribution operation allows the system to close the entire optimality gap to UAP .
Additionally, the improvement percentages above a single proportional distribution ID are quite
significant; %IP improvements range from 5 to over 25%.
R V ID ρID IHR L, I ρIHR %BG %GC %IP ρIH %IHI Time
(hrs)
13
500
Pro
3,851
AG 5
4,832 43.7% 20.7% 25.4% 4,030 19.9% 0.6
1,000 6,344 7,778 9.5% 63.9% 22.7% 6,644 17.1% 1.4
1,500 7,652 8,561 0.3% 96.9% 11.9% 7,862 8.9% 3.8
2,000 8,163 8,590 0.0% 100.0% 5.2% 8,445 1.7% 2.5
20
500
Pro
3,978
AG 5
4,894 43.2% 19.8% 23.2% 4,309 13.6% 0.6
1,000 6,109 7,581 12.0% 58.7% 24.0% 6,643 14.1% 0.9
1,500 7,117 8,559 0.6% 96.5% 20.2% 7,828 9.3% 3.7
2,000 7,692 8,612 0.0% 100.0% 11.9% 8,437 2.1% 3.1
50
500
Pro
4,012
AG 5
4,576 46.3% 12.4% 13.9% 4,239 7.9% 1.5
1,000 6,755 7,418 12.9% 37.6% 9.8% 6,850 8.3% 0.7
1,500 7,572 8,410 1.3% 88.4% 11.1% 7,871 6.9% 5.0
2,000 7,936 8,514 0.1% 99.0% 7.3% 8,358 1.9% 5.0
100
500
Pro
3,275
AG 10
5,616 35.1% 27.6% 25.9% 4,950 13.4% 5.0
1,000 6,825 7,657 11.5% 45.3% 12.0% 6,985 9.6% 2.9
1,500 7,760 8,393 3.0% 71.3% 8.2% 7,973 5.3% 5.0
2,000 8,195 8,568 0.9% 82.3% 4.6% 8,396 2.0% 5.0
Table 5 Improved performance when incorporating two mid-day redistribution operations using AG to improve
a proportional initial distribution.
Lastly, results in the table also show large improvements by redistribution when compared to
the earlier, single-distribution improvement heuristic results given by the sample averages ρIH. The
%IHI improvements here range from about 2 to nearly 20%, and thus for many reasonable systems
it likely makes sense to consider mid-day vehicle repositioning to capture more average daily trip
demand. Of course, there are significant costs to repositioning that create an interesting tradeoff
here.
6. Future Work
This paper has shown that there are effective heuristics for finding initial vehicle distributions, or
midday target vehicle distributions, for free-floating micromobility systems. The full multi-stage
optimization problem is very difficult to solve, but simulation-based construction and improvement
heuristics can find very good suboptimal solutions to even large problem instances and effective
upper bounding techniques provide quality guarantees for these approaches. For some problems, a
heuristic based on solving a two-stage approximation optimization model can also find reasonable
solutions when using a small set of sample scenarios.
Once good target vehicle distributions are identified, the next step in managing free-floating
systems is to effectively deploy vehicles into the target distribution. While using independent
30 Greening, Erera: Heuristics for Distributing Vehicles
contractors to charge and locate vehicles eliminates solving overnight repositioning problems, a new
set of concerns arises. One key challenge is finding effective approaches to both give independent
chargers choices in redistributing their vehicles each day while also achieving a system target
distribution. Contract chargers may have vehicle release location preferences that are difficult
to know in advance and may need to be learned over time; they will likely strongly prefer to
release vehicles near their home or on their way to work. Limiting choices to autonomous chargers
may lead to better distributions, but can also frustrate contractors and reduce their participation
or increase the costs of finding willing system participants. Another interesting challenge is to
design approaches to relocate vehicles during the operating day using a decentralized and semi-
autonomous set of contractors.
Greening, Erera: Heuristics for Distributing Vehicles 31
Appendix A: Improvement Heuristic Pseudocode
Algorithm 3: First Improvement
Input : vc,u
Output: Improved starting distribution v
1Initialize v←− vc,ρ←− P M (v) ;
2while u > 0do
3for i∈ R do
4for j6=i∈ R do
5n←− min{u, vj};
6if ρ0←− PM(v+nei−nej)> ρ then
7v←− v+nei−nej,ρ←− ρ0;
8go to 2;
9end
10 end
11 end
12 u←− u−1;
13 end
Algorithm 4: First-Continuous Improvement
Input : vc,u
Output: Improved starting distribution v
1Initialize v←− vc,ρ←− P M (v) ;
2while u > 0do
3update ←− false;
4for i∈ R do
5for j6=i∈ R do
6n←− min{u, vj};
7if ρ0←− PM(v+nei−nej)> ρ then
8v←− v+nei−nej,ρ←− ρ0;
9update ←− true;
10 end
11 end
12 end
13 if update = false then
14 u←− u−1;
15 end
16 end
32 Greening, Erera: Heuristics for Distributing Vehicles
Algorithm 5: Best Improvement
Input : vc,u
Output: Improved starting configuration v
1Initialize v←− vc,ρ←− P M (v) ;
2while u > 0do
3ρmax ←− 0;
4for i∈ R do
5for j6=i∈ R do
6n←− min{u, vj};
7if ρ0←− PM(v+nei−nej)> ρmax then
8Region+←− i, Region−←− j,ρmax ←− ρ0;
9end
10 end
11 end
12 if ρmax > ρ then
13 v←− v+neRegion+−neRegion−,ρ←− ρmax;
14 else
15 u←− u−1;
16 end
17 end
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